Spatial Characterization Of Two-Photon States - GAP-Optique

DOCTORAL THESIS IN PHOTONICS
ICFO BARCELONA 2010
Spatial
Characterization Of
Two-Photon States
CLARA INÉS OSORIO TAMAYO
Advisor: Juan P. Torres

Spatial Characterization
Of Two-Photon States
By
Clara Inés Osorio Tamayo
ICFO - Institut de Ciències Fotòniques
Universitat Politècnica de Catalunya
Barcelona, September 2009

Spatial Characterization
Of Two-Photon States
Clara Inés Osorio Tamayo
under the supervision of
Dr. Juan P. Torres
submitted this thesis in partial fulfillment
of the requirements for the degree of
Doctor
by the
Universitat Politècnica de Catalunya
Barcelona, September 2009

A Luz Stella y Luis Alfonso, mis papás.

Contents
Contents vii
Acknowledgements ix
Abstract xi
Introduction xv
List of Publications xvii
1 General description of two-photon states 1
1.1 Spontaneous parametric down-conversion . . . . . . . . . . . . 2
1.2 Two-photon state . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Approximations and other considerations . . . . . . . . . . . . 5
1.4 The mode function in matrix form . . . . . . . . . . . . . . . . 12
2 Correlations and entanglement 15
2.1 The purity as a correlation indicator . . . . . . . . . . . . . . . 16
2.2 Correlations between space and frequency . . . . . . . . . . . . 18
2.3 Correlations between signal and idler . . . . . . . . . . . . . . . 23
3 Spatial correlations and OAM transfer 29
3.1 Laguerre-Gaussian modes and OAM content . . . . . . . . . . . 30
3.2 OAM transfer in general SPDC configurations . . . . . . . . . . 31
3.3 OAM transfer in collinear configurations . . . . . . . . . . . . . 34
4 OAM transfer in noncollinear configurations 37
4.1 Ellipticity in noncollinear configurations . . . . . . . . . . . . . 38
4.2 Effect of the pump beam waist on the OAM transfer . . . . . . 39
4.3 Effect of the Poynting vector walk-off on the OAM transfer . . 44
5 Spatial correlations in Raman transitions 51
5.1 The quantum state of Stokes and anti-Stokes photon pairs . . . 52
5.2 Orbital angular momentum correlations . . . . . . . . . . . . . 55
5.3 Spatial entanglement . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Summary 61
A The matrix form of the mode function 63
vii

Contents
B Integrals of the matrix mode function 69
C Methods for OAM measurements 71
Bibliography 73
viii

Acknowledgements
This thesis compiles the result of five years of work at icfo. But it is, in some
sense, the result of a longer process that started many years ago in Medellín.
I would like to show my gratitude to all the people that helped me during all
this time even though I cannot mention everybody explicitly.
I would like to thank my advisor Juan P. Torres for inviting me to his
group, and for being very patient with my impatience. I am grateful to Alejandra
Valencia and Xiaojuan Shi for their generous and unconditional support.
I wish to thank all the people that provide a stimulating and fun environment
in icfo, especially Maurizio Righini, Laura Grau, Carsten Schuck, Masood
Ghotbi, Xavier Vidal, Sibylle Braungardt, Jorge Luis Domínguez-Juárez,
Petru Ghenuche, Rafael Betancurt, Philipp Hauke, Alessandro Ferraro, Agata
Checinska, and Martin Hendrych. I would like to thank Niek van Hulst and
his group, for adopting me at lunch time and for all party matters; and Morgan
Mitchell and his group, especially Marco Koschorreck, Mario Napolitano,
Florian Wolfgramm, Alessandro Cere and Brice Dubost. Logistical support by
Nuria Segù, Maria del Mar Gil, Anne Gstöttner, Jose Carlos Cifuentes, and
the electronic workshop have also been most helpful. I am extremely grateful
to Miguel Navascués that has been like a little brother to me since I came to
Barcelona, and to Artur García that welcomed me in his office every time that
I needed to talk (and that was very often).
I want to thank my “Barcelonian” friends for their caring, their emotional
and gastronomical support, Mauricio Álvarez, Carolina Mora, Friman Sánchez,
Tania de la Paz, Maria Teresa Vasco and Jose Uribe. Regardless the distance,
many of my friends made available their support in a number of ways, I am
grateful to Carlos Molina, Jose Palacios, Catalina López, Elizabeth Agudelo,
Esteban Silva, Sebastían Patrón, Jaime Hincapie, Wolfgang Niedenzu, Andrew
Hilliard, Juan C. Muñoz (y familia), Jaime Forero and Javier Moreno. It is a
pleasure to thank Boudewijn Taminiau, Ied van Oorschot, and all the Taminiau
family, since their visits and hospitality have made this years much nicer.
Agradezco hasta el cielo a mi familia. A mis papás que me han inspirado
siempre con su creatividad y fortaleza para resolver los problemas, a mis hermanas
que siempre me apoyan, y a mi hermanito Felipe, que me ha trasmitido
todo su amor por la ciencia, su curiosidad y su escepticismo.
Finally, this thesis would not have been possible without the constant support
of Tim Taminiau, his careful corrections of this manuscript and the nice
3D images of the cone and the phase fronts that he made for me. I also thank
Tim for giving me my first bicycle, for teaching me how to use it, and for
pushing me up the mountain every day when we were going to work.
ix

When he [Kepler] found that his long
cherished beliefs did not agree with the
most precise observations, he accepted
the uncomfortable facts, he preferred the
hard truth to his dearest illusions.
That is the heart of science.
Cosmos - Carl Sagan

Abstract
In the same way that electronics is based on measuring and controlling the state
of electrons, the technological applications of quantum optics will be based
on our ability to generate and characterize photonic states. The generation
of photonic states is traditionally associated to nonlinear optics, where the
interaction of a beam and a nonlinear material results in the generation of
multi-photon states. The most common process is spontaneous parametric
down-conversion (spdc), which is used as a source of pairs of photons not only
for quantum optics applications but also for quantum information and quantum
cryptography [1, 2].
The popularity of spdc lies in the relative simplicity of its experimental realization,
and in the variety of quantum features that down-converted photons
exhibit. For instance, a pair of photons generated via spdc can be entangled
in polarization [3, 4], frequency [5, 6], or in the equivalent degrees of freedom of
orbital angular momentum, space, and transverse momentum [7, 8, 9, 10, 11].
Standard spdc applications focus on a single degree of freedom, wasting the
entanglement in other degrees of freedom and the correlations between them.
Among the few configurations using more than one degree of freedom are hyperentanglement
[12, 13], spatial entanglement distillation using polarization
[14], or control of the joined spectrum using the pump’s spatial properties [6].
This thesis describes the spatial properties of the two-photon state generated
via spdc, considering the different parameters of the process, and the
correlations between space and frequency. To achieve this goal, I use the purity
to quantify the correlations between the photons, and between the degrees
of freedom. Additionally, I study the spatial correlations by describing the
transfer of orbital angular momentum (oam) from the pump to the signal and
idler photons, taking into account the pump, the detection system and other
parameters of the process.
This thesis is composed of six chapters. Chapter 1 introduces the mode
function, used throughout the thesis to describe the two-photon state in space
and frequency. Chapter 2 describes the correlations between degrees of freedom
or photons in the two-photon state, using the purity to quantify such correlations.
Chapter 3 explains the mechanism of the oam transfer from pump
to signal and idler photons. Chapter 4 describes the effect of different spdc
parameters on the oam transfer in noncollinear configurations, both theoretically
and experimentally. In analogy with the downconverted case, chapter 5
discusses the two-photon state generated via Raman transition, by describing
its mode function, the correlations between different parts of the state, and the
oam transfer in the process. Finally, chapter 6 summarizes the main results
presented by this thesis.
xi

Abstract
The matrix notation, introduced here to describe the two-photon mode
function, reduces the calculation time for several features of the state. In
particular, this notation allows to calculate the purity of different parts of
the state analytically. This analytical solution reveals the effect of each spdc
parameter on the internal correlations, and shows the necessary conditions to
suppress the correlations, or to maximize them.
The description of the oam transfer mechanism shows that the pump oam
is totally transfer to the generated photons. But if only a portion of the generated
photons is detected their oam may not be equal to the pump’s oam. The
experiments described in the thesis show that the amount of oam transfer in
the noncollinear case is tailored by the parameters of the spdc. The analysis
of the spdc case can be extended to other nonlinear processes, such as Raman
transitions, where the specific characteristics of the process determine the
correlations and the oam transfer mechanism.
The results of this thesis contribute to a full description of the correlations
inside the two-photon state. Such a description allows to use the correlations
as a tool to modify the spatial state of the photons. This spatial information,
translated into oam modes, provides a multidimensional and continuum degree
of freedom, useful for certain tasks where the polarization, discrete and bidimensional,
is not enough. To make such future applications possible, it will be
necessary to optimize the tools for the detection of oam states at the single
photon level [15, 16].
xii

Resumen
De la misma manera que la electrónica se basa en medir y controlar el estado
de los electrones, las aplicaciones tecnológicas de la óptica cuántica se basarán
en nuestra habilidad para generar estados fotónicos bien caracterizados. La
generación de estos estados está tradicionalmente asociada a la óptica no lineal,
donde la interacción de un haz con un material no lineal da lugar a la
generación de estados de múltiples fotones. El proceso no lineal más popular
es la conversión paramétrica descendente, o spdc por su sigla en inglés, que
es usada como fuente de pares de fotones no sólo en aplicaciones de óptica
cuántica si no también para información y criptografía cuánticas [1, 2].
La popularidad de spdc se debe a la relativa simplicidad de su realización
experimental, y a la variedad de fenómenos cuánticos exhibidos por los pares
de fotones generados. Por ejemplo, estos pares pueden estar entrelazados en
polarización [3, 4], frecuencia [5, 6], o en los grados de libertad espacial: momento
angular orbital o momento transversal [7, 9, 10, 11]. Las aplicaciones
usuales de spdc usan sólo un grado de libertad, perdiendo la información contenida
en los otros grados de libertad y en las correlaciones entre ellos. Entre
las únicas aplicaciones que usan más de un grado de libertad se encuentran el
hiperentrelazamiento [12, 13], la destilación de entrelazamiento espacial usando
la polarización [14], o el control de la distribución espectral conjunta usando
las propiedades espaciales del haz generador del spdc [6].
Esta tesis describe las características espaciales de los pares de fotones generados
en spdc, teniendo en cuenta el efecto de los otros grados de libertad,
especialmente de la frecuencia. Para ello, usaré la pureza como cuantificador de
las correlaciones entre los grados de libertad, y entre los fotones. Además, usaré
la transferencia de momento angular orbital (oam), asociada a la distribución
espacial de los fotones, para estudiar el efecto de diferentes parámetros del
spdc sobre el estado espacial generado.
Esta tesis esta compuesta de seis capítulos. El capítulo 1 introduce la
función de modo, que es usada en toda la tesis para describir los estados de dos
fotones en espacio y frecuencia. El capítulo 2 describe las correlaciones entre
los grados de libertad, y entre los fotones, usando la pureza para cuantificar
estas correlaciones. El capítulo 3 describe la transferencia de oam desde el haz
generador hasta los fotones generados. El capítulo 4 demuestra teórica y experimentalmente,
el efecto de diferentes parámetros del spdc sobre la transferencia
de momento angular orbital en configuraciones no colineales. El capítulo
5 discute la generación de estados de dos fotones en transiciones Raman, caracterizando
estos estados de una manera análoga a la utilizada para aquellos
generados en spdc. Por último, el capítulo 6 resume las contribuciones más
importantes de esta tesis.
xiii

Abstract
La notación matricial introducida para describir la función de modo de los
pares generados, reduce considerablemente el tiempo de cálculo de diferentes
características del estado. En particular, ésta notación hace posible calcular
analíticamente la pureza de diferentes partes del estado, y estudiar el efecto de
cada parámetro del spdc sobre las correlaciones entre los grados de libertad o
entre los fotones. Hace posible, además, encontrar las condiciones necesarias
para suprimir las correlaciones entre los grados de libertad o entre los fotones,
y distinguir en que casos estas correlaciones se hacen mas relevantes.
El estudio del mecanismo de transferencia del momento angular orbital,
revela que éste es transferido totalmente del haz generador a los fotones generados.
Si sólo una porción de estos fotones es considerada, el oam de éstos no
da cuenta del momento oam total invertido por el haz generador. Los experimentos
descritos en esta tesis muestran que la cantidad de oam transferido
a una porción de los fotones puede ser controlada modificando el tamaño de
esa porción, ya sea cambiando el ancho del haz incidente, el largo del cristal u
otro parámetro. En el caso de la generación de pares de fotones en otros procesos
no lineales, como las transiciones Raman, tanto las correlaciones como la
transferencia de oam, son determinadas por las características especificas del
proceso.
Los resultados de esta tesis contribuyen a completar la descripción de las
correlaciones dentro del estado de dos fotones. Esta descripción permite el
uso de las correlaciones como herramienta para modificar el estado espacial de
los fotones. La información espacial, traducida a modos de oam, ofrece un
grado de libertad infinito dimensional y continuo, útil en ciertas tareas donde
la polarización, bidimensional y discreta, no es suficiente. Para hacer estas
tareas posibles, es necesario optimizar las herramientas para la detección de
estados de oam para un sólo fotón [15, 16].
xiv

Introduction
The role of photons in quantum physics and technology is growing [1, 2, 17].
Frequently, those applications are based on two dimensional systems, such as
the two orthogonal polarization states of a photon, losing all information in
other degrees of freedom. These applications only use a portion of the total
quantum state of the light.
The polarization itself is related to a broader degree of freedom. The angular
momentum of the photons contains a spin contribution associated with
the polarization, as well as an orbital contribution associated with the spatial
distribution of the light (its intensity and phase). In general, the spin and
the orbital angular momentum cannot be considered separately. However, in
the paraxial regime both contributions are independent [18]. In this regime it
is possible to exploit the possibilities offered by the infinite dimensions of the
orbital angular momentum (oam) of the light.
Currently available technology offers different possibilities to work with
oam. Computer generated holograms are widely used in classical and quantum
optics for generating and detecting different oam states [19, 20]. The efficiency
of these processes has increased with the design of spatial light modulators
capable of real time hologram generation [21].
Of special interest is the generation of paired photons entangled in oam.
Entanglement is a quantum feature with no analogue in classical physics. Spontaneous
parametric down-conversion (spdc) is a reliable source for the generation
of pairs of photons entangled in different degrees of freedom, including
oam [22]. The existence of correlations in the oam of pairs of photons generated
in spdc was proven experimentally by Mair and coworkers in 2001. Other
nonlinear processes can be used to generate pairs of photons with correlations
in their oam content. In reference [23], the authors show the generation of
spatially entangled pairs of photons by the excitation of Raman transitions in
cold atomic ensembles.
Several studies illustrate the potential offered by higher-dimensional quantum
systems. For instance, reference [24] reports a Bell inequality violation
by a two-photon three-dimensional system, confirming the existence of oam
entanglement in the system. Reference [25] introduced a quantum coin tossing
protocol based on oam states. In reference [26], the authors present a quantum
key distribution scheme using entangled qutrits, which are encoded into
the oam of pairs of photons generated in spdc. And, in reference [12], the
authors report the generation of hyperentangled quantum states by using the
combination of the degrees of freedom of polarization, time-energy and oam.
All the previously mentioned applications are based on specific oam correlations
between the photons. However, there has been some controversy about
xv

Introduction
the transfer of oam from the generating pump beam to the down-converted
photons. Several studies have reported a compleate transfer of the oam of the
pump [7, 8, 22, 27, 28, 29], while other studies have reported a partial transfer
[30, 31, 32, 33]. The elucidation of the oam spectra of down-converted photons,
and their relation with the spectra of the pump, is a key step for developing
new applications based on this degree of freedom.
This thesis addresses the mentioned oam controversy, as well as the characterization
of the different correlations present in two-photon states. For both
processes, spdc and Raman transitions in cold atoms, this thesis describes the
generated two-photon state, the correlations between different degrees of freedom
and between photons, and the oam transfer. A general goal of the thesis
is to contribute to the characterization of the two-photon sources, making it
possible to exploit a bigger portion of the total quantum state of the light in
future applications.
xvi

This thesis is based on the following papers:
List of Publications
Orbital angular momentum correlations of entangled paired photons.
C. I. Osorio, G. Molina-Terriza, J. P. Torres.
J. Opt. A: Pure Appl. Opt. 11, 094013 (2009).
Chapters 3 and 5
Spatial entanglement of paired photons generated in cold atomic ensembles
C. I. Osorio, S. Barreiro, M. W. Mitchell, and J. P. Torres.
Phys. Rev. A 78, 052301 (2008). arXiv:0804.3257v2 [quant-ph].
Chapter 5
Spatiotemporal correlations in entangled photons generated by spontaneous parametric
down conversion.
C. I. Osorio, A. Valencia, and J. P. Torres.
New J. Phys. 10, 113012 (2008). arXiv:0804.2425v2 [quant-ph].
Chapters 1 and 2
Correlations in orbital angular momentum of spatially entangled paired photons generated
in parametric downconversion.
C. I. Osorio, G. Molina-Terriza, and J. P. Torres.
Phys. Rev. A 77, 015810 (2008). arXiv:0711.4500v1 [quant-ph].
Chapters 3
Azimuthal distinguishability of entangled photons generated in spontaneous parametric
down-conversion.
C. I. Osorio, G. Molina-Terriza, B. Font, and J. P. Torres.
Opt. Express 15, 14636 (2007). arXiv:0709.3437v1 [quant-ph].
Chapter 4
Control of the shape of the spatial mode function of photons generated in noncollinear
spontaneous parametric down-conversion.
G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych, and J. P.
Torres.
Phys. Rev. A 72, 065802 (2005). arXiv:quant-ph/0508058v1.
Chapter 4
Orbital angular momentum of entangled counterpropagating photons.
J. P. Torres, C. I. Osorio, and L. Torner.
Opt. Lett. 29, 1939 (2004).
Chapters 3 and 4
xvii

CHAPTER 1
General description of
Two-photon states
Experimental implementations of spontaneous parametric down-conversion (spdc)
share three main components: A laser beam used as a pump, a nonlinear material
that induces the down-conversion, and a detection system that measures
the generated state. The properties of the resulting two-photon state depend
on the interaction, and on the individual properties of those components. This
chapter derives the two-photon mode function that characterizes the generated
state, using first order perturbation theory. The chapter is divided in four
sections. Section 1.1 describes the spdc process in general terms. Section 1.2
introduces a general expression for the mode function. Section 1.3 lists a series
of common approximations used to reduce the complexity of the mode function.
Finally, section 1.4 introduces a novel matrix notation for the simplified
mode function. One of the advantages of this new notation is that many characteristics
of the two-photon state can be analytically calculated by using it.
The following chapters will use the notation and results introduced here as a
starting point to describe the correlations between the generated photons, and
between their spatial states.
1

1. General description of two-photon states
virtual state
pump
ground state
signal
idler
Figure 1.1: In spontaneous parametric down conversion the interaction of a molecule
with the pump results in the annihilation of the pump photon and the creation of
two new photons.
1.1 Spontaneous parametric down-conversion
Spontaneous parametric down-conversion is one of the possible resulting processes
of the interaction of a pump photon and a molecule, in the presence of
the vacuum field. In spdc, the pump-molecule interaction leads to the generation
of a single physical system composed of two photons: signal and idler, as
figure 1.1 shows.
The name of the process reveals some of its characteristics. spdc is a
parametric process since the incident energy totally transfers to the generated
photons, and not to the molecule. And, it is a down-conversion process since
each of the generated photons has a lower energy than the incident photon.
The conservation of energy and momentum establish the relations between the
frequencies (ωp,s,i) and wave vectors (kp,s,i) of the photons,
ωp = ωs + ωi
kp = ks + ki. (1.1)
where the subscripts p, s, i stand for pump, signal and idler. Macroscopically,
spdc results from the interaction of a field with a nonlinear medium. Commonly
used mediums include uniaxial birefringent crystals. The conditions in
equation 1.1 can be achieved for this kind of crystals, since their refractive index
changes with the frequency and the polarization of an incident field [34, 35].
This thesis considers type-i configurations in uniaxial birefringent crystals.
In such configuration the pump beam polarization is extraordinary since it is
contained in the principal plane, defined by the crystal axis and the wave vector
of the incident field. The polarization of the generated photons is ordinary,
which means that it is orthogonal to the principal plane.
For a given material, the conditions imposed by equation 1.1 determine the
characteristics of the generated pair. The second line in equation 1.1 defines the
spdc geometrical configuration, and it is known as a phase matching condition.
The generated photons are emitted in two cones coaxial to the pump, as shown
in figure 1.2 (b). The apertures of the cone are given by the emission angles ϕs,i
associated to the frequencies ωs,i. Once a single frequency or emission angle is
selected, only one of all possible cones is considered. In degenerate spdc, both
photons are emitted at the same angle ϕs = ϕi and the cones overlap, as in
figure 1.2 (c). There are two kinds of degenerate spdc processes: noncollinear
2

y
y
x
z
z
pump
s
i
(a)
signal
idler
(c)
1.2. Two-photon state
Figure 1.2: (a) Phase matching conditions impose certain propagation directions for
the emitted photons at frequencies ωs and ωi. (b-c) This directions define two cones,
that collapse into one in the degenerate case. (d) In the collinear configuration, the
aperture of the cones tends to zero as the direction of emission is parallel to the pump
propagation direction.
in which the signal and idler are not parallel to the pump, and collinear in
which the aperture of the cones tends to zero as the photons propagate almost
parallel to the pump, as shown in figure 1.2 (d).
Even though the phase matching condition defines the main characteristics
of the generated two-photon state, other important factors influence the measured
state of the photons, for example the detection system. The next section
describes the two-photon state mathematically, taking into account all these
factors.
1.2 Two-photon state
When an electromagnetic wave propagates inside a medium, the electric field
acts over each particle (electrons, atoms, or molecules) displacing the positive
charges in the direction of the field and the negative charges in the opposite
direction. The resulting separation between positive and negative charges of
the material generates a global dipolar moment in each unit volume known as
polarization, and defined as
(b)
(d)
P = ɛ0(χ (1) E + χ (2) EE + χ (3) EEE + · · · ) (1.2)
where ɛ0 is the vacuum electric permittivity, and χ (n) are the electric susceptibility
tensors of order n [34, 35].
For small field amplitudes, as in linear optics, the polarization is approximately
linear,
P ≈ ɛ0χ (1) E. (1.3)
3

1. General description of two-photon states
When the amplitude of the electric field increases, the higher orders term in
equation 1.2 become relevant, and then a nonlinear response of the material
to the field appears. Optical nonlinear phenomena resulting from this kind
of interaction include the generation of harmonics, the Kerr effect, Raman
scattering, self-phase modulation, and cross-phase modulation [34].
Spontaneous parametric down-conversion, and other second order nonlinear
processes, result from the second order polarization, defined as the first
nonlinear term in the polarization tensor
P (2) = ɛ0χ (2) EE. (1.4)
The quantization of the electromagnetic field leads to a quantization of the
second order polarization, so that the nonlinear polarization operator ˆ P (2)
becomes
ˆP (2) = ɛ0χ (2) ( Ê(+) + Ê(−) )( Ê(+) + Ê(−) ), (1.5)
where Ê(+) and Ê(−) are the positive and negative frequency parts of the field
operator [36]. The positive frequency part of the electric field operator is a
function of the annihilation operator â(k), and is defined at position rn
xnˆxn + ynˆyn + znˆzn and time t as
=
Ê (+)
1/2
ωn
n (rn, t) = ien
2ɛ0v
dkn exp [ikn · rn − iωnt]â(kn), (1.6)
where the magnitude of the wave vector k satisfies k 2 = ωn/c. The volume
v contains the field, and en is the unitary polarization vector. The negative
frequency part of the field is the Hermitian conjugate of the positive part,
Ê (−)
n (rn, t) = Ê(+)† n (rn, t). Thus, the negative frequency part is a function of
the creation operator â † (k).
Following references [37] and [38], in first order perturbation theory, the
interaction of a volume V of a material with a nonlinear polarization ˆ P (2) and
the field Êp(rp, t), produces a system described by the state
|ΨT 〉 ∝ |1〉p|0〉s|0〉i − i
τ
0
dt ˆ HI|1〉p|0〉s|0〉i, (1.7)
where τ is the interaction time, |1〉p|0〉s|0〉i is the initial state of the field, and
the interaction Hamiltonian reads
ˆHI = − dV ˆ P (2) Êp(rp, t). (1.8)
V
The first term at the right side of equation 1.7 describes a one photon system,
while the second term describes a two-photon system. In what follows we will
consider only the second term as we are mainly interested in the generation of
pairs of photons. The two-photon system state is given by
|Ψ〉 = i
τ
0
dt ˆ HI|1〉p|0〉s|0〉i, (1.9)
or ˆρ = |Ψ〉〈Ψ| in the density matrix formalism. Seven of the eight terms
that compose the interaction Hamiltonian vanish when they are applied over
the state |1〉p|0〉s|0〉i. The remaining term is proportional to the annihilation
4

1.3. Approximations and other considerations
operator in mode p, and the creation operators in modes s and i. Assuming
constant χ (2) , the two-photon state reduces to
(2) τ
iɛ0χ
|Ψ〉 = dt dV
0 V
Ê(+) p (rp, ωp) Ê(−) s (rs, ωs) Ê(−) i (ri, ωi)|1〉p|0〉s|0〉i,
(1.10)
which can be written as
|Ψ〉 ∝ dks dkiΦ(ks, ωs, ki, ωi)a † (ks)a † (ki)|0〉s|0〉i; (1.11)
where Φ(ks, ωs, ki, ωi) is known as the mode function and, assuming that the
pump has certain spatial distribution Ep(kp), is given by
τ
Φ(ks, ωs, ki, ωi) ∝ dt dV dkpEp(kp)
0
V
× exp [ikp · rp − iks · rs − iki · ri − i(ωp − ωs − ωi)t]
× â(kp)|1〉p. (1.12)
The mode function contains all the information about the generated two-photon
system in space and frequency, not only about their individual state but about
their correlations. This function is therefore highly complex, and to calculate
any feature of the two-photon state it is necessary to simplify it, as the next
section shows.
1.3 Approximations and other considerations
Analytical calculations of any features of the down converted photons require
simplification of the mode function in equation 1.12. This section lists the most
important approximations used in this thesis, as well as the restrictions that
they impose.
Separation of transversal and longitudinal components
A field with a wave vector k almost parallel to its propagation direction spreads
only a little in the transversal direction. It is then possible, to consider the
longitudinal and transversal components of the wave vector separately,
k = kzˆz + q. (1.13)
As the wave vector’s magnitude k = ωn/c is bigger than the transversal component’s
magnitude q = |q| = (k 2 x + k 2 y) 1/2 , the magnitude of the longitudinal
component kz is always a real number,
kz = k 2 1
− q
2 2 . (1.14)
Assuming that the pump, the signal, and the idler are such fields, the mode
function becomes
τ
Φ(qs, ωs, qi, ωi) ∝
0
dt dV
V
dqpEp(qp) exp [ik z pzp − ik z szs − ik z i zi]
× exp [iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i − i(ωp − ωs − ωi)t]
× â(q p)|1〉p. (1.15)
where r ⊥ n = xnˆxn + ynˆyn is the transversal position vector.
5

1. General description of two-photon states
The semiclassical approximation
As a consequence of the low efficiency of the nonlinear process, the incident
field is several orders of magnitude stronger than the generated fields. In that
case, it is possible to consider the pump as a classical field, while the signal
and idler are considered as quantum fields, this is known as the semiclassical
approximation.
By defining the pump as a classical field, with a spatial amplitude distribution
Ep(q p) and a spectral distribution Fp(ωp):
Ep(rp, t) ∝ dqp dωpEp(qp)Fp(ωp) exp [iqp · r ⊥ p + ik z pzp − iωpt], (1.16)
the mode function reduces to
τ
Φ(qs, ωs, qi, ωi) ∝ dt dV dqp dωpEp(qp)Fp(ωp) Gaussian beam approximation
0 V
× exp [ik z pzp − ik z szs − ik z i zi + iqp · r ⊥ p − iqs · r ⊥ s − iqi · r ⊥ i ]
× exp [−i(ωp − ωs − ωi)t]. (1.17)
According to equation 1.17, the mode function depends on the spatial and
temporal profiles of the pump beam. To write these profiles explicitly we
assume a pump beam with a Gaussian temporal distribution,
Fp(ωp) = exp − T 2 0
4 ω2
p
(1.18)
where T0 is the pulse duration (standard deviation), which tends to infinity for
continuous wave beams. Also, we assume that the pump beam is an optical
vortex with orbital angular momentum lp per photon. As will be described
in section 3.1, under these conditions the pump spatial profile is given by the
Laguerre-Gaussian polynomials
1
2
wp
Ep(qp) =
2π|lp|!
|lp|
−iwp
√ qp exp −
2 w2 p
4 q2
p + ilpθp
(1.19)
that are functions of the pump beam waist wp, and the transversal vector
magnitude qp and phase θp, given by
6
qp =
θp = tan −1
(q x p ) 2 + (q y p) 2
q y p
q x p
.
(1.20)

x
1.3. Approximations and other considerations
pump
signal
beam s
y z xi
idler
z
Figure 1.3: The propagation direction of the pump beam defines the z direction of
the general coordinate system. The generated pair of photons propagates with angles
ϕs and ϕi with respect to the pump beam in the yz plane. Each photon coordinate
system transforms to the general coordinate system through the relations in equation
1.23.
For the special case of a Gaussian pump beam with lp = 0 the mode function
becomes
τ
Φ(qs, ωs, qi, ωi) ∝ dt dV
0 V
i
y
dq p exp [− w2 p
4 q2 p − T 2 0
4 ω2 p]
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ]
× exp [−i(ωp − ωs − ωi)t], (1.21)
which assumes that the interaction time τ is longer than the spontaneous emission
life time of the material.
Frequency bandwidth
Fields generated experimentally always contain a distribution of frequencies,
and totally monochromatic fields only exist in theory. This thesis considers the
frequency ω of each field as the sum of a constant central frequency ω 0 , and a
small deviation from that frequency Ω, so that ω = ω 0 + Ω.
As a result of integrating over the interaction time taking into account the
conservation of energy, the mode function becomes
Φ(qs, Ωs, qi, Ωi) ∝ dV
V
Coordinate transformation
dq p exp [− w2 p
4 q2 p − T 2 0
4 (Ωs + Ωi) 2 ]
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ].
(1.22)
The mode function depends on three position vectors, rp, rs, and ri. Each of
them is defined in a coordinate system with the z axis parallel to the propagation
direction of each field, as shown in figure 1.3. Defining all position vectors
in the same coordinate system simplifies the integration of the mode function
over volume.
i
i
xs
ys
zs
7

1. General description of two-photon states
pump
polarization
y
Figure 1.4: The azimuthal angle α between the pump beam polarization and the x
axis defines the position of a single pair of photons on the cone. The x axis is by
definition normal to the plane of emission.
With the origin at the crystal’s center, the z direction parallel to the pump
propagation direction, and the yz plane containing the emitted photons, the
unitary vectors in the coordinate systems of the generated photons transform
as
ˆxs =ˆx
ˆys =ˆy cos ϕs + ˆz sin ϕs
x
ˆzs = − ˆy sin ϕs + ˆz cos ϕs
ˆxi =ˆx
ˆyi =ˆy cos ϕi − ˆz sin ϕi
ˆzi =ˆy sin ϕi + ˆz cos ϕi. (1.23)
As a single pair of photons defines the yz plane, the coordinate transformations
consider only one of the possible directions of propagation on the cone. The
angle α between the x axis and the pump polarization defines the transverse
position on the cone for one photon pair, as figure 1.4 shows. The pump
polarization is normal to the plane in which the generated photons propagate
when α = 0 ◦ or α = 180 ◦ , and it is contained in the yz plane when α = 90 ◦ or
α = 270 ◦ .
Poynting vector walk-off
This thesis considers an extraordinary polarized pump beam and ordinary polarized
generated photons, an eoo configuration. Therefore, while the refractive
index does not change with the direction for the generated photons, it changes
for the pump beam with the angle between the pump wave vector and the axis
of the crystal. Figure 1.5 shows how the energy flux direction of the pump,
given by the Poynting vector, is rotated from the direction of the wave vector
by an angle ρ0 given by
ρ0 = − 1 ∂ne
. (1.24)
∂θ
ne
where ne is the refractive index for the extraordinary pump beam, and θ is
the angle between the optical axis and the pump’s wave vector. The Poynting
8

x
Wave fronts
1.3. Approximations and other considerations
Poynting vector
0
Wave vector
Figure 1.5: As a consequence of the birefringence, the Poynting vector is no longer
parallel to the wave vector. The energy flows in an angle ρ0 with respect to propagation
direction.
x
y z
Figure 1.6: The Poynting vector moves away from the wave vector in the direction of
the pump beam polarization. The angles α and ρ0 characterize the displacement. A
nonradial effect such as the Poynting vector walk-off inevitably brakes the azimuthal
symmetry of the properties of the photons on the cone.
vector walk-off displaces the effective transversal shape of the pump beam
inside the crystal in the pump polarization direction, as shown in figure 1.6.
The vector p = z tan ρ0 cos αˆx + z tan ρ0 sin αˆy, describes the magnitude and
direction of this displacement.
By including the Poynting vector walk-off, and by using the same coordinate
system for all the fields, the mode function becomes
Φ(qs, Ωs, qi, Ωi) ∝ dV dqp exp −
V
w2 p
4 q2 p − T0
4 (Ωs + Ωi) 2
× exp i(q x p − q x s − q x i )x
0
× exp [i(q y p + k z s sin ϕs − k z i sin ϕi − q y s cos ϕs − q y
i cos ϕi)y]
× exp [i(k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y
i sin ϕi)z]
× exp [i(q x p tan ρ0 cos α + q y p tan ρ0 sin α)z]. (1.25)
p
z
9

1. General description of two-photon states
Interaction volume approximation
In the experimental cases studied here, the crystals are cuboids with two square
faces. While typical crystal faces have areas of about 1cm 2 , the pump beam
waist is only some hundreds of micrometers. By assuming that the transversal
dimensions of the crystal are much larger than the pump, the integral over the
volume becomes
V
dV →
∞
−∞
∞ L/2
dx dy dz, (1.26)
−∞ −L/2
where L is the length of the crystal. After integrating over x and y, the mode
function reduces to
where
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆20 − w2 p
4 ∆21 − T0
4 (Ωs + Ωi) 2
L/2
× dz exp [i∆kz] (1.27)
−L/2
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi − k z s sin ϕs + k z i sin ϕi
∆k =k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y
i sin ϕi
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.28)
Wave vector and group velocity
The use of a Taylor series expansion of kz s,i , simplifies the delta factors defined
in the previous paragraph. Because the wave vector magnitude is a function
of the frequency deviation Ω, the expansion around the origin reads
k z n = k z0 ∂k
n + Ωn
z n
+ Ω 2 ∂
n
2kz n
∂2Ωn ∂Ωn
+ . . . , (1.29)
where k z0
n is the magnitude of the wave vector at the central frequency ω 0 n, and
the first partial derivative of the wave vector magnitude with respect to the
frequency is the group velocity Nn.
Taking only the first order terms of the Taylor expansion, and considering
momentum conservation, the delta factors become
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi − NsΩs sin ϕs + NiΩi sin ϕi
∆k =Np(Ωs + Ωi) − NsΩs cos ϕs − NiΩi cos ϕi − q y s sin ϕs + q y
i sin ϕi
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.30)
Exponential approximation of the sinc function
The mode function in equation 1.27, depends on the integral of an exponential
function of z. By integrating the exponential over the length of the crystal, the
10

1
0
-0.2
1.3. Approximations and other considerations
Gaussian Sine Cardinal
0
FWHM
Figure 1.7: An exponential function is chosen as an approximation for the sine cardinal
function. The parameters of the exponential were chosen in such a way that
both functions have the same full width at half maximum (fwhm).
mode function becomes
Φ(q s, Ωs, q i, Ωi) ∝ exp
− w2 p
4 ∆2 0 − w2 p
4 ∆2 1 − T0
4 (Ωs + Ωi) 2
sinc
L
2 ∆k
(1.31)
where the sine cardinal function is defined as sinc(a) = sin a/a.
In chapter 2, the sine cardinal function sinc(a) is approximated by a Gaussian
function exp[−(γa) 2 ]. With γ = 0.4393 the two functions have the same
full width at half maximum as figure 1.7 shows. In the rest of the thesis the
sinc function will be consider without approximations.
Detection systems as Gaussian filters
The specific optical system used to detect the photons at the output face of
the crystal, acts as a filter in space and frequency. The effect of these filters
is modeled by multiplying the mode function by a spatial collection function
Cspatial(qn) and a frequency filter function Ffrequency(Ωn). For the sake of
simplicity, both functions are assumed to be Gaussian functions so that
Cspatial(qn) ∝ exp − w2 n
2 q2
n , (1.32)
and
Ffrequency(Ωn) ∝ exp − 1
2B2 Ω
n
2
n , (1.33)
where n labels each of the generated photons, wn is the spatial collection mode
width, and Bn is the frequency bandwidth.
The angular acceptance θn is related to the vector qn by the equation qn =
knθn. Therefore, for a given collection mode wn the acceptance is θn = 1/knwn
(the half width at 1/e 2 ).
11

