Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
Spatial Characterization Of Two-Photon States - GAP-Optique
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DOCTORAL THESIS IN PHOTONICS<br />
ICFO BARCELONA 2010<br />
<strong>Spatial</strong><br />
<strong>Characterization</strong> <strong>Of</strong><br />
<strong>Two</strong>-<strong>Photon</strong> <strong>States</strong><br />
CLARA INÉS OSORIO TAMAYO<br />
Advisor: Juan P. Torres
<strong>Spatial</strong> <strong>Characterization</strong><br />
<strong>Of</strong> <strong>Two</strong>-<strong>Photon</strong> <strong>States</strong><br />
By<br />
Clara Inés Osorio Tamayo<br />
ICFO - Institut de Ciències Fotòniques<br />
Universitat Politècnica de Catalunya<br />
Barcelona, September 2009
<strong>Spatial</strong> <strong>Characterization</strong><br />
<strong>Of</strong> <strong>Two</strong>-<strong>Photon</strong> <strong>States</strong><br />
Clara Inés Osorio Tamayo<br />
under the supervision of<br />
Dr. Juan P. Torres<br />
submitted this thesis in partial fulfillment<br />
of the requirements for the degree of<br />
Doctor<br />
by the<br />
Universitat Politècnica de Catalunya<br />
Barcelona, September 2009
A Luz Stella y Luis Alfonso, mis papás.
Contents<br />
Contents vii<br />
Acknowledgements ix<br />
Abstract xi<br />
Introduction xv<br />
List of Publications xvii<br />
1 General description of two-photon states 1<br />
1.1 Spontaneous parametric down-conversion . . . . . . . . . . . . 2<br />
1.2 <strong>Two</strong>-photon state . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
1.3 Approximations and other considerations . . . . . . . . . . . . 5<br />
1.4 The mode function in matrix form . . . . . . . . . . . . . . . . 12<br />
2 Correlations and entanglement 15<br />
2.1 The purity as a correlation indicator . . . . . . . . . . . . . . . 16<br />
2.2 Correlations between space and frequency . . . . . . . . . . . . 18<br />
2.3 Correlations between signal and idler . . . . . . . . . . . . . . . 23<br />
3 <strong>Spatial</strong> correlations and OAM transfer 29<br />
3.1 Laguerre-Gaussian modes and OAM content . . . . . . . . . . . 30<br />
3.2 OAM transfer in general SPDC configurations . . . . . . . . . . 31<br />
3.3 OAM transfer in collinear configurations . . . . . . . . . . . . . 34<br />
4 OAM transfer in noncollinear configurations 37<br />
4.1 Ellipticity in noncollinear configurations . . . . . . . . . . . . . 38<br />
4.2 Effect of the pump beam waist on the OAM transfer . . . . . . 39<br />
4.3 Effect of the Poynting vector walk-off on the OAM transfer . . 44<br />
5 <strong>Spatial</strong> correlations in Raman transitions 51<br />
5.1 The quantum state of Stokes and anti-Stokes photon pairs . . . 52<br />
5.2 Orbital angular momentum correlations . . . . . . . . . . . . . 55<br />
5.3 <strong>Spatial</strong> entanglement . . . . . . . . . . . . . . . . . . . . . . . . 57<br />
6 Summary 61<br />
A The matrix form of the mode function 63<br />
vii
Contents<br />
B Integrals of the matrix mode function 69<br />
C Methods for OAM measurements 71<br />
Bibliography 73<br />
viii
Acknowledgements<br />
This thesis compiles the result of five years of work at icfo. But it is, in some<br />
sense, the result of a longer process that started many years ago in Medellín.<br />
I would like to show my gratitude to all the people that helped me during all<br />
this time even though I cannot mention everybody explicitly.<br />
I would like to thank my advisor Juan P. Torres for inviting me to his<br />
group, and for being very patient with my impatience. I am grateful to Alejandra<br />
Valencia and Xiaojuan Shi for their generous and unconditional support.<br />
I wish to thank all the people that provide a stimulating and fun environment<br />
in icfo, especially Maurizio Righini, Laura Grau, Carsten Schuck, Masood<br />
Ghotbi, Xavier Vidal, Sibylle Braungardt, Jorge Luis Domínguez-Juárez,<br />
Petru Ghenuche, Rafael Betancurt, Philipp Hauke, Alessandro Ferraro, Agata<br />
Checinska, and Martin Hendrych. I would like to thank Niek van Hulst and<br />
his group, for adopting me at lunch time and for all party matters; and Morgan<br />
Mitchell and his group, especially Marco Koschorreck, Mario Napolitano,<br />
Florian Wolfgramm, Alessandro Cere and Brice Dubost. Logistical support by<br />
Nuria Segù, Maria del Mar Gil, Anne Gstöttner, Jose Carlos Cifuentes, and<br />
the electronic workshop have also been most helpful. I am extremely grateful<br />
to Miguel Navascués that has been like a little brother to me since I came to<br />
Barcelona, and to Artur García that welcomed me in his office every time that<br />
I needed to talk (and that was very often).<br />
I want to thank my “Barcelonian” friends for their caring, their emotional<br />
and gastronomical support, Mauricio Álvarez, Carolina Mora, Friman Sánchez,<br />
Tania de la Paz, Maria Teresa Vasco and Jose Uribe. Regardless the distance,<br />
many of my friends made available their support in a number of ways, I am<br />
grateful to Carlos Molina, Jose Palacios, Catalina López, Elizabeth Agudelo,<br />
Esteban Silva, Sebastían Patrón, Jaime Hincapie, Wolfgang Niedenzu, Andrew<br />
Hilliard, Juan C. Muñoz (y familia), Jaime Forero and Javier Moreno. It is a<br />
pleasure to thank Boudewijn Taminiau, Ied van Oorschot, and all the Taminiau<br />
family, since their visits and hospitality have made this years much nicer.<br />
Agradezco hasta el cielo a mi familia. A mis papás que me han inspirado<br />
siempre con su creatividad y fortaleza para resolver los problemas, a mis hermanas<br />
que siempre me apoyan, y a mi hermanito Felipe, que me ha trasmitido<br />
todo su amor por la ciencia, su curiosidad y su escepticismo.<br />
Finally, this thesis would not have been possible without the constant support<br />
of Tim Taminiau, his careful corrections of this manuscript and the nice<br />
3D images of the cone and the phase fronts that he made for me. I also thank<br />
Tim for giving me my first bicycle, for teaching me how to use it, and for<br />
pushing me up the mountain every day when we were going to work.<br />
ix
When he [Kepler] found that his long<br />
cherished beliefs did not agree with the<br />
most precise observations, he accepted<br />
the uncomfortable facts, he preferred the<br />
hard truth to his dearest illusions.<br />
That is the heart of science.<br />
Cosmos - Carl Sagan
Abstract<br />
In the same way that electronics is based on measuring and controlling the state<br />
of electrons, the technological applications of quantum optics will be based<br />
on our ability to generate and characterize photonic states. The generation<br />
of photonic states is traditionally associated to nonlinear optics, where the<br />
interaction of a beam and a nonlinear material results in the generation of<br />
multi-photon states. The most common process is spontaneous parametric<br />
down-conversion (spdc), which is used as a source of pairs of photons not only<br />
for quantum optics applications but also for quantum information and quantum<br />
cryptography [1, 2].<br />
The popularity of spdc lies in the relative simplicity of its experimental realization,<br />
and in the variety of quantum features that down-converted photons<br />
exhibit. For instance, a pair of photons generated via spdc can be entangled<br />
in polarization [3, 4], frequency [5, 6], or in the equivalent degrees of freedom of<br />
orbital angular momentum, space, and transverse momentum [7, 8, 9, 10, 11].<br />
Standard spdc applications focus on a single degree of freedom, wasting the<br />
entanglement in other degrees of freedom and the correlations between them.<br />
Among the few configurations using more than one degree of freedom are hyperentanglement<br />
[12, 13], spatial entanglement distillation using polarization<br />
[14], or control of the joined spectrum using the pump’s spatial properties [6].<br />
This thesis describes the spatial properties of the two-photon state generated<br />
via spdc, considering the different parameters of the process, and the<br />
correlations between space and frequency. To achieve this goal, I use the purity<br />
to quantify the correlations between the photons, and between the degrees<br />
of freedom. Additionally, I study the spatial correlations by describing the<br />
transfer of orbital angular momentum (oam) from the pump to the signal and<br />
idler photons, taking into account the pump, the detection system and other<br />
parameters of the process.<br />
This thesis is composed of six chapters. Chapter 1 introduces the mode<br />
function, used throughout the thesis to describe the two-photon state in space<br />
and frequency. Chapter 2 describes the correlations between degrees of freedom<br />
or photons in the two-photon state, using the purity to quantify such correlations.<br />
Chapter 3 explains the mechanism of the oam transfer from pump<br />
to signal and idler photons. Chapter 4 describes the effect of different spdc<br />
parameters on the oam transfer in noncollinear configurations, both theoretically<br />
and experimentally. In analogy with the downconverted case, chapter 5<br />
discusses the two-photon state generated via Raman transition, by describing<br />
its mode function, the correlations between different parts of the state, and the<br />
oam transfer in the process. Finally, chapter 6 summarizes the main results<br />
presented by this thesis.<br />
xi
Abstract<br />
The matrix notation, introduced here to describe the two-photon mode<br />
function, reduces the calculation time for several features of the state. In<br />
particular, this notation allows to calculate the purity of different parts of<br />
the state analytically. This analytical solution reveals the effect of each spdc<br />
parameter on the internal correlations, and shows the necessary conditions to<br />
suppress the correlations, or to maximize them.<br />
The description of the oam transfer mechanism shows that the pump oam<br />
is totally transfer to the generated photons. But if only a portion of the generated<br />
photons is detected their oam may not be equal to the pump’s oam. The<br />
experiments described in the thesis show that the amount of oam transfer in<br />
the noncollinear case is tailored by the parameters of the spdc. The analysis<br />
of the spdc case can be extended to other nonlinear processes, such as Raman<br />
transitions, where the specific characteristics of the process determine the<br />
correlations and the oam transfer mechanism.<br />
The results of this thesis contribute to a full description of the correlations<br />
inside the two-photon state. Such a description allows to use the correlations<br />
as a tool to modify the spatial state of the photons. This spatial information,<br />
translated into oam modes, provides a multidimensional and continuum degree<br />
of freedom, useful for certain tasks where the polarization, discrete and bidimensional,<br />
is not enough. To make such future applications possible, it will be<br />
necessary to optimize the tools for the detection of oam states at the single<br />
photon level [15, 16].<br />
xii
Resumen<br />
De la misma manera que la electrónica se basa en medir y controlar el estado<br />
de los electrones, las aplicaciones tecnológicas de la óptica cuántica se basarán<br />
en nuestra habilidad para generar estados fotónicos bien caracterizados. La<br />
generación de estos estados está tradicionalmente asociada a la óptica no lineal,<br />
donde la interacción de un haz con un material no lineal da lugar a la<br />
generación de estados de múltiples fotones. El proceso no lineal más popular<br />
es la conversión paramétrica descendente, o spdc por su sigla en inglés, que<br />
es usada como fuente de pares de fotones no sólo en aplicaciones de óptica<br />
cuántica si no también para información y criptografía cuánticas [1, 2].<br />
La popularidad de spdc se debe a la relativa simplicidad de su realización<br />
experimental, y a la variedad de fenómenos cuánticos exhibidos por los pares<br />
de fotones generados. Por ejemplo, estos pares pueden estar entrelazados en<br />
polarización [3, 4], frecuencia [5, 6], o en los grados de libertad espacial: momento<br />
angular orbital o momento transversal [7, 9, 10, 11]. Las aplicaciones<br />
usuales de spdc usan sólo un grado de libertad, perdiendo la información contenida<br />
en los otros grados de libertad y en las correlaciones entre ellos. Entre<br />
las únicas aplicaciones que usan más de un grado de libertad se encuentran el<br />
hiperentrelazamiento [12, 13], la destilación de entrelazamiento espacial usando<br />
la polarización [14], o el control de la distribución espectral conjunta usando<br />
las propiedades espaciales del haz generador del spdc [6].<br />
Esta tesis describe las características espaciales de los pares de fotones generados<br />
en spdc, teniendo en cuenta el efecto de los otros grados de libertad,<br />
especialmente de la frecuencia. Para ello, usaré la pureza como cuantificador de<br />
las correlaciones entre los grados de libertad, y entre los fotones. Además, usaré<br />
la transferencia de momento angular orbital (oam), asociada a la distribución<br />
espacial de los fotones, para estudiar el efecto de diferentes parámetros del<br />
spdc sobre el estado espacial generado.<br />
Esta tesis esta compuesta de seis capítulos. El capítulo 1 introduce la<br />
función de modo, que es usada en toda la tesis para describir los estados de dos<br />
fotones en espacio y frecuencia. El capítulo 2 describe las correlaciones entre<br />
los grados de libertad, y entre los fotones, usando la pureza para cuantificar<br />
estas correlaciones. El capítulo 3 describe la transferencia de oam desde el haz<br />
generador hasta los fotones generados. El capítulo 4 demuestra teórica y experimentalmente,<br />
el efecto de diferentes parámetros del spdc sobre la transferencia<br />
de momento angular orbital en configuraciones no colineales. El capítulo<br />
5 discute la generación de estados de dos fotones en transiciones Raman, caracterizando<br />
estos estados de una manera análoga a la utilizada para aquellos<br />
generados en spdc. Por último, el capítulo 6 resume las contribuciones más<br />
importantes de esta tesis.<br />
xiii
Abstract<br />
La notación matricial introducida para describir la función de modo de los<br />
pares generados, reduce considerablemente el tiempo de cálculo de diferentes<br />
características del estado. En particular, ésta notación hace posible calcular<br />
analíticamente la pureza de diferentes partes del estado, y estudiar el efecto de<br />
cada parámetro del spdc sobre las correlaciones entre los grados de libertad o<br />
entre los fotones. Hace posible, además, encontrar las condiciones necesarias<br />
para suprimir las correlaciones entre los grados de libertad o entre los fotones,<br />
y distinguir en que casos estas correlaciones se hacen mas relevantes.<br />
El estudio del mecanismo de transferencia del momento angular orbital,<br />
revela que éste es transferido totalmente del haz generador a los fotones generados.<br />
Si sólo una porción de estos fotones es considerada, el oam de éstos no<br />
da cuenta del momento oam total invertido por el haz generador. Los experimentos<br />
descritos en esta tesis muestran que la cantidad de oam transferido<br />
a una porción de los fotones puede ser controlada modificando el tamaño de<br />
esa porción, ya sea cambiando el ancho del haz incidente, el largo del cristal u<br />
otro parámetro. En el caso de la generación de pares de fotones en otros procesos<br />
no lineales, como las transiciones Raman, tanto las correlaciones como la<br />
transferencia de oam, son determinadas por las características especificas del<br />
proceso.<br />
Los resultados de esta tesis contribuyen a completar la descripción de las<br />
correlaciones dentro del estado de dos fotones. Esta descripción permite el<br />
uso de las correlaciones como herramienta para modificar el estado espacial de<br />
los fotones. La información espacial, traducida a modos de oam, ofrece un<br />
grado de libertad infinito dimensional y continuo, útil en ciertas tareas donde<br />
la polarización, bidimensional y discreta, no es suficiente. Para hacer estas<br />
tareas posibles, es necesario optimizar las herramientas para la detección de<br />
estados de oam para un sólo fotón [15, 16].<br />
xiv
Introduction<br />
The role of photons in quantum physics and technology is growing [1, 2, 17].<br />
Frequently, those applications are based on two dimensional systems, such as<br />
the two orthogonal polarization states of a photon, losing all information in<br />
other degrees of freedom. These applications only use a portion of the total<br />
quantum state of the light.<br />
The polarization itself is related to a broader degree of freedom. The angular<br />
momentum of the photons contains a spin contribution associated with<br />
the polarization, as well as an orbital contribution associated with the spatial<br />
distribution of the light (its intensity and phase). In general, the spin and<br />
the orbital angular momentum cannot be considered separately. However, in<br />
the paraxial regime both contributions are independent [18]. In this regime it<br />
is possible to exploit the possibilities offered by the infinite dimensions of the<br />
orbital angular momentum (oam) of the light.<br />
Currently available technology offers different possibilities to work with<br />
oam. Computer generated holograms are widely used in classical and quantum<br />
optics for generating and detecting different oam states [19, 20]. The efficiency<br />
of these processes has increased with the design of spatial light modulators<br />
capable of real time hologram generation [21].<br />
<strong>Of</strong> special interest is the generation of paired photons entangled in oam.<br />
Entanglement is a quantum feature with no analogue in classical physics. Spontaneous<br />
parametric down-conversion (spdc) is a reliable source for the generation<br />
of pairs of photons entangled in different degrees of freedom, including<br />
oam [22]. The existence of correlations in the oam of pairs of photons generated<br />
in spdc was proven experimentally by Mair and coworkers in 2001. Other<br />
nonlinear processes can be used to generate pairs of photons with correlations<br />
in their oam content. In reference [23], the authors show the generation of<br />
spatially entangled pairs of photons by the excitation of Raman transitions in<br />
cold atomic ensembles.<br />
Several studies illustrate the potential offered by higher-dimensional quantum<br />
systems. For instance, reference [24] reports a Bell inequality violation<br />
by a two-photon three-dimensional system, confirming the existence of oam<br />
entanglement in the system. Reference [25] introduced a quantum coin tossing<br />
protocol based on oam states. In reference [26], the authors present a quantum<br />
key distribution scheme using entangled qutrits, which are encoded into<br />
the oam of pairs of photons generated in spdc. And, in reference [12], the<br />
authors report the generation of hyperentangled quantum states by using the<br />
combination of the degrees of freedom of polarization, time-energy and oam.<br />
All the previously mentioned applications are based on specific oam correlations<br />
between the photons. However, there has been some controversy about<br />
xv
Introduction<br />
the transfer of oam from the generating pump beam to the down-converted<br />
photons. Several studies have reported a compleate transfer of the oam of the<br />
pump [7, 8, 22, 27, 28, 29], while other studies have reported a partial transfer<br />
[30, 31, 32, 33]. The elucidation of the oam spectra of down-converted photons,<br />
and their relation with the spectra of the pump, is a key step for developing<br />
new applications based on this degree of freedom.<br />
This thesis addresses the mentioned oam controversy, as well as the characterization<br />
of the different correlations present in two-photon states. For both<br />
processes, spdc and Raman transitions in cold atoms, this thesis describes the<br />
generated two-photon state, the correlations between different degrees of freedom<br />
and between photons, and the oam transfer. A general goal of the thesis<br />
is to contribute to the characterization of the two-photon sources, making it<br />
possible to exploit a bigger portion of the total quantum state of the light in<br />
future applications.<br />
xvi
This thesis is based on the following papers:<br />
List of Publications<br />
Orbital angular momentum correlations of entangled paired photons.<br />
C. I. Osorio, G. Molina-Terriza, J. P. Torres.<br />
J. Opt. A: Pure Appl. Opt. 11, 094013 (2009).<br />
Chapters 3 and 5<br />
<strong>Spatial</strong> entanglement of paired photons generated in cold atomic ensembles<br />
C. I. Osorio, S. Barreiro, M. W. Mitchell, and J. P. Torres.<br />
Phys. Rev. A 78, 052301 (2008). arXiv:0804.3257v2 [quant-ph].<br />
Chapter 5<br />
Spatiotemporal correlations in entangled photons generated by spontaneous parametric<br />
down conversion.<br />
C. I. Osorio, A. Valencia, and J. P. Torres.<br />
New J. Phys. 10, 113012 (2008). arXiv:0804.2425v2 [quant-ph].<br />
Chapters 1 and 2<br />
Correlations in orbital angular momentum of spatially entangled paired photons generated<br />
in parametric downconversion.<br />
C. I. Osorio, G. Molina-Terriza, and J. P. Torres.<br />
Phys. Rev. A 77, 015810 (2008). arXiv:0711.4500v1 [quant-ph].<br />
Chapters 3<br />
Azimuthal distinguishability of entangled photons generated in spontaneous parametric<br />
down-conversion.<br />
C. I. Osorio, G. Molina-Terriza, B. Font, and J. P. Torres.<br />
Opt. Express 15, 14636 (2007). arXiv:0709.3437v1 [quant-ph].<br />
Chapter 4<br />
Control of the shape of the spatial mode function of photons generated in noncollinear<br />
spontaneous parametric down-conversion.<br />
G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych, and J. P.<br />
Torres.<br />
Phys. Rev. A 72, 065802 (2005). arXiv:quant-ph/0508058v1.<br />
Chapter 4<br />
Orbital angular momentum of entangled counterpropagating photons.<br />
J. P. Torres, C. I. Osorio, and L. Torner.<br />
Opt. Lett. 29, 1939 (2004).<br />
Chapters 3 and 4<br />
xvii
CHAPTER 1<br />
General description of<br />
<strong>Two</strong>-photon states<br />
Experimental implementations of spontaneous parametric down-conversion (spdc)<br />
share three main components: A laser beam used as a pump, a nonlinear material<br />
that induces the down-conversion, and a detection system that measures<br />
the generated state. The properties of the resulting two-photon state depend<br />
on the interaction, and on the individual properties of those components. This<br />
chapter derives the two-photon mode function that characterizes the generated<br />
state, using first order perturbation theory. The chapter is divided in four<br />
sections. Section 1.1 describes the spdc process in general terms. Section 1.2<br />
introduces a general expression for the mode function. Section 1.3 lists a series<br />
of common approximations used to reduce the complexity of the mode function.<br />
Finally, section 1.4 introduces a novel matrix notation for the simplified<br />
mode function. One of the advantages of this new notation is that many characteristics<br />
of the two-photon state can be analytically calculated by using it.<br />
The following chapters will use the notation and results introduced here as a<br />
starting point to describe the correlations between the generated photons, and<br />
between their spatial states.<br />
1
1. General description of two-photon states<br />
virtual state<br />
pump<br />
ground state<br />
signal<br />
idler<br />
Figure 1.1: In spontaneous parametric down conversion the interaction of a molecule<br />
with the pump results in the annihilation of the pump photon and the creation of<br />
two new photons.<br />
1.1 Spontaneous parametric down-conversion<br />
Spontaneous parametric down-conversion is one of the possible resulting processes<br />
of the interaction of a pump photon and a molecule, in the presence of<br />
the vacuum field. In spdc, the pump-molecule interaction leads to the generation<br />
of a single physical system composed of two photons: signal and idler, as<br />
figure 1.1 shows.<br />
The name of the process reveals some of its characteristics. spdc is a<br />
parametric process since the incident energy totally transfers to the generated<br />
photons, and not to the molecule. And, it is a down-conversion process since<br />
each of the generated photons has a lower energy than the incident photon.<br />
The conservation of energy and momentum establish the relations between the<br />
frequencies (ωp,s,i) and wave vectors (kp,s,i) of the photons,<br />
ωp = ωs + ωi<br />
kp = ks + ki. (1.1)<br />
where the subscripts p, s, i stand for pump, signal and idler. Macroscopically,<br />
spdc results from the interaction of a field with a nonlinear medium. Commonly<br />
used mediums include uniaxial birefringent crystals. The conditions in<br />
equation 1.1 can be achieved for this kind of crystals, since their refractive index<br />
changes with the frequency and the polarization of an incident field [34, 35].<br />
This thesis considers type-i configurations in uniaxial birefringent crystals.<br />
In such configuration the pump beam polarization is extraordinary since it is<br />
contained in the principal plane, defined by the crystal axis and the wave vector<br />
of the incident field. The polarization of the generated photons is ordinary,<br />
which means that it is orthogonal to the principal plane.<br />
For a given material, the conditions imposed by equation 1.1 determine the<br />
characteristics of the generated pair. The second line in equation 1.1 defines the<br />
spdc geometrical configuration, and it is known as a phase matching condition.<br />
The generated photons are emitted in two cones coaxial to the pump, as shown<br />
in figure 1.2 (b). The apertures of the cone are given by the emission angles ϕs,i<br />
associated to the frequencies ωs,i. Once a single frequency or emission angle is<br />
selected, only one of all possible cones is considered. In degenerate spdc, both<br />
photons are emitted at the same angle ϕs = ϕi and the cones overlap, as in<br />
figure 1.2 (c). There are two kinds of degenerate spdc processes: noncollinear<br />
2
y<br />
y<br />
x<br />
z<br />
z<br />
pump<br />
s<br />
<br />
i<br />
(a)<br />
signal<br />
idler<br />
(c)<br />
1.2. <strong>Two</strong>-photon state<br />
Figure 1.2: (a) Phase matching conditions impose certain propagation directions for<br />
the emitted photons at frequencies ωs and ωi. (b-c) This directions define two cones,<br />
that collapse into one in the degenerate case. (d) In the collinear configuration, the<br />
aperture of the cones tends to zero as the direction of emission is parallel to the pump<br />
propagation direction.<br />
in which the signal and idler are not parallel to the pump, and collinear in<br />
which the aperture of the cones tends to zero as the photons propagate almost<br />
parallel to the pump, as shown in figure 1.2 (d).<br />
Even though the phase matching condition defines the main characteristics<br />
of the generated two-photon state, other important factors influence the measured<br />
state of the photons, for example the detection system. The next section<br />
describes the two-photon state mathematically, taking into account all these<br />
factors.<br />
1.2 <strong>Two</strong>-photon state<br />
When an electromagnetic wave propagates inside a medium, the electric field<br />
acts over each particle (electrons, atoms, or molecules) displacing the positive<br />
charges in the direction of the field and the negative charges in the opposite<br />
direction. The resulting separation between positive and negative charges of<br />
the material generates a global dipolar moment in each unit volume known as<br />
polarization, and defined as<br />
(b)<br />
(d)<br />
P = ɛ0(χ (1) E + χ (2) EE + χ (3) EEE + · · · ) (1.2)<br />
where ɛ0 is the vacuum electric permittivity, and χ (n) are the electric susceptibility<br />
tensors of order n [34, 35].<br />
For small field amplitudes, as in linear optics, the polarization is approximately<br />
linear,<br />
P ≈ ɛ0χ (1) E. (1.3)<br />
3
1. General description of two-photon states<br />
When the amplitude of the electric field increases, the higher orders term in<br />
equation 1.2 become relevant, and then a nonlinear response of the material<br />
to the field appears. Optical nonlinear phenomena resulting from this kind<br />
of interaction include the generation of harmonics, the Kerr effect, Raman<br />
scattering, self-phase modulation, and cross-phase modulation [34].<br />
Spontaneous parametric down-conversion, and other second order nonlinear<br />
processes, result from the second order polarization, defined as the first<br />
nonlinear term in the polarization tensor<br />
P (2) = ɛ0χ (2) EE. (1.4)<br />
The quantization of the electromagnetic field leads to a quantization of the<br />
second order polarization, so that the nonlinear polarization operator ˆ P (2)<br />
becomes<br />
ˆP (2) = ɛ0χ (2) ( Ê(+) + Ê(−) )( Ê(+) + Ê(−) ), (1.5)<br />
where Ê(+) and Ê(−) are the positive and negative frequency parts of the field<br />
operator [36]. The positive frequency part of the electric field operator is a<br />
function of the annihilation operator â(k), and is defined at position rn<br />
xnˆxn + ynˆyn + znˆzn and time t as<br />
=<br />
Ê (+)<br />
1/2 <br />
ωn<br />
n (rn, t) = ien<br />
2ɛ0v<br />
dkn exp [ikn · rn − iωnt]â(kn), (1.6)<br />
where the magnitude of the wave vector k satisfies k 2 = ωn/c. The volume<br />
v contains the field, and en is the unitary polarization vector. The negative<br />
frequency part of the field is the Hermitian conjugate of the positive part,<br />
Ê (−)<br />
n (rn, t) = Ê(+)† n (rn, t). Thus, the negative frequency part is a function of<br />
the creation operator â † (k).<br />
Following references [37] and [38], in first order perturbation theory, the<br />
interaction of a volume V of a material with a nonlinear polarization ˆ P (2) and<br />
the field Êp(rp, t), produces a system described by the state<br />
|ΨT 〉 ∝ |1〉p|0〉s|0〉i − i<br />
<br />
τ<br />
0<br />
dt ˆ HI|1〉p|0〉s|0〉i, (1.7)<br />
where τ is the interaction time, |1〉p|0〉s|0〉i is the initial state of the field, and<br />
the interaction Hamiltonian reads<br />
<br />
ˆHI = − dV ˆ P (2) Êp(rp, t). (1.8)<br />
V<br />
The first term at the right side of equation 1.7 describes a one photon system,<br />
while the second term describes a two-photon system. In what follows we will<br />
consider only the second term as we are mainly interested in the generation of<br />
pairs of photons. The two-photon system state is given by<br />
|Ψ〉 = i<br />
<br />
τ<br />
0<br />
dt ˆ HI|1〉p|0〉s|0〉i, (1.9)<br />
or ˆρ = |Ψ〉〈Ψ| in the density matrix formalism. Seven of the eight terms<br />
that compose the interaction Hamiltonian vanish when they are applied over<br />
the state |1〉p|0〉s|0〉i. The remaining term is proportional to the annihilation<br />
4
1.3. Approximations and other considerations<br />
operator in mode p, and the creation operators in modes s and i. Assuming<br />
constant χ (2) , the two-photon state reduces to<br />
(2) τ <br />
iɛ0χ<br />
|Ψ〉 = dt dV<br />
0 V<br />
Ê(+) p (rp, ωp) Ê(−) s (rs, ωs) Ê(−) i (ri, ωi)|1〉p|0〉s|0〉i,<br />
(1.10)<br />
which can be written as<br />
<br />
|Ψ〉 ∝ dks dkiΦ(ks, ωs, ki, ωi)a † (ks)a † (ki)|0〉s|0〉i; (1.11)<br />
where Φ(ks, ωs, ki, ωi) is known as the mode function and, assuming that the<br />
pump has certain spatial distribution Ep(kp), is given by<br />
τ <br />
Φ(ks, ωs, ki, ωi) ∝ dt dV dkpEp(kp)<br />
0<br />
V<br />
× exp [ikp · rp − iks · rs − iki · ri − i(ωp − ωs − ωi)t]<br />
× â(kp)|1〉p. (1.12)<br />
The mode function contains all the information about the generated two-photon<br />
system in space and frequency, not only about their individual state but about<br />
their correlations. This function is therefore highly complex, and to calculate<br />
any feature of the two-photon state it is necessary to simplify it, as the next<br />
section shows.<br />
1.3 Approximations and other considerations<br />
Analytical calculations of any features of the down converted photons require<br />
simplification of the mode function in equation 1.12. This section lists the most<br />
important approximations used in this thesis, as well as the restrictions that<br />
they impose.<br />
Separation of transversal and longitudinal components<br />
A field with a wave vector k almost parallel to its propagation direction spreads<br />
only a little in the transversal direction. It is then possible, to consider the<br />
longitudinal and transversal components of the wave vector separately,<br />
k = kzˆz + q. (1.13)<br />
As the wave vector’s magnitude k = ωn/c is bigger than the transversal component’s<br />
magnitude q = |q| = (k 2 x + k 2 y) 1/2 , the magnitude of the longitudinal<br />
component kz is always a real number,<br />
kz = k 2 1<br />
− q<br />
2 2 . (1.14)<br />
Assuming that the pump, the signal, and the idler are such fields, the mode<br />
function becomes<br />
τ <br />
Φ(qs, ωs, qi, ωi) ∝<br />
0<br />
dt dV<br />
V<br />
dqpEp(qp) exp [ik z pzp − ik z szs − ik z i zi]<br />
× exp [iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i − i(ωp − ωs − ωi)t]<br />
× â(q p)|1〉p. (1.15)<br />
where r ⊥ n = xnˆxn + ynˆyn is the transversal position vector.<br />
5
1. General description of two-photon states<br />
The semiclassical approximation<br />
As a consequence of the low efficiency of the nonlinear process, the incident<br />
field is several orders of magnitude stronger than the generated fields. In that<br />
case, it is possible to consider the pump as a classical field, while the signal<br />
and idler are considered as quantum fields, this is known as the semiclassical<br />
approximation.<br />
By defining the pump as a classical field, with a spatial amplitude distribution<br />
Ep(q p) and a spectral distribution Fp(ωp):<br />
<br />
Ep(rp, t) ∝ dqp dωpEp(qp)Fp(ωp) exp [iqp · r ⊥ p + ik z pzp − iωpt], (1.16)<br />
the mode function reduces to<br />
τ <br />
Φ(qs, ωs, qi, ωi) ∝ dt dV dqp dωpEp(qp)Fp(ωp) Gaussian beam approximation<br />
0 V<br />
× exp [ik z pzp − ik z szs − ik z i zi + iqp · r ⊥ p − iqs · r ⊥ s − iqi · r ⊥ i ]<br />
× exp [−i(ωp − ωs − ωi)t]. (1.17)<br />
According to equation 1.17, the mode function depends on the spatial and<br />
temporal profiles of the pump beam. To write these profiles explicitly we<br />
assume a pump beam with a Gaussian temporal distribution,<br />
<br />
Fp(ωp) = exp − T 2 0<br />
4 ω2 <br />
p<br />
(1.18)<br />
where T0 is the pulse duration (standard deviation), which tends to infinity for<br />
continuous wave beams. Also, we assume that the pump beam is an optical<br />
vortex with orbital angular momentum lp per photon. As will be described<br />
in section 3.1, under these conditions the pump spatial profile is given by the<br />
Laguerre-Gaussian polynomials<br />
1<br />
2<br />
wp<br />
Ep(qp) =<br />
2π|lp|!<br />
<br />
|lp|<br />
−iwp<br />
√ qp exp −<br />
2 w2 p<br />
4 q2 <br />
p + ilpθp<br />
(1.19)<br />
that are functions of the pump beam waist wp, and the transversal vector<br />
magnitude qp and phase θp, given by<br />
6<br />
<br />
qp =<br />
θp = tan −1<br />
(q x p ) 2 + (q y p) 2<br />
q y p<br />
q x p<br />
<br />
.<br />
(1.20)
x<br />
1.3. Approximations and other considerations<br />
pump<br />
signal<br />
beam s<br />
y z xi<br />
idler<br />
z<br />
Figure 1.3: The propagation direction of the pump beam defines the z direction of<br />
the general coordinate system. The generated pair of photons propagates with angles<br />
ϕs and ϕi with respect to the pump beam in the yz plane. Each photon coordinate<br />
system transforms to the general coordinate system through the relations in equation<br />
1.23.<br />
For the special case of a Gaussian pump beam with lp = 0 the mode function<br />
becomes<br />
τ <br />
Φ(qs, ωs, qi, ωi) ∝ dt dV<br />
0 V<br />
i<br />
y<br />
dq p exp [− w2 p<br />
4 q2 p − T 2 0<br />
4 ω2 p]<br />
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ]<br />
× exp [−i(ωp − ωs − ωi)t], (1.21)<br />
which assumes that the interaction time τ is longer than the spontaneous emission<br />
life time of the material.<br />
Frequency bandwidth<br />
Fields generated experimentally always contain a distribution of frequencies,<br />
and totally monochromatic fields only exist in theory. This thesis considers the<br />
frequency ω of each field as the sum of a constant central frequency ω 0 , and a<br />
small deviation from that frequency Ω, so that ω = ω 0 + Ω.<br />
As a result of integrating over the interaction time taking into account the<br />
conservation of energy, the mode function becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ dV<br />
V<br />
Coordinate transformation<br />
dq p exp [− w2 p<br />
4 q2 p − T 2 0<br />
4 (Ωs + Ωi) 2 ]<br />
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ].<br />
(1.22)<br />
The mode function depends on three position vectors, rp, rs, and ri. Each of<br />
them is defined in a coordinate system with the z axis parallel to the propagation<br />
direction of each field, as shown in figure 1.3. Defining all position vectors<br />
in the same coordinate system simplifies the integration of the mode function<br />
over volume.<br />
i<br />
i<br />
xs<br />
ys<br />
zs<br />
7
1. General description of two-photon states<br />
pump<br />
polarization<br />
y<br />
Figure 1.4: The azimuthal angle α between the pump beam polarization and the x<br />
axis defines the position of a single pair of photons on the cone. The x axis is by<br />
definition normal to the plane of emission.<br />
With the origin at the crystal’s center, the z direction parallel to the pump<br />
propagation direction, and the yz plane containing the emitted photons, the<br />
unitary vectors in the coordinate systems of the generated photons transform<br />
as<br />
ˆxs =ˆx<br />
<br />
ˆys =ˆy cos ϕs + ˆz sin ϕs<br />
x<br />
ˆzs = − ˆy sin ϕs + ˆz cos ϕs<br />
ˆxi =ˆx<br />
ˆyi =ˆy cos ϕi − ˆz sin ϕi<br />
ˆzi =ˆy sin ϕi + ˆz cos ϕi. (1.23)<br />
As a single pair of photons defines the yz plane, the coordinate transformations<br />
consider only one of the possible directions of propagation on the cone. The<br />
angle α between the x axis and the pump polarization defines the transverse<br />
position on the cone for one photon pair, as figure 1.4 shows. The pump<br />
polarization is normal to the plane in which the generated photons propagate<br />
when α = 0 ◦ or α = 180 ◦ , and it is contained in the yz plane when α = 90 ◦ or<br />
α = 270 ◦ .<br />
Poynting vector walk-off<br />
This thesis considers an extraordinary polarized pump beam and ordinary polarized<br />
generated photons, an eoo configuration. Therefore, while the refractive<br />
index does not change with the direction for the generated photons, it changes<br />
for the pump beam with the angle between the pump wave vector and the axis<br />
of the crystal. Figure 1.5 shows how the energy flux direction of the pump,<br />
given by the Poynting vector, is rotated from the direction of the wave vector<br />
by an angle ρ0 given by<br />
ρ0 = − 1 ∂ne<br />
. (1.24)<br />
∂θ<br />
ne<br />
where ne is the refractive index for the extraordinary pump beam, and θ is<br />
the angle between the optical axis and the pump’s wave vector. The Poynting<br />
8
x<br />
Wave fronts<br />
1.3. Approximations and other considerations<br />
Poynting vector<br />
0<br />
Wave vector<br />
Figure 1.5: As a consequence of the birefringence, the Poynting vector is no longer<br />
parallel to the wave vector. The energy flows in an angle ρ0 with respect to propagation<br />
direction.<br />
x<br />
<br />
y z<br />
Figure 1.6: The Poynting vector moves away from the wave vector in the direction of<br />
the pump beam polarization. The angles α and ρ0 characterize the displacement. A<br />
nonradial effect such as the Poynting vector walk-off inevitably brakes the azimuthal<br />
symmetry of the properties of the photons on the cone.<br />
vector walk-off displaces the effective transversal shape of the pump beam<br />
inside the crystal in the pump polarization direction, as shown in figure 1.6.<br />
The vector p = z tan ρ0 cos αˆx + z tan ρ0 sin αˆy, describes the magnitude and<br />
direction of this displacement.<br />
By including the Poynting vector walk-off, and by using the same coordinate<br />
system for all the fields, the mode function becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ dV dqp exp −<br />
V<br />
w2 p<br />
4 q2 p − T0<br />
4 (Ωs + Ωi) 2<br />
<br />
× exp i(q x p − q x s − q x i )x <br />
<br />
0<br />
× exp [i(q y p + k z s sin ϕs − k z i sin ϕi − q y s cos ϕs − q y<br />
i cos ϕi)y]<br />
× exp [i(k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y<br />
i sin ϕi)z]<br />
× exp [i(q x p tan ρ0 cos α + q y p tan ρ0 sin α)z]. (1.25)<br />
p<br />
z<br />
9
1. General description of two-photon states<br />
Interaction volume approximation<br />
In the experimental cases studied here, the crystals are cuboids with two square<br />
faces. While typical crystal faces have areas of about 1cm 2 , the pump beam<br />
waist is only some hundreds of micrometers. By assuming that the transversal<br />
dimensions of the crystal are much larger than the pump, the integral over the<br />
volume becomes<br />
<br />
V<br />
dV →<br />
∞<br />
−∞<br />
∞ L/2<br />
dx dy dz, (1.26)<br />
−∞ −L/2<br />
where L is the length of the crystal. After integrating over x and y, the mode<br />
function reduces to<br />
<br />
<br />
where<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p<br />
4 ∆20 − w2 p<br />
4 ∆21 − T0<br />
4 (Ωs + Ωi) 2<br />
L/2<br />
× dz exp [i∆kz] (1.27)<br />
−L/2<br />
∆0 =q x s + q x i<br />
∆1 =q y s cos ϕs + q y<br />
i cos ϕi − k z s sin ϕs + k z i sin ϕi<br />
∆k =k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y<br />
i sin ϕi<br />
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.28)<br />
Wave vector and group velocity<br />
The use of a Taylor series expansion of kz s,i , simplifies the delta factors defined<br />
in the previous paragraph. Because the wave vector magnitude is a function<br />
of the frequency deviation Ω, the expansion around the origin reads<br />
k z n = k z0 ∂k<br />
n + Ωn<br />
z n<br />
+ Ω 2 ∂<br />
n<br />
2kz n<br />
∂2Ωn ∂Ωn<br />
+ . . . , (1.29)<br />
where k z0<br />
n is the magnitude of the wave vector at the central frequency ω 0 n, and<br />
the first partial derivative of the wave vector magnitude with respect to the<br />
frequency is the group velocity Nn.<br />
Taking only the first order terms of the Taylor expansion, and considering<br />
momentum conservation, the delta factors become<br />
