Spatial Characterization Of Two-Photon States - GAP-Optique

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
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