1. General description of two-photon states
In order to obtain more convenient units, the filter in momentum is defined
in a different way than the filter in frequency. While Bs,i → 0 implies a single
frequency collection, the condition for a single q vector collection is ws,i → ∞.
Finally, after all the approximations and taking into account the filters, the
mode function equation 1.12 becomes
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆20 − w2 p
4 ∆21
× exp − (γL)2
4 ∆2k − T 2 0
4 (Ωs + Ωi) 2
× exp − w2 s
2 q2 s − w2 i
2 q2 i − 1
2B2 Ω
s
2 s − 1
2B2 Ω
i
2
i . (1.34)
The approximations listed in this chapter were introduced by several authors,
and can be found for example in references [39, 40, 30]. In reference [41],
we introduced a novel matrix notation based on the simplified mode function
equation 1.34. The next section explains the details and the advantages of that
notation.
1.4 The mode function in matrix form
The argument of the exponential function in equation 1.34 is a second order
polynomial of the mode function variables. Such a function can be written in
matrix form, as is shown in appendix A. The mode function then becomes
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1
2 xt
Ax
(1.35)
where all the parameters of the spdc process are contained in A, a positivedefinite
real 6 × 6 matrix given by
A = 1
⎛
a
⎜ h
⎜ i
2 ⎜ j
⎝ k
h
b
m
n
p
i
m
c
s
t
j
n
s
d
v
k
p
t
v
f
l
r
u
w
z
⎞
⎟ ,
⎟
⎠
(1.36)
l r u w z g
x is the vector including all variables of the mode function, defined as
⎛ ⎞
⎜
x = ⎜
⎝
q x s
q y s
q x i
q y
i
Ωs
Ωi
⎟ , (1.37)
⎟
⎠
and x t is the transpose of x. This matrix representation of the mode function,
introduced in reference [41], is an important result of this thesis. The use of
this notation reduces the calculation time for integrals over the mode function,
12

1.4. The mode function in matrix form
and allows to solve some of those integrals analytically. For instance, the mode
function normalization requires six integrals over the modulus square of the
mode function. As appendix B shows, the matrix notation provides a way to
calculate the normalization constant analytically,
dx exp
− 1
2 xt
(2A)x
and thus the normalized mode function reads
=
Φ(qs, Ωs, qi, Ωi) = [det(2A)]1/4
(2π) 3/2
(2π) 3
, (1.38)
[det(2A)] 1/2
exp − 1
2 xt
Ax . (1.39)
In the same way, many other integrals over the mode function can be solved
analytically using the matrix representation of the mode function.
Conclusion
The two-photon mode function can be written in a matrix form after a series
of approximations. The matrix notation makes it possible to calculate several
features of the down-conversion photons analytically, and reduces the numerical
calculation time of others.
The next chapter describes the different correlations in the two-photon state
using the purity as a correlation indicator. The use of the approximate mode
function in matrix notation makes it possible to find an analytical expression
for the purity, and to study the effect of the different parameters on it.
13

CHAPTER 2
Correlations and
entanglement
The two-photon system described in chapter 1 consists of two photons with two
degrees of freedom. Although the mode function deduced fully characterizes the
two-photon state, the degree of correlations between each of its parts (different
subsystems) are not explicit. In this chapter, I use the purity of a subsystem
state to indicate the presence of correlations between it and the rest of the
two-photon state, as shown in figure 2.1. This chapter has three sections.
Section 2.1 describes the main characteristics of the purity, and explains why
it can be used to characterize correlations in composed systems. Section 2.2
describes the spatial part of the two-photon state, and section 2.3 describes
the state of the signal photon. In both sections a discussion about the origin
of the correlations is followed by analytical and numerical calculations of the
purity. The calculations clarify the role of each spdc parameter in the internal
correlations of the two-photon state. By engineering the spdc process, it is
possible to tailor, and even to suppress correlations between different degrees
of freedom or between different photons. Chapter 3 considers the spatial part
of the two-photon state, after the correlations between space and frequency are
suppressed.
15

2. Correlations and entanglement
(a)
Space Frequency Signal
q
q
s
i
s
i
qs
qi
Idler
Figure 2.1: The frequency part, gray in (a), is traced out in section 2.2, to calculate
the state of the subsystem composed by the two-photon state in the spatial degree
of freedom. Calculating the signal photon state in space and frequency, section 2.3,
requires to trace out the idler photon, gray in (b).
2.1 The purity as a correlation indicator
When a physical system is used to implement a task, its state should be characterized.
For quantum states, the reconstruction of the state density matrix
demands a tomography [1]. This process requires an infinite amount of copies
of the state in order to eliminate statistical errors [42]. In an experimental
implementation, the number of copies of the state available is finite, but even
under this condition an experimental tomography is a complex process. Instead
when just a certain feature of the system is relevant for the task, it is preferable
to use a figure of merit to characterize that part of its quantum state.
For a system in a quantum state given by the density operator ˆρ, the purity
P = T r[ˆρ 2 ] is a figure of merit that describes its ability to be correlated with
others. That is, the purity indicates if correlations can or cannot exist. Since
the purity is a second order function of the density operator, measuring it does
not require a full tomography, but just a simultaneous measurement over two
copies of the system [43, 44].
To illustrate other characteristics of the purity, consider as an example
a complete orthonormal basis for the electromagnetic field, where each basis
vector |n〉 describes a mode of the field [36]. A pure state |R〉 is any state
written as a superposition of basis vectors:
|R〉 =
Cn|n〉, (2.1)
n
where the mode amplitude and the relative phases between modes, given by
the coefficients Cn, are fixed. This pure state can be described by the density
operator ˆρ = |R〉〈R|, which has the maximum possible value for the purity
T r[ˆρ 2 ] = 1.
A mixed state of the electromagnetic field is a state that can not be written
as in equation 2.1. For instance, in the case of a statistical mixture of field
modes, where there are no fixed amplitude or fixed relative phases associated
16
s
i
(b)

2.1. The purity as a correlation indicator
to each vector of the base. There is, however, a fixed probability pR of finding
the electromagnetic field in a state |R〉. Using the density operator formalism,
the probability distribution is given by
ˆρ =
pR|R〉〈R|. (2.2)
R
The purity of the state described by ˆρ quantifies how close it is to a pure
state. If the state is pure then T r[ˆρ 2 ] = 1, and if it is maximally mixed then
T r[ˆρ 2 ] = 1/n, where n is de dimension of the Hilbert space in which the state is
expanded. Maximally mixed states of the electromagnetic field, in the infinite
dimensional spatial or temporal degrees of freedom have a purity T r[ˆρ 2 ] = 0.
To extend the discussion about the purity to the kind of states generated
by spdc, consider the pure bipartite system described by |ψ〉 and composed by
the fields A and B. The Schmidt decomposition guarantees that orthonormal
states |RA〉 and |RB〉 exist for the fields A and B, so that
|ψ〉 =
λR|RA〉|RB〉 (2.3)
R
where λR are non-negative real numbers that satisfy
R λR = 1, and are
known as Schmidt coefficients [1, 17]. The average of the nonzero Schmidt
coefficients is a common entanglement quantifier [11] known as the Schmidt
number and given by
K =
1
R λ2 R
. (2.4)
While equation 2.3 describes the state of the whole bipartite system, the state
of each part is calculated by tracing out the other part. Thus, the states of the
fields A and B are given by the density operators
ˆρA =
λR|RA〉〈RA| (2.5)
and
R
ˆρB =
λR|RB〉〈RB|. (2.6)
R
Since both operators have equal eigenvalues λ2 R , the purity of both states is
equal, and given by the inverse of the Schmidt number K
T r[(ˆρ A ) 2 ] = T r[(ˆρ B ) 2 ] = 1
K
= λ 2 R. (2.7)
Thus, when the field A is in a pure state, the field B is in a pure state too,
and
R λ2
R = 1. Since R λR = 1, the Schmidt coefficients satisfy the new
condition only if all except one of them are equal to zero. If that is the case,
the bipartite system in equation 2.3 is a product state, and no correlations
exist between A and B. In other words, if there are correlations, the purity of
A (and B) is smaller than 1.
In conclusion, when considering a composed global system, the purity of
each subsystem measures the strength of the correlation between such subsystem
and the rest. This is always true independently of how the subsystems are
R
17

2. Correlations and entanglement
chosen. In the particular case of a two-photon system, the purity of the spatial
part can be used to study correlations between the degrees of freedom, or the
purity of the signal photon can be used to study the entanglement between the
photons. The next sections explore both these approaches.
2.2 Correlations between space and frequency
In type-i spdc, the photons generated have a polarization orthogonal to the
polarization of the incident beam. Therefore, the two-photon state generated
in a type-i process has only two degrees of freedom: frequency and transversal
spatial distribution. In this section, I study the origin and the characteristics
of the correlations between those degrees of freedom by using the purity of the
spatial part as a correlation indicator.
Except for hyperentanglement configurations [12, 13] and some other novel
configurations [6, 14], most applications of type-i spdc processes use just one of
the degrees of freedom, ignoring the correlation that may exist between space
and frequency. That is, those configurations assume a high degree of spatial
purity. This section explores the conditions in which this assumption is valid.
The first part of this section contains a discussion about the origin and
suppression of the correlations. The second part includes the calculations for
the spatial purity that is used as correlation indicator. The third part shows
the results of numerical calculation of the spatial purity in several cases.
2.2.1 Origin of the correlations
According to the two-photon mode function, equation 1.34, the correlations
between space and frequency appear as cross terms in the variables associated
to those degrees of freedom, as figure 2.2 shows. The correlations between
q s x
q s y
q i x
y
qi s
i
qs x
a
h
i
j
k
l
q s
y
h
b
m
n
p
r
qi y
i
m
c
s
t
u
Figure 2.2: Cross terms in matrix A generate correlations between space and frequency.
Without those terms, the matrix A is composed by two nonzero block matrices,
one for each degree of freedom.
space and time are always relevant, if at least one of the cross terms k, l, p,
r, t, u, v, w is comparable with another term of matrix A. Therefore, it is
possible to suppress the correlations by making all the cross terms negligible,
or by increasing the values of the other terms.
The use of ultra-narrow filters in one of the degrees of freedom will suppress
most of the information available in it, destroying any possible correlation with
18
q i
x
j
n
s
d
v
w
s
k
p
t
v
f
z
i
l
r
u
w
z
g

2.2. Correlations between space and frequency
other degrees of freedom. Filtering makes the diagonal terms of matrix A larger
than the rest of the terms, including the terms responsible for the correlations.
But while they remove the correlations, the filters reduce the amount of photons
available for any measurement.
An appropriate design of the source makes it possible to suppress cross
terms without using filters. Such a design should fulfill the conditions
Np − Ns cos ϕs
Np − Ni cos ϕi
Np − Ns cos ϕs
Np − Ni cos ϕi
Np − Ns cos ϕs = 0
Np − Ni cos ϕi = 0
1 + w2 p
γ2L2 1 + w2 p
γ 2 L 2
1 − w2 p
γ 2 L 2
1 − w2 p
γ 2 L 2
Ns sin ϕs
tan ϕi
Ni sin ϕi
tan ϕs
= 0
= 0
= 0
= 0. (2.8)
These conditions can be met for example by using a long crystal, a highly
focused pump beam, for which wp/γL → 0, and emission angles such that
cos ϕs = Np/Ns and cos ϕi = Np/Ni.
Even though these conditions have been deduced in a simplified regime,
reference [45] shows that these conditions are sufficient to remove the correlations
when considering the general mode function of equation 1.12. Once the
correlations are suppressed, the two-photon mode function can be written as a
product of a spatial and a frequency two-photon mode function like
Φ(q s, ωs, q i, ωi) = Φq(q s, q i)ΦΩ(Ωs, Ωi). (2.9)
Since the global two-photon system in space and frequency is in a pure state,
the presence of correlations can be confirmed by using the purity. The next
section calculates the purity of the spatial state, which can be used as an
indicator of correlations between space and frequency.
2.2.2 Two-photon spatial state
To study space/frequency correlations I will describe the purity of the spatial
part of the two-photon state. As the purity of the spatial and frequency parts
of the state are equal, all results derived for the spatial two-photon state extend
to a frequency two-photon state.
After tracing-out the frequency from the two-photon state, the remaining
state describes the spatial part of the state. The reduced density matrix for
19

2. Correlations and entanglement
the spatial two-photon state is
ˆρq = T rΩ[ρ]
=
=
dΩ ′′
s dΩ ′′
i 〈Ω ′′
s , Ω ′′
i |ˆρ|Ω ′′
s , Ω ′′
i 〉
dqsdΩsdqidΩidq ′ sdq ′ i
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)|qs, qi〉〈q ′ s, q ′ i|, (2.10)
while the purity of this state, defined as T r[ˆρ 2 q], is given by
T r[ˆρ 2
q] =
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (2.11)
It is possible to solve these integrals analytically by using the exponential
character of the mode function given by equation (1.35). The purity becomes
T r[ˆρ 2 q] = det(2A)
det(B) . (2.12)
As seen in appendix A, B is a positive-definite real 12 × 12 matrix
B = 1
⎛
⎜
2 ⎜
⎝
2a
2h
2i
2j
k
l
0
0
0
0
k
2h
2b
2m
2n
p
r
0
0
0
0
p
2i
2m
2c
2s
t
u
0
0
0
0
t
2j
2n
2s
2d
v
w
0
0
0
0
v
k
p
t
v
2f
2z
k
p
t
v
0
l
r
u
w
2z
2g
l
r
u
w
0
0
0
0
0
k
l
2a
2h
2i
2j
k
0
0
0
0
p
r
2h
2b
2m
2n
p
0
0
0
0
t
u
2i
2m
2c
2s
t
0
0
0
0
v
w
2j
2n
2s
2d
v
k
p
t
v
0
0
k
p
t
v
2f
l
r
u
w
0
0
l
r
u
w
2z
l r u w 0 0 l r u w 2z 2g
defined by the equation
N 4
exp − 1
2 Xt
BX = Φ(qs, Ωs, qi, Ωi)
⎞
⎟ ,
⎟
⎠
(2.13)
× 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)
where vector X is the result of concatenation of x and x ′ .
In order to see the effect of each of the spdc parameters on the spatiofrequency
correlations, some typical values are used in next subsection to calculate
T r[ˆρ 2 q] numerically.
2.2.3 Numerical calculations
As first example, consider a degenerate type-i spdc process characterized by the
parameters in the second column of table 2.1. A pump beam, with wavelength
20

2.2. Correlations between space and frequency
λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium iodate (liio3)
crystal with length L = 1 mm, and negligible Poynting vector walk-off (ρ0 = 0).
The crystal emits signal and idler photons with a wavelength λ0 s = λ0 i = 810
nm at an angle ϕs,i = 10◦ .
Under the above conditions, figure 2.3 shows the purity of the spatial state
T r[ρ2 q] given by equation 2.12, as a function of the spatial filter width for different
frequency filter bandwidths. The half width at 1/e of the frequency filters
∆λn, is Bn = πc∆λn/(λ2 √
n ln 2). Figure 2.3 shows the presence of correlations
between space and frequency in all the situations for which the purity is smaller
than one.
According to figure 2.3, the spatial purity of the two-photon state gets closer
to one as ws (= wi) increase. Narrow filters in space (infinitely large ws,i) makes
the two-photon state separable in frequency and space. The separability and
the lack of correlations between frequency and space appear also in the“narrow
band” limit for the frequency. If ∆λs = ∆λi → 0 nm, then the purity is always
1 for any value of ws = wi.
When using typical interference filters, ∆λs = ∆λi ≈ 1 nm, the purity decreases
indicating space/frequency correlations. As the width of the frequency
filters increases, the purity converges quickly, the case of ∆λs = ∆λi = 10 nm
is almost equivalent to the case without frequency filters ∆λs, ∆λi → ∞.
In order to compare the theoretical results of this section with previous
experimental work, figure 2.4 shows the purity of the spatial state as a function
of the collection mode for three reported experiments. Figure 2.4 (a) shows the
purity for the data reported in reference [6]. The authors of that paper studied
a type-i spdc process in a 1 mm long liio3 crystal. They used a diode laser
with wavelength λp = 405 nm and bandwidth ∆λp = 0.4 nm as a pump beam.
And, by using monochromators with ∆λs,i = 0.2 nm, they detected pairs of
photons emitted at ϕs,i = 17◦ . Table 2.1 resumes these parameters.
The paper analyzes two pump waist values: wp = 30 µm and wp = 462 µm.
For those values, the authors demonstrated control of the frequency correlations
by changing the spatial properties of the pump beam. As the pump beam
influences the spatial shape of the generated photons, they implicitly reported
a correlation between space and frequency. This spatial to spectral mapping
exists at their collection mode ws = 133.48 µm for which the correlation indeed
appears according with our model.
Figure 2.4(b) shows T r[ρ2 q] for the case described in reference [46]. The
authors used a continuous wave pump beam with λp = 405 nm to generate
photons with λs,i = 810 nm in a 1.5 mm length bbo crystal. They detected the
photons after filtering with ∆λs,i = 10 nm interference filters. The pump beam
waist satisfied the condition wp >> L, and the generated photons propagated
at ϕs,i = 0◦ .
The experiment reported in reference [46] does not show evidence of spacefrequency
correlations. In agreement with this fact, the figure shows a lack
of correlation for the collinear case. However, under the conditions reported,
but in a non collinear configuration, it is not possible to neglect the space and
frequency correlations.
Finally figure 2.4 (c) plots T r[ˆρ 2 q] for the case described in reference [31].
In this paper, the authors illuminated a 2 mm long bbo crystal with a pump
beam with λp = 351.1 nm and wp ≈ 20 µm. Their crystal generated photons
at λs,i = 702 nm that propagated at 4o . They collected the photons after using
21

2. Correlations and entanglement
Filters
0nm
1nm
10nm
→∞
0.1 Collection mode 1mm
Spatial
purity
Figure 2.3: The spatial purity increases by filtering the state in space or frequency.
For the parameters chosen here, the bandwidth of the generated photons is smaller
than 10 nm, making 10 nm filters equivalent to infinitely broad ones. The parameters
used to generate this figure are listed in table 2.1.
Spatial
purity
1
0
0.1
Pump
waist
30m 462m Emission
angle
1mm 0.1 Collection mode 1mm
1
0.2
1º
5º
(a) (b) (c)
0º
0.1 1mm
Pump
waist
20m 500m Figure 2.4: For the values reported in reference [6], figure (a) shows that the purity
is smaller than 1. This fact is in agreement with the implicit spatio-temporal
correlations reported in the reference. For the values reported in references [46] and
[31], figures (b) and (c) show that the purity has its maximum value. There was no
evidence of spatio-temporal correlations in the results reported in these references.
In all three cases, it is possible to modify the value of the purity by tailoring the spdc
parameters. The parameters used to generate this figure are listed in table 2.1.
10 nm interference filters.
For the noncollinear configuration in reference [31], the highly focused pump
beam is responsible for the lack of correlations. The figure shows that for lessfocused
pump beams the purity decreases as the correlations between space
and frequency become more important.
The correlations between frequency and space in the two-photon state discussed
in this section, suggest that a complete description of the correlations
between the generated photons, should take into account both degrees of freedom,
as the next section discusses.
22

2.3. Correlations between signal and idler
Table 2.1: The parameters used in figures 2.3 and 2.4
Parameter Figure 2.3 Figure 2.4(a) Figure 2.4(b) Figure 2.4(c)
Crystal liio3 liio3 bbo bbo
L 1 mm 1 mm 1.5 mm 2 mm
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦
T0 → ∞ → ∞ → ∞ → ∞
wp 400 µm 30 / 462 µm wp ≫ L 20, 500 µm
λp 405 nm 405 nm 405 nm 351.1 nm
∆λp 0 0.4 nm
λ 0 s 810 nm 810 nm 810 nm 702 nm
∆λs 0.2 nm 10 nm 10 nm
ϕs 10 ◦ 17 ◦ 0, 1, 5 ◦ 4 ◦
2.3 Correlations between signal and idler
In contrast to the previous section, here the two-photon system is considered
as composed by two photons, each described by space and time variables, as
figure 2.1 (b) shows. Since the global state in equation 1.9 is pure, the purity
of the signal photon calculated in this section is a measurement of the degree of
spatiotemporal entanglement between signal and idler photons, as in references
[5, 10, 11, 47, 48].
spdc as a source of two-photon states can be used to generate pairs of
photons maximally entangled or two single photons [5, 49]. The purity of the
signal photon in the first case should be equal to 0 and in the second case equal
to 1. This section explores the necessary conditions to be in each regime.
This section is divided in three parts, the first one discusses the origin of
the correlations and the strategies to suppress them. The second part describes
how to obtain an analytical expression for the signal photon purity. And the
third part shows the results of numerical calculation of that purity.
2.3.1 Origin of the correlations between photons
The presence of cross terms in variables of both signal and idler in equation
1.34 indicates correlations between the generated photons. Figure 2.5 shows
those terms: i, j, l, m, n, r, t, v, and z.
In contrast to the last section, filtering one degree of freedom is not enough
to suppress the correlations between signal and idler. For example, by using a
frequency filter, the photons can be still correlated in space. The total suppression
of the correlation requires infinite narrow filters in both degrees of freedom
for one of the photons, as shown experimentally in reference [50]. But that kind
of filtering reduces the number of available pairs of photons substantially.
The other possibility to generate uncorrelated photons is to tailor the parameters
of the spdc process to make the cross terms negligible simultaneously.
According with appendix A, when α = 0 ◦ , this condition is equivalent to mak-
23

2. Correlations and entanglement
q s x
q s y
q i x
y
qi s
i
qs x
a
h
i
j
k
l
q s
y
h
b
m
n
p
r
qi y
i
m
c
s
t
u
Figure 2.5: The signal-idler cross terms in matrix A are responsible for the correlations
between the generated photons. By suppressing these cross terms, and by making
permutations over columns and rows, the matrix becomes a two block matrix, where
each block contains the information of one photon.
ing the following terms negligible
i =w 2 p + γ 2 L 2 tan ρ0 2
j =γ 2 L 2 sin ϕi tan ρ0
l =γ 2 L 2 tan ρ0(Np − Ni cos ϕi)
m = − γ 2 L 2 sin ϕs tan ρ0
q i
x
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi
t =γ 2 L 2 tan ρ0(Np − Ns cos ϕs)
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs
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
j
n
s
d
v
w
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi. (2.15)
As an example, a configuration with a negligible Poynting walk-off fulfills this
condition always when
cos ϕs = Np
2Ns
cos ϕi = Np
2Ni
w 2 p
γ 2 L 2 = tan ϕs tan ϕi
T 2 0 =N 2 p γ 2 L 2
wp ≪ws, wi
s
k
p
t
v
f
z
i
l
r
u
w
tan ϕs 2 tan ϕi 2 − 1
4
z
g
(2.16)
When the correlations are totally suppressed by satisfying these or other conditions,
the two-photon state is separable, and each photon is in a pure state.
Therefore, the two-photon mode function can be written as a product of two
mode functions, one for each photon,
24
Φ(q s, ωs, q i, ωi) = Φs(q s, Ωs)Φi(q i, Ωi). (2.17)

2.3. Correlations between signal and idler
In any other case, correlations between signal and idler are unavoidable. To
evaluate their strength, the next section deduces an analytical expression for
the purity of the signal photon state.
2.3.2 Spatio temporal state of signal photon
A single photon state can be generated in spdc by ignoring any information
about one of the generated photons. The single photon state is calculated by
tracing out the other photon from the two-photon state. For example, after
a partial trace over the idler photon, the reduced density matrix in space and
frequency for the signal photon is
ˆρsignal =T ridler[ρ]
=
=
and its purity is given by
dq ′′
i dΩ ′′
i 〈q ′′
i , Ω ′′
i |ˆρ|q ′′
i , Ω ′′
i 〉
dqsdΩsdqidΩidq ′ sdΩ ′ s
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)|qs, Ωs〉〈q ′ s, Ω ′ s|, (2.18)
T r[ˆρ 2 signal] =
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (2.19)
Recalling the exponential character of the mode function described by equation
1.35, equation 2.19 writes
T r[ˆρ 2 signal] = det(2A)
det(C) , (2.20)
where C is a positive-definite real 12 × 12 matrix defined by
N 4
exp − 1
2 Xt
CX = Φ(qs, Ωs, qi, Ωi)
and given by
⎛
C = 1
2
⎜
⎝
× 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)
2a 2h i j 2k l 0 0 i j 0 l
2h 2b m n 2p r 0 0 m n 0 r
i m 2c 2s t 2u i m 0 0 t 0
j n 2s 2d v 2w j n 0 0 v 0
2k 2p t v 2f z 0 0 t v 0 z
l r 2u 2w z 2g l r 0 0 z 0
0 0 i j 0 l 2a 2h i j 2k l
0 0 m n 0 r 2h 2b m n 2p r
i m 0 0 t 0 i m 2c 2s t 2u
j n 0 0 v 0 j n 2s 2d v 2w
0 0 t v 0 z 2k 2p t v 2f z
l r 0 0 z 0 l r 2u 2w z 2g
⎞
⎟ .
⎟
⎠
(2.22)
25

2. Correlations and entanglement
Signal
purity Filters
1
0
0.1 Collection mode 1mm
0nm
1nm
10nm
→∞
Figure 2.6: The correlations between photons can not be suppressed by filtering
in only one degree of freedom. The purity of the signal state is only maximum
when using ultra-narrow spatial and frequency filters. In this case, the 1 nm filters
are broad enough to allow the correlations, and the 10 nm filters are equivalent to
infinitely broad ones. The parameters used to generate this figure are listed in table
2.2.
The differences between the matrix C and matrix B in the last section, are
due to the order of the primed and unprimed variables in the arguments of the
mode functions in equations 2.14 and 2.21.
The next part of this section uses equation 2.20, to calculate the values of
the purity of the signal photon for different spdc configurations.
2.3.3 Numerical calculations
Figure 2.6 shows the space-frequency purity of the signal photon as a function
of the spatial filter width for different values of the frequency filter width.
Consider a degenerate type-i spdc configuration, where a pump beam with
wavelength λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium
iodate (liio3) crystal with length L = 1 mm, and negligible Poynting vector
walk-off (ρ0 = 0). The crystal emits signal and idler photons with a wavelength
λ 0 s = λ 0 i = 810 nm at an angle ϕs,i = 10 ◦ , and the collection modes for signal
and idler are assumed to be equal (ws = wi). All these parameters are listed
in the first column of table 2.2.
As was discussed in the first part of this section, it is possible to achieve
maximal separability between the photons by using infinitely narrow filters
in both space and frequency. In the region of small values of ws (= wi) a
considerable correlation between signal and idler exists even in the case of
infinitely narrow frequency filters. Different values for the signal photon purity
can be achieved by changing the filter width, as shown by figure 2.6.
Additionally, the figure shows how the purity T r[ˆρ 2 signal ] is confined between
the values obtained for ∆λs = ∆λi → 0 nm and ∆λs, ∆λi → ∞. This limits
can be tailored by modifying the other parameters of the spdc configuration.
Figure 2.7 shows T r[ˆρ 2 signal ] as a function of the pump beam waist wp for
26

Signal
purity
1
0
(a)
0.1 3
2.3. Correlations between signal and idler
(b)
0.1 Pump waist 3
(c)
0.1
Emission
angle
Figure 2.7: The effect of the emission angle on the signal purity changes depending
on the filters used in space and frequency. When both filters have a finite size like in
(a), the purity increases with the angle of emission, and has a maximum for a fixed
value of the pump beam waist. When the frequency filters are infinitely narrow like
in (b), the dependence of the purity on the angle of emission disappears. When using
narrow filters in space, as in (c), the purity is equal to 1 for a certain pump beam
waist that changes with the angle of emission. The parameters used to generate this
figure are listed in table 2.2.
Table 2.2: The parameters used in figures 2.6 and 2.7
Parameter Figure 2.6 Figure 2.7(a) Figure 2.7(b) Figure 2.7(c)
Crystal liio3 liio3 liio3 liio3
L 1 mm 1 mm 1 mm 1 mm
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦
T0 → ∞ → ∞ → ∞ → ∞
wp 400 µm 400 µm 400 µm 400 µm
λp 405 nm 405 nm 405 nm 405 nm
∆λp → 0 → 0 → 0 → 0
ws variable → ∞ 400 µm 400 µm
λ 0 s 810 nm 810 nm 810 nm 810 nm
∆λs 0, 1, 10 nm → ∞ 10 nm → 0 10 nm
ϕs 10 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦
various values of the emission angle ϕs = ϕi, in degenerate type-i spdc configurations
described by the parameters listed in the second, third and fourth
columns of table 2.2.
Figure 2.7 (a) shows the case of finite spatial and temporal filters. The
purity of the signal photon is always smaller than one, increases with the emission
angle, and has a maximum for a given value of the pump beam waist. For
a very narrow band frequency filter with a ws = wi = 400 µm spatial filter,
figure 2.7 (b) shows how the correlation between the photons is minimal for
small pump beams. The emission angle for this particular crystal length is
irrelevant. Finally, figure 2.7 (c) considers a frequency filter ∆λs = ∆λi = 10
nm with infinitely narrow spatial filters. In this case, maximal purity appears
for each emission angle at a particular value of the pump beam waist.
Conclusion
Two photons with two degrees of freedom compose the two-photon state generated
in spdc. The correlations between all four elements affect the state of
single photons or two-photon states with one degree of freedom. Using the pu-
3mm
5º
10º
20º
27

2. Correlations and entanglement
rity as correlation indicator it is possible to determine the conditions in which
the specific correlations are suppressed, without requiring the use of infinitely
narrow filters.
The next chapter continues with the description of the correlations between
photons, in the case where only their spatial state is relevant. The chapter describes
the spatial correlations in terms of orbital angular momentum transfer.
28

CHAPTER 3
Spatial correlations and
orbital angular momentum
transfer
The previous chapter describes the correlations between the signal and idler
photons considering space and frequency, as well as the correlations between
those degrees of freedom. This chapter focuses on the cases where the degrees
of freedom are not correlated, and the only relevant correlations are
those between the photons in the spatial degree of freedom. To characterize
those correlations I use the orbital angular momentum (oam) content of
the generated photons, which is directly associated to their spatial distribution.
The spatial correlations between the photons determine the mechanism
of oam transfer from the pump to the signal and idler. Most studies of the
oam correlations in spdc can be divided in two categories: those reporting a
full transference of the pump’s oam [7, 8, 22, 27, 28, 29], and those reporting
a partial transference [30, 31, 32, 33]. This chapter describes the oam transfer
mechanism, and shows the difference between both regimes. The chapter is
divided in three sections. Section 3.1 introduces the eigenstates of the oam
operator: the Laguerre-Gaussian modes. This section shows how to calculate
and measure the oam content of the photons. Section 3.2 describes how the
oam is transferred in spdc. As an example, section 3.3 shows how this transfer
happens in the collinear case, where all the emitted photons propagate in the
same direction. The main result of this chapter is a selection rule that summarizes
the oam transfer mechanism: the oam carried by the pump is completely
transferred to all the signal and idler photons emitted over the cone.
29

3. Spatial correlations and OAM transfer
-1
0 1 2
OAM content
in ħ units
Phase front
Intensity
profile
Figure 3.1: The Laguerre-Gaussian modes are characterized by their phase front
distribution and intensity profile. The figure shows the phase fronts and the intensity
profiles for the modes with oam contents from −1 to 2.
3.1 Laguerre-Gaussian modes and OAM content
In an analogous way to the decomposition of an electromagnetic field as a
series of planes waves, it is possible to decompose the field in other bases. For
instance, paraxial fields can be decomposed as a sum of Laguerre-Gaussian (lg)
modes. This basis is especially convenient since the lg modes are eigenstates of
the orbital angular momentum (oam) operator. That is, the state of a photon
in a lg mode has a well defined oam value [51]. This section describes the
properties of the lg modes, and shows how to calculate the weight of each
mode in a given decomposition.
Photons carrying oam different from zero have at least one phase singularity
(or vortex) in the electromagnetic field, a region in the wavefront where the
intensity vanishes. This is precisely one of the most distinctive characteristics
of Lagurre-Gaussian modes as figure 3.1 shows. Each lg mode is defined by
the number p of non-axial vortices, and the number l of 2π-phase shifts along
a close path around the beam center. The index l also describes the helical
structure of the phase front around the singularity, and more important here,
l determines the orbital angular momentum carried by the photon in units.
The state of a single photon in a lg mode is
|lp〉 = dqLGlp(q)â † (q)|0〉, (3.1)
where the mode function lglp(q) is given by Laguerre-Gaussian polynomials
1
2 wp!
LGlp(q) =
2π(|l| + p)!
|l|
wq
√2 L |l|
2 2 w q
p
2
× exp − w2q2
exp ilθ + iπ(p −
4
|l|
2 )
(3.2)
as a function of the beam waist w, the modulus q and the phase θ of the
transversal vector, and the associated Laguerre polynomials L |l|
p defined as
30
L |l|
p [x] =
p
i=0
i
l + p (−x)
. (3.3)
p − i i!