∆0 =q x s + q x i<br />
∆1 =q y s cos ϕs + q y<br />
i cos ϕi − NsΩs sin ϕs + NiΩi sin ϕi<br />
∆k =Np(Ωs + Ωi) − NsΩs cos ϕs − NiΩi cos ϕi − q y s sin ϕs + q y<br />
i sin ϕi<br />
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.30)<br />
Exponential approximation of the sinc function<br />
The mode function in equation 1.27, depends on the integral of an exponential<br />
function of z. By integrating the exponential over the length of the crystal, the<br />
10
1<br />
0<br />
-0.2<br />
1.3. Approximations and other considerations<br />
Gaussian Sine Cardinal<br />
0<br />
FWHM<br />
Figure 1.7: An exponential function is chosen as an approximation for the sine cardinal<br />
function. The parameters of the exponential were chosen in such a way that<br />
both functions have the same full width at half maximum (fwhm).<br />
mode function becomes<br />
Φ(q s, Ωs, q i, Ωi) ∝ exp<br />
<br />
− w2 p<br />
4 ∆2 0 − w2 p<br />
4 ∆2 1 − T0<br />
4 (Ωs + Ωi) 2<br />
<br />
sinc<br />
<br />
L<br />
2 ∆k<br />
<br />
(1.31)<br />
where the sine cardinal function is defined as sinc(a) = sin a/a.<br />
In chapter 2, the sine cardinal function sinc(a) is approximated by a Gaussian<br />
function exp[−(γa) 2 ]. With γ = 0.4393 the two functions have the same<br />
full width at half maximum as figure 1.7 shows. In the rest of the thesis the<br />
sinc function will be consider without approximations.<br />
Detection systems as Gaussian filters<br />
The specific optical system used to detect the photons at the output face of<br />
the crystal, acts as a filter in space and frequency. The effect of these filters<br />
is modeled by multiplying the mode function by a spatial collection function<br />
Cspatial(qn) and a frequency filter function Ffrequency(Ωn). For the sake of<br />
simplicity, both functions are assumed to be Gaussian functions so that<br />
<br />
Cspatial(qn) ∝ exp − w2 n<br />
2 q2 <br />
n , (1.32)<br />
and<br />
<br />
Ffrequency(Ωn) ∝ exp − 1<br />
2B2 Ω<br />
n<br />
2 <br />
n , (1.33)<br />
where n labels each of the generated photons, wn is the spatial collection mode<br />
width, and Bn is the frequency bandwidth.<br />
The angular acceptance θn is related to the vector qn by the equation qn =<br />
knθn. Therefore, for a given collection mode wn the acceptance is θn = 1/knwn<br />
(the half width at 1/e 2 ).<br />
11
1. General description of two-photon states<br />
In order to obtain more convenient units, the filter in momentum is defined<br />
in a different way than the filter in frequency. While Bs,i → 0 implies a single<br />
frequency collection, the condition for a single q vector collection is ws,i → ∞.<br />
Finally, after all the approximations and taking into account the filters, the<br />
mode function equation 1.12 becomes<br />
<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p<br />
4 ∆20 − w2 p<br />
4 ∆21 <br />
× exp − (γL)2<br />
4 ∆2k − T 2 0<br />
4 (Ωs + Ωi) 2<br />
<br />
<br />
× exp − w2 s<br />
2 q2 s − w2 i<br />
2 q2 i − 1<br />
2B2 Ω<br />
s<br />
2 s − 1<br />
2B2 Ω<br />
i<br />
2 <br />
i . (1.34)<br />
The approximations listed in this chapter were introduced by several authors,<br />
and can be found for example in references [39, 40, 30]. In reference [41],<br />
we introduced a novel matrix notation based on the simplified mode function<br />
equation 1.34. The next section explains the details and the advantages of that<br />
notation.<br />
1.4 The mode function in matrix form<br />
The argument of the exponential function in equation 1.34 is a second order<br />
polynomial of the mode function variables. Such a function can be written in<br />
matrix form, as is shown in appendix A. The mode function then becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1<br />
2 xt <br />
Ax<br />
(1.35)<br />
where all the parameters of the spdc process are contained in A, a positivedefinite<br />
real 6 × 6 matrix given by<br />
A = 1<br />
⎛<br />
a<br />
⎜ h<br />
⎜ i<br />
2 ⎜ j<br />
⎝ k<br />
h<br />
b<br />
m<br />
n<br />
p<br />
i<br />
m<br />
c<br />
s<br />
t<br />
j<br />
n<br />
s<br />
d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
f<br />
l<br />
r<br />
u<br />
w<br />
z<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(1.36)<br />
l r u w z g<br />
x is the vector including all variables of the mode function, defined as<br />
⎛ ⎞<br />
⎜<br />
x = ⎜<br />
⎝<br />
q x s<br />
q y s<br />
q x i<br />
q y<br />
i<br />
Ωs<br />
Ωi<br />
⎟ , (1.37)<br />
⎟<br />
⎠<br />
and x t is the transpose of x. This matrix representation of the mode function,<br />
introduced in reference [41], is an important result of this thesis. The use of<br />
this notation reduces the calculation time for integrals over the mode function,<br />
12
1.4. The mode function in matrix form<br />
and allows to solve some of those integrals analytically. For instance, the mode<br />
function normalization requires six integrals over the modulus square of the<br />
mode function. As appendix B shows, the matrix notation provides a way to<br />
calculate the normalization constant analytically,<br />
<br />
dx exp<br />
<br />
− 1<br />
2 xt <br />
(2A)x<br />
and thus the normalized mode function reads<br />
=<br />
Φ(qs, Ωs, qi, Ωi) = [det(2A)]1/4<br />
(2π) 3/2<br />
(2π) 3<br />
, (1.38)<br />
[det(2A)] 1/2<br />
<br />
exp − 1<br />
2 xt <br />
Ax . (1.39)<br />
In the same way, many other integrals over the mode function can be solved<br />
analytically using the matrix representation of the mode function.<br />
Conclusion<br />
The two-photon mode function can be written in a matrix form after a series<br />
of approximations. The matrix notation makes it possible to calculate several<br />
features of the down-conversion photons analytically, and reduces the numerical<br />
calculation time of others.<br />
The next chapter describes the different correlations in the two-photon state<br />
using the purity as a correlation indicator. The use of the approximate mode<br />
function in matrix notation makes it possible to find an analytical expression<br />
for the purity, and to study the effect of the different parameters on it.<br />
13
CHAPTER 2<br />
Correlations and<br />
entanglement<br />
The two-photon system described in chapter 1 consists of two photons with two<br />
degrees of freedom. Although the mode function deduced fully characterizes the<br />
two-photon state, the degree of correlations between each of its parts (different<br />
subsystems) are not explicit. In this chapter, I use the purity of a subsystem<br />
state to indicate the presence of correlations between it and the rest of the<br />
two-photon state, as shown in figure 2.1. This chapter has three sections.<br />
Section 2.1 describes the main characteristics of the purity, and explains why<br />
it can be used to characterize correlations in composed systems. Section 2.2<br />
describes the spatial part of the two-photon state, and section 2.3 describes<br />
the state of the signal photon. In both sections a discussion about the origin<br />
of the correlations is followed by analytical and numerical calculations of the<br />
purity. The calculations clarify the role of each spdc parameter in the internal<br />
correlations of the two-photon state. By engineering the spdc process, it is<br />
possible to tailor, and even to suppress correlations between different degrees<br />
of freedom or between different photons. Chapter 3 considers the spatial part<br />
of the two-photon state, after the correlations between space and frequency are<br />
suppressed.<br />
15
2. Correlations and entanglement<br />
(a)<br />
Space Frequency Signal<br />
q<br />
q<br />
s<br />
i<br />
<br />
<br />
s<br />
i<br />
qs<br />
qi<br />
Idler<br />
Figure 2.1: The frequency part, gray in (a), is traced out in section 2.2, to calculate<br />
the state of the subsystem composed by the two-photon state in the spatial degree<br />
of freedom. Calculating the signal photon state in space and frequency, section 2.3,<br />
requires to trace out the idler photon, gray in (b).<br />
2.1 The purity as a correlation indicator<br />
When a physical system is used to implement a task, its state should be characterized.<br />
For quantum states, the reconstruction of the state density matrix<br />
demands a tomography [1]. This process requires an infinite amount of copies<br />
of the state in order to eliminate statistical errors [42]. In an experimental<br />
implementation, the number of copies of the state available is finite, but even<br />
under this condition an experimental tomography is a complex process. Instead<br />
when just a certain feature of the system is relevant for the task, it is preferable<br />
to use a figure of merit to characterize that part of its quantum state.<br />
For a system in a quantum state given by the density operator ˆρ, the purity<br />
P = T r[ˆρ 2 ] is a figure of merit that describes its ability to be correlated with<br />
others. That is, the purity indicates if correlations can or cannot exist. Since<br />
the purity is a second order function of the density operator, measuring it does<br />
not require a full tomography, but just a simultaneous measurement over two<br />
copies of the system [43, 44].<br />
To illustrate other characteristics of the purity, consider as an example<br />
a complete orthonormal basis for the electromagnetic field, where each basis<br />
vector |n〉 describes a mode of the field [36]. A pure state |R〉 is any state<br />
written as a superposition of basis vectors:<br />
|R〉 = <br />
Cn|n〉, (2.1)<br />
n<br />
where the mode amplitude and the relative phases between modes, given by<br />
the coefficients Cn, are fixed. This pure state can be described by the density<br />
operator ˆρ = |R〉〈R|, which has the maximum possible value for the purity<br />
T r[ˆρ 2 ] = 1.<br />
A mixed state of the electromagnetic field is a state that can not be written<br />
as in equation 2.1. For instance, in the case of a statistical mixture of field<br />
modes, where there are no fixed amplitude or fixed relative phases associated<br />
16<br />
<br />
<br />
s<br />
i<br />
(b)
2.1. The purity as a correlation indicator<br />
to each vector of the base. There is, however, a fixed probability pR of finding<br />
the electromagnetic field in a state |R〉. Using the density operator formalism,<br />
the probability distribution is given by<br />
ˆρ = <br />
pR|R〉〈R|. (2.2)<br />
R<br />
The purity of the state described by ˆρ quantifies how close it is to a pure<br />
state. If the state is pure then T r[ˆρ 2 ] = 1, and if it is maximally mixed then<br />
T r[ˆρ 2 ] = 1/n, where n is de dimension of the Hilbert space in which the state is<br />
expanded. Maximally mixed states of the electromagnetic field, in the infinite<br />
dimensional spatial or temporal degrees of freedom have a purity T r[ˆρ 2 ] = 0.<br />
To extend the discussion about the purity to the kind of states generated<br />
by spdc, consider the pure bipartite system described by |ψ〉 and composed by<br />
the fields A and B. The Schmidt decomposition guarantees that orthonormal<br />
states |RA〉 and |RB〉 exist for the fields A and B, so that<br />
|ψ〉 = <br />
λR|RA〉|RB〉 (2.3)<br />
R<br />
where λR are non-negative real numbers that satisfy <br />
R λR = 1, and are<br />
known as Schmidt coefficients [1, 17]. The average of the nonzero Schmidt<br />
coefficients is a common entanglement quantifier [11] known as the Schmidt<br />
number and given by<br />
K =<br />
1<br />
<br />
R λ2 R<br />
. (2.4)<br />
While equation 2.3 describes the state of the whole bipartite system, the state<br />
of each part is calculated by tracing out the other part. Thus, the states of the<br />
fields A and B are given by the density operators<br />
ˆρA = <br />
λR|RA〉〈RA| (2.5)<br />
and<br />
R<br />
ˆρB = <br />
λR|RB〉〈RB|. (2.6)<br />
R<br />
Since both operators have equal eigenvalues λ2 R , the purity of both states is<br />
equal, and given by the inverse of the Schmidt number K<br />
T r[(ˆρ A ) 2 ] = T r[(ˆρ B ) 2 ] = 1<br />
K<br />
<br />
= λ 2 R. (2.7)<br />
Thus, when the field A is in a pure state, the field B is in a pure state too,<br />
and <br />
R λ2 <br />
R = 1. Since R λR = 1, the Schmidt coefficients satisfy the new<br />
condition only if all except one of them are equal to zero. If that is the case,<br />
the bipartite system in equation 2.3 is a product state, and no correlations<br />
exist between A and B. In other words, if there are correlations, the purity of<br />
A (and B) is smaller than 1.<br />
In conclusion, when considering a composed global system, the purity of<br />
each subsystem measures the strength of the correlation between such subsystem<br />
and the rest. This is always true independently of how the subsystems are<br />
R<br />
17
2. Correlations and entanglement<br />
chosen. In the particular case of a two-photon system, the purity of the spatial<br />
part can be used to study correlations between the degrees of freedom, or the<br />
purity of the signal photon can be used to study the entanglement between the<br />
photons. The next sections explore both these approaches.<br />
2.2 Correlations between space and frequency<br />
In type-i spdc, the photons generated have a polarization orthogonal to the<br />
polarization of the incident beam. Therefore, the two-photon state generated<br />
in a type-i process has only two degrees of freedom: frequency and transversal<br />
spatial distribution. In this section, I study the origin and the characteristics<br />
of the correlations between those degrees of freedom by using the purity of the<br />
spatial part as a correlation indicator.<br />
Except for hyperentanglement configurations [12, 13] and some other novel<br />
configurations [6, 14], most applications of type-i spdc processes use just one of<br />
the degrees of freedom, ignoring the correlation that may exist between space<br />
and frequency. That is, those configurations assume a high degree of spatial<br />
purity. This section explores the conditions in which this assumption is valid.<br />
The first part of this section contains a discussion about the origin and<br />
suppression of the correlations. The second part includes the calculations for<br />
the spatial purity that is used as correlation indicator. The third part shows<br />
the results of numerical calculation of the spatial purity in several cases.<br />
2.2.1 Origin of the correlations<br />
According to the two-photon mode function, equation 1.34, the correlations<br />
between space and frequency appear as cross terms in the variables associated<br />
to those degrees of freedom, as figure 2.2 shows. The correlations between<br />
q s x<br />
q s y<br />
q i x<br />
y<br />
qi s<br />
i<br />
qs x<br />
a<br />
h<br />
i<br />
j<br />
k<br />
l<br />
q s<br />
y<br />
h<br />
b<br />
m<br />
n<br />
p<br />
r<br />
qi y<br />
i<br />
m<br />
c<br />
s<br />
t<br />
u<br />
Figure 2.2: Cross terms in matrix A generate correlations between space and frequency.<br />
Without those terms, the matrix A is composed by two nonzero block matrices,<br />
one for each degree of freedom.<br />
space and time are always relevant, if at least one of the cross terms k, l, p,<br />
r, t, u, v, w is comparable with another term of matrix A. Therefore, it is<br />
possible to suppress the correlations by making all the cross terms negligible,<br />
or by increasing the values of the other terms.<br />
The use of ultra-narrow filters in one of the degrees of freedom will suppress<br />
most of the information available in it, destroying any possible correlation with<br />
18<br />
q i<br />
x<br />
j<br />
n<br />
s<br />
d<br />
v<br />
w<br />
s<br />
k<br />
p<br />
t<br />
v<br />
f<br />
z<br />
i<br />
l<br />
r<br />
u<br />
w<br />
z<br />
g
2.2. Correlations between space and frequency<br />
other degrees of freedom. Filtering makes the diagonal terms of matrix A larger<br />
than the rest of the terms, including the terms responsible for the correlations.<br />
But while they remove the correlations, the filters reduce the amount of photons<br />
available for any measurement.<br />
An appropriate design of the source makes it possible to suppress cross<br />
terms without using filters. Such a design should fulfill the conditions<br />
Np − Ns cos ϕs<br />
Np − Ni cos ϕi<br />
Np − Ns cos ϕs<br />
Np − Ni cos ϕi<br />
<br />
<br />
Np − Ns cos ϕs = 0<br />
Np − Ni cos ϕi = 0<br />
<br />
<br />
1 + w2 p<br />
γ2L2 1 + w2 p<br />
γ 2 L 2<br />
1 − w2 p<br />
γ 2 L 2<br />
1 − w2 p<br />
γ 2 L 2<br />
Ns sin ϕs<br />
tan ϕi<br />
Ni sin ϕi<br />
tan ϕs<br />
<br />
<br />
<br />
= 0<br />
= 0<br />
= 0<br />
= 0. (2.8)<br />
These conditions can be met for example by using a long crystal, a highly<br />
focused pump beam, for which wp/γL → 0, and emission angles such that<br />
cos ϕs = Np/Ns and cos ϕi = Np/Ni.<br />
Even though these conditions have been deduced in a simplified regime,<br />
reference [45] shows that these conditions are sufficient to remove the correlations<br />
when considering the general mode function of equation 1.12. Once the<br />
correlations are suppressed, the two-photon mode function can be written as a<br />
product of a spatial and a frequency two-photon mode function like<br />
Φ(q s, ωs, q i, ωi) = Φq(q s, q i)ΦΩ(Ωs, Ωi). (2.9)<br />
Since the global two-photon system in space and frequency is in a pure state,<br />
the presence of correlations can be confirmed by using the purity. The next<br />
section calculates the purity of the spatial state, which can be used as an<br />
indicator of correlations between space and frequency.<br />
2.2.2 <strong>Two</strong>-photon spatial state<br />
To study space/frequency correlations I will describe the purity of the spatial<br />
part of the two-photon state. As the purity of the spatial and frequency parts<br />
of the state are equal, all results derived for the spatial two-photon state extend<br />
to a frequency two-photon state.<br />
After tracing-out the frequency from the two-photon state, the remaining<br />
state describes the spatial part of the state. The reduced density matrix for<br />
19
2. Correlations and entanglement<br />
the spatial two-photon state is<br />
ˆρq = T rΩ[ρ]<br />
<br />
=<br />
<br />
=<br />
dΩ ′′<br />
s dΩ ′′<br />
i 〈Ω ′′<br />
s , Ω ′′<br />
i |ˆρ|Ω ′′<br />
s , Ω ′′<br />
i 〉<br />
dqsdΩsdqidΩidq ′ sdq ′ i<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)|qs, qi〉〈q ′ s, q ′ i|, (2.10)<br />
while the purity of this state, defined as T r[ˆρ 2 q], is given by<br />
T r[ˆρ 2 <br />
q] =<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i). (2.11)<br />
It is possible to solve these integrals analytically by using the exponential<br />
character of the mode function given by equation (1.35). The purity becomes<br />
T r[ˆρ 2 q] = det(2A)<br />
det(B) . (2.12)<br />
As seen in appendix A, B is a positive-definite real 12 × 12 matrix<br />
B = 1<br />
⎛<br />
⎜<br />
2 ⎜<br />
⎝<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
l<br />
0<br />
0<br />
0<br />
0<br />
k<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
r<br />
0<br />
0<br />
0<br />
0<br />
p<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
u<br />
0<br />
0<br />
0<br />
0<br />
t<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
w<br />
0<br />
0<br />
0<br />
0<br />
v<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
2z<br />
k<br />
p<br />
t<br />
v<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
2g<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
0<br />
0<br />
0<br />
k<br />
l<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
0<br />
0<br />
0<br />
0<br />
p<br />
r<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
0<br />
0<br />
0<br />
0<br />
t<br />
u<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
0<br />
0<br />
0<br />
0<br />
v<br />
w<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
0<br />
0<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
l r u w 0 0 l r u w 2z 2g<br />
defined by the equation<br />
N 4 <br />
exp − 1<br />
2 Xt <br />
BX = Φ(qs, Ωs, qi, Ωi)<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(2.13)<br />
× Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i), (2.14)<br />
where vector X is the result of concatenation of x and x ′ .<br />
In order to see the effect of each of the spdc parameters on the spatiofrequency<br />
correlations, some typical values are used in next subsection to calculate<br />
T r[ˆρ 2 q] numerically.<br />
2.2.3 Numerical calculations<br />
As first example, consider a degenerate type-i spdc process characterized by the<br />
parameters in the second column of table 2.1. A pump beam, with wavelength<br />
20
2.2. Correlations between space and frequency<br />
λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium iodate (liio3)<br />
crystal with length L = 1 mm, and negligible Poynting vector walk-off (ρ0 = 0).<br />
The crystal emits signal and idler photons with a wavelength λ0 s = λ0 i = 810<br />
nm at an angle ϕs,i = 10◦ .<br />
Under the above conditions, figure 2.3 shows the purity of the spatial state<br />
T r[ρ2 q] given by equation 2.12, as a function of the spatial filter width for different<br />
frequency filter bandwidths. The half width at 1/e of the frequency filters<br />
∆λn, is Bn = πc∆λn/(λ2 √<br />
n ln 2). Figure 2.3 shows the presence of correlations<br />
between space and frequency in all the situations for which the purity is smaller<br />
than one.<br />
According to figure 2.3, the spatial purity of the two-photon state gets closer<br />
to one as ws (= wi) increase. Narrow filters in space (infinitely large ws,i) makes<br />
the two-photon state separable in frequency and space. The separability and<br />
the lack of correlations between frequency and space appear also in the“narrow<br />
band” limit for the frequency. If ∆λs = ∆λi → 0 nm, then the purity is always<br />
1 for any value of ws = wi.<br />
When using typical interference filters, ∆λs = ∆λi ≈ 1 nm, the purity decreases<br />
indicating space/frequency correlations. As the width of the frequency<br />
filters increases, the purity converges quickly, the case of ∆λs = ∆λi = 10 nm<br />
is almost equivalent to the case without frequency filters ∆λs, ∆λi → ∞.<br />
In order to compare the theoretical results of this section with previous<br />
experimental work, figure 2.4 shows the purity of the spatial state as a function<br />
of the collection mode for three reported experiments. Figure 2.4 (a) shows the<br />
purity for the data reported in reference [6]. The authors of that paper studied<br />
a type-i spdc process in a 1 mm long liio3 crystal. They used a diode laser<br />
with wavelength λp = 405 nm and bandwidth ∆λp = 0.4 nm as a pump beam.<br />
And, by using monochromators with ∆λs,i = 0.2 nm, they detected pairs of<br />
photons emitted at ϕs,i = 17◦ . Table 2.1 resumes these parameters.<br />
The paper analyzes two pump waist values: wp = 30 µm and wp = 462 µm.<br />
For those values, the authors demonstrated control of the frequency correlations<br />
by changing the spatial properties of the pump beam. As the pump beam<br />
influences the spatial shape of the generated photons, they implicitly reported<br />
a correlation between space and frequency. This spatial to spectral mapping<br />
exists at their collection mode ws = 133.48 µm for which the correlation indeed<br />
appears according with our model.<br />
Figure 2.4(b) shows T r[ρ2 q] for the case described in reference [46]. The<br />
authors used a continuous wave pump beam with λp = 405 nm to generate<br />
photons with λs,i = 810 nm in a 1.5 mm length bbo crystal. They detected the<br />
photons after filtering with ∆λs,i = 10 nm interference filters. The pump beam<br />
waist satisfied the condition wp >> L, and the generated photons propagated<br />
at ϕs,i = 0◦ .<br />
The experiment reported in reference [46] does not show evidence of spacefrequency<br />
correlations. In agreement with this fact, the figure shows a lack<br />
of correlation for the collinear case. However, under the conditions reported,<br />
but in a non collinear configuration, it is not possible to neglect the space and<br />
frequency correlations.<br />
Finally figure 2.4 (c) plots T r[ˆρ 2 q] for the case described in reference [31].<br />
In this paper, the authors illuminated a 2 mm long bbo crystal with a pump<br />
beam with λp = 351.1 nm and wp ≈ 20 µm. Their crystal generated photons<br />
at λs,i = 702 nm that propagated at 4o . They collected the photons after using<br />
21
2. Correlations and entanglement<br />
Filters<br />
0nm<br />
1nm<br />
10nm<br />
→∞<br />
0.1 Collection mode 1mm<br />
<strong>Spatial</strong><br />
purity<br />
Figure 2.3: The spatial purity increases by filtering the state in space or frequency.<br />
For the parameters chosen here, the bandwidth of the generated photons is smaller<br />
than 10 nm, making 10 nm filters equivalent to infinitely broad ones. The parameters<br />
used to generate this figure are listed in table 2.1.<br />
<strong>Spatial</strong><br />
purity<br />
1<br />
0<br />
0.1<br />
Pump<br />
waist<br />
30m 462m Emission<br />
angle<br />
1mm 0.1 Collection mode 1mm<br />
1<br />
0.2<br />
1º<br />
5º<br />
(a) (b) (c)<br />
0º<br />
0.1 1mm<br />
Pump<br />
waist<br />
20m 500m Figure 2.4: For the values reported in reference [6], figure (a) shows that the purity<br />
is smaller than 1. This fact is in agreement with the implicit spatio-temporal<br />
correlations reported in the reference. For the values reported in references [46] and<br />
[31], figures (b) and (c) show that the purity has its maximum value. There was no<br />
evidence of spatio-temporal correlations in the results reported in these references.<br />
In all three cases, it is possible to modify the value of the purity by tailoring the spdc<br />
parameters. The parameters used to generate this figure are listed in table 2.1.<br />
10 nm interference filters.<br />
For the noncollinear configuration in reference [31], the highly focused pump<br />
beam is responsible for the lack of correlations. The figure shows that for lessfocused<br />
pump beams the purity decreases as the correlations between space<br />
and frequency become more important.<br />
The correlations between frequency and space in the two-photon state discussed<br />
in this section, suggest that a complete description of the correlations<br />
between the generated photons, should take into account both degrees of freedom,<br />
as the next section discusses.<br />
22
2.3. Correlations between signal and idler<br />
Table 2.1: The parameters used in figures 2.3 and 2.4<br />
Parameter Figure 2.3 Figure 2.4(a) Figure 2.4(b) Figure 2.4(c)<br />
Crystal liio3 liio3 bbo bbo<br />
L 1 mm 1 mm 1.5 mm 2 mm<br />
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦<br />
T0 → ∞ → ∞ → ∞ → ∞<br />
wp 400 µm 30 / 462 µm wp ≫ L 20, 500 µm<br />
λp 405 nm 405 nm 405 nm 351.1 nm<br />
∆λp 0 0.4 nm<br />
λ 0 s 810 nm 810 nm 810 nm 702 nm<br />
∆λs 0.2 nm 10 nm 10 nm<br />
ϕs 10 ◦ 17 ◦ 0, 1, 5 ◦ 4 ◦<br />
2.3 Correlations between signal and idler<br />
In contrast to the previous section, here the two-photon system is considered<br />
as composed by two photons, each described by space and time variables, as<br />
figure 2.1 (b) shows. Since the global state in equation 1.9 is pure, the purity<br />
of the signal photon calculated in this section is a measurement of the degree of<br />
spatiotemporal entanglement between signal and idler photons, as in references<br />
[5, 10, 11, 47, 48].<br />
spdc as a source of two-photon states can be used to generate pairs of<br />
photons maximally entangled or two single photons [5, 49]. The purity of the<br />
signal photon in the first case should be equal to 0 and in the second case equal<br />
to 1. This section explores the necessary conditions to be in each regime.<br />
This section is divided in three parts, the first one discusses the origin of<br />
the correlations and the strategies to suppress them. The second part describes<br />
how to obtain an analytical expression for the signal photon purity. And the<br />
third part shows the results of numerical calculation of that purity.<br />
2.3.1 Origin of the correlations between photons<br />
The presence of cross terms in variables of both signal and idler in equation<br />
1.34 indicates correlations between the generated photons. Figure 2.5 shows<br />
those terms: i, j, l, m, n, r, t, v, and z.<br />
In contrast to the last section, filtering one degree of freedom is not enough<br />
to suppress the correlations between signal and idler. For example, by using a<br />
frequency filter, the photons can be still correlated in space. The total suppression<br />
of the correlation requires infinite narrow filters in both degrees of freedom<br />
for one of the photons, as shown experimentally in reference [50]. But that kind<br />
of filtering reduces the number of available pairs of photons substantially.<br />
The other possibility to generate uncorrelated photons is to tailor the parameters<br />
of the spdc process to make the cross terms negligible simultaneously.<br />
According with appendix A, when α = 0 ◦ , this condition is equivalent to mak-<br />
23
2. Correlations and entanglement<br />
q s x<br />
q s y<br />
q i x<br />
y<br />
qi s<br />
i<br />
qs x<br />
a<br />
h<br />
i<br />
j<br />
k<br />
l<br />
q s<br />
y<br />
h<br />
b<br />
m<br />
n<br />
p<br />
r<br />
qi y<br />
i<br />
m<br />
c<br />
s<br />
t<br />
u<br />
Figure 2.5: The signal-idler cross terms in matrix A are responsible for the correlations<br />
between the generated photons. By suppressing these cross terms, and by making<br />
permutations over columns and rows, the matrix becomes a two block matrix, where<br />
each block contains the information of one photon.<br />
ing the following terms negligible<br />
i =w 2 p + γ 2 L 2 tan ρ0 2<br />
j =γ 2 L 2 sin ϕi tan ρ0<br />
l =γ 2 L 2 tan ρ0(Np − Ni cos ϕi)<br />
m = − γ 2 L 2 sin ϕs tan ρ0<br />
q i<br />
x<br />
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi<br />
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi<br />
t =γ 2 L 2 tan ρ0(Np − Ns cos ϕs)<br />
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs<br />
z =γ 2 L 2 N 2 p − γ 2 L 2 Np(Ni cos ϕi + Ns cos ϕs) + γ 2 L 2 NsNi cos ϕs cos ϕi<br />
j<br />
n<br />
s<br />
d<br />
v<br />
w<br />
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi. (2.15)<br />
As an example, a configuration with a negligible Poynting walk-off fulfills this<br />
condition always when<br />
cos ϕs = Np<br />
2Ns<br />
cos ϕi = Np<br />
2Ni<br />
w 2 p<br />
γ 2 L 2 = tan ϕs tan ϕi<br />
T 2 0 =N 2 p γ 2 L 2<br />
wp ≪ws, wi<br />
s<br />
k<br />
p<br />
t<br />
v<br />
f<br />
z<br />
i<br />
l<br />
r<br />
u<br />
w<br />
<br />
tan ϕs 2 tan ϕi 2 − 1<br />
<br />
4<br />
z<br />
g<br />
(2.16)<br />
When the correlations are totally suppressed by satisfying these or other conditions,<br />
the two-photon state is separable, and each photon is in a pure state.<br />
Therefore, the two-photon mode function can be written as a product of two<br />
mode functions, one for each photon,<br />
24<br />
Φ(q s, ωs, q i, ωi) = Φs(q s, Ωs)Φi(q i, Ωi). (2.17)
2.3. Correlations between signal and idler<br />
In any other case, correlations between signal and idler are unavoidable. To<br />
evaluate their strength, the next section deduces an analytical expression for<br />
the purity of the signal photon state.<br />
2.3.2 Spatio temporal state of signal photon<br />
A single photon state can be generated in spdc by ignoring any information<br />
about one of the generated photons. The single photon state is calculated by<br />
tracing out the other photon from the two-photon state. For example, after<br />
a partial trace over the idler photon, the reduced density matrix in space and<br />
frequency for the signal photon is<br />
ˆρsignal =T ridler[ρ]<br />
<br />
=<br />
<br />
=<br />
and its purity is given by<br />
dq ′′<br />
i dΩ ′′<br />
i 〈q ′′<br />
i , Ω ′′<br />
i |ˆρ|q ′′<br />
i , Ω ′′<br />
i 〉<br />
dqsdΩsdqidΩidq ′ sdΩ ′ s<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)|qs, Ωs〉〈q ′ s, Ω ′ s|, (2.18)<br />
T r[ˆρ 2 signal] =<br />
<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i). (2.19)<br />
Recalling the exponential character of the mode function described by equation<br />
1.35, equation 2.19 writes<br />
T r[ˆρ 2 signal] = det(2A)<br />
det(C) , (2.20)<br />
where C is a positive-definite real 12 × 12 matrix defined by<br />
N 4 <br />
exp − 1<br />
2 Xt <br />
CX = Φ(qs, Ωs, qi, Ωi)<br />
and given by<br />
⎛<br />
C = 1<br />
2<br />
⎜<br />
⎝<br />
× Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i), (2.21)<br />
2a 2h i j 2k l 0 0 i j 0 l<br />
2h 2b m n 2p r 0 0 m n 0 r<br />
i m 2c 2s t 2u i m 0 0 t 0<br />
j n 2s 2d v 2w j n 0 0 v 0<br />
2k 2p t v 2f z 0 0 t v 0 z<br />
l r 2u 2w z 2g l r 0 0 z 0<br />
0 0 i j 0 l 2a 2h i j 2k l<br />
0 0 m n 0 r 2h 2b m n 2p r<br />
i m 0 0 t 0 i m 2c 2s t 2u<br />
j n 0 0 v 0 j n 2s 2d v 2w<br />
0 0 t v 0 z 2k 2p t v 2f z<br />
l r 0 0 z 0 l r 2u 2w z 2g<br />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
(2.22)<br />
25
2. Correlations and entanglement<br />
Signal<br />
purity Filters<br />
1<br />
0<br />
0.1 Collection mode 1mm<br />
0nm<br />
1nm<br />
10nm<br />
→∞<br />
Figure 2.6: The correlations between photons can not be suppressed by filtering<br />
in only one degree of freedom. The purity of the signal state is only maximum<br />
when using ultra-narrow spatial and frequency filters. In this case, the 1 nm filters<br />
are broad enough to allow the correlations, and the 10 nm filters are equivalent to<br />
infinitely broad ones. The parameters used to generate this figure are listed in table<br />
2.2.<br />
The differences between the matrix C and matrix B in the last section, are<br />
due to the order of the primed and unprimed variables in the arguments of the<br />
mode functions in equations 2.14 and 2.21.<br />
The next part of this section uses equation 2.20, to calculate the values of<br />
the purity of the signal photon for different spdc configurations.<br />
2.3.3 Numerical calculations<br />
Figure 2.6 shows the space-frequency purity of the signal photon as a function<br />
of the spatial filter width for different values of the frequency filter width.<br />
Consider a degenerate type-i spdc configuration, where a pump beam with<br />
wavelength λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium<br />
iodate (liio3) crystal with length L = 1 mm, and negligible Poynting vector<br />
walk-off (ρ0 = 0). The crystal emits signal and idler photons with a wavelength<br />
λ 0 s = λ 0 i = 810 nm at an angle ϕs,i = 10 ◦ , and the collection modes for signal<br />
and idler are assumed to be equal (ws = wi). All these parameters are listed<br />
in the first column of table 2.2.<br />
As was discussed in the first part of this section, it is possible to achieve<br />
maximal separability between the photons by using infinitely narrow filters<br />
in both space and frequency. In the region of small values of ws (= wi) a<br />
considerable correlation between signal and idler exists even in the case of<br />
infinitely narrow frequency filters. Different values for the signal photon purity<br />
can be achieved by changing the filter width, as shown by figure 2.6.<br />
Additionally, the figure shows how the purity T r[ˆρ 2 signal ] is confined between<br />
the values obtained for ∆λs = ∆λi → 0 nm and ∆λs, ∆λi → ∞. This limits<br />
can be tailored by modifying the other parameters of the spdc configuration.<br />
Figure 2.7 shows T r[ˆρ 2 signal ] as a function of the pump beam waist wp for<br />
26
Signal<br />
purity<br />
1<br />
0<br />
(a)<br />
0.1 3<br />
2.3. Correlations between signal and idler<br />
(b)<br />
0.1 Pump waist 3<br />
(c)<br />
0.1<br />
Emission<br />
angle<br />
Figure 2.7: The effect of the emission angle on the signal purity changes depending<br />
on the filters used in space and frequency. When both filters have a finite size like in<br />
(a), the purity increases with the angle of emission, and has a maximum for a fixed<br />
value of the pump beam waist. When the frequency filters are infinitely narrow like<br />
in (b), the dependence of the purity on the angle of emission disappears. When using<br />
narrow filters in space, as in (c), the purity is equal to 1 for a certain pump beam<br />
waist that changes with the angle of emission. The parameters used to generate this<br />
figure are listed in table 2.2.<br />
Table 2.2: The parameters used in figures 2.6 and 2.7<br />
Parameter Figure 2.6 Figure 2.7(a) Figure 2.7(b) Figure 2.7(c)<br />
Crystal liio3 liio3 liio3 liio3<br />
L 1 mm 1 mm 1 mm 1 mm<br />
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦<br />
T0 → ∞ → ∞ → ∞ → ∞<br />
wp 400 µm 400 µm 400 µm 400 µm<br />
λp 405 nm 405 nm 405 nm 405 nm<br />
∆λp → 0 → 0 → 0 → 0<br />
ws variable → ∞ 400 µm 400 µm<br />
λ 0 s 810 nm 810 nm 810 nm 810 nm<br />
∆λs 0, 1, 10 nm → ∞ 10 nm → 0 10 nm<br />
ϕs 10 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦<br />
various values of the emission angle ϕs = ϕi, in degenerate type-i spdc configurations<br />
described by the parameters listed in the second, third and fourth<br />
columns of table 2.2.<br />
Figure 2.7 (a) shows the case of finite spatial and temporal filters. The<br />
purity of the signal photon is always smaller than one, increases with the emission<br />
angle, and has a maximum for a given value of the pump beam waist. For<br />
a very narrow band frequency filter with a ws = wi = 400 µm spatial filter,<br />
figure 2.7 (b) shows how the correlation between the photons is minimal for<br />
small pump beams. The emission angle for this particular crystal length is<br />
irrelevant. Finally, figure 2.7 (c) considers a frequency filter ∆λs = ∆λi = 10<br />
nm with infinitely narrow spatial filters. In this case, maximal purity appears<br />
for each emission angle at a particular value of the pump beam waist.<br />
Conclusion<br />
<strong>Two</strong> photons with two degrees of freedom compose the two-photon state generated<br />
in spdc. The correlations between all four elements affect the state of<br />
single photons or two-photon states with one degree of freedom. Using the pu-<br />
3mm<br />
5º<br />
10º<br />
20º<br />
27
2. Correlations and entanglement<br />
rity as correlation indicator it is possible to determine the conditions in which<br />
the specific correlations are suppressed, without requiring the use of infinitely<br />
narrow filters.<br />
The next chapter continues with the description of the correlations between<br />
photons, in the case where only their spatial state is relevant. The chapter describes<br />
the spatial correlations in terms of orbital angular momentum transfer.<br />
28
CHAPTER 3<br />
<strong>Spatial</strong> correlations and<br />
orbital angular momentum<br />
transfer<br />
The previous chapter describes the correlations between the signal and idler<br />
photons considering space and frequency, as well as the correlations between<br />
those degrees of freedom. This chapter focuses on the cases where the degrees<br />
of freedom are not correlated, and the only relevant correlations are<br />
those between the photons in the spatial degree of freedom. To characterize<br />
those correlations I use the orbital angular momentum (oam) content of<br />
the generated photons, which is directly associated to their spatial distribution.<br />
The spatial correlations between the photons determine the mechanism<br />
of oam transfer from the pump to the signal and idler. Most studies of the<br />
oam correlations in spdc can be divided in two categories: those reporting a<br />
full transference of the pump’s oam [7, 8, 22, 27, 28, 29], and those reporting<br />
a partial transference [30, 31, 32, 33]. This chapter describes the oam transfer<br />
mechanism, and shows the difference between both regimes. The chapter is<br />
divided in three sections. Section 3.1 introduces the eigenstates of the oam<br />
operator: the Laguerre-Gaussian modes. This section shows how to calculate<br />
and measure the oam content of the photons. Section 3.2 describes how the<br />
oam is transferred in spdc. As an example, section 3.3 shows how this transfer<br />
happens in the collinear case, where all the emitted photons propagate in the<br />
same direction. The main result of this chapter is a selection rule that summarizes<br />
the oam transfer mechanism: the oam carried by the pump is completely<br />
transferred to all the signal and idler photons emitted over the cone.<br />
29
3. <strong>Spatial</strong> correlations and OAM transfer<br />
-1<br />
0 1 2<br />
OAM content<br />
in ħ units<br />
Phase front<br />
Intensity<br />
profile<br />
Figure 3.1: The Laguerre-Gaussian modes are characterized by their phase front<br />
distribution and intensity profile. The figure shows the phase fronts and the intensity<br />
profiles for the modes with oam contents from −1 to 2.<br />
3.1 Laguerre-Gaussian modes and OAM content<br />
In an analogous way to the decomposition of an electromagnetic field as a<br />
series of planes waves, it is possible to decompose the field in other bases. For<br />
instance, paraxial fields can be decomposed as a sum of Laguerre-Gaussian (lg)<br />
modes. This basis is especially convenient since the lg modes are eigenstates of<br />
the orbital angular momentum (oam) operator. That is, the state of a photon<br />
in a lg mode has a well defined oam value [51]. This section describes the<br />
properties of the lg modes, and shows how to calculate the weight of each<br />
mode in a given decomposition.<br />
<strong>Photon</strong>s carrying oam different from zero have at least one phase singularity<br />
(or vortex) in the electromagnetic field, a region in the wavefront where the<br />
intensity vanishes. This is precisely one of the most distinctive characteristics<br />
of Lagurre-Gaussian modes as figure 3.1 shows. Each lg mode is defined by<br />
the number p of non-axial vortices, and the number l of 2π-phase shifts along<br />
a close path around the beam center. The index l also describes the helical<br />
structure of the phase front around the singularity, and more important here,<br />
l determines the orbital angular momentum carried by the photon in units.<br />
The state of a single photon in a lg mode is<br />
<br />
|lp〉 = dqLGlp(q)â † (q)|0〉, (3.1)<br />
where the mode function lglp(q) is given by Laguerre-Gaussian polynomials<br />
<br />
1<br />
2 wp!<br />
LGlp(q) =<br />
2π(|l| + p)!<br />
|l|<br />
wq<br />
√2 L |l|<br />
2 2 w q<br />
p<br />
2<br />
<br />
× exp − w2q2 <br />
exp ilθ + iπ(p −<br />
4<br />
|l|<br />
2 )<br />
<br />
(3.2)<br />
as a function of the beam waist w, the modulus q and the phase θ of the<br />
transversal vector, and the associated Laguerre polynomials L |l|<br />
p defined as<br />
30<br />
L |l|<br />
p [x] =<br />
p<br />
i=0<br />
i<br />
l + p (−x)<br />
. (3.3)<br />
p − i i!