3.2. OAM transfer in general SPDC configurations
A special case of the lg modes is the zero-order mode that does not carry oam.
Since the zero-order Laguerre polynomial L 0 0 = 1, the zero-order Laguerre-
Gaussian lg00 = 1 is the Gaussian mode given by
w
1
2
LG00(q) = exp
2π
− w2 q 2
4
. (3.4)
This is not, however, the only mode with oam equal to zero, the same is true
for all other modes lg0p. Spiral harmonics are defined to collect all the modes
with the same oam value, regardless of the value of p, these modes are defined
as
LGl(q) =
LGlp(q) (3.5)
p
=al(q) exp [ilθ].
The phase dependence on l, shown by each spiral harmonic mode, is exploited
to determine the photon’s oam content [52]. Consider, for instance, a photon
with a spatial distribution given by Φ(q), which using the spiral harmonic
modes can be written as
∞
Φ(q) = al(q) exp [ilθ], (3.6)
l=−∞
so that each mode in the decomposition has a well defined oam of l per
photon. Therefore, the probability Cl of having a photon with oam equal to l
is the weight of the corresponding mode in the distribution:
where
Cl =
al(q) = 1
√ 2
∞
0
2π
0
dq|al(q)| 2 q (3.7)
dθΦ(q, θ) exp (−ilθ). (3.8)
If the photon has a well defined oam of l0, the weight of the corresponding
mode Cl0 = 1 and the weights of the other modes Cl=l 0 = 0. If Cl = 0 for
different values of l, the photon state is a superposition of those modes with
different oam values. Compared to the relative simplicity of these calculations,
measuring the oam content is a more complicated task described in appendix
C.
Now that the techniques for the calculation of the oam content are introduced,
the next section will use the spherical harmonics and the oam decomposition
to study the transfer of oam from the pump photon to the signal and
idler.
3.2 OAM transfer in general SPDC configurations
The oam carried by a photon is directly associated to its spatial shape. In
order to simplify the description of the oam transfer mechanisms, this section
considers only the spatial part of the mode function in a spdc configuration
31

3. Spatial correlations and OAM transfer
in which the space and the frequency are not correlated. By writing the mode
function in a different coordinate system, it will be possible to show that a
selection rule exists for the oam transfer in spdc.
According to equation 1.6, all the fields have been defined in terms of a
coordinate system related with their propagation direction. To study the oam
transfer, we start by defining the fields in terms of a unique coordinate system,
the one of the pump. This coordinate transformation is possible since there is
a vector Kn such that kn · rn = Kn · rp.
This coordinate transformation is more general than the one of section 1.3,
where the plane that contained all photons was defined as the yz plane. As
a consequence this section considers all possible emission directions over the
cone, while section 1.3 considered only one.
As was done for the vector kn in section 1.3, it is useful to separate the
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
function in the semiclassical approximation can be written as
∆kL
Φq(Qs, Qi) ∝ Ep(Qs + Qi)sinc
(3.9)
2
where the delta factor is given by
with
∆k = K z p(Qs + Qi) − K z s (Qi) − K z i (Qi). (3.10)
K z j (Q) =
ω 2 j n2 j
c 2 − |Q|2 . (3.11)
To study the oam transfer, consider the two functions in equation 3.9 separately.
Consider first the sinc function. Due to its dependence on the delta
factor, the sinc function depends only on the magnitudes |Qs +Qi|, Qs = |Qs|,
Qi = |Qi|. And, since
|Qs + Qi| 2 = Q 2 s + Q 2 i + 2QsQi cos(Θs − Θi), (3.12)
the only angular dependence of the function is through the periodical term
cos(Θs − Θi), therefore, using Fourier analysis it can be written in a very
general form as
∆kL
sinc =
2
∞
m=−∞
Fm(Qs, Qi) exp [im(Θs − Θi)], (3.13)
where, importantly, Fm(Qs, Qi) does not depend on the angles Θs and Θi.
On the other hand, if we consider the pump beam as a Laguerre-Gaussian
beam with a oam content of lp per photon, its mode function can be written
as
32
Ep(Qs + Qi) ∝ exp
− w2 p|Qs + Qi| 2
4
(Q x s + Q x i ) + i(Q y s + Q y
i )
l , (3.14)

3.2. OAM transfer in general SPDC configurations
where the identity Q exp[iθ] = Q x + iQ y was used. Using equation 3.12, the
last equation can be written as
Ep(Qs + Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
4
l × Qs exp (iΘs) + Qi exp (iΘi)
and by using the binomial theorem it becomes
(3.15)
lp
lp
Ep(Qs + Qi) ∝ Q
l
l=0
l sQ lp−l
i exp [ilΘs + ilpΘi − ilΘi]
× exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
. (3.16)
4
By replacing equations 3.16 and 3.13 into equation 3.9, the two-photon spatial
mode function becomes
Φ(Qs, Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
4
∞
× Gn(Qs, Qi) exp [inΘs + i(lp − n)Θi], (3.17)
n=−∞
where the value of Gn(Qs, Qi) does not depend on Θs and Θi. The index
(lp − n) associated to Θi is fixed for each value of lp and ls, according to the
phase dependence of the last equation. This relation can be written as the
selection rule
lp = ls + li. (3.18)
Therefore, the angular momentum carried by the pump is completely transferred
to the signal and idler photons.
The geometry of the spdc configuration restricts the validity of the selection
rule since it was deduced from equation 3.9, and that equation describes
only the configuration in which all possible emission directions are taken into
account. Collinear configurations achieve this condition by definition, but in
the type-I noncollinear configurations, all the experiments up to now have measured
only a certain portion of the whole cone. Under this condition the transfer
of oam from the pump to the signal and idler is not governed necessarily by
equation 3.18, in agreement with previous reports [32, 30, 31].
Using numerical methods to calculate the oam content of the downconverted
photons, the next section explores the oam transfer mechanisms in
collinear configurations. This configurations are special cases where the generated
photons propagate in the same direction and therefore, the detection
system detects all photons from the emission cone.
33

3. Spatial correlations and OAM transfer
Configuration
Weight
1
0
-4 4
0
Spatial
-4 4
distribution OAM content
1
Mode
Figure 3.2: In a collinear configuration a pump beam with oam equal to 1 transfers
its oam into the signal photon when the idler is projected into a Gaussian mode. The
figure shows the pump in blue and the signal and idler in pink. Additionally, it shows
the spatial distribution and the oam content of the signal photon for co-propagating
and counter-propagating configurations.
3.3 OAM transfer in collinear configurations
The aperture of the down-conversion cone is given by the angle of emission
of the photons. In the collinear configuration this angle is zero, and therefore
the cone collapses onto its axis, as shown in figure 1.2 (d). As each photon
propagates in the same direction as all the other photons, equation 3.18 is
valid.
To analyze the transfer of oam from the pump beam to the generated photons,
we will calculate the signal oam content, assuming a lg1 pump beam so
that lp = 1, and an idler photon projected into a Gaussian state li = 0. We
will consider two kinds of collinear configurations: co-propagating and counterpropagating,
both shown in figure 3.2. The collinear counter-propagating generation
of photons requires fulfilling a special phase-matching condition, which
can be achieved by quasi-phase-matching in a periodic structure. Such counterpropagating
generation has been demonstrated in second harmonic generation
[53], parametric fluorescence [54], and spdc in fibers [55]. Chapter 5 explores
another way to generate counter-propagating two-photon states: Raman scattering.
Figure 3.2 shows the calculated signal spatial distribution and oam content
in two cases. The first row shows the collinear co-propagating case. The second
row shows the collinear counterpropagating case.
According to figure 3.2, the collinear case fulfills the selection rule, except
for a minus sign in the counterpropagating case. The phase appears because
the photon’s oam content is evaluated with respect to the pump propagation
direction, as figure 3.3 shows. To take this effect into account equation 3.18
34

Phase front
3.3. OAM transfer in collinear configurations
OAM respect to z
direction
Figure 3.3: The direction of rotation of the phase shift defines the sign of the mode’s
oam content. For a Laguerre-Gaussian mode corresponding to positive oam content,
the phase shift rotates clockwise with respect to the propagation direction. If the
same mode propagates backwards, the phase shift rotates anticlockwise with respect
to the original propagation direction, and the mode has a negative oam content.
can be generalized to
lp = dsls + d1li
+2
-2
-2
(3.19)
where ds,i = ±1 corresponds to forward and backward propagation of the
corresponding photon.
Conclusion
The oam content of the pump is completely transferred to the signal and idler
photons, emitted all over a cone. This makes it possible to introduce a selection
rule that always holds if all possible emission directions are considered, but not
necessarily if only a small part is detected. This result clarifies the apparent
contradiction between several works that support the selection rule [7, 28, 56],
and those that report that it is not valid [32, 31, 57].
When only a portion of all generated photons are considered there is no
simple relationship between lp, ls and li. The next chapter studies the oam
transfer in those cases, describing the effect of different spdc parameters on
the amount of oam that is transferred to the subset of the photons considered.
35

CHAPTER 4
OAM transfer in
noncollinear configurations
The previous chapter describes the oam transfer from the pump to the signal
and idler photons considering all the possible emission directions of those
photons. However, in non collinear configurations, the photons detected are
just a subset of the total emission cone; noncollinear configurations generate
correlated photons, which are naturally spatially separated, which is exactly
what makes noncollinear so important. The change in the geometry of the
process, imposed by the detection system, has a strong effect on the oam of
the detected photons. This chapter focuses on the description of the oam that
is transferred to a small portion of the emitted photons in noncollinear configurations.
To characterize the transfer, I study a particular spdc process in
which the pump is a Gaussian beam and one of the photons is projected into a
Gaussian mode. Therefore, the oam content of the other photon will indicate
how close the configuration is to satisfying the oam selection rule. This chapter
is divided in three sections. Section 4.1 shows a simple example in which the
selection rule does not apply as just a small section of the cone is considered.
In a more general scenario, section 4.2 shows that the violation of the selection
rule is not only mediated by the emission angle, but also by the pump beam
waist. Finally, section 4.3 shows the effect of the Poynting vector walk-off on
the oam of the noncollinear photons. As a main result, this chapter explains
how the pump beam waist and the Poynting vector walk-off affect the oam
transfer, in the cases where the selection rule does not apply. By tailoring both
parameters it is possible to generate photons with specific spatial shapes. For
instance, it is possible to generate photons in Gaussian modes that are more
efficiently coupled into single mode fibers.
37

4. OAM transfer in noncollinear configurations
4.1 Ellipticity in noncollinear configurations
In the experimental implementation of spdc, the detection system selects the
photons that are emitted in a certain spatial direction. In a collinear configuration,
the selection is not a problem since all the photons are emitted in the
same direction. But, in a noncollinear configuration, the photons are emitted in
different azimuthal directions described by the angle α, therefore the detection
system selects only a section of the full cone. As the symmetry is broken when
only a portion of the cone is considered, the oam transfer mechanism can not
be described by the selection rule in 3.18. This section studies this mechanism
in a simple spdc configuration by calculating the spatial distribution of the
signal photon after fixing the oam content of the pump and idler photons.
As a first example, consider a Gaussian pump beam lp = 0 and an idler
photon projected into a Gaussian mode li = 0, given by
u(qi) = Ni exp − w2 i
4 (qx2 i + q y2
i )
, (4.1)
the spatial distribution of the signal photon is given by the normalized mode
function
Φs(qs) = Ns dqiΦq(qs, qi)u(qi). (4.2)
This integral has an analytical solution for simple spdc configurations. Consider
a degenerate spdc process with negligible Poynting vector walk-off. After
suppressing the correlations between space and frequency the spatial part of
the two-photon state is given by
with
Φq(qs, qi) = exp
− w2 p
4 ∆2 0 − w2 p
4 ∆2 1
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi
∆k = − q y s sin ϕs + q y
i
sin ϕi.
L
sinc
2 ∆k
(4.3)
(4.4)
Therefore, using the sinc to exponential approximation, the signal mode function
defined by equation 4.2 is given by
Φs(qs) =Ns exp
× exp
−
w2 pw2 i
4(w2 p + w2 qx2 s
i )2
− 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
4(w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs 2 + w 2 i )
q y2
s
(4.5)
When the coefficients of the variables q x s and q y s are equal, the signal mode
function reduces to a Gaussian and the selection rule is fulfilled. These coefficients
are equal in collinear configurations, or in noncollinear configurations in
38
.

4.2. Effect of the pump beam waist on the OAM transfer
which the spdc parameters satisfy the relationship
γ 2 L 2 w
=
2 pw4 i
w4 i + 4w2 p(w2 i + w2 p) cos ϕ2 s
. (4.6)
In any other case, the coefficients of the variables q x s and q y s are different, and
the signal mode function becomes elliptical. The ellipticity of the spatial profile
implies the presence of non-Gaussian modes, and therefore it can be used as a
qualitative probe that the selection rule is not fulfilled (lp = ls + li).
As the ellipticity is an effect of the partial detection, it can be controlled
by changing the total shape of the cone, or by changing the detector angular
acceptance. The phase matching conditions define the shape of the cone as a
function of the emission angle ϕs, the pump beam waist wp, and the length of
the crystal L; the detector angular acceptance is a function of the waist of the
spatial modes ws, wi. The next two sections use both the mode decomposition
and the ellipticity to describe the effect of all these parameters on the signal
oam content in a more general scenario than considered in this section.
4.2 Effect of the pump beam waist on the OAM transfer
Of all the spdc parameters that affect the signal ellipticity, the easiest to
control is the pump beam waist. To change the angle of emission or the crystal
length implies changing of the geometrical configuration. While changing the
pump beam waist just requires adding lenses in the beam path. In the first
part of this section, numerical calculations show the role of the pump beam
waist on the signal oam content. The second part describes the experimental
corroboration of this effect.
4.2.1 Theoretical calculations
Consider a spdc configuration as the one in equation 1.31. With a Gaussian
pump beam and an idler photon projected into a Gaussian mode, the oam content
of the signal photon can be used to describe the oam transfer mechanism
in spdc. The selection rule is fullfilled only if ls = 0. Equivalently, one could
choose to use the idler photon to study the oam transfer after projecting the
signal into a Gaussian mode.
In a degenerate type-i spdc process characterized by the parameters in
table 4.1, a Gaussian pump beam, with wavelength λ 0 p = 405 nm illuminates
a 10 mm ppktp crystal. The crystal emits signal and idler photons with a
wavelength λ 0 s = λ 0 i = 810 nm. Both photons propagate at ϕs,i = 1 ◦ , after
the crystal they traverse a 2f system, and finally the idler photon is projected
into a Gaussian mode with wi → ∞, so that only idler photons with qi = 0
are considered.
Figure 4.1 shows the signal oam content for two values of the pump beam
waist: wp = 100 µm in the left and wp = 1000 µm in the right. In the
distributions, each bar represents a mode ls, with a weight in the distribution
Cls given by the height of the bar. In the left part of the figure, where the
pump beam waist is smaller, the distribution shows several modes. For larger
waists, the Gaussian mode becomes the only important mode, as seen in the
right part of the figure.
39

4. OAM transfer in noncollinear configurations
Weight
1
0
(a) (b)
-4 Mode 4 -4 Mode 4
Figure 4.1: As the pump beam waist increases, the Gaussian mode in the signal oam
distribution becomes more important. The left part of the figure, where wp = 100 µm,
shows several non-Gaussian modes in blue, while the right part, where wp = 1000 µm,
shows only a Gaussian mode in gray. The other parameters used to generate this
figure are listed in table 4.1.
Table 4.1: The parameters used in the theoretical calculations in section 4.2.
Parameter Value
Crystal ppktp
L 10 mm
ρ0
0◦ Laser cw diode
wp 100 − 1000 µm
λp
405 nm
λ0 s
ϕs
810 nm
1◦ Figure 4.2 shows the probability of finding a non-Gaussian mode as a function
of the pump beam waist. The presence of non-Gaussian modes implies a
violation of the selection rule in the configuration considered here. The figure
shows the probability Cls = 0 for emission angles 1, 5, 10 ◦ . As the collinear
angle ϕ becomes larger, the probability to detect signal photons with ls = 0
increases. In a highly noncollinear configuration a large pump waist does not
guarantee the generation of a Gaussian mode, that is, it does not imply the
validity of the selection rule.
Reference [30] introduced the noncollinear length defined as Lnc = wp/ sin ϕ.
Lnc quantifies the strength of the violation of the selection rule due to the angle
of emission and the pump beam waist. The authors defined this quantity
based on the fact that when Lq y sin ϕ is small the sinc function (that introduces
ellipticity) tends to one. As q y is of the order of 1/wp the condition can be
written as L sin ϕ/wp ≪ 1, or L ≪ Lnc. In this regime the ellipticity of the
mode function is small, and thus the selection rule lp = ls + li is fulfilled. This
is the case for nearly all the experiments using the oam of photons generated
in spdc [7, 12, 28, 56, 58, 25, 26, 29]. Instead, if the crystal length is larger
than or equal to the noncollinear length L ≥ Lnc, a strong violation of the
selection rule is expected. The experiments reported in references [31, 32, 57]
are in this regime since the authors used highly focused pump beams or long
crystals.
40

1.0
0.0
4.2. Effect of the pump beam waist on the OAM transfer
Nongaussian
mode probability
0.8
0.4
Emission
angle
10º
0.1 Pump beam width 1mm
Figure 4.2: The violation of the selection rule, given by the weight of the non-Gaussian
modes, becomes more important as the collinear angle increases, or as the pump beam
is more focalized. The parameters used to generate this figure are listed in table 4.1.
4.2.2 Experiment
Figure 4.3 depicts the experimental set-up that we used to measure the effect
of the pump beam waist over the signal photon spatial shape. A laser beam
with wavelength λ 0 p = 405 nm, and bandwidth ∆λp = 0.6 nm illuminated a
lithium iodate (liio3) crystal with length L = 5 mm. The crystal was cut in a
configuration for which non of the interacting waves exhibited Poynting vector
walk-off, so ρ0 = 0 ◦ . The signal and idler photons are emitted with wavelengths
λ 0 s = λ 0 i = 810 nm at ϕs,i = 17.1 ◦ . Table 4.2 summarizes the most relevant
experimental parameters.
A spatial filter after the laser provided an approximately Gaussian beam
with a waist of 500 µm. By adding lenses before the crystal, this waist could
be changed in the range of 32 − 500 µm.
Each generated photon passed through a 2 − f system (f = 250 mm),
that provided an image of the spatial shape of the two-photon state. Later,
the photons passed through a broadband colored filter, that removed the scattered
radiation at 405 nm, and through small pinholes to increase the spatial
resolution.
Multimode fibers mounted in xy translation stages collected the photons,
and carried them into two single photon counting modules. A coincidence
circuit counted the number of times that a signal photon was detected within
8 ns of the detection of an idler photon. We used a data acquisition card to
transfer the circuit’s output into a computer.
We measured the signal and idler counts, as well as the number of coincidences
for a fixed position of the idler in two orthogonal directions in the
transverse signal plane as shown by figure 4.3. During the experiments it was
checked that the use of narrow band filters of ∆λs = 10 nm does not modify
5º
1º
41

4. OAM transfer in noncollinear configurations
Laser and
spatial filter lenses
crystal
lenses
filter
collection
system
pinhole
x
idler
pinhole
signal
Figure 4.3: A spatially filtered cw laser is focused into a nonlinear crystal. A coincidence
circuit measures the correlations between the photon counts in a fixed position
for the idler photon with the counts of photons in two orthogonal directions in the
signal transverse plane. The values of the different experimental parameters are listed
in table 4.2.
the measured shape of the coincidence rate, although it does modify the single
counts spatial shape.
Table 4.2: The parameters of the experiment described in section 4.2.
Parameter Value
Crystal liio3
L 5 mm
ρ0
0◦ Laser cw diode
wp 32 − 500 µm
λp 405 nm
∆λp 0.6 µm
λ0 s
∆λ
810 nm
0 s
ϕs
10 nm
17.1◦ The left side of figure 4.4 shows the coincidence measurements in the x and
y directions for a focalized pump beam. As the pump beam waist is wp = 32
µm, the noncollinear length Lnc = 108.8 µm is much smaller than the crystal
length. According with reference [30], in this regime the ellipticity of the signal
photon should be appreciable.
By fitting the result to a Gaussian function, the waist in the x direction
is wx 960 µm, while in the y direction it is wy 150 µm. The waist in x
is about six times larger than the waist in y, and therefore the signal spatial
distribution is elliptical. As a reference, the figure shows the singles counts for
the signal coupler.
The right side of the figure shows the coincidence measurements for a larger
pump beam, wp 500 µm. In this case, the noncollinear length Lnc = 1.7 mm
is of the same order of magnitude as the crystal length. When fitting the
42
y

Coincidences
800
0
-4
4.2. Effect of the pump beam waist on the OAM transfer
x
y
4mm
Singles
6x10 5
~
-1
y x
Singles
5x10 4
~
1mm
Figure 4.4: The ellipticity decreases as the pump beam waist increases from wp = 30
µm in the left to 500 µm in the right image. As the pump beam waist affects
the efficiency of the process, the coincidences on the left were collected over 600
seconds and the ones in the right over 20 seconds. Solid lines are the best fit to the
experimental data shown as points. The values of the experimental parameters are
listed in table 4.2.
Coincidences
region width
2.5m
0.5
0
0
x dimension
y dimension
L nc=1.1mm
Pump width 330 500m Figure 4.5: The pump beam waist controls the width of the profile in the x direction,
while it does not affect the width in the y direction. The width in both directions
becomes similar when the noncollinear length is of the same order of magnitude as
the crystal length. The vertical line shows the point where, according to the fits, both
dimensions are equal. In this point the noncollinear length is 1.1 mm. The values of
the experimental parameters are listed in table 4.2.
transverse cuts to Gaussian functions, the waist in the x direction is wx 120
µm and the waist in the y direction is wy 180 µm. The waist in the x
direction is much smaller than before and of the same order as the waist in y
which implies a decrease of the ellipticity.
Figure 4.4 illustrates the changes in the width in the x and y directions as a
function of pump beam waist. In the range where the waist varies from 32−500
µm, the width in y remains almost constant while the width in x decreases by
80%. After a certain threshold the width in x becomes stable at a value close
43

4. OAM transfer in noncollinear configurations
100µ m
1mm
pump crystal
-4 4
-4 4
1
0
1
0
after 1mm
-4
-4
output mode
Figure 4.6: Due to the crystal birefringence, the oam content of a beam changes as
the beam passes trough the material. The number of new oam modes introduced
by the birefringence increases for more focused beams. The figure shows the mode
content of a Gaussian beam after traveling 1 mm in the crystal, and at the output of
the 5- mm-long crystal.
to the value of the width in y, and therefore the ellipticity disappears.
4.3 Effect of the Poynting vector walk-off on the OAM
transfer
Up to now, this chapter only considered spdc configurations where the Poynting
vector walk-off was not relevant. However, since the selective detection of
one section of the cone affects the oam transfer, the distinguishability introduced
by the walk-off should affect it as well.
To understand the effect of the walk-off on the oam transfer, consider that
the spatial shape of the pump is modified as it passes trough the crystal due to
the birefringence. A Gaussian photon thus acquires more modes when traveling
in a crystal. This section explains this effect using theoretical calculations and
experimental results.
4.3.1 Theoretical calculations
The displacement introduced by the walk-off, explained in section 1.3, changes
the pump beam spatial distribution and therefore its oam content. The transverse
profile of the pump, at each position z inside the nonlinear crystal, can
be written as
Ep (qp, z) = E0 exp
−q 2 p
w 2 p
4
z
+ i
2k0
∞
Jn (zqp tan ρ0) exp {inθp}
p n=−∞
(4.7)
where Jn are Bessel functions of the first kind. Based on this expression, figure
4.6 shows the oam content of a Gaussian pump beam after traveling through
a 5 mm crystal. New modes appear and become important as the pump beam
gets narrower.
The oam content of the pump with respect to z is different at different
positions inside the crystal. Pairs of photons produced at the beginning or at
the end of the crystal are effectively generated by a pump beam with different
spatial properties, as figure 4.6 shows. This change in the pump beam is
44

Weight
1
0
4.3. Effect of the Poynting vector walk-off on the OAM transfer
0º 90º Azimuthal angle 360º
other
modes
Gaussian
mode
0º 90º Azimuthal angle 360º
Gaussian
mode
other
modes
Figure 4.7: The probability of generating a Gaussian signal photon varies with the
azimuthal angle, and has a maximum at α = 90 ◦ where the noncollinearity effect
compensates the Poynting vector walk-off. At other angles, like α = 360 ◦ the probability
of generating a Gaussian signal decreases and other modes become important
in the distribution. Those non-Gaussian modes are more numerous for more focused
beams, as seen by comparing the left part of the figure, where wp = 100 µm, to the
right part, where wp = 600 µm.
Signal
purity
0.3
0.1
0.0
without
walk-off
with
walk-off
azimuthal
0º 90º 180º 270º 360º angle
Figure 4.8: Like the signal oam content, the correlations between the photons change
with the azimuthal angle. The walk-off not only introduces an azimuthal variation
but increases the correlations.
translated into a change in the oam content of the generated pair with respect
to the z axis.
Additionally, because of the walk-off the generated photons are not symmetric
with respect to the displaced pump beam, and their azimuthal position
α becomes relevant. Figure 4.7 shows the weight of the mode ls = 0, and the
weight of all other oam modes, as a function of the angle α for two different
pump beam widths. In the left part, the pump beam waist is 100 µm, while
in the right part it is 600 µm. The oam correlations of the two-photon state
change over the down-conversion cone due to the azimuthal symmetry break-
45

4. OAM transfer in noncollinear configurations
ing induced by the spatial walk-off. The symmetry breaking implies that the
correlations between oam modes do not follow the relationship lp = ls + li.
Figure 4.7 also shows that for larger pump beams the azimuthal changes are
smoothened out.
The insets in figure 4.7 show the oam decomposition at α = 90 ◦ and at
α = 360 ◦ . At α = 90 ◦ , the walk-off effect compensates the noncollinear effect,
and the weight of the ls = 0 mode is larger than the weight of any other
mode. This angle is optimal for the generation of heralded single photons with
a Gaussian shape.
The degree of spatial entanglement between the photons also exhibits az-
imuthal variations depending on their emission direction. Figure 4.8 shows the
] as a function of the azimuthal angle α with and with-
signal purity T r[ρ2 signal
out walk-off, for a pump beam waist wp = 100 µm with collection modes of
ws = wi = 50 µm. When considering the walk-off, the degree of entanglement
increases and becomes a function of the azimuthal angle. The purity has a
minimum for α = 0◦ and α = 180◦ and maximum for α = 90◦ and α = 270◦ .
The spatial azimuthal dependence affects especially those experimental configurations
where photons from different parts of the cone are used, as in the
case in the experiment reported in reference [12] for the generation of photons
entangled in polarization. Using two identical type-i spdc crystals with the
optical axes rotated 90◦ with respect to each other, the authors generated a
space-frequency quantum state given by
|Ψ〉 = 1
√ dqsdqi Φ
2
1 q(qs, qi)|H〉s|H〉i + Φ 2
q(qs, qi)|V 〉s|V 〉i
(4.8)
where Φ 1 q(qs, qi) is the spatial mode function of the photons generated in the
first crystal
Φ 1 q(qs, qi) = Φq(qs, qi, α = 0 ◦ ) exp [iL tan ρ0(q y s + q y
i )], (4.9)
and Φ 2 q(qs, qi) is the spatial mode function of the photons generated in the
second crystal
Φ 2 q(qs, qi) = Φq(qs, qi, α = 90 ◦ ). (4.10)
Since the photons generated in the first crystal are affected by the walk-off as
they pass by the second crystal, the mode functions are different.
Following chapter 2, the polarization state of the generated photons is calculated
by tracing out the spatial variables from equation 4.8. The resulting
state is described by the density matrix
where
46
ˆρp = 1
|H〉s|H〉i〈H|s〈H|i + |V 〉s|V 〉i〈V |s〈V |i
2
+c|H〉s|H〉i〈V |s〈V |i + |V 〉s|V 〉i〈H|s〈H|i
(4.11)
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
Length
0.5mm
2mm
beam
waist
Concurrence
1
50 500m Figure 4.9: The concurrence of the polarization entangled state given by equation
4.8 decreases when the effect of the azimuthal variation is stronger, that is for small
pump beam waist or for large crystals.
The degree of correlation between space and polarization is given by the purity
of the polarization state
T r[ρ 2 p] =
0.5
0.0
1 + |c|2
. (4.13)
2
If the walk-off effect is negligible T r[ρ 2 p] = 1, then space and polarization are
not correlated. If the walk-off is not negligible, the azimuthal changes in the
spatial shape are correlated with the polarization of the photons.
Figure 4.9 indirectly shows the effect of the azimuthal spatial information
over the polarization entanglement. The entanglement between the photons
is quantified using the concurrence C = |c|, which is equal to 1 for maximally
entangled states. The figure shows the variation of C as a function of the pump
beam waist for two crystal lengths. For small pump beam waist, the azimuthal
effect is stronger and the concurrence decreases. As the waist increases, the
walk-off effect becomes weaker and the value of the concurrence increases. This
effect is more important for longer crystals, where the walk-off modifies the
pump beam shape more.
4.3.2 Experiment
To experimentally corroborate the predicted azimuthal changes in the signal
spatial shape we set up the experiment described by the parameters listed in
table 4.3 and shown in figure 4.10. We used a continuous wave diode laser
emitting at λp = 405 nm and with an approximate Gaussian spatial profile,
obtained by using a spatial filter. A half wave plate (hwp) controlled the
direction of polarization of the beam, while a lense focalized it on the input
face of the crystal, with a beam waist of wp = 136 µm. The 5 mm liio3 crystal
produced pairs of photons at 810 nm that propagated at ϕs,i = 4 ◦ . Due to
the crystal birefringence, the pump beam exhibited a Pointing vector walk-off
47

4. OAM transfer in noncollinear configurations
Laser and
spatial filter HWP lenses
crystal
lenses
filter
collection
system
camera
pinhole
Figure 4.10: By rotating the crystal and the pump beam polarization, it was possible
to measure the coincidences between pairs of photons at different positions over
the emission cone. The photons were collected using multimode fibers and 300 µm
pinholes to increase the resolution. The values of the experimental parameters are
listed in table 4.3.
given by the angle ρ0 = 4.9 ◦ , while the generated photons did not exhibit
spatial walk-off.
Table 4.3: The parameters of the experiment described in section 4.3.
Parameter Value
Crystal liio3
L 5 mm
ρ0 4.9◦ Laser cw diode
wp 136 µm
λp 405 nm
∆λp 0.4 nm
λ0 s
ϕs
810 nm
4◦ We measured the relative position of the pump beam in the xy plane at
the input and output faces of the nonlinear crystal using a ccd camera. At
the output of the crystal, the beam displacement due to the Poynting vector
walk-off allowed us to determine the direction of the crystal’s optical axis, and
to fix its direction parallel to the pump polarization, as shown in figure 4.11.
Right after the crystal, each of the generated photons passed through a
2 − f system with focal length f=50 cm. Cut-off spectral filters removed the
remaining pump beam radiation. After the filters, the photons were coupled
into multimode fibers. In order to increase the spatial resolution, we used small
pinholes with a diameter of 300 µm.
We kept the idler pinhole fixed and measured the coincidence rate while
mapping the signal photon’s transversal shape by scanning the detector with
a motorized xy translation stage, as in the right part of figure 4.3. Finally,
we rotated the crystal and the pump beam polarization to measure the spatial
correlations at different azimuthal positions on the down-conversion cone. After
48

Before
the crystal
4.3. Effect of the Poynting vector walk-off on the OAM transfer
After
the crystal
Polarization Optical axis
Figure 4.11: The images obtained with a ccd camera show that before the crystal,
the pump beam has a circular shape, and that after passing the crystal part of the
beam gets displaced in the direction of the optical axis of the crystal. The displaced
part corresponds to the portion of the beam polarized orthogonal to the crystal axis.
In the case of the last image almost all the beam is displaced. The values of the
experimental parameters are listed in table 4.3.
every rotation, the tilt of the crystal was adjusted to achieve the generation of
photons at the same noncollinear angle in all cases.
The bottom row of figure 4.12 presents a sample of images taken at different
azimuthal sections of the cone, while the upper row shows the expected shape
predicted by the theory. Each column shows the coincidence rate for different
angles, α = 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . Each point of these images corresponds to
the recording of a 10 seconds measurement. The typical maximum number of
coincidences is around 10 coincidences per second. Each image is 10 × 10 mm,
and its resolution is 50 × 50 points.
Theory
Experiment
0º 90º 180º 270º
Figure 4.12: The coincidence measurements of the mode function, as well as the
theoretical calculations, show that the ellipticity of the mode function changes for
different positions on the cone. The ellipticity is minimal for α = 90 ◦ . The values of
the experimental parameters are listed in table 4.3.
49

4. OAM transfer in noncollinear configurations
The different spatial shapes in figure 4.12 clearly show that the downconversion
cone does not posses azimuthal symmetry. As was predicted by the
theoretical calculations, the coincidence measurement for α = 90 ◦ , presents a
nearly Gaussian shape, while the other cases are highly elliptical.
The slight discrepancies between experimental data and theoretical predictions
observed might be due to the small (but not negligible) bandwidth of the
pump beam, and due to the fact that the resolution of our system is limited
by the detection pinhole size.
Conclusion
The spdc parameters and the detection system determine the portion of the
cone that is detected in a noncollinear configuration. This chapter explains how
the pump beam waist and the Poynting vector walk-off affect the oam transfer.
By tailoring both parameters it is possible to generate photons with specific
spatial shapes. The walk-off affects especially those configurations where pairs
of photons with different α are used.
The next chapter extends the analysis of the spatial correlations to pairs
of photons generated in Raman transitions. The chapter describes how the
specific characteristics of that source are translated into the two-photon spatial
state.
50

CHAPTER 5
Spatial correlations
in Raman transitions
Two-photon states can be generated in different nonlinear processes, and in every
case the oam transfer will depend on the particular configuration. Raman
transition is an alternative method for the generation of two-photon states.
Several authors have proven the generation of correlated photons in polarization
[59], frequency [60] and oam [23] via Raman transitions. Typical configurations
involve the partial detection of the generated photons [61, 62, 63] in
quasi-collinear configurations [64]. Just as in the case of spdc, the geometrical
conditions determine the oam transfer. This chapter analyses the oam
transfer in Raman transitions using the techniques of the previous chapters.
This chapter is divided in three sections. Section 5.1 describes the generated
two-photon state by introducing its mode function. Section 5.2 studies the
oam content of one of the photons with the oam of the other photons fixed.
Section 5.3 describes the effect of the geometry of the process on the spatial
entanglement between the photons. Numerical calculations show the effect of
the geometrical configuration on the oam transfer mechanism. The finite size
of the nonlinear medium results in new effects that do not appear in spdc.
51

5. Spatial correlations in Raman transitions
pump
g
e
s
anti
Stokes Stokes
control
Figure 5.1: One atom with a Λ-type energy level configuration, can produce Stokes
and anti-Stokes photons by the interaction with the pump and control beams.
5.1 The quantum state of Stokes and anti-Stokes photon
pairs
This section describes the Stokes and anti-Stokes state generated by Raman
transitions in cold atomic ensembles in an analogous way to the description of
the two-photon state generated via spdc in chapter 1. The section discusses the
general characteristics of the nonlinear process, and introduces the two-photon
mode function.
Consider as a nonlinear medium an ensemble of n identical Λ−type cold
atoms trapped in a magneto-optical trap (mot). The atoms have an energy
level configuration with one excited state: |e〉 and two hyperfine ground states:
|s〉, and |g〉. This is the case, for example, in the d2 hyperfine transition of
87rb. In the initial state of the cloud all atoms are in the ground state |g〉, and
after emission all atoms return to their initial state, as figure figure 5.1 shows.
The two-photon generation results from the interaction of a single atom of
the cloud with two counter-propagating classical beams in a four step process.
In the first step, the atom gets excited by the interaction with the pump beam
far detuned from the |g〉 → |e〉 transition. In the second step, the excited atom
decays into the |s〉 state by emitting one Stokes photon in the direction zs as
shown in figure 5.2. In the third step, the atom is re-excited by the interaction
with the control beam far detuned from the |s〉 → |e〉 transition. In the last
step, the atom decays to the ground state by emitting an anti-Stokes photon
in the zas direction.
If ω 0 i
is the central angular frequency for the photons involved in the pro-
cess (i = p, c, s, as), and k 0 i is the corresponding wave number at the central
frequencies, energy and momentum conservation implies
52
and
g
e
ω 0 p + ω 0 c = ω 0 s + ω 0 as, (5.1)
k 0 p − k 0 c = k 0 s cos ϕs − k 0 as cos ϕas, (5.2)
k 0 s sin ϕs = k 0 as sin ϕas. (5.3)
s

x
y
z
5.1. The quantum state of Stokes and anti-Stokes photon pairs
zas yas
xas
pump
anti Stokes
Stokes
ys
control
Figure 5.2: According to energy and momentum conservation, the generated photons
counterpropagate. Equation 5.7 describes the relation between their propagation
direction and the propagation direction of the pump and control beams, where due
to the phase matching the angle of emission of the anti-Stokes photon is ϕas = π−ϕs.
Because we consider a pump and a control beam with the same central frequency
and k0 s k0 as, the phase matching conditions allow any angle of emission
ϕas = π − ϕs, if it is not forbidden by the transition matrix elements [60].
This assumption is valid only for cold atoms, since for warm atoms the process
is highly directional, that is all photons are emitted along a preferred direction
as proven in reference [61].
There are two ways to describe the generated two-photon quantum state: it
can be described by using two coupled equations in the slowly varying envelope
approximation for the Stokes and anti-Stokes electric fields [65], or alternatively
by using an effective Hamiltonian of interaction and first order perturbation
theory [66, 67]. As the latter approach is analogous to the formalism used in
chapter 1, it will be used in what follows to calculate the Stokes and anti-Stokes
state.
The effective Hamiltonian in the interaction picture HI describes the photonatom
interaction, and is given by
HI = ɛ0
ϕas
ϕ s
xs
zs
zc
yc
xc
dV χ (3) Ê − as Ê− s Ê+ c Ê+ p + h.c. (5.4)
where χ (3) is the effective nonlinearity, independent of the beam intensity since
the pump and the control are non-resonant [65]. Assuming a Gaussian distribution
of atoms in the cloud the effective nonlinearity χ (3) can be written
as
χ (3)
(x, y, z) ∝ exp − x2 + y2 R2 z2
−
L2
(5.5)
where R is the size of the cloud of atoms in the transverse plane (x, y) and L
is the size in the longitudinal direction.
53