3.2. OAM transfer in general SPDC configurations<br />
A special case of the lg modes is the zero-order mode that does not carry oam.<br />
Since the zero-order Laguerre polynomial L 0 0 = 1, the zero-order Laguerre-<br />
Gaussian lg00 = 1 is the Gaussian mode given by<br />
<br />
w<br />
1<br />
2<br />
LG00(q) = exp<br />
2π<br />
<br />
− w2 q 2<br />
4<br />
<br />
. (3.4)<br />
This is not, however, the only mode with oam equal to zero, the same is true<br />
for all other modes lg0p. Spiral harmonics are defined to collect all the modes<br />
with the same oam value, regardless of the value of p, these modes are defined<br />
as<br />
LGl(q) = <br />
LGlp(q) (3.5)<br />
p<br />
=al(q) exp [ilθ].<br />
The phase dependence on l, shown by each spiral harmonic mode, is exploited<br />
to determine the photon’s oam content [52]. Consider, for instance, a photon<br />
with a spatial distribution given by Φ(q), which using the spiral harmonic<br />
modes can be written as<br />
∞<br />
Φ(q) = al(q) exp [ilθ], (3.6)<br />
l=−∞<br />
so that each mode in the decomposition has a well defined oam of l per<br />
photon. Therefore, the probability Cl of having a photon with oam equal to l<br />
is the weight of the corresponding mode in the distribution:<br />
where<br />
Cl =<br />
al(q) = 1<br />
√ 2<br />
∞<br />
0<br />
2π<br />
0<br />
dq|al(q)| 2 q (3.7)<br />
dθΦ(q, θ) exp (−ilθ). (3.8)<br />
If the photon has a well defined oam of l0, the weight of the corresponding<br />
mode Cl0 = 1 and the weights of the other modes Cl=l 0 = 0. If Cl = 0 for<br />
different values of l, the photon state is a superposition of those modes with<br />
different oam values. Compared to the relative simplicity of these calculations,<br />
measuring the oam content is a more complicated task described in appendix<br />
C.<br />
Now that the techniques for the calculation of the oam content are introduced,<br />
the next section will use the spherical harmonics and the oam decomposition<br />
to study the transfer of oam from the pump photon to the signal and<br />
idler.<br />
3.2 OAM transfer in general SPDC configurations<br />
The oam carried by a photon is directly associated to its spatial shape. In<br />
order to simplify the description of the oam transfer mechanisms, this section<br />
considers only the spatial part of the mode function in a spdc configuration<br />
31
3. <strong>Spatial</strong> correlations and OAM transfer<br />
in which the space and the frequency are not correlated. By writing the mode<br />
function in a different coordinate system, it will be possible to show that a<br />
selection rule exists for the oam transfer in spdc.<br />
According to equation 1.6, all the fields have been defined in terms of a<br />
coordinate system related with their propagation direction. To study the oam<br />
transfer, we start by defining the fields in terms of a unique coordinate system,<br />
the one of the pump. This coordinate transformation is possible since there is<br />
a vector Kn such that kn · rn = Kn · rp.<br />
This coordinate transformation is more general than the one of section 1.3,<br />
where the plane that contained all photons was defined as the yz plane. As<br />
a consequence this section considers all possible emission directions over the<br />
cone, while section 1.3 considered only one.<br />
As was done for the vector kn in section 1.3, it is useful to separate the<br />
vector Kn in its longitudinal component Kz n and transversal component Qn. Taking into account the change of coordinates, the spatial part of the mode<br />
function in the semiclassical approximation can be written as<br />
<br />
∆kL<br />
Φq(Qs, Qi) ∝ Ep(Qs + Qi)sinc<br />
(3.9)<br />
2<br />
where the delta factor is given by<br />
with<br />
∆k = K z p(Qs + Qi) − K z s (Qi) − K z i (Qi). (3.10)<br />
K z j (Q) =<br />
<br />
ω 2 j n2 j<br />
c 2 − |Q|2 . (3.11)<br />
To study the oam transfer, consider the two functions in equation 3.9 separately.<br />
Consider first the sinc function. Due to its dependence on the delta<br />
factor, the sinc function depends only on the magnitudes |Qs +Qi|, Qs = |Qs|,<br />
Qi = |Qi|. And, since<br />
|Qs + Qi| 2 = Q 2 s + Q 2 i + 2QsQi cos(Θs − Θi), (3.12)<br />
the only angular dependence of the function is through the periodical term<br />
cos(Θs − Θi), therefore, using Fourier analysis it can be written in a very<br />
general form as<br />
<br />
∆kL<br />
sinc =<br />
2<br />
∞<br />
m=−∞<br />
Fm(Qs, Qi) exp [im(Θs − Θi)], (3.13)<br />
where, importantly, Fm(Qs, Qi) does not depend on the angles Θs and Θi.<br />
On the other hand, if we consider the pump beam as a Laguerre-Gaussian<br />
beam with a oam content of lp per photon, its mode function can be written<br />
as<br />
32<br />
Ep(Qs + Qi) ∝ exp<br />
<br />
− w2 p|Qs + Qi| 2<br />
4<br />
<br />
<br />
(Q x s + Q x i ) + i(Q y s + Q y<br />
i )<br />
l , (3.14)
3.2. OAM transfer in general SPDC configurations<br />
where the identity Q exp[iθ] = Q x + iQ y was used. Using equation 3.12, the<br />
last equation can be written as<br />
<br />
Ep(Qs + Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi<br />
<br />
cos(Θs − Θi))<br />
4<br />
<br />
l × Qs exp (iΘs) + Qi exp (iΘi)<br />
and by using the binomial theorem it becomes<br />
(3.15)<br />
lp<br />
<br />
lp<br />
Ep(Qs + Qi) ∝ Q<br />
l<br />
l=0<br />
l sQ lp−l<br />
i exp [ilΘs + ilpΘi − ilΘi]<br />
<br />
× exp − w2 p(Q2 s + Q2 i + 2QsQi<br />
<br />
cos(Θs − Θi))<br />
. (3.16)<br />
4<br />
By replacing equations 3.16 and 3.13 into equation 3.9, the two-photon spatial<br />
mode function becomes<br />
<br />
Φ(Qs, Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi<br />
<br />
cos(Θs − Θi))<br />
4<br />
∞<br />
× Gn(Qs, Qi) exp [inΘs + i(lp − n)Θi], (3.17)<br />
n=−∞<br />
where the value of Gn(Qs, Qi) does not depend on Θs and Θi. The index<br />
(lp − n) associated to Θi is fixed for each value of lp and ls, according to the<br />
phase dependence of the last equation. This relation can be written as the<br />
selection rule<br />
lp = ls + li. (3.18)<br />
Therefore, the angular momentum carried by the pump is completely transferred<br />
to the signal and idler photons.<br />
The geometry of the spdc configuration restricts the validity of the selection<br />
rule since it was deduced from equation 3.9, and that equation describes<br />
only the configuration in which all possible emission directions are taken into<br />
account. Collinear configurations achieve this condition by definition, but in<br />
the type-I noncollinear configurations, all the experiments up to now have measured<br />
only a certain portion of the whole cone. Under this condition the transfer<br />
of oam from the pump to the signal and idler is not governed necessarily by<br />
equation 3.18, in agreement with previous reports [32, 30, 31].<br />
Using numerical methods to calculate the oam content of the downconverted<br />
photons, the next section explores the oam transfer mechanisms in<br />
collinear configurations. This configurations are special cases where the generated<br />
photons propagate in the same direction and therefore, the detection<br />
system detects all photons from the emission cone.<br />
33
3. <strong>Spatial</strong> correlations and OAM transfer<br />
Configuration<br />
Weight<br />
1<br />
0<br />
-4 4<br />
0<br />
<strong>Spatial</strong><br />
-4 4<br />
distribution OAM content<br />
1<br />
Mode<br />
Figure 3.2: In a collinear configuration a pump beam with oam equal to 1 transfers<br />
its oam into the signal photon when the idler is projected into a Gaussian mode. The<br />
figure shows the pump in blue and the signal and idler in pink. Additionally, it shows<br />
the spatial distribution and the oam content of the signal photon for co-propagating<br />
and counter-propagating configurations.<br />
3.3 OAM transfer in collinear configurations<br />
The aperture of the down-conversion cone is given by the angle of emission<br />
of the photons. In the collinear configuration this angle is zero, and therefore<br />
the cone collapses onto its axis, as shown in figure 1.2 (d). As each photon<br />
propagates in the same direction as all the other photons, equation 3.18 is<br />
valid.<br />
To analyze the transfer of oam from the pump beam to the generated photons,<br />
we will calculate the signal oam content, assuming a lg1 pump beam so<br />
that lp = 1, and an idler photon projected into a Gaussian state li = 0. We<br />
will consider two kinds of collinear configurations: co-propagating and counterpropagating,<br />
both shown in figure 3.2. The collinear counter-propagating generation<br />
of photons requires fulfilling a special phase-matching condition, which<br />
can be achieved by quasi-phase-matching in a periodic structure. Such counterpropagating<br />
generation has been demonstrated in second harmonic generation<br />
[53], parametric fluorescence [54], and spdc in fibers [55]. Chapter 5 explores<br />
another way to generate counter-propagating two-photon states: Raman scattering.<br />
Figure 3.2 shows the calculated signal spatial distribution and oam content<br />
in two cases. The first row shows the collinear co-propagating case. The second<br />
row shows the collinear counterpropagating case.<br />
According to figure 3.2, the collinear case fulfills the selection rule, except<br />
for a minus sign in the counterpropagating case. The phase appears because<br />
the photon’s oam content is evaluated with respect to the pump propagation<br />
direction, as figure 3.3 shows. To take this effect into account equation 3.18<br />
34
Phase front<br />
3.3. OAM transfer in collinear configurations<br />
OAM respect to z<br />
direction<br />
Figure 3.3: The direction of rotation of the phase shift defines the sign of the mode’s<br />
oam content. For a Laguerre-Gaussian mode corresponding to positive oam content,<br />
the phase shift rotates clockwise with respect to the propagation direction. If the<br />
same mode propagates backwards, the phase shift rotates anticlockwise with respect<br />
to the original propagation direction, and the mode has a negative oam content.<br />
can be generalized to<br />
lp = dsls + d1li<br />
+2<br />
-2<br />
-2<br />
(3.19)<br />
where ds,i = ±1 corresponds to forward and backward propagation of the<br />
corresponding photon.<br />
Conclusion<br />
The oam content of the pump is completely transferred to the signal and idler<br />
photons, emitted all over a cone. This makes it possible to introduce a selection<br />
rule that always holds if all possible emission directions are considered, but not<br />
necessarily if only a small part is detected. This result clarifies the apparent<br />
contradiction between several works that support the selection rule [7, 28, 56],<br />
and those that report that it is not valid [32, 31, 57].<br />
When only a portion of all generated photons are considered there is no<br />
simple relationship between lp, ls and li. The next chapter studies the oam<br />
transfer in those cases, describing the effect of different spdc parameters on<br />
the amount of oam that is transferred to the subset of the photons considered.<br />
35
CHAPTER 4<br />
OAM transfer in<br />
noncollinear configurations<br />
The previous chapter describes the oam transfer from the pump to the signal<br />
and idler photons considering all the possible emission directions of those<br />
photons. However, in non collinear configurations, the photons detected are<br />
just a subset of the total emission cone; noncollinear configurations generate<br />
correlated photons, which are naturally spatially separated, which is exactly<br />
what makes noncollinear so important. The change in the geometry of the<br />
process, imposed by the detection system, has a strong effect on the oam of<br />
the detected photons. This chapter focuses on the description of the oam that<br />
is transferred to a small portion of the emitted photons in noncollinear configurations.<br />
To characterize the transfer, I study a particular spdc process in<br />
which the pump is a Gaussian beam and one of the photons is projected into a<br />
Gaussian mode. Therefore, the oam content of the other photon will indicate<br />
how close the configuration is to satisfying the oam selection rule. This chapter<br />
is divided in three sections. Section 4.1 shows a simple example in which the<br />
selection rule does not apply as just a small section of the cone is considered.<br />
In a more general scenario, section 4.2 shows that the violation of the selection<br />
rule is not only mediated by the emission angle, but also by the pump beam<br />
waist. Finally, section 4.3 shows the effect of the Poynting vector walk-off on<br />
the oam of the noncollinear photons. As a main result, this chapter explains<br />
how the pump beam waist and the Poynting vector walk-off affect the oam<br />
transfer, in the cases where the selection rule does not apply. By tailoring both<br />
parameters it is possible to generate photons with specific spatial shapes. For<br />
instance, it is possible to generate photons in Gaussian modes that are more<br />
efficiently coupled into single mode fibers.<br />
37
4. OAM transfer in noncollinear configurations<br />
4.1 Ellipticity in noncollinear configurations<br />
In the experimental implementation of spdc, the detection system selects the<br />
photons that are emitted in a certain spatial direction. In a collinear configuration,<br />
the selection is not a problem since all the photons are emitted in the<br />
same direction. But, in a noncollinear configuration, the photons are emitted in<br />
different azimuthal directions described by the angle α, therefore the detection<br />
system selects only a section of the full cone. As the symmetry is broken when<br />
only a portion of the cone is considered, the oam transfer mechanism can not<br />
be described by the selection rule in 3.18. This section studies this mechanism<br />
in a simple spdc configuration by calculating the spatial distribution of the<br />
signal photon after fixing the oam content of the pump and idler photons.<br />
As a first example, consider a Gaussian pump beam lp = 0 and an idler<br />
photon projected into a Gaussian mode li = 0, given by<br />
<br />
u(qi) = Ni exp − w2 i<br />
4 (qx2 i + q y2<br />
i )<br />
<br />
, (4.1)<br />
the spatial distribution of the signal photon is given by the normalized mode<br />
function<br />
<br />
Φs(qs) = Ns dqiΦq(qs, qi)u(qi). (4.2)<br />
This integral has an analytical solution for simple spdc configurations. Consider<br />
a degenerate spdc process with negligible Poynting vector walk-off. After<br />
suppressing the correlations between space and frequency the spatial part of<br />
the two-photon state is given by<br />
<br />
<br />
with<br />
Φq(qs, qi) = exp<br />
− w2 p<br />
4 ∆2 0 − w2 p<br />
4 ∆2 1<br />
∆0 =q x s + q x i<br />
∆1 =q y s cos ϕs + q y<br />
i cos ϕi<br />
∆k = − q y s sin ϕs + q y<br />
i<br />
sin ϕi.<br />
<br />
L<br />
sinc<br />
2 ∆k<br />
<br />
(4.3)<br />
(4.4)<br />
Therefore, using the sinc to exponential approximation, the signal mode function<br />
defined by equation 4.2 is given by<br />
<br />
<br />
Φs(qs) =Ns exp<br />
× exp<br />
<br />
−<br />
w2 pw2 i<br />
4(w2 p + w2 qx2 s<br />
i )2<br />
− 4w2 pγ 2 L 2 cos ϕs 2 sin ϕs 2 + (w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs)w 2 i<br />
4(w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs 2 + w 2 i )<br />
q y2<br />
s<br />
<br />
(4.5)<br />
When the coefficients of the variables q x s and q y s are equal, the signal mode<br />
function reduces to a Gaussian and the selection rule is fulfilled. These coefficients<br />
are equal in collinear configurations, or in noncollinear configurations in<br />
38<br />
.
4.2. Effect of the pump beam waist on the OAM transfer<br />
which the spdc parameters satisfy the relationship<br />
γ 2 L 2 w<br />
=<br />
2 pw4 i<br />
w4 i + 4w2 p(w2 i + w2 p) cos ϕ2 s<br />
. (4.6)<br />
In any other case, the coefficients of the variables q x s and q y s are different, and<br />
the signal mode function becomes elliptical. The ellipticity of the spatial profile<br />
implies the presence of non-Gaussian modes, and therefore it can be used as a<br />
qualitative probe that the selection rule is not fulfilled (lp = ls + li).<br />
As the ellipticity is an effect of the partial detection, it can be controlled<br />
by changing the total shape of the cone, or by changing the detector angular<br />
acceptance. The phase matching conditions define the shape of the cone as a<br />
function of the emission angle ϕs, the pump beam waist wp, and the length of<br />
the crystal L; the detector angular acceptance is a function of the waist of the<br />
spatial modes ws, wi. The next two sections use both the mode decomposition<br />
and the ellipticity to describe the effect of all these parameters on the signal<br />
oam content in a more general scenario than considered in this section.<br />
4.2 Effect of the pump beam waist on the OAM transfer<br />
<strong>Of</strong> all the spdc parameters that affect the signal ellipticity, the easiest to<br />
control is the pump beam waist. To change the angle of emission or the crystal<br />
length implies changing of the geometrical configuration. While changing the<br />
pump beam waist just requires adding lenses in the beam path. In the first<br />
part of this section, numerical calculations show the role of the pump beam<br />
waist on the signal oam content. The second part describes the experimental<br />
corroboration of this effect.<br />
4.2.1 Theoretical calculations<br />
Consider a spdc configuration as the one in equation 1.31. With a Gaussian<br />
pump beam and an idler photon projected into a Gaussian mode, the oam content<br />
of the signal photon can be used to describe the oam transfer mechanism<br />
in spdc. The selection rule is fullfilled only if ls = 0. Equivalently, one could<br />
choose to use the idler photon to study the oam transfer after projecting the<br />
signal into a Gaussian mode.<br />
In a degenerate type-i spdc process characterized by the parameters in<br />
table 4.1, a Gaussian pump beam, with wavelength λ 0 p = 405 nm illuminates<br />
a 10 mm ppktp crystal. The crystal emits signal and idler photons with a<br />
wavelength λ 0 s = λ 0 i = 810 nm. Both photons propagate at ϕs,i = 1 ◦ , after<br />
the crystal they traverse a 2f system, and finally the idler photon is projected<br />
into a Gaussian mode with wi → ∞, so that only idler photons with qi = 0<br />
are considered.<br />
Figure 4.1 shows the signal oam content for two values of the pump beam<br />
waist: wp = 100 µm in the left and wp = 1000 µm in the right. In the<br />
distributions, each bar represents a mode ls, with a weight in the distribution<br />
Cls given by the height of the bar. In the left part of the figure, where the<br />
pump beam waist is smaller, the distribution shows several modes. For larger<br />
waists, the Gaussian mode becomes the only important mode, as seen in the<br />
right part of the figure.<br />
39
4. OAM transfer in noncollinear configurations<br />
Weight<br />
1<br />
0<br />
(a) (b)<br />
-4 Mode 4 -4 Mode 4<br />
Figure 4.1: As the pump beam waist increases, the Gaussian mode in the signal oam<br />
distribution becomes more important. The left part of the figure, where wp = 100 µm,<br />
shows several non-Gaussian modes in blue, while the right part, where wp = 1000 µm,<br />
shows only a Gaussian mode in gray. The other parameters used to generate this<br />
figure are listed in table 4.1.<br />
Table 4.1: The parameters used in the theoretical calculations in section 4.2.<br />
Parameter Value<br />
Crystal ppktp<br />
L 10 mm<br />
ρ0<br />
0◦ Laser cw diode<br />
wp 100 − 1000 µm<br />
λp<br />
405 nm<br />
λ0 s<br />
ϕs<br />
810 nm<br />
1◦ Figure 4.2 shows the probability of finding a non-Gaussian mode as a function<br />
of the pump beam waist. The presence of non-Gaussian modes implies a<br />
violation of the selection rule in the configuration considered here. The figure<br />
shows the probability Cls = 0 for emission angles 1, 5, 10 ◦ . As the collinear<br />
angle ϕ becomes larger, the probability to detect signal photons with ls = 0<br />
increases. In a highly noncollinear configuration a large pump waist does not<br />
guarantee the generation of a Gaussian mode, that is, it does not imply the<br />
validity of the selection rule.<br />
Reference [30] introduced the noncollinear length defined as Lnc = wp/ sin ϕ.<br />
Lnc quantifies the strength of the violation of the selection rule due to the angle<br />
of emission and the pump beam waist. The authors defined this quantity<br />
based on the fact that when Lq y sin ϕ is small the sinc function (that introduces<br />
ellipticity) tends to one. As q y is of the order of 1/wp the condition can be<br />
written as L sin ϕ/wp ≪ 1, or L ≪ Lnc. In this regime the ellipticity of the<br />
mode function is small, and thus the selection rule lp = ls + li is fulfilled. This<br />
is the case for nearly all the experiments using the oam of photons generated<br />
in spdc [7, 12, 28, 56, 58, 25, 26, 29]. Instead, if the crystal length is larger<br />
than or equal to the noncollinear length L ≥ Lnc, a strong violation of the<br />
selection rule is expected. The experiments reported in references [31, 32, 57]<br />
are in this regime since the authors used highly focused pump beams or long<br />
crystals.<br />
40
1.0<br />
0.0<br />
4.2. Effect of the pump beam waist on the OAM transfer<br />
Nongaussian<br />
mode probability<br />
0.8<br />
0.4<br />
Emission<br />
angle<br />
10º<br />
0.1 Pump beam width 1mm<br />
Figure 4.2: The violation of the selection rule, given by the weight of the non-Gaussian<br />
modes, becomes more important as the collinear angle increases, or as the pump beam<br />
is more focalized. The parameters used to generate this figure are listed in table 4.1.<br />
4.2.2 Experiment<br />
Figure 4.3 depicts the experimental set-up that we used to measure the effect<br />
of the pump beam waist over the signal photon spatial shape. A laser beam<br />
with wavelength λ 0 p = 405 nm, and bandwidth ∆λp = 0.6 nm illuminated a<br />
lithium iodate (liio3) crystal with length L = 5 mm. The crystal was cut in a<br />
configuration for which non of the interacting waves exhibited Poynting vector<br />
walk-off, so ρ0 = 0 ◦ . The signal and idler photons are emitted with wavelengths<br />
λ 0 s = λ 0 i = 810 nm at ϕs,i = 17.1 ◦ . Table 4.2 summarizes the most relevant<br />
experimental parameters.<br />
A spatial filter after the laser provided an approximately Gaussian beam<br />
with a waist of 500 µm. By adding lenses before the crystal, this waist could<br />
be changed in the range of 32 − 500 µm.<br />
Each generated photon passed through a 2 − f system (f = 250 mm),<br />
that provided an image of the spatial shape of the two-photon state. Later,<br />
the photons passed through a broadband colored filter, that removed the scattered<br />
radiation at 405 nm, and through small pinholes to increase the spatial<br />
resolution.<br />
Multimode fibers mounted in xy translation stages collected the photons,<br />
and carried them into two single photon counting modules. A coincidence<br />
circuit counted the number of times that a signal photon was detected within<br />
8 ns of the detection of an idler photon. We used a data acquisition card to<br />
transfer the circuit’s output into a computer.<br />
We measured the signal and idler counts, as well as the number of coincidences<br />
for a fixed position of the idler in two orthogonal directions in the<br />
transverse signal plane as shown by figure 4.3. During the experiments it was<br />
checked that the use of narrow band filters of ∆λs = 10 nm does not modify<br />
5º<br />
1º<br />
41
4. OAM transfer in noncollinear configurations<br />
Laser and<br />
spatial filter lenses<br />
crystal<br />
lenses<br />
filter<br />
collection<br />
system<br />
pinhole<br />
x<br />
idler<br />
pinhole<br />
signal<br />
Figure 4.3: A spatially filtered cw laser is focused into a nonlinear crystal. A coincidence<br />
circuit measures the correlations between the photon counts in a fixed position<br />
for the idler photon with the counts of photons in two orthogonal directions in the<br />
signal transverse plane. The values of the different experimental parameters are listed<br />
in table 4.2.<br />
the measured shape of the coincidence rate, although it does modify the single<br />
counts spatial shape.<br />
Table 4.2: The parameters of the experiment described in section 4.2.<br />
Parameter Value<br />
Crystal liio3<br />
L 5 mm<br />
ρ0<br />
0◦ Laser cw diode<br />
wp 32 − 500 µm<br />
λp 405 nm<br />
∆λp 0.6 µm<br />
λ0 s<br />
∆λ<br />
810 nm<br />
0 s<br />
ϕs<br />
10 nm<br />
17.1◦ The left side of figure 4.4 shows the coincidence measurements in the x and<br />
y directions for a focalized pump beam. As the pump beam waist is wp = 32<br />
µm, the noncollinear length Lnc = 108.8 µm is much smaller than the crystal<br />
length. According with reference [30], in this regime the ellipticity of the signal<br />
photon should be appreciable.<br />
By fitting the result to a Gaussian function, the waist in the x direction<br />
is wx 960 µm, while in the y direction it is wy 150 µm. The waist in x<br />
is about six times larger than the waist in y, and therefore the signal spatial<br />
distribution is elliptical. As a reference, the figure shows the singles counts for<br />
the signal coupler.<br />
The right side of the figure shows the coincidence measurements for a larger<br />
pump beam, wp 500 µm. In this case, the noncollinear length Lnc = 1.7 mm<br />
is of the same order of magnitude as the crystal length. When fitting the<br />
42<br />
y
Coincidences<br />
800<br />
0<br />
-4<br />
4.2. Effect of the pump beam waist on the OAM transfer<br />
x<br />
y<br />
4mm<br />
Singles<br />
6x10 5<br />
~<br />
-1<br />
y x<br />
Singles<br />
5x10 4<br />
~<br />
1mm<br />
Figure 4.4: The ellipticity decreases as the pump beam waist increases from wp = 30<br />
µm in the left to 500 µm in the right image. As the pump beam waist affects<br />
the efficiency of the process, the coincidences on the left were collected over 600<br />
seconds and the ones in the right over 20 seconds. Solid lines are the best fit to the<br />
experimental data shown as points. The values of the experimental parameters are<br />
listed in table 4.2.<br />
Coincidences<br />
region width<br />
2.5m<br />
0.5<br />
0<br />
0<br />
x dimension<br />
y dimension<br />
L nc=1.1mm<br />
Pump width 330 500m Figure 4.5: The pump beam waist controls the width of the profile in the x direction,<br />
while it does not affect the width in the y direction. The width in both directions<br />
becomes similar when the noncollinear length is of the same order of magnitude as<br />
the crystal length. The vertical line shows the point where, according to the fits, both<br />
dimensions are equal. In this point the noncollinear length is 1.1 mm. The values of<br />
the experimental parameters are listed in table 4.2.<br />
transverse cuts to Gaussian functions, the waist in the x direction is wx 120<br />
µm and the waist in the y direction is wy 180 µm. The waist in the x<br />
direction is much smaller than before and of the same order as the waist in y<br />
which implies a decrease of the ellipticity.<br />
Figure 4.4 illustrates the changes in the width in the x and y directions as a<br />
function of pump beam waist. In the range where the waist varies from 32−500<br />
µm, the width in y remains almost constant while the width in x decreases by<br />
80%. After a certain threshold the width in x becomes stable at a value close<br />
43
4. OAM transfer in noncollinear configurations<br />
100µ m<br />
1mm<br />
pump crystal<br />
-4 4<br />
-4 4<br />
1<br />
0<br />
1<br />
0<br />
after 1mm<br />
-4<br />
-4<br />
output mode<br />
Figure 4.6: Due to the crystal birefringence, the oam content of a beam changes as<br />
the beam passes trough the material. The number of new oam modes introduced<br />
by the birefringence increases for more focused beams. The figure shows the mode<br />
content of a Gaussian beam after traveling 1 mm in the crystal, and at the output of<br />
the 5- mm-long crystal.<br />
to the value of the width in y, and therefore the ellipticity disappears.<br />
4.3 Effect of the Poynting vector walk-off on the OAM<br />
transfer<br />
Up to now, this chapter only considered spdc configurations where the Poynting<br />
vector walk-off was not relevant. However, since the selective detection of<br />
one section of the cone affects the oam transfer, the distinguishability introduced<br />
by the walk-off should affect it as well.<br />
To understand the effect of the walk-off on the oam transfer, consider that<br />
the spatial shape of the pump is modified as it passes trough the crystal due to<br />
the birefringence. A Gaussian photon thus acquires more modes when traveling<br />
in a crystal. This section explains this effect using theoretical calculations and<br />
experimental results.<br />
4.3.1 Theoretical calculations<br />
The displacement introduced by the walk-off, explained in section 1.3, changes<br />
the pump beam spatial distribution and therefore its oam content. The transverse<br />
profile of the pump, at each position z inside the nonlinear crystal, can<br />
be written as<br />
Ep (qp, z) = E0 exp<br />
<br />
−q 2 p<br />
w 2 p<br />
4<br />
z<br />
+ i<br />
2k0 <br />
∞<br />
Jn (zqp tan ρ0) exp {inθp}<br />
p n=−∞<br />
(4.7)<br />
where Jn are Bessel functions of the first kind. Based on this expression, figure<br />
4.6 shows the oam content of a Gaussian pump beam after traveling through<br />
a 5 mm crystal. New modes appear and become important as the pump beam<br />
gets narrower.<br />
The oam content of the pump with respect to z is different at different<br />
positions inside the crystal. Pairs of photons produced at the beginning or at<br />
the end of the crystal are effectively generated by a pump beam with different<br />
spatial properties, as figure 4.6 shows. This change in the pump beam is<br />
44
Weight<br />
1<br />
0<br />
4.3. Effect of the Poynting vector walk-off on the OAM transfer<br />
0º 90º Azimuthal angle 360º<br />
other<br />
modes<br />
Gaussian<br />
mode<br />
0º 90º Azimuthal angle 360º<br />
Gaussian<br />
mode<br />
other<br />
modes<br />
Figure 4.7: The probability of generating a Gaussian signal photon varies with the<br />
azimuthal angle, and has a maximum at α = 90 ◦ where the noncollinearity effect<br />
compensates the Poynting vector walk-off. At other angles, like α = 360 ◦ the probability<br />
of generating a Gaussian signal decreases and other modes become important<br />
in the distribution. Those non-Gaussian modes are more numerous for more focused<br />
beams, as seen by comparing the left part of the figure, where wp = 100 µm, to the<br />
right part, where wp = 600 µm.<br />
Signal<br />
purity<br />
0.3<br />
0.1<br />
0.0<br />
without<br />
walk-off<br />
with<br />
walk-off<br />
azimuthal<br />
0º 90º 180º 270º 360º angle<br />
Figure 4.8: Like the signal oam content, the correlations between the photons change<br />
with the azimuthal angle. The walk-off not only introduces an azimuthal variation<br />
but increases the correlations.<br />
translated into a change in the oam content of the generated pair with respect<br />
to the z axis.<br />
Additionally, because of the walk-off the generated photons are not symmetric<br />
with respect to the displaced pump beam, and their azimuthal position<br />
α becomes relevant. Figure 4.7 shows the weight of the mode ls = 0, and the<br />
weight of all other oam modes, as a function of the angle α for two different<br />
pump beam widths. In the left part, the pump beam waist is 100 µm, while<br />
in the right part it is 600 µm. The oam correlations of the two-photon state<br />
change over the down-conversion cone due to the azimuthal symmetry break-<br />
45
4. OAM transfer in noncollinear configurations<br />
ing induced by the spatial walk-off. The symmetry breaking implies that the<br />
correlations between oam modes do not follow the relationship lp = ls + li.<br />
Figure 4.7 also shows that for larger pump beams the azimuthal changes are<br />
smoothened out.<br />
The insets in figure 4.7 show the oam decomposition at α = 90 ◦ and at<br />
α = 360 ◦ . At α = 90 ◦ , the walk-off effect compensates the noncollinear effect,<br />
and the weight of the ls = 0 mode is larger than the weight of any other<br />
mode. This angle is optimal for the generation of heralded single photons with<br />
a Gaussian shape.<br />
The degree of spatial entanglement between the photons also exhibits az-<br />
imuthal variations depending on their emission direction. Figure 4.8 shows the<br />
] as a function of the azimuthal angle α with and with-<br />
signal purity T r[ρ2 signal<br />
out walk-off, for a pump beam waist wp = 100 µm with collection modes of<br />
ws = wi = 50 µm. When considering the walk-off, the degree of entanglement<br />
increases and becomes a function of the azimuthal angle. The purity has a<br />
minimum for α = 0◦ and α = 180◦ and maximum for α = 90◦ and α = 270◦ .<br />
The spatial azimuthal dependence affects especially those experimental configurations<br />
where photons from different parts of the cone are used, as in the<br />
case in the experiment reported in reference [12] for the generation of photons<br />
entangled in polarization. Using two identical type-i spdc crystals with the<br />
optical axes rotated 90◦ with respect to each other, the authors generated a<br />
space-frequency quantum state given by<br />
|Ψ〉 = 1<br />
<br />
√ dqsdqi Φ<br />
2<br />
1 q(qs, qi)|H〉s|H〉i + Φ 2 <br />
q(qs, qi)|V 〉s|V 〉i<br />
(4.8)<br />
where Φ 1 q(qs, qi) is the spatial mode function of the photons generated in the<br />
first crystal<br />
Φ 1 q(qs, qi) = Φq(qs, qi, α = 0 ◦ ) exp [iL tan ρ0(q y s + q y<br />
i )], (4.9)<br />
and Φ 2 q(qs, qi) is the spatial mode function of the photons generated in the<br />
second crystal<br />
Φ 2 q(qs, qi) = Φq(qs, qi, α = 90 ◦ ). (4.10)<br />
Since the photons generated in the first crystal are affected by the walk-off as<br />
they pass by the second crystal, the mode functions are different.<br />
Following chapter 2, the polarization state of the generated photons is calculated<br />
by tracing out the spatial variables from equation 4.8. The resulting<br />
state is described by the density matrix<br />
where<br />
46<br />
ˆρp = 1<br />
<br />
|H〉s|H〉i〈H|s〈H|i + |V 〉s|V 〉i〈V |s〈V |i<br />
2<br />
<br />
+c|H〉s|H〉i〈V |s〈V |i + |V 〉s|V 〉i〈H|s〈H|i<br />
(4.11)<br />
<br />
c = dqsdqiΦ 1 q(qs, qi)Φ 2 q(qs, qi). (4.12)
4.3. Effect of the Poynting vector walk-off on the OAM transfer<br />
Length<br />
0.5mm<br />
2mm<br />
beam<br />
waist<br />
Concurrence<br />
1<br />
50 500m Figure 4.9: The concurrence of the polarization entangled state given by equation<br />
4.8 decreases when the effect of the azimuthal variation is stronger, that is for small<br />
pump beam waist or for large crystals.<br />
The degree of correlation between space and polarization is given by the purity<br />
of the polarization state<br />
T r[ρ 2 p] =<br />
0.5<br />
0.0<br />
1 + |c|2<br />
. (4.13)<br />
2<br />
If the walk-off effect is negligible T r[ρ 2 p] = 1, then space and polarization are<br />
not correlated. If the walk-off is not negligible, the azimuthal changes in the<br />
spatial shape are correlated with the polarization of the photons.<br />
Figure 4.9 indirectly shows the effect of the azimuthal spatial information<br />
over the polarization entanglement. The entanglement between the photons<br />
is quantified using the concurrence C = |c|, which is equal to 1 for maximally<br />
entangled states. The figure shows the variation of C as a function of the pump<br />
beam waist for two crystal lengths. For small pump beam waist, the azimuthal<br />
effect is stronger and the concurrence decreases. As the waist increases, the<br />
walk-off effect becomes weaker and the value of the concurrence increases. This<br />
effect is more important for longer crystals, where the walk-off modifies the<br />
pump beam shape more.<br />
4.3.2 Experiment<br />
To experimentally corroborate the predicted azimuthal changes in the signal<br />
spatial shape we set up the experiment described by the parameters listed in<br />
table 4.3 and shown in figure 4.10. We used a continuous wave diode laser<br />
emitting at λp = 405 nm and with an approximate Gaussian spatial profile,<br />
obtained by using a spatial filter. A half wave plate (hwp) controlled the<br />
direction of polarization of the beam, while a lense focalized it on the input<br />
face of the crystal, with a beam waist of wp = 136 µm. The 5 mm liio3 crystal<br />
produced pairs of photons at 810 nm that propagated at ϕs,i = 4 ◦ . Due to<br />
the crystal birefringence, the pump beam exhibited a Pointing vector walk-off<br />
47
4. OAM transfer in noncollinear configurations<br />
Laser and<br />
spatial filter HWP lenses<br />
crystal<br />
lenses<br />
filter<br />
collection<br />
system<br />
camera<br />
pinhole<br />
Figure 4.10: By rotating the crystal and the pump beam polarization, it was possible<br />
to measure the coincidences between pairs of photons at different positions over<br />
the emission cone. The photons were collected using multimode fibers and 300 µm<br />
pinholes to increase the resolution. The values of the experimental parameters are<br />
listed in table 4.3.<br />
given by the angle ρ0 = 4.9 ◦ , while the generated photons did not exhibit<br />
spatial walk-off.<br />
Table 4.3: The parameters of the experiment described in section 4.3.<br />
Parameter Value<br />
Crystal liio3<br />
L 5 mm<br />
ρ0 4.9◦ Laser cw diode<br />
wp 136 µm<br />
λp 405 nm<br />
∆λp 0.4 nm<br />
λ0 s<br />
ϕs<br />
810 nm<br />
4◦ We measured the relative position of the pump beam in the xy plane at<br />
the input and output faces of the nonlinear crystal using a ccd camera. At<br />
the output of the crystal, the beam displacement due to the Poynting vector<br />
walk-off allowed us to determine the direction of the crystal’s optical axis, and<br />
to fix its direction parallel to the pump polarization, as shown in figure 4.11.<br />
Right after the crystal, each of the generated photons passed through a<br />
2 − f system with focal length f=50 cm. Cut-off spectral filters removed the<br />
remaining pump beam radiation. After the filters, the photons were coupled<br />
into multimode fibers. In order to increase the spatial resolution, we used small<br />
pinholes with a diameter of 300 µm.<br />
We kept the idler pinhole fixed and measured the coincidence rate while<br />
mapping the signal photon’s transversal shape by scanning the detector with<br />
a motorized xy translation stage, as in the right part of figure 4.3. Finally,<br />
we rotated the crystal and the pump beam polarization to measure the spatial<br />
correlations at different azimuthal positions on the down-conversion cone. After<br />
48
Before<br />
the crystal<br />
4.3. Effect of the Poynting vector walk-off on the OAM transfer<br />
After<br />
the crystal<br />
Polarization Optical axis<br />
Figure 4.11: The images obtained with a ccd camera show that before the crystal,<br />
the pump beam has a circular shape, and that after passing the crystal part of the<br />
beam gets displaced in the direction of the optical axis of the crystal. The displaced<br />
part corresponds to the portion of the beam polarized orthogonal to the crystal axis.<br />
In the case of the last image almost all the beam is displaced. The values of the<br />
experimental parameters are listed in table 4.3.<br />
every rotation, the tilt of the crystal was adjusted to achieve the generation of<br />
photons at the same noncollinear angle in all cases.<br />
The bottom row of figure 4.12 presents a sample of images taken at different<br />
azimuthal sections of the cone, while the upper row shows the expected shape<br />
predicted by the theory. Each column shows the coincidence rate for different<br />
angles, α = 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . Each point of these images corresponds to<br />
the recording of a 10 seconds measurement. The typical maximum number of<br />
coincidences is around 10 coincidences per second. Each image is 10 × 10 mm,<br />
and its resolution is 50 × 50 points.<br />
Theory<br />
Experiment<br />
0º 90º 180º 270º<br />
Figure 4.12: The coincidence measurements of the mode function, as well as the<br />
theoretical calculations, show that the ellipticity of the mode function changes for<br />
different positions on the cone. The ellipticity is minimal for α = 90 ◦ . The values of<br />
the experimental parameters are listed in table 4.3.<br />
49
4. OAM transfer in noncollinear configurations<br />
The different spatial shapes in figure 4.12 clearly show that the downconversion<br />
cone does not posses azimuthal symmetry. As was predicted by the<br />
theoretical calculations, the coincidence measurement for α = 90 ◦ , presents a<br />
nearly Gaussian shape, while the other cases are highly elliptical.<br />
The slight discrepancies between experimental data and theoretical predictions<br />
observed might be due to the small (but not negligible) bandwidth of the<br />
pump beam, and due to the fact that the resolution of our system is limited<br />
by the detection pinhole size.<br />
Conclusion<br />
The spdc parameters and the detection system determine the portion of the<br />
cone that is detected in a noncollinear configuration. This chapter explains how<br />
the pump beam waist and the Poynting vector walk-off affect the oam transfer.<br />
By tailoring both parameters it is possible to generate photons with specific<br />
spatial shapes. The walk-off affects especially those configurations where pairs<br />
of photons with different α are used.<br />
The next chapter extends the analysis of the spatial correlations to pairs<br />
of photons generated in Raman transitions. The chapter describes how the<br />
specific characteristics of that source are translated into the two-photon spatial<br />
state.<br />
50
CHAPTER 5<br />
<strong>Spatial</strong> correlations<br />
in Raman transitions<br />
<strong>Two</strong>-photon states can be generated in different nonlinear processes, and in every<br />
case the oam transfer will depend on the particular configuration. Raman<br />
transition is an alternative method for the generation of two-photon states.<br />
Several authors have proven the generation of correlated photons in polarization<br />
[59], frequency [60] and oam [23] via Raman transitions. Typical configurations<br />
involve the partial detection of the generated photons [61, 62, 63] in<br />
quasi-collinear configurations [64]. Just as in the case of spdc, the geometrical<br />
conditions determine the oam transfer. This chapter analyses the oam<br />
transfer in Raman transitions using the techniques of the previous chapters.<br />
This chapter is divided in three sections. Section 5.1 describes the generated<br />
two-photon state by introducing its mode function. Section 5.2 studies the<br />
oam content of one of the photons with the oam of the other photons fixed.<br />
Section 5.3 describes the effect of the geometry of the process on the spatial<br />
entanglement between the photons. Numerical calculations show the effect of<br />
the geometrical configuration on the oam transfer mechanism. The finite size<br />
of the nonlinear medium results in new effects that do not appear in spdc.<br />
51
5. <strong>Spatial</strong> correlations in Raman transitions<br />
pump<br />
g<br />
e<br />
s<br />
anti<br />
Stokes Stokes<br />
control<br />
Figure 5.1: One atom with a Λ-type energy level configuration, can produce Stokes<br />
and anti-Stokes photons by the interaction with the pump and control beams.<br />
5.1 The quantum state of Stokes and anti-Stokes photon<br />
pairs<br />
This section describes the Stokes and anti-Stokes state generated by Raman<br />
transitions in cold atomic ensembles in an analogous way to the description of<br />
the two-photon state generated via spdc in chapter 1. The section discusses the<br />
general characteristics of the nonlinear process, and introduces the two-photon<br />
mode function.<br />
Consider as a nonlinear medium an ensemble of n identical Λ−type cold<br />
atoms trapped in a magneto-optical trap (mot). The atoms have an energy<br />
level configuration with one excited state: |e〉 and two hyperfine ground states:<br />
|s〉, and |g〉. This is the case, for example, in the d2 hyperfine transition of<br />
87rb. In the initial state of the cloud all atoms are in the ground state |g〉, and<br />
after emission all atoms return to their initial state, as figure figure 5.1 shows.<br />
The two-photon generation results from the interaction of a single atom of<br />
the cloud with two counter-propagating classical beams in a four step process.<br />
In the first step, the atom gets excited by the interaction with the pump beam<br />
far detuned from the |g〉 → |e〉 transition. In the second step, the excited atom<br />
decays into the |s〉 state by emitting one Stokes photon in the direction zs as<br />
shown in figure 5.2. In the third step, the atom is re-excited by the interaction<br />
with the control beam far detuned from the |s〉 → |e〉 transition. In the last<br />
step, the atom decays to the ground state by emitting an anti-Stokes photon<br />
in the zas direction.<br />
If ω 0 i<br />
is the central angular frequency for the photons involved in the pro-<br />
cess (i = p, c, s, as), and k 0 i is the corresponding wave number at the central<br />
frequencies, energy and momentum conservation implies<br />
52<br />
and<br />
g<br />
e<br />
ω 0 p + ω 0 c = ω 0 s + ω 0 as, (5.1)<br />
k 0 p − k 0 c = k 0 s cos ϕs − k 0 as cos ϕas, (5.2)<br />
k 0 s sin ϕs = k 0 as sin ϕas. (5.3)<br />
s
x<br />
y<br />
z<br />
5.1. The quantum state of Stokes and anti-Stokes photon pairs<br />
zas yas<br />
xas<br />
pump<br />
anti Stokes<br />
Stokes<br />
ys<br />
control<br />
Figure 5.2: According to energy and momentum conservation, the generated photons<br />
counterpropagate. Equation 5.7 describes the relation between their propagation<br />
direction and the propagation direction of the pump and control beams, where due<br />
to the phase matching the angle of emission of the anti-Stokes photon is ϕas = π−ϕs.<br />
Because we consider a pump and a control beam with the same central frequency<br />
and k0 s k0 as, the phase matching conditions allow any angle of emission<br />
ϕas = π − ϕs, if it is not forbidden by the transition matrix elements [60].<br />
This assumption is valid only for cold atoms, since for warm atoms the process<br />
is highly directional, that is all photons are emitted along a preferred direction<br />
as proven in reference [61].<br />
There are two ways to describe the generated two-photon quantum state: it<br />
can be described by using two coupled equations in the slowly varying envelope<br />
approximation for the Stokes and anti-Stokes electric fields [65], or alternatively<br />
by using an effective Hamiltonian of interaction and first order perturbation<br />
theory [66, 67]. As the latter approach is analogous to the formalism used in<br />
chapter 1, it will be used in what follows to calculate the Stokes and anti-Stokes<br />
state.<br />
The effective Hamiltonian in the interaction picture HI describes the photonatom<br />
interaction, and is given by<br />
<br />
HI = ɛ0<br />
ϕas<br />
ϕ s<br />
xs<br />
zs<br />
zc<br />
yc<br />
xc<br />
dV χ (3) Ê − as Ê− s Ê+ c Ê+ p + h.c. (5.4)<br />
where χ (3) is the effective nonlinearity, independent of the beam intensity since<br />
the pump and the control are non-resonant [65]. Assuming a Gaussian distribution<br />
of atoms in the cloud the effective nonlinearity χ (3) can be written<br />
as<br />
χ (3) <br />
(x, y, z) ∝ exp − x2 + y2 R2 z2<br />
−<br />
L2 <br />
(5.5)<br />
where R is the size of the cloud of atoms in the transverse plane (x, y) and L<br />
is the size in the longitudinal direction.<br />
53
5. <strong>Spatial</strong> correlations in Raman transitions<br />
According to equation 1.6, the electric field operators Ên in equation 5.4,<br />
are given by<br />
Ê (+)<br />
<br />
n (rn, t) = dkn exp [ikn · rn − iωnt]â(kn) (5.6)<br />
where, as in equation 1.23, we are using a more convenient set of transverse<br />
wave vector coordinates given by<br />
ˆxs,as =ˆx<br />
ˆys,as =ˆy cos ϕs,as + ˆz sin ϕs,as<br />
ˆzs,as = − ˆy sin ϕs,as + ˆz cos ϕs,as.<br />
(5.7)<br />
Under these conditions, at first order perturbation theory, the spatial quantum<br />
state of the generated pair of photons is<br />
<br />
|Ψ〉 =<br />
(5.8)<br />
dqsdqasΦ (qs, qas) |qs〉s|qas〉s<br />
where the mode function Φ (qs, qas) of the two-photon state is<br />
<br />
<br />
Φ (qs, ωs, qas, ωas) = dqpdqcEp (qp) Ec (qc) exp − ∆20R 2<br />
4 − ∆21R 2<br />
4 − ∆22L 2 <br />
4<br />
(5.9)<br />
with the delta factors defined as<br />
∆0 = q x s + q x as<br />
∆1 = (ks − kas) sin ϕs + (q y s − q y as) cos ϕs<br />
∆2 = kp − kc − (ks + kas) cos ϕ + (q y s − q y as) sin ϕs<br />
and the longitudinal wave vector of the pump beam given by<br />
ωpnp kp =<br />
c<br />
2<br />
− ∆ 2 0 − ∆ 2 1<br />
1/2<br />
(5.10)<br />
. (5.11)<br />
In the case of the Stokes and anti-Stokes two-photon state, there are no correlations<br />
between space and frequency due to the narrow bandwidth (∼ GHz)<br />
of the generated photons [60]. Therefore, to analyze the spatial shape of the<br />
mode function Φq (qs, qas), we can consider ωs = ω 0 s and ωas = ω 0 as.<br />
The effect of the unavoidable spatial filtering produced by the specific op-<br />
tical detection system used is described by Gaussian filters. The angular acceptance<br />
of the single photon detection system is 1/ k0 <br />
sws,as . In most experimental<br />
configurations, ws ≈ 50-150 µm and the length of the cloud is a few<br />
millimeters or less. For simplicity, we will assume that ws = was.<br />
In the calculations, the pump and the control are Gaussian beams with<br />
the same waist wp at the center of the cloud. As the waist is typically about<br />
200-500 µm, the Rayleigh range of the pump, Stokes and anti-Stokes modes<br />
Lp = πw2 p/λp and Ls,as = πw2 s/λs,as satisfy L ≪ Lp, Ls,as. This condition<br />
allows us to neglect the transverse wavenumber dependence of all longitudinal<br />
wave vectors in equations 5.9 and 5.10.<br />
54
where<br />
5.2. Orbital angular momentum correlations<br />
Under these conditions, the mode function Φq (qs, qas) can be written as<br />
Φq (qs, qas) = (ABCD)1/4<br />
<br />
π<br />
× exp − A<br />
4 (qx s + q x as) 2 − B<br />
4 (qx s − q x as) 2<br />
<br />
<br />
× exp − C<br />
4 (qy s + q y as) 2 − D<br />
4 (qy s − q y as) 2<br />
<br />
A = w2 pR2 2R2 + w2 +<br />
p<br />
w2 s<br />
2<br />
B = w2 s<br />
2<br />
C = w2 s<br />
2<br />
D = w2 pR 2 cos 2 ϕs<br />
2R 2 + w 2 p<br />
(5.12)<br />
+ L 2 sin 2 ϕs + w2 s<br />
. (5.13)<br />
2<br />
The specific characteristics of the state are determined by the size of the atomic<br />
cloud in the longitudinal and transverse planes, L and R; by the beam waist of<br />
the pump and control beams, wp,c; by the waist of the Stokes and anti-Stokes<br />
modes ws; and by the angle of emission ϕs. The next section describes the<br />
effect off all these parameters on the oam transfer from the pump and the<br />
control beams to the pair Stokes and anti-Stokes.<br />
5.2 Orbital angular momentum correlations<br />
Since the pump and control beams are Gaussian beams, lp = lc = 0, the<br />
analysis of the oam transfer reduces to the study of the oam content of the<br />
Stokes and anti-Stokes pair. The Stokes oam content becomes the only free<br />
variable, after projecting the anti-Stokes into a Gaussian mode<br />
u(qas) = Nas exp<br />
<br />
− w2 g<br />
4 (qx2 as + q y2<br />
as)<br />
<br />
(5.14)<br />
with beam width wg at the center of the cloud. The Stokes mode function<br />
defined as<br />
<br />
Φs (qs) = dqasΦ (qs, qas) u(qas) (5.15)<br />
becomes<br />
<br />
(F G)(1/4)<br />
Φs (qs) = exp −<br />
1/2<br />
(2π) F<br />
4 (qx s ) 2 − G<br />
4 (qy s ) 2<br />
<br />
, (5.16)<br />
55
5. <strong>Spatial</strong> correlations in Raman transitions<br />
with<br />
F = 4AB + (A + B) w2 g<br />
A + B + w 2 g<br />
G = 4CD + (C + D) w2 g<br />
C + D + w2 . (5.17)<br />
g<br />
As in the previous chapter, this mode function can be written as a superposition<br />
of spherical harmonics<br />
Φs (qs, θs) = (2π)<br />
−1/2 <br />
ls<br />
als (qs) exp (ilsθs) . (5.18)<br />
The probability of having a Stokes photon with oam equal to ls is given by the<br />
weight of each spiral mode<br />
<br />
F + G<br />
C2ls = (F G)1/2 dqsqs exp − q<br />
4<br />
2 <br />
s I 2 <br />
G − F<br />
ls q<br />
8<br />
2 <br />
s (5.19)<br />
where Ils are the Bessel functions of the second kind. Only even modes appear<br />
in the distribution as a consequence of the symmetry of the Stokes mode<br />
function.<br />
Figure 5.3 shows the weight of the mode ls = 0 as a function of the angle<br />
of emission for different values of the length of the cloud of atoms. For nearly<br />
collinear configurations, the probability of having a Gaussian Stokes photon is<br />
one, therefore, in this case the process satisfies the relation lp+lc = ls +las. For<br />
these kind of configurations, A = D in equation 5.12, and therefore the Stokes<br />
spatial mode function shows cylindrical symmetry in the transverse planes<br />
(xs, ys) and (xas, yas). This is the case in most experimental configurations<br />
[59, 64, 68], and in fact, reference [23] experimentally proves the relationship<br />
lp + lc = ls + las for collinear configurations.<br />
The probability of having a Gaussian Stokes photon decreases as other<br />
modes appear in the distribution in highly noncollinear configurations. The<br />
length of the cloud controls the importance of the other modes and it is possible<br />
to obtain a spatial mode with cylindrical symmetry for any emission angle, by<br />
fulfilling the condition<br />
<br />
L = R 1 + 2R2<br />
w2 −1/2<br />
. (5.20)<br />
p<br />
This condition reduces to L = R, when the beam waist is much larger than the<br />
transverse size of the cloud. Any deviation from a spherical volume of interaction<br />
(described by this condition) introduces ellipticity in the mode function.<br />
For this reason, a highly elliptical configuration, as the one described in reference<br />
[60], will not satisfy the oam selection rule.<br />
Figure 5.4 shows the weight of the mode ls = 0 as a function of the length<br />
of the cloud for different values of the emission angle. For any angle, the<br />
probability of having a Gaussian mode is maximum at the length given by<br />
equation 5.20. In a collinear configuration, the Stokes photon always has a<br />
Gaussian distribution, independently of the length of the cloud. As the angle<br />
of emission increases, the length of the cloud becomes more important. The<br />
mode weight is weakly affected by the change of the cloud length when the<br />
length is much longer than the other relevant parameters: ws, wg and R. This<br />
is especially evident for an angle of emission ϕ = 90◦ .<br />
56
weight<br />
1<br />
0.6<br />
-180º<br />
-90º<br />
0º 90º 180º<br />
5.3. <strong>Spatial</strong> entanglement<br />
length<br />
0.2mm<br />
0.4mm<br />
1mm<br />
2mm<br />
angle<br />
Figure 5.3: The weight of the Gaussian mode in the oam decomposition for the Stokes<br />
photon has a maximum in the collinear configuration. In noncollinear configurations<br />
the probability of a Gaussian Stokes photon decreases as the length of the cloud<br />
increases. Table 5.1 lists the parameters used to generate this figure.<br />
weight angle<br />
1<br />
0.6<br />
0.2 Length 1.5 mm<br />
Figure 5.4: The probability of having a Gaussian Stokes photon is maximum in a<br />
collinear configuration independently of the cloud length. In the noncollinear cases<br />
the maximum appears at the length given by equation 5.20, in these configurations<br />
the probability of having a Gaussian Stokes photon changes drastically with the cloud<br />
length and the emission angle. Table 5.1 lists the parameters used to generate this<br />
figure.<br />
5.3 <strong>Spatial</strong> entanglement<br />
The spatial correlations between the generated photons inherit the angular<br />
dependence from the Stokes oam content. To explore this phenomena, this<br />
0º<br />
10º<br />
90º<br />
57
5. <strong>Spatial</strong> correlations in Raman transitions<br />
Schmidt number<br />
40<br />
20<br />
1<br />
-180º<br />
0º 180º<br />
lenght spatial filter<br />
2 mm<br />
1 mm<br />
400 m<br />
200 m<br />
7<br />
4<br />
1<br />
-180º<br />
0º 180º<br />
100 m<br />
200 m<br />
500 m<br />
angle<br />
Figure 5.5: The length of the cloud controls the Schmidt number angular dependence.<br />
By modifying this length it is possible to change the position of the maximum and<br />
minimum values of the Schmidt number. The angular dependence can be removed<br />
completely by tailoring the parameters of the process, or by filtering. Table 5.1 lists<br />
the parameters used to generate this figure.<br />
Table 5.1: The parameters used in figures 5.3, 5.4 and 5.5<br />
Parameter Figure 5.3 Figure 5.4 Figure 5.5 (left) Figure 5.5 (right)<br />
L 0.2, 0.4, 1, 2 mm [0.2, 1.5] mm 0.2, 0.4, 1, 2 mm 200 µm<br />
wp 100 µm 100 µm 500 µm 500 µm<br />
R 400µm 400µm 1000 µm 1000 µm<br />
ws 100 µm 100 µm 100 µm 100, 200, 500 µm<br />
wg 500 µm 500 µm<br />
ϕs [−180, 180] ◦ 0, 10, 90 ◦ [−180, 180] ◦ [−180, 180] ◦<br />
section shows the effects of changing the emission angle on the Schmidt number<br />
K defined in section 2.1, is a common entanglement quantifier.<br />
The Schmidt number of the two-photon state described by equation 5.12<br />
quantifies the entanglement between the generated photons. According to references<br />
[69, 70] it is given by<br />
(A + B) (C + D)<br />
K =<br />
4 (ABCD) 1/2<br />
. (5.21)<br />
Figure 5.5 shows the Schmidt number as a function of the emission angle for<br />
different values of the atomic cloud length and the pump beam waist. The<br />
function extrema are located at 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . At each of these<br />
values the function can have a maximum or a minimum depending on the<br />
other parameters of process. For instance, if<br />
L < R<br />
<br />
1 + 2R2<br />
w2 −1/2<br />
p<br />
(5.22)<br />
the entanglement is maximum at collinear configurations where ϕ = 0 ◦ , 180 ◦ ,<br />
and minimum at transverse configurations ϕ = 90 ◦ , 270 ◦ . In the left part of<br />
figure 5.5, the condition in equation 5.22 is achieved at cloud lengths smaller<br />
than 333 µm.<br />
As the length of the cloud increases, the variation of the entanglement with<br />
the angle is smoothed out. In the left part of figure 5.5 the relation in equation<br />
58
5.3. <strong>Spatial</strong> entanglement<br />
5.20 is satisfied when L = 333 µm; therefore, the amount of entanglement is<br />
constant. If the length of the cloud increases even more, the position for the<br />
maxima and the minima get inverted. When<br />
<br />
L > R 1 + 2R2<br />
w2 −1/2<br />
p<br />
(5.23)<br />
the entanglement is maximum for transverse emitting configurations ϕ = 90 ◦ , 270 ◦ ,<br />
and minimum for collinear configurations ϕ = 0 ◦ , 180 ◦ . This variation is possible<br />
in Raman transitions where the transversal size of the cloud is comparable<br />
to the longitudinal size.<br />
Another parameter that plays a role in the variation of the amount of<br />
entanglement is the Stokes spatial filter ws. The right side of figure 5.5 shows<br />
the effect of changing the filtering over the amount of entanglement. Very<br />
narrow spatial filters (ws → ∞) diminish both the amount of entanglement<br />
and its azimuthal variability.<br />
Conclusion<br />
As in spdc, the geometrical configuration of the Raman transitions determines<br />
the oam content and the spatial correlations of the generated Stokes and anti-<br />
Stokes photons. The size and shape of the cloud defines the emission angles<br />
for which the correlations are maximum and minimum.<br />
59
CHAPTER 6<br />
Summary<br />
This thesis characterizes the correlations in two-photon quantum states, both<br />
between photons, and between the spatial and frequency degrees of freedom.<br />
The photon pairs are generated by spontaneous parametric down-conversion<br />
spdc, or by the excitation of Raman transitions in cold atomic ensembles.<br />
Special attention is given to the entanglement of the photon pair in the spatial<br />
degree of freedom, associated to the orbital angular momentum oam. The<br />
main contributions or the thesis are:<br />
A novel matrix formalism to describe the two-photon mode function.<br />
This formalism makes it possible to analytically calculate several features<br />
of the down-converted photons, and reduces the numerical calculation time of<br />
other features, as shown in chapter 1 and reference [41].<br />
Analytical expressions for the purity of the subsystems formed by<br />
a single photon in space and frequency, or by a spatial two-photon<br />
state. These expressions quantify the correlations in the two-photon state,<br />
and make the effect of the various parameters of the spdc process on these<br />
correlations explicit. The thesis presents a set of conditions to suppress or<br />
enhance the correlations between the photons, or between degrees of freedom<br />
in the two-photon state. With these conditions, it is possible to design spdc<br />
sources with specific levels of correlations, for example, sources that generate<br />
pure heralded single photons, or alternatively, maximally entangled states; as<br />
shown in chapter 2 and in references [41, 45].<br />
A selection rule for oam transfer. This selection rule explains several<br />
contradictory measurements of the oam transfer from the pump beam to the<br />
signal and idler photons in spdc. We showed that the oam content of the<br />
pump is completely transferred to the photons emitted in all directions. This<br />
selection rule always holds if all possible emission directions are considered,<br />
for example in collinear configurations, but not necessarily if only a part of<br />
all emission directions is detected. These conditions for the validity of the<br />
selection rule give clear guidelines for the design of sources for protocols that<br />
exploit the multidimensionality of oam, as shown in chapter 3 and references<br />
[71, 72].<br />
61
6. Summary<br />
Experiments that explain oam transfer in noncollinear spdc. In<br />
noncollinear configurations only a subset of emission directions is detected. The<br />
transfer of oam to the detected photons is strongly affected by the change in<br />
geometry imposed by the detection system, and depends on the angle of detection,<br />
the pump-beam waist and the Poynting-vector walk-off. The thesis shows<br />
that, by tailoring these parameters, it is possible to design noncollinear sources<br />
that naturally generate spatially-separated photons with specific desired spatial<br />
shapes, as shown in chapter 4 and references [39, 33, 32].<br />
An analysis of the spatial correlations generated by Raman transitions<br />
in cold atomic ensembles. Like for spdc, the spatial correlations<br />
between the Stokes and anti-Stokes photons depend on the geometrical configuration<br />
of the pump beam, the control beam, and the Stokes and anti-Stokes<br />
photons. This thesis shows how the size and shape of the atom cloud define<br />
the emission angles for which the correlations are maximum and minimum, as<br />
shown in chapter 5 and reference [73].<br />
62
APPENDIX A<br />
The matrix form<br />
of the mode function<br />
According to section 1.3, the normalized two-photon mode function, after some<br />
approximations, reads<br />
<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p<br />
4 ∆2 0 − w2 p<br />
4 ∆21 <br />
× exp − (γL)2<br />
4 ∆2k − T 2 0<br />
4 (Ωs + Ωi) 2<br />
<br />
<br />
× exp − w2 s<br />
2 |qs| 2 − w2 i<br />
2 |qi| 2 − 1<br />
2B2 Ω<br />
s<br />
2 s − 1<br />
2B2 Ω<br />
i<br />
2 <br />
i . (A.1)<br />
The argument of the exponential function is a second order polynomial. Each<br />
term is the product of at most two variables (qx s , qy s , qx i , qy i , Ωs, Ωi) and a coefficient<br />
f = a<br />
4 qx2 s + b<br />
4 qy2 s + c<br />
4 qx2 i + d<br />
4 qy2<br />
h<br />
i + . . . +<br />
2 qx s q y s + . . . + z<br />
2 ΩsΩi. (A.2)<br />
Such a polynomial can be written as the product of a matrix A with the<br />
coefficients as elements, and a vector x of the variables<br />
f = 1 <br />
x qs 4<br />
qy s qx i q y<br />
i Ωs Ωs<br />
⎛<br />
⎜<br />
⎜<br />
⎝<br />
a<br />
h<br />
i<br />
j<br />
k<br />
h<br />
b<br />
m<br />
n<br />
p<br />
i<br />
m<br />
c<br />
s<br />
t<br />
j<br />
n<br />
s<br />
d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
f<br />
l<br />
r<br />
u<br />
w<br />
z<br />
⎞ ⎛<br />
q<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎠ ⎝<br />
l r u w z g<br />
x s<br />
qy s<br />
qx i<br />
q y<br />
i<br />
Ωs<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
Ωi<br />
(A.3)<br />
therefore, the mode function can be written using matrix notation as<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1<br />
2 xt <br />
Ax . (A.4)<br />
63
A. The matrix form of the mode function<br />
Each of the terms of the matrix is defined by comparing the product of the<br />
polynomial f with the argument of the exponential in equation 1.34. The<br />
elements of matrix A are<br />
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0 cos α 2<br />
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2 − 2γ 2 L 2 sin ϕs cos ϕs sin α tan ρ0<br />
+ γ 2 L 2 cos ϕs 2 tan ρ0 2 sin α 2<br />
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0 cos α 2<br />
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2 + 2γ 2 L 2 sin ϕi cos ϕi tan ρ0 sin α<br />
+ γ 2 L 2 cos ϕi 2 tan ρ0 2 sin α 2<br />
f =2B −2<br />
s + T 2 0 + γ 2 L 2 N 2 p + γ 2 L 2 N 2 s cos ϕs 2 + w 2 pN 2 s sin ϕs 2 − 2γ 2 L 2 NpNs cos ϕs<br />
− 2γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + 2γ 2 L 2 N 2 s cos ϕs sin ϕs sin α tan ρ0<br />
+ γ 2 L 2 N 2 s sin ϕs 2 tan ρ0 2 sin α 2<br />
g =2B −2<br />
i<br />
+ T 2 0 + γ 2 L 2 N 2 p + γ 2 L 2 N 2 i cos ϕi 2 + w 2 pN 2 i sin ϕi 2 − 2γ 2 L 2 NpNi cos ϕi<br />
+ 2γ 2 L 2 NpNi sin ϕi tan ρ0 sin α − 2γ 2 L 2 N 2 i cos ϕi sin ϕi tan ρ0 sin α<br />
+ γ 2 L 2 N 2 i sin ϕi 2 tan ρ0 2 sin α 2<br />
h = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α 2<br />
i =w 2 p + γ 2 L 2 tan ρ0 2 cos α<br />
j =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α<br />
k =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ns cos ϕs tan ρ0 cos α<br />
− γ 2 L 2 Ns sin ϕs tan ρ0 2 sin α cos α<br />
l =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ni cos ϕi tan ρ0 cos α<br />
+ γ 2 L 2 Ni sin ϕi tan ρ0 2 sin α cos α<br />
m = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α<br />
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi + γ 2 L 2 cos ϕs sin ϕi sin α tan ρ0<br />
− γ 2 L 2 sin ϕs cos ϕi sin α tan ρ0 + γ 2 L 2 cos ϕs cos ϕi tan ρ0 2 sin α 2<br />
p = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ns cos ϕs sin ϕs − w 2 pNs cos ϕs sin ϕs<br />
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ns cos ϕs 2 tan ρ0 sin α<br />
+ γ 2 L 2 Ns sin ϕs 2 tan ρ0 sin α − γ 2 L 2 Ns sin ϕs cos ϕs tan ρ0 2 sin α 2<br />
r = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ni sin ϕs cos ϕi + w 2 pNi cos ϕs sin ϕi<br />
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ni cos ϕs cos ϕi tan ρ0 sin α<br />
− γ 2 L 2 Ni sin ϕs sin ϕi tan ρ0 sin α + γ 2 L 2 Ni cos ϕs sin ϕi tan ρ0 2 sin α 2<br />
s =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α<br />
t =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ns cos ϕs tan ρ0 cos α − γ 2 L 2 Ns sin ϕs tan ρ0 2 sin α cos α<br />
u =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ni cos ϕi tan ρ0 cos α + γ 2 L 2 Ni sin ϕi tan ρ0 2 sin α cos α<br />
64
v =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ns cos ϕs sin ϕi − w 2 pNs sin ϕs cos ϕi<br />
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕs cos ϕi tan ρ0 sin α<br />
− γ 2 L 2 Ns sin ϕs sin ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕi sin ϕs tan ρ0 2 sin α 2<br />
w =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ni sin ϕi cos ϕi + w 2 pNi sin ϕi cos ϕi<br />
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ni cos ϕi 2 tan ρ0 sin α<br />
+ γ 2 L 2 Ni sin ϕi 2 tan ρ0 sin α + γ 2 L 2 Ni sin ϕi cos ϕi tan ρ0 2 sin α 2<br />
z =γ 2 L 2 N 2 p − γ 2 L 2 NpNi cos ϕi − γ 2 L 2 NpNs cos ϕs + γ 2 L 2 NsNi cos ϕs cos ϕi<br />
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi − γ 2 L 2 NsNi cos ϕs sin ϕi tan ρ0 sin α<br />
− γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + γ 2 L 2 NsNi cos ϕi sin ϕs tan ρ0 sin α<br />
− γ 2 L 2 NsNi sin ϕs sin ϕi tan ρ0 2 sin α 2 + γ 2 L 2 NpNi sin ϕi tan ρ0 sin α.<br />
(A.5)<br />
This set of expressions is far more useful than compact. The matrix terms<br />
become simpler in some particular spdc configurations that are treated in<br />
chapters 2 and 4. For instance, when the pump polarization is parallel to the<br />
x axis, the matrix terms become<br />
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0<br />
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2<br />
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0<br />
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2<br />
f = 2<br />
B2 s<br />
g = 2<br />
B2 i<br />
+ T 2 0 + γ 2 L 2 (Np − Ns cos ϕs) 2 + w 2 pN 2 s sin ϕs 2<br />
+ T 2 0 + γ 2 L 2 (Np − Ni cos ϕi) 2 + w 2 pN 2 i sin ϕi 2<br />
h =m = −γ 2 L 2 sin ϕs tan ρ0<br />
i =w 2 p + γ 2 L 2 tan ρ0 2<br />
j =s = γ 2 L 2 sin ϕi tan ρ0<br />
k =t = γ 2 L 2 tan ρ0(Np − Ns cos ϕs)<br />
l =u = γ 2 L 2 tan ρ0(Np − Ni cos ϕi)<br />
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi<br />
p = − γ 2 L 2 sin ϕs(Np − Ns cos ϕs) − w 2 pNs cos ϕs sin ϕs<br />
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi<br />
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs<br />
w =γ 2 L 2 sin ϕi(Np − Ni cos ϕi) + w 2 pNi cos ϕi sin ϕi<br />
z =γ 2 L 2 N 2 p − γ 2 L 2 Np(Ni cos ϕi + Ns cos ϕs) + γ 2 L 2 NsNi cos ϕs cos ϕi<br />
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi.<br />
(A.6)<br />
The matrix notation extends to functions of the mode function. For instance,<br />
the purity of the spatial part of the two-photon state, given by equation 2.11,<br />
65
A. The matrix form of the mode function<br />
writes<br />
T r[ρ 2 <br />
q] =<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i). (A.7)<br />
where the dimension increases as new primed variables appear. Using the<br />
matrix notation the integrand becomes<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i)<br />
= N 4 <br />
exp − 1<br />
2 Xt <br />
BX , (A.8)<br />
where the vector X is the result of concatenation of x and x ′ , such that<br />
⎛<br />
⎜<br />
X = ⎜<br />
⎝<br />
↑<br />
x<br />
↓<br />
↑<br />
x ′<br />
↓<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(A.9)<br />
and the new matrix B is given by<br />
B = 1<br />
⎛<br />
⎜<br />
2 ⎜<br />
⎝<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
l<br />
0<br />
0<br />
0<br />
0<br />
k<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
r<br />
0<br />
0<br />
0<br />
0<br />
p<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
u<br />
0<br />
0<br />
0<br />
0<br />
t<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
w<br />
0<br />
0<br />
0<br />
0<br />
v<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
2z<br />
k<br />
p<br />
t<br />
v<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
2g<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
0<br />
0<br />
0<br />
k<br />
l<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
0<br />
0<br />
0<br />
0<br />
p<br />
r<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
0<br />
0<br />
0<br />
0<br />
t<br />
u<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
0<br />
0<br />
0<br />
0<br />
v<br />
w<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
0<br />
0<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
l r u w 0 0 l r u w 2z 2g<br />
(A.10)<br />
In an analogous way, the integrand on the expression for the signal photon<br />
purity<br />
<br />
T r[ρ 2 signal] =<br />
is written in a matrix notation as<br />
66<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i). (A.11)<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i)<br />
= N 4 exp<br />
<br />
− 1<br />
2 Xt <br />
CX . (A.12)
The matrix C is given by<br />
C = 1<br />
⎛<br />
2a<br />
⎜ 2h<br />
⎜ i<br />
⎜ j<br />
⎜ 2k<br />
⎜ l<br />
2 ⎜ 0<br />
⎜ 0<br />
⎜ i<br />
⎜ j<br />
⎝ 0<br />
2h<br />
2b<br />
m<br />
n<br />
2p<br />
r<br />
0<br />
0<br />
m<br />
n<br />
0<br />
i<br />
m<br />
2c<br />
2s<br />
t<br />
2u<br />
i<br />
m<br />
0<br />
0<br />
t<br />
j<br />
n<br />
2s<br />
2d<br />
v<br />
2w<br />
j<br />
n<br />
0<br />
0<br />
v<br />
2k<br />
2p<br />
t<br />
v<br />
2f<br />
z<br />
0<br />
0<br />
t<br />
v<br />
0<br />
l<br />
r<br />
2u<br />
2w<br />
z<br />
2g<br />
l<br />
r<br />
0<br />
0<br />
z<br />
0<br />
0<br />
i<br />
j<br />
0<br />
l<br />
2a<br />
2h<br />
i<br />
j<br />
2k<br />
0<br />
0<br />
m<br />
n<br />
0<br />
r<br />
2h<br />
2b<br />
m<br />
n<br />
2p<br />
i<br />
m<br />
0<br />
0<br />
t<br />
0<br />
i<br />
m<br />
2c<br />
2s<br />
t<br />
j<br />
n<br />
0<br />
0<br />
v<br />
0<br />
j<br />
n<br />
2s<br />
2d<br />
v<br />
0<br />
0<br />
t<br />
v<br />
0<br />
z<br />
2k<br />
2p<br />
t<br />
v<br />
2f<br />
l<br />
r<br />
0<br />
0<br />
z<br />
0<br />
l<br />
r<br />
2u<br />
2w<br />
z<br />
l r 0 0 z 0 l r 2u 2w z 2g<br />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
(A.13)<br />
67
APPENDIX B<br />
Integrals of<br />
the matrix mode function<br />
Given a n×n symmetric and positive definite matrix A, and two n order vectors<br />
x and b<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = (2π)n/2<br />
<br />
exp −<br />
det(A) btA−1 <br />
b<br />
.<br />
2<br />
(B.1)<br />
To proof this result consider that, since A is positive definite, there exist another<br />
matrix O such that OAO t = D where D is a diagonal matrix. With the<br />
transformation y = Ox, the integral in equation B.1 becomes<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = dy exp<br />
<br />
− yt Dy<br />
2 + iyt b ′<br />
<br />
,<br />
(B.2)<br />
with b ′ = O −1 b.<br />
As the argument of the exponential, in the right side of equation B.1, is<br />
given by the matrix product<br />
− 1 <br />
y1<br />
2<br />
y2 . . . yn<br />
⎛<br />
d1<br />
⎜ 0<br />
⎜<br />
⎝ .<br />
0<br />
d2<br />
.<br />
. . .<br />
. . .<br />
. ..<br />
0<br />
0<br />
.<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎠ ⎝<br />
0 0 0 dn<br />
y1<br />
y2<br />
.<br />
yn<br />
⎞<br />
⎛<br />
⎟<br />
⎠ + i ⎜<br />
y1 y2 . . . yn ⎜<br />
⎝<br />
the integral can be written in a polynomial form as<br />
<br />
dx exp − xtAx 2 + ixt <br />
b<br />
<br />
<br />
= dy1dy2...dyn exp − y2 1d1<br />
2 + iy1b ′ 1 − y2 2d2<br />
2 + iy2b ′ 2... − y2 ndn<br />
2 + iynb ′ <br />
n<br />
(B.3)<br />
(B.4)<br />
69<br />
b ′ 1<br />
b ′ 2<br />
.<br />
b ′ n<br />
⎞<br />
⎟<br />
⎠ ,
B. Integrals of the matrix mode function<br />
where the integrals in each variable are independent, so it is possible to write<br />
them as<br />
<br />
dx exp − xtAx <br />
=<br />
2 + ixt <br />
b<br />
<br />
dy1 exp − y2 1d1<br />
2 + iy1b ′ 1<br />
<br />
dy2<br />
<br />
− y2 2d2<br />
2 + iy2b ′ <br />
2 ...<br />
which, after solving each integral separately, becomes<br />
<br />
<br />
dx exp − xtAx 2 + ixt <br />
b =<br />
n<br />
j=1<br />
<br />
2π<br />
dj<br />
exp<br />
<br />
dyn<br />
− b′ 2<br />
j<br />
2dj<br />
<br />
− y2 ndn<br />
2 + iynb ′ <br />
n ,<br />
(B.5)<br />
<br />
. (B.6)<br />
Because D is a diagonal matrix, 1/dj are the elements of D−1 , and n j=1 dj =<br />
det(D); thus it is possible to write the last expression as<br />
<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = (2π)n/2<br />
exp<br />
det(D)<br />
<br />
− b′ t D −1 b ′<br />
2<br />
<br />
. (B.7)<br />
Finally, since det(A) = det(D) and b ′ t D −1 b ′ = b t A −1 b, the integral becomes<br />
<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = (2π)n/2<br />
<br />
exp −<br />
det(A) btA−1 <br />
b<br />
. (B.8)<br />
2<br />
This proof can be applied to the special case in which all the elements of the<br />
vector b are equal to 0. In that case the integral reduces to<br />
<br />
dx exp − xt <br />
Ax<br />
=<br />
2<br />
(2π)n/2<br />
. (B.9)<br />
det(A)<br />
70
APPENDIX C<br />
Methods for OAM<br />
measurements<br />
A real world oam application will require a compact tool able to separate the<br />
different modes at the single photon level. A tool analogous to the beam splitter<br />
for polarization, or to a diffraction grating for frequency. Such a tool probably<br />
will be based on the two current methods to determine the oam of photons:<br />
holographic and interferometric methods.<br />
In a very simplistic way, an hologram is a record of the interference pattern<br />
of two beams. When the hologram is illuminated with one of the beams,<br />
the other beam can be reconstructed. Based on this principle, a computer<br />
generated hologram of the interference of a lg mode with a Gaussian beam,<br />
in conjunction with a single mode fiber can be used to detect that particular<br />
lg mode [7, 51]. Even though this method works at single photon level, it<br />
is restricted to a two-value response, reducing the effective dimensions of the<br />
oam to two, as shown in figure C.1.<br />
Reference [15] introduced an analyzer able to sort photons into even and<br />
odd oam modes, using a Mach-Zender interferometer. With a phase shift in<br />
one of the interferometer arms, the constructive and destructive interference<br />
were so that odd modes go to one of the outputs, and even modes to the other,<br />
as figure C.2 shows. By nesting several interferometers, and using holograms,<br />
the authors proved that was possible to separate up to 2 n modes with (2 n − 1)<br />
interferometers. This method works at single photon level, and can distinguish<br />
many different oam modes, but it relies on the simultaneous stabilization of<br />
different interferometers, which is challenging experimentally.<br />
71
C. Methods for OAM measurements<br />
Input hologram<br />
lens<br />
Gaussian<br />
mode<br />
non-Gaussian<br />
mode<br />
Figure C.1: A hologram that records the interference of a lg mode with a Gaussian<br />
beam, can be used to detect that particular lg mode. Only if the generating lg<br />
mode is incident on the hologram, a Gaussian beam is recovered. If another mode is<br />
incident, the hologram will generate non-Gaussian beams. Single mode optical fibers<br />
distinguish between Gaussian and non-Gaussian beams.<br />
Input<br />
beam splitter<br />
Dove prism mirror<br />
even modes<br />
odd modes<br />
Figure C.2: <strong>Two</strong> Dove prisms rotated with respect to each other by an angle π/2<br />
induce a relative rotation of π between the two arms of a Mach-Zender interferometer.<br />
As a consequence of the rotation, odd and even lg modes leave the interferometer<br />
at different outputs after the second beam splitter. Reference [15] introduces this<br />
principle as a base of an oam sorter.<br />
72
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78
DOCTORAL THESIS IN PHOTONICS<br />
ICFO BARCELONA 2010<br />
ICFO · THE INSTITUTE OF PHOTONIC SCIENCES<br />
AV. CANAL OLÍMPIC, S/N · CASTELLDEFELS · BARCELONA<br />
&<br />
UPC · UNIVERSITAT POLITÈCNICA DE CATALUNYA<br />
CAMPUS NORD · BARCELONA
<strong>Spatial</strong> <strong>Characterization</strong><br />
<strong>Of</strong> <strong>Two</strong>-<strong>Photon</strong> <strong>States</strong><br />
By<br />
Clara Inés Osorio Tamayo<br />
ICFO - Institut de Ciències Fotòniques<br />
Universitat Politècnica de Catalunya<br />
Barcelona, September 2009
<strong>Spatial</strong> <strong>Characterization</strong><br />
<strong>Of</strong> <strong>Two</strong>-<strong>Photon</strong> <strong>States</strong><br />
Clara Inés Osorio Tamayo<br />
under the supervision of<br />
Dr. Juan P. Torres<br />
submitted this thesis in partial fulfillment<br />
of the requirements for the degree of<br />
Doctor<br />
by the<br />
Universitat Politècnica de Catalunya<br />
Barcelona, September 2009
A Luz Stella y Luis Alfonso, mis papás.