5. Spatial correlations in Raman transitions
According to equation 1.6, the electric field operators Ên in equation 5.4,
are given by
Ê (+)
n (rn, t) = dkn exp [ikn · rn − iωnt]â(kn) (5.6)
where, as in equation 1.23, we are using a more convenient set of transverse
wave vector coordinates given by
ˆxs,as =ˆx
ˆys,as =ˆy cos ϕs,as + ˆz sin ϕs,as
ˆzs,as = − ˆy sin ϕs,as + ˆz cos ϕs,as.
(5.7)
Under these conditions, at first order perturbation theory, the spatial quantum
state of the generated pair of photons is
|Ψ〉 =
(5.8)
dqsdqasΦ (qs, qas) |qs〉s|qas〉s
where the mode function Φ (qs, qas) of the two-photon state is
Φ (qs, ωs, qas, ωas) = dqpdqcEp (qp) Ec (qc) exp − ∆20R 2
4 − ∆21R 2
4 − ∆22L 2
4
(5.9)
with the delta factors defined as
∆0 = q x s + q x as
∆1 = (ks − kas) sin ϕs + (q y s − q y as) cos ϕs
∆2 = kp − kc − (ks + kas) cos ϕ + (q y s − q y as) sin ϕs
and the longitudinal wave vector of the pump beam given by
ωpnp kp =
c
2
− ∆ 2 0 − ∆ 2 1
1/2
(5.10)
. (5.11)
In the case of the Stokes and anti-Stokes two-photon state, there are no correlations
between space and frequency due to the narrow bandwidth (∼ GHz)
of the generated photons [60]. Therefore, to analyze the spatial shape of the
mode function Φq (qs, qas), we can consider ωs = ω 0 s and ωas = ω 0 as.
The effect of the unavoidable spatial filtering produced by the specific op-
tical detection system used is described by Gaussian filters. The angular acceptance
of the single photon detection system is 1/ k0
sws,as . In most experimental
configurations, ws ≈ 50-150 µm and the length of the cloud is a few
millimeters or less. For simplicity, we will assume that ws = was.
In the calculations, the pump and the control are Gaussian beams with
the same waist wp at the center of the cloud. As the waist is typically about
200-500 µm, the Rayleigh range of the pump, Stokes and anti-Stokes modes
Lp = πw2 p/λp and Ls,as = πw2 s/λs,as satisfy L ≪ Lp, Ls,as. This condition
allows us to neglect the transverse wavenumber dependence of all longitudinal
wave vectors in equations 5.9 and 5.10.
54

where
5.2. Orbital angular momentum correlations
Under these conditions, the mode function Φq (qs, qas) can be written as
Φq (qs, qas) = (ABCD)1/4
π
× exp − A
4 (qx s + q x as) 2 − B
4 (qx s − q x as) 2
× exp − C
4 (qy s + q y as) 2 − D
4 (qy s − q y as) 2
A = w2 pR2 2R2 + w2 +
p
w2 s
2
B = w2 s
2
C = w2 s
2
D = w2 pR 2 cos 2 ϕs
2R 2 + w 2 p
(5.12)
+ L 2 sin 2 ϕs + w2 s
. (5.13)
2
The specific characteristics of the state are determined by the size of the atomic
cloud in the longitudinal and transverse planes, L and R; by the beam waist of
the pump and control beams, wp,c; by the waist of the Stokes and anti-Stokes
modes ws; and by the angle of emission ϕs. The next section describes the
effect off all these parameters on the oam transfer from the pump and the
control beams to the pair Stokes and anti-Stokes.
5.2 Orbital angular momentum correlations
Since the pump and control beams are Gaussian beams, lp = lc = 0, the
analysis of the oam transfer reduces to the study of the oam content of the
Stokes and anti-Stokes pair. The Stokes oam content becomes the only free
variable, after projecting the anti-Stokes into a Gaussian mode
u(qas) = Nas exp
− w2 g
4 (qx2 as + q y2
as)
(5.14)
with beam width wg at the center of the cloud. The Stokes mode function
defined as
Φs (qs) = dqasΦ (qs, qas) u(qas) (5.15)
becomes
(F G)(1/4)
Φs (qs) = exp −
1/2
(2π) F
4 (qx s ) 2 − G
4 (qy s ) 2
, (5.16)
55

5. Spatial correlations in Raman transitions
with
F = 4AB + (A + B) w2 g
A + B + w 2 g
G = 4CD + (C + D) w2 g
C + D + w2 . (5.17)
g
As in the previous chapter, this mode function can be written as a superposition
of spherical harmonics
Φs (qs, θs) = (2π)
−1/2
ls
als (qs) exp (ilsθs) . (5.18)
The probability of having a Stokes photon with oam equal to ls is given by the
weight of each spiral mode
F + G
C2ls = (F G)1/2 dqsqs exp − q
4
2
s I 2
G − F
ls q
8
2
s (5.19)
where Ils are the Bessel functions of the second kind. Only even modes appear
in the distribution as a consequence of the symmetry of the Stokes mode
function.
Figure 5.3 shows the weight of the mode ls = 0 as a function of the angle
of emission for different values of the length of the cloud of atoms. For nearly
collinear configurations, the probability of having a Gaussian Stokes photon is
one, therefore, in this case the process satisfies the relation lp+lc = ls +las. For
these kind of configurations, A = D in equation 5.12, and therefore the Stokes
spatial mode function shows cylindrical symmetry in the transverse planes
(xs, ys) and (xas, yas). This is the case in most experimental configurations
[59, 64, 68], and in fact, reference [23] experimentally proves the relationship
lp + lc = ls + las for collinear configurations.
The probability of having a Gaussian Stokes photon decreases as other
modes appear in the distribution in highly noncollinear configurations. The
length of the cloud controls the importance of the other modes and it is possible
to obtain a spatial mode with cylindrical symmetry for any emission angle, by
fulfilling the condition
L = R 1 + 2R2
w2 −1/2
. (5.20)
p
This condition reduces to L = R, when the beam waist is much larger than the
transverse size of the cloud. Any deviation from a spherical volume of interaction
(described by this condition) introduces ellipticity in the mode function.
For this reason, a highly elliptical configuration, as the one described in reference
[60], will not satisfy the oam selection rule.
Figure 5.4 shows the weight of the mode ls = 0 as a function of the length
of the cloud for different values of the emission angle. For any angle, the
probability of having a Gaussian mode is maximum at the length given by
equation 5.20. In a collinear configuration, the Stokes photon always has a
Gaussian distribution, independently of the length of the cloud. As the angle
of emission increases, the length of the cloud becomes more important. The
mode weight is weakly affected by the change of the cloud length when the
length is much longer than the other relevant parameters: ws, wg and R. This
is especially evident for an angle of emission ϕ = 90◦ .
56

weight
1
0.6
-180º
-90º
0º 90º 180º
5.3. Spatial entanglement
length
0.2mm
0.4mm
1mm
2mm
angle
Figure 5.3: The weight of the Gaussian mode in the oam decomposition for the Stokes
photon has a maximum in the collinear configuration. In noncollinear configurations
the probability of a Gaussian Stokes photon decreases as the length of the cloud
increases. Table 5.1 lists the parameters used to generate this figure.
weight angle
1
0.6
0.2 Length 1.5 mm
Figure 5.4: The probability of having a Gaussian Stokes photon is maximum in a
collinear configuration independently of the cloud length. In the noncollinear cases
the maximum appears at the length given by equation 5.20, in these configurations
the probability of having a Gaussian Stokes photon changes drastically with the cloud
length and the emission angle. Table 5.1 lists the parameters used to generate this
figure.
5.3 Spatial entanglement
The spatial correlations between the generated photons inherit the angular
dependence from the Stokes oam content. To explore this phenomena, this
0º
10º
90º
57

5. Spatial correlations in Raman transitions
Schmidt number
40
20
1
-180º
0º 180º
lenght spatial filter
2 mm
1 mm
400 m
200 m
7
4
1
-180º
0º 180º
100 m
200 m
500 m
angle
Figure 5.5: The length of the cloud controls the Schmidt number angular dependence.
By modifying this length it is possible to change the position of the maximum and
minimum values of the Schmidt number. The angular dependence can be removed
completely by tailoring the parameters of the process, or by filtering. Table 5.1 lists
the parameters used to generate this figure.
Table 5.1: The parameters used in figures 5.3, 5.4 and 5.5
Parameter Figure 5.3 Figure 5.4 Figure 5.5 (left) Figure 5.5 (right)
L 0.2, 0.4, 1, 2 mm [0.2, 1.5] mm 0.2, 0.4, 1, 2 mm 200 µm
wp 100 µm 100 µm 500 µm 500 µm
R 400µm 400µm 1000 µm 1000 µm
ws 100 µm 100 µm 100 µm 100, 200, 500 µm
wg 500 µm 500 µm
ϕs [−180, 180] ◦ 0, 10, 90 ◦ [−180, 180] ◦ [−180, 180] ◦
section shows the effects of changing the emission angle on the Schmidt number
K defined in section 2.1, is a common entanglement quantifier.
The Schmidt number of the two-photon state described by equation 5.12
quantifies the entanglement between the generated photons. According to references
[69, 70] it is given by
(A + B) (C + D)
K =
4 (ABCD) 1/2
. (5.21)
Figure 5.5 shows the Schmidt number as a function of the emission angle for
different values of the atomic cloud length and the pump beam waist. The
function extrema are located at 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . At each of these
values the function can have a maximum or a minimum depending on the
other parameters of process. For instance, if
L < R
1 + 2R2
w2 −1/2
p
(5.22)
the entanglement is maximum at collinear configurations where ϕ = 0 ◦ , 180 ◦ ,
and minimum at transverse configurations ϕ = 90 ◦ , 270 ◦ . In the left part of
figure 5.5, the condition in equation 5.22 is achieved at cloud lengths smaller
than 333 µm.
As the length of the cloud increases, the variation of the entanglement with
the angle is smoothed out. In the left part of figure 5.5 the relation in equation
58

5.3. Spatial entanglement
5.20 is satisfied when L = 333 µm; therefore, the amount of entanglement is
constant. If the length of the cloud increases even more, the position for the
maxima and the minima get inverted. When
L > R 1 + 2R2
w2 −1/2
p
(5.23)
the entanglement is maximum for transverse emitting configurations ϕ = 90 ◦ , 270 ◦ ,
and minimum for collinear configurations ϕ = 0 ◦ , 180 ◦ . This variation is possible
in Raman transitions where the transversal size of the cloud is comparable
to the longitudinal size.
Another parameter that plays a role in the variation of the amount of
entanglement is the Stokes spatial filter ws. The right side of figure 5.5 shows
the effect of changing the filtering over the amount of entanglement. Very
narrow spatial filters (ws → ∞) diminish both the amount of entanglement
and its azimuthal variability.
Conclusion
As in spdc, the geometrical configuration of the Raman transitions determines
the oam content and the spatial correlations of the generated Stokes and anti-
Stokes photons. The size and shape of the cloud defines the emission angles
for which the correlations are maximum and minimum.
59

CHAPTER 6
Summary
This thesis characterizes the correlations in two-photon quantum states, both
between photons, and between the spatial and frequency degrees of freedom.
The photon pairs are generated by spontaneous parametric down-conversion
spdc, or by the excitation of Raman transitions in cold atomic ensembles.
Special attention is given to the entanglement of the photon pair in the spatial
degree of freedom, associated to the orbital angular momentum oam. The
main contributions or the thesis are:
A novel matrix formalism to describe the two-photon mode function.
This formalism makes it possible to analytically calculate several features
of the down-converted photons, and reduces the numerical calculation time of
other features, as shown in chapter 1 and reference [41].
Analytical expressions for the purity of the subsystems formed by
a single photon in space and frequency, or by a spatial two-photon
state. These expressions quantify the correlations in the two-photon state,
and make the effect of the various parameters of the spdc process on these
correlations explicit. The thesis presents a set of conditions to suppress or
enhance the correlations between the photons, or between degrees of freedom
in the two-photon state. With these conditions, it is possible to design spdc
sources with specific levels of correlations, for example, sources that generate
pure heralded single photons, or alternatively, maximally entangled states; as
shown in chapter 2 and in references [41, 45].
A selection rule for oam transfer. This selection rule explains several
contradictory measurements of the oam transfer from the pump beam to the
signal and idler photons in spdc. We showed that the oam content of the
pump is completely transferred to the photons emitted in all directions. This
selection rule always holds if all possible emission directions are considered,
for example in collinear configurations, but not necessarily if only a part of
all emission directions is detected. These conditions for the validity of the
selection rule give clear guidelines for the design of sources for protocols that
exploit the multidimensionality of oam, as shown in chapter 3 and references
[71, 72].
61

6. Summary
Experiments that explain oam transfer in noncollinear spdc. In
noncollinear configurations only a subset of emission directions is detected. The
transfer of oam to the detected photons is strongly affected by the change in
geometry imposed by the detection system, and depends on the angle of detection,
the pump-beam waist and the Poynting-vector walk-off. The thesis shows
that, by tailoring these parameters, it is possible to design noncollinear sources
that naturally generate spatially-separated photons with specific desired spatial
shapes, as shown in chapter 4 and references [39, 33, 32].
An analysis of the spatial correlations generated by Raman transitions
in cold atomic ensembles. Like for spdc, the spatial correlations
between the Stokes and anti-Stokes photons depend on the geometrical configuration
of the pump beam, the control beam, and the Stokes and anti-Stokes
photons. This thesis shows how the size and shape of the atom cloud define
the emission angles for which the correlations are maximum and minimum, as
shown in chapter 5 and reference [73].
62

APPENDIX A
The matrix form
of the mode function
According to section 1.3, the normalized two-photon mode function, after some
approximations, reads
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆2 0 − w2 p
4 ∆21
× exp − (γL)2
4 ∆2k − T 2 0
4 (Ωs + Ωi) 2
× exp − w2 s
2 |qs| 2 − w2 i
2 |qi| 2 − 1
2B2 Ω
s
2 s − 1
2B2 Ω
i
2
i . (A.1)
The argument of the exponential function is a second order polynomial. Each
term is the product of at most two variables (qx s , qy s , qx i , qy i , Ωs, Ωi) and a coefficient
f = a
4 qx2 s + b
4 qy2 s + c
4 qx2 i + d
4 qy2
h
i + . . . +
2 qx s q y s + . . . + z
2 ΩsΩi. (A.2)
Such a polynomial can be written as the product of a matrix A with the
coefficients as elements, and a vector x of the variables
f = 1
x qs 4
qy s qx i q y
i Ωs Ωs
⎛
⎜
⎜
⎝
a
h
i
j
k
h
b
m
n
p
i
m
c
s
t
j
n
s
d
v
k
p
t
v
f
l
r
u
w
z
⎞ ⎛
q
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
l r u w z g
x s
qy s
qx i
q y
i
Ωs
⎞
⎟ ,
⎟
⎠
Ωi
(A.3)
therefore, the mode function can be written using matrix notation as
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1
2 xt
Ax . (A.4)
63

A. The matrix form of the mode function
Each of the terms of the matrix is defined by comparing the product of the
polynomial f with the argument of the exponential in equation 1.34. The
elements of matrix A are
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0 cos α 2
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
+ γ 2 L 2 cos ϕs 2 tan ρ0 2 sin α 2
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0 cos α 2
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
+ γ 2 L 2 cos ϕi 2 tan ρ0 2 sin α 2
f =2B −2
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
− 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
+ γ 2 L 2 N 2 s sin ϕs 2 tan ρ0 2 sin α 2
g =2B −2
i
+ 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
+ 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
+ γ 2 L 2 N 2 i sin ϕi 2 tan ρ0 2 sin α 2
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
i =w 2 p + γ 2 L 2 tan ρ0 2 cos α
j =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α
k =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
l =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
m = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α
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
− 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
p = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ns cos ϕs sin ϕs − w 2 pNs cos ϕs sin ϕs
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ns cos ϕs 2 tan ρ0 sin α
+ 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
r = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ni sin ϕs cos ϕi + w 2 pNi cos ϕs sin ϕi
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ni cos ϕs cos ϕi tan ρ0 sin α
− 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
s =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α
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
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
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
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕs cos ϕi tan ρ0 sin α
− 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
w =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ni sin ϕi cos ϕi + w 2 pNi sin ϕi cos ϕi
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ni cos ϕi 2 tan ρ0 sin α
+ 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
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
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi − γ 2 L 2 NsNi cos ϕs sin ϕi tan ρ0 sin α
− γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + γ 2 L 2 NsNi cos ϕi sin ϕs tan ρ0 sin α
− 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.
(A.5)
This set of expressions is far more useful than compact. The matrix terms
become simpler in some particular spdc configurations that are treated in
chapters 2 and 4. For instance, when the pump polarization is parallel to the
x axis, the matrix terms become
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2
f = 2
B2 s
g = 2
B2 i
+ T 2 0 + γ 2 L 2 (Np − Ns cos ϕs) 2 + w 2 pN 2 s sin ϕs 2
+ T 2 0 + γ 2 L 2 (Np − Ni cos ϕi) 2 + w 2 pN 2 i sin ϕi 2
h =m = −γ 2 L 2 sin ϕs tan ρ0
i =w 2 p + γ 2 L 2 tan ρ0 2
j =s = γ 2 L 2 sin ϕi tan ρ0
k =t = γ 2 L 2 tan ρ0(Np − Ns cos ϕs)
l =u = γ 2 L 2 tan ρ0(Np − Ni cos ϕi)
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi
p = − γ 2 L 2 sin ϕs(Np − Ns cos ϕs) − w 2 pNs cos ϕs sin ϕs
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs
w =γ 2 L 2 sin ϕi(Np − Ni cos ϕi) + w 2 pNi cos ϕi sin ϕi
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
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi.
(A.6)
The matrix notation extends to functions of the mode function. For instance,
the purity of the spatial part of the two-photon state, given by equation 2.11,
65

A. The matrix form of the mode function
writes
T r[ρ 2
q] =
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (A.7)
where the dimension increases as new primed variables appear. Using the
matrix notation the integrand becomes
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)
= N 4
exp − 1
2 Xt
BX , (A.8)
where the vector X is the result of concatenation of x and x ′ , such that
⎛
⎜
X = ⎜
⎝
↑
x
↓
↑
x ′
↓
⎞
⎟ ,
⎟
⎠
(A.9)
and the new matrix B is given by
B = 1
⎛
⎜
2 ⎜
⎝
2a
2h
2i
2j
k
l
0
0
0
0
k
2h
2b
2m
2n
p
r
0
0
0
0
p
2i
2m
2c
2s
t
u
0
0
0
0
t
2j
2n
2s
2d
v
w
0
0
0
0
v
k
p
t
v
2f
2z
k
p
t
v
0
l
r
u
w
2z
2g
l
r
u
w
0
0
0
0
0
k
l
2a
2h
2i
2j
k
0
0
0
0
p
r
2h
2b
2m
2n
p
0
0
0
0
t
u
2i
2m
2c
2s
t
0
0
0
0
v
w
2j
2n
2s
2d
v
k
p
t
v
0
0
k
p
t
v
2f
l
r
u
w
0
0
l
r
u
w
2z
⎞
⎟ .
⎟
⎠
l r u w 0 0 l r u w 2z 2g
(A.10)
In an analogous way, the integrand on the expression for the signal photon
purity
T r[ρ 2 signal] =
is written in a matrix notation as
66
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (A.11)
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)
= N 4 exp
− 1
2 Xt
CX . (A.12)

The matrix C is given by
C = 1
⎛
2a
⎜ 2h
⎜ i
⎜ j
⎜ 2k
⎜ l
2 ⎜ 0
⎜ 0
⎜ i
⎜ j
⎝ 0
2h
2b
m
n
2p
r
0
0
m
n
0
i
m
2c
2s
t
2u
i
m
0
0
t
j
n
2s
2d
v
2w
j
n
0
0
v
2k
2p
t
v
2f
z
0
0
t
v
0
l
r
2u
2w
z
2g
l
r
0
0
z
0
0
i
j
0
l
2a
2h
i
j
2k
0
0
m
n
0
r
2h
2b
m
n
2p
i
m
0
0
t
0
i
m
2c
2s
t
j
n
0
0
v
0
j
n
2s
2d
v
0
0
t
v
0
z
2k
2p
t
v
2f
l
r
0
0
z
0
l
r
2u
2w
z
l r 0 0 z 0 l r 2u 2w z 2g
⎞
⎟ .
⎟
⎠
(A.13)
67

APPENDIX B
Integrals of
the matrix mode function
Given a n×n symmetric and positive definite matrix A, and two n order vectors
x and b
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp −
det(A) btA−1
b
.
2
(B.1)
To proof this result consider that, since A is positive definite, there exist another
matrix O such that OAO t = D where D is a diagonal matrix. With the
transformation y = Ox, the integral in equation B.1 becomes
dx exp − xtAx 2 + ixt
b = dy exp
− yt Dy
2 + iyt b ′
,
(B.2)
with b ′ = O −1 b.
As the argument of the exponential, in the right side of equation B.1, is
given by the matrix product
− 1
y1
2
y2 . . . yn
⎛
d1
⎜ 0
⎜
⎝ .
0
d2
.
. . .
. . .
. ..
0
0
.
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
0 0 0 dn
y1
y2
.
yn
⎞
⎛
⎟
⎠ + i ⎜
y1 y2 . . . yn ⎜
⎝
the integral can be written in a polynomial form as
dx exp − xtAx 2 + ixt
b
= dy1dy2...dyn exp − y2 1d1
2 + iy1b ′ 1 − y2 2d2
2 + iy2b ′ 2... − y2 ndn
2 + iynb ′
n
(B.3)
(B.4)
69
b ′ 1
b ′ 2
.
b ′ n
⎞
⎟
⎠ ,

B. Integrals of the matrix mode function
where the integrals in each variable are independent, so it is possible to write
them as
dx exp − xtAx
=
2 + ixt
b
dy1 exp − y2 1d1
2 + iy1b ′ 1
dy2
− y2 2d2
2 + iy2b ′
2 ...
which, after solving each integral separately, becomes
dx exp − xtAx 2 + ixt
b =
n
j=1
2π
dj
exp
dyn
− b′ 2
j
2dj
− y2 ndn
2 + iynb ′
n ,
(B.5)
. (B.6)
Because D is a diagonal matrix, 1/dj are the elements of D−1 , and n j=1 dj =
det(D); thus it is possible to write the last expression as
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp
det(D)
− b′ t D −1 b ′
2
. (B.7)
Finally, since det(A) = det(D) and b ′ t D −1 b ′ = b t A −1 b, the integral becomes
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp −
det(A) btA−1
b
. (B.8)
2
This proof can be applied to the special case in which all the elements of the
vector b are equal to 0. In that case the integral reduces to
dx exp − xt
Ax
=
2
(2π)n/2
. (B.9)
det(A)
70

APPENDIX C
Methods for OAM
measurements
A real world oam application will require a compact tool able to separate the
different modes at the single photon level. A tool analogous to the beam splitter
for polarization, or to a diffraction grating for frequency. Such a tool probably
will be based on the two current methods to determine the oam of photons:
holographic and interferometric methods.
In a very simplistic way, an hologram is a record of the interference pattern
of two beams. When the hologram is illuminated with one of the beams,
the other beam can be reconstructed. Based on this principle, a computer
generated hologram of the interference of a lg mode with a Gaussian beam,
in conjunction with a single mode fiber can be used to detect that particular
lg mode [7, 51]. Even though this method works at single photon level, it
is restricted to a two-value response, reducing the effective dimensions of the
oam to two, as shown in figure C.1.
Reference [15] introduced an analyzer able to sort photons into even and
odd oam modes, using a Mach-Zender interferometer. With a phase shift in
one of the interferometer arms, the constructive and destructive interference
were so that odd modes go to one of the outputs, and even modes to the other,
as figure C.2 shows. By nesting several interferometers, and using holograms,
the authors proved that was possible to separate up to 2 n modes with (2 n − 1)
interferometers. This method works at single photon level, and can distinguish
many different oam modes, but it relies on the simultaneous stabilization of
different interferometers, which is challenging experimentally.
71

C. Methods for OAM measurements
Input hologram
lens
Gaussian
mode
non-Gaussian
mode
Figure C.1: A hologram that records the interference of a lg mode with a Gaussian
beam, can be used to detect that particular lg mode. Only if the generating lg
mode is incident on the hologram, a Gaussian beam is recovered. If another mode is
incident, the hologram will generate non-Gaussian beams. Single mode optical fibers
distinguish between Gaussian and non-Gaussian beams.
Input
beam splitter
Dove prism mirror
even modes
odd modes
Figure C.2: Two Dove prisms rotated with respect to each other by an angle π/2
induce a relative rotation of π between the two arms of a Mach-Zender interferometer.
As a consequence of the rotation, odd and even lg modes leave the interferometer
at different outputs after the second beam splitter. Reference [15] introduces this
principle as a base of an oam sorter.
72

Bibliography
[1] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information.
Cambridge, UK: Cambridge University Press, 2000.
[2] M. Fox, Quantum Optics, An Introduction. Oxford, UK: Oxford University
Press, 2006.
[3] S. P. Walborn, S. Pádua, and C. H. Monken, “Hyperentanglement-assisted
bell-state analysis,” Phys. Rev. A, vol. 68, p. 042313, 2003.
[4] P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Y. Chiao, “Proposal
for a loophole-free bell inequality experiment,” Phys. Rev. A, vol. 49,
pp. 3209–3220, 1994.
[5] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren,
C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single
photons in pure quantum states,” Phys. Rev. Lett., vol. 100, p. 133601,
2008.
[6] A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping
the waveform of entangled photons,” Phys. Rev. Lett., vol. 99, p. 243601,
2007.
[7] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the
orbital angular momentum states of photons,” Nature, vol. 412, pp. 313–
316, 2001.
[8] S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Twophoton
entanglement of orbital angular momentum states,” Phys. Rev. A,
vol. 65, p. 033823, 2002.
[9] G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature
Phys., vol. 3, p. 305, 2007.
[10] S. P. Walborn and C. H. Monken, “Transverse spatial entanglement in
parametric down-conversion,” Phys. Rev. A, vol. 76, p. 062305, 2007.
[11] C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse
entanglement in optical parametric down conversion,” Phys. Rev. Lett.,
vol. 92, p. 127903, 2004.
[12] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation
of hyperentangled photon pairs,” Phys. Rev. Lett., vol. 95, p. 260501, 2005.
73

Bibliography
[13] M. Barbieri, C. Cinelli, P. Mataloni, and F. D. Martini, “Polarizationmomentum
hyperentangled states: Realization and characterization,”
Phys. Rev. A, vol. 72, p. 052110, 2005.
[14] D. P. Caetano and P. H. S. Ribeiro, “Quantum distillation of position
entanglement with the polarization degrees of freedom,” Opt. Commun.,
vol. 211, p. 265, 2002.
[15] J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial,
“Measuring the orbital angular momentum of a single photon,” Phys. Rev.
Lett., vol. 88, p. 257901, 2002.
[16] H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold,
E. Yao, and J. Courtial, “Simplified measurement of the orbital angular
momentum of single photons,” Opt. Commun., vol. 223, pp. 117 – 122,
2003.
[17] S. Haroche and J.-M. Raimond, Exploring the quantum. Oxford, UK:
Oxford University Press, 2006.
[18] J. D. Jackson, Classical electrodynamics. New York, US: John Wiley and
sons, 1999.
[19] V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with
screw dislocations in their wavefonts,” JETP Lett, vol. 52, p. 429, 1990.
[20] N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “‘generation
of optical phase singularities by computer generated holograms,” Opt.
Lett., vol. 17, p. 221, 1992.
[21] E. Yao, S. Franke-Arnold, J. Courtial, M. J. Padgett, and S. M. Barnett,
“Observation of quantum entanglement using spatial light modulators,”
Opt. Express, vol. 14, pp. 13089–13094, 2006.
[22] H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum
of single photons and entangled pairs of photons generated by
parametric down-conversion,” Phys. Rev. Lett., vol. 85, pp. 286–289, Jul
2000.
[23] R. Inoue, N. Kanai, T. Yonehara, Y. Miyamoto, M. Koashi, and
M. Kozuma, “Entanglement of orbital angular momentum states between
an ensemble of cold atoms and a photon,” Phys. Rev. A, vol. 74, p. 053809,
2006.
[24] A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three dimensional
entanglement for quantum communications,” Phys. Rev. Lett.,
vol. 89, p. 240401, 2002.
[25] G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental
quantum coin tossing,” Phys. Rev. Lett., vol. 94, p. 040501, 2005.
[26] S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental
quantum cryptography with qutrits,” New J. Phys., vol. 8,
p. 75, 2006.
74

Bibliography
[27] J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth
of entangled two-photon states,” Phys. Rev. A, vol. 68, p. 050301, Nov
2003.
[28] S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement
and conservation of orbital angular momentum in spontaneous
parametric down-conversion,” Phys. Rev. A, vol. 69, p. 023811, 2004.
[29] G. Molina-Terriza, A. Vaziri, J. ˇ Reháček, Z. Hradil, and A. Zeilinger,
“Triggered qutrits for quantum communication protocols,” Phys. Rev.
Lett., vol. 92, p. 167903, 2004.
[30] J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of
entangled photon states generated in non-collinear, walking parametric
down conversion,” J. Opt. B: Quantum Semiclass. Opt., vol. 7, pp. 235–
239, 2005.
[31] A. R. Altman, K. G. Köprülü, E. Corndorf, P. Kumar, and G. A. Barbosa,
“Quantum imaging of nonlocal spatial correlations induced by orbital angular
momentum,” Phys. Rev. Lett., vol. 94, p. 123601, 2005.
[32] G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych,
and J. P. Torres, “Control of the shape of the spatial mode function of photons
generated in noncollinear spontaneous parametric down-conversion,”
Phys. Rev. A, vol. 72, p. 065802, 2005.
[33] C. I. Osorio, G. Molina-Terriza, B. G. Font, and J. P. Torres, “Azimuthal
distinguishability of entangled photons generated in spontaneous parametric
down-conversion,” Opt. Express, vol. 15, pp. 14636–14643, 2007.
[34] R. W. Boyd, Nonlinear Optics. London, UK: Academic Press - Elsevier,
1977.
[35] A. Yariv, Quantum Electronics. New York, US: Wiley-VCH, 1987.
[36] R. Loudon, The quantum theory of light. Oxford, UK: Oxford University
Press, 2000.
[37] C. K. Hong and L. Mandel, “Theory of parametric frequency downconversion
of light,” Phys. Rev. A, vol. 31, pp. 2409–2418, 1985.
[38] A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence
properties of entangled light beams generated by parametric downconversion,”
Phys. Rev. A, vol. 50, pp. 3349–3361, 1994.
[39] J. P. Torres, C. I. Osorio, and L. Torner, “Orbital angular momentum of
entangled counterpropagating photons,” Opt. Lett., vol. 29, pp. 1939–1941,
2004.
[40] A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence
properties of entangled light beams generated by parametric downconversion:
Theory and experiment,” Phys. Rev. A, vol. 53, pp. 4360–4371,
1996.
75

Bibliography
[41] C. I. Osorio, A. Valencia, and J. P. Torres, “Spatiotemporal correlations
in entangled photons generated by spontaneous parametric down conversion,”
New J. Phys., vol. 10, p. 113012, 2008.
[42] J. Altepeter, E. Jeffrey, and P. Kwiat, “Photonic state tomography,” in Advances
in Atomic, Molecular and Optical Physics (P. Berman and C. Lin,
eds.), vol. 52, p. 107, Elsevier, 2005.
[43] T. A. Brun, “Measuring polynomial functions of states,” Quantum Inf.
Comput., vol. 4, pp. 401–408, 2004.
[44] R. B. A. Adamson, L. K. Shalm, and A. M. Steinberg, “Preparation of
pure and mixed polarization qubits and the direct measurement of figures
of merit,” Phys. Rev. A, vol. 75, p. 012104, 2007.
[45] L. E. Vicent, A. B. U’Ren, C. I. Osorio, and J. P. Torres, “Design of bright,
fiber-coupled and fully factorable photon pair sources,” To be submitted,
2009.
[46] T. Yarnall, A. F. Abouraddy, B. E. Saleh, and M. C. Teich, “Spatial
coherence effects on second- andfourth-order temporal interference,” Opt.
Express, vol. 16, pp. 7634–7640, 2008.
[47] M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P.
Woerdman, “Effect of spatial filtering on the schmidt decomposition of
entangled photons,” Phys. Rev. A, vol. 74, p. 012309, 2006.
[48] G. Adesso, A. Serafini, and F. Illuminati, “Determination of continuous
variable entanglement by purity measurements,” Phys. Rev. Lett., vol. 92,
p. 087901, 2004.
[49] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren,
C. Silberhorn, and I. A. Walmsley, “Conditional preparation of single photons
using parametric downconversion: a recipe for purity,” New J. Phys.,
vol. 10, p. 093011, 2008.
[50] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and
S. Schiller, “Quantum state reconstruction of the single-photon fock state,”
Phys. Rev. Lett., vol. 87, p. 050402, 2001.
[51] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman,
“Orbital angular momentum of light and the transformation of laguerregaussian
laser modes,” Phys. Rev. A, vol. 45, pp. 8185–8189, 1992.
[52] G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular
momentum of light: Preparation of photons in multidimensional vector
states of angular momentum,” Phys. Rev. Lett., vol. 88, p. 013601, 2001.
[53] J. U. Kang, Y. J. Ding, W. K. Burns, and J. S. Melinger, “Backward
second-harmonic generation in periodically poled bulk linbo3,” Opt. Lett.,
vol. 22, pp. 862–864, 1997.
[54] A. De Rossi and V. Berger, “Counterpropagating twin photons by parametric
fluorescence,” Phys. Rev. Lett., vol. 88, p. 043901, 2002.
76

Bibliography
[55] M. C. Booth, M. Atatüre, G. Di Giuseppe, B. E. A. Saleh, A. V. Sergienko,
and M. C. Teich, “Counterpropagating entangled photons from a waveguide
with periodic nonlinearity,” Phys. Rev. A, vol. 66, p. 023815, 2002.
[56] N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde,
A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled
qutrits and their use for quantum bit commitment,” Phys. Rev. Lett.,
vol. 93, p. 053601, 2004.
[57] D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in
parametric production of optical photon pairs,” Phys. Rev. Lett., vol. 25,
pp. 84–87, 1970.
[58] A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration
of higher dimensional entanglement: Qutrits of photon orbital
angular momentum,” Phys. Rev. Lett., vol. 91, p. 227902, 2003.
[59] D. N. Matsukevich, T. Chanelière, M. Bhattacharya, S.-Y. Lan, S. D.
Jenkins, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of a photon
and a collective atomic excitation,” Phys. Rev. Lett., vol. 95, p. 040405,
2005.
[60] V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation
of paired photons with controllable waveforms,” Phys. Rev. Lett., vol. 94,
p. 183601, 2005.
[61] L. M. Duan, J. I. Cirac, and P. Zoller, “Three-dimensional theory for
interaction between atomic ensembles and free-space light,” Phys. Rev. A,
vol. 66, p. 023818, 2002.
[62] J. H. Eberly, “Emission of one photon in an electric dipole transition of
one among n atoms,” J. Phys. B, vol. 39, no. 15, pp. S599–S604, 2006.
[63] M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed
spontaneous emission from an extended ensemble of n atoms: Timing is
everything,” Phys. Rev. Lett., vol. 96, p. 010501, Jan 2006.
[64] A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M.
Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable
quantum communication with atomic ensembles,” Nature, vol. 423,
pp. 731–734, 2003.
[65] D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency
mixing using electromagnetically induced transparency in cold atoms,”
Phys. Rev. Lett., vol. 93, p. 183601, 2004.
[66] J. Wen and M. H. Rubin, “Transverse effects in paired-photon generation
via an electromagnetically induced transparency medium. i. perturbation
theory,” Phys. Rev. A, vol. 74, p. 023808, 2006.
[67] J. Wen and M. H. Rubin, “Transverse effects in paired-photon generation
via an electromagnetically induced transparency medium. ii. beyond
perturbation theory,” Phys. Rev. A, vol. 74, p. 023809, 2006.
77

Bibliography
[68] S. Chen, Y.-A. Chen, B. Zhao, Z.-S. Yuan, J. Schmiedmayer, and J.-W.
Pan, “Demonstration of a stable atom-photon entanglement source for
quantum repeaters,” Physical Review Letters, vol. 99, p. 180505, 2007.
[69] P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping
affects type-II spontaneous parametric down-conversion,” Phys. Rev. A,
vol. 72, p. 033803, 2005.
[70] K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement
migration in hilbert space,” Phys. Rev. A, vol. 75, p. 050101, 2007.
[71] C. I. Osorio, G. Molina-Terriza, and J. P. Torres, “Correlations in orbital
angular momentum of spatially entangled paired photons generated in
parametric downconversion.,” Phys. Rev. A, vol. 77, p. 015810, 2008.
[72] C. I. Osorio, G. Molina-Terriza, and J. P. Torres, “Orbital angular momentum
correlations of entangled paired photons,” J. Opt. A: Pure Appl.
Opt., vol. 11, p. 094013, 2009.
[73] C. I. Osorio, S. Barreiro, M. W. Mitchell, and J. P. Torres., “Spatial entanglement
of paired photons generated in cold atomic ensembles,” Phys.
Rev. A, vol. 78, p. 052301, 2008.
78

DOCTORAL THESIS IN PHOTONICS
ICFO BARCELONA 2010
ICFO · THE INSTITUTE OF PHOTONIC SCIENCES
AV. CANAL OLÍMPIC, S/N · CASTELLDEFELS · BARCELONA
&
UPC · UNIVERSITAT POLITÈCNICA DE CATALUNYA
CAMPUS NORD · BARCELONA

Spatial Characterization
Of Two-Photon States
By
Clara Inés Osorio Tamayo
ICFO - Institut de Ciències Fotòniques
Universitat Politècnica de Catalunya
Barcelona, September 2009

Spatial Characterization
Of Two-Photon States
Clara Inés Osorio Tamayo
under the supervision of
Dr. Juan P. Torres
submitted this thesis in partial fulfillment
of the requirements for the degree of
Doctor
by the
Universitat Politècnica de Catalunya
Barcelona, September 2009

A Luz Stella y Luis Alfonso, mis papás.