Contents<br />
Contents vii<br />
Acknowledgements ix<br />
Abstract xi<br />
Introduction xv<br />
List of Publications xvii<br />
1 General description of two-photon states 1<br />
1.1 Spontaneous parametric down-conversion . . . . . . . . . . . . 2<br />
1.2 <strong>Two</strong>-photon state . . . . . . . . . . . . . . . . . . . . . . . . . . 3<br />
1.3 Approximations and other considerations . . . . . . . . . . . . 5<br />
1.4 The mode function in matrix form . . . . . . . . . . . . . . . . 12<br />
2 Correlations and entanglement 15<br />
2.1 The purity as a correlation indicator . . . . . . . . . . . . . . . 16<br />
2.2 Correlations between space and frequency . . . . . . . . . . . . 18<br />
2.3 Correlations between signal and idler . . . . . . . . . . . . . . . 23<br />
3 <strong>Spatial</strong> correlations and OAM transfer 29<br />
3.1 Laguerre-Gaussian modes and OAM content . . . . . . . . . . . 30<br />
3.2 OAM transfer in general SPDC configurations . . . . . . . . . . 31<br />
3.3 OAM transfer in collinear configurations . . . . . . . . . . . . . 34<br />
4 OAM transfer in noncollinear configurations 37<br />
4.1 Ellipticity in noncollinear configurations . . . . . . . . . . . . . 38<br />
4.2 Effect of the pump beam waist on the OAM transfer . . . . . . 39<br />
4.3 Effect of the Poynting vector walk-off on the OAM transfer . . 44<br />
5 <strong>Spatial</strong> correlations in Raman transitions 51<br />
5.1 The quantum state of Stokes and anti-Stokes photon pairs . . . 52<br />
5.2 Orbital angular momentum correlations . . . . . . . . . . . . . 55<br />
5.3 <strong>Spatial</strong> entanglement . . . . . . . . . . . . . . . . . . . . . . . . 57<br />
6 Summary 61<br />
A The matrix form of the mode function 63<br />
vii
Contents<br />
B Integrals of the matrix mode function 69<br />
C Methods for OAM measurements 71<br />
Bibliography 73<br />
viii
Acknowledgements<br />
This thesis compiles the result of five years of work at icfo. But it is, in some<br />
sense, the result of a longer process that started many years ago in Medellín.<br />
I would like to show my gratitude to all the people that helped me during all<br />
this time even though I cannot mention everybody explicitly.<br />
I would like to thank my advisor Juan P. Torres for inviting me to his<br />
group, and for being very patient with my impatience. I am grateful to Alejandra<br />
Valencia and Xiaojuan Shi for their generous and unconditional support.<br />
I wish to thank all the people that provide a stimulating and fun environment<br />
in icfo, especially Maurizio Righini, Laura Grau, Carsten Schuck, Masood<br />
Ghotbi, Xavier Vidal, Sibylle Braungardt, Jorge Luis Domínguez-Juárez,<br />
Petru Ghenuche, Rafael Betancurt, Philipp Hauke, Alessandro Ferraro, Agata<br />
Checinska, and Martin Hendrych. I would like to thank Niek van Hulst and<br />
his group, for adopting me at lunch time and for all party matters; and Morgan<br />
Mitchell and his group, especially Marco Koschorreck, Mario Napolitano,<br />
Florian Wolfgramm, Alessandro Cere and Brice Dubost. Logistical support by<br />
Nuria Segù, Maria del Mar Gil, Anne Gstöttner, Jose Carlos Cifuentes, and<br />
the electronic workshop have also been most helpful. I am extremely grateful<br />
to Miguel Navascués that has been like a little brother to me since I came to<br />
Barcelona, and to Artur García that welcomed me in his office every time that<br />
I needed to talk (and that was very often).<br />
I want to thank my “Barcelonian” friends for their caring, their emotional<br />
and gastronomical support, Mauricio Álvarez, Carolina Mora, Friman Sánchez,<br />
Tania de la Paz, Maria Teresa Vasco and Jose Uribe. Regardless the distance,<br />
many of my friends made available their support in a number of ways, I am<br />
grateful to Carlos Molina, Jose Palacios, Catalina López, Elizabeth Agudelo,<br />
Esteban Silva, Sebastían Patrón, Jaime Hincapie, Wolfgang Niedenzu, Andrew<br />
Hilliard, Juan C. Muñoz (y familia), Jaime Forero and Javier Moreno. It is a<br />
pleasure to thank Boudewijn Taminiau, Ied van Oorschot, and all the Taminiau<br />
family, since their visits and hospitality have made this years much nicer.<br />
Agradezco hasta el cielo a mi familia. A mis papás que me han inspirado<br />
siempre con su creatividad y fortaleza para resolver los problemas, a mis hermanas<br />
que siempre me apoyan, y a mi hermanito Felipe, que me ha trasmitido<br />
todo su amor por la ciencia, su curiosidad y su escepticismo.<br />
Finally, this thesis would not have been possible without the constant support<br />
of Tim Taminiau, his careful corrections of this manuscript and the nice<br />
3D images of the cone and the phase fronts that he made for me. I also thank<br />
Tim for giving me my first bicycle, for teaching me how to use it, and for<br />
pushing me up the mountain every day when we were going to work.<br />
ix
When he [Kepler] found that his long<br />
cherished beliefs did not agree with the<br />
most precise observations, he accepted<br />
the uncomfortable facts, he preferred the<br />
hard truth to his dearest illusions.<br />
That is the heart of science.<br />
Cosmos - Carl Sagan
Abstract<br />
In the same way that electronics is based on measuring and controlling the state<br />
of electrons, the technological applications of quantum optics will be based<br />
on our ability to generate and characterize photonic states. The generation<br />
of photonic states is traditionally associated to nonlinear optics, where the<br />
interaction of a beam and a nonlinear material results in the generation of<br />
multi-photon states. The most common process is spontaneous parametric<br />
down-conversion (spdc), which is used as a source of pairs of photons not only<br />
for quantum optics applications but also for quantum information and quantum<br />
cryptography [1, 2].<br />
The popularity of spdc lies in the relative simplicity of its experimental realization,<br />
and in the variety of quantum features that down-converted photons<br />
exhibit. For instance, a pair of photons generated via spdc can be entangled<br />
in polarization [3, 4], frequency [5, 6], or in the equivalent degrees of freedom of<br />
orbital angular momentum, space, and transverse momentum [7, 8, 9, 10, 11].<br />
Standard spdc applications focus on a single degree of freedom, wasting the<br />
entanglement in other degrees of freedom and the correlations between them.<br />
Among the few configurations using more than one degree of freedom are hyperentanglement<br />
[12, 13], spatial entanglement distillation using polarization<br />
[14], or control of the joined spectrum using the pump’s spatial properties [6].<br />
This thesis describes the spatial properties of the two-photon state generated<br />
via spdc, considering the different parameters of the process, and the<br />
correlations between space and frequency. To achieve this goal, I use the purity<br />
to quantify the correlations between the photons, and between the degrees<br />
of freedom. Additionally, I study the spatial correlations by describing the<br />
transfer of orbital angular momentum (oam) from the pump to the signal and<br />
idler photons, taking into account the pump, the detection system and other<br />
parameters of the process.<br />
This thesis is composed of six chapters. Chapter 1 introduces the mode<br />
function, used throughout the thesis to describe the two-photon state in space<br />
and frequency. Chapter 2 describes the correlations between degrees of freedom<br />
or photons in the two-photon state, using the purity to quantify such correlations.<br />
Chapter 3 explains the mechanism of the oam transfer from pump<br />
to signal and idler photons. Chapter 4 describes the effect of different spdc<br />
parameters on the oam transfer in noncollinear configurations, both theoretically<br />
and experimentally. In analogy with the downconverted case, chapter 5<br />
discusses the two-photon state generated via Raman transition, by describing<br />
its mode function, the correlations between different parts of the state, and the<br />
oam transfer in the process. Finally, chapter 6 summarizes the main results<br />
presented by this thesis.<br />
xi
Abstract<br />
The matrix notation, introduced here to describe the two-photon mode<br />
function, reduces the calculation time for several features of the state. In<br />
particular, this notation allows to calculate the purity of different parts of<br />
the state analytically. This analytical solution reveals the effect of each spdc<br />
parameter on the internal correlations, and shows the necessary conditions to<br />
suppress the correlations, or to maximize them.<br />
The description of the oam transfer mechanism shows that the pump oam<br />
is totally transfer to the generated photons. But if only a portion of the generated<br />
photons is detected their oam may not be equal to the pump’s oam. The<br />
experiments described in the thesis show that the amount of oam transfer in<br />
the noncollinear case is tailored by the parameters of the spdc. The analysis<br />
of the spdc case can be extended to other nonlinear processes, such as Raman<br />
transitions, where the specific characteristics of the process determine the<br />
correlations and the oam transfer mechanism.<br />
The results of this thesis contribute to a full description of the correlations<br />
inside the two-photon state. Such a description allows to use the correlations<br />
as a tool to modify the spatial state of the photons. This spatial information,<br />
translated into oam modes, provides a multidimensional and continuum degree<br />
of freedom, useful for certain tasks where the polarization, discrete and bidimensional,<br />
is not enough. To make such future applications possible, it will be<br />
necessary to optimize the tools for the detection of oam states at the single<br />
photon level [15, 16].<br />
xii
Resumen<br />
De la misma manera que la electrónica se basa en medir y controlar el estado<br />
de los electrones, las aplicaciones tecnológicas de la óptica cuántica se basarán<br />
en nuestra habilidad para generar estados fotónicos bien caracterizados. La<br />
generación de estos estados está tradicionalmente asociada a la óptica no lineal,<br />
donde la interacción de un haz con un material no lineal da lugar a la<br />
generación de estados de múltiples fotones. El proceso no lineal más popular<br />
es la conversión paramétrica descendente, o spdc por su sigla en inglés, que<br />
es usada como fuente de pares de fotones no sólo en aplicaciones de óptica<br />
cuántica si no también para información y criptografía cuánticas [1, 2].<br />
La popularidad de spdc se debe a la relativa simplicidad de su realización<br />
experimental, y a la variedad de fenómenos cuánticos exhibidos por los pares<br />
de fotones generados. Por ejemplo, estos pares pueden estar entrelazados en<br />
polarización [3, 4], frecuencia [5, 6], o en los grados de libertad espacial: momento<br />
angular orbital o momento transversal [7, 9, 10, 11]. Las aplicaciones<br />
usuales de spdc usan sólo un grado de libertad, perdiendo la información contenida<br />
en los otros grados de libertad y en las correlaciones entre ellos. Entre<br />
las únicas aplicaciones que usan más de un grado de libertad se encuentran el<br />
hiperentrelazamiento [12, 13], la destilación de entrelazamiento espacial usando<br />
la polarización [14], o el control de la distribución espectral conjunta usando<br />
las propiedades espaciales del haz generador del spdc [6].<br />
Esta tesis describe las características espaciales de los pares de fotones generados<br />
en spdc, teniendo en cuenta el efecto de los otros grados de libertad,<br />
especialmente de la frecuencia. Para ello, usaré la pureza como cuantificador de<br />
las correlaciones entre los grados de libertad, y entre los fotones. Además, usaré<br />
la transferencia de momento angular orbital (oam), asociada a la distribución<br />
espacial de los fotones, para estudiar el efecto de diferentes parámetros del<br />
spdc sobre el estado espacial generado.<br />
Esta tesis esta compuesta de seis capítulos. El capítulo 1 introduce la<br />
función de modo, que es usada en toda la tesis para describir los estados de dos<br />
fotones en espacio y frecuencia. El capítulo 2 describe las correlaciones entre<br />
los grados de libertad, y entre los fotones, usando la pureza para cuantificar<br />
estas correlaciones. El capítulo 3 describe la transferencia de oam desde el haz<br />
generador hasta los fotones generados. El capítulo 4 demuestra teórica y experimentalmente,<br />
el efecto de diferentes parámetros del spdc sobre la transferencia<br />
de momento angular orbital en configuraciones no colineales. El capítulo<br />
5 discute la generación de estados de dos fotones en transiciones Raman, caracterizando<br />
estos estados de una manera análoga a la utilizada para aquellos<br />
generados en spdc. Por último, el capítulo 6 resume las contribuciones más<br />
importantes de esta tesis.<br />
xiii
Abstract<br />
La notación matricial introducida para describir la función de modo de los<br />
pares generados, reduce considerablemente el tiempo de cálculo de diferentes<br />
características del estado. En particular, ésta notación hace posible calcular<br />
analíticamente la pureza de diferentes partes del estado, y estudiar el efecto de<br />
cada parámetro del spdc sobre las correlaciones entre los grados de libertad o<br />
entre los fotones. Hace posible, además, encontrar las condiciones necesarias<br />
para suprimir las correlaciones entre los grados de libertad o entre los fotones,<br />
y distinguir en que casos estas correlaciones se hacen mas relevantes.<br />
El estudio del mecanismo de transferencia del momento angular orbital,<br />
revela que éste es transferido totalmente del haz generador a los fotones generados.<br />
Si sólo una porción de estos fotones es considerada, el oam de éstos no<br />
da cuenta del momento oam total invertido por el haz generador. Los experimentos<br />
descritos en esta tesis muestran que la cantidad de oam transferido<br />
a una porción de los fotones puede ser controlada modificando el tamaño de<br />
esa porción, ya sea cambiando el ancho del haz incidente, el largo del cristal u<br />
otro parámetro. En el caso de la generación de pares de fotones en otros procesos<br />
no lineales, como las transiciones Raman, tanto las correlaciones como la<br />
transferencia de oam, son determinadas por las características especificas del<br />
proceso.<br />
Los resultados de esta tesis contribuyen a completar la descripción de las<br />
correlaciones dentro del estado de dos fotones. Esta descripción permite el<br />
uso de las correlaciones como herramienta para modificar el estado espacial de<br />
los fotones. La información espacial, traducida a modos de oam, ofrece un<br />
grado de libertad infinito dimensional y continuo, útil en ciertas tareas donde<br />
la polarización, bidimensional y discreta, no es suficiente. Para hacer estas<br />
tareas posibles, es necesario optimizar las herramientas para la detección de<br />
estados de oam para un sólo fotón [15, 16].<br />
xiv
Introduction<br />
The role of photons in quantum physics and technology is growing [1, 2, 17].<br />
Frequently, those applications are based on two dimensional systems, such as<br />
the two orthogonal polarization states of a photon, losing all information in<br />
other degrees of freedom. These applications only use a portion of the total<br />
quantum state of the light.<br />
The polarization itself is related to a broader degree of freedom. The angular<br />
momentum of the photons contains a spin contribution associated with<br />
the polarization, as well as an orbital contribution associated with the spatial<br />
distribution of the light (its intensity and phase). In general, the spin and<br />
the orbital angular momentum cannot be considered separately. However, in<br />
the paraxial regime both contributions are independent [18]. In this regime it<br />
is possible to exploit the possibilities offered by the infinite dimensions of the<br />
orbital angular momentum (oam) of the light.<br />
Currently available technology offers different possibilities to work with<br />
oam. Computer generated holograms are widely used in classical and quantum<br />
optics for generating and detecting different oam states [19, 20]. The efficiency<br />
of these processes has increased with the design of spatial light modulators<br />
capable of real time hologram generation [21].<br />
<strong>Of</strong> special interest is the generation of paired photons entangled in oam.<br />
Entanglement is a quantum feature with no analogue in classical physics. Spontaneous<br />
parametric down-conversion (spdc) is a reliable source for the generation<br />
of pairs of photons entangled in different degrees of freedom, including<br />
oam [22]. The existence of correlations in the oam of pairs of photons generated<br />
in spdc was proven experimentally by Mair and coworkers in 2001. Other<br />
nonlinear processes can be used to generate pairs of photons with correlations<br />
in their oam content. In reference [23], the authors show the generation of<br />
spatially entangled pairs of photons by the excitation of Raman transitions in<br />
cold atomic ensembles.<br />
Several studies illustrate the potential offered by higher-dimensional quantum<br />
systems. For instance, reference [24] reports a Bell inequality violation<br />
by a two-photon three-dimensional system, confirming the existence of oam<br />
entanglement in the system. Reference [25] introduced a quantum coin tossing<br />
protocol based on oam states. In reference [26], the authors present a quantum<br />
key distribution scheme using entangled qutrits, which are encoded into<br />
the oam of pairs of photons generated in spdc. And, in reference [12], the<br />
authors report the generation of hyperentangled quantum states by using the<br />
combination of the degrees of freedom of polarization, time-energy and oam.<br />
All the previously mentioned applications are based on specific oam correlations<br />
between the photons. However, there has been some controversy about<br />
xv
Introduction<br />
the transfer of oam from the generating pump beam to the down-converted<br />
photons. Several studies have reported a compleate transfer of the oam of the<br />
pump [7, 8, 22, 27, 28, 29], while other studies have reported a partial transfer<br />
[30, 31, 32, 33]. The elucidation of the oam spectra of down-converted photons,<br />
and their relation with the spectra of the pump, is a key step for developing<br />
new applications based on this degree of freedom.<br />
This thesis addresses the mentioned oam controversy, as well as the characterization<br />
of the different correlations present in two-photon states. For both<br />
processes, spdc and Raman transitions in cold atoms, this thesis describes the<br />
generated two-photon state, the correlations between different degrees of freedom<br />
and between photons, and the oam transfer. A general goal of the thesis<br />
is to contribute to the characterization of the two-photon sources, making it<br />
possible to exploit a bigger portion of the total quantum state of the light in<br />
future applications.<br />
xvi
This thesis is based on the following papers:<br />
List of Publications<br />
Orbital angular momentum correlations of entangled paired photons.<br />
C. I. Osorio, G. Molina-Terriza, J. P. Torres.<br />
J. Opt. A: Pure Appl. Opt. 11, 094013 (2009).<br />
Chapters 3 and 5<br />
<strong>Spatial</strong> entanglement of paired photons generated in cold atomic ensembles<br />
C. I. Osorio, S. Barreiro, M. W. Mitchell, and J. P. Torres.<br />
Phys. Rev. A 78, 052301 (2008). arXiv:0804.3257v2 [quant-ph].<br />
Chapter 5<br />
Spatiotemporal correlations in entangled photons generated by spontaneous parametric<br />
down conversion.<br />
C. I. Osorio, A. Valencia, and J. P. Torres.<br />
New J. Phys. 10, 113012 (2008). arXiv:0804.2425v2 [quant-ph].<br />
Chapters 1 and 2<br />
Correlations in orbital angular momentum of spatially entangled paired photons generated<br />
in parametric downconversion.<br />
C. I. Osorio, G. Molina-Terriza, and J. P. Torres.<br />
Phys. Rev. A 77, 015810 (2008). arXiv:0711.4500v1 [quant-ph].<br />
Chapters 3<br />
Azimuthal distinguishability of entangled photons generated in spontaneous parametric<br />
down-conversion.<br />
C. I. Osorio, G. Molina-Terriza, B. Font, and J. P. Torres.<br />
Opt. Express 15, 14636 (2007). arXiv:0709.3437v1 [quant-ph].<br />
Chapter 4<br />
Control of the shape of the spatial mode function of photons generated in noncollinear<br />
spontaneous parametric down-conversion.<br />
G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych, and J. P.<br />
Torres.<br />
Phys. Rev. A 72, 065802 (2005). arXiv:quant-ph/0508058v1.<br />
Chapter 4<br />
Orbital angular momentum of entangled counterpropagating photons.<br />
J. P. Torres, C. I. Osorio, and L. Torner.<br />
Opt. Lett. 29, 1939 (2004).<br />
Chapters 3 and 4<br />
xvii
CHAPTER 1<br />
General description of<br />
<strong>Two</strong>-photon states<br />
Experimental implementations of spontaneous parametric down-conversion (spdc)<br />
share three main components: A laser beam used as a pump, a nonlinear material<br />
that induces the down-conversion, and a detection system that measures<br />
the generated state. The properties of the resulting two-photon state depend<br />
on the interaction, and on the individual properties of those components. This<br />
chapter derives the two-photon mode function that characterizes the generated<br />
state, using first order perturbation theory. The chapter is divided in four<br />
sections. Section 1.1 describes the spdc process in general terms. Section 1.2<br />
introduces a general expression for the mode function. Section 1.3 lists a series<br />
of common approximations used to reduce the complexity of the mode function.<br />
Finally, section 1.4 introduces a novel matrix notation for the simplified<br />
mode function. One of the advantages of this new notation is that many characteristics<br />
of the two-photon state can be analytically calculated by using it.<br />
The following chapters will use the notation and results introduced here as a<br />
starting point to describe the correlations between the generated photons, and<br />
between their spatial states.<br />
1
1. General description of two-photon states<br />
virtual state<br />
pump<br />
ground state<br />
signal<br />
idler<br />
Figure 1.1: In spontaneous parametric down conversion the interaction of a molecule<br />
with the pump results in the annihilation of the pump photon and the creation of<br />
two new photons.<br />
1.1 Spontaneous parametric down-conversion<br />
Spontaneous parametric down-conversion is one of the possible resulting processes<br />
of the interaction of a pump photon and a molecule, in the presence of<br />
the vacuum field. In spdc, the pump-molecule interaction leads to the generation<br />
of a single physical system composed of two photons: signal and idler, as<br />
figure 1.1 shows.<br />
The name of the process reveals some of its characteristics. spdc is a<br />
parametric process since the incident energy totally transfers to the generated<br />
photons, and not to the molecule. And, it is a down-conversion process since<br />
each of the generated photons has a lower energy than the incident photon.<br />
The conservation of energy and momentum establish the relations between the<br />
frequencies (ωp,s,i) and wave vectors (kp,s,i) of the photons,<br />
ωp = ωs + ωi<br />
kp = ks + ki. (1.1)<br />
where the subscripts p, s, i stand for pump, signal and idler. Macroscopically,<br />
spdc results from the interaction of a field with a nonlinear medium. Commonly<br />
used mediums include uniaxial birefringent crystals. The conditions in<br />
equation 1.1 can be achieved for this kind of crystals, since their refractive index<br />
changes with the frequency and the polarization of an incident field [34, 35].<br />
This thesis considers type-i configurations in uniaxial birefringent crystals.<br />
In such configuration the pump beam polarization is extraordinary since it is<br />
contained in the principal plane, defined by the crystal axis and the wave vector<br />
of the incident field. The polarization of the generated photons is ordinary,<br />
which means that it is orthogonal to the principal plane.<br />
For a given material, the conditions imposed by equation 1.1 determine the<br />
characteristics of the generated pair. The second line in equation 1.1 defines the<br />
spdc geometrical configuration, and it is known as a phase matching condition.<br />
The generated photons are emitted in two cones coaxial to the pump, as shown<br />
in figure 1.2 (b). The apertures of the cone are given by the emission angles ϕs,i<br />
associated to the frequencies ωs,i. Once a single frequency or emission angle is<br />
selected, only one of all possible cones is considered. In degenerate spdc, both<br />
photons are emitted at the same angle ϕs = ϕi and the cones overlap, as in<br />
figure 1.2 (c). There are two kinds of degenerate spdc processes: noncollinear<br />
2
y<br />
y<br />
x<br />
z<br />
z<br />
pump<br />
s<br />
<br />
i<br />
(a)<br />
signal<br />
idler<br />
(c)<br />
1.2. <strong>Two</strong>-photon state<br />
Figure 1.2: (a) Phase matching conditions impose certain propagation directions for<br />
the emitted photons at frequencies ωs and ωi. (b-c) This directions define two cones,<br />
that collapse into one in the degenerate case. (d) In the collinear configuration, the<br />
aperture of the cones tends to zero as the direction of emission is parallel to the pump<br />
propagation direction.<br />
in which the signal and idler are not parallel to the pump, and collinear in<br />
which the aperture of the cones tends to zero as the photons propagate almost<br />
parallel to the pump, as shown in figure 1.2 (d).<br />
Even though the phase matching condition defines the main characteristics<br />
of the generated two-photon state, other important factors influence the measured<br />
state of the photons, for example the detection system. The next section<br />
describes the two-photon state mathematically, taking into account all these<br />
factors.<br />
1.2 <strong>Two</strong>-photon state<br />
When an electromagnetic wave propagates inside a medium, the electric field<br />
acts over each particle (electrons, atoms, or molecules) displacing the positive<br />
charges in the direction of the field and the negative charges in the opposite<br />
direction. The resulting separation between positive and negative charges of<br />
the material generates a global dipolar moment in each unit volume known as<br />
polarization, and defined as<br />
(b)<br />
(d)<br />
P = ɛ0(χ (1) E + χ (2) EE + χ (3) EEE + · · · ) (1.2)<br />
where ɛ0 is the vacuum electric permittivity, and χ (n) are the electric susceptibility<br />
tensors of order n [34, 35].<br />
For small field amplitudes, as in linear optics, the polarization is approximately<br />
linear,<br />
P ≈ ɛ0χ (1) E. (1.3)<br />
3
1. General description of two-photon states<br />
When the amplitude of the electric field increases, the higher orders term in<br />
equation 1.2 become relevant, and then a nonlinear response of the material<br />
to the field appears. Optical nonlinear phenomena resulting from this kind<br />
of interaction include the generation of harmonics, the Kerr effect, Raman<br />
scattering, self-phase modulation, and cross-phase modulation [34].<br />
Spontaneous parametric down-conversion, and other second order nonlinear<br />
processes, result from the second order polarization, defined as the first<br />
nonlinear term in the polarization tensor<br />
P (2) = ɛ0χ (2) EE. (1.4)<br />
The quantization of the electromagnetic field leads to a quantization of the<br />
second order polarization, so that the nonlinear polarization operator ˆ P (2)<br />
becomes<br />
ˆP (2) = ɛ0χ (2) ( Ê(+) + Ê(−) )( Ê(+) + Ê(−) ), (1.5)<br />
where Ê(+) and Ê(−) are the positive and negative frequency parts of the field<br />
operator [36]. The positive frequency part of the electric field operator is a<br />
function of the annihilation operator â(k), and is defined at position rn<br />
xnˆxn + ynˆyn + znˆzn and time t as<br />
=<br />
Ê (+)<br />
1/2 <br />
ωn<br />
n (rn, t) = ien<br />
2ɛ0v<br />
dkn exp [ikn · rn − iωnt]â(kn), (1.6)<br />
where the magnitude of the wave vector k satisfies k 2 = ωn/c. The volume<br />
v contains the field, and en is the unitary polarization vector. The negative<br />
frequency part of the field is the Hermitian conjugate of the positive part,<br />
Ê (−)<br />
n (rn, t) = Ê(+)† n (rn, t). Thus, the negative frequency part is a function of<br />
the creation operator â † (k).<br />
Following references [37] and [38], in first order perturbation theory, the<br />
interaction of a volume V of a material with a nonlinear polarization ˆ P (2) and<br />
the field Êp(rp, t), produces a system described by the state<br />
|ΨT 〉 ∝ |1〉p|0〉s|0〉i − i<br />
<br />
τ<br />
0<br />
dt ˆ HI|1〉p|0〉s|0〉i, (1.7)<br />
where τ is the interaction time, |1〉p|0〉s|0〉i is the initial state of the field, and<br />
the interaction Hamiltonian reads<br />
<br />
ˆHI = − dV ˆ P (2) Êp(rp, t). (1.8)<br />
V<br />
The first term at the right side of equation 1.7 describes a one photon system,<br />
while the second term describes a two-photon system. In what follows we will<br />
consider only the second term as we are mainly interested in the generation of<br />
pairs of photons. The two-photon system state is given by<br />
|Ψ〉 = i<br />
<br />
τ<br />
0<br />
dt ˆ HI|1〉p|0〉s|0〉i, (1.9)<br />
or ˆρ = |Ψ〉〈Ψ| in the density matrix formalism. Seven of the eight terms<br />
that compose the interaction Hamiltonian vanish when they are applied over<br />
the state |1〉p|0〉s|0〉i. The remaining term is proportional to the annihilation<br />
4
1.3. Approximations and other considerations<br />
operator in mode p, and the creation operators in modes s and i. Assuming<br />
constant χ (2) , the two-photon state reduces to<br />
(2) τ <br />
iɛ0χ<br />
|Ψ〉 = dt dV<br />
0 V<br />
Ê(+) p (rp, ωp) Ê(−) s (rs, ωs) Ê(−) i (ri, ωi)|1〉p|0〉s|0〉i,<br />
(1.10)<br />
which can be written as<br />
<br />
|Ψ〉 ∝ dks dkiΦ(ks, ωs, ki, ωi)a † (ks)a † (ki)|0〉s|0〉i; (1.11)<br />
where Φ(ks, ωs, ki, ωi) is known as the mode function and, assuming that the<br />
pump has certain spatial distribution Ep(kp), is given by<br />
τ <br />
Φ(ks, ωs, ki, ωi) ∝ dt dV dkpEp(kp)<br />
0<br />
V<br />
× exp [ikp · rp − iks · rs − iki · ri − i(ωp − ωs − ωi)t]<br />
× â(kp)|1〉p. (1.12)<br />
The mode function contains all the information about the generated two-photon<br />
system in space and frequency, not only about their individual state but about<br />
their correlations. This function is therefore highly complex, and to calculate<br />
any feature of the two-photon state it is necessary to simplify it, as the next<br />
section shows.<br />
1.3 Approximations and other considerations<br />
Analytical calculations of any features of the down converted photons require<br />
simplification of the mode function in equation 1.12. This section lists the most<br />
important approximations used in this thesis, as well as the restrictions that<br />
they impose.<br />
Separation of transversal and longitudinal components<br />
A field with a wave vector k almost parallel to its propagation direction spreads<br />
only a little in the transversal direction. It is then possible, to consider the<br />
longitudinal and transversal components of the wave vector separately,<br />
k = kzˆz + q. (1.13)<br />
As the wave vector’s magnitude k = ωn/c is bigger than the transversal component’s<br />
magnitude q = |q| = (k 2 x + k 2 y) 1/2 , the magnitude of the longitudinal<br />
component kz is always a real number,<br />
kz = k 2 1<br />
− q<br />
2 2 . (1.14)<br />
Assuming that the pump, the signal, and the idler are such fields, the mode<br />
function becomes<br />
τ <br />
Φ(qs, ωs, qi, ωi) ∝<br />
0<br />
dt dV<br />
V<br />
dqpEp(qp) exp [ik z pzp − ik z szs − ik z i zi]<br />
× exp [iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i − i(ωp − ωs − ωi)t]<br />
× â(q p)|1〉p. (1.15)<br />
where r ⊥ n = xnˆxn + ynˆyn is the transversal position vector.<br />
5
1. General description of two-photon states<br />
The semiclassical approximation<br />
As a consequence of the low efficiency of the nonlinear process, the incident<br />
field is several orders of magnitude stronger than the generated fields. In that<br />
case, it is possible to consider the pump as a classical field, while the signal<br />
and idler are considered as quantum fields, this is known as the semiclassical<br />
approximation.<br />
By defining the pump as a classical field, with a spatial amplitude distribution<br />
Ep(q p) and a spectral distribution Fp(ωp):<br />
<br />
Ep(rp, t) ∝ dqp dωpEp(qp)Fp(ωp) exp [iqp · r ⊥ p + ik z pzp − iωpt], (1.16)<br />
the mode function reduces to<br />
τ <br />
Φ(qs, ωs, qi, ωi) ∝ dt dV dqp dωpEp(qp)Fp(ωp) Gaussian beam approximation<br />
0 V<br />
× exp [ik z pzp − ik z szs − ik z i zi + iqp · r ⊥ p − iqs · r ⊥ s − iqi · r ⊥ i ]<br />
× exp [−i(ωp − ωs − ωi)t]. (1.17)<br />
According to equation 1.17, the mode function depends on the spatial and<br />
temporal profiles of the pump beam. To write these profiles explicitly we<br />
assume a pump beam with a Gaussian temporal distribution,<br />
<br />
Fp(ωp) = exp − T 2 0<br />
4 ω2 <br />
p<br />
(1.18)<br />
where T0 is the pulse duration (standard deviation), which tends to infinity for<br />
continuous wave beams. Also, we assume that the pump beam is an optical<br />
vortex with orbital angular momentum lp per photon. As will be described<br />
in section 3.1, under these conditions the pump spatial profile is given by the<br />
Laguerre-Gaussian polynomials<br />
1<br />
2<br />
wp<br />
Ep(qp) =<br />
2π|lp|!<br />
<br />
|lp|<br />
−iwp<br />
√ qp exp −<br />
2 w2 p<br />
4 q2 <br />
p + ilpθp<br />
(1.19)<br />
that are functions of the pump beam waist wp, and the transversal vector<br />
magnitude qp and phase θp, given by<br />
6<br />
<br />
qp =<br />
θp = tan −1<br />
(q x p ) 2 + (q y p) 2<br />
q y p<br />
q x p<br />
<br />
.<br />
(1.20)
x<br />
1.3. Approximations and other considerations<br />
pump<br />
signal<br />
beam s<br />
y z xi<br />
idler<br />
z<br />
Figure 1.3: The propagation direction of the pump beam defines the z direction of<br />
the general coordinate system. The generated pair of photons propagates with angles<br />
ϕs and ϕi with respect to the pump beam in the yz plane. Each photon coordinate<br />
system transforms to the general coordinate system through the relations in equation<br />
1.23.<br />
For the special case of a Gaussian pump beam with lp = 0 the mode function<br />
becomes<br />
τ <br />
Φ(qs, ωs, qi, ωi) ∝ dt dV<br />
0 V<br />
i<br />
y<br />
dq p exp [− w2 p<br />
4 q2 p − T 2 0<br />
4 ω2 p]<br />
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ]<br />
× exp [−i(ωp − ωs − ωi)t], (1.21)<br />
which assumes that the interaction time τ is longer than the spontaneous emission<br />
life time of the material.<br />
Frequency bandwidth<br />
Fields generated experimentally always contain a distribution of frequencies,<br />
and totally monochromatic fields only exist in theory. This thesis considers the<br />
frequency ω of each field as the sum of a constant central frequency ω 0 , and a<br />
small deviation from that frequency Ω, so that ω = ω 0 + Ω.<br />
As a result of integrating over the interaction time taking into account the<br />
conservation of energy, the mode function becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ dV<br />
V<br />
Coordinate transformation<br />
dq p exp [− w2 p<br />
4 q2 p − T 2 0<br />
4 (Ωs + Ωi) 2 ]<br />
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ].<br />
(1.22)<br />
The mode function depends on three position vectors, rp, rs, and ri. Each of<br />
them is defined in a coordinate system with the z axis parallel to the propagation<br />
direction of each field, as shown in figure 1.3. Defining all position vectors<br />
in the same coordinate system simplifies the integration of the mode function<br />
over volume.<br />
i<br />
i<br />
xs<br />
ys<br />
zs<br />
7
1. General description of two-photon states<br />
pump<br />
polarization<br />
y<br />
Figure 1.4: The azimuthal angle α between the pump beam polarization and the x<br />
axis defines the position of a single pair of photons on the cone. The x axis is by<br />
definition normal to the plane of emission.<br />
With the origin at the crystal’s center, the z direction parallel to the pump<br />
propagation direction, and the yz plane containing the emitted photons, the<br />
unitary vectors in the coordinate systems of the generated photons transform<br />
as<br />
ˆxs =ˆx<br />
<br />
ˆys =ˆy cos ϕs + ˆz sin ϕs<br />
x<br />
ˆzs = − ˆy sin ϕs + ˆz cos ϕs<br />
ˆxi =ˆx<br />
ˆyi =ˆy cos ϕi − ˆz sin ϕi<br />
ˆzi =ˆy sin ϕi + ˆz cos ϕi. (1.23)<br />
As a single pair of photons defines the yz plane, the coordinate transformations<br />
consider only one of the possible directions of propagation on the cone. The<br />
angle α between the x axis and the pump polarization defines the transverse<br />
position on the cone for one photon pair, as figure 1.4 shows. The pump<br />
polarization is normal to the plane in which the generated photons propagate<br />
when α = 0 ◦ or α = 180 ◦ , and it is contained in the yz plane when α = 90 ◦ or<br />
α = 270 ◦ .<br />
Poynting vector walk-off<br />
This thesis considers an extraordinary polarized pump beam and ordinary polarized<br />
generated photons, an eoo configuration. Therefore, while the refractive<br />
index does not change with the direction for the generated photons, it changes<br />
for the pump beam with the angle between the pump wave vector and the axis<br />
of the crystal. Figure 1.5 shows how the energy flux direction of the pump,<br />
given by the Poynting vector, is rotated from the direction of the wave vector<br />
by an angle ρ0 given by<br />
ρ0 = − 1 ∂ne<br />
. (1.24)<br />
∂θ<br />
ne<br />
where ne is the refractive index for the extraordinary pump beam, and θ is<br />
the angle between the optical axis and the pump’s wave vector. The Poynting<br />
8
x<br />
Wave fronts<br />
1.3. Approximations and other considerations<br />
Poynting vector<br />
0<br />
Wave vector<br />
Figure 1.5: As a consequence of the birefringence, the Poynting vector is no longer<br />
parallel to the wave vector. The energy flows in an angle ρ0 with respect to propagation<br />
direction.<br />
x<br />
<br />
y z<br />
Figure 1.6: The Poynting vector moves away from the wave vector in the direction of<br />
the pump beam polarization. The angles α and ρ0 characterize the displacement. A<br />
nonradial effect such as the Poynting vector walk-off inevitably brakes the azimuthal<br />
symmetry of the properties of the photons on the cone.<br />
vector walk-off displaces the effective transversal shape of the pump beam<br />
inside the crystal in the pump polarization direction, as shown in figure 1.6.<br />
The vector p = z tan ρ0 cos αˆx + z tan ρ0 sin αˆy, describes the magnitude and<br />
direction of this displacement.<br />
By including the Poynting vector walk-off, and by using the same coordinate<br />
system for all the fields, the mode function becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ dV dqp exp −<br />
V<br />
w2 p<br />
4 q2 p − T0<br />
4 (Ωs + Ωi) 2<br />
<br />
× exp i(q x p − q x s − q x i )x <br />
<br />
0<br />
× exp [i(q y p + k z s sin ϕs − k z i sin ϕi − q y s cos ϕs − q y<br />
i cos ϕi)y]<br />
× exp [i(k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y<br />
i sin ϕi)z]<br />
× exp [i(q x p tan ρ0 cos α + q y p tan ρ0 sin α)z]. (1.25)<br />
p<br />
z<br />
9
1. General description of two-photon states<br />
Interaction volume approximation<br />
In the experimental cases studied here, the crystals are cuboids with two square<br />
faces. While typical crystal faces have areas of about 1cm 2 , the pump beam<br />
waist is only some hundreds of micrometers. By assuming that the transversal<br />
dimensions of the crystal are much larger than the pump, the integral over the<br />
volume becomes<br />
<br />
V<br />
dV →<br />
∞<br />
−∞<br />
∞ L/2<br />
dx dy dz, (1.26)<br />
−∞ −L/2<br />
where L is the length of the crystal. After integrating over x and y, the mode<br />
function reduces to<br />
<br />
<br />
where<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p<br />
4 ∆20 − w2 p<br />
4 ∆21 − T0<br />
4 (Ωs + Ωi) 2<br />
L/2<br />
× dz exp [i∆kz] (1.27)<br />
−L/2<br />
∆0 =q x s + q x i<br />
∆1 =q y s cos ϕs + q y<br />
i cos ϕi − k z s sin ϕs + k z i sin ϕi<br />
∆k =k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y<br />
i sin ϕi<br />
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.28)<br />
Wave vector and group velocity<br />
The use of a Taylor series expansion of kz s,i , simplifies the delta factors defined<br />
in the previous paragraph. Because the wave vector magnitude is a function<br />
of the frequency deviation Ω, the expansion around the origin reads<br />
k z n = k z0 ∂k<br />
n + Ωn<br />
z n<br />
+ Ω 2 ∂<br />
n<br />
2kz n<br />
∂2Ωn ∂Ωn<br />
+ . . . , (1.29)<br />
where k z0<br />
n is the magnitude of the wave vector at the central frequency ω 0 n, and<br />
the first partial derivative of the wave vector magnitude with respect to the<br />
frequency is the group velocity Nn.<br />
Taking only the first order terms of the Taylor expansion, and considering<br />
momentum conservation, the delta factors become<br />
∆0 =q x s + q x i<br />
∆1 =q y s cos ϕs + q y<br />
i cos ϕi − NsΩs sin ϕs + NiΩi sin ϕi<br />
∆k =Np(Ωs + Ωi) − NsΩs cos ϕs − NiΩi cos ϕi − q y s sin ϕs + q y<br />
i sin ϕi<br />
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.30)<br />
Exponential approximation of the sinc function<br />
The mode function in equation 1.27, depends on the integral of an exponential<br />
function of z. By integrating the exponential over the length of the crystal, the<br />
10
1<br />
0<br />
-0.2<br />
1.3. Approximations and other considerations<br />
Gaussian Sine Cardinal<br />
0<br />
FWHM<br />