Contents
Contents vii
Acknowledgements ix
Abstract xi
Introduction xv
List of Publications xvii
1 General description of two-photon states 1
1.1 Spontaneous parametric down-conversion . . . . . . . . . . . . 2
1.2 Two-photon state . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Approximations and other considerations . . . . . . . . . . . . 5
1.4 The mode function in matrix form . . . . . . . . . . . . . . . . 12
2 Correlations and entanglement 15
2.1 The purity as a correlation indicator . . . . . . . . . . . . . . . 16
2.2 Correlations between space and frequency . . . . . . . . . . . . 18
2.3 Correlations between signal and idler . . . . . . . . . . . . . . . 23
3 Spatial correlations and OAM transfer 29
3.1 Laguerre-Gaussian modes and OAM content . . . . . . . . . . . 30
3.2 OAM transfer in general SPDC configurations . . . . . . . . . . 31
3.3 OAM transfer in collinear configurations . . . . . . . . . . . . . 34
4 OAM transfer in noncollinear configurations 37
4.1 Ellipticity in noncollinear configurations . . . . . . . . . . . . . 38
4.2 Effect of the pump beam waist on the OAM transfer . . . . . . 39
4.3 Effect of the Poynting vector walk-off on the OAM transfer . . 44
5 Spatial correlations in Raman transitions 51
5.1 The quantum state of Stokes and anti-Stokes photon pairs . . . 52
5.2 Orbital angular momentum correlations . . . . . . . . . . . . . 55
5.3 Spatial entanglement . . . . . . . . . . . . . . . . . . . . . . . . 57
6 Summary 61
A The matrix form of the mode function 63
vii

Contents
B Integrals of the matrix mode function 69
C Methods for OAM measurements 71
Bibliography 73
viii

Acknowledgements
This thesis compiles the result of five years of work at icfo. But it is, in some
sense, the result of a longer process that started many years ago in Medellín.
I would like to show my gratitude to all the people that helped me during all
this time even though I cannot mention everybody explicitly.
I would like to thank my advisor Juan P. Torres for inviting me to his
group, and for being very patient with my impatience. I am grateful to Alejandra
Valencia and Xiaojuan Shi for their generous and unconditional support.
I wish to thank all the people that provide a stimulating and fun environment
in icfo, especially Maurizio Righini, Laura Grau, Carsten Schuck, Masood
Ghotbi, Xavier Vidal, Sibylle Braungardt, Jorge Luis Domínguez-Juárez,
Petru Ghenuche, Rafael Betancurt, Philipp Hauke, Alessandro Ferraro, Agata
Checinska, and Martin Hendrych. I would like to thank Niek van Hulst and
his group, for adopting me at lunch time and for all party matters; and Morgan
Mitchell and his group, especially Marco Koschorreck, Mario Napolitano,
Florian Wolfgramm, Alessandro Cere and Brice Dubost. Logistical support by
Nuria Segù, Maria del Mar Gil, Anne Gstöttner, Jose Carlos Cifuentes, and
the electronic workshop have also been most helpful. I am extremely grateful
to Miguel Navascués that has been like a little brother to me since I came to
Barcelona, and to Artur García that welcomed me in his office every time that
I needed to talk (and that was very often).
I want to thank my “Barcelonian” friends for their caring, their emotional
and gastronomical support, Mauricio Álvarez, Carolina Mora, Friman Sánchez,
Tania de la Paz, Maria Teresa Vasco and Jose Uribe. Regardless the distance,
many of my friends made available their support in a number of ways, I am
grateful to Carlos Molina, Jose Palacios, Catalina López, Elizabeth Agudelo,
Esteban Silva, Sebastían Patrón, Jaime Hincapie, Wolfgang Niedenzu, Andrew
Hilliard, Juan C. Muñoz (y familia), Jaime Forero and Javier Moreno. It is a
pleasure to thank Boudewijn Taminiau, Ied van Oorschot, and all the Taminiau
family, since their visits and hospitality have made this years much nicer.
Agradezco hasta el cielo a mi familia. A mis papás que me han inspirado
siempre con su creatividad y fortaleza para resolver los problemas, a mis hermanas
que siempre me apoyan, y a mi hermanito Felipe, que me ha trasmitido
todo su amor por la ciencia, su curiosidad y su escepticismo.
Finally, this thesis would not have been possible without the constant support
of Tim Taminiau, his careful corrections of this manuscript and the nice
3D images of the cone and the phase fronts that he made for me. I also thank
Tim for giving me my first bicycle, for teaching me how to use it, and for
pushing me up the mountain every day when we were going to work.
ix

When he [Kepler] found that his long
cherished beliefs did not agree with the
most precise observations, he accepted
the uncomfortable facts, he preferred the
hard truth to his dearest illusions.
That is the heart of science.
Cosmos - Carl Sagan

Abstract
In the same way that electronics is based on measuring and controlling the state
of electrons, the technological applications of quantum optics will be based
on our ability to generate and characterize photonic states. The generation
of photonic states is traditionally associated to nonlinear optics, where the
interaction of a beam and a nonlinear material results in the generation of
multi-photon states. The most common process is spontaneous parametric
down-conversion (spdc), which is used as a source of pairs of photons not only
for quantum optics applications but also for quantum information and quantum
cryptography [1, 2].
The popularity of spdc lies in the relative simplicity of its experimental realization,
and in the variety of quantum features that down-converted photons
exhibit. For instance, a pair of photons generated via spdc can be entangled
in polarization [3, 4], frequency [5, 6], or in the equivalent degrees of freedom of
orbital angular momentum, space, and transverse momentum [7, 8, 9, 10, 11].
Standard spdc applications focus on a single degree of freedom, wasting the
entanglement in other degrees of freedom and the correlations between them.
Among the few configurations using more than one degree of freedom are hyperentanglement
[12, 13], spatial entanglement distillation using polarization
[14], or control of the joined spectrum using the pump’s spatial properties [6].
This thesis describes the spatial properties of the two-photon state generated
via spdc, considering the different parameters of the process, and the
correlations between space and frequency. To achieve this goal, I use the purity
to quantify the correlations between the photons, and between the degrees
of freedom. Additionally, I study the spatial correlations by describing the
transfer of orbital angular momentum (oam) from the pump to the signal and
idler photons, taking into account the pump, the detection system and other
parameters of the process.
This thesis is composed of six chapters. Chapter 1 introduces the mode
function, used throughout the thesis to describe the two-photon state in space
and frequency. Chapter 2 describes the correlations between degrees of freedom
or photons in the two-photon state, using the purity to quantify such correlations.
Chapter 3 explains the mechanism of the oam transfer from pump
to signal and idler photons. Chapter 4 describes the effect of different spdc
parameters on the oam transfer in noncollinear configurations, both theoretically
and experimentally. In analogy with the downconverted case, chapter 5
discusses the two-photon state generated via Raman transition, by describing
its mode function, the correlations between different parts of the state, and the
oam transfer in the process. Finally, chapter 6 summarizes the main results
presented by this thesis.
xi

Abstract
The matrix notation, introduced here to describe the two-photon mode
function, reduces the calculation time for several features of the state. In
particular, this notation allows to calculate the purity of different parts of
the state analytically. This analytical solution reveals the effect of each spdc
parameter on the internal correlations, and shows the necessary conditions to
suppress the correlations, or to maximize them.
The description of the oam transfer mechanism shows that the pump oam
is totally transfer to the generated photons. But if only a portion of the generated
photons is detected their oam may not be equal to the pump’s oam. The
experiments described in the thesis show that the amount of oam transfer in
the noncollinear case is tailored by the parameters of the spdc. The analysis
of the spdc case can be extended to other nonlinear processes, such as Raman
transitions, where the specific characteristics of the process determine the
correlations and the oam transfer mechanism.
The results of this thesis contribute to a full description of the correlations
inside the two-photon state. Such a description allows to use the correlations
as a tool to modify the spatial state of the photons. This spatial information,
translated into oam modes, provides a multidimensional and continuum degree
of freedom, useful for certain tasks where the polarization, discrete and bidimensional,
is not enough. To make such future applications possible, it will be
necessary to optimize the tools for the detection of oam states at the single
photon level [15, 16].
xii

Resumen
De la misma manera que la electrónica se basa en medir y controlar el estado
de los electrones, las aplicaciones tecnológicas de la óptica cuántica se basarán
en nuestra habilidad para generar estados fotónicos bien caracterizados. La
generación de estos estados está tradicionalmente asociada a la óptica no lineal,
donde la interacción de un haz con un material no lineal da lugar a la
generación de estados de múltiples fotones. El proceso no lineal más popular
es la conversión paramétrica descendente, o spdc por su sigla en inglés, que
es usada como fuente de pares de fotones no sólo en aplicaciones de óptica
cuántica si no también para información y criptografía cuánticas [1, 2].
La popularidad de spdc se debe a la relativa simplicidad de su realización
experimental, y a la variedad de fenómenos cuánticos exhibidos por los pares
de fotones generados. Por ejemplo, estos pares pueden estar entrelazados en
polarización [3, 4], frecuencia [5, 6], o en los grados de libertad espacial: momento
angular orbital o momento transversal [7, 9, 10, 11]. Las aplicaciones
usuales de spdc usan sólo un grado de libertad, perdiendo la información contenida
en los otros grados de libertad y en las correlaciones entre ellos. Entre
las únicas aplicaciones que usan más de un grado de libertad se encuentran el
hiperentrelazamiento [12, 13], la destilación de entrelazamiento espacial usando
la polarización [14], o el control de la distribución espectral conjunta usando
las propiedades espaciales del haz generador del spdc [6].
Esta tesis describe las características espaciales de los pares de fotones generados
en spdc, teniendo en cuenta el efecto de los otros grados de libertad,
especialmente de la frecuencia. Para ello, usaré la pureza como cuantificador de
las correlaciones entre los grados de libertad, y entre los fotones. Además, usaré
la transferencia de momento angular orbital (oam), asociada a la distribución
espacial de los fotones, para estudiar el efecto de diferentes parámetros del
spdc sobre el estado espacial generado.
Esta tesis esta compuesta de seis capítulos. El capítulo 1 introduce la
función de modo, que es usada en toda la tesis para describir los estados de dos
fotones en espacio y frecuencia. El capítulo 2 describe las correlaciones entre
los grados de libertad, y entre los fotones, usando la pureza para cuantificar
estas correlaciones. El capítulo 3 describe la transferencia de oam desde el haz
generador hasta los fotones generados. El capítulo 4 demuestra teórica y experimentalmente,
el efecto de diferentes parámetros del spdc sobre la transferencia
de momento angular orbital en configuraciones no colineales. El capítulo
5 discute la generación de estados de dos fotones en transiciones Raman, caracterizando
estos estados de una manera análoga a la utilizada para aquellos
generados en spdc. Por último, el capítulo 6 resume las contribuciones más
importantes de esta tesis.
xiii

Abstract
La notación matricial introducida para describir la función de modo de los
pares generados, reduce considerablemente el tiempo de cálculo de diferentes
características del estado. En particular, ésta notación hace posible calcular
analíticamente la pureza de diferentes partes del estado, y estudiar el efecto de
cada parámetro del spdc sobre las correlaciones entre los grados de libertad o
entre los fotones. Hace posible, además, encontrar las condiciones necesarias
para suprimir las correlaciones entre los grados de libertad o entre los fotones,
y distinguir en que casos estas correlaciones se hacen mas relevantes.
El estudio del mecanismo de transferencia del momento angular orbital,
revela que éste es transferido totalmente del haz generador a los fotones generados.
Si sólo una porción de estos fotones es considerada, el oam de éstos no
da cuenta del momento oam total invertido por el haz generador. Los experimentos
descritos en esta tesis muestran que la cantidad de oam transferido
a una porción de los fotones puede ser controlada modificando el tamaño de
esa porción, ya sea cambiando el ancho del haz incidente, el largo del cristal u
otro parámetro. En el caso de la generación de pares de fotones en otros procesos
no lineales, como las transiciones Raman, tanto las correlaciones como la
transferencia de oam, son determinadas por las características especificas del
proceso.
Los resultados de esta tesis contribuyen a completar la descripción de las
correlaciones dentro del estado de dos fotones. Esta descripción permite el
uso de las correlaciones como herramienta para modificar el estado espacial de
los fotones. La información espacial, traducida a modos de oam, ofrece un
grado de libertad infinito dimensional y continuo, útil en ciertas tareas donde
la polarización, bidimensional y discreta, no es suficiente. Para hacer estas
tareas posibles, es necesario optimizar las herramientas para la detección de
estados de oam para un sólo fotón [15, 16].
xiv

Introduction
The role of photons in quantum physics and technology is growing [1, 2, 17].
Frequently, those applications are based on two dimensional systems, such as
the two orthogonal polarization states of a photon, losing all information in
other degrees of freedom. These applications only use a portion of the total
quantum state of the light.
The polarization itself is related to a broader degree of freedom. The angular
momentum of the photons contains a spin contribution associated with
the polarization, as well as an orbital contribution associated with the spatial
distribution of the light (its intensity and phase). In general, the spin and
the orbital angular momentum cannot be considered separately. However, in
the paraxial regime both contributions are independent [18]. In this regime it
is possible to exploit the possibilities offered by the infinite dimensions of the
orbital angular momentum (oam) of the light.
Currently available technology offers different possibilities to work with
oam. Computer generated holograms are widely used in classical and quantum
optics for generating and detecting different oam states [19, 20]. The efficiency
of these processes has increased with the design of spatial light modulators
capable of real time hologram generation [21].
Of special interest is the generation of paired photons entangled in oam.
Entanglement is a quantum feature with no analogue in classical physics. Spontaneous
parametric down-conversion (spdc) is a reliable source for the generation
of pairs of photons entangled in different degrees of freedom, including
oam [22]. The existence of correlations in the oam of pairs of photons generated
in spdc was proven experimentally by Mair and coworkers in 2001. Other
nonlinear processes can be used to generate pairs of photons with correlations
in their oam content. In reference [23], the authors show the generation of
spatially entangled pairs of photons by the excitation of Raman transitions in
cold atomic ensembles.
Several studies illustrate the potential offered by higher-dimensional quantum
systems. For instance, reference [24] reports a Bell inequality violation
by a two-photon three-dimensional system, confirming the existence of oam
entanglement in the system. Reference [25] introduced a quantum coin tossing
protocol based on oam states. In reference [26], the authors present a quantum
key distribution scheme using entangled qutrits, which are encoded into
the oam of pairs of photons generated in spdc. And, in reference [12], the
authors report the generation of hyperentangled quantum states by using the
combination of the degrees of freedom of polarization, time-energy and oam.
All the previously mentioned applications are based on specific oam correlations
between the photons. However, there has been some controversy about
xv

Introduction
the transfer of oam from the generating pump beam to the down-converted
photons. Several studies have reported a compleate transfer of the oam of the
pump [7, 8, 22, 27, 28, 29], while other studies have reported a partial transfer
[30, 31, 32, 33]. The elucidation of the oam spectra of down-converted photons,
and their relation with the spectra of the pump, is a key step for developing
new applications based on this degree of freedom.
This thesis addresses the mentioned oam controversy, as well as the characterization
of the different correlations present in two-photon states. For both
processes, spdc and Raman transitions in cold atoms, this thesis describes the
generated two-photon state, the correlations between different degrees of freedom
and between photons, and the oam transfer. A general goal of the thesis
is to contribute to the characterization of the two-photon sources, making it
possible to exploit a bigger portion of the total quantum state of the light in
future applications.
xvi

This thesis is based on the following papers:
List of Publications
Orbital angular momentum correlations of entangled paired photons.
C. I. Osorio, G. Molina-Terriza, J. P. Torres.
J. Opt. A: Pure Appl. Opt. 11, 094013 (2009).
Chapters 3 and 5
Spatial entanglement of paired photons generated in cold atomic ensembles
C. I. Osorio, S. Barreiro, M. W. Mitchell, and J. P. Torres.
Phys. Rev. A 78, 052301 (2008). arXiv:0804.3257v2 [quant-ph].
Chapter 5
Spatiotemporal correlations in entangled photons generated by spontaneous parametric
down conversion.
C. I. Osorio, A. Valencia, and J. P. Torres.
New J. Phys. 10, 113012 (2008). arXiv:0804.2425v2 [quant-ph].
Chapters 1 and 2
Correlations in orbital angular momentum of spatially entangled paired photons generated
in parametric downconversion.
C. I. Osorio, G. Molina-Terriza, and J. P. Torres.
Phys. Rev. A 77, 015810 (2008). arXiv:0711.4500v1 [quant-ph].
Chapters 3
Azimuthal distinguishability of entangled photons generated in spontaneous parametric
down-conversion.
C. I. Osorio, G. Molina-Terriza, B. Font, and J. P. Torres.
Opt. Express 15, 14636 (2007). arXiv:0709.3437v1 [quant-ph].
Chapter 4
Control of the shape of the spatial mode function of photons generated in noncollinear
spontaneous parametric down-conversion.
G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych, and J. P.
Torres.
Phys. Rev. A 72, 065802 (2005). arXiv:quant-ph/0508058v1.
Chapter 4
Orbital angular momentum of entangled counterpropagating photons.
J. P. Torres, C. I. Osorio, and L. Torner.
Opt. Lett. 29, 1939 (2004).
Chapters 3 and 4
xvii

CHAPTER 1
General description of
Two-photon states
Experimental implementations of spontaneous parametric down-conversion (spdc)
share three main components: A laser beam used as a pump, a nonlinear material
that induces the down-conversion, and a detection system that measures
the generated state. The properties of the resulting two-photon state depend
on the interaction, and on the individual properties of those components. This
chapter derives the two-photon mode function that characterizes the generated
state, using first order perturbation theory. The chapter is divided in four
sections. Section 1.1 describes the spdc process in general terms. Section 1.2
introduces a general expression for the mode function. Section 1.3 lists a series
of common approximations used to reduce the complexity of the mode function.
Finally, section 1.4 introduces a novel matrix notation for the simplified
mode function. One of the advantages of this new notation is that many characteristics
of the two-photon state can be analytically calculated by using it.
The following chapters will use the notation and results introduced here as a
starting point to describe the correlations between the generated photons, and
between their spatial states.
1

1. General description of two-photon states
virtual state
pump
ground state
signal
idler
Figure 1.1: In spontaneous parametric down conversion the interaction of a molecule
with the pump results in the annihilation of the pump photon and the creation of
two new photons.
1.1 Spontaneous parametric down-conversion
Spontaneous parametric down-conversion is one of the possible resulting processes
of the interaction of a pump photon and a molecule, in the presence of
the vacuum field. In spdc, the pump-molecule interaction leads to the generation
of a single physical system composed of two photons: signal and idler, as
figure 1.1 shows.
The name of the process reveals some of its characteristics. spdc is a
parametric process since the incident energy totally transfers to the generated
photons, and not to the molecule. And, it is a down-conversion process since
each of the generated photons has a lower energy than the incident photon.
The conservation of energy and momentum establish the relations between the
frequencies (ωp,s,i) and wave vectors (kp,s,i) of the photons,
ωp = ωs + ωi
kp = ks + ki. (1.1)
where the subscripts p, s, i stand for pump, signal and idler. Macroscopically,
spdc results from the interaction of a field with a nonlinear medium. Commonly
used mediums include uniaxial birefringent crystals. The conditions in
equation 1.1 can be achieved for this kind of crystals, since their refractive index
changes with the frequency and the polarization of an incident field [34, 35].
This thesis considers type-i configurations in uniaxial birefringent crystals.
In such configuration the pump beam polarization is extraordinary since it is
contained in the principal plane, defined by the crystal axis and the wave vector
of the incident field. The polarization of the generated photons is ordinary,
which means that it is orthogonal to the principal plane.
For a given material, the conditions imposed by equation 1.1 determine the
characteristics of the generated pair. The second line in equation 1.1 defines the
spdc geometrical configuration, and it is known as a phase matching condition.
The generated photons are emitted in two cones coaxial to the pump, as shown
in figure 1.2 (b). The apertures of the cone are given by the emission angles ϕs,i
associated to the frequencies ωs,i. Once a single frequency or emission angle is
selected, only one of all possible cones is considered. In degenerate spdc, both
photons are emitted at the same angle ϕs = ϕi and the cones overlap, as in
figure 1.2 (c). There are two kinds of degenerate spdc processes: noncollinear
2

y
y
x
z
z
pump
s
i
(a)
signal
idler
(c)
1.2. Two-photon state
Figure 1.2: (a) Phase matching conditions impose certain propagation directions for
the emitted photons at frequencies ωs and ωi. (b-c) This directions define two cones,
that collapse into one in the degenerate case. (d) In the collinear configuration, the
aperture of the cones tends to zero as the direction of emission is parallel to the pump
propagation direction.
in which the signal and idler are not parallel to the pump, and collinear in
which the aperture of the cones tends to zero as the photons propagate almost
parallel to the pump, as shown in figure 1.2 (d).
Even though the phase matching condition defines the main characteristics
of the generated two-photon state, other important factors influence the measured
state of the photons, for example the detection system. The next section
describes the two-photon state mathematically, taking into account all these
factors.
1.2 Two-photon state
When an electromagnetic wave propagates inside a medium, the electric field
acts over each particle (electrons, atoms, or molecules) displacing the positive
charges in the direction of the field and the negative charges in the opposite
direction. The resulting separation between positive and negative charges of
the material generates a global dipolar moment in each unit volume known as
polarization, and defined as
(b)
(d)
P = ɛ0(χ (1) E + χ (2) EE + χ (3) EEE + · · · ) (1.2)
where ɛ0 is the vacuum electric permittivity, and χ (n) are the electric susceptibility
tensors of order n [34, 35].
For small field amplitudes, as in linear optics, the polarization is approximately
linear,
P ≈ ɛ0χ (1) E. (1.3)
3

1. General description of two-photon states
When the amplitude of the electric field increases, the higher orders term in
equation 1.2 become relevant, and then a nonlinear response of the material
to the field appears. Optical nonlinear phenomena resulting from this kind
of interaction include the generation of harmonics, the Kerr effect, Raman
scattering, self-phase modulation, and cross-phase modulation [34].
Spontaneous parametric down-conversion, and other second order nonlinear
processes, result from the second order polarization, defined as the first
nonlinear term in the polarization tensor
P (2) = ɛ0χ (2) EE. (1.4)
The quantization of the electromagnetic field leads to a quantization of the
second order polarization, so that the nonlinear polarization operator ˆ P (2)
becomes
ˆP (2) = ɛ0χ (2) ( Ê(+) + Ê(−) )( Ê(+) + Ê(−) ), (1.5)
where Ê(+) and Ê(−) are the positive and negative frequency parts of the field
operator [36]. The positive frequency part of the electric field operator is a
function of the annihilation operator â(k), and is defined at position rn
xnˆxn + ynˆyn + znˆzn and time t as
=
Ê (+)
1/2
ωn
n (rn, t) = ien
2ɛ0v
dkn exp [ikn · rn − iωnt]â(kn), (1.6)
where the magnitude of the wave vector k satisfies k 2 = ωn/c. The volume
v contains the field, and en is the unitary polarization vector. The negative
frequency part of the field is the Hermitian conjugate of the positive part,
Ê (−)
n (rn, t) = Ê(+)† n (rn, t). Thus, the negative frequency part is a function of
the creation operator â † (k).
Following references [37] and [38], in first order perturbation theory, the
interaction of a volume V of a material with a nonlinear polarization ˆ P (2) and
the field Êp(rp, t), produces a system described by the state
|ΨT 〉 ∝ |1〉p|0〉s|0〉i − i
τ
0
dt ˆ HI|1〉p|0〉s|0〉i, (1.7)
where τ is the interaction time, |1〉p|0〉s|0〉i is the initial state of the field, and
the interaction Hamiltonian reads
ˆHI = − dV ˆ P (2) Êp(rp, t). (1.8)
V
The first term at the right side of equation 1.7 describes a one photon system,
while the second term describes a two-photon system. In what follows we will
consider only the second term as we are mainly interested in the generation of
pairs of photons. The two-photon system state is given by
|Ψ〉 = i
τ
0
dt ˆ HI|1〉p|0〉s|0〉i, (1.9)
or ˆρ = |Ψ〉〈Ψ| in the density matrix formalism. Seven of the eight terms
that compose the interaction Hamiltonian vanish when they are applied over
the state |1〉p|0〉s|0〉i. The remaining term is proportional to the annihilation
4

1.3. Approximations and other considerations
operator in mode p, and the creation operators in modes s and i. Assuming
constant χ (2) , the two-photon state reduces to
(2) τ
iɛ0χ
|Ψ〉 = dt dV
0 V
Ê(+) p (rp, ωp) Ê(−) s (rs, ωs) Ê(−) i (ri, ωi)|1〉p|0〉s|0〉i,
(1.10)
which can be written as
|Ψ〉 ∝ dks dkiΦ(ks, ωs, ki, ωi)a † (ks)a † (ki)|0〉s|0〉i; (1.11)
where Φ(ks, ωs, ki, ωi) is known as the mode function and, assuming that the
pump has certain spatial distribution Ep(kp), is given by
τ
Φ(ks, ωs, ki, ωi) ∝ dt dV dkpEp(kp)
0
V
× exp [ikp · rp − iks · rs − iki · ri − i(ωp − ωs − ωi)t]
× â(kp)|1〉p. (1.12)
The mode function contains all the information about the generated two-photon
system in space and frequency, not only about their individual state but about
their correlations. This function is therefore highly complex, and to calculate
any feature of the two-photon state it is necessary to simplify it, as the next
section shows.
1.3 Approximations and other considerations
Analytical calculations of any features of the down converted photons require
simplification of the mode function in equation 1.12. This section lists the most
important approximations used in this thesis, as well as the restrictions that
they impose.
Separation of transversal and longitudinal components
A field with a wave vector k almost parallel to its propagation direction spreads
only a little in the transversal direction. It is then possible, to consider the
longitudinal and transversal components of the wave vector separately,
k = kzˆz + q. (1.13)
As the wave vector’s magnitude k = ωn/c is bigger than the transversal component’s
magnitude q = |q| = (k 2 x + k 2 y) 1/2 , the magnitude of the longitudinal
component kz is always a real number,
kz = k 2 1
− q
2 2 . (1.14)
Assuming that the pump, the signal, and the idler are such fields, the mode
function becomes
τ
Φ(qs, ωs, qi, ωi) ∝
0
dt dV
V
dqpEp(qp) exp [ik z pzp − ik z szs − ik z i zi]
× exp [iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i − i(ωp − ωs − ωi)t]
× â(q p)|1〉p. (1.15)
where r ⊥ n = xnˆxn + ynˆyn is the transversal position vector.
5

1. General description of two-photon states
The semiclassical approximation
As a consequence of the low efficiency of the nonlinear process, the incident
field is several orders of magnitude stronger than the generated fields. In that
case, it is possible to consider the pump as a classical field, while the signal
and idler are considered as quantum fields, this is known as the semiclassical
approximation.
By defining the pump as a classical field, with a spatial amplitude distribution
Ep(q p) and a spectral distribution Fp(ωp):
Ep(rp, t) ∝ dqp dωpEp(qp)Fp(ωp) exp [iqp · r ⊥ p + ik z pzp − iωpt], (1.16)
the mode function reduces to
τ
Φ(qs, ωs, qi, ωi) ∝ dt dV dqp dωpEp(qp)Fp(ωp) Gaussian beam approximation
0 V
× exp [ik z pzp − ik z szs − ik z i zi + iqp · r ⊥ p − iqs · r ⊥ s − iqi · r ⊥ i ]
× exp [−i(ωp − ωs − ωi)t]. (1.17)
According to equation 1.17, the mode function depends on the spatial and
temporal profiles of the pump beam. To write these profiles explicitly we
assume a pump beam with a Gaussian temporal distribution,
Fp(ωp) = exp − T 2 0
4 ω2
p
(1.18)
where T0 is the pulse duration (standard deviation), which tends to infinity for
continuous wave beams. Also, we assume that the pump beam is an optical
vortex with orbital angular momentum lp per photon. As will be described
in section 3.1, under these conditions the pump spatial profile is given by the
Laguerre-Gaussian polynomials
1
2
wp
Ep(qp) =
2π|lp|!
|lp|
−iwp
√ qp exp −
2 w2 p
4 q2
p + ilpθp
(1.19)
that are functions of the pump beam waist wp, and the transversal vector
magnitude qp and phase θp, given by
6
qp =
θp = tan −1
(q x p ) 2 + (q y p) 2
q y p
q x p
.
(1.20)

x
1.3. Approximations and other considerations
pump
signal
beam s
y z xi
idler
z
Figure 1.3: The propagation direction of the pump beam defines the z direction of
the general coordinate system. The generated pair of photons propagates with angles
ϕs and ϕi with respect to the pump beam in the yz plane. Each photon coordinate
system transforms to the general coordinate system through the relations in equation
1.23.
For the special case of a Gaussian pump beam with lp = 0 the mode function
becomes
τ
Φ(qs, ωs, qi, ωi) ∝ dt dV
0 V
i
y
dq p exp [− w2 p
4 q2 p − T 2 0
4 ω2 p]
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ]
× exp [−i(ωp − ωs − ωi)t], (1.21)
which assumes that the interaction time τ is longer than the spontaneous emission
life time of the material.
Frequency bandwidth
Fields generated experimentally always contain a distribution of frequencies,
and totally monochromatic fields only exist in theory. This thesis considers the
frequency ω of each field as the sum of a constant central frequency ω 0 , and a
small deviation from that frequency Ω, so that ω = ω 0 + Ω.
As a result of integrating over the interaction time taking into account the
conservation of energy, the mode function becomes
Φ(qs, Ωs, qi, Ωi) ∝ dV
V
Coordinate transformation
dq p exp [− w2 p
4 q2 p − T 2 0
4 (Ωs + Ωi) 2 ]
× exp [ik z pzp − ik z szs − ik z i zi + iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i ].
(1.22)
The mode function depends on three position vectors, rp, rs, and ri. Each of
them is defined in a coordinate system with the z axis parallel to the propagation
direction of each field, as shown in figure 1.3. Defining all position vectors
in the same coordinate system simplifies the integration of the mode function
over volume.
i
i
xs
ys
zs
7

1. General description of two-photon states
pump
polarization
y
Figure 1.4: The azimuthal angle α between the pump beam polarization and the x
axis defines the position of a single pair of photons on the cone. The x axis is by
definition normal to the plane of emission.
With the origin at the crystal’s center, the z direction parallel to the pump
propagation direction, and the yz plane containing the emitted photons, the
unitary vectors in the coordinate systems of the generated photons transform
as
ˆxs =ˆx
ˆys =ˆy cos ϕs + ˆz sin ϕs
x
ˆzs = − ˆy sin ϕs + ˆz cos ϕs
ˆxi =ˆx
ˆyi =ˆy cos ϕi − ˆz sin ϕi
ˆzi =ˆy sin ϕi + ˆz cos ϕi. (1.23)
As a single pair of photons defines the yz plane, the coordinate transformations
consider only one of the possible directions of propagation on the cone. The
angle α between the x axis and the pump polarization defines the transverse
position on the cone for one photon pair, as figure 1.4 shows. The pump
polarization is normal to the plane in which the generated photons propagate
when α = 0 ◦ or α = 180 ◦ , and it is contained in the yz plane when α = 90 ◦ or
α = 270 ◦ .
Poynting vector walk-off
This thesis considers an extraordinary polarized pump beam and ordinary polarized
generated photons, an eoo configuration. Therefore, while the refractive
index does not change with the direction for the generated photons, it changes
for the pump beam with the angle between the pump wave vector and the axis
of the crystal. Figure 1.5 shows how the energy flux direction of the pump,
given by the Poynting vector, is rotated from the direction of the wave vector
by an angle ρ0 given by
ρ0 = − 1 ∂ne
. (1.24)
∂θ
ne
where ne is the refractive index for the extraordinary pump beam, and θ is
the angle between the optical axis and the pump’s wave vector. The Poynting
8

x
Wave fronts
1.3. Approximations and other considerations
Poynting vector
0
Wave vector
Figure 1.5: As a consequence of the birefringence, the Poynting vector is no longer
parallel to the wave vector. The energy flows in an angle ρ0 with respect to propagation
direction.
x
y z
Figure 1.6: The Poynting vector moves away from the wave vector in the direction of
the pump beam polarization. The angles α and ρ0 characterize the displacement. A
nonradial effect such as the Poynting vector walk-off inevitably brakes the azimuthal
symmetry of the properties of the photons on the cone.
vector walk-off displaces the effective transversal shape of the pump beam
inside the crystal in the pump polarization direction, as shown in figure 1.6.
The vector p = z tan ρ0 cos αˆx + z tan ρ0 sin αˆy, describes the magnitude and
direction of this displacement.
By including the Poynting vector walk-off, and by using the same coordinate
system for all the fields, the mode function becomes
Φ(qs, Ωs, qi, Ωi) ∝ dV dqp exp −
V
w2 p
4 q2 p − T0
4 (Ωs + Ωi) 2
× exp i(q x p − q x s − q x i )x
0
× exp [i(q y p + k z s sin ϕs − k z i sin ϕi − q y s cos ϕs − q y
i cos ϕi)y]
× exp [i(k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y
i sin ϕi)z]
× exp [i(q x p tan ρ0 cos α + q y p tan ρ0 sin α)z]. (1.25)
p
z
9

1. General description of two-photon states
Interaction volume approximation
In the experimental cases studied here, the crystals are cuboids with two square
faces. While typical crystal faces have areas of about 1cm 2 , the pump beam
waist is only some hundreds of micrometers. By assuming that the transversal
dimensions of the crystal are much larger than the pump, the integral over the
volume becomes
V
dV →
∞
−∞
∞ L/2
dx dy dz, (1.26)
−∞ −L/2
where L is the length of the crystal. After integrating over x and y, the mode
function reduces to
where
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆20 − w2 p
4 ∆21 − T0
4 (Ωs + Ωi) 2
L/2
× dz exp [i∆kz] (1.27)
−L/2
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi − k z s sin ϕs + k z i sin ϕi
∆k =k z p − k z s cos ϕs − k z i cos ϕi − q y s sin ϕs + q y
i sin ϕi
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.28)
Wave vector and group velocity
The use of a Taylor series expansion of kz s,i , simplifies the delta factors defined
in the previous paragraph. Because the wave vector magnitude is a function
of the frequency deviation Ω, the expansion around the origin reads
k z n = k z0 ∂k
n + Ωn
z n
+ Ω 2 ∂
n
2kz n
∂2Ωn ∂Ωn
+ . . . , (1.29)
where k z0
n is the magnitude of the wave vector at the central frequency ω 0 n, and
the first partial derivative of the wave vector magnitude with respect to the
frequency is the group velocity Nn.
Taking only the first order terms of the Taylor expansion, and considering
momentum conservation, the delta factors become
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi − NsΩs sin ϕs + NiΩi sin ϕi
∆k =Np(Ωs + Ωi) − NsΩs cos ϕs − NiΩi cos ϕi − q y s sin ϕs + q y
i sin ϕi
+ ∆0 tan ρ0 cos α + ∆1 tan ρ0 sin α. (1.30)
Exponential approximation of the sinc function
The mode function in equation 1.27, depends on the integral of an exponential
function of z. By integrating the exponential over the length of the crystal, the
10

1
0
-0.2
1.3. Approximations and other considerations
Gaussian Sine Cardinal
0
FWHM
Figure 1.7: An exponential function is chosen as an approximation for the sine cardinal
function. The parameters of the exponential were chosen in such a way that
both functions have the same full width at half maximum (fwhm).
mode function becomes
Φ(q s, Ωs, q i, Ωi) ∝ exp
− w2 p
4 ∆2 0 − w2 p
4 ∆2 1 − T0
4 (Ωs + Ωi) 2
sinc
L
2 ∆k
(1.31)
where the sine cardinal function is defined as sinc(a) = sin a/a.
In chapter 2, the sine cardinal function sinc(a) is approximated by a Gaussian
function exp[−(γa) 2 ]. With γ = 0.4393 the two functions have the same
full width at half maximum as figure 1.7 shows. In the rest of the thesis the
sinc function will be consider without approximations.
Detection systems as Gaussian filters
The specific optical system used to detect the photons at the output face of
the crystal, acts as a filter in space and frequency. The effect of these filters
is modeled by multiplying the mode function by a spatial collection function
Cspatial(qn) and a frequency filter function Ffrequency(Ωn). For the sake of
simplicity, both functions are assumed to be Gaussian functions so that
Cspatial(qn) ∝ exp − w2 n
2 q2
n , (1.32)
and
Ffrequency(Ωn) ∝ exp − 1
2B2 Ω
n
2
n , (1.33)
where n labels each of the generated photons, wn is the spatial collection mode
width, and Bn is the frequency bandwidth.
The angular acceptance θn is related to the vector qn by the equation qn =
knθn. Therefore, for a given collection mode wn the acceptance is θn = 1/knwn
(the half width at 1/e 2 ).
11

1. General description of two-photon states
In order to obtain more convenient units, the filter in momentum is defined
in a different way than the filter in frequency. While Bs,i → 0 implies a single
frequency collection, the condition for a single q vector collection is ws,i → ∞.
Finally, after all the approximations and taking into account the filters, the
mode function equation 1.12 becomes
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆20 − w2 p
4 ∆21
× exp − (γL)2
4 ∆2k − T 2 0
4 (Ωs + Ωi) 2
× exp − w2 s
2 q2 s − w2 i
2 q2 i − 1
2B2 Ω
s
2 s − 1
2B2 Ω
i
2
i . (1.34)
The approximations listed in this chapter were introduced by several authors,
and can be found for example in references [39, 40, 30]. In reference [41],
we introduced a novel matrix notation based on the simplified mode function
equation 1.34. The next section explains the details and the advantages of that
notation.
1.4 The mode function in matrix form
The argument of the exponential function in equation 1.34 is a second order
polynomial of the mode function variables. Such a function can be written in
matrix form, as is shown in appendix A. The mode function then becomes
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1
2 xt
Ax
(1.35)
where all the parameters of the spdc process are contained in A, a positivedefinite
real 6 × 6 matrix given by
A = 1
⎛
a
⎜ h
⎜ i
2 ⎜ j
⎝ k
h
b
m
n
p
i
m
c
s
t
j
n
s
d
v
k
p
t
v
f
l
r
u
w
z
⎞
⎟ ,
⎟
⎠
(1.36)
l r u w z g
x is the vector including all variables of the mode function, defined as
⎛ ⎞
⎜
x = ⎜
⎝
q x s
q y s
q x i
q y
i
Ωs
Ωi
⎟ , (1.37)
⎟
⎠
and x t is the transpose of x. This matrix representation of the mode function,
introduced in reference [41], is an important result of this thesis. The use of
this notation reduces the calculation time for integrals over the mode function,
12

1.4. The mode function in matrix form
and allows to solve some of those integrals analytically. For instance, the mode
function normalization requires six integrals over the modulus square of the
mode function. As appendix B shows, the matrix notation provides a way to
calculate the normalization constant analytically,
dx exp
− 1
2 xt
(2A)x
and thus the normalized mode function reads
=
Φ(qs, Ωs, qi, Ωi) = [det(2A)]1/4
(2π) 3/2
(2π) 3
, (1.38)
[det(2A)] 1/2
exp − 1
2 xt
Ax . (1.39)
In the same way, many other integrals over the mode function can be solved
analytically using the matrix representation of the mode function.
Conclusion
The two-photon mode function can be written in a matrix form after a series
of approximations. The matrix notation makes it possible to calculate several
features of the down-conversion photons analytically, and reduces the numerical
calculation time of others.
The next chapter describes the different correlations in the two-photon state
using the purity as a correlation indicator. The use of the approximate mode
function in matrix notation makes it possible to find an analytical expression
for the purity, and to study the effect of the different parameters on it.
13

CHAPTER 2
Correlations and
entanglement
The two-photon system described in chapter 1 consists of two photons with two
degrees of freedom. Although the mode function deduced fully characterizes the
two-photon state, the degree of correlations between each of its parts (different
subsystems) are not explicit. In this chapter, I use the purity of a subsystem
state to indicate the presence of correlations between it and the rest of the
two-photon state, as shown in figure 2.1. This chapter has three sections.
Section 2.1 describes the main characteristics of the purity, and explains why
it can be used to characterize correlations in composed systems. Section 2.2
describes the spatial part of the two-photon state, and section 2.3 describes
the state of the signal photon. In both sections a discussion about the origin
of the correlations is followed by analytical and numerical calculations of the
purity. The calculations clarify the role of each spdc parameter in the internal
correlations of the two-photon state. By engineering the spdc process, it is
possible to tailor, and even to suppress correlations between different degrees
of freedom or between different photons. Chapter 3 considers the spatial part
of the two-photon state, after the correlations between space and frequency are
suppressed.
15

2. Correlations and entanglement
(a)
Space Frequency Signal
q
q
s
i
s
i
qs
qi
Idler
Figure 2.1: The frequency part, gray in (a), is traced out in section 2.2, to calculate
the state of the subsystem composed by the two-photon state in the spatial degree
of freedom. Calculating the signal photon state in space and frequency, section 2.3,
requires to trace out the idler photon, gray in (b).
2.1 The purity as a correlation indicator
When a physical system is used to implement a task, its state should be characterized.
For quantum states, the reconstruction of the state density matrix
demands a tomography [1]. This process requires an infinite amount of copies
of the state in order to eliminate statistical errors [42]. In an experimental
implementation, the number of copies of the state available is finite, but even
under this condition an experimental tomography is a complex process. Instead
when just a certain feature of the system is relevant for the task, it is preferable
to use a figure of merit to characterize that part of its quantum state.
For a system in a quantum state given by the density operator ˆρ, the purity
P = T r[ˆρ 2 ] is a figure of merit that describes its ability to be correlated with
others. That is, the purity indicates if correlations can or cannot exist. Since
the purity is a second order function of the density operator, measuring it does
not require a full tomography, but just a simultaneous measurement over two
copies of the system [43, 44].
To illustrate other characteristics of the purity, consider as an example
a complete orthonormal basis for the electromagnetic field, where each basis
vector |n〉 describes a mode of the field [36]. A pure state |R〉 is any state
written as a superposition of basis vectors:
|R〉 =
Cn|n〉, (2.1)
n
where the mode amplitude and the relative phases between modes, given by
the coefficients Cn, are fixed. This pure state can be described by the density
operator ˆρ = |R〉〈R|, which has the maximum possible value for the purity
T r[ˆρ 2 ] = 1.
A mixed state of the electromagnetic field is a state that can not be written
as in equation 2.1. For instance, in the case of a statistical mixture of field
modes, where there are no fixed amplitude or fixed relative phases associated
16
s
i
(b)

2.1. The purity as a correlation indicator
to each vector of the base. There is, however, a fixed probability pR of finding
the electromagnetic field in a state |R〉. Using the density operator formalism,
the probability distribution is given by
ˆρ =
pR|R〉〈R|. (2.2)
R
The purity of the state described by ˆρ quantifies how close it is to a pure
state. If the state is pure then T r[ˆρ 2 ] = 1, and if it is maximally mixed then
T r[ˆρ 2 ] = 1/n, where n is de dimension of the Hilbert space in which the state is
expanded. Maximally mixed states of the electromagnetic field, in the infinite
dimensional spatial or temporal degrees of freedom have a purity T r[ˆρ 2 ] = 0.
To extend the discussion about the purity to the kind of states generated
by spdc, consider the pure bipartite system described by |ψ〉 and composed by
the fields A and B. The Schmidt decomposition guarantees that orthonormal
states |RA〉 and |RB〉 exist for the fields A and B, so that
|ψ〉 =
λR|RA〉|RB〉 (2.3)
R
where λR are non-negative real numbers that satisfy
R λR = 1, and are
known as Schmidt coefficients [1, 17]. The average of the nonzero Schmidt
coefficients is a common entanglement quantifier [11] known as the Schmidt
number and given by
K =
1
R λ2 R
. (2.4)
While equation 2.3 describes the state of the whole bipartite system, the state
of each part is calculated by tracing out the other part. Thus, the states of the
fields A and B are given by the density operators
ˆρA =
λR|RA〉〈RA| (2.5)
and
R
ˆρB =
λR|RB〉〈RB|. (2.6)
R
Since both operators have equal eigenvalues λ2 R , the purity of both states is
equal, and given by the inverse of the Schmidt number K
T r[(ˆρ A ) 2 ] = T r[(ˆρ B ) 2 ] = 1
K
= λ 2 R. (2.7)
Thus, when the field A is in a pure state, the field B is in a pure state too,
and
R λ2
R = 1. Since R λR = 1, the Schmidt coefficients satisfy the new
condition only if all except one of them are equal to zero. If that is the case,
the bipartite system in equation 2.3 is a product state, and no correlations
exist between A and B. In other words, if there are correlations, the purity of
A (and B) is smaller than 1.
In conclusion, when considering a composed global system, the purity of
each subsystem measures the strength of the correlation between such subsystem
and the rest. This is always true independently of how the subsystems are
R
17

2. Correlations and entanglement
chosen. In the particular case of a two-photon system, the purity of the spatial
part can be used to study correlations between the degrees of freedom, or the
purity of the signal photon can be used to study the entanglement between the
photons. The next sections explore both these approaches.
2.2 Correlations between space and frequency
In type-i spdc, the photons generated have a polarization orthogonal to the
polarization of the incident beam. Therefore, the two-photon state generated
in a type-i process has only two degrees of freedom: frequency and transversal
spatial distribution. In this section, I study the origin and the characteristics
of the correlations between those degrees of freedom by using the purity of the
spatial part as a correlation indicator.
Except for hyperentanglement configurations [12, 13] and some other novel
configurations [6, 14], most applications of type-i spdc processes use just one of
the degrees of freedom, ignoring the correlation that may exist between space
and frequency. That is, those configurations assume a high degree of spatial
purity. This section explores the conditions in which this assumption is valid.
The first part of this section contains a discussion about the origin and
suppression of the correlations. The second part includes the calculations for
the spatial purity that is used as correlation indicator. The third part shows
the results of numerical calculation of the spatial purity in several cases.
2.2.1 Origin of the correlations
According to the two-photon mode function, equation 1.34, the correlations
between space and frequency appear as cross terms in the variables associated
to those degrees of freedom, as figure 2.2 shows. The correlations between
q s x
q s y
q i x
y
qi s
i
qs x
a
h
i
j
k
l
q s
y
h
b
m
n
p
r
qi y
i
m
c
s
t
u
Figure 2.2: Cross terms in matrix A generate correlations between space and frequency.
Without those terms, the matrix A is composed by two nonzero block matrices,
one for each degree of freedom.
space and time are always relevant, if at least one of the cross terms k, l, p,
r, t, u, v, w is comparable with another term of matrix A. Therefore, it is
possible to suppress the correlations by making all the cross terms negligible,
or by increasing the values of the other terms.
The use of ultra-narrow filters in one of the degrees of freedom will suppress
most of the information available in it, destroying any possible correlation with
18
q i
x
j
n
s
d
v
w
s
k
p
t
v
f
z
i
l
r
u
w
z
g

2.2. Correlations between space and frequency
other degrees of freedom. Filtering makes the diagonal terms of matrix A larger
than the rest of the terms, including the terms responsible for the correlations.
But while they remove the correlations, the filters reduce the amount of photons
available for any measurement.
An appropriate design of the source makes it possible to suppress cross
terms without using filters. Such a design should fulfill the conditions
Np − Ns cos ϕs
Np − Ni cos ϕi
Np − Ns cos ϕs
Np − Ni cos ϕi
Np − Ns cos ϕs = 0
Np − Ni cos ϕi = 0
1 + w2 p
γ2L2 1 + w2 p
γ 2 L 2
1 − w2 p
γ 2 L 2
1 − w2 p
γ 2 L 2
Ns sin ϕs
tan ϕi
Ni sin ϕi
tan ϕs
= 0
= 0
= 0
= 0. (2.8)
These conditions can be met for example by using a long crystal, a highly
focused pump beam, for which wp/γL → 0, and emission angles such that
cos ϕs = Np/Ns and cos ϕi = Np/Ni.
Even though these conditions have been deduced in a simplified regime,
reference [45] shows that these conditions are sufficient to remove the correlations
when considering the general mode function of equation 1.12. Once the
correlations are suppressed, the two-photon mode function can be written as a
product of a spatial and a frequency two-photon mode function like
Φ(q s, ωs, q i, ωi) = Φq(q s, q i)ΦΩ(Ωs, Ωi). (2.9)
Since the global two-photon system in space and frequency is in a pure state,
the presence of correlations can be confirmed by using the purity. The next
section calculates the purity of the spatial state, which can be used as an
indicator of correlations between space and frequency.
2.2.2 Two-photon spatial state
To study space/frequency correlations I will describe the purity of the spatial
part of the two-photon state. As the purity of the spatial and frequency parts
of the state are equal, all results derived for the spatial two-photon state extend
to a frequency two-photon state.
After tracing-out the frequency from the two-photon state, the remaining
state describes the spatial part of the state. The reduced density matrix for
19

2. Correlations and entanglement
the spatial two-photon state is
ˆρq = T rΩ[ρ]
=
=
dΩ ′′
s dΩ ′′
i 〈Ω ′′
s , Ω ′′
i |ˆρ|Ω ′′
s , Ω ′′
i 〉
dqsdΩsdqidΩidq ′ sdq ′ i
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)|qs, qi〉〈q ′ s, q ′ i|, (2.10)
while the purity of this state, defined as T r[ˆρ 2 q], is given by
T r[ˆρ 2
q] =
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (2.11)
It is possible to solve these integrals analytically by using the exponential
character of the mode function given by equation (1.35). The purity becomes
T r[ˆρ 2 q] = det(2A)
det(B) . (2.12)
As seen in appendix A, B is a positive-definite real 12 × 12 matrix
B = 1
⎛
⎜
2 ⎜
⎝
2a
2h
2i
2j
k
l
0
0
0
0
k
2h
2b
2m
2n
p
r
0
0
0
0
p
2i
2m
2c
2s
t
u
0
0
0
0
t
2j
2n
2s
2d
v
w
0
0
0
0
v
k
p
t
v
2f
2z
k
p
t
v
0
l
r
u
w
2z
2g
l
r
u
w
0
0
0
0
0
k
l
2a
2h
2i
2j
k
0
0
0
0
p
r
2h
2b
2m
2n
p
0
0
0
0
t
u
2i
2m
2c
2s
t
0
0
0
0
v
w
2j
2n
2s
2d
v
k
p
t
v
0
0
k
p
t
v
2f
l
r
u
w
0
0
l
r
u
w
2z
l r u w 0 0 l r u w 2z 2g
defined by the equation
N 4
exp − 1
2 Xt
BX = Φ(qs, Ωs, qi, Ωi)
⎞
⎟ ,
⎟
⎠
(2.13)
× 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)
where vector X is the result of concatenation of x and x ′ .
In order to see the effect of each of the spdc parameters on the spatiofrequency
correlations, some typical values are used in next subsection to calculate
T r[ˆρ 2 q] numerically.
2.2.3 Numerical calculations
As first example, consider a degenerate type-i spdc process characterized by the
parameters in the second column of table 2.1. A pump beam, with wavelength
20

2.2. Correlations between space and frequency
λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium iodate (liio3)
crystal with length L = 1 mm, and negligible Poynting vector walk-off (ρ0 = 0).
The crystal emits signal and idler photons with a wavelength λ0 s = λ0 i = 810
nm at an angle ϕs,i = 10◦ .
Under the above conditions, figure 2.3 shows the purity of the spatial state
T r[ρ2 q] given by equation 2.12, as a function of the spatial filter width for different
frequency filter bandwidths. The half width at 1/e of the frequency filters
∆λn, is Bn = πc∆λn/(λ2 √
n ln 2). Figure 2.3 shows the presence of correlations
between space and frequency in all the situations for which the purity is smaller
than one.
According to figure 2.3, the spatial purity of the two-photon state gets closer
to one as ws (= wi) increase. Narrow filters in space (infinitely large ws,i) makes
the two-photon state separable in frequency and space. The separability and
the lack of correlations between frequency and space appear also in the“narrow
band” limit for the frequency. If ∆λs = ∆λi → 0 nm, then the purity is always
1 for any value of ws = wi.
When using typical interference filters, ∆λs = ∆λi ≈ 1 nm, the purity decreases
indicating space/frequency correlations. As the width of the frequency
filters increases, the purity converges quickly, the case of ∆λs = ∆λi = 10 nm
is almost equivalent to the case without frequency filters ∆λs, ∆λi → ∞.
In order to compare the theoretical results of this section with previous
experimental work, figure 2.4 shows the purity of the spatial state as a function
of the collection mode for three reported experiments. Figure 2.4 (a) shows the
purity for the data reported in reference [6]. The authors of that paper studied
a type-i spdc process in a 1 mm long liio3 crystal. They used a diode laser
with wavelength λp = 405 nm and bandwidth ∆λp = 0.4 nm as a pump beam.
And, by using monochromators with ∆λs,i = 0.2 nm, they detected pairs of
photons emitted at ϕs,i = 17◦ . Table 2.1 resumes these parameters.
The paper analyzes two pump waist values: wp = 30 µm and wp = 462 µm.
For those values, the authors demonstrated control of the frequency correlations
by changing the spatial properties of the pump beam. As the pump beam
influences the spatial shape of the generated photons, they implicitly reported
a correlation between space and frequency. This spatial to spectral mapping
exists at their collection mode ws = 133.48 µm for which the correlation indeed
appears according with our model.
Figure 2.4(b) shows T r[ρ2 q] for the case described in reference [46]. The
authors used a continuous wave pump beam with λp = 405 nm to generate
photons with λs,i = 810 nm in a 1.5 mm length bbo crystal. They detected the
photons after filtering with ∆λs,i = 10 nm interference filters. The pump beam
waist satisfied the condition wp >> L, and the generated photons propagated
at ϕs,i = 0◦ .
The experiment reported in reference [46] does not show evidence of spacefrequency
correlations. In agreement with this fact, the figure shows a lack
of correlation for the collinear case. However, under the conditions reported,
but in a non collinear configuration, it is not possible to neglect the space and
frequency correlations.
Finally figure 2.4 (c) plots T r[ˆρ 2 q] for the case described in reference [31].
In this paper, the authors illuminated a 2 mm long bbo crystal with a pump
beam with λp = 351.1 nm and wp ≈ 20 µm. Their crystal generated photons
at λs,i = 702 nm that propagated at 4o . They collected the photons after using
21

2. Correlations and entanglement
Filters
0nm
1nm
10nm
→∞
0.1 Collection mode 1mm
Spatial
purity
Figure 2.3: The spatial purity increases by filtering the state in space or frequency.
For the parameters chosen here, the bandwidth of the generated photons is smaller
than 10 nm, making 10 nm filters equivalent to infinitely broad ones. The parameters
used to generate this figure are listed in table 2.1.
Spatial
purity
1
0
0.1
Pump
waist
30m 462m Emission
angle
1mm 0.1 Collection mode 1mm
1
0.2
1º
5º
(a) (b) (c)
0º
0.1 1mm
Pump
waist
20m 500m Figure 2.4: For the values reported in reference [6], figure (a) shows that the purity
is smaller than 1. This fact is in agreement with the implicit spatio-temporal
correlations reported in the reference. For the values reported in references [46] and
[31], figures (b) and (c) show that the purity has its maximum value. There was no
evidence of spatio-temporal correlations in the results reported in these references.
In all three cases, it is possible to modify the value of the purity by tailoring the spdc
parameters. The parameters used to generate this figure are listed in table 2.1.
10 nm interference filters.
For the noncollinear configuration in reference [31], the highly focused pump
beam is responsible for the lack of correlations. The figure shows that for lessfocused
pump beams the purity decreases as the correlations between space
and frequency become more important.
The correlations between frequency and space in the two-photon state discussed
in this section, suggest that a complete description of the correlations
between the generated photons, should take into account both degrees of freedom,
as the next section discusses.
22

2.3. Correlations between signal and idler
Table 2.1: The parameters used in figures 2.3 and 2.4
Parameter Figure 2.3 Figure 2.4(a) Figure 2.4(b) Figure 2.4(c)
Crystal liio3 liio3 bbo bbo
L 1 mm 1 mm 1.5 mm 2 mm
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦
T0 → ∞ → ∞ → ∞ → ∞
wp 400 µm 30 / 462 µm wp ≫ L 20, 500 µm
λp 405 nm 405 nm 405 nm 351.1 nm
∆λp 0 0.4 nm
λ 0 s 810 nm 810 nm 810 nm 702 nm
∆λs 0.2 nm 10 nm 10 nm
ϕs 10 ◦ 17 ◦ 0, 1, 5 ◦ 4 ◦
2.3 Correlations between signal and idler
In contrast to the previous section, here the two-photon system is considered
as composed by two photons, each described by space and time variables, as
figure 2.1 (b) shows. Since the global state in equation 1.9 is pure, the purity
of the signal photon calculated in this section is a measurement of the degree of
spatiotemporal entanglement between signal and idler photons, as in references
[5, 10, 11, 47, 48].
spdc as a source of two-photon states can be used to generate pairs of
photons maximally entangled or two single photons [5, 49]. The purity of the
signal photon in the first case should be equal to 0 and in the second case equal
to 1. This section explores the necessary conditions to be in each regime.
This section is divided in three parts, the first one discusses the origin of
the correlations and the strategies to suppress them. The second part describes
how to obtain an analytical expression for the signal photon purity. And the
third part shows the results of numerical calculation of that purity.
2.3.1 Origin of the correlations between photons
The presence of cross terms in variables of both signal and idler in equation
1.34 indicates correlations between the generated photons. Figure 2.5 shows
those terms: i, j, l, m, n, r, t, v, and z.
In contrast to the last section, filtering one degree of freedom is not enough
to suppress the correlations between signal and idler. For example, by using a
frequency filter, the photons can be still correlated in space. The total suppression
of the correlation requires infinite narrow filters in both degrees of freedom
for one of the photons, as shown experimentally in reference [50]. But that kind
of filtering reduces the number of available pairs of photons substantially.
The other possibility to generate uncorrelated photons is to tailor the parameters
of the spdc process to make the cross terms negligible simultaneously.
According with appendix A, when α = 0 ◦ , this condition is equivalent to mak-
23

2. Correlations and entanglement
q s x
q s y
q i x
y
qi s
i
qs x
a
h
i
j
k
l
q s
y
h
b
m
n
p
r
qi y
i
m
c
s
t
u
Figure 2.5: The signal-idler cross terms in matrix A are responsible for the correlations
between the generated photons. By suppressing these cross terms, and by making
permutations over columns and rows, the matrix becomes a two block matrix, where
each block contains the information of one photon.
ing the following terms negligible
i =w 2 p + γ 2 L 2 tan ρ0 2
j =γ 2 L 2 sin ϕi tan ρ0
l =γ 2 L 2 tan ρ0(Np − Ni cos ϕi)
m = − γ 2 L 2 sin ϕs tan ρ0
q i
x
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi
t =γ 2 L 2 tan ρ0(Np − Ns cos ϕs)
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs
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
j
n
s
d
v
w
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi. (2.15)
As an example, a configuration with a negligible Poynting walk-off fulfills this
condition always when
cos ϕs = Np
2Ns
cos ϕi = Np
2Ni
w 2 p
γ 2 L 2 = tan ϕs tan ϕi
T 2 0 =N 2 p γ 2 L 2
wp ≪ws, wi
s
k
p
t
v
f
z
i
l
r
u
w
tan ϕs 2 tan ϕi 2 − 1
4
z
g
(2.16)
When the correlations are totally suppressed by satisfying these or other conditions,
the two-photon state is separable, and each photon is in a pure state.
Therefore, the two-photon mode function can be written as a product of two
mode functions, one for each photon,
24
Φ(q s, ωs, q i, ωi) = Φs(q s, Ωs)Φi(q i, Ωi). (2.17)

2.3. Correlations between signal and idler
In any other case, correlations between signal and idler are unavoidable. To
evaluate their strength, the next section deduces an analytical expression for
the purity of the signal photon state.
2.3.2 Spatio temporal state of signal photon
A single photon state can be generated in spdc by ignoring any information
about one of the generated photons. The single photon state is calculated by
tracing out the other photon from the two-photon state. For example, after
a partial trace over the idler photon, the reduced density matrix in space and
frequency for the signal photon is
ˆρsignal =T ridler[ρ]
=
=
and its purity is given by
dq ′′
i dΩ ′′
i 〈q ′′
i , Ω ′′
i |ˆρ|q ′′
i , Ω ′′
i 〉
dqsdΩsdqidΩidq ′ sdΩ ′ s
Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)|qs, Ωs〉〈q ′ s, Ω ′ s|, (2.18)
T r[ˆρ 2 signal] =
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (2.19)
Recalling the exponential character of the mode function described by equation
1.35, equation 2.19 writes
T r[ˆρ 2 signal] = det(2A)
det(C) , (2.20)
where C is a positive-definite real 12 × 12 matrix defined by
N 4
exp − 1
2 Xt
CX = Φ(qs, Ωs, qi, Ωi)
and given by
⎛
C = 1
2
⎜
⎝
× 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)
2a 2h i j 2k l 0 0 i j 0 l
2h 2b m n 2p r 0 0 m n 0 r
i m 2c 2s t 2u i m 0 0 t 0
j n 2s 2d v 2w j n 0 0 v 0
2k 2p t v 2f z 0 0 t v 0 z
l r 2u 2w z 2g l r 0 0 z 0
0 0 i j 0 l 2a 2h i j 2k l
0 0 m n 0 r 2h 2b m n 2p r
i m 0 0 t 0 i m 2c 2s t 2u
j n 0 0 v 0 j n 2s 2d v 2w
0 0 t v 0 z 2k 2p t v 2f z
l r 0 0 z 0 l r 2u 2w z 2g
⎞
⎟ .
⎟
⎠
(2.22)
25

2. Correlations and entanglement
Signal
purity Filters
1
0
0.1 Collection mode 1mm
0nm
1nm
10nm
→∞
Figure 2.6: The correlations between photons can not be suppressed by filtering
in only one degree of freedom. The purity of the signal state is only maximum
when using ultra-narrow spatial and frequency filters. In this case, the 1 nm filters
are broad enough to allow the correlations, and the 10 nm filters are equivalent to
infinitely broad ones. The parameters used to generate this figure are listed in table
2.2.
The differences between the matrix C and matrix B in the last section, are
due to the order of the primed and unprimed variables in the arguments of the
mode functions in equations 2.14 and 2.21.
The next part of this section uses equation 2.20, to calculate the values of
the purity of the signal photon for different spdc configurations.
2.3.3 Numerical calculations
Figure 2.6 shows the space-frequency purity of the signal photon as a function
of the spatial filter width for different values of the frequency filter width.
Consider a degenerate type-i spdc configuration, where a pump beam with
wavelength λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium
iodate (liio3) crystal with length L = 1 mm, and negligible Poynting vector
walk-off (ρ0 = 0). The crystal emits signal and idler photons with a wavelength
λ 0 s = λ 0 i = 810 nm at an angle ϕs,i = 10 ◦ , and the collection modes for signal
and idler are assumed to be equal (ws = wi). All these parameters are listed
in the first column of table 2.2.
As was discussed in the first part of this section, it is possible to achieve
maximal separability between the photons by using infinitely narrow filters
in both space and frequency. In the region of small values of ws (= wi) a
considerable correlation between signal and idler exists even in the case of
infinitely narrow frequency filters. Different values for the signal photon purity
can be achieved by changing the filter width, as shown by figure 2.6.
Additionally, the figure shows how the purity T r[ˆρ 2 signal ] is confined between
the values obtained for ∆λs = ∆λi → 0 nm and ∆λs, ∆λi → ∞. This limits
can be tailored by modifying the other parameters of the spdc configuration.
Figure 2.7 shows T r[ˆρ 2 signal ] as a function of the pump beam waist wp for
26

Signal
purity
1
0
(a)
0.1 3
2.3. Correlations between signal and idler
(b)
0.1 Pump waist 3
(c)
0.1
Emission
angle
Figure 2.7: The effect of the emission angle on the signal purity changes depending
on the filters used in space and frequency. When both filters have a finite size like in
(a), the purity increases with the angle of emission, and has a maximum for a fixed
value of the pump beam waist. When the frequency filters are infinitely narrow like
in (b), the dependence of the purity on the angle of emission disappears. When using
narrow filters in space, as in (c), the purity is equal to 1 for a certain pump beam
waist that changes with the angle of emission. The parameters used to generate this
figure are listed in table 2.2.
Table 2.2: The parameters used in figures 2.6 and 2.7
Parameter Figure 2.6 Figure 2.7(a) Figure 2.7(b) Figure 2.7(c)
Crystal liio3 liio3 liio3 liio3
L 1 mm 1 mm 1 mm 1 mm
ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦
T0 → ∞ → ∞ → ∞ → ∞
wp 400 µm 400 µm 400 µm 400 µm
λp 405 nm 405 nm 405 nm 405 nm
∆λp → 0 → 0 → 0 → 0
ws variable → ∞ 400 µm 400 µm
λ 0 s 810 nm 810 nm 810 nm 810 nm
∆λs 0, 1, 10 nm → ∞ 10 nm → 0 10 nm
ϕs 10 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦
various values of the emission angle ϕs = ϕi, in degenerate type-i spdc configurations
described by the parameters listed in the second, third and fourth
columns of table 2.2.
Figure 2.7 (a) shows the case of finite spatial and temporal filters. The
purity of the signal photon is always smaller than one, increases with the emission
angle, and has a maximum for a given value of the pump beam waist. For
a very narrow band frequency filter with a ws = wi = 400 µm spatial filter,
figure 2.7 (b) shows how the correlation between the photons is minimal for
small pump beams. The emission angle for this particular crystal length is
irrelevant. Finally, figure 2.7 (c) considers a frequency filter ∆λs = ∆λi = 10
nm with infinitely narrow spatial filters. In this case, maximal purity appears
for each emission angle at a particular value of the pump beam waist.
Conclusion
Two photons with two degrees of freedom compose the two-photon state generated
in spdc. The correlations between all four elements affect the state of
single photons or two-photon states with one degree of freedom. Using the pu-
3mm
5º
10º
20º
27

2. Correlations and entanglement
rity as correlation indicator it is possible to determine the conditions in which
the specific correlations are suppressed, without requiring the use of infinitely
narrow filters.
The next chapter continues with the description of the correlations between
photons, in the case where only their spatial state is relevant. The chapter describes
the spatial correlations in terms of orbital angular momentum transfer.
28

CHAPTER 3
Spatial correlations and
orbital angular momentum
transfer
The previous chapter describes the correlations between the signal and idler
photons considering space and frequency, as well as the correlations between
those degrees of freedom. This chapter focuses on the cases where the degrees
of freedom are not correlated, and the only relevant correlations are
those between the photons in the spatial degree of freedom. To characterize
those correlations I use the orbital angular momentum (oam) content of
the generated photons, which is directly associated to their spatial distribution.
The spatial correlations between the photons determine the mechanism
of oam transfer from the pump to the signal and idler. Most studies of the
oam correlations in spdc can be divided in two categories: those reporting a
full transference of the pump’s oam [7, 8, 22, 27, 28, 29], and those reporting
a partial transference [30, 31, 32, 33]. This chapter describes the oam transfer
mechanism, and shows the difference between both regimes. The chapter is
divided in three sections. Section 3.1 introduces the eigenstates of the oam
operator: the Laguerre-Gaussian modes. This section shows how to calculate
and measure the oam content of the photons. Section 3.2 describes how the
oam is transferred in spdc. As an example, section 3.3 shows how this transfer
happens in the collinear case, where all the emitted photons propagate in the
same direction. The main result of this chapter is a selection rule that summarizes
the oam transfer mechanism: the oam carried by the pump is completely
transferred to all the signal and idler photons emitted over the cone.
29

3. Spatial correlations and OAM transfer
-1
0 1 2
OAM content
in ħ units
Phase front
Intensity
profile
Figure 3.1: The Laguerre-Gaussian modes are characterized by their phase front
distribution and intensity profile. The figure shows the phase fronts and the intensity
profiles for the modes with oam contents from −1 to 2.
3.1 Laguerre-Gaussian modes and OAM content
In an analogous way to the decomposition of an electromagnetic field as a
series of planes waves, it is possible to decompose the field in other bases. For
instance, paraxial fields can be decomposed as a sum of Laguerre-Gaussian (lg)
modes. This basis is especially convenient since the lg modes are eigenstates of
the orbital angular momentum (oam) operator. That is, the state of a photon
in a lg mode has a well defined oam value [51]. This section describes the
properties of the lg modes, and shows how to calculate the weight of each
mode in a given decomposition.
Photons carrying oam different from zero have at least one phase singularity
(or vortex) in the electromagnetic field, a region in the wavefront where the
intensity vanishes. This is precisely one of the most distinctive characteristics
of Lagurre-Gaussian modes as figure 3.1 shows. Each lg mode is defined by
the number p of non-axial vortices, and the number l of 2π-phase shifts along
a close path around the beam center. The index l also describes the helical
structure of the phase front around the singularity, and more important here,
l determines the orbital angular momentum carried by the photon in units.
The state of a single photon in a lg mode is
|lp〉 = dqLGlp(q)â † (q)|0〉, (3.1)
where the mode function lglp(q) is given by Laguerre-Gaussian polynomials
1
2 wp!
LGlp(q) =
2π(|l| + p)!
|l|
wq
√2 L |l|
2 2 w q
p
2
× exp − w2q2
exp ilθ + iπ(p −
4
|l|
2 )
(3.2)
as a function of the beam waist w, the modulus q and the phase θ of the
transversal vector, and the associated Laguerre polynomials L |l|
p defined as
30
L |l|
p [x] =
p
i=0
i
l + p (−x)
. (3.3)
p − i i!