Figure 1.7: An exponential function is chosen as an approximation for the sine cardinal<br />
function. The parameters of the exponential were chosen in such a way that<br />
both functions have the same full width at half maximum (fwhm).<br />
mode function becomes<br />
Φ(q s, Ωs, q i, Ωi) ∝ exp<br />
<br />
− w2 p<br />
4 ∆2 0 − w2 p<br />
4 ∆2 1 − T0<br />
4 (Ωs + Ωi) 2<br />
<br />
sinc<br />
<br />
L<br />
2 ∆k<br />
<br />
(1.31)<br />
where the sine cardinal function is defined as sinc(a) = sin a/a.<br />
In chapter 2, the sine cardinal function sinc(a) is approximated by a Gaussian<br />
function exp[−(γa) 2 ]. With γ = 0.4393 the two functions have the same<br />
full width at half maximum as figure 1.7 shows. In the rest of the thesis the<br />
sinc function will be consider without approximations.<br />
Detection systems as Gaussian filters<br />
The specific optical system used to detect the photons at the output face of<br />
the crystal, acts as a filter in space and frequency. The effect of these filters<br />
is modeled by multiplying the mode function by a spatial collection function<br />
Cspatial(qn) and a frequency filter function Ffrequency(Ωn). For the sake of<br />
simplicity, both functions are assumed to be Gaussian functions so that<br />
<br />
Cspatial(qn) ∝ exp − w2 n<br />
2 q2 <br />
n , (1.32)<br />
and<br />
<br />
Ffrequency(Ωn) ∝ exp − 1<br />
2B2 Ω<br />
n<br />
2 <br />
n , (1.33)<br />
where n labels each of the generated photons, wn is the spatial collection mode<br />
width, and Bn is the frequency bandwidth.<br />
The angular acceptance θn is related to the vector qn by the equation qn =<br />
knθn. Therefore, for a given collection mode wn the acceptance is θn = 1/knwn<br />
(the half width at 1/e 2 ).<br />
11
1. General description of two-photon states<br />
In order to obtain more convenient units, the filter in momentum is defined<br />
in a different way than the filter in frequency. While Bs,i → 0 implies a single<br />
frequency collection, the condition for a single q vector collection is ws,i → ∞.<br />
Finally, after all the approximations and taking into account the filters, the<br />
mode function equation 1.12 becomes<br />
<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p<br />
4 ∆20 − w2 p<br />
4 ∆21 <br />
× exp − (γL)2<br />
4 ∆2k − T 2 0<br />
4 (Ωs + Ωi) 2<br />
<br />
<br />
× exp − w2 s<br />
2 q2 s − w2 i<br />
2 q2 i − 1<br />
2B2 Ω<br />
s<br />
2 s − 1<br />
2B2 Ω<br />
i<br />
2 <br />
i . (1.34)<br />
The approximations listed in this chapter were introduced by several authors,<br />
and can be found for example in references [39, 40, 30]. In reference [41],<br />
we introduced a novel matrix notation based on the simplified mode function<br />
equation 1.34. The next section explains the details and the advantages of that<br />
notation.<br />
1.4 The mode function in matrix form<br />
The argument of the exponential function in equation 1.34 is a second order<br />
polynomial of the mode function variables. Such a function can be written in<br />
matrix form, as is shown in appendix A. The mode function then becomes<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1<br />
2 xt <br />
Ax<br />
(1.35)<br />
where all the parameters of the spdc process are contained in A, a positivedefinite<br />
real 6 × 6 matrix given by<br />
A = 1<br />
⎛<br />
a<br />
⎜ h<br />
⎜ i<br />
2 ⎜ j<br />
⎝ k<br />
h<br />
b<br />
m<br />
n<br />
p<br />
i<br />
m<br />
c<br />
s<br />
t<br />
j<br />
n<br />
s<br />
d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
f<br />
l<br />
r<br />
u<br />
w<br />
z<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(1.36)<br />
l r u w z g<br />
x is the vector including all variables of the mode function, defined as<br />
⎛ ⎞<br />
⎜<br />
x = ⎜<br />
⎝<br />
q x s<br />
q y s<br />
q x i<br />
q y<br />
i<br />
Ωs<br />
Ωi<br />
⎟ , (1.37)<br />
⎟<br />
⎠<br />
and x t is the transpose of x. This matrix representation of the mode function,<br />
introduced in reference [41], is an important result of this thesis. The use of<br />
this notation reduces the calculation time for integrals over the mode function,<br />
12
1.4. The mode function in matrix form<br />
and allows to solve some of those integrals analytically. For instance, the mode<br />
function normalization requires six integrals over the modulus square of the<br />
mode function. As appendix B shows, the matrix notation provides a way to<br />
calculate the normalization constant analytically,<br />
<br />
dx exp<br />
<br />
− 1<br />
2 xt <br />
(2A)x<br />
and thus the normalized mode function reads<br />
=<br />
Φ(qs, Ωs, qi, Ωi) = [det(2A)]1/4<br />
(2π) 3/2<br />
(2π) 3<br />
, (1.38)<br />
[det(2A)] 1/2<br />
<br />
exp − 1<br />
2 xt <br />
Ax . (1.39)<br />
In the same way, many other integrals over the mode function can be solved<br />
analytically using the matrix representation of the mode function.<br />
Conclusion<br />
The two-photon mode function can be written in a matrix form after a series<br />
of approximations. The matrix notation makes it possible to calculate several<br />
features of the down-conversion photons analytically, and reduces the numerical<br />
calculation time of others.<br />
The next chapter describes the different correlations in the two-photon state<br />
using the purity as a correlation indicator. The use of the approximate mode<br />
function in matrix notation makes it possible to find an analytical expression<br />
for the purity, and to study the effect of the different parameters on it.<br />
13
CHAPTER 2<br />
Correlations and<br />
entanglement<br />
The two-photon system described in chapter 1 consists of two photons with two<br />
degrees of freedom. Although the mode function deduced fully characterizes the<br />
two-photon state, the degree of correlations between each of its parts (different<br />
subsystems) are not explicit. In this chapter, I use the purity of a subsystem<br />
state to indicate the presence of correlations between it and the rest of the<br />
two-photon state, as shown in figure 2.1. This chapter has three sections.<br />
Section 2.1 describes the main characteristics of the purity, and explains why<br />
it can be used to characterize correlations in composed systems. Section 2.2<br />
describes the spatial part of the two-photon state, and section 2.3 describes<br />
the state of the signal photon. In both sections a discussion about the origin<br />
of the correlations is followed by analytical and numerical calculations of the<br />
purity. The calculations clarify the role of each spdc parameter in the internal<br />
correlations of the two-photon state. By engineering the spdc process, it is<br />
possible to tailor, and even to suppress correlations between different degrees<br />
of freedom or between different photons. Chapter 3 considers the spatial part<br />
of the two-photon state, after the correlations between space and frequency are<br />
suppressed.<br />
15
2. Correlations and entanglement<br />
(a)<br />
Space Frequency Signal<br />
q<br />
q<br />
s<br />
i<br />
<br />
<br />
s<br />
i<br />
qs<br />
qi<br />
Idler<br />
Figure 2.1: The frequency part, gray in (a), is traced out in section 2.2, to calculate<br />
the state of the subsystem composed by the two-photon state in the spatial degree<br />
of freedom. Calculating the signal photon state in space and frequency, section 2.3,<br />
requires to trace out the idler photon, gray in (b).<br />
2.1 The purity as a correlation indicator<br />
When a physical system is used to implement a task, its state should be characterized.<br />
For quantum states, the reconstruction of the state density matrix<br />
demands a tomography [1]. This process requires an infinite amount of copies<br />
of the state in order to eliminate statistical errors [42]. In an experimental<br />
implementation, the number of copies of the state available is finite, but even<br />
under this condition an experimental tomography is a complex process. Instead<br />
when just a certain feature of the system is relevant for the task, it is preferable<br />
to use a figure of merit to characterize that part of its quantum state.<br />
For a system in a quantum state given by the density operator ˆρ, the purity<br />
P = T r[ˆρ 2 ] is a figure of merit that describes its ability to be correlated with<br />
others. That is, the purity indicates if correlations can or cannot exist. Since<br />
the purity is a second order function of the density operator, measuring it does<br />
not require a full tomography, but just a simultaneous measurement over two<br />
copies of the system [43, 44].<br />
To illustrate other characteristics of the purity, consider as an example<br />
a complete orthonormal basis for the electromagnetic field, where each basis<br />
vector |n〉 describes a mode of the field [36]. A pure state |R〉 is any state<br />
written as a superposition of basis vectors:<br />
|R〉 = <br />
Cn|n〉, (2.1)<br />
n<br />
where the mode amplitude and the relative phases between modes, given by<br />
the coefficients Cn, are fixed. This pure state can be described by the density<br />
operator ˆρ = |R〉〈R|, which has the maximum possible value for the purity<br />
T r[ˆρ 2 ] = 1.<br />
A mixed state of the electromagnetic field is a state that can not be written<br />
as in equation 2.1. For instance, in the case of a statistical mixture of field<br />
modes, where there are no fixed amplitude or fixed relative phases associated<br />
16<br />
<br />
<br />
s<br />
i<br />
(b)
2.1. The purity as a correlation indicator<br />
to each vector of the base. There is, however, a fixed probability pR of finding<br />
the electromagnetic field in a state |R〉. Using the density operator formalism,<br />
the probability distribution is given by<br />
ˆρ = <br />
pR|R〉〈R|. (2.2)<br />
R<br />
The purity of the state described by ˆρ quantifies how close it is to a pure<br />
state. If the state is pure then T r[ˆρ 2 ] = 1, and if it is maximally mixed then<br />
T r[ˆρ 2 ] = 1/n, where n is de dimension of the Hilbert space in which the state is<br />
expanded. Maximally mixed states of the electromagnetic field, in the infinite<br />
dimensional spatial or temporal degrees of freedom have a purity T r[ˆρ 2 ] = 0.<br />
To extend the discussion about the purity to the kind of states generated<br />
by spdc, consider the pure bipartite system described by |ψ〉 and composed by<br />
the fields A and B. The Schmidt decomposition guarantees that orthonormal<br />
states |RA〉 and |RB〉 exist for the fields A and B, so that<br />
|ψ〉 = <br />
λR|RA〉|RB〉 (2.3)<br />
R<br />
where λR are non-negative real numbers that satisfy <br />
R λR = 1, and are<br />
known as Schmidt coefficients [1, 17]. The average of the nonzero Schmidt<br />
coefficients is a common entanglement quantifier [11] known as the Schmidt<br />
number and given by<br />
K =<br />
1<br />
<br />
R λ2 R<br />
. (2.4)<br />
While equation 2.3 describes the state of the whole bipartite system, the state<br />
of each part is calculated by tracing out the other part. Thus, the states of the<br />
fields A and B are given by the density operators<br />
ˆρA = <br />
λR|RA〉〈RA| (2.5)<br />
and<br />
R<br />
ˆρB = <br />
λR|RB〉〈RB|. (2.6)<br />
R<br />
Since both operators have equal eigenvalues λ2 R , the purity of both states is<br />
equal, and given by the inverse of the Schmidt number K<br />
T r[(ˆρ A ) 2 ] = T r[(ˆρ B ) 2 ] = 1<br />
K<br />
<br />
= λ 2 R. (2.7)<br />
Thus, when the field A is in a pure state, the field B is in a pure state too,<br />
and <br />
R λ2 <br />
R = 1. Since R λR = 1, the Schmidt coefficients satisfy the new<br />
condition only if all except one of them are equal to zero. If that is the case,<br />
the bipartite system in equation 2.3 is a product state, and no correlations<br />
exist between A and B. In other words, if there are correlations, the purity of<br />
A (and B) is smaller than 1.<br />
In conclusion, when considering a composed global system, the purity of<br />
each subsystem measures the strength of the correlation between such subsystem<br />
and the rest. This is always true independently of how the subsystems are<br />
R<br />
17
2. Correlations and entanglement<br />
chosen. In the particular case of a two-photon system, the purity of the spatial<br />
part can be used to study correlations between the degrees of freedom, or the<br />
purity of the signal photon can be used to study the entanglement between the<br />
photons. The next sections explore both these approaches.<br />
2.2 Correlations between space and frequency<br />
In type-i spdc, the photons generated have a polarization orthogonal to the<br />
polarization of the incident beam. Therefore, the two-photon state generated<br />
in a type-i process has only two degrees of freedom: frequency and transversal<br />
spatial distribution. In this section, I study the origin and the characteristics<br />
of the correlations between those degrees of freedom by using the purity of the<br />
spatial part as a correlation indicator.<br />
Except for hyperentanglement configurations [12, 13] and some other novel<br />
configurations [6, 14], most applications of type-i spdc processes use just one of<br />
the degrees of freedom, ignoring the correlation that may exist between space<br />
and frequency. That is, those configurations assume a high degree of spatial<br />
purity. This section explores the conditions in which this assumption is valid.<br />
The first part of this section contains a discussion about the origin and<br />
suppression of the correlations. The second part includes the calculations for<br />
the spatial purity that is used as correlation indicator. The third part shows<br />
the results of numerical calculation of the spatial purity in several cases.<br />
2.2.1 Origin of the correlations<br />
According to the two-photon mode function, equation 1.34, the correlations<br />
between space and frequency appear as cross terms in the variables associated<br />
to those degrees of freedom, as figure 2.2 shows. The correlations between<br />
q s x<br />
q s y<br />
q i x<br />
y<br />
qi s<br />
i<br />
qs x<br />
a<br />
h<br />
i<br />
j<br />
k<br />
l<br />
q s<br />
y<br />
h<br />
b<br />
m<br />
n<br />
p<br />
r<br />
qi y<br />
i<br />
m<br />
c<br />
s<br />
t<br />
u<br />
Figure 2.2: Cross terms in matrix A generate correlations between space and frequency.<br />
Without those terms, the matrix A is composed by two nonzero block matrices,<br />
one for each degree of freedom.<br />
space and time are always relevant, if at least one of the cross terms k, l, p,<br />
r, t, u, v, w is comparable with another term of matrix A. Therefore, it is<br />
possible to suppress the correlations by making all the cross terms negligible,<br />
or by increasing the values of the other terms.<br />
The use of ultra-narrow filters in one of the degrees of freedom will suppress<br />
most of the information available in it, destroying any possible correlation with<br />
18<br />
q i<br />
x<br />
j<br />
n<br />
s<br />
d<br />
v<br />
w<br />
s<br />
k<br />
p<br />
t<br />
v<br />
f<br />
z<br />
i<br />
l<br />
r<br />
u<br />
w<br />
z<br />
g
2.2. Correlations between space and frequency<br />
other degrees of freedom. Filtering makes the diagonal terms of matrix A larger<br />
than the rest of the terms, including the terms responsible for the correlations.<br />
But while they remove the correlations, the filters reduce the amount of photons<br />
available for any measurement.<br />
An appropriate design of the source makes it possible to suppress cross<br />
terms without using filters. Such a design should fulfill the conditions<br />
Np − Ns cos ϕs<br />
Np − Ni cos ϕi<br />
Np − Ns cos ϕs<br />
Np − Ni cos ϕi<br />
<br />
<br />
Np − Ns cos ϕs = 0<br />
Np − Ni cos ϕi = 0<br />
<br />
<br />
1 + w2 p<br />
γ2L2 1 + w2 p<br />
γ 2 L 2<br />
1 − w2 p<br />
γ 2 L 2<br />
1 − w2 p<br />
γ 2 L 2<br />
Ns sin ϕs<br />
tan ϕi<br />
Ni sin ϕi<br />
tan ϕs<br />
<br />
<br />
<br />
= 0<br />
= 0<br />
= 0<br />
= 0. (2.8)<br />
These conditions can be met for example by using a long crystal, a highly<br />
focused pump beam, for which wp/γL → 0, and emission angles such that<br />
cos ϕs = Np/Ns and cos ϕi = Np/Ni.<br />
Even though these conditions have been deduced in a simplified regime,<br />
reference [45] shows that these conditions are sufficient to remove the correlations<br />
when considering the general mode function of equation 1.12. Once the<br />
correlations are suppressed, the two-photon mode function can be written as a<br />
product of a spatial and a frequency two-photon mode function like<br />
Φ(q s, ωs, q i, ωi) = Φq(q s, q i)ΦΩ(Ωs, Ωi). (2.9)<br />
Since the global two-photon system in space and frequency is in a pure state,<br />
the presence of correlations can be confirmed by using the purity. The next<br />
section calculates the purity of the spatial state, which can be used as an<br />
indicator of correlations between space and frequency.<br />
2.2.2 <strong>Two</strong>-photon spatial state<br />
To study space/frequency correlations I will describe the purity of the spatial<br />
part of the two-photon state. As the purity of the spatial and frequency parts<br />
of the state are equal, all results derived for the spatial two-photon state extend<br />
to a frequency two-photon state.<br />
After tracing-out the frequency from the two-photon state, the remaining<br />
state describes the spatial part of the state. The reduced density matrix for<br />
19
2. Correlations and entanglement<br />
the spatial two-photon state is<br />
ˆρq = T rΩ[ρ]<br />
<br />
=<br />
<br />
=<br />
dΩ ′′<br />
s dΩ ′′<br />
i 〈Ω ′′<br />
s , Ω ′′<br />
i |ˆρ|Ω ′′<br />
s , Ω ′′<br />
i 〉<br />
dqsdΩsdqidΩidq ′ sdq ′ i<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)|qs, qi〉〈q ′ s, q ′ i|, (2.10)<br />
while the purity of this state, defined as T r[ˆρ 2 q], is given by<br />
T r[ˆρ 2 <br />
q] =<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i). (2.11)<br />
It is possible to solve these integrals analytically by using the exponential<br />
character of the mode function given by equation (1.35). The purity becomes<br />
T r[ˆρ 2 q] = det(2A)<br />
det(B) . (2.12)<br />
As seen in appendix A, B is a positive-definite real 12 × 12 matrix<br />
B = 1<br />
⎛<br />
⎜<br />
2 ⎜<br />
⎝<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
l<br />
0<br />
0<br />
0<br />
0<br />
k<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
r<br />
0<br />
0<br />
0<br />
0<br />
p<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
u<br />
0<br />
0<br />
0<br />
0<br />
t<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
w<br />
0<br />
0<br />
0<br />
0<br />
v<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
2z<br />
k<br />
p<br />
t<br />
v<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
2g<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
0<br />
0<br />
0<br />
k<br />
l<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
0<br />
0<br />
0<br />
0<br />
p<br />
r<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
0<br />
0<br />
0<br />
0<br />
t<br />
u<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
0<br />
0<br />
0<br />
0<br />
v<br />
w<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
0<br />
0<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
l r u w 0 0 l r u w 2z 2g<br />
defined by the equation<br />
N 4 <br />
exp − 1<br />
2 Xt <br />
BX = Φ(qs, Ωs, qi, Ωi)<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(2.13)<br />
× Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i), (2.14)<br />
where vector X is the result of concatenation of x and x ′ .<br />
In order to see the effect of each of the spdc parameters on the spatiofrequency<br />
correlations, some typical values are used in next subsection to calculate<br />
T r[ˆρ 2 q] numerically.<br />
2.2.3 Numerical calculations<br />
As first example, consider a degenerate type-i spdc process characterized by the<br />
parameters in the second column of table 2.1. A pump beam, with wavelength<br />
20
2.2. Correlations between space and frequency<br />
λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium iodate (liio3)<br />
crystal with length L = 1 mm, and negligible Poynting vector walk-off (ρ0 = 0).<br />
The crystal emits signal and idler photons with a wavelength λ0 s = λ0 i = 810<br />
nm at an angle ϕs,i = 10◦ .<br />
Under the above conditions, figure 2.3 shows the purity of the spatial state<br />
T r[ρ2 q] given by equation 2.12, as a function of the spatial filter width for different<br />
frequency filter bandwidths. The half width at 1/e of the frequency filters<br />
∆λn, is Bn = πc∆λn/(λ2 √<br />
n ln 2). Figure 2.3 shows the presence of correlations<br />
between space and frequency in all the situations for which the purity is smaller<br />
than one.<br />
According to figure 2.3, the spatial purity of the two-photon state gets closer<br />
to one as ws (= wi) increase. Narrow filters in space (infinitely large ws,i) makes<br />
the two-photon state separable in frequency and space. The separability and<br />
the lack of correlations between frequency and space appear also in the“narrow<br />
band” limit for the frequency. If ∆λs = ∆λi → 0 nm, then the purity is always<br />
1 for any value of ws = wi.<br />
When using typical interference filters, ∆λs = ∆λi ≈ 1 nm, the purity decreases<br />
indicating space/frequency correlations. As the width of the frequency<br />
filters increases, the purity converges quickly, the case of ∆λs = ∆λi = 10 nm<br />
is almost equivalent to the case without frequency filters ∆λs, ∆λi → ∞.<br />
In order to compare the theoretical results of this section with previous<br />
experimental work, figure 2.4 shows the purity of the spatial state as a function<br />
of the collection mode for three reported experiments. Figure 2.4 (a) shows the<br />
purity for the data reported in reference [6]. The authors of that paper studied<br />
a type-i spdc process in a 1 mm long liio3 crystal. They used a diode laser<br />
with wavelength λp = 405 nm and bandwidth ∆λp = 0.4 nm as a pump beam.<br />
And, by using monochromators with ∆λs,i = 0.2 nm, they detected pairs of<br />
photons emitted at ϕs,i = 17◦ . Table 2.1 resumes these parameters.<br />
The paper analyzes two pump waist values: wp = 30 µm and wp = 462 µm.<br />
For those values, the authors demonstrated control of the frequency correlations<br />
by changing the spatial properties of the pump beam. As the pump beam<br />
influences the spatial shape of the generated photons, they implicitly reported<br />
a correlation between space and frequency. This spatial to spectral mapping<br />
exists at their collection mode ws = 133.48 µm for which the correlation indeed<br />
appears according with our model.<br />
Figure 2.4(b) shows T r[ρ2 q] for the case described in reference [46]. The<br />
authors used a continuous wave pump beam with λp = 405 nm to generate<br />
photons with λs,i = 810 nm in a 1.5 mm length bbo crystal. They detected the<br />
photons after filtering with ∆λs,i = 10 nm interference filters. The pump beam<br />
waist satisfied the condition wp >> L, and the generated photons propagated<br />
at ϕs,i = 0◦ .<br />
The experiment reported in reference [46] does not show evidence of spacefrequency<br />
correlations. In agreement with this fact, the figure shows a lack<br />
of correlation for the collinear case. However, under the conditions reported,<br />
but in a non collinear configuration, it is not possible to neglect the space and<br />
frequency correlations.<br />
Finally figure 2.4 (c) plots T r[ˆρ 2 q] for the case described in reference [31].<br />
In this paper, the authors illuminated a 2 mm long bbo crystal with a pump<br />
beam with λp = 351.1 nm and wp ≈ 20 µm. Their crystal generated photons<br />
at λs,i = 702 nm that propagated at 4o . They collected the photons after using<br />
21
2. Correlations and entanglement<br />
Filters<br />
0nm<br />
1nm<br />
10nm<br />
→∞<br />
0.1 Collection mode 1mm<br />
<strong>Spatial</strong><br />
purity<br />
Figure 2.3: The spatial purity increases by filtering the state in space or frequency.<br />
For the parameters chosen here, the bandwidth of the generated photons is smaller<br />
than 10 nm, making 10 nm filters equivalent to infinitely broad ones. The parameters<br />
used to generate this figure are listed in table 2.1.<br />
<strong>Spatial</strong><br />
purity<br />
1<br />
0<br />
0.1<br />
Pump<br />
waist<br />
30m 462m Emission<br />
angle<br />
1mm 0.1 Collection mode 1mm<br />
1<br />
0.2<br />
1º<br />
5º<br />
(a) (b) (c)<br />
0º<br />
0.1 1mm<br />
Pump<br />
waist<br />
20m 500m Figure 2.4: For the values reported in reference [6], figure (a) shows that the purity<br />
is smaller than 1. This fact is in agreement with the implicit spatio-temporal<br />
correlations reported in the reference. For the values reported in references [46] and<br />
[31], figures (b) and (c) show that the purity has its maximum value. There was no<br />
evidence of spatio-temporal correlations in the results reported in these references.<br />
In all three cases, it is possible to modify the value of the purity by tailoring the spdc<br />
parameters. The parameters used to generate this figure are listed in table 2.1.<br />
10 nm interference filters.<br />
For the noncollinear configuration in reference [31], the highly focused pump<br />
beam is responsible for the lack of correlations. The figure shows that for lessfocused<br />
pump beams the purity decreases as the correlations between space<br />
and frequency become more important.<br />
The correlations between frequency and space in the two-photon state discussed<br />
in this section, suggest that a complete description of the correlations<br />
between the generated photons, should take into account both degrees of freedom,<br />
as the next section discusses.<br />
22
2.3. Correlations between signal and idler<br />
Table 2.1: The parameters used in figures 2.3 and 2.4<br />
Parameter Figure 2.3 Figure 2.4(a) Figure 2.4(b) Figure 2.4(c)<br />
Crystal liio3 liio3 bbo bbo<br />
L 1 mm 1 mm 1.5 mm 2 mm<br />
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦<br />
T0 → ∞ → ∞ → ∞ → ∞<br />
wp 400 µm 30 / 462 µm wp ≫ L 20, 500 µm<br />
λp 405 nm 405 nm 405 nm 351.1 nm<br />
∆λp 0 0.4 nm<br />
λ 0 s 810 nm 810 nm 810 nm 702 nm<br />
∆λs 0.2 nm 10 nm 10 nm<br />
ϕs 10 ◦ 17 ◦ 0, 1, 5 ◦ 4 ◦<br />
2.3 Correlations between signal and idler<br />
In contrast to the previous section, here the two-photon system is considered<br />
as composed by two photons, each described by space and time variables, as<br />
figure 2.1 (b) shows. Since the global state in equation 1.9 is pure, the purity<br />
of the signal photon calculated in this section is a measurement of the degree of<br />
spatiotemporal entanglement between signal and idler photons, as in references<br />
[5, 10, 11, 47, 48].<br />
spdc as a source of two-photon states can be used to generate pairs of<br />
photons maximally entangled or two single photons [5, 49]. The purity of the<br />
signal photon in the first case should be equal to 0 and in the second case equal<br />
to 1. This section explores the necessary conditions to be in each regime.<br />
This section is divided in three parts, the first one discusses the origin of<br />
the correlations and the strategies to suppress them. The second part describes<br />
how to obtain an analytical expression for the signal photon purity. And the<br />
third part shows the results of numerical calculation of that purity.<br />
2.3.1 Origin of the correlations between photons<br />
The presence of cross terms in variables of both signal and idler in equation<br />
1.34 indicates correlations between the generated photons. Figure 2.5 shows<br />
those terms: i, j, l, m, n, r, t, v, and z.<br />
In contrast to the last section, filtering one degree of freedom is not enough<br />
to suppress the correlations between signal and idler. For example, by using a<br />
frequency filter, the photons can be still correlated in space. The total suppression<br />
of the correlation requires infinite narrow filters in both degrees of freedom<br />
for one of the photons, as shown experimentally in reference [50]. But that kind<br />
of filtering reduces the number of available pairs of photons substantially.<br />
The other possibility to generate uncorrelated photons is to tailor the parameters<br />
of the spdc process to make the cross terms negligible simultaneously.<br />
According with appendix A, when α = 0 ◦ , this condition is equivalent to mak-<br />
23
2. Correlations and entanglement<br />
q s x<br />
q s y<br />
q i x<br />
y<br />
qi s<br />
i<br />
qs x<br />
a<br />
h<br />
i<br />
j<br />
k<br />
l<br />
q s<br />
y<br />
h<br />
b<br />
m<br />
n<br />
p<br />
r<br />
qi y<br />
i<br />
m<br />
c<br />
s<br />
t<br />
u<br />
Figure 2.5: The signal-idler cross terms in matrix A are responsible for the correlations<br />
between the generated photons. By suppressing these cross terms, and by making<br />
permutations over columns and rows, the matrix becomes a two block matrix, where<br />
each block contains the information of one photon.<br />
ing the following terms negligible<br />
i =w 2 p + γ 2 L 2 tan ρ0 2<br />
j =γ 2 L 2 sin ϕi tan ρ0<br />
l =γ 2 L 2 tan ρ0(Np − Ni cos ϕi)<br />
m = − γ 2 L 2 sin ϕs tan ρ0<br />
q i<br />
x<br />
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi<br />
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi<br />
t =γ 2 L 2 tan ρ0(Np − Ns cos ϕs)<br />
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs<br />
z =γ 2 L 2 N 2 p − γ 2 L 2 Np(Ni cos ϕi + Ns cos ϕs) + γ 2 L 2 NsNi cos ϕs cos ϕi<br />
j<br />
n<br />
s<br />
d<br />
v<br />
w<br />
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi. (2.15)<br />
As an example, a configuration with a negligible Poynting walk-off fulfills this<br />
condition always when<br />
cos ϕs = Np<br />
2Ns<br />
cos ϕi = Np<br />
2Ni<br />
w 2 p<br />
γ 2 L 2 = tan ϕs tan ϕi<br />
T 2 0 =N 2 p γ 2 L 2<br />
wp ≪ws, wi<br />
s<br />
k<br />
p<br />
t<br />
v<br />
f<br />
z<br />
i<br />
l<br />
r<br />
u<br />
w<br />
<br />
tan ϕs 2 tan ϕi 2 − 1<br />
<br />
4<br />
z<br />
g<br />
(2.16)<br />
When the correlations are totally suppressed by satisfying these or other conditions,<br />
the two-photon state is separable, and each photon is in a pure state.<br />
Therefore, the two-photon mode function can be written as a product of two<br />
mode functions, one for each photon,<br />
24<br />
Φ(q s, ωs, q i, ωi) = Φs(q s, Ωs)Φi(q i, Ωi). (2.17)
2.3. Correlations between signal and idler<br />
In any other case, correlations between signal and idler are unavoidable. To<br />
evaluate their strength, the next section deduces an analytical expression for<br />
the purity of the signal photon state.<br />
2.3.2 Spatio temporal state of signal photon<br />
A single photon state can be generated in spdc by ignoring any information<br />
about one of the generated photons. The single photon state is calculated by<br />
tracing out the other photon from the two-photon state. For example, after<br />
a partial trace over the idler photon, the reduced density matrix in space and<br />
frequency for the signal photon is<br />
ˆρsignal =T ridler[ρ]<br />
<br />
=<br />
<br />
=<br />
and its purity is given by<br />
dq ′′<br />
i dΩ ′′<br />
i 〈q ′′<br />
i , Ω ′′<br />
i |ˆρ|q ′′<br />
i , Ω ′′<br />
i 〉<br />
dqsdΩsdqidΩidq ′ sdΩ ′ s<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)|qs, Ωs〉〈q ′ s, Ω ′ s|, (2.18)<br />
T r[ˆρ 2 signal] =<br />
<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i). (2.19)<br />
Recalling the exponential character of the mode function described by equation<br />
1.35, equation 2.19 writes<br />
T r[ˆρ 2 signal] = det(2A)<br />
det(C) , (2.20)<br />
where C is a positive-definite real 12 × 12 matrix defined by<br />
N 4 <br />
exp − 1<br />
2 Xt <br />
CX = Φ(qs, Ωs, qi, Ωi)<br />
and given by<br />
⎛<br />
C = 1<br />
2<br />
⎜<br />
⎝<br />
× Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i), (2.21)<br />
2a 2h i j 2k l 0 0 i j 0 l<br />
2h 2b m n 2p r 0 0 m n 0 r<br />
i m 2c 2s t 2u i m 0 0 t 0<br />
j n 2s 2d v 2w j n 0 0 v 0<br />
2k 2p t v 2f z 0 0 t v 0 z<br />
l r 2u 2w z 2g l r 0 0 z 0<br />
0 0 i j 0 l 2a 2h i j 2k l<br />
0 0 m n 0 r 2h 2b m n 2p r<br />
i m 0 0 t 0 i m 2c 2s t 2u<br />
j n 0 0 v 0 j n 2s 2d v 2w<br />
0 0 t v 0 z 2k 2p t v 2f z<br />
l r 0 0 z 0 l r 2u 2w z 2g<br />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
(2.22)<br />
25
2. Correlations and entanglement<br />
Signal<br />
purity Filters<br />
1<br />
0<br />
0.1 Collection mode 1mm<br />
0nm<br />
1nm<br />
10nm<br />
→∞<br />
Figure 2.6: The correlations between photons can not be suppressed by filtering<br />
in only one degree of freedom. The purity of the signal state is only maximum<br />
when using ultra-narrow spatial and frequency filters. In this case, the 1 nm filters<br />
are broad enough to allow the correlations, and the 10 nm filters are equivalent to<br />
infinitely broad ones. The parameters used to generate this figure are listed in table<br />
2.2.<br />
The differences between the matrix C and matrix B in the last section, are<br />
due to the order of the primed and unprimed variables in the arguments of the<br />
mode functions in equations 2.14 and 2.21.<br />
The next part of this section uses equation 2.20, to calculate the values of<br />
the purity of the signal photon for different spdc configurations.<br />
2.3.3 Numerical calculations<br />
Figure 2.6 shows the space-frequency purity of the signal photon as a function<br />
of the spatial filter width for different values of the frequency filter width.<br />
Consider a degenerate type-i spdc configuration, where a pump beam with<br />
wavelength λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium<br />
iodate (liio3) crystal with length L = 1 mm, and negligible Poynting vector<br />
walk-off (ρ0 = 0). The crystal emits signal and idler photons with a wavelength<br />
λ 0 s = λ 0 i = 810 nm at an angle ϕs,i = 10 ◦ , and the collection modes for signal<br />
and idler are assumed to be equal (ws = wi). All these parameters are listed<br />
in the first column of table 2.2.<br />
As was discussed in the first part of this section, it is possible to achieve<br />
maximal separability between the photons by using infinitely narrow filters<br />
in both space and frequency. In the region of small values of ws (= wi) a<br />
considerable correlation between signal and idler exists even in the case of<br />
infinitely narrow frequency filters. Different values for the signal photon purity<br />
can be achieved by changing the filter width, as shown by figure 2.6.<br />
Additionally, the figure shows how the purity T r[ˆρ 2 signal ] is confined between<br />
the values obtained for ∆λs = ∆λi → 0 nm and ∆λs, ∆λi → ∞. This limits<br />
can be tailored by modifying the other parameters of the spdc configuration.<br />
Figure 2.7 shows T r[ˆρ 2 signal ] as a function of the pump beam waist wp for<br />
26
Signal<br />
purity<br />
1<br />
0<br />
(a)<br />
0.1 3<br />
2.3. Correlations between signal and idler<br />
(b)<br />
0.1 Pump waist 3<br />
(c)<br />
0.1<br />
Emission<br />
angle<br />
Figure 2.7: The effect of the emission angle on the signal purity changes depending<br />
on the filters used in space and frequency. When both filters have a finite size like in<br />
(a), the purity increases with the angle of emission, and has a maximum for a fixed<br />
value of the pump beam waist. When the frequency filters are infinitely narrow like<br />
in (b), the dependence of the purity on the angle of emission disappears. When using<br />
narrow filters in space, as in (c), the purity is equal to 1 for a certain pump beam<br />
waist that changes with the angle of emission. The parameters used to generate this<br />
figure are listed in table 2.2.<br />
Table 2.2: The parameters used in figures 2.6 and 2.7<br />
Parameter Figure 2.6 Figure 2.7(a) Figure 2.7(b) Figure 2.7(c)<br />
Crystal liio3 liio3 liio3 liio3<br />
L 1 mm 1 mm 1 mm 1 mm<br />
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦<br />
T0 → ∞ → ∞ → ∞ → ∞<br />
wp 400 µm 400 µm 400 µm 400 µm<br />
λp 405 nm 405 nm 405 nm 405 nm<br />
∆λp → 0 → 0 → 0 → 0<br />
ws variable → ∞ 400 µm 400 µm<br />
λ 0 s 810 nm 810 nm 810 nm 810 nm<br />
∆λs 0, 1, 10 nm → ∞ 10 nm → 0 10 nm<br />
ϕs 10 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦<br />
various values of the emission angle ϕs = ϕi, in degenerate type-i spdc configurations<br />
described by the parameters listed in the second, third and fourth<br />
columns of table 2.2.<br />
Figure 2.7 (a) shows the case of finite spatial and temporal filters. The<br />
purity of the signal photon is always smaller than one, increases with the emission<br />
angle, and has a maximum for a given value of the pump beam waist. For<br />
a very narrow band frequency filter with a ws = wi = 400 µm spatial filter,<br />
figure 2.7 (b) shows how the correlation between the photons is minimal for<br />
small pump beams. The emission angle for this particular crystal length is<br />
irrelevant. Finally, figure 2.7 (c) considers a frequency filter ∆λs = ∆λi = 10<br />
nm with infinitely narrow spatial filters. In this case, maximal purity appears<br />
for each emission angle at a particular value of the pump beam waist.<br />
Conclusion<br />
<strong>Two</strong> photons with two degrees of freedom compose the two-photon state generated<br />
in spdc. The correlations between all four elements affect the state of<br />
single photons or two-photon states with one degree of freedom. Using the pu-<br />
3mm<br />
5º<br />
10º<br />
20º<br />
27
2. Correlations and entanglement<br />
rity as correlation indicator it is possible to determine the conditions in which<br />
the specific correlations are suppressed, without requiring the use of infinitely<br />
narrow filters.<br />
The next chapter continues with the description of the correlations between<br />
photons, in the case where only their spatial state is relevant. The chapter describes<br />
the spatial correlations in terms of orbital angular momentum transfer.<br />
28
CHAPTER 3<br />
<strong>Spatial</strong> correlations and<br />
orbital angular momentum<br />
transfer<br />
The previous chapter describes the correlations between the signal and idler<br />
photons considering space and frequency, as well as the correlations between<br />
those degrees of freedom. This chapter focuses on the cases where the degrees<br />
of freedom are not correlated, and the only relevant correlations are<br />
those between the photons in the spatial degree of freedom. To characterize<br />
those correlations I use the orbital angular momentum (oam) content of<br />
the generated photons, which is directly associated to their spatial distribution.<br />
The spatial correlations between the photons determine the mechanism<br />
of oam transfer from the pump to the signal and idler. Most studies of the<br />
oam correlations in spdc can be divided in two categories: those reporting a<br />
full transference of the pump’s oam [7, 8, 22, 27, 28, 29], and those reporting<br />
a partial transference [30, 31, 32, 33]. This chapter describes the oam transfer<br />
mechanism, and shows the difference between both regimes. The chapter is<br />
divided in three sections. Section 3.1 introduces the eigenstates of the oam<br />
operator: the Laguerre-Gaussian modes. This section shows how to calculate<br />
and measure the oam content of the photons. Section 3.2 describes how the<br />
oam is transferred in spdc. As an example, section 3.3 shows how this transfer<br />
happens in the collinear case, where all the emitted photons propagate in the<br />
same direction. The main result of this chapter is a selection rule that summarizes<br />
the oam transfer mechanism: the oam carried by the pump is completely<br />
transferred to all the signal and idler photons emitted over the cone.<br />
29
3. <strong>Spatial</strong> correlations and OAM transfer<br />
-1<br />
0 1 2<br />
OAM content<br />
in ħ units<br />
Phase front<br />
Intensity<br />
profile<br />
Figure 3.1: The Laguerre-Gaussian modes are characterized by their phase front<br />
distribution and intensity profile. The figure shows the phase fronts and the intensity<br />
profiles for the modes with oam contents from −1 to 2.<br />
3.1 Laguerre-Gaussian modes and OAM content<br />
In an analogous way to the decomposition of an electromagnetic field as a<br />
series of planes waves, it is possible to decompose the field in other bases. For<br />
instance, paraxial fields can be decomposed as a sum of Laguerre-Gaussian (lg)<br />
modes. This basis is especially convenient since the lg modes are eigenstates of<br />
the orbital angular momentum (oam) operator. That is, the state of a photon<br />
in a lg mode has a well defined oam value [51]. This section describes the<br />
properties of the lg modes, and shows how to calculate the weight of each<br />
mode in a given decomposition.<br />
<strong>Photon</strong>s carrying oam different from zero have at least one phase singularity<br />
(or vortex) in the electromagnetic field, a region in the wavefront where the<br />
intensity vanishes. This is precisely one of the most distinctive characteristics<br />
of Lagurre-Gaussian modes as figure 3.1 shows. Each lg mode is defined by<br />
the number p of non-axial vortices, and the number l of 2π-phase shifts along<br />
a close path around the beam center. The index l also describes the helical<br />
structure of the phase front around the singularity, and more important here,<br />
l determines the orbital angular momentum carried by the photon in units.<br />
The state of a single photon in a lg mode is<br />
<br />
|lp〉 = dqLGlp(q)â † (q)|0〉, (3.1)<br />
where the mode function lglp(q) is given by Laguerre-Gaussian polynomials<br />
<br />
1<br />
2 wp!<br />
LGlp(q) =<br />
2π(|l| + p)!<br />
|l|<br />
wq<br />
√2 L |l|<br />
2 2 w q<br />
p<br />
2<br />
<br />
× exp − w2q2 <br />
exp ilθ + iπ(p −<br />
4<br />
|l|<br />
2 )<br />
<br />
(3.2)<br />
as a function of the beam waist w, the modulus q and the phase θ of the<br />
transversal vector, and the associated Laguerre polynomials L |l|<br />
p defined as<br />
30<br />
L |l|<br />
p [x] =<br />
p<br />
i=0<br />
i<br />
l + p (−x)<br />
. (3.3)<br />
p − i i!