3.2. OAM transfer in general SPDC configurations
A special case of the lg modes is the zero-order mode that does not carry oam.
Since the zero-order Laguerre polynomial L 0 0 = 1, the zero-order Laguerre-
Gaussian lg00 = 1 is the Gaussian mode given by
w
1
2
LG00(q) = exp
2π
− w2 q 2
4
. (3.4)
This is not, however, the only mode with oam equal to zero, the same is true
for all other modes lg0p. Spiral harmonics are defined to collect all the modes
with the same oam value, regardless of the value of p, these modes are defined
as
LGl(q) =
LGlp(q) (3.5)
p
=al(q) exp [ilθ].
The phase dependence on l, shown by each spiral harmonic mode, is exploited
to determine the photon’s oam content [52]. Consider, for instance, a photon
with a spatial distribution given by Φ(q), which using the spiral harmonic
modes can be written as
∞
Φ(q) = al(q) exp [ilθ], (3.6)
l=−∞
so that each mode in the decomposition has a well defined oam of l per
photon. Therefore, the probability Cl of having a photon with oam equal to l
is the weight of the corresponding mode in the distribution:
where
Cl =
al(q) = 1
√ 2
∞
0
2π
0
dq|al(q)| 2 q (3.7)
dθΦ(q, θ) exp (−ilθ). (3.8)
If the photon has a well defined oam of l0, the weight of the corresponding
mode Cl0 = 1 and the weights of the other modes Cl=l 0 = 0. If Cl = 0 for
different values of l, the photon state is a superposition of those modes with
different oam values. Compared to the relative simplicity of these calculations,
measuring the oam content is a more complicated task described in appendix
C.
Now that the techniques for the calculation of the oam content are introduced,
the next section will use the spherical harmonics and the oam decomposition
to study the transfer of oam from the pump photon to the signal and
idler.
3.2 OAM transfer in general SPDC configurations
The oam carried by a photon is directly associated to its spatial shape. In
order to simplify the description of the oam transfer mechanisms, this section
considers only the spatial part of the mode function in a spdc configuration
31

3. Spatial correlations and OAM transfer
in which the space and the frequency are not correlated. By writing the mode
function in a different coordinate system, it will be possible to show that a
selection rule exists for the oam transfer in spdc.
According to equation 1.6, all the fields have been defined in terms of a
coordinate system related with their propagation direction. To study the oam
transfer, we start by defining the fields in terms of a unique coordinate system,
the one of the pump. This coordinate transformation is possible since there is
a vector Kn such that kn · rn = Kn · rp.
This coordinate transformation is more general than the one of section 1.3,
where the plane that contained all photons was defined as the yz plane. As
a consequence this section considers all possible emission directions over the
cone, while section 1.3 considered only one.
As was done for the vector kn in section 1.3, it is useful to separate the
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
function in the semiclassical approximation can be written as
∆kL
Φq(Qs, Qi) ∝ Ep(Qs + Qi)sinc
(3.9)
2
where the delta factor is given by
with
∆k = K z p(Qs + Qi) − K z s (Qi) − K z i (Qi). (3.10)
K z j (Q) =
ω 2 j n2 j
c 2 − |Q|2 . (3.11)
To study the oam transfer, consider the two functions in equation 3.9 separately.
Consider first the sinc function. Due to its dependence on the delta
factor, the sinc function depends only on the magnitudes |Qs +Qi|, Qs = |Qs|,
Qi = |Qi|. And, since
|Qs + Qi| 2 = Q 2 s + Q 2 i + 2QsQi cos(Θs − Θi), (3.12)
the only angular dependence of the function is through the periodical term
cos(Θs − Θi), therefore, using Fourier analysis it can be written in a very
general form as
∆kL
sinc =
2
∞
m=−∞
Fm(Qs, Qi) exp [im(Θs − Θi)], (3.13)
where, importantly, Fm(Qs, Qi) does not depend on the angles Θs and Θi.
On the other hand, if we consider the pump beam as a Laguerre-Gaussian
beam with a oam content of lp per photon, its mode function can be written
as
32
Ep(Qs + Qi) ∝ exp
− w2 p|Qs + Qi| 2
4
(Q x s + Q x i ) + i(Q y s + Q y
i )
l , (3.14)

3.2. OAM transfer in general SPDC configurations
where the identity Q exp[iθ] = Q x + iQ y was used. Using equation 3.12, the
last equation can be written as
Ep(Qs + Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
4
l × Qs exp (iΘs) + Qi exp (iΘi)
and by using the binomial theorem it becomes
(3.15)
lp
lp
Ep(Qs + Qi) ∝ Q
l
l=0
l sQ lp−l
i exp [ilΘs + ilpΘi − ilΘi]
× exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
. (3.16)
4
By replacing equations 3.16 and 3.13 into equation 3.9, the two-photon spatial
mode function becomes
Φ(Qs, Qi) ∝ exp − w2 p(Q2 s + Q2 i + 2QsQi
cos(Θs − Θi))
4
∞
× Gn(Qs, Qi) exp [inΘs + i(lp − n)Θi], (3.17)
n=−∞
where the value of Gn(Qs, Qi) does not depend on Θs and Θi. The index
(lp − n) associated to Θi is fixed for each value of lp and ls, according to the
phase dependence of the last equation. This relation can be written as the
selection rule
lp = ls + li. (3.18)
Therefore, the angular momentum carried by the pump is completely transferred
to the signal and idler photons.
The geometry of the spdc configuration restricts the validity of the selection
rule since it was deduced from equation 3.9, and that equation describes
only the configuration in which all possible emission directions are taken into
account. Collinear configurations achieve this condition by definition, but in
the type-I noncollinear configurations, all the experiments up to now have measured
only a certain portion of the whole cone. Under this condition the transfer
of oam from the pump to the signal and idler is not governed necessarily by
equation 3.18, in agreement with previous reports [32, 30, 31].
Using numerical methods to calculate the oam content of the downconverted
photons, the next section explores the oam transfer mechanisms in
collinear configurations. This configurations are special cases where the generated
photons propagate in the same direction and therefore, the detection
system detects all photons from the emission cone.
33

3. Spatial correlations and OAM transfer
Configuration
Weight
1
0
-4 4
0
Spatial
-4 4
distribution OAM content
1
Mode
Figure 3.2: In a collinear configuration a pump beam with oam equal to 1 transfers
its oam into the signal photon when the idler is projected into a Gaussian mode. The
figure shows the pump in blue and the signal and idler in pink. Additionally, it shows
the spatial distribution and the oam content of the signal photon for co-propagating
and counter-propagating configurations.
3.3 OAM transfer in collinear configurations
The aperture of the down-conversion cone is given by the angle of emission
of the photons. In the collinear configuration this angle is zero, and therefore
the cone collapses onto its axis, as shown in figure 1.2 (d). As each photon
propagates in the same direction as all the other photons, equation 3.18 is
valid.
To analyze the transfer of oam from the pump beam to the generated photons,
we will calculate the signal oam content, assuming a lg1 pump beam so
that lp = 1, and an idler photon projected into a Gaussian state li = 0. We
will consider two kinds of collinear configurations: co-propagating and counterpropagating,
both shown in figure 3.2. The collinear counter-propagating generation
of photons requires fulfilling a special phase-matching condition, which
can be achieved by quasi-phase-matching in a periodic structure. Such counterpropagating
generation has been demonstrated in second harmonic generation
[53], parametric fluorescence [54], and spdc in fibers [55]. Chapter 5 explores
another way to generate counter-propagating two-photon states: Raman scattering.
Figure 3.2 shows the calculated signal spatial distribution and oam content
in two cases. The first row shows the collinear co-propagating case. The second
row shows the collinear counterpropagating case.
According to figure 3.2, the collinear case fulfills the selection rule, except
for a minus sign in the counterpropagating case. The phase appears because
the photon’s oam content is evaluated with respect to the pump propagation
direction, as figure 3.3 shows. To take this effect into account equation 3.18
34

Phase front
3.3. OAM transfer in collinear configurations
OAM respect to z
direction
Figure 3.3: The direction of rotation of the phase shift defines the sign of the mode’s
oam content. For a Laguerre-Gaussian mode corresponding to positive oam content,
the phase shift rotates clockwise with respect to the propagation direction. If the
same mode propagates backwards, the phase shift rotates anticlockwise with respect
to the original propagation direction, and the mode has a negative oam content.
can be generalized to
lp = dsls + d1li
+2
-2
-2
(3.19)
where ds,i = ±1 corresponds to forward and backward propagation of the
corresponding photon.
Conclusion
The oam content of the pump is completely transferred to the signal and idler
photons, emitted all over a cone. This makes it possible to introduce a selection
rule that always holds if all possible emission directions are considered, but not
necessarily if only a small part is detected. This result clarifies the apparent
contradiction between several works that support the selection rule [7, 28, 56],
and those that report that it is not valid [32, 31, 57].
When only a portion of all generated photons are considered there is no
simple relationship between lp, ls and li. The next chapter studies the oam
transfer in those cases, describing the effect of different spdc parameters on
the amount of oam that is transferred to the subset of the photons considered.
35

CHAPTER 4
OAM transfer in
noncollinear configurations
The previous chapter describes the oam transfer from the pump to the signal
and idler photons considering all the possible emission directions of those
photons. However, in non collinear configurations, the photons detected are
just a subset of the total emission cone; noncollinear configurations generate
correlated photons, which are naturally spatially separated, which is exactly
what makes noncollinear so important. The change in the geometry of the
process, imposed by the detection system, has a strong effect on the oam of
the detected photons. This chapter focuses on the description of the oam that
is transferred to a small portion of the emitted photons in noncollinear configurations.
To characterize the transfer, I study a particular spdc process in
which the pump is a Gaussian beam and one of the photons is projected into a
Gaussian mode. Therefore, the oam content of the other photon will indicate
how close the configuration is to satisfying the oam selection rule. This chapter
is divided in three sections. Section 4.1 shows a simple example in which the
selection rule does not apply as just a small section of the cone is considered.
In a more general scenario, section 4.2 shows that the violation of the selection
rule is not only mediated by the emission angle, but also by the pump beam
waist. Finally, section 4.3 shows the effect of the Poynting vector walk-off on
the oam of the noncollinear photons. As a main result, this chapter explains
how the pump beam waist and the Poynting vector walk-off affect the oam
transfer, in the cases where the selection rule does not apply. By tailoring both
parameters it is possible to generate photons with specific spatial shapes. For
instance, it is possible to generate photons in Gaussian modes that are more
efficiently coupled into single mode fibers.
37

4. OAM transfer in noncollinear configurations
4.1 Ellipticity in noncollinear configurations
In the experimental implementation of spdc, the detection system selects the
photons that are emitted in a certain spatial direction. In a collinear configuration,
the selection is not a problem since all the photons are emitted in the
same direction. But, in a noncollinear configuration, the photons are emitted in
different azimuthal directions described by the angle α, therefore the detection
system selects only a section of the full cone. As the symmetry is broken when
only a portion of the cone is considered, the oam transfer mechanism can not
be described by the selection rule in 3.18. This section studies this mechanism
in a simple spdc configuration by calculating the spatial distribution of the
signal photon after fixing the oam content of the pump and idler photons.
As a first example, consider a Gaussian pump beam lp = 0 and an idler
photon projected into a Gaussian mode li = 0, given by
u(qi) = Ni exp − w2 i
4 (qx2 i + q y2
i )
, (4.1)
the spatial distribution of the signal photon is given by the normalized mode
function
Φs(qs) = Ns dqiΦq(qs, qi)u(qi). (4.2)
This integral has an analytical solution for simple spdc configurations. Consider
a degenerate spdc process with negligible Poynting vector walk-off. After
suppressing the correlations between space and frequency the spatial part of
the two-photon state is given by
with
Φq(qs, qi) = exp
− w2 p
4 ∆2 0 − w2 p
4 ∆2 1
∆0 =q x s + q x i
∆1 =q y s cos ϕs + q y
i cos ϕi
∆k = − q y s sin ϕs + q y
i
sin ϕi.
L
sinc
2 ∆k
(4.3)
(4.4)
Therefore, using the sinc to exponential approximation, the signal mode function
defined by equation 4.2 is given by
Φs(qs) =Ns exp
× exp
−
w2 pw2 i
4(w2 p + w2 qx2 s
i )2
− 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
4(w 2 p cos ϕs 2 + γ 2 L 2 sin ϕs 2 + w 2 i )
q y2
s
(4.5)
When the coefficients of the variables q x s and q y s are equal, the signal mode
function reduces to a Gaussian and the selection rule is fulfilled. These coefficients
are equal in collinear configurations, or in noncollinear configurations in
38
.

4.2. Effect of the pump beam waist on the OAM transfer
which the spdc parameters satisfy the relationship
γ 2 L 2 w
=
2 pw4 i
w4 i + 4w2 p(w2 i + w2 p) cos ϕ2 s
. (4.6)
In any other case, the coefficients of the variables q x s and q y s are different, and
the signal mode function becomes elliptical. The ellipticity of the spatial profile
implies the presence of non-Gaussian modes, and therefore it can be used as a
qualitative probe that the selection rule is not fulfilled (lp = ls + li).
As the ellipticity is an effect of the partial detection, it can be controlled
by changing the total shape of the cone, or by changing the detector angular
acceptance. The phase matching conditions define the shape of the cone as a
function of the emission angle ϕs, the pump beam waist wp, and the length of
the crystal L; the detector angular acceptance is a function of the waist of the
spatial modes ws, wi. The next two sections use both the mode decomposition
and the ellipticity to describe the effect of all these parameters on the signal
oam content in a more general scenario than considered in this section.
4.2 Effect of the pump beam waist on the OAM transfer
Of all the spdc parameters that affect the signal ellipticity, the easiest to
control is the pump beam waist. To change the angle of emission or the crystal
length implies changing of the geometrical configuration. While changing the
pump beam waist just requires adding lenses in the beam path. In the first
part of this section, numerical calculations show the role of the pump beam
waist on the signal oam content. The second part describes the experimental
corroboration of this effect.
4.2.1 Theoretical calculations
Consider a spdc configuration as the one in equation 1.31. With a Gaussian
pump beam and an idler photon projected into a Gaussian mode, the oam content
of the signal photon can be used to describe the oam transfer mechanism
in spdc. The selection rule is fullfilled only if ls = 0. Equivalently, one could
choose to use the idler photon to study the oam transfer after projecting the
signal into a Gaussian mode.
In a degenerate type-i spdc process characterized by the parameters in
table 4.1, a Gaussian pump beam, with wavelength λ 0 p = 405 nm illuminates
a 10 mm ppktp crystal. The crystal emits signal and idler photons with a
wavelength λ 0 s = λ 0 i = 810 nm. Both photons propagate at ϕs,i = 1 ◦ , after
the crystal they traverse a 2f system, and finally the idler photon is projected
into a Gaussian mode with wi → ∞, so that only idler photons with qi = 0
are considered.
Figure 4.1 shows the signal oam content for two values of the pump beam
waist: wp = 100 µm in the left and wp = 1000 µm in the right. In the
distributions, each bar represents a mode ls, with a weight in the distribution
Cls given by the height of the bar. In the left part of the figure, where the
pump beam waist is smaller, the distribution shows several modes. For larger
waists, the Gaussian mode becomes the only important mode, as seen in the
right part of the figure.
39

4. OAM transfer in noncollinear configurations
Weight
1
0
(a) (b)
-4 Mode 4 -4 Mode 4
Figure 4.1: As the pump beam waist increases, the Gaussian mode in the signal oam
distribution becomes more important. The left part of the figure, where wp = 100 µm,
shows several non-Gaussian modes in blue, while the right part, where wp = 1000 µm,
shows only a Gaussian mode in gray. The other parameters used to generate this
figure are listed in table 4.1.
Table 4.1: The parameters used in the theoretical calculations in section 4.2.
Parameter Value
Crystal ppktp
L 10 mm
ρ0
0◦ Laser cw diode
wp 100 − 1000 µm
λp
405 nm
λ0 s
ϕs
810 nm
1◦ Figure 4.2 shows the probability of finding a non-Gaussian mode as a function
of the pump beam waist. The presence of non-Gaussian modes implies a
violation of the selection rule in the configuration considered here. The figure
shows the probability Cls = 0 for emission angles 1, 5, 10 ◦ . As the collinear
angle ϕ becomes larger, the probability to detect signal photons with ls = 0
increases. In a highly noncollinear configuration a large pump waist does not
guarantee the generation of a Gaussian mode, that is, it does not imply the
validity of the selection rule.
Reference [30] introduced the noncollinear length defined as Lnc = wp/ sin ϕ.
Lnc quantifies the strength of the violation of the selection rule due to the angle
of emission and the pump beam waist. The authors defined this quantity
based on the fact that when Lq y sin ϕ is small the sinc function (that introduces
ellipticity) tends to one. As q y is of the order of 1/wp the condition can be
written as L sin ϕ/wp ≪ 1, or L ≪ Lnc. In this regime the ellipticity of the
mode function is small, and thus the selection rule lp = ls + li is fulfilled. This
is the case for nearly all the experiments using the oam of photons generated
in spdc [7, 12, 28, 56, 58, 25, 26, 29]. Instead, if the crystal length is larger
than or equal to the noncollinear length L ≥ Lnc, a strong violation of the
selection rule is expected. The experiments reported in references [31, 32, 57]
are in this regime since the authors used highly focused pump beams or long
crystals.
40

1.0
0.0
4.2. Effect of the pump beam waist on the OAM transfer
Nongaussian
mode probability
0.8
0.4
Emission
angle
10º
0.1 Pump beam width 1mm
Figure 4.2: The violation of the selection rule, given by the weight of the non-Gaussian
modes, becomes more important as the collinear angle increases, or as the pump beam
is more focalized. The parameters used to generate this figure are listed in table 4.1.
4.2.2 Experiment
Figure 4.3 depicts the experimental set-up that we used to measure the effect
of the pump beam waist over the signal photon spatial shape. A laser beam
with wavelength λ 0 p = 405 nm, and bandwidth ∆λp = 0.6 nm illuminated a
lithium iodate (liio3) crystal with length L = 5 mm. The crystal was cut in a
configuration for which non of the interacting waves exhibited Poynting vector
walk-off, so ρ0 = 0 ◦ . The signal and idler photons are emitted with wavelengths
λ 0 s = λ 0 i = 810 nm at ϕs,i = 17.1 ◦ . Table 4.2 summarizes the most relevant
experimental parameters.
A spatial filter after the laser provided an approximately Gaussian beam
with a waist of 500 µm. By adding lenses before the crystal, this waist could
be changed in the range of 32 − 500 µm.
Each generated photon passed through a 2 − f system (f = 250 mm),
that provided an image of the spatial shape of the two-photon state. Later,
the photons passed through a broadband colored filter, that removed the scattered
radiation at 405 nm, and through small pinholes to increase the spatial
resolution.
Multimode fibers mounted in xy translation stages collected the photons,
and carried them into two single photon counting modules. A coincidence
circuit counted the number of times that a signal photon was detected within
8 ns of the detection of an idler photon. We used a data acquisition card to
transfer the circuit’s output into a computer.
We measured the signal and idler counts, as well as the number of coincidences
for a fixed position of the idler in two orthogonal directions in the
transverse signal plane as shown by figure 4.3. During the experiments it was
checked that the use of narrow band filters of ∆λs = 10 nm does not modify
5º
1º
41

4. OAM transfer in noncollinear configurations
Laser and
spatial filter lenses
crystal
lenses
filter
collection
system
pinhole
x
idler
pinhole
signal
Figure 4.3: A spatially filtered cw laser is focused into a nonlinear crystal. A coincidence
circuit measures the correlations between the photon counts in a fixed position
for the idler photon with the counts of photons in two orthogonal directions in the
signal transverse plane. The values of the different experimental parameters are listed
in table 4.2.
the measured shape of the coincidence rate, although it does modify the single
counts spatial shape.
Table 4.2: The parameters of the experiment described in section 4.2.
Parameter Value
Crystal liio3
L 5 mm
ρ0
0◦ Laser cw diode
wp 32 − 500 µm
λp 405 nm
∆λp 0.6 µm
λ0 s
∆λ
810 nm
0 s
ϕs
10 nm
17.1◦ The left side of figure 4.4 shows the coincidence measurements in the x and
y directions for a focalized pump beam. As the pump beam waist is wp = 32
µm, the noncollinear length Lnc = 108.8 µm is much smaller than the crystal
length. According with reference [30], in this regime the ellipticity of the signal
photon should be appreciable.
By fitting the result to a Gaussian function, the waist in the x direction
is wx 960 µm, while in the y direction it is wy 150 µm. The waist in x
is about six times larger than the waist in y, and therefore the signal spatial
distribution is elliptical. As a reference, the figure shows the singles counts for
the signal coupler.
The right side of the figure shows the coincidence measurements for a larger
pump beam, wp 500 µm. In this case, the noncollinear length Lnc = 1.7 mm
is of the same order of magnitude as the crystal length. When fitting the
42
y

Coincidences
800
0
-4
4.2. Effect of the pump beam waist on the OAM transfer
x
y
4mm
Singles
6x10 5
~
-1
y x
Singles
5x10 4
~
1mm
Figure 4.4: The ellipticity decreases as the pump beam waist increases from wp = 30
µm in the left to 500 µm in the right image. As the pump beam waist affects
the efficiency of the process, the coincidences on the left were collected over 600
seconds and the ones in the right over 20 seconds. Solid lines are the best fit to the
experimental data shown as points. The values of the experimental parameters are
listed in table 4.2.
Coincidences
region width
2.5m
0.5
0
0
x dimension
y dimension
L nc=1.1mm
Pump width 330 500m Figure 4.5: The pump beam waist controls the width of the profile in the x direction,
while it does not affect the width in the y direction. The width in both directions
becomes similar when the noncollinear length is of the same order of magnitude as
the crystal length. The vertical line shows the point where, according to the fits, both
dimensions are equal. In this point the noncollinear length is 1.1 mm. The values of
the experimental parameters are listed in table 4.2.
transverse cuts to Gaussian functions, the waist in the x direction is wx 120
µm and the waist in the y direction is wy 180 µm. The waist in the x
direction is much smaller than before and of the same order as the waist in y
which implies a decrease of the ellipticity.
Figure 4.4 illustrates the changes in the width in the x and y directions as a
function of pump beam waist. In the range where the waist varies from 32−500
µm, the width in y remains almost constant while the width in x decreases by
80%. After a certain threshold the width in x becomes stable at a value close
43

4. OAM transfer in noncollinear configurations
100µ m
1mm
pump crystal
-4 4
-4 4
1
0
1
0
after 1mm
-4 4
-4 4
0.1
0
1
0
output mode
Figure 4.6: Due to the crystal birefringence, the oam content of a beam changes as
the beam passes trough the material. The number of new oam modes introduced
by the birefringence increases for more focused beams. The figure shows the mode
content of a Gaussian beam after traveling 1 mm in the crystal, and at the output of
the 5- mm-long crystal.
to the value of the width in y, and therefore the ellipticity disappears.
4.3 Effect of the Poynting vector walk-off on the OAM
transfer
Up to now, this chapter only considered spdc configurations where the Poynting
vector walk-off was not relevant. However, since the selective detection of
one section of the cone affects the oam transfer, the distinguishability introduced
by the walk-off should affect it as well.
To understand the effect of the walk-off on the oam transfer, consider that
the spatial shape of the pump is modified as it passes trough the crystal due to
the birefringence. A Gaussian photon thus acquires more modes when traveling
in a crystal. This section explains this effect using theoretical calculations and
experimental results.
4.3.1 Theoretical calculations
The displacement introduced by the walk-off, explained in section 1.3, changes
the pump beam spatial distribution and therefore its oam content. The transverse
profile of the pump, at each position z inside the nonlinear crystal, can
be written as
Ep (qp,z)=E0 exp
−q 2 p
w 2 p
4
z
+ i
2k0
∞
Jn (zqp tan ρ0) exp {inθp}
p n=−∞
(4.7)
where Jn are Bessel functions of the first kind. Based on this expression, figure
4.6 shows the oam content of a Gaussian pump beam after traveling through
a 5 mm crystal. New modes appear and become important as the pump beam
gets narrower.
The oam content of the pump with respect to z is different at different
positions inside the crystal. Pairs of photons produced at the beginning or at
the end of the crystal are effectively generated by a pump beam with different
spatial properties, as figure 4.6 shows. This change in the pump beam is
44

Weight
1
0
4.3. Effect of the Poynting vector walk-off on the OAM transfer
0º 90º Azimuthal angle 360º
other
modes
Gaussian
mode
0º 90º Azimuthal angle 360º
Gaussian
mode
other
modes
Figure 4.7: The probability of generating a Gaussian signal photon varies with the
azimuthal angle, and has a maximum at α = 90 ◦ where the noncollinearity effect
compensates the Poynting vector walk-off. At other angles, like α = 360 ◦ the probability
of generating a Gaussian signal decreases and other modes become important
in the distribution. Those non-Gaussian modes are more numerous for more focused
beams, as seen by comparing the left part of the figure, where wp = 100 µm, to the
right part, where wp = 600 µm.
Signal
purity
0.3
0.1
0.0
without
walk-off
with
walk-off
azimuthal
0º 90º 180º 270º 360º angle
Figure 4.8: Like the signal oam content, the correlations between the photons change
with the azimuthal angle. The walk-off not only introduces an azimuthal variation
but increases the correlations.
translated into a change in the oam content of the generated pair with respect
to the z axis.
Additionally, because of the walk-off the generated photons are not symmetric
with respect to the displaced pump beam, and their azimuthal position
α becomes relevant. Figure 4.7 shows the weight of the mode ls = 0, and the
weight of all other oam modes, as a function of the angle α for two different
pump beam widths. In the left part, the pump beam waist is 100 µm, while
in the right part it is 600 µm. The oam correlations of the two-photon state
change over the down-conversion cone due to the azimuthal symmetry break-
45

4. OAM transfer in noncollinear configurations
ing induced by the spatial walk-off. The symmetry breaking implies that the
correlations between oam modes do not follow the relationship lp = ls + li.
Figure 4.7 also shows that for larger pump beams the azimuthal changes are
smoothened out.
The insets in figure 4.7 show the oam decomposition at α = 90 ◦ and at
α = 360 ◦ . At α = 90 ◦ , the walk-off effect compensates the noncollinear effect,
and the weight of the ls = 0 mode is larger than the weight of any other
mode. This angle is optimal for the generation of heralded single photons with
a Gaussian shape.
The degree of spatial entanglement between the photons also exhibits az-
imuthal variations depending on their emission direction. Figure 4.8 shows the
] as a function of the azimuthal angle α with and with-
signal purity T r[ρ2 signal
out walk-off, for a pump beam waist wp = 100 µm with collection modes of
ws = wi = 50 µm. When considering the walk-off, the degree of entanglement
increases and becomes a function of the azimuthal angle. The purity has a
minimum for α = 0◦ and α = 180◦ and maximum for α = 90◦ and α = 270◦ .
The spatial azimuthal dependence affects especially those experimental configurations
where photons from different parts of the cone are used, as in the
case in the experiment reported in reference [12] for the generation of photons
entangled in polarization. Using two identical type-i spdc crystals with the
optical axes rotated 90◦ with respect to each other, the authors generated a
space-frequency quantum state given by
|Ψ〉 = 1
√ dqsdqi Φ
2
1 q(qs, qi)|H〉s|H〉i + Φ 2
q(qs, qi)|V 〉s|V 〉i
(4.8)
where Φ 1 q(qs, qi) is the spatial mode function of the photons generated in the
first crystal
Φ 1 q(qs, qi) = Φq(qs, qi, α = 0 ◦ ) exp [iL tan ρ0(q y s + q y
i )], (4.9)
and Φ 2 q(qs, qi) is the spatial mode function of the photons generated in the
second crystal
Φ 2 q(qs, qi) = Φq(qs, qi, α = 90 ◦ ). (4.10)
Since the photons generated in the first crystal are affected by the walk-off as
they pass by the second crystal, the mode functions are different.
Following chapter 2, the polarization state of the generated photons is calculated
by tracing out the spatial variables from equation 4.8. The resulting
state is described by the density matrix
where
46
ˆρp = 1
|H〉s|H〉i〈H|s〈H|i + |V 〉s|V 〉i〈V |s〈V |i
2
+c|H〉s|H〉i〈V |s〈V |i + |V 〉s|V 〉i〈H|s〈H|i
(4.11)
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
Length
0.5mm
2mm
beam
waist
Concurrence
1
50 500m Figure 4.9: The concurrence of the polarization entangled state given by equation
4.8 decreases when the effect of the azimuthal variation is stronger, that is for small
pump beam waist or for large crystals.
The degree of correlation between space and polarization is given by the purity
of the polarization state
T r[ρ 2 p] =
0.5
0.0
1 + |c|2
. (4.13)
2
If the walk-off effect is negligible T r[ρ 2 p] = 1, then space and polarization are
not correlated. If the walk-off is not negligible, the azimuthal changes in the
spatial shape are correlated with the polarization of the photons.
Figure 4.9 indirectly shows the effect of the azimuthal spatial information
over the polarization entanglement. The entanglement between the photons
is quantified using the concurrence C = |c|, which is equal to 1 for maximally
entangled states. The figure shows the variation of C as a function of the pump
beam waist for two crystal lengths. For small pump beam waist, the azimuthal
effect is stronger and the concurrence decreases. As the waist increases, the
walk-off effect becomes weaker and the value of the concurrence increases. This
effect is more important for longer crystals, where the walk-off modifies the
pump beam shape more.
4.3.2 Experiment
To experimentally corroborate the predicted azimuthal changes in the signal
spatial shape we set up the experiment described by the parameters listed in
table 4.3 and shown in figure 4.10. We used a continuous wave diode laser
emitting at λp = 405 nm and with an approximate Gaussian spatial profile,
obtained by using a spatial filter. A half wave plate (hwp) controlled the
direction of polarization of the beam, while a lense focalized it on the input
face of the crystal, with a beam waist of wp = 136 µm. The 5 mm liio3 crystal
produced pairs of photons at 810 nm that propagated at ϕs,i = 4 ◦ . Due to
the crystal birefringence, the pump beam exhibited a Pointing vector walk-off
47

4. OAM transfer in noncollinear configurations
Laser and
spatial filter HWP lenses
crystal
lenses
filter
collection
system
camera
pinhole
Figure 4.10: By rotating the crystal and the pump beam polarization, it was possible
to measure the coincidences between pairs of photons at different positions over
the emission cone. The photons were collected using multimode fibers and 300 µm
pinholes to increase the resolution. The values of the experimental parameters are
listed in table 4.3.
given by the angle ρ0 = 4.9 ◦ , while the generated photons did not exhibit
spatial walk-off.
Table 4.3: The parameters of the experiment described in section 4.3.
Parameter Value
Crystal liio3
L 5 mm
ρ0 4.9◦ Laser cw diode
wp 136 µm
λp 405 nm
∆λp 0.4 nm
λ0 s
ϕs
810 nm
4◦ We measured the relative position of the pump beam in the xy plane at
the input and output faces of the nonlinear crystal using a ccd camera. At
the output of the crystal, the beam displacement due to the Poynting vector
walk-off allowed us to determine the direction of the crystal’s optical axis, and
to fix its direction parallel to the pump polarization, as shown in figure 4.11.
Right after the crystal, each of the generated photons passed through a
2 − f system with focal length f=50 cm. Cut-off spectral filters removed the
remaining pump beam radiation. After the filters, the photons were coupled
into multimode fibers. In order to increase the spatial resolution, we used small
pinholes with a diameter of 300 µm.
We kept the idler pinhole fixed and measured the coincidence rate while
mapping the signal photon’s transversal shape by scanning the detector with
a motorized xy translation stage, as in the right part of figure 4.3. Finally,
we rotated the crystal and the pump beam polarization to measure the spatial
correlations at different azimuthal positions on the down-conversion cone. After
48

Before
the crystal
4.3. Effect of the Poynting vector walk-off on the OAM transfer
After
the crystal
Polarization Optical axis
Figure 4.11: The images obtained with a ccd camera show that before the crystal,
the pump beam has a circular shape, and that after passing the crystal part of the
beam gets displaced in the direction of the optical axis of the crystal. The displaced
part corresponds to the portion of the beam polarized orthogonal to the crystal axis.
In the case of the last image almost all the beam is displaced. The values of the
experimental parameters are listed in table 4.3.
every rotation, the tilt of the crystal was adjusted to achieve the generation of
photons at the same noncollinear angle in all cases.
The bottom row of figure 4.12 presents a sample of images taken at different
azimuthal sections of the cone, while the upper row shows the expected shape
predicted by the theory. Each column shows the coincidence rate for different
angles, α = 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . Each point of these images corresponds to
the recording of a 10 seconds measurement. The typical maximum number of
coincidences is around 10 coincidences per second. Each image is 10 × 10 mm,
and its resolution is 50 × 50 points.
Theory
Experiment
0º 90º 180º 270º
Figure 4.12: The coincidence measurements of the mode function, as well as the
theoretical calculations, show that the ellipticity of the mode function changes for
different positions on the cone. The ellipticity is minimal for α = 90 ◦ . The values of
the experimental parameters are listed in table 4.3.
49

4. OAM transfer in noncollinear configurations
The different spatial shapes in figure 4.12 clearly show that the downconversion
cone does not posses azimuthal symmetry. As was predicted by the
theoretical calculations, the coincidence measurement for α = 90 ◦ , presents a
nearly Gaussian shape, while the other cases are highly elliptical.
The slight discrepancies between experimental data and theoretical predictions
observed might be due to the small (but not negligible) bandwidth of the
pump beam, and due to the fact that the resolution of our system is limited
by the detection pinhole size.
Conclusion
The spdc parameters and the detection system determine the portion of the
cone that is detected in a noncollinear configuration. This chapter explains how
the pump beam waist and the Poynting vector walk-off affect the oam transfer.
By tailoring both parameters it is possible to generate photons with specific
spatial shapes. The walk-off affects especially those configurations where pairs
of photons with different α are used.
The next chapter extends the analysis of the spatial correlations to pairs
of photons generated in Raman transitions. The chapter describes how the
specific characteristics of that source are translated into the two-photon spatial
state.
50

CHAPTER 5
Spatial correlations
in Raman transitions
Two-photon states can be generated in different nonlinear processes, and in every
case the oam transfer will depend on the particular configuration. Raman
transition is an alternative method for the generation of two-photon states.
Several authors have proven the generation of correlated photons in polarization
[59], frequency [60] and oam [23] via Raman transitions. Typical configurations
involve the partial detection of the generated photons [61, 62, 63] in
quasi-collinear configurations [64]. Just as in the case of spdc, the geometrical
conditions determine the oam transfer. This chapter analyses the oam
transfer in Raman transitions using the techniques of the previous chapters.
This chapter is divided in three sections. Section 5.1 describes the generated
two-photon state by introducing its mode function. Section 5.2 studies the
oam content of one of the photons with the oam of the other photons fixed.
Section 5.3 describes the effect of the geometry of the process on the spatial
entanglement between the photons. Numerical calculations show the effect of
the geometrical configuration on the oam transfer mechanism. The finite size
of the nonlinear medium results in new effects that do not appear in spdc.
51

5. Spatial correlations in Raman transitions
pump
g
e
s
anti
Stokes Stokes
control
Figure 5.1: One atom with a Λ-type energy level configuration, can produce Stokes
and anti-Stokes photons by the interaction with the pump and control beams.
5.1 The quantum state of Stokes and anti-Stokes photon
pairs
This section describes the Stokes and anti-Stokes state generated by Raman
transitions in cold atomic ensembles in an analogous way to the description of
the two-photon state generated via spdc in chapter 1. The section discusses the
general characteristics of the nonlinear process, and introduces the two-photon
mode function.
Consider as a nonlinear medium an ensemble of n identical Λ−type cold
atoms trapped in a magneto-optical trap (mot). The atoms have an energy
level configuration with one excited state: |e〉 and two hyperfine ground states:
|s〉, and |g〉. This is the case, for example, in the d2 hyperfine transition of
87rb. In the initial state of the cloud all atoms are in the ground state |g〉, and
after emission all atoms return to their initial state, as figure figure 5.1 shows.
The two-photon generation results from the interaction of a single atom of
the cloud with two counter-propagating classical beams in a four step process.
In the first step, the atom gets excited by the interaction with the pump beam
far detuned from the |g〉 → |e〉 transition. In the second step, the excited atom
decays into the |s〉 state by emitting one Stokes photon in the direction zs as
shown in figure 5.2. In the third step, the atom is re-excited by the interaction
with the control beam far detuned from the |s〉 → |e〉 transition. In the last
step, the atom decays to the ground state by emitting an anti-Stokes photon
in the zas direction.
If ω 0 i
is the central angular frequency for the photons involved in the pro-
cess (i = p, c, s, as), and k 0 i is the corresponding wave number at the central
frequencies, energy and momentum conservation implies
52
and
g
e
ω 0 p + ω 0 c = ω 0 s + ω 0 as, (5.1)
k 0 p − k 0 c = k 0 s cos ϕs − k 0 as cos ϕas, (5.2)
k 0 s sin ϕs = k 0 as sin ϕas. (5.3)
s

x
y
z
5.1. The quantum state of Stokes and anti-Stokes photon pairs
zas yas
xas
pump
anti Stokes
Stokes
ys
control
Figure 5.2: According to energy and momentum conservation, the generated photons
counterpropagate. Equation 5.7 describes the relation between their propagation
direction and the propagation direction of the pump and control beams, where due
to the phase matching the angle of emission of the anti-Stokes photon is ϕas = π−ϕs.
Because we consider a pump and a control beam with the same central frequency
and k0 s k0 as, the phase matching conditions allow any angle of emission
ϕas = π − ϕs, if it is not forbidden by the transition matrix elements [60].
This assumption is valid only for cold atoms, since for warm atoms the process
is highly directional, that is all photons are emitted along a preferred direction
as proven in reference [61].
There are two ways to describe the generated two-photon quantum state: it
can be described by using two coupled equations in the slowly varying envelope
approximation for the Stokes and anti-Stokes electric fields [65], or alternatively
by using an effective Hamiltonian of interaction and first order perturbation
theory [66, 67]. As the latter approach is analogous to the formalism used in
chapter 1, it will be used in what follows to calculate the Stokes and anti-Stokes
state.
The effective Hamiltonian in the interaction picture HI describes the photonatom
interaction, and is given by
HI = ɛ0
ϕas
ϕ s
xs
zs
zc
yc
xc
dV χ (3) Ê − as Ê− s Ê+ c Ê+ p + h.c. (5.4)
where χ (3) is the effective nonlinearity, independent of the beam intensity since
the pump and the control are non-resonant [65]. Assuming a Gaussian distribution
of atoms in the cloud the effective nonlinearity χ (3) can be written
as
χ (3)
(x, y, z) ∝ exp − x2 + y2 R2 z2
−
L2
(5.5)
where R is the size of the cloud of atoms in the transverse plane (x, y) and L
is the size in the longitudinal direction.
53

5. Spatial correlations in Raman transitions
According to equation 1.6, the electric field operators Ên in equation 5.4,
are given by
Ê (+)
n (rn, t) = dkn exp [ikn · rn − iωnt]â(kn) (5.6)
where, as in equation 1.23, we are using a more convenient set of transverse
wave vector coordinates given by
ˆxs,as =ˆx
ˆys,as =ˆy cos ϕs,as + ˆz sin ϕs,as
ˆzs,as = − ˆy sin ϕs,as + ˆz cos ϕs,as.
(5.7)
Under these conditions, at first order perturbation theory, the spatial quantum
state of the generated pair of photons is
|Ψ〉 =
(5.8)
dqsdqasΦ (qs, qas) |qs〉s|qas〉s
where the mode function Φ (qs, qas) of the two-photon state is
Φ (qs, ωs, qas, ωas) = dqpdqcEp (qp) Ec (qc) exp − ∆20R 2
4 − ∆21R 2
4 − ∆22L 2
4
(5.9)
with the delta factors defined as
∆0 = q x s + q x as
∆1 = (ks − kas) sin ϕs + (q y s − q y as) cos ϕs
∆2 = kp − kc − (ks + kas) cos ϕ + (q y s − q y as) sin ϕs
and the longitudinal wave vector of the pump beam given by
ωpnp kp =
c
2
− ∆ 2 0 − ∆ 2 1
1/2
(5.10)
. (5.11)
In the case of the Stokes and anti-Stokes two-photon state, there are no correlations
between space and frequency due to the narrow bandwidth (∼ GHz)
of the generated photons [60]. Therefore, to analyze the spatial shape of the
mode function Φq (qs, qas), we can consider ωs = ω 0 s and ωas = ω 0 as.
The effect of the unavoidable spatial filtering produced by the specific op-
tical detection system used is described by Gaussian filters. The angular acceptance
of the single photon detection system is 1/ k0
sws,as . In most experimental
configurations, ws ≈ 50-150 µm and the length of the cloud is a few
millimeters or less. For simplicity, we will assume that ws = was.
In the calculations, the pump and the control are Gaussian beams with
the same waist wp at the center of the cloud. As the waist is typically about
200-500 µm, the Rayleigh range of the pump, Stokes and anti-Stokes modes
Lp = πw2 p/λp and Ls,as = πw2 s/λs,as satisfy L ≪ Lp, Ls,as. This condition
allows us to neglect the transverse wavenumber dependence of all longitudinal
wave vectors in equations 5.9 and 5.10.
54

where
5.2. Orbital angular momentum correlations
Under these conditions, the mode function Φq (qs, qas) can be written as
Φq (qs, qas) = (ABCD)1/4
π
× exp − A
4 (qx s + q x as) 2 − B
4 (qx s − q x as) 2
× exp − C
4 (qy s + q y as) 2 − D
4 (qy s − q y as) 2
A = w2 pR2 2R2 + w2 +
p
w2 s
2
B = w2 s
2
C = w2 s
2
D = w2 pR 2 cos 2 ϕs
2R 2 + w 2 p
(5.12)
+ L 2 sin 2 ϕs + w2 s
. (5.13)
2
The specific characteristics of the state are determined by the size of the atomic
cloud in the longitudinal and transverse planes, L and R; by the beam waist of
the pump and control beams, wp,c; by the waist of the Stokes and anti-Stokes
modes ws; and by the angle of emission ϕs. The next section describes the
effect off all these parameters on the oam transfer from the pump and the
control beams to the pair Stokes and anti-Stokes.
5.2 Orbital angular momentum correlations
Since the pump and control beams are Gaussian beams, lp = lc = 0, the
analysis of the oam transfer reduces to the study of the oam content of the
Stokes and anti-Stokes pair. The Stokes oam content becomes the only free
variable, after projecting the anti-Stokes into a Gaussian mode
u(qas) = Nas exp
− w2 g
4 (qx2 as + q y2
as)
(5.14)
with beam width wg at the center of the cloud. The Stokes mode function
defined as
Φs (qs) = dqasΦ (qs, qas) u(qas) (5.15)
becomes
(F G)(1/4)
Φs (qs) = exp −
1/2
(2π) F
4 (qx s ) 2 − G
4 (qy s ) 2
, (5.16)
55

5. Spatial correlations in Raman transitions
with
F = 4AB + (A + B) w2 g
A + B + w 2 g
G = 4CD + (C + D) w2 g
C + D + w2 . (5.17)
g
As in the previous chapter, this mode function can be written as a superposition
of spherical harmonics
Φs (qs, θs) = (2π)
−1/2
ls
als (qs) exp (ilsθs) . (5.18)
The probability of having a Stokes photon with oam equal to ls is given by the
weight of each spiral mode
F + G
C2ls = (F G)1/2 dqsqs exp − q
4
2
s I 2
G − F
ls q
8
2
s (5.19)
where Ils are the Bessel functions of the second kind. Only even modes appear
in the distribution as a consequence of the symmetry of the Stokes mode
function.
Figure 5.3 shows the weight of the mode ls = 0 as a function of the angle
of emission for different values of the length of the cloud of atoms. For nearly
collinear configurations, the probability of having a Gaussian Stokes photon is
one, therefore, in this case the process satisfies the relation lp+lc = ls +las. For
these kind of configurations, A = D in equation 5.12, and therefore the Stokes
spatial mode function shows cylindrical symmetry in the transverse planes
(xs, ys) and (xas, yas). This is the case in most experimental configurations
[59, 64, 68], and in fact, reference [23] experimentally proves the relationship
lp + lc = ls + las for collinear configurations.
The probability of having a Gaussian Stokes photon decreases as other
modes appear in the distribution in highly noncollinear configurations. The
length of the cloud controls the importance of the other modes and it is possible
to obtain a spatial mode with cylindrical symmetry for any emission angle, by
fulfilling the condition
L = R 1 + 2R2
w2 −1/2
. (5.20)
p
This condition reduces to L = R, when the beam waist is much larger than the
transverse size of the cloud. Any deviation from a spherical volume of interaction
(described by this condition) introduces ellipticity in the mode function.
For this reason, a highly elliptical configuration, as the one described in reference
[60], will not satisfy the oam selection rule.
Figure 5.4 shows the weight of the mode ls = 0 as a function of the length
of the cloud for different values of the emission angle. For any angle, the
probability of having a Gaussian mode is maximum at the length given by
equation 5.20. In a collinear configuration, the Stokes photon always has a
Gaussian distribution, independently of the length of the cloud. As the angle
of emission increases, the length of the cloud becomes more important. The
mode weight is weakly affected by the change of the cloud length when the
length is much longer than the other relevant parameters: ws, wg and R. This
is especially evident for an angle of emission ϕ = 90◦ .
56

weight
1
0.6
-180º
-90º
0º 90º 180º
5.3. Spatial entanglement
length
0.2mm
0.4mm
1mm
2mm
angle
Figure 5.3: The weight of the Gaussian mode in the oam decomposition for the Stokes
photon has a maximum in the collinear configuration. In noncollinear configurations
the probability of a Gaussian Stokes photon decreases as the length of the cloud
increases. Table 5.1 lists the parameters used to generate this figure.
weight angle
1
0.6
0.2 Length 1.5 mm
Figure 5.4: The probability of having a Gaussian Stokes photon is maximum in a
collinear configuration independently of the cloud length. In the noncollinear cases
the maximum appears at the length given by equation 5.20, in these configurations
the probability of having a Gaussian Stokes photon changes drastically with the cloud
length and the emission angle. Table 5.1 lists the parameters used to generate this
figure.
5.3 Spatial entanglement
The spatial correlations between the generated photons inherit the angular
dependence from the Stokes oam content. To explore this phenomena, this
0º
10º
90º
57

5. Spatial correlations in Raman transitions
Schmidt number
40
20
1
-180º
0º 180º
lenght spatial filter
2 mm
1 mm
400 m
200 m
7
4
1
-180º
0º 180º
100 m
200 m
500 m
angle
Figure 5.5: The length of the cloud controls the Schmidt number angular dependence.
By modifying this length it is possible to change the position of the maximum and
minimum values of the Schmidt number. The angular dependence can be removed
completely by tailoring the parameters of the process, or by filtering. Table 5.1 lists
the parameters used to generate this figure.
Table 5.1: The parameters used in figures 5.3, 5.4 and 5.5
Parameter Figure 5.3 Figure 5.4 Figure 5.5 (left) Figure 5.5 (right)
L 0.2, 0.4, 1, 2 mm [0.2, 1.5] mm 0.2, 0.4, 1, 2 mm 200 µm
wp 100 µm 100 µm 500 µm 500 µm
R 400µm 400µm 1000 µm 1000 µm
ws 100 µm 100 µm 100 µm 100, 200, 500 µm
wg 500 µm 500 µm
ϕs [−180, 180] ◦ 0, 10, 90 ◦ [−180, 180] ◦ [−180, 180] ◦
section shows the effects of changing the emission angle on the Schmidt number
K defined in section 2.1, is a common entanglement quantifier.
The Schmidt number of the two-photon state described by equation 5.12
quantifies the entanglement between the generated photons. According to references
[69, 70] it is given by
(A + B) (C + D)
K =
4 (ABCD) 1/2
. (5.21)
Figure 5.5 shows the Schmidt number as a function of the emission angle for
different values of the atomic cloud length and the pump beam waist. The
function extrema are located at 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . At each of these
values the function can have a maximum or a minimum depending on the
other parameters of process. For instance, if
L < R
1 + 2R2
w2 −1/2
p
(5.22)
the entanglement is maximum at collinear configurations where ϕ = 0 ◦ , 180 ◦ ,
and minimum at transverse configurations ϕ = 90 ◦ , 270 ◦ . In the left part of
figure 5.5, the condition in equation 5.22 is achieved at cloud lengths smaller
than 333 µm.
As the length of the cloud increases, the variation of the entanglement with
the angle is smoothed out. In the left part of figure 5.5 the relation in equation
58

5.3. Spatial entanglement
5.20 is satisfied when L = 333 µm; therefore, the amount of entanglement is
constant. If the length of the cloud increases even more, the position for the
maxima and the minima get inverted. When
L > R 1 + 2R2
w2 −1/2
p
(5.23)
the entanglement is maximum for transverse emitting configurations ϕ = 90 ◦ , 270 ◦ ,
and minimum for collinear configurations ϕ = 0 ◦ , 180 ◦ . This variation is possible
in Raman transitions where the transversal size of the cloud is comparable
to the longitudinal size.
Another parameter that plays a role in the variation of the amount of
entanglement is the Stokes spatial filter ws. The right side of figure 5.5 shows
the effect of changing the filtering over the amount of entanglement. Very
narrow spatial filters (ws → ∞) diminish both the amount of entanglement
and its azimuthal variability.
Conclusion
As in spdc, the geometrical configuration of the Raman transitions determines
the oam content and the spatial correlations of the generated Stokes and anti-
Stokes photons. The size and shape of the cloud defines the emission angles
for which the correlations are maximum and minimum.
59

CHAPTER 6
Summary
This thesis characterizes the correlations in two-photon quantum states, both
between photons, and between the spatial and frequency degrees of freedom.
The photon pairs are generated by spontaneous parametric down-conversion
spdc, or by the excitation of Raman transitions in cold atomic ensembles.
Special attention is given to the entanglement of the photon pair in the spatial
degree of freedom, associated to the orbital angular momentum oam. The
main contributions or the thesis are:
A novel matrix formalism to describe the two-photon mode function.
This formalism makes it possible to analytically calculate several features
of the down-converted photons, and reduces the numerical calculation time of
other features, as shown in chapter 1 and reference [41].
Analytical expressions for the purity of the subsystems formed by
a single photon in space and frequency, or by a spatial two-photon
state. These expressions quantify the correlations in the two-photon state,
and make the effect of the various parameters of the spdc process on these
correlations explicit. The thesis presents a set of conditions to suppress or
enhance the correlations between the photons, or between degrees of freedom
in the two-photon state. With these conditions, it is possible to design spdc
sources with specific levels of correlations, for example, sources that generate
pure heralded single photons, or alternatively, maximally entangled states; as
shown in chapter 2 and in references [41, 45].
A selection rule for oam transfer. This selection rule explains several
contradictory measurements of the oam transfer from the pump beam to the
signal and idler photons in spdc. We showed that the oam content of the
pump is completely transferred to the photons emitted in all directions. This
selection rule always holds if all possible emission directions are considered,
for example in collinear configurations, but not necessarily if only a part of
all emission directions is detected. These conditions for the validity of the
selection rule give clear guidelines for the design of sources for protocols that
exploit the multidimensionality of oam, as shown in chapter 3 and references
[71, 72].
61

6. Summary
Experiments that explain oam transfer in noncollinear spdc. In
noncollinear configurations only a subset of emission directions is detected. The
transfer of oam to the detected photons is strongly affected by the change in
geometry imposed by the detection system, and depends on the angle of detection,
the pump-beam waist and the Poynting-vector walk-off. The thesis shows
that, by tailoring these parameters, it is possible to design noncollinear sources
that naturally generate spatially-separated photons with specific desired spatial
shapes, as shown in chapter 4 and references [39, 33, 32].
An analysis of the spatial correlations generated by Raman transitions
in cold atomic ensembles. Like for spdc, the spatial correlations
between the Stokes and anti-Stokes photons depend on the geometrical configuration
of the pump beam, the control beam, and the Stokes and anti-Stokes
photons. This thesis shows how the size and shape of the atom cloud define
the emission angles for which the correlations are maximum and minimum, as
shown in chapter 5 and reference [73].
62

APPENDIX A
The matrix form
of the mode function
According to section 1.3, the normalized two-photon mode function, after some
approximations, reads
Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p
4 ∆2 0 − w2 p
4 ∆21
× exp − (γL)2
4 ∆2k − T 2 0
4 (Ωs + Ωi) 2
× exp − w2 s
2 |qs| 2 − w2 i
2 |qi| 2 − 1
2B2 Ω
s
2 s − 1
2B2 Ω
i
2
i . (A.1)
The argument of the exponential function is a second order polynomial. Each
term is the product of at most two variables (qx s , qy s , qx i , qy i , Ωs, Ωi) and a coefficient
f = a
4 qx2 s + b
4 qy2 s + c
4 qx2 i + d
4 qy2
h
i + . . . +
2 qx s q y s + . . . + z
2 ΩsΩi. (A.2)
Such a polynomial can be written as the product of a matrix A with the
coefficients as elements, and a vector x of the variables
f = 1
x qs 4
qy s qx i q y
i Ωs Ωs
⎛
⎜
⎜
⎝
a
h
i
j
k
h
b
m
n
p
i
m
c
s
t
j
n
s
d
v
k
p
t
v
f
l
r
u
w
z
⎞ ⎛
q
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
l r u w z g
x s
qy s
qx i
q y
i
Ωs
⎞
⎟ ,
⎟
⎠
Ωi
(A.3)
therefore, the mode function can be written using matrix notation as
Φ(qs, Ωs, qi, Ωi) ∝ exp − 1
2 xt
Ax . (A.4)
63

A. The matrix form of the mode function
Each of the terms of the matrix is defined by comparing the product of the
polynomial f with the argument of the exponential in equation 1.34. The
elements of matrix A are
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0 cos α 2
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
+ γ 2 L 2 cos ϕs 2 tan ρ0 2 sin α 2
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0 cos α 2
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
+ γ 2 L 2 cos ϕi 2 tan ρ0 2 sin α 2
f =2B −2
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
− 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
+ γ 2 L 2 N 2 s sin ϕs 2 tan ρ0 2 sin α 2
g =2B −2
i
+ 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
+ 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
+ γ 2 L 2 N 2 i sin ϕi 2 tan ρ0 2 sin α 2
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
i =w 2 p + γ 2 L 2 tan ρ0 2 cos α
j =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α
k =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
l =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
m = − γ 2 L 2 sin ϕs tan ρ0 cos α + γ 2 L 2 cos ϕs tan ρ0 2 sin α cos α
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
− 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
p = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ns cos ϕs sin ϕs − w 2 pNs cos ϕs sin ϕs
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ns cos ϕs 2 tan ρ0 sin α
+ 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
r = − γ 2 L 2 Np sin ϕs + γ 2 L 2 Ni sin ϕs cos ϕi + w 2 pNi cos ϕs sin ϕi
+ γ 2 L 2 Np cos ϕs tan ρ0 sin α − γ 2 L 2 Ni cos ϕs cos ϕi tan ρ0 sin α
− 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
s =γ 2 L 2 sin ϕi tan ρ0 cos α + γ 2 L 2 cos ϕi tan ρ0 2 sin α cos α
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
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
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
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ns cos ϕs cos ϕi tan ρ0 sin α
− 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
w =γ 2 L 2 Np sin ϕi − γ 2 L 2 Ni sin ϕi cos ϕi + w 2 pNi sin ϕi cos ϕi
+ γ 2 L 2 Np cos ϕi tan ρ0 sin α − γ 2 L 2 Ni cos ϕi 2 tan ρ0 sin α
+ 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
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
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi − γ 2 L 2 NsNi cos ϕs sin ϕi tan ρ0 sin α
− γ 2 L 2 NpNs sin ϕs tan ρ0 sin α + γ 2 L 2 NsNi cos ϕi sin ϕs tan ρ0 sin α
− 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.
(A.5)
This set of expressions is far more useful than compact. The matrix terms
become simpler in some particular spdc configurations that are treated in
chapters 2 and 4. For instance, when the pump polarization is parallel to the
x axis, the matrix terms become
a =w 2 p + 2w 2 s + γ 2 L 2 tan ρ 2 0
b =w 2 p cos ϕs 2 + 2w 2 s + γ 2 L 2 sin ϕs 2
c =w 2 p + 2w 2 i + γ 2 L 2 tan ρ 2 0
d =w 2 p cos ϕi 2 + 2w 2 i + γ 2 L 2 sin ϕi 2
f = 2
B2 s
g = 2
B2 i
+ T 2 0 + γ 2 L 2 (Np − Ns cos ϕs) 2 + w 2 pN 2 s sin ϕs 2
+ T 2 0 + γ 2 L 2 (Np − Ni cos ϕi) 2 + w 2 pN 2 i sin ϕi 2
h =m = −γ 2 L 2 sin ϕs tan ρ0
i =w 2 p + γ 2 L 2 tan ρ0 2
j =s = γ 2 L 2 sin ϕi tan ρ0
k =t = γ 2 L 2 tan ρ0(Np − Ns cos ϕs)
l =u = γ 2 L 2 tan ρ0(Np − Ni cos ϕi)
n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi
p = − γ 2 L 2 sin ϕs(Np − Ns cos ϕs) − w 2 pNs cos ϕs sin ϕs
r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi
v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs
w =γ 2 L 2 sin ϕi(Np − Ni cos ϕi) + w 2 pNi cos ϕi sin ϕi
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
+ T 2 0 − w 2 pNsNi sin ϕs sin ϕi.
(A.6)
The matrix notation extends to functions of the mode function. For instance,
the purity of the spatial part of the two-photon state, given by equation 2.11,
65

A. The matrix form of the mode function
writes
T r[ρ 2
q] =
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (A.7)
where the dimension increases as new primed variables appear. Using the
matrix notation the integrand becomes
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)
= N 4
exp − 1
2 Xt
BX , (A.8)
where the vector X is the result of concatenation of x and x ′ , such that
⎛
⎜
X = ⎜
⎝
↑
x
↓
↑
x ′
↓
⎞
⎟ ,
⎟
⎠
(A.9)
and the new matrix B is given by
B = 1
⎛
⎜
2 ⎜
⎝
2a
2h
2i
2j
k
l
0
0
0
0
k
2h
2b
2m
2n
p
r
0
0
0
0
p
2i
2m
2c
2s
t
u
0
0
0
0
t
2j
2n
2s
2d
v
w
0
0
0
0
v
k
p
t
v
2f
2z
k
p
t
v
0
l
r
u
w
2z
2g
l
r
u
w
0
0
0
0
0
k
l
2a
2h
2i
2j
k
0
0
0
0
p
r
2h
2b
2m
2n
p
0
0
0
0
t
u
2i
2m
2c
2s
t
0
0
0
0
v
w
2j
2n
2s
2d
v
k
p
t
v
0
0
k
p
t
v
2f
l
r
u
w
0
0
l
r
u
w
2z
⎞
⎟ .
⎟
⎠
l r u w 0 0 l r u w 2z 2g
(A.10)
In an analogous way, the integrand on the expression for the signal photon
purity
T r[ρ 2 signal] =
is written in a matrix notation as
66
dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i
× 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). (A.11)
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)
= N 4 exp
− 1
2 Xt
CX . (A.12)

The matrix C is given by
C = 1
⎛
2a
⎜ 2h
⎜ i
⎜ j
⎜ 2k
⎜ l
2 ⎜ 0
⎜ 0
⎜ i
⎜ j
⎝ 0
2h
2b
m
n
2p
r
0
0
m
n
0
i
m
2c
2s
t
2u
i
m
0
0
t
j
n
2s
2d
v
2w
j
n
0
0
v
2k
2p
t
v
2f
z
0
0
t
v
0
l
r
2u
2w
z
2g
l
r
0
0
z
0
0
i
j
0
l
2a
2h
i
j
2k
0
0
m
n
0
r
2h
2b
m
n
2p
i
m
0
0
t
0
i
m
2c
2s
t
j
n
0
0
v
0
j
n
2s
2d
v
0
0
t
v
0
z
2k
2p
t
v
2f
l
r
0
0
z
0
l
r
2u
2w
z
l r 0 0 z 0 l r 2u 2w z 2g
⎞
⎟ .
⎟
⎠
(A.13)
67

APPENDIX B
Integrals of
the matrix mode function
Given a n×n symmetric and positive definite matrix A, and two n order vectors
x and b
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp −
det(A) btA−1
b
.
2
(B.1)
To proof this result consider that, since A is positive definite, there exist another
matrix O such that OAO t = D where D is a diagonal matrix. With the
transformation y = Ox, the integral in equation B.1 becomes
dx exp − xtAx 2 + ixt
b = dy exp
− yt Dy
2 + iyt b ′
,
(B.2)
with b ′ = O −1 b.
As the argument of the exponential, in the right side of equation B.1, is
given by the matrix product
− 1
y1
2
y2 . . . yn
⎛
d1
⎜ 0
⎜
⎝ .
0
d2
.
. . .
. . .
. ..
0
0
.
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
0 0 0 dn
y1
y2
.
yn
⎞
⎛
⎟
⎠ + i ⎜
y1 y2 . . . yn ⎜
⎝
the integral can be written in a polynomial form as
dx exp − xtAx 2 + ixt
b
= dy1dy2...dyn exp − y2 1d1
2 + iy1b ′ 1 − y2 2d2
2 + iy2b ′ 2... − y2 ndn
2 + iynb ′
n
(B.3)
(B.4)
69
b ′ 1
b ′ 2
.
b ′ n
⎞
⎟
⎠ ,

B. Integrals of the matrix mode function
where the integrals in each variable are independent, so it is possible to write
them as
dx exp − xtAx
=
2 + ixt
b
dy1 exp − y2 1d1
2 + iy1b ′ 1
dy2
− y2 2d2
2 + iy2b ′
2 ...
which, after solving each integral separately, becomes
dx exp − xtAx 2 + ixt
b =
n
j=1
2π
dj
exp
dyn
− b′ 2
j
2dj
− y2 ndn
2 + iynb ′
n ,
(B.5)
. (B.6)
Because D is a diagonal matrix, 1/dj are the elements of D−1 , and n j=1 dj =
det(D); thus it is possible to write the last expression as
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp
det(D)
− b′ t D −1 b ′
2
. (B.7)
Finally, since det(A) = det(D) and b ′ t D −1 b ′ = b t A −1 b, the integral becomes
dx exp − xtAx 2 + ixt
b = (2π)n/2
exp −
det(A) btA−1
b
. (B.8)
2
This proof can be applied to the special case in which all the elements of the
vector b are equal to 0. In that case the integral reduces to
dx exp − xt
Ax
=
2
(2π)n/2
. (B.9)
det(A)
70

APPENDIX C
Methods for OAM
measurements
A real world oam application will require a compact tool able to separate the
different modes at the single photon level. A tool analogous to the beam splitter
for polarization, or to a diffraction grating for frequency. Such a tool probably
will be based on the two current methods to determine the oam of photons:
holographic and interferometric methods.
In a very simplistic way, an hologram is a record of the interference pattern
of two beams. When the hologram is illuminated with one of the beams,
the other beam can be reconstructed. Based on this principle, a computer
generated hologram of the interference of a lg mode with a Gaussian beam,
in conjunction with a single mode fiber can be used to detect that particular
lg mode [7, 51]. Even though this method works at single photon level, it
is restricted to a two-value response, reducing the effective dimensions of the
oam to two, as shown in figure C.1.
Reference [15] introduced an analyzer able to sort photons into even and
odd oam modes, using a Mach-Zender interferometer. With a phase shift in
one of the interferometer arms, the constructive and destructive interference
were so that odd modes go to one of the outputs, and even modes to the other,
as figure C.2 shows. By nesting several interferometers, and using holograms,
the authors proved that was possible to separate up to 2 n modes with (2 n − 1)
interferometers. This method works at single photon level, and can distinguish
many different oam modes, but it relies on the simultaneous stabilization of
different interferometers, which is challenging experimentally.
71

C. Methods for OAM measurements
Input hologram
lens
Gaussian
mode
non-Gaussian
mode
Figure C.1: A hologram that records the interference of a lg mode with a Gaussian
beam, can be used to detect that particular lg mode. Only if the generating lg
mode is incident on the hologram, a Gaussian beam is recovered. If another mode is
incident, the hologram will generate non-Gaussian beams. Single mode optical fibers
distinguish between Gaussian and non-Gaussian beams.
Input
beam splitter
Dove prism mirror
even modes
odd modes
Figure C.2: Two Dove prisms rotated with respect to each other by an angle π/2
induce a relative rotation of π between the two arms of a Mach-Zender interferometer.
As a consequence of the rotation, odd and even lg modes leave the interferometer
at different outputs after the second beam splitter. Reference [15] introduces this
principle as a base of an oam sorter.
72

Bibliography
[1] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information.
Cambridge, UK: Cambridge University Press, 2000.
[2] M. Fox, Quantum Optics, An Introduction. Oxford, UK: Oxford University
Press, 2006.
[3] S. P. Walborn, S. Pádua, and C. H. Monken, “Hyperentanglement-assisted
bell-state analysis,” Phys. Rev. A, vol. 68, p. 042313, 2003.
[4] P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Y. Chiao, “Proposal
for a loophole-free bell inequality experiment,” Phys. Rev. A, vol. 49,
pp. 3209–3220, 1994.
[5] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren,
C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single
photons in pure quantum states,” Phys. Rev. Lett., vol. 100, p. 133601,
2008.
[6] A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping
the waveform of entangled photons,” Phys. Rev. Lett., vol. 99, p. 243601,
2007.
[7] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the
orbital angular momentum states of photons,” Nature, vol. 412, pp. 313–
316, 2001.
[8] S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Twophoton
entanglement of orbital angular momentum states,” Phys. Rev. A,
vol. 65, p. 033823, 2002.
[9] G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature
Phys., vol. 3, p. 305, 2007.
[10] S. P. Walborn and C. H. Monken, “Transverse spatial entanglement in
parametric down-conversion,” Phys. Rev. A, vol. 76, p. 062305, 2007.
[11] C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse
entanglement in optical parametric down conversion,” Phys. Rev. Lett.,
vol. 92, p. 127903, 2004.
[12] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation
of hyperentangled photon pairs,” Phys. Rev. Lett., vol. 95, p. 260501, 2005.
73

Bibliography
[13] M. Barbieri, C. Cinelli, P. Mataloni, and F. D. Martini, “Polarizationmomentum
hyperentangled states: Realization and characterization,”
Phys. Rev. A, vol. 72, p. 052110, 2005.
[14] D. P. Caetano and P. H. S. Ribeiro, “Quantum distillation of position
entanglement with the polarization degrees of freedom,” Opt. Commun.,
vol. 211, p. 265, 2002.
[15] J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial,
“Measuring the orbital angular momentum of a single photon,” Phys. Rev.
Lett., vol. 88, p. 257901, 2002.
[16] H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold,
E. Yao, and J. Courtial, “Simplified measurement of the orbital angular
momentum of single photons,” Opt. Commun., vol. 223, pp. 117 – 122,
2003.
[17] S. Haroche and J.-M. Raimond, Exploring the quantum. Oxford, UK:
Oxford University Press, 2006.
[18] J. D. Jackson, Classical electrodynamics. New York, US: John Wiley and
sons, 1999.
[19] V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with
screw dislocations in their wavefonts,” JETP Lett, vol. 52, p. 429, 1990.
[20] N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “‘generation
of optical phase singularities by computer generated holograms,” Opt.
Lett., vol. 17, p. 221, 1992.
[21] E. Yao, S. Franke-Arnold, J. Courtial, M. J. Padgett, and S. M. Barnett,
“Observation of quantum entanglement using spatial light modulators,”
Opt. Express, vol. 14, pp. 13089–13094, 2006.
[22] H. H. Arnaut and G. A. Barbosa, “Orbital and intrinsic angular momentum
of single photons and entangled pairs of photons generated by
parametric down-conversion,” Phys. Rev. Lett., vol. 85, pp. 286–289, Jul
2000.
[23] R. Inoue, N. Kanai, T. Yonehara, Y. Miyamoto, M. Koashi, and
M. Kozuma, “Entanglement of orbital angular momentum states between
an ensemble of cold atoms and a photon,” Phys. Rev. A, vol. 74, p. 053809,
2006.
[24] A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three dimensional
entanglement for quantum communications,” Phys. Rev. Lett.,
vol. 89, p. 240401, 2002.
[25] G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “Experimental
quantum coin tossing,” Phys. Rev. Lett., vol. 94, p. 040501, 2005.
[26] S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental
quantum cryptography with qutrits,” New J. Phys., vol. 8,
p. 75, 2006.
74

Bibliography
[27] J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth
of entangled two-photon states,” Phys. Rev. A, vol. 68, p. 050301, Nov
2003.
[28] S. P. Walborn, A. N. de Oliveira, R. S. Thebaldi, and C. H. Monken, “Entanglement
and conservation of orbital angular momentum in spontaneous
parametric down-conversion,” Phys. Rev. A, vol. 69, p. 023811, 2004.
[29] G. Molina-Terriza, A. Vaziri, J. ˇ Reháček, Z. Hradil, and A. Zeilinger,
“Triggered qutrits for quantum communication protocols,” Phys. Rev.
Lett., vol. 92, p. 167903, 2004.
[30] J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of
entangled photon states generated in non-collinear, walking parametric
down conversion,” J. Opt. B: Quantum Semiclass. Opt., vol. 7, pp. 235–
239, 2005.
[31] A. R. Altman, K. G. Köprülü, E. Corndorf, P. Kumar, and G. A. Barbosa,
“Quantum imaging of nonlocal spatial correlations induced by orbital angular
momentum,” Phys. Rev. Lett., vol. 94, p. 123601, 2005.
[32] G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych,
and J. P. Torres, “Control of the shape of the spatial mode function of photons
generated in noncollinear spontaneous parametric down-conversion,”
Phys. Rev. A, vol. 72, p. 065802, 2005.
[33] C. I. Osorio, G. Molina-Terriza, B. G. Font, and J. P. Torres, “Azimuthal
distinguishability of entangled photons generated in spontaneous parametric
down-conversion,” Opt. Express, vol. 15, pp. 14636–14643, 2007.
[34] R. W. Boyd, Nonlinear Optics. London, UK: Academic Press - Elsevier,
1977.
[35] A. Yariv, Quantum Electronics. New York, US: Wiley-VCH, 1987.
[36] R. Loudon, The quantum theory of light. Oxford, UK: Oxford University
Press, 2000.
[37] C. K. Hong and L. Mandel, “Theory of parametric frequency downconversion
of light,” Phys. Rev. A, vol. 31, pp. 2409–2418, 1985.
[38] A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence
properties of entangled light beams generated by parametric downconversion,”
Phys. Rev. A, vol. 50, pp. 3349–3361, 1994.
[39] J. P. Torres, C. I. Osorio, and L. Torner, “Orbital angular momentum of
entangled counterpropagating photons,” Opt. Lett., vol. 29, pp. 1939–1941,
2004.
[40] A. Joobeur, B. E. A. Saleh, T. S. Larchuk, and M. C. Teich, “Coherence
properties of entangled light beams generated by parametric downconversion:
Theory and experiment,” Phys. Rev. A, vol. 53, pp. 4360–4371,
1996.
75

Bibliography
[41] C. I. Osorio, A. Valencia, and J. P. Torres, “Spatiotemporal correlations
in entangled photons generated by spontaneous parametric down conversion,”
New J. Phys., vol. 10, p. 113012, 2008.
[42] J. Altepeter, E. Jeffrey, and P. Kwiat, “Photonic state tomography,” in Advances
in Atomic, Molecular and Optical Physics (P. Berman and C. Lin,
eds.), vol. 52, p. 107, Elsevier, 2005.
[43] T. A. Brun, “Measuring polynomial functions of states,” Quantum Inf.
Comput., vol. 4, pp. 401–408, 2004.
[44] R. B. A. Adamson, L. K. Shalm, and A. M. Steinberg, “Preparation of
pure and mixed polarization qubits and the direct measurement of figures
of merit,” Phys. Rev. A, vol. 75, p. 012104, 2007.
[45] L. E. Vicent, A. B. U’Ren, C. I. Osorio, and J. P. Torres, “Design of bright,
fiber-coupled and fully factorable photon pair sources,” To be submitted,
2009.
[46] T. Yarnall, A. F. Abouraddy, B. E. Saleh, and M. C. Teich, “Spatial
coherence effects on second- andfourth-order temporal interference,” Opt.
Express, vol. 16, pp. 7634–7640, 2008.
[47] M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P.
Woerdman, “Effect of spatial filtering on the schmidt decomposition of
entangled photons,” Phys. Rev. A, vol. 74, p. 012309, 2006.
[48] G. Adesso, A. Serafini, and F. Illuminati, “Determination of continuous
variable entanglement by purity measurements,” Phys. Rev. Lett., vol. 92,
p. 087901, 2004.
[49] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren,
C. Silberhorn, and I. A. Walmsley, “Conditional preparation of single photons
using parametric downconversion: a recipe for purity,” New J. Phys.,
vol. 10, p. 093011, 2008.
[50] A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and
S. Schiller, “Quantum state reconstruction of the single-photon fock state,”
Phys. Rev. Lett., vol. 87, p. 050402, 2001.
[51] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman,
“Orbital angular momentum of light and the transformation of laguerregaussian
laser modes,” Phys. Rev. A, vol. 45, pp. 8185–8189, 1992.
[52] G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular
momentum of light: Preparation of photons in multidimensional vector
states of angular momentum,” Phys. Rev. Lett., vol. 88, p. 013601, 2001.
[53] J. U. Kang, Y. J. Ding, W. K. Burns, and J. S. Melinger, “Backward
second-harmonic generation in periodically poled bulk linbo3,” Opt. Lett.,
vol. 22, pp. 862–864, 1997.
[54] A. De Rossi and V. Berger, “Counterpropagating twin photons by parametric
fluorescence,” Phys. Rev. Lett., vol. 88, p. 043901, 2002.
76

Bibliography
[55] M. C. Booth, M. Atatüre, G. Di Giuseppe, B. E. A. Saleh, A. V. Sergienko,
and M. C. Teich, “Counterpropagating entangled photons from a waveguide
with periodic nonlinearity,” Phys. Rev. A, vol. 66, p. 023815, 2002.
[56] N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde,
A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled
qutrits and their use for quantum bit commitment,” Phys. Rev. Lett.,
vol. 93, p. 053601, 2004.
[57] D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in
parametric production of optical photon pairs,” Phys. Rev. Lett., vol. 25,
pp. 84–87, 1970.
[58] A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration
of higher dimensional entanglement: Qutrits of photon orbital
angular momentum,” Phys. Rev. Lett., vol. 91, p. 227902, 2003.
[59] D. N. Matsukevich, T. Chanelière, M. Bhattacharya, S.-Y. Lan, S. D.
Jenkins, T. A. B. Kennedy, and A. Kuzmich, “Entanglement of a photon
and a collective atomic excitation,” Phys. Rev. Lett., vol. 95, p. 040405,
2005.
[60] V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, “Generation
of paired photons with controllable waveforms,” Phys. Rev. Lett., vol. 94,
p. 183601, 2005.
[61] L. M. Duan, J. I. Cirac, and P. Zoller, “Three-dimensional theory for
interaction between atomic ensembles and free-space light,” Phys. Rev. A,
vol. 66, p. 023818, 2002.
[62] J. H. Eberly, “Emission of one photon in an electric dipole transition of
one among n atoms,” J. Phys. B, vol. 39, no. 15, pp. S599–S604, 2006.
[63] M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. Wódkiewicz, “Directed
spontaneous emission from an extended ensemble of n atoms: Timing is
everything,” Phys. Rev. Lett., vol. 96, p. 010501, Jan 2006.
[64] A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M.
Duan, and H. J. Kimble, “Generation of nonclassical photon pairs for scalable
quantum communication with atomic ensembles,” Nature, vol. 423,
pp. 731–734, 2003.
[65] D. A. Braje, V. Balić, S. Goda, G. Y. Yin, and S. E. Harris, “Frequency
mixing using electromagnetically induced transparency in cold atoms,”
Phys. Rev. Lett., vol. 93, p. 183601, 2004.
[66] J. Wen and M. H. Rubin, “Transverse effects in paired-photon generation
via an electromagnetically induced transparency medium. i. perturbation
theory,” Phys. Rev. A, vol. 74, p. 023808, 2006.
[67] J. Wen and M. H. Rubin, “Transverse effects in paired-photon generation
via an electromagnetically induced transparency medium. ii. beyond
perturbation theory,” Phys. Rev. A, vol. 74, p. 023809, 2006.
77

Bibliography
[68] S. Chen, Y.-A. Chen, B. Zhao, Z.-S. Yuan, J. Schmiedmayer, and J.-W.
Pan, “Demonstration of a stable atom-photon entanglement source for
quantum repeaters,” Physical Review Letters, vol. 99, p. 180505, 2007.
[69] P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping
affects type-II spontaneous parametric down-conversion,” Phys. Rev. A,
vol. 72, p. 033803, 2005.
[70] K. W. Chan, J. P. Torres, and J. H. Eberly, “Transverse entanglement
migration in hilbert space,” Phys. Rev. A, vol. 75, p. 050101, 2007.
[71] C. I. Osorio, G. Molina-Terriza, and J. P. Torres, “Correlations in orbital
angular momentum of spatially entangled paired photons generated in
parametric downconversion.,” Phys. Rev. A, vol. 77, p. 015810, 2008.
[72] C. I. Osorio, G. Molina-Terriza, and J. P. Torres, “Orbital angular momentum
correlations of entangled paired photons,” J. Opt. A: Pure Appl.
Opt., vol. 11, p. 094013, 2009.
[73] C. I. Osorio, S. Barreiro, M. W. Mitchell, and J. P. Torres., “Spatial entanglement
of paired photons generated in cold atomic ensembles,” Phys.
Rev. A, vol. 78, p. 052301, 2008.
78