3.2. OAM transfer in general SPDC configurations<br />
A special case of the lg modes is the zero-order mode that does not carry oam.<br />
Since the zero-order Laguerre polynomial L 0 0 = 1, the zero-order Laguerre-<br />
Gaussian lg00 = 1 is the Gaussian mode given by<br />
<br />
w<br />
1<br />
2<br />
LG00(q) = exp<br />
2π<br />
<br />
− w2 q 2<br />
4<br />
<br />
. (3.4)<br />
This is not, however, the only mode with oam equal to zero, the same is true<br />
for all other modes lg0p. Spiral harmonics are defined to collect all the modes<br />
with the same oam value, regardless of the value of p, these modes are defined<br />
as<br />
LGl(q) = <br />
LGlp(q) (3.5)<br />
p<br />
=al(q) exp [ilθ].<br />
The phase dependence on l, shown by each spiral harmonic mode, is exploited<br />
to determine the photon’s oam content [52]. Consider, for instance, a photon<br />
with a spatial distribution given by Φ(q), which using the spiral harmonic<br />
modes can be written as<br />
∞<br />
Φ(q) = al(q) exp [ilθ], (3.6)<br />
l=−∞<br />
so that each mode in the decomposition has a well defined oam of l per<br />
photon. Therefore, the probability Cl of having a photon with oam equal to l<br />
is the weight of the corresponding mode in the distribution:<br />
where<br />
Cl =<br />
al(q) = 1<br />
√ 2<br />
∞<br />
0<br />
2π<br />
0<br />
dq|al(q)| 2 q (3.7)<br />
dθΦ(q, θ) exp (−ilθ). (3.8)<br />
If the photon has a well defined oam of l0, the weight of the corresponding<br />
mode Cl0 = 1 and the weights of the other modes Cl=l 0 = 0. If Cl = 0 for<br />
different values of l, the photon state is a superposition of those modes with<br />
different oam values. Compared to the relative simplicity of these calculations,<br />
measuring the oam content is a more complicated task described in appendix<br />
C.<br />
Now that the techniques for the calculation of the oam content are introduced,<br />
the next section will use the spherical harmonics and the oam decomposition<br />
to study the transfer of oam from the pump photon to the signal and<br />
idler.<br />
3.2 OAM transfer in general SPDC configurations<br />
The oam carried by a photon is directly associated to its spatial shape. In<br />
order to simplify the description of the oam transfer mechanisms, this section<br />
considers only the spatial part of the mode function in a spdc configuration<br />
31
3. <strong>Spatial</strong> correlations and OAM transfer<br />
in which the space and the frequency are not correlated. By writing the mode<br />
function in a different coordinate system, it will be possible to show that a<br />
selection rule exists for the oam transfer in spdc.<br />
According to equation 1.6, all the fields have been defined in terms of a<br />
coordinate system related with their propagation direction. To study the oam<br />
transfer, we start by defining the fields in terms of a unique coordinate system,<br />
the one of the pump. This coordinate transformation is possible since there is<br />
a vector Kn such that kn · rn = Kn · rp.<br />
This coordinate transformation is more general than the one of section 1.3,<br />
where the plane that contained all photons was defined as the yz plane. As<br />
a consequence this section considers all possible emission directions over the<br />
cone, while section 1.3 considered only one.<br />
As was done for the vector kn in section 1.3, it is useful to separate the<br />
vector Kn in its longitudinal component Kz n and transversal component Qn. Taking into account the change of coordinates, the spatial part of the mode<br />
function in the semiclassical approximation can be written as<br />
<br />
∆kL<br />
Φq(Qs, Qi) ∝ Ep(Qs + Qi)sinc<br />
(3.9)<br />
2<br />
where the delta factor is given by<br />
with<br />
∆k = K z p(Qs + Qi) − K z s (Qi) − K z i (Qi). (3.10)<br />
K z j (Q) =<br />
<br />
ω 2 j n2 j<br />
c 2 − |Q|2 . (3.11)<br />
To study the oam transfer, consider the two functions in equation 3.9 separately.<br />
Consider first the sinc function. Due to its dependence on the delta<br />
factor, the sinc function depends only on the magnitudes |Qs +Qi|, Qs = |Qs|,<br />
Qi = |Qi|. And, since<br />
|Qs + Qi| 2 = Q 2 s + Q 2 i + 2QsQi cos(Θs − Θi), (3.12)<br />
the only angular dependence of the function is through the periodical term<br />
cos(Θs − Θi), therefore, using Fourier analysis it can be written in a very<br />
general form as<br />
<br />
∆kL<br />
sinc =<br />
2<br />
∞<br />
m=−∞<br />
Fm(Qs, Qi) exp [im(Θs − Θi)], (3.13)<br />
where, importantly, Fm(Qs, Qi) does not depend on the angles Θs and Θi.<br />
On the other hand, if we consider the pump beam as a Laguerre-Gaussian<br />
beam with a oam content of lp per photon, its mode function can be written<br />
as<br />
32<br />
Ep(Qs + Qi) ∝ exp<br />
<br />
− w2 p|Qs + Qi| 2<br />
4<br />
<br />
<br />
(Q x s + Q x i ) + i(Q y s + Q y<br />
i )<br />
l , (3.14)
3.2. OAM transfer in general SPDC configurations<br />
where the identity Q exp[iθ] = Q x + iQ y was used. Using equation 3.12, the<br />
last equation can be written as<br />
<br />
Ep(Qs + Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi<br />
<br />
cos(Θs − Θi))<br />
4<br />
<br />
l × Qs exp (iΘs) + Qi exp (iΘi)<br />
and by using the binomial theorem it becomes<br />
(3.15)<br />
lp<br />
<br />
lp<br />
Ep(Qs + Qi) ∝ Q<br />
l<br />
l=0<br />
l sQ lp−l<br />
i exp [ilΘs + ilpΘi − ilΘi]<br />
<br />
× exp − w2 p(Q2 s + Q2 i + 2QsQi<br />
<br />
cos(Θs − Θi))<br />
. (3.16)<br />
4<br />
By replacing equations 3.16 and 3.13 into equation 3.9, the two-photon spatial<br />
mode function becomes<br />
<br />
Φ(Qs, Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi<br />
<br />
cos(Θs − Θi))<br />
4<br />
∞<br />
× Gn(Qs, Qi) exp [inΘs + i(lp − n)Θi], (3.17)<br />
n=−∞<br />
where the value of Gn(Qs, Qi) does not depend on Θs and Θi. The index<br />
(lp − n) associated to Θi is fixed for each value of lp and ls, according to the<br />
phase dependence of the last equation. This relation can be written as the<br />
selection rule<br />
lp = ls + li. (3.18)<br />
Therefore, the angular momentum carried by the pump is completely transferred<br />
to the signal and idler photons.<br />
The geometry of the spdc configuration restricts the validity of the selection<br />
rule since it was deduced from equation 3.9, and that equation describes<br />
only the configuration in which all possible emission directions are taken into<br />
account. Collinear configurations achieve this condition by definition, but in<br />
the type-I noncollinear configurations, all the experiments up to now have measured<br />
only a certain portion of the whole cone. Under this condition the transfer<br />
of oam from the pump to the signal and idler is not governed necessarily by<br />
equation 3.18, in agreement with previous reports [32, 30, 31].<br />
Using numerical methods to calculate the oam content of the downconverted<br />
photons, the next section explores the oam transfer mechanisms in<br />
collinear configurations. This configurations are special cases where the generated<br />
photons propagate in the same direction and therefore, the detection<br />
system detects all photons from the emission cone.<br />
33
3. <strong>Spatial</strong> correlations and OAM transfer<br />
Configuration<br />
Weight<br />
1<br />
0<br />
-4 4<br />
0<br />
<strong>Spatial</strong><br />
-4 4<br />
distribution OAM content<br />
1<br />
Mode<br />
Figure 3.2: In a collinear configuration a pump beam with oam equal to 1 transfers<br />
its oam into the signal photon when the idler is projected into a Gaussian mode. The<br />
figure shows the pump in blue and the signal and idler in pink. Additionally, it shows<br />
the spatial distribution and the oam content of the signal photon for co-propagating<br />
and counter-propagating configurations.<br />
3.3 OAM transfer in collinear configurations<br />
The aperture of the down-conversion cone is given by the angle of emission<br />
of the photons. In the collinear configuration this angle is zero, and therefore<br />
the cone collapses onto its axis, as shown in figure 1.2 (d). As each photon<br />
propagates in the same direction as all the other photons, equation 3.18 is<br />
valid.<br />
To analyze the transfer of oam from the pump beam to the generated photons,<br />
we will calculate the signal oam content, assuming a lg1 pump beam so<br />
that lp = 1, and an idler photon projected into a Gaussian state li = 0. We<br />
will consider two kinds of collinear configurations: co-propagating and counterpropagating,<br />
both shown in figure 3.2. The collinear counter-propagating generation<br />
of photons requires fulfilling a special phase-matching condition, which<br />
can be achieved by quasi-phase-matching in a periodic structure. Such counterpropagating<br />
generation has been demonstrated in second harmonic generation<br />
[53], parametric fluorescence [54], and spdc in fibers [55]. Chapter 5 explores<br />
another way to generate counter-propagating two-photon states: Raman scattering.<br />
Figure 3.2 shows the calculated signal spatial distribution and oam content<br />
in two cases. The first row shows the collinear co-propagating case. The second<br />
row shows the collinear counterpropagating case.<br />
According to figure 3.2, the collinear case fulfills the selection rule, except<br />
for a minus sign in the counterpropagating case. The phase appears because<br />
the photon’s oam content is evaluated with respect to the pump propagation<br />
direction, as figure 3.3 shows. To take this effect into account equation 3.18<br />
34
Phase front<br />
3.3. OAM transfer in collinear configurations<br />
OAM respect to z<br />
direction<br />
Figure 3.3: The direction of rotation of the phase shift defines the sign of the mode’s<br />
oam content. For a Laguerre-Gaussian mode corresponding to positive oam content,<br />
the phase shift rotates clockwise with respect to the propagation direction. If the<br />
same mode propagates backwards, the phase shift rotates anticlockwise with respect<br />
to the original propagation direction, and the mode has a negative oam content.<br />
can be generalized to<br />
lp = dsls + d1li<br />
+2<br />
-2<br />
-2<br />
(3.19)<br />
where ds,i = ±1 corresponds to forward and backward propagation of the<br />
corresponding photon.<br />
Conclusion<br />
The oam content of the pump is completely transferred to the signal and idler<br />
photons, emitted all over a cone. This makes it possible to introduce a selection<br />
rule that always holds if all possible emission directions are considered, but not<br />
necessarily if only a small part is detected. This result clarifies the apparent<br />
contradiction between several works that support the selection rule [7, 28, 56],<br />
and those that report that it is not valid [32, 31, 57].<br />
When only a portion of all generated photons are considered there is no<br />
simple relationship between lp, ls and li. The next chapter studies the oam<br />
transfer in those cases, describing the effect of different spdc parameters on<br />
the amount of oam that is transferred to the subset of the photons considered.<br />
35
CHAPTER 4<br />
OAM transfer in<br />
noncollinear configurations<br />
The previous chapter describes the oam transfer from the pump to the signal<br />
and idler photons considering all the possible emission directions of those<br />
photons. However, in non collinear configurations, the photons detected are<br />
just a subset of the total emission cone; noncollinear configurations generate<br />
correlated photons, which are naturally spatially separated, which is exactly<br />
what makes noncollinear so important. The change in the geometry of the<br />
process, imposed by the detection system, has a strong effect on the oam of<br />
the detected photons. This chapter focuses on the description of the oam that<br />
is transferred to a small portion of the emitted photons in noncollinear configurations.<br />
To characterize the transfer, I study a particular spdc process in<br />
which the pump is a Gaussian beam and one of the photons is projected into a<br />
Gaussian mode. Therefore, the oam content of the other photon will indicate<br />
how close the configuration is to satisfying the oam selection rule. This chapter<br />
is divided in three sections. Section 4.1 shows a simple example in which the<br />
selection rule does not apply as just a small section of the cone is considered.<br />
In a more general scenario, section 4.2 shows that the violation of the selection<br />
rule is not only mediated by the emission angle, but also by the pump beam<br />
waist. Finally, section 4.3 shows the effect of the Poynting vector walk-off on<br />
the oam of the noncollinear photons. As a main result, this chapter explains<br />
how the pump beam waist and the Poynting vector walk-off affect the oam<br />
transfer, in the cases where the selection rule does not apply. By tailoring both<br />
parameters it is possible to generate photons with specific spatial shapes. For<br />
instance, it is possible to generate photons in Gaussian modes that are more<br />
efficiently coupled into single mode fibers.<br />
37
4. OAM transfer in noncollinear configurations<br />
4.1 Ellipticity in noncollinear configurations<br />
In the experimental implementation of spdc, the detection system selects the<br />
photons that are emitted in a certain spatial direction. In a collinear configuration,<br />
the selection is not a problem since all the photons are emitted in the<br />
same direction. But, in a noncollinear configuration, the photons are emitted in<br />
different azimuthal directions described by the angle α, therefore the detection<br />
system selects only a section of the full cone. As the symmetry is broken when<br />
only a portion of the cone is considered, the oam transfer mechanism can not<br />
be described by the selection rule in 3.18. This section studies this mechanism<br />
in a simple spdc configuration by calculating the spatial distribution of the<br />
signal photon after fixing the oam content of the pump and idler photons.<br />
As a first example, consider a Gaussian pump beam lp = 0 and an idler<br />
photon projected into a Gaussian mode li = 0, given by<br />
<br />
u(qi) = Ni exp − w2 i<br />
4 (qx2 i + q y2<br />
i )<br />
<br />
, (4.1)<br />
the spatial distribution of the signal photon is given by the normalized mode<br />
function<br />
<br />
Φs(qs) = Ns dqiΦq(qs, qi)u(qi). (4.2)<br />
This integral has an analytical solution for simple spdc configurations. Consider<br />
a degenerate spdc process with negligible Poynting vector walk-off. After<br />
suppressing the correlations between space and frequency the spatial part of<br />
the two-photon state is given by<br />
<br />
<br />
with<br />
Φq(qs, qi) = exp<br />
− w2 p<br />
4 ∆2 0 − w2 p<br />
4 ∆2 1<br />
∆0 =q x s + q x i<br />
∆1 =q y s cos ϕs + q y<br />
i cos ϕi<br />
∆k = − q y s sin ϕs + q y<br />
i<br />
sin ϕi.<br />
<br />
L<br />
sinc<br />
2 ∆k<br />
<br />
(4.3)<br />
(4.4)<br />
Therefore, using the sinc to exponential approximation, the signal mode function<br />
defined by equation 4.2 is given by<br />
<br />
<br />
Φs(qs) =Ns exp<br />
× exp<br />
<br />
−<br />
w2 pw2 i<br />
4(w2 p + w2 qx2 s<br />
i )2<br />
− 4w2 pγ 2 L 2 cos ϕs 2 sin ϕs 2 + (w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs)w 2 i<br />
4(w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs 2 + w 2 i )<br />
q y2<br />
s<br />
<br />
(4.5)<br />
When the coefficients of the variables q x s and q y s are equal, the signal mode<br />
function reduces to a Gaussian and the selection rule is fulfilled. These coefficients<br />
are equal in collinear configurations, or in noncollinear configurations in<br />
38<br />
.
4.2. Effect of the pump beam waist on the OAM transfer<br />
which the spdc parameters satisfy the relationship<br />
γ 2 L 2 w<br />
=<br />
2 pw4 i<br />
w4 i + 4w2 p(w2 i + w2 p) cos ϕ2 s<br />
. (4.6)<br />
In any other case, the coefficients of the variables q x s and q y s are different, and<br />
the signal mode function becomes elliptical. The ellipticity of the spatial profile<br />
implies the presence of non-Gaussian modes, and therefore it can be used as a<br />
qualitative probe that the selection rule is not fulfilled (lp = ls + li).<br />
As the ellipticity is an effect of the partial detection, it can be controlled<br />
by changing the total shape of the cone, or by changing the detector angular<br />
acceptance. The phase matching conditions define the shape of the cone as a<br />
function of the emission angle ϕs, the pump beam waist wp, and the length of<br />
the crystal L; the detector angular acceptance is a function of the waist of the<br />
spatial modes ws, wi. The next two sections use both the mode decomposition<br />
and the ellipticity to describe the effect of all these parameters on the signal<br />
oam content in a more general scenario than considered in this section.<br />
4.2 Effect of the pump beam waist on the OAM transfer<br />
<strong>Of</strong> all the spdc parameters that affect the signal ellipticity, the easiest to<br />
control is the pump beam waist. To change the angle of emission or the crystal<br />
length implies changing of the geometrical configuration. While changing the<br />
pump beam waist just requires adding lenses in the beam path. In the first<br />
part of this section, numerical calculations show the role of the pump beam<br />
waist on the signal oam content. The second part describes the experimental<br />
corroboration of this effect.<br />
4.2.1 Theoretical calculations<br />
Consider a spdc configuration as the one in equation 1.31. With a Gaussian<br />
pump beam and an idler photon projected into a Gaussian mode, the oam content<br />
of the signal photon can be used to describe the oam transfer mechanism<br />
in spdc. The selection rule is fullfilled only if ls = 0. Equivalently, one could<br />
choose to use the idler photon to study the oam transfer after projecting the<br />
signal into a Gaussian mode.<br />
In a degenerate type-i spdc process characterized by the parameters in<br />
table 4.1, a Gaussian pump beam, with wavelength λ 0 p = 405 nm illuminates<br />
a 10 mm ppktp crystal. The crystal emits signal and idler photons with a<br />
wavelength λ 0 s = λ 0 i = 810 nm. Both photons propagate at ϕs,i = 1 ◦ , after<br />
the crystal they traverse a 2f system, and finally the idler photon is projected<br />
into a Gaussian mode with wi → ∞, so that only idler photons with qi = 0<br />
are considered.<br />
Figure 4.1 shows the signal oam content for two values of the pump beam<br />
waist: wp = 100 µm in the left and wp = 1000 µm in the right. In the<br />
distributions, each bar represents a mode ls, with a weight in the distribution<br />
Cls given by the height of the bar. In the left part of the figure, where the<br />
pump beam waist is smaller, the distribution shows several modes. For larger<br />
waists, the Gaussian mode becomes the only important mode, as seen in the<br />
right part of the figure.<br />
39
4. OAM transfer in noncollinear configurations<br />
Weight<br />
1<br />
0<br />
(a) (b)<br />
-4 Mode 4 -4 Mode 4<br />
Figure 4.1: As the pump beam waist increases, the Gaussian mode in the signal oam<br />
distribution becomes more important. The left part of the figure, where wp = 100 µm,<br />
shows several non-Gaussian modes in blue, while the right part, where wp = 1000 µm,<br />
shows only a Gaussian mode in gray. The other parameters used to generate this<br />
figure are listed in table 4.1.<br />
Table 4.1: The parameters used in the theoretical calculations in section 4.2.<br />
Parameter Value<br />
Crystal ppktp<br />
L 10 mm<br />
ρ0<br />
0◦ Laser cw diode<br />
wp 100 − 1000 µm<br />
λp<br />
405 nm<br />
λ0 s<br />
ϕs<br />
810 nm<br />
1◦ Figure 4.2 shows the probability of finding a non-Gaussian mode as a function<br />
of the pump beam waist. The presence of non-Gaussian modes implies a<br />
violation of the selection rule in the configuration considered here. The figure<br />
shows the probability Cls = 0 for emission angles 1, 5, 10 ◦ . As the collinear<br />
angle ϕ becomes larger, the probability to detect signal photons with ls = 0<br />
increases. In a highly noncollinear configuration a large pump waist does not<br />
guarantee the generation of a Gaussian mode, that is, it does not imply the<br />
validity of the selection rule.<br />
Reference [30] introduced the noncollinear length defined as Lnc = wp/ sin ϕ.<br />
Lnc quantifies the strength of the violation of the selection rule due to the angle<br />
of emission and the pump beam waist. The authors defined this quantity<br />
based on the fact that when Lq y sin ϕ is small the sinc function (that introduces<br />
ellipticity) tends to one. As q y is of the order of 1/wp the condition can be<br />
written as L sin ϕ/wp ≪ 1, or L ≪ Lnc. In this regime the ellipticity of the<br />
mode function is small, and thus the selection rule lp = ls + li is fulfilled. This<br />
is the case for nearly all the experiments using the oam of photons generated<br />
in spdc [7, 12, 28, 56, 58, 25, 26, 29]. Instead, if the crystal length is larger<br />
than or equal to the noncollinear length L ≥ Lnc, a strong violation of the<br />
selection rule is expected. The experiments reported in references [31, 32, 57]<br />
are in this regime since the authors used highly focused pump beams or long<br />
crystals.<br />
40
1.0<br />
0.0<br />
4.2. Effect of the pump beam waist on the OAM transfer<br />
Nongaussian<br />
mode probability<br />
0.8<br />
0.4<br />
Emission<br />
angle<br />
10º<br />
0.1 Pump beam width 1mm<br />
Figure 4.2: The violation of the selection rule, given by the weight of the non-Gaussian<br />
modes, becomes more important as the collinear angle increases, or as the pump beam<br />
is more focalized. The parameters used to generate this figure are listed in table 4.1.<br />
4.2.2 Experiment<br />
Figure 4.3 depicts the experimental set-up that we used to measure the effect<br />
of the pump beam waist over the signal photon spatial shape. A laser beam<br />
with wavelength λ 0 p = 405 nm, and bandwidth ∆λp = 0.6 nm illuminated a<br />
lithium iodate (liio3) crystal with length L = 5 mm. The crystal was cut in a<br />
configuration for which non of the interacting waves exhibited Poynting vector<br />
walk-off, so ρ0 = 0 ◦ . The signal and idler photons are emitted with wavelengths<br />
λ 0 s = λ 0 i = 810 nm at ϕs,i = 17.1 ◦ . Table 4.2 summarizes the most relevant<br />
experimental parameters.<br />
A spatial filter after the laser provided an approximately Gaussian beam<br />
with a waist of 500 µm. By adding lenses before the crystal, this waist could<br />
be changed in the range of 32 − 500 µm.<br />
Each generated photon passed through a 2 − f system (f = 250 mm),<br />
that provided an image of the spatial shape of the two-photon state. Later,<br />
the photons passed through a broadband colored filter, that removed the scattered<br />
radiation at 405 nm, and through small pinholes to increase the spatial<br />
resolution.<br />
Multimode fibers mounted in xy translation stages collected the photons,<br />
and carried them into two single photon counting modules. A coincidence<br />
circuit counted the number of times that a signal photon was detected within<br />
8 ns of the detection of an idler photon. We used a data acquisition card to<br />
transfer the circuit’s output into a computer.<br />
We measured the signal and idler counts, as well as the number of coincidences<br />
for a fixed position of the idler in two orthogonal directions in the<br />
transverse signal plane as shown by figure 4.3. During the experiments it was<br />
checked that the use of narrow band filters of ∆λs = 10 nm does not modify<br />
5º<br />
1º<br />
41
4. OAM transfer in noncollinear configurations<br />
Laser and<br />
spatial filter lenses<br />
crystal<br />
lenses<br />
filter<br />
collection<br />
system<br />
pinhole<br />
x<br />
idler<br />
pinhole<br />
signal<br />
Figure 4.3: A spatially filtered cw laser is focused into a nonlinear crystal. A coincidence<br />
circuit measures the correlations between the photon counts in a fixed position<br />
for the idler photon with the counts of photons in two orthogonal directions in the<br />
signal transverse plane. The values of the different experimental parameters are listed<br />
in table 4.2.<br />
the measured shape of the coincidence rate, although it does modify the single<br />
counts spatial shape.<br />
Table 4.2: The parameters of the experiment described in section 4.2.<br />
Parameter Value<br />
Crystal liio3<br />
L 5 mm<br />
ρ0<br />
0◦ Laser cw diode<br />
wp 32 − 500 µm<br />
λp 405 nm<br />
∆λp 0.6 µm<br />
λ0 s<br />
∆λ<br />
810 nm<br />
0 s<br />
ϕs<br />
10 nm<br />
17.1◦ The left side of figure 4.4 shows the coincidence measurements in the x and<br />
y directions for a focalized pump beam. As the pump beam waist is wp = 32<br />
µm, the noncollinear length Lnc = 108.8 µm is much smaller than the crystal<br />
length. According with reference [30], in this regime the ellipticity of the signal<br />
photon should be appreciable.<br />
By fitting the result to a Gaussian function, the waist in the x direction<br />
is wx 960 µm, while in the y direction it is wy 150 µm. The waist in x<br />
is about six times larger than the waist in y, and therefore the signal spatial<br />
distribution is elliptical. As a reference, the figure shows the singles counts for<br />
the signal coupler.<br />
The right side of the figure shows the coincidence measurements for a larger<br />
pump beam, wp 500 µm. In this case, the noncollinear length Lnc = 1.7 mm<br />
is of the same order of magnitude as the crystal length. When fitting the<br />
42<br />
y
Coincidences<br />
800<br />
0<br />
-4<br />
4.2. Effect of the pump beam waist on the OAM transfer<br />
x<br />
y<br />
4mm<br />
Singles<br />
6x10 5<br />
~<br />
-1<br />
y x<br />
Singles<br />
5x10 4<br />
~<br />
1mm<br />
Figure 4.4: The ellipticity decreases as the pump beam waist increases from wp = 30<br />
µm in the left to 500 µm in the right image. As the pump beam waist affects<br />
the efficiency of the process, the coincidences on the left were collected over 600<br />
seconds and the ones in the right over 20 seconds. Solid lines are the best fit to the<br />
experimental data shown as points. The values of the experimental parameters are<br />
listed in table 4.2.<br />
Coincidences<br />
region width<br />
2.5m<br />
0.5<br />
0<br />
0<br />
x dimension<br />
y dimension<br />
L nc=1.1mm<br />
Pump width 330 500m Figure 4.5: The pump beam waist controls the width of the profile in the x direction,<br />
while it does not affect the width in the y direction. The width in both directions<br />
becomes similar when the noncollinear length is of the same order of magnitude as<br />
the crystal length. The vertical line shows the point where, according to the fits, both<br />
dimensions are equal. In this point the noncollinear length is 1.1 mm. The values of<br />
the experimental parameters are listed in table 4.2.<br />
transverse cuts to Gaussian functions, the waist in the x direction is wx 120<br />
µm and the waist in the y direction is wy 180 µm. The waist in the x<br />
direction is much smaller than before and of the same order as the waist in y<br />
which implies a decrease of the ellipticity.<br />
Figure 4.4 illustrates the changes in the width in the x and y directions as a<br />
function of pump beam waist. In the range where the waist varies from 32−500<br />
µm, the width in y remains almost constant while the width in x decreases by<br />
80%. After a certain threshold the width in x becomes stable at a value close<br />
43
4. OAM transfer in noncollinear configurations<br />
100µ m<br />
1mm<br />
pump crystal<br />
-4 4<br />
-4 4<br />
1<br />
0<br />
1<br />
0<br />
after 1mm<br />
-4 4<br />
-4 4<br />
0.1<br />
0<br />
1<br />
0<br />
output mode<br />
Figure 4.6: Due to the crystal birefringence, the oam content of a beam changes as<br />
the beam passes trough the material. The number of new oam modes introduced<br />
by the birefringence increases for more focused beams. The figure shows the mode<br />
content of a Gaussian beam after traveling 1 mm in the crystal, and at the output of<br />
the 5- mm-long crystal.<br />
to the value of the width in y, and therefore the ellipticity disappears.<br />
4.3 Effect of the Poynting vector walk-off on the OAM<br />
transfer<br />
Up to now, this chapter only considered spdc configurations where the Poynting<br />
vector walk-off was not relevant. However, since the selective detection of<br />
one section of the cone affects the oam transfer, the distinguishability introduced<br />
by the walk-off should affect it as well.<br />
To understand the effect of the walk-off on the oam transfer, consider that<br />
the spatial shape of the pump is modified as it passes trough the crystal due to<br />
the birefringence. A Gaussian photon thus acquires more modes when traveling<br />
in a crystal. This section explains this effect using theoretical calculations and<br />
experimental results.<br />
4.3.1 Theoretical calculations<br />
The displacement introduced by the walk-off, explained in section 1.3, changes<br />
the pump beam spatial distribution and therefore its oam content. The transverse<br />
profile of the pump, at each position z inside the nonlinear crystal, can<br />
be written as<br />
Ep (qp,z)=E0 exp<br />
<br />
−q 2 p<br />
w 2 p<br />
4<br />
z<br />
+ i<br />
2k0 <br />
∞<br />
Jn (zqp tan ρ0) exp {inθp}<br />
p n=−∞<br />
(4.7)<br />
where Jn are Bessel functions of the first kind. Based on this expression, figure<br />
4.6 shows the oam content of a Gaussian pump beam after traveling through<br />
a 5 mm crystal. New modes appear and become important as the pump beam<br />
gets narrower.<br />
The oam content of the pump with respect to z is different at different<br />
positions inside the crystal. Pairs of photons produced at the beginning or at<br />
the end of the crystal are effectively generated by a pump beam with different<br />
spatial properties, as figure 4.6 shows. This change in the pump beam is<br />
44
Weight<br />
1<br />
0<br />
4.3. Effect of the Poynting vector walk-off on the OAM transfer<br />
0º 90º Azimuthal angle 360º<br />
other<br />
modes<br />
Gaussian<br />
mode<br />
0º 90º Azimuthal angle 360º<br />
Gaussian<br />
mode<br />
other<br />
modes<br />
Figure 4.7: The probability of generating a Gaussian signal photon varies with the<br />
azimuthal angle, and has a maximum at α = 90 ◦ where the noncollinearity effect<br />
compensates the Poynting vector walk-off. At other angles, like α = 360 ◦ the probability<br />
of generating a Gaussian signal decreases and other modes become important<br />
in the distribution. Those non-Gaussian modes are more numerous for more focused<br />
beams, as seen by comparing the left part of the figure, where wp = 100 µm, to the<br />
right part, where wp = 600 µm.<br />
Signal<br />
purity<br />
0.3<br />
0.1<br />
0.0<br />
without<br />
walk-off<br />
with<br />
walk-off<br />
azimuthal<br />
0º 90º 180º 270º 360º angle<br />
Figure 4.8: Like the signal oam content, the correlations between the photons change<br />
with the azimuthal angle. The walk-off not only introduces an azimuthal variation<br />
but increases the correlations.<br />
translated into a change in the oam content of the generated pair with respect<br />
to the z axis.<br />
Additionally, because of the walk-off the generated photons are not symmetric<br />
with respect to the displaced pump beam, and their azimuthal position<br />
α becomes relevant. Figure 4.7 shows the weight of the mode ls = 0, and the<br />
weight of all other oam modes, as a function of the angle α for two different<br />
pump beam widths. In the left part, the pump beam waist is 100 µm, while<br />
in the right part it is 600 µm. The oam correlations of the two-photon state<br />
change over the down-conversion cone due to the azimuthal symmetry break-<br />
45
4. OAM transfer in noncollinear configurations<br />
ing induced by the spatial walk-off. The symmetry breaking implies that the<br />
correlations between oam modes do not follow the relationship lp = ls + li.<br />
Figure 4.7 also shows that for larger pump beams the azimuthal changes are<br />
smoothened out.<br />
The insets in figure 4.7 show the oam decomposition at α = 90 ◦ and at<br />
α = 360 ◦ . At α = 90 ◦ , the walk-off effect compensates the noncollinear effect,<br />
and the weight of the ls = 0 mode is larger than the weight of any other<br />
mode. This angle is optimal for the generation of heralded single photons with<br />
a Gaussian shape.<br />
The degree of spatial entanglement between the photons also exhibits az-<br />
imuthal variations depending on their emission direction. Figure 4.8 shows the<br />
] as a function of the azimuthal angle α with and with-<br />
signal purity T r[ρ2 signal<br />
out walk-off, for a pump beam waist wp = 100 µm with collection modes of<br />
ws = wi = 50 µm. When considering the walk-off, the degree of entanglement<br />
increases and becomes a function of the azimuthal angle. The purity has a<br />
minimum for α = 0◦ and α = 180◦ and maximum for α = 90◦ and α = 270◦ .<br />
The spatial azimuthal dependence affects especially those experimental configurations<br />
where photons from different parts of the cone are used, as in the<br />
case in the experiment reported in reference [12] for the generation of photons<br />
entangled in polarization. Using two identical type-i spdc crystals with the<br />
optical axes rotated 90◦ with respect to each other, the authors generated a<br />
space-frequency quantum state given by<br />
|Ψ〉 = 1<br />
<br />
√ dqsdqi Φ<br />
2<br />
1 q(qs, qi)|H〉s|H〉i + Φ 2 <br />
q(qs, qi)|V 〉s|V 〉i<br />
(4.8)<br />
where Φ 1 q(qs, qi) is the spatial mode function of the photons generated in the<br />
first crystal<br />
Φ 1 q(qs, qi) = Φq(qs, qi, α = 0 ◦ ) exp [iL tan ρ0(q y s + q y<br />
i )], (4.9)<br />
and Φ 2 q(qs, qi) is the spatial mode function of the photons generated in the<br />
second crystal<br />
Φ 2 q(qs, qi) = Φq(qs, qi, α = 90 ◦ ). (4.10)<br />
Since the photons generated in the first crystal are affected by the walk-off as<br />
they pass by the second crystal, the mode functions are different.<br />
Following chapter 2, the polarization state of the generated photons is calculated<br />
by tracing out the spatial variables from equation 4.8. The resulting<br />
state is described by the density matrix<br />
where<br />
46<br />
ˆρp = 1<br />
<br />
|H〉s|H〉i〈H|s〈H|i + |V 〉s|V 〉i〈V |s〈V |i<br />
2<br />
<br />
+c|H〉s|H〉i〈V |s〈V |i + |V 〉s|V 〉i〈H|s〈H|i<br />
(4.11)<br />
<br />
c = dqsdqiΦ 1 q(qs, qi)Φ 2 q(qs, qi). (4.12)
4.3. Effect of the Poynting vector walk-off on the OAM transfer<br />
Length<br />
0.5mm<br />
2mm<br />
beam<br />
waist<br />
Concurrence<br />
1<br />
50 500m Figure 4.9: The concurrence of the polarization entangled state given by equation<br />
4.8 decreases when the effect of the azimuthal variation is stronger, that is for small<br />
pump beam waist or for large crystals.<br />
The degree of correlation between space and polarization is given by the purity<br />
of the polarization state<br />
T r[ρ 2 p] =<br />
0.5<br />
0.0<br />
1 + |c|2<br />
. (4.13)<br />
2<br />
If the walk-off effect is negligible T r[ρ 2 p] = 1, then space and polarization are<br />
not correlated. If the walk-off is not negligible, the azimuthal changes in the<br />
spatial shape are correlated with the polarization of the photons.<br />
Figure 4.9 indirectly shows the effect of the azimuthal spatial information<br />
over the polarization entanglement. The entanglement between the photons<br />
is quantified using the concurrence C = |c|, which is equal to 1 for maximally<br />
entangled states. The figure shows the variation of C as a function of the pump<br />
beam waist for two crystal lengths. For small pump beam waist, the azimuthal<br />
effect is stronger and the concurrence decreases. As the waist increases, the<br />
walk-off effect becomes weaker and the value of the concurrence increases. This<br />
effect is more important for longer crystals, where the walk-off modifies the<br />
pump beam shape more.<br />
4.3.2 Experiment<br />
To experimentally corroborate the predicted azimuthal changes in the signal<br />
spatial shape we set up the experiment described by the parameters listed in<br />
table 4.3 and shown in figure 4.10. We used a continuous wave diode laser<br />
emitting at λp = 405 nm and with an approximate Gaussian spatial profile,<br />
obtained by using a spatial filter. A half wave plate (hwp) controlled the<br />
direction of polarization of the beam, while a lense focalized it on the input<br />
face of the crystal, with a beam waist of wp = 136 µm. The 5 mm liio3 crystal<br />
produced pairs of photons at 810 nm that propagated at ϕs,i = 4 ◦ . Due to<br />
the crystal birefringence, the pump beam exhibited a Pointing vector walk-off<br />
47
4. OAM transfer in noncollinear configurations<br />
Laser and<br />
spatial filter HWP lenses<br />
crystal<br />
lenses<br />
filter<br />
collection<br />
system<br />
camera<br />
pinhole<br />
Figure 4.10: By rotating the crystal and the pump beam polarization, it was possible<br />
to measure the coincidences between pairs of photons at different positions over<br />
the emission cone. The photons were collected using multimode fibers and 300 µm<br />
pinholes to increase the resolution. The values of the experimental parameters are<br />
listed in table 4.3.<br />
given by the angle ρ0 = 4.9 ◦ , while the generated photons did not exhibit<br />
spatial walk-off.<br />
Table 4.3: The parameters of the experiment described in section 4.3.<br />
Parameter Value<br />
Crystal liio3<br />
L 5 mm<br />
ρ0 4.9◦ Laser cw diode<br />
wp 136 µm<br />
λp 405 nm<br />
∆λp 0.4 nm<br />
λ0 s<br />
ϕs<br />
810 nm<br />
4◦ We measured the relative position of the pump beam in the xy plane at<br />
the input and output faces of the nonlinear crystal using a ccd camera. At<br />
the output of the crystal, the beam displacement due to the Poynting vector<br />
walk-off allowed us to determine the direction of the crystal’s optical axis, and<br />
to fix its direction parallel to the pump polarization, as shown in figure 4.11.<br />
Right after the crystal, each of the generated photons passed through a<br />
2 − f system with focal length f=50 cm. Cut-off spectral filters removed the<br />
remaining pump beam radiation. After the filters, the photons were coupled<br />
into multimode fibers. In order to increase the spatial resolution, we used small<br />
pinholes with a diameter of 300 µm.<br />
We kept the idler pinhole fixed and measured the coincidence rate while<br />
mapping the signal photon’s transversal shape by scanning the detector with<br />
a motorized xy translation stage, as in the right part of figure 4.3. Finally,<br />
we rotated the crystal and the pump beam polarization to measure the spatial<br />
correlations at different azimuthal positions on the down-conversion cone. After<br />
48
Before<br />
the crystal<br />
4.3. Effect of the Poynting vector walk-off on the OAM transfer<br />
After<br />
the crystal<br />
Polarization Optical axis<br />
Figure 4.11: The images obtained with a ccd camera show that before the crystal,<br />
the pump beam has a circular shape, and that after passing the crystal part of the<br />
beam gets displaced in the direction of the optical axis of the crystal. The displaced<br />
part corresponds to the portion of the beam polarized orthogonal to the crystal axis.<br />
In the case of the last image almost all the beam is displaced. The values of the<br />
experimental parameters are listed in table 4.3.<br />
every rotation, the tilt of the crystal was adjusted to achieve the generation of<br />
photons at the same noncollinear angle in all cases.<br />
The bottom row of figure 4.12 presents a sample of images taken at different<br />
azimuthal sections of the cone, while the upper row shows the expected shape<br />
predicted by the theory. Each column shows the coincidence rate for different<br />
angles, α = 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . Each point of these images corresponds to<br />
the recording of a 10 seconds measurement. The typical maximum number of<br />
coincidences is around 10 coincidences per second. Each image is 10 × 10 mm,<br />
and its resolution is 50 × 50 points.<br />
Theory<br />
Experiment<br />
0º 90º 180º 270º<br />
Figure 4.12: The coincidence measurements of the mode function, as well as the<br />
theoretical calculations, show that the ellipticity of the mode function changes for<br />
different positions on the cone. The ellipticity is minimal for α = 90 ◦ . The values of<br />
the experimental parameters are listed in table 4.3.<br />
49
4. OAM transfer in noncollinear configurations<br />
The different spatial shapes in figure 4.12 clearly show that the downconversion<br />
cone does not posses azimuthal symmetry. As was predicted by the<br />
theoretical calculations, the coincidence measurement for α = 90 ◦ , presents a<br />
nearly Gaussian shape, while the other cases are highly elliptical.<br />
The slight discrepancies between experimental data and theoretical predictions<br />
observed might be due to the small (but not negligible) bandwidth of the<br />
pump beam, and due to the fact that the resolution of our system is limited<br />
by the detection pinhole size.<br />
Conclusion<br />
The spdc parameters and the detection system determine the portion of the<br />
cone that is detected in a noncollinear configuration. This chapter explains how<br />
the pump beam waist and the Poynting vector walk-off affect the oam transfer.<br />
By tailoring both parameters it is possible to generate photons with specific<br />
spatial shapes. The walk-off affects especially those configurations where pairs<br />
of photons with different α are used.<br />
The next chapter extends the analysis of the spatial correlations to pairs<br />
of photons generated in Raman transitions. The chapter describes how the<br />
specific characteristics of that source are translated into the two-photon spatial<br />
state.<br />
50
CHAPTER 5<br />
<strong>Spatial</strong> correlations<br />
in Raman transitions<br />
<strong>Two</strong>-photon states can be generated in different nonlinear processes, and in every<br />
case the oam transfer will depend on the particular configuration. Raman<br />
transition is an alternative method for the generation of two-photon states.<br />
Several authors have proven the generation of correlated photons in polarization<br />
[59], frequency [60] and oam [23] via Raman transitions. Typical configurations<br />
involve the partial detection of the generated photons [61, 62, 63] in<br />
quasi-collinear configurations [64]. Just as in the case of spdc, the geometrical<br />
conditions determine the oam transfer. This chapter analyses the oam<br />
transfer in Raman transitions using the techniques of the previous chapters.<br />
This chapter is divided in three sections. Section 5.1 describes the generated<br />
two-photon state by introducing its mode function. Section 5.2 studies the<br />
oam content of one of the photons with the oam of the other photons fixed.<br />
Section 5.3 describes the effect of the geometry of the process on the spatial<br />
entanglement between the photons. Numerical calculations show the effect of<br />
the geometrical configuration on the oam transfer mechanism. The finite size<br />
of the nonlinear medium results in new effects that do not appear in spdc.<br />
51
5. <strong>Spatial</strong> correlations in Raman transitions<br />
pump<br />
g<br />
e<br />
s<br />
anti<br />
Stokes Stokes<br />
control<br />
Figure 5.1: One atom with a Λ-type energy level configuration, can produce Stokes<br />
and anti-Stokes photons by the interaction with the pump and control beams.<br />
5.1 The quantum state of Stokes and anti-Stokes photon<br />
pairs<br />
This section describes the Stokes and anti-Stokes state generated by Raman<br />
transitions in cold atomic ensembles in an analogous way to the description of<br />
the two-photon state generated via spdc in chapter 1. The section discusses the<br />
general characteristics of the nonlinear process, and introduces the two-photon<br />
mode function.<br />
Consider as a nonlinear medium an ensemble of n identical Λ−type cold<br />
atoms trapped in a magneto-optical trap (mot). The atoms have an energy<br />
level configuration with one excited state: |e〉 and two hyperfine ground states:<br />
|s〉, and |g〉. This is the case, for example, in the d2 hyperfine transition of<br />
87rb. In the initial state of the cloud all atoms are in the ground state |g〉, and<br />
after emission all atoms return to their initial state, as figure figure 5.1 shows.<br />
The two-photon generation results from the interaction of a single atom of<br />
the cloud with two counter-propagating classical beams in a four step process.<br />
In the first step, the atom gets excited by the interaction with the pump beam<br />
far detuned from the |g〉 → |e〉 transition. In the second step, the excited atom<br />
decays into the |s〉 state by emitting one Stokes photon in the direction zs as<br />
shown in figure 5.2. In the third step, the atom is re-excited by the interaction<br />
with the control beam far detuned from the |s〉 → |e〉 transition. In the last<br />
step, the atom decays to the ground state by emitting an anti-Stokes photon<br />
in the zas direction.<br />
If ω 0 i<br />
is the central angular frequency for the photons involved in the pro-<br />
cess (i = p, c, s, as), and k 0 i is the corresponding wave number at the central<br />
frequencies, energy and momentum conservation implies<br />
52<br />
and<br />
g<br />
e<br />
ω 0 p + ω 0 c = ω 0 s + ω 0 as, (5.1)<br />
k 0 p − k 0 c = k 0 s cos ϕs − k 0 as cos ϕas, (5.2)<br />
k 0 s sin ϕs = k 0 as sin ϕas. (5.3)<br />
s
x<br />
y<br />
z<br />
5.1. The quantum state of Stokes and anti-Stokes photon pairs<br />
zas yas<br />
xas<br />
pump<br />
anti Stokes<br />
Stokes<br />
ys<br />
control<br />
Figure 5.2: According to energy and momentum conservation, the generated photons<br />
counterpropagate. Equation 5.7 describes the relation between their propagation<br />
direction and the propagation direction of the pump and control beams, where due<br />
to the phase matching the angle of emission of the anti-Stokes photon is ϕas = π−ϕs.<br />
Because we consider a pump and a control beam with the same central frequency<br />
and k0 s k0 as, the phase matching conditions allow any angle of emission<br />
ϕas = π − ϕs, if it is not forbidden by the transition matrix elements [60].<br />
This assumption is valid only for cold atoms, since for warm atoms the process<br />
is highly directional, that is all photons are emitted along a preferred direction<br />
as proven in reference [61].<br />
There are two ways to describe the generated two-photon quantum state: it<br />
can be described by using two coupled equations in the slowly varying envelope<br />
approximation for the Stokes and anti-Stokes electric fields [65], or alternatively<br />
by using an effective Hamiltonian of interaction and first order perturbation<br />
theory [66, 67]. As the latter approach is analogous to the formalism used in<br />
chapter 1, it will be used in what follows to calculate the Stokes and anti-Stokes<br />
state.<br />
The effective Hamiltonian in the interaction picture HI describes the photonatom<br />
interaction, and is given by<br />
<br />
HI = ɛ0<br />
ϕas<br />
ϕ s<br />
xs<br />
zs<br />
zc<br />
yc<br />
xc<br />
dV χ (3) Ê − as Ê− s Ê+ c Ê+ p + h.c. (5.4)<br />
where χ (3) is the effective nonlinearity, independent of the beam intensity since<br />
the pump and the control are non-resonant [65]. Assuming a Gaussian distribution<br />
of atoms in the cloud the effective nonlinearity χ (3) can be written<br />
as<br />
χ (3) <br />
(x, y, z) ∝ exp − x2 + y2 R2 z2<br />
−<br />
L2 <br />
(5.5)<br />
where R is the size of the cloud of atoms in the transverse plane (x, y) and L<br />
is the size in the longitudinal direction.<br />
53
5. <strong>Spatial</strong> correlations in Raman transitions<br />
According to equation 1.6, the electric field operators Ên in equation 5.4,<br />
are given by<br />
Ê (+)<br />
<br />
n (rn, t) = dkn exp [ikn · rn − iωnt]â(kn) (5.6)<br />
where, as in equation 1.23, we are using a more convenient set of transverse<br />
wave vector coordinates given by<br />
ˆxs,as =ˆx<br />
ˆys,as =ˆy cos ϕs,as + ˆz sin ϕs,as<br />
ˆzs,as = − ˆy sin ϕs,as + ˆz cos ϕs,as.<br />
(5.7)<br />
Under these conditions, at first order perturbation theory, the spatial quantum<br />
state of the generated pair of photons is<br />
<br />
|Ψ〉 =<br />
(5.8)<br />
dqsdqasΦ (qs, qas) |qs〉s|qas〉s<br />
where the mode function Φ (qs, qas) of the two-photon state is<br />
<br />
<br />
Φ (qs, ωs, qas, ωas) = dqpdqcEp (qp) Ec (qc) exp − ∆20R 2<br />
4 − ∆21R 2<br />
4 − ∆22L 2 <br />
4<br />
(5.9)<br />
with the delta factors defined as<br />
∆0 = q x s + q x as<br />
∆1 = (ks − kas) sin ϕs + (q y s − q y as) cos ϕs<br />
∆2 = kp − kc − (ks + kas) cos ϕ + (q y s − q y as) sin ϕs<br />
and the longitudinal wave vector of the pump beam given by<br />
ωpnp kp =<br />
c<br />
2<br />
− ∆ 2 0 − ∆ 2 1<br />
1/2<br />
(5.10)<br />
. (5.11)<br />
In the case of the Stokes and anti-Stokes two-photon state, there are no correlations<br />
between space and frequency due to the narrow bandwidth (∼ GHz)<br />
of the generated photons [60]. Therefore, to analyze the spatial shape of the<br />
mode function Φq (qs, qas), we can consider ωs = ω 0 s and ωas = ω 0 as.<br />
The effect of the unavoidable spatial filtering produced by the specific op-<br />
tical detection system used is described by Gaussian filters. The angular acceptance<br />
of the single photon detection system is 1/ k0 <br />
sws,as . In most experimental<br />
configurations, ws ≈ 50-150 µm and the length of the cloud is a few<br />
millimeters or less. For simplicity, we will assume that ws = was.<br />
In the calculations, the pump and the control are Gaussian beams with<br />
the same waist wp at the center of the cloud. As the waist is typically about<br />
200-500 µm, the Rayleigh range of the pump, Stokes and anti-Stokes modes<br />
Lp = πw2 p/λp and Ls,as = πw2 s/λs,as satisfy L ≪ Lp, Ls,as. This condition<br />
allows us to neglect the transverse wavenumber dependence of all longitudinal<br />
wave vectors in equations 5.9 and 5.10.<br />
54
where<br />
5.2. Orbital angular momentum correlations<br />
Under these conditions, the mode function Φq (qs, qas) can be written as<br />
Φq (qs, qas) = (ABCD)1/4<br />
<br />
π<br />
× exp − A<br />
4 (qx s + q x as) 2 − B<br />
4 (qx s − q x as) 2<br />
<br />
<br />
× exp − C<br />
4 (qy s + q y as) 2 − D<br />
4 (qy s − q y as) 2<br />
<br />
A = w2 pR2 2R2 + w2 +<br />
p<br />
w2 s<br />
2<br />
B = w2 s<br />
2<br />
C = w2 s<br />
2<br />
D = w2 pR 2 cos 2 ϕs<br />
2R 2 + w 2 p<br />
(5.12)<br />
+ L 2 sin 2 ϕs + w2 s<br />
. (5.13)<br />
2<br />
The specific characteristics of the state are determined by the size of the atomic<br />
cloud in the longitudinal and transverse planes, L and R; by the beam waist of<br />
the pump and control beams, wp,c; by the waist of the Stokes and anti-Stokes<br />
modes ws; and by the angle of emission ϕs. The next section describes the<br />
effect off all these parameters on the oam transfer from the pump and the<br />
control beams to the pair Stokes and anti-Stokes.<br />
5.2 Orbital angular momentum correlations<br />
Since the pump and control beams are Gaussian beams, lp = lc = 0, the<br />
analysis of the oam transfer reduces to the study of the oam content of the<br />
Stokes and anti-Stokes pair. The Stokes oam content becomes the only free<br />
variable, after projecting the anti-Stokes into a Gaussian mode<br />
u(qas) = Nas exp<br />
<br />
− w2 g<br />
4 (qx2 as + q y2<br />
as)<br />
<br />
(5.14)<br />
with beam width wg at the center of the cloud. The Stokes mode function<br />
defined as<br />
<br />
Φs (qs) = dqasΦ (qs, qas) u(qas) (5.15)<br />
becomes<br />
<br />
(F G)(1/4)<br />
Φs (qs) = exp −<br />
1/2<br />
(2π) F<br />
4 (qx s ) 2 − G<br />
4 (qy s ) 2<br />
<br />
, (5.16)<br />
55
5. <strong>Spatial</strong> correlations in Raman transitions<br />
with<br />
F = 4AB + (A + B) w2 g<br />
A + B + w 2 g<br />
G = 4CD + (C + D) w2 g<br />
C + D + w2 . (5.17)<br />
g<br />
As in the previous chapter, this mode function can be written as a superposition<br />
of spherical harmonics<br />
Φs (qs, θs) = (2π)<br />
−1/2 <br />
ls<br />
als (qs) exp (ilsθs) . (5.18)<br />
The probability of having a Stokes photon with oam equal to ls is given by the<br />
weight of each spiral mode<br />
<br />
F + G<br />
C2ls = (F G)1/2 dqsqs exp − q<br />
4<br />
2 <br />
s I 2 <br />
G − F<br />
ls q<br />
8<br />
2 <br />
s (5.19)<br />
where Ils are the Bessel functions of the second kind. Only even modes appear<br />
in the distribution as a consequence of the symmetry of the Stokes mode<br />
function.<br />
Figure 5.3 shows the weight of the mode ls = 0 as a function of the angle<br />
of emission for different values of the length of the cloud of atoms. For nearly<br />
collinear configurations, the probability of having a Gaussian Stokes photon is<br />
one, therefore, in this case the process satisfies the relation lp+lc = ls +las. For<br />
these kind of configurations, A = D in equation 5.12, and therefore the Stokes<br />
spatial mode function shows cylindrical symmetry in the transverse planes<br />
(xs, ys) and (xas, yas). This is the case in most experimental configurations<br />
[59, 64, 68], and in fact, reference [23] experimentally proves the relationship<br />
lp + lc = ls + las for collinear configurations.<br />
The probability of having a Gaussian Stokes photon decreases as other<br />
modes appear in the distribution in highly noncollinear configurations. The<br />
length of the cloud controls the importance of the other modes and it is possible<br />
to obtain a spatial mode with cylindrical symmetry for any emission angle, by<br />
fulfilling the condition<br />
<br />
L = R 1 + 2R2<br />
w2 −1/2<br />
. (5.20)<br />
p<br />
This condition reduces to L = R, when the beam waist is much larger than the<br />
transverse size of the cloud. Any deviation from a spherical volume of interaction<br />
(described by this condition) introduces ellipticity in the mode function.<br />
For this reason, a highly elliptical configuration, as the one described in reference<br />
[60], will not satisfy the oam selection rule.<br />
Figure 5.4 shows the weight of the mode ls = 0 as a function of the length<br />
of the cloud for different values of the emission angle. For any angle, the<br />
probability of having a Gaussian mode is maximum at the length given by<br />
equation 5.20. In a collinear configuration, the Stokes photon always has a<br />
Gaussian distribution, independently of the length of the cloud. As the angle<br />
of emission increases, the length of the cloud becomes more important. The<br />
mode weight is weakly affected by the change of the cloud length when the<br />
length is much longer than the other relevant parameters: ws, wg and R. This<br />
is especially evident for an angle of emission ϕ = 90◦ .<br />
56
weight<br />
1<br />
0.6<br />
-180º<br />
-90º<br />
0º 90º 180º<br />
5.3. <strong>Spatial</strong> entanglement<br />
length<br />
0.2mm<br />
0.4mm<br />
1mm<br />
2mm<br />
angle<br />
Figure 5.3: The weight of the Gaussian mode in the oam decomposition for the Stokes<br />
photon has a maximum in the collinear configuration. In noncollinear configurations<br />
the probability of a Gaussian Stokes photon decreases as the length of the cloud<br />
increases. Table 5.1 lists the parameters used to generate this figure.<br />
weight angle<br />
1<br />
0.6<br />
0.2 Length 1.5 mm<br />
Figure 5.4: The probability of having a Gaussian Stokes photon is maximum in a<br />
collinear configuration independently of the cloud length. In the noncollinear cases<br />
the maximum appears at the length given by equation 5.20, in these configurations<br />
the probability of having a Gaussian Stokes photon changes drastically with the cloud<br />
length and the emission angle. Table 5.1 lists the parameters used to generate this<br />
figure.<br />
5.3 <strong>Spatial</strong> entanglement<br />
The spatial correlations between the generated photons inherit the angular<br />
dependence from the Stokes oam content. To explore this phenomena, this<br />
0º<br />
10º<br />
90º<br />
57
5. <strong>Spatial</strong> correlations in Raman transitions<br />
Schmidt number<br />
40<br />
20<br />
1<br />
-180º<br />
0º 180º<br />
lenght spatial filter<br />
2 mm<br />
1 mm<br />
400 m<br />
200 m<br />
7<br />
4<br />
1<br />
-180º<br />
0º 180º<br />
100 m<br />
200 m<br />
500 m<br />
angle<br />
Figure 5.5: The length of the cloud controls the Schmidt number angular dependence.<br />
By modifying this length it is possible to change the position of the maximum and<br />
minimum values of the Schmidt number. The angular dependence can be removed<br />
completely by tailoring the parameters of the process, or by filtering. Table 5.1 lists<br />
the parameters used to generate this figure.<br />
Table 5.1: The parameters used in figures 5.3, 5.4 and 5.5<br />
Parameter Figure 5.3 Figure 5.4 Figure 5.5 (left) Figure 5.5 (right)<br />
L 0.2, 0.4, 1, 2 mm [0.2, 1.5] mm 0.2, 0.4, 1, 2 mm 200 µm<br />
wp 100 µm 100 µm 500 µm 500 µm<br />
R 400µm 400µm 1000 µm 1000 µm<br />
ws 100 µm 100 µm 100 µm 100, 200, 500 µm<br />
wg 500 µm 500 µm<br />
ϕs [−180, 180] ◦ 0, 10, 90 ◦ [−180, 180] ◦ [−180, 180] ◦<br />
section shows the effects of changing the emission angle on the Schmidt number<br />
K defined in section 2.1, is a common entanglement quantifier.<br />
The Schmidt number of the two-photon state described by equation 5.12<br />
quantifies the entanglement between the generated photons. According to references<br />
[69, 70] it is given by<br />
(A + B) (C + D)<br />
K =<br />
4 (ABCD) 1/2<br />
. (5.21)<br />
Figure 5.5 shows the Schmidt number as a function of the emission angle for<br />
different values of the atomic cloud length and the pump beam waist. The<br />
function extrema are located at 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . At each of these<br />
values the function can have a maximum or a minimum depending on the<br />
other parameters of process. For instance, if<br />
L < R<br />
<br />
1 + 2R2<br />
w2 −1/2<br />
p<br />
(5.22)<br />
the entanglement is maximum at collinear configurations where ϕ = 0 ◦ , 180 ◦ ,<br />
and minimum at transverse configurations ϕ = 90 ◦ , 270 ◦ . In the left part of<br />
figure 5.5, the condition in equation 5.22 is achieved at cloud lengths smaller<br />
than 333 µm.<br />
As the length of the cloud increases, the variation of the entanglement with<br />
the angle is smoothed out. In the left part of figure 5.5 the relation in equation<br />
58
5.3. <strong>Spatial</strong> entanglement<br />
5.20 is satisfied when L = 333 µm; therefore, the amount of entanglement is<br />
constant. If the length of the cloud increases even more, the position for the<br />
maxima and the minima get inverted. When<br />
<br />
L > R 1 + 2R2<br />
w2 −1/2<br />
p<br />
(5.23)<br />
the entanglement is maximum for transverse emitting configurations ϕ = 90 ◦ , 270 ◦ ,<br />
and minimum for collinear configurations ϕ = 0 ◦ , 180 ◦ . This variation is possible<br />
in Raman transitions where the transversal size of the cloud is comparable<br />
to the longitudinal size.<br />
Another parameter that plays a role in the variation of the amount of<br />
entanglement is the Stokes spatial filter ws. The right side of figure 5.5 shows<br />
the effect of changing the filtering over the amount of entanglement. Very<br />
narrow spatial filters (ws → ∞) diminish both the amount of entanglement<br />
and its azimuthal variability.<br />
Conclusion<br />
As in spdc, the geometrical configuration of the Raman transitions determines<br />
the oam content and the spatial correlations of the generated Stokes and anti-<br />
Stokes photons. The size and shape of the cloud defines the emission angles<br />
for which the correlations are maximum and minimum.<br />
59
CHAPTER 6<br />
Summary<br />
This thesis characterizes the correlations in two-photon quantum states, both<br />
between photons, and between the spatial and frequency degrees of freedom.<br />
The photon pairs are generated by spontaneous parametric down-conversion<br />
spdc, or by the excitation of Raman transitions in cold atomic ensembles.<br />
Special attention is given to the entanglement of the photon pair in the spatial<br />
degree of freedom, associated to the orbital angular momentum oam. The<br />
main contributions or the thesis are:<br />
A novel matrix formalism to describe the two-photon mode function.<br />
This formalism makes it possible to analytically calculate several features<br />
of the down-converted photons, and reduces the numerical calculation time of<br />
other features, as shown in chapter 1 and reference [41].<br />
Analytical expressions for the purity of the subsystems formed by<br />
a single photon in space and frequency, or by a spatial two-photon<br />
state. These expressions quantify the correlations in the two-photon state,<br />
and make the effect of the various parameters of the spdc process on these<br />
correlations explicit. The thesis presents a set of conditions to suppress or<br />
enhance the correlations between the photons, or between degrees of freedom<br />
in the two-photon state. With these conditions, it is possible to design spdc<br />
sources with specific levels of correlations, for example, sources that generate<br />
pure heralded single photons, or alternatively, maximally entangled states; as<br />
shown in chapter 2 and in references [41, 45].<br />
A selection rule for oam transfer. This selection rule explains several<br />
contradictory measurements of the oam transfer from the pump beam to the<br />
signal and idler photons in spdc. We showed that the oam content of the<br />
pump is completely transferred to the photons emitted in all directions. This<br />
selection rule always holds if all possible emission directions are considered,<br />
for example in collinear configurations, but not necessarily if only a part of<br />
all emission directions is detected. These conditions for the validity of the<br />
selection rule give clear guidelines for the design of sources for protocols that<br />
exploit the multidimensionality of oam, as shown in chapter 3 and references<br />
[71, 72].<br />
61
6. Summary<br />
Experiments that explain oam transfer in noncollinear spdc. In<br />
noncollinear configurations only a subset of emission directions is detected. The<br />
transfer of oam to the detected photons is strongly affected by the change in<br />
geometry imposed by the detection system, and depends on the angle of detection,<br />
the pump-beam waist and the Poynting-vector walk-off. The thesis shows<br />
that, by tailoring these parameters, it is possible to design noncollinear sources<br />
that naturally generate spatially-separated photons with specific desired spatial<br />
shapes, as shown in chapter 4 and references [39, 33, 32].<br />
An analysis of the spatial correlations generated by Raman transitions<br />
in cold atomic ensembles. Like for spdc, the spatial correlations<br />
between the Stokes and anti-Stokes photons depend on the geometrical configuration<br />
of the pump beam, the control beam, and the Stokes and anti-Stokes<br />
photons. This thesis shows how the size and shape of the atom cloud define<br />
the emission angles for which the correlations are maximum and minimum, as<br />
shown in chapter 5 and reference [73].<br />
62
APPENDIX A<br />
The matrix form<br />
of the mode function<br />
According to section 1.3, the normalized two-photon mode function, after some<br />
approximations, reads<br />
<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p<br />
4 ∆2 0 − w2 p<br />
4 ∆21 <br />
× exp − (γL)2<br />
4 ∆2k − T 2 0<br />
4 (Ωs + Ωi) 2<br />
<br />
<br />
× exp − w2 s<br />
2 |qs| 2 − w2 i<br />
2 |qi| 2 − 1<br />
2B2 Ω<br />
s<br />
2 s − 1<br />
2B2 Ω<br />
i<br />
2 <br />
i . (A.1)<br />
The argument of the exponential function is a second order polynomial. Each<br />
term is the product of at most two variables (qx s , qy s , qx i , qy i , Ωs, Ωi) and a coefficient<br />
f = a<br />
4 qx2 s + b<br />
4 qy2 s + c<br />
4 qx2 i + d<br />
4 qy2<br />
h<br />
i + . . . +<br />
2 qx s q y s + . . . + z<br />
2 ΩsΩi. (A.2)<br />
Such a polynomial can be written as the product of a matrix A with the<br />
coefficients as elements, and a vector x of the variables<br />
f = 1 <br />
x qs 4<br />
qy s qx i q y<br />
i Ωs Ωs<br />
⎛<br />
⎜<br />
⎜<br />
⎝<br />
a<br />
h<br />
i<br />
j<br />
k<br />
h<br />
b<br />
m<br />
n<br />
p<br />
i<br />
m<br />
c<br />
s<br />
t<br />
j<br />
n<br />
s<br />
d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
f<br />
l<br />
r<br />
u<br />
w<br />
z<br />
⎞ ⎛<br />
q<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎠ ⎝<br />
l r u w z g<br />
x s<br />
qy s<br />
qx i<br />
q y<br />
i<br />
Ωs<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
Ωi<br />
(A.3)<br />
therefore, the mode function can be written using matrix notation as<br />
<br />
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1<br />
2 xt <br />
Ax . (A.4)<br />
63
A. The matrix form of the mode function<br />
Each of the terms of the matrix is defined by comparing the product of the<br />
polynomial f with the argument of the exponential in equation 1.34. The<br />
elements of matrix A are<br />
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0 cos α 2<br />
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2 − 2γ 2 L 2 sin ϕs cos ϕs sin α tan ρ0<br />
+ γ 2 L 2 cos ϕs 2 tan ρ0 2 sin α 2<br />
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0 cos α 2<br />
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2 + 2γ 2 L 2 sin ϕi cos ϕi tan ρ0 sin α<br />
+ γ 2 L 2 cos ϕi 2 tan ρ0 2 sin α 2<br />
f =2B −2<br />
s + T 2 0 + γ 2 L 2 N 2 p + γ 2 L 2 N 2 s cos ϕs 2 + w 2 pN 2 s sin ϕs 2 − 2γ 2 L 2 NpNs cos ϕs<br />
− 2γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + 2γ 2 L 2 N 2 s cos ϕs sin ϕs sin α tan ρ0<br />
+ γ 2 L 2 N 2 s sin ϕs 2 tan ρ0 2 sin α 2<br />
g =2B −2<br />
i<br />
+ T 2 0 + γ 2 L 2 N 2 p + γ 2 L 2 N 2 i cos ϕi 2 + w 2 pN 2 i sin ϕi 2 − 2γ 2 L 2 NpNi cos ϕi<br />
+ 2γ 2 L 2 NpNi sin ϕi tan ρ0 sin α − 2γ 2 L 2 N 2 i cos ϕi sin ϕi tan ρ0 sin α<br />
+ γ 2 L 2 N 2 i sin ϕi 2 tan ρ0 2 sin α 2<br />
h = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α 2<br />
i =w 2 p + γ 2 L 2 tan ρ0 2 cos α<br />
j =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α<br />
k =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ns cos ϕs tan ρ0 cos α<br />
− γ 2 L 2 Ns sin ϕs tan ρ0 2 sin α cos α<br />
l =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ni cos ϕi tan ρ0 cos α<br />
+ γ 2 L 2 Ni sin ϕi tan ρ0 2 sin α cos α<br />
m = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α<br />
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi + γ 2 L 2 cos ϕs sin ϕi sin α tan ρ0<br />
− γ 2 L 2 sin ϕs cos ϕi sin α tan ρ0 + γ 2 L 2 cos ϕs cos ϕi tan ρ0 2 sin α 2<br />
p = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ns cos ϕs sin ϕs − w 2 pNs cos ϕs sin ϕs<br />
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ns cos ϕs 2 tan ρ0 sin α<br />
+ γ 2 L 2 Ns sin ϕs 2 tan ρ0 sin α − γ 2 L 2 Ns sin ϕs cos ϕs tan ρ0 2 sin α 2<br />
r = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ni sin ϕs cos ϕi + w 2 pNi cos ϕs sin ϕi<br />
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ni cos ϕs cos ϕi tan ρ0 sin α<br />
− γ 2 L 2 Ni sin ϕs sin ϕi tan ρ0 sin α + γ 2 L 2 Ni cos ϕs sin ϕi tan ρ0 2 sin α 2<br />
s =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α<br />
t =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ns cos ϕs tan ρ0 cos α − γ 2 L 2 Ns sin ϕs tan ρ0 2 sin α cos α<br />
u =γ 2 L 2 Np tan ρ0 cos α − γ 2 L 2 Ni cos ϕi tan ρ0 cos α + γ 2 L 2 Ni sin ϕi tan ρ0 2 sin α cos α<br />
64
v =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ns cos ϕs sin ϕi − w 2 pNs sin ϕs cos ϕi<br />
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕs cos ϕi tan ρ0 sin α<br />
− γ 2 L 2 Ns sin ϕs sin ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕi sin ϕs tan ρ0 2 sin α 2<br />
w =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ni sin ϕi cos ϕi + w 2 pNi sin ϕi cos ϕi<br />
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ni cos ϕi 2 tan ρ0 sin α<br />
+ γ 2 L 2 Ni sin ϕi 2 tan ρ0 sin α + γ 2 L 2 Ni sin ϕi cos ϕi tan ρ0 2 sin α 2<br />
z =γ 2 L 2 N 2 p − γ 2 L 2 NpNi cos ϕi − γ 2 L 2 NpNs cos ϕs + γ 2 L 2 NsNi cos ϕs cos ϕi<br />
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi − γ 2 L 2 NsNi cos ϕs sin ϕi tan ρ0 sin α<br />
− γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + γ 2 L 2 NsNi cos ϕi sin ϕs tan ρ0 sin α<br />
− γ 2 L 2 NsNi sin ϕs sin ϕi tan ρ0 2 sin α 2 + γ 2 L 2 NpNi sin ϕi tan ρ0 sin α.<br />
(A.5)<br />
This set of expressions is far more useful than compact. The matrix terms<br />
become simpler in some particular spdc configurations that are treated in<br />
chapters 2 and 4. For instance, when the pump polarization is parallel to the<br />
x axis, the matrix terms become<br />
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0<br />
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2<br />
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0<br />
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2<br />
f = 2<br />
B2 s<br />
g = 2<br />
B2 i<br />
+ T 2 0 + γ 2 L 2 (Np − Ns cos ϕs) 2 + w 2 pN 2 s sin ϕs 2<br />
+ T 2 0 + γ 2 L 2 (Np − Ni cos ϕi) 2 + w 2 pN 2 i sin ϕi 2<br />
h =m = −γ 2 L 2 sin ϕs tan ρ0<br />
i =w 2 p + γ 2 L 2 tan ρ0 2<br />
j =s = γ 2 L 2 sin ϕi tan ρ0<br />
k =t = γ 2 L 2 tan ρ0(Np − Ns cos ϕs)<br />
l =u = γ 2 L 2 tan ρ0(Np − Ni cos ϕi)<br />
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi<br />
p = − γ 2 L 2 sin ϕs(Np − Ns cos ϕs) − w 2 pNs cos ϕs sin ϕs<br />
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi<br />
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs<br />
w =γ 2 L 2 sin ϕi(Np − Ni cos ϕi) + w 2 pNi cos ϕi sin ϕi<br />
z =γ 2 L 2 N 2 p − γ 2 L 2 Np(Ni cos ϕi + Ns cos ϕs) + γ 2 L 2 NsNi cos ϕs cos ϕi<br />
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi.<br />
(A.6)<br />
The matrix notation extends to functions of the mode function. For instance,<br />
the purity of the spatial part of the two-photon state, given by equation 2.11,<br />
65
A. The matrix form of the mode function<br />
writes<br />
T r[ρ 2 <br />
q] =<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i). (A.7)<br />
where the dimension increases as new primed variables appear. Using the<br />
matrix notation the integrand becomes<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i)<br />
= N 4 <br />
exp − 1<br />
2 Xt <br />
BX , (A.8)<br />
where the vector X is the result of concatenation of x and x ′ , such that<br />
⎛<br />
⎜<br />
X = ⎜<br />
⎝<br />
↑<br />
x<br />
↓<br />
↑<br />
x ′<br />
↓<br />
⎞<br />
⎟ ,<br />
⎟<br />
⎠<br />
(A.9)<br />
and the new matrix B is given by<br />
B = 1<br />
⎛<br />
⎜<br />
2 ⎜<br />
⎝<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
l<br />
0<br />
0<br />
0<br />
0<br />
k<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
r<br />
0<br />
0<br />
0<br />
0<br />
p<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
u<br />
0<br />
0<br />
0<br />
0<br />
t<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
w<br />
0<br />
0<br />
0<br />
0<br />
v<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
2z<br />
k<br />
p<br />
t<br />
v<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
2g<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
0<br />
0<br />
0<br />
k<br />
l<br />
2a<br />
2h<br />
2i<br />
2j<br />
k<br />
0<br />
0<br />
0<br />
0<br />
p<br />
r<br />
2h<br />
2b<br />
2m<br />
2n<br />
p<br />
0<br />
0<br />
0<br />
0<br />
t<br />
u<br />
2i<br />
2m<br />
2c<br />
2s<br />
t<br />
0<br />
0<br />
0<br />
0<br />
v<br />
w<br />
2j<br />
2n<br />
2s<br />
2d<br />
v<br />
k<br />
p<br />
t<br />
v<br />
0<br />
0<br />
k<br />
p<br />
t<br />
v<br />
2f<br />
l<br />
r<br />
u<br />
w<br />
0<br />
0<br />
l<br />
r<br />
u<br />
w<br />
2z<br />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
l r u w 0 0 l r u w 2z 2g<br />
(A.10)<br />
In an analogous way, the integrand on the expression for the signal photon<br />
purity<br />
<br />
T r[ρ 2 signal] =<br />
is written in a matrix notation as<br />
66<br />
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i<br />
× Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)<br />
× Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i). (A.11)<br />
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i)<br />
= N 4 exp<br />
<br />
− 1<br />
2 Xt <br />
CX . (A.12)
The matrix C is given by<br />
C = 1<br />
⎛<br />
2a<br />
⎜ 2h<br />
⎜ i<br />
⎜ j<br />
⎜ 2k<br />
⎜ l<br />
2 ⎜ 0<br />
⎜ 0<br />
⎜ i<br />
⎜ j<br />
⎝ 0<br />
2h<br />
2b<br />
m<br />
n<br />
2p<br />
r<br />
0<br />
0<br />
m<br />
n<br />
0<br />
i<br />
m<br />
2c<br />
2s<br />
t<br />
2u<br />
i<br />
m<br />
0<br />
0<br />
t<br />
j<br />
n<br />
2s<br />
2d<br />
v<br />
2w<br />
j<br />
n<br />
0<br />
0<br />
v<br />
2k<br />
2p<br />
t<br />
v<br />
2f<br />
z<br />
0<br />
0<br />
t<br />
v<br />
0<br />
l<br />
r<br />
2u<br />
2w<br />
z<br />
2g<br />
l<br />
r<br />
0<br />
0<br />
z<br />
0<br />
0<br />
i<br />
j<br />
0<br />
l<br />
2a<br />
2h<br />
i<br />
j<br />
2k<br />
0<br />
0<br />
m<br />
n<br />
0<br />
r<br />
2h<br />
2b<br />
m<br />
n<br />
2p<br />
i<br />
m<br />
0<br />
0<br />
t<br />
0<br />
i<br />
m<br />
2c<br />
2s<br />
t<br />
j<br />
n<br />
0<br />
0<br />
v<br />
0<br />
j<br />
n<br />
2s<br />
2d<br />
v<br />
0<br />
0<br />
t<br />
v<br />
0<br />
z<br />
2k<br />
2p<br />
t<br />
v<br />
2f<br />
l<br />
r<br />
0<br />
0<br />
z<br />
0<br />
l<br />
r<br />
2u<br />
2w<br />
z<br />
l r 0 0 z 0 l r 2u 2w z 2g<br />
⎞<br />
⎟ .<br />
⎟<br />
⎠<br />
(A.13)<br />
67
APPENDIX B<br />
Integrals of<br />
the matrix mode function<br />
Given a n×n symmetric and positive definite matrix A, and two n order vectors<br />
x and b<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = (2π)n/2<br />
<br />
exp −<br />
det(A) btA−1 <br />
b<br />
.<br />
2<br />
(B.1)<br />
To proof this result consider that, since A is positive definite, there exist another<br />
matrix O such that OAO t = D where D is a diagonal matrix. With the<br />
transformation y = Ox, the integral in equation B.1 becomes<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = dy exp<br />
<br />
− yt Dy<br />
2 + iyt b ′<br />
<br />
,<br />
(B.2)<br />
with b ′ = O −1 b.<br />
As the argument of the exponential, in the right side of equation B.1, is<br />
given by the matrix product<br />
− 1 <br />
y1<br />
2<br />
y2 . . . yn<br />
⎛<br />
d1<br />
⎜ 0<br />
⎜<br />
⎝ .<br />
0<br />
d2<br />
.<br />
. . .<br />
. . .<br />
. ..<br />
0<br />
0<br />
.<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎠ ⎝<br />
0 0 0 dn<br />
y1<br />
y2<br />
.<br />
yn<br />
⎞<br />
⎛<br />
⎟<br />
⎠ + i ⎜<br />
y1 y2 . . . yn ⎜<br />
⎝<br />
the integral can be written in a polynomial form as<br />
<br />
dx exp − xtAx 2 + ixt <br />
b<br />
<br />
<br />
= dy1dy2...dyn exp − y2 1d1<br />
2 + iy1b ′ 1 − y2 2d2<br />
2 + iy2b ′ 2... − y2 ndn<br />
2 + iynb ′ <br />
n<br />
(B.3)<br />
(B.4)<br />
69<br />
b ′ 1<br />
b ′ 2<br />
.<br />
b ′ n<br />
⎞<br />
⎟<br />
⎠ ,
B. Integrals of the matrix mode function<br />
where the integrals in each variable are independent, so it is possible to write<br />
them as<br />
<br />
dx exp − xtAx <br />
=<br />
2 + ixt <br />
b<br />
<br />
dy1 exp − y2 1d1<br />
2 + iy1b ′ 1<br />
<br />
dy2<br />
<br />
− y2 2d2<br />
2 + iy2b ′ <br />
2 ...<br />
which, after solving each integral separately, becomes<br />
<br />
<br />
dx exp − xtAx 2 + ixt <br />
b =<br />
n<br />
j=1<br />
<br />
2π<br />
dj<br />
exp<br />
<br />
dyn<br />
− b′ 2<br />
j<br />
2dj<br />
<br />
− y2 ndn<br />
2 + iynb ′ <br />
n ,<br />
(B.5)<br />
<br />
. (B.6)<br />
Because D is a diagonal matrix, 1/dj are the elements of D−1 , and n j=1 dj =<br />
det(D); thus it is possible to write the last expression as<br />
<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = (2π)n/2<br />
exp<br />
det(D)<br />
<br />
− b′ t D −1 b ′<br />
2<br />
<br />
. (B.7)<br />
Finally, since det(A) = det(D) and b ′ t D −1 b ′ = b t A −1 b, the integral becomes<br />
<br />
<br />
dx exp − xtAx 2 + ixt <br />
b = (2π)n/2<br />
<br />
exp −<br />
det(A) btA−1 <br />
b<br />
. (B.8)<br />
2<br />
This proof can be applied to the special case in which all the elements of the<br />
vector b are equal to 0. In that case the integral reduces to<br />
<br />
dx exp − xt <br />
Ax<br />
=<br />
2<br />
(2π)n/2<br />
. (B.9)<br />
det(A)<br />
70
APPENDIX C<br />
Methods for OAM<br />
measurements<br />
A real world oam application will require a compact tool able to separate the<br />
different modes at the single photon level. A tool analogous to the beam splitter<br />
for polarization, or to a diffraction grating for frequency. Such a tool probably<br />
will be based on the two current methods to determine the oam of photons:<br />
holographic and interferometric methods.<br />
In a very simplistic way, an hologram is a record of the interference pattern<br />
of two beams. When the hologram is illuminated with one of the beams,<br />
the other beam can be reconstructed. Based on this principle, a computer<br />
generated hologram of the interference of a lg mode with a Gaussian beam,<br />
in conjunction with a single mode fiber can be used to detect that particular<br />
lg mode [7, 51]. Even though this method works at single photon level, it<br />
is restricted to a two-value response, reducing the effective dimensions of the<br />
oam to two, as shown in figure C.1.<br />
Reference [15] introduced an analyzer able to sort photons into even and<br />
odd oam modes, using a Mach-Zender interferometer. With a phase shift in<br />
one of the interferometer arms, the constructive and destructive interference<br />
were so that odd modes go to one of the outputs, and even modes to the other,<br />
as figure C.2 shows. By nesting several interferometers, and using holograms,<br />
the authors proved that was possible to separate up to 2 n modes with (2 n − 1)<br />
interferometers. This method works at single photon level, and can distinguish<br />
many different oam modes, but it relies on the simultaneous stabilization of<br />
different interferometers, which is challenging experimentally.<br />
71
C. Methods for OAM measurements<br />
Input hologram<br />
lens<br />
Gaussian<br />
mode<br />
non-Gaussian<br />
mode<br />
Figure C.1: A hologram that records the interference of a lg mode with a Gaussian<br />
beam, can be used to detect that particular lg mode. Only if the generating lg<br />
mode is incident on the hologram, a Gaussian beam is recovered. If another mode is<br />
incident, the hologram will generate non-Gaussian beams. Single mode optical fibers<br />
distinguish between Gaussian and non-Gaussian beams.<br />
Input<br />
beam splitter<br />
Dove prism mirror<br />
even modes<br />
odd modes<br />
Figure C.2: <strong>Two</strong> Dove prisms rotated with respect to each other by an angle π/2<br />
induce a relative rotation of π between the two arms of a Mach-Zender interferometer.<br />
As a consequence of the rotation, odd and even lg modes leave the interferometer<br />
at different outputs after the second beam splitter. Reference [15] introduces this<br />
principle as a base of an oam sorter.<br />
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