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Journal of Optics A Pure and Applied Optics

Orbital angular momentum correlations of entangled paired photons

Clara I. Osorio, Gabriel Molina‐Terriza, Juan P. Torres

Published Aug 5, 2009• 14 citations
Publisher

Abstract

The generation of paired photons entangled in the spatial degree of freedom, i.e., in orbital angular momentum, offers a convenient physical resource to investigate the nature of entanglement in a multidimensional Hilbert space with controllable dimensionality. The two main physical processes that generate pairs of photons which show correlations in orbital angular momentum are (a) spontaneous parametric down-conversion (SPDC), and (b) Raman transitions induced in atomic ensembles. One question naturally arises: what kinds of correlations exist between the orbital angular momentum of the generated photons? The answer might be different if we consider the whole quantum state of the generated photons, i.e., all possible directions where the pairs of photons can be emitted, or if we consider only a small section of the full set of directions.

DOI
10.1088/1464-4258/11/9/094013

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The generation of paired photons entangled in the spatial degree of freedom, i.e., in orbital angular momentum, offers a convenient physical resource to investigate the nature of entanglement in a multidimensional Hilbert space with controllable dimensionality. The two main physical processes that generate pairs of photons which show correlations in orbital angular momentum are (a) spontaneous parametric down-conversion (SPDC), and (b) Raman transitions induced in atomic ensembles. One question naturally arises: what kinds of correlations exist between the orbital angular momentum of the generated photons? The answer might be different if we consider the whole quantum state of the generated photons, i.e., all possible directions where the pairs of photons can be emitted, or if we consider only a small section of the full set of directions.
The generation of paired photons entangled in orbital angular momentum (OAM), provides a new degree of freedom that is increasingly being used as a source of quantum states. Elucidation of the OAM spectrum of OAM spectrum of the generated photons is of paramount importance, since quantum information applications require the ability to generate arbitrary entangled states with the appropriate OAM correlations. Here we discuss the OAM spectrum of the photon generated via SPDC in two complementary scenarios. On one hand, we should consider the whole geometry of the nonlinear process, taking into account azimuthal variations of the nonlinear coefficient or the phase matching conditions. On the other hand, all relevant experiments reported to date detect only a small section of the full down-conversion cone. In this scenario, the measured OAM correlations depend of the emission angle of the photons and the strength of the Poynting vector walk-off. We will present experiments in SPDC where this dependence is clearly revealed.
We elucidate the paraxial orbital angular momentum of entangled photon pairs generated by spontaneous parametric down-conversion (SPDC) in different non-collinear geometries. To date, most investigations addressed SPDC in nearly collinear phase-matching geometries, where the pump, the signal and idler photons propagate coaxially almost along the same direction. However, non-collinear geometries introduce a variety of new features. The OAM of the entangled photons strongly depend on the propagation direction of the photons. Here we show that locally paraxial measurements of the OAM conducted with entangled photons generated in non collinear geometries, they do not comply with the known selection rules for the spiral index of the pump, signal and idler mode functions (Mair et al., Nature 412, 313 (2001)). In particular, we find the orbital angular momentum of entangled pairs generated in purely transverse-emitting configurations, where the entangled photons counter-propagate perpendicularly to the direction of propagation of the pump beam. In transverse emitting configurations, the spatial shape of the down converted in one transverse dimensions strongly depends on the corresponding spatial shape of the input pump beam, while in the other transverse dimension, the shape is tailored by the longitudinal phase matching. The spatial walk-off of all interacting waves in the parametric process also determines the OAM content of the down-converted photons, and here its influence is also revealed.
The bandwidth of any communication system, classical or quantum, is limited by the number of orthogonal states in which the information can be encoded. Quantum key distribution systems available commercially rely on the two-dimensional polarisation state of photons. Quantum computation has also been largely designed on the basis of qubits. However, a photon is endowed with other degrees of freedom, such as orbital angular momentum (OAM). OAM is an attractive basis to be used for quantum information because it is discrete and theoretically infinite-dimensional. This promises a higher information capacity per photon which can lead to more complex quantum computation protocols and more security and robustness for quantum cryptography. Entanglement of OAM naturally arises from spontaneous parametric down-conversion (SPDC). However, any practical experiment utilising the innately high-dimensional entanglement of the orbital angular momentum (OAM) state space of photons is subject to the modal capacity of the detection system. Only a finite subset of this space is accessible experimentally. Given such a constraint, we show that the number of measured, entangled OAM modes in photon pairs generated by SPDC can be increased by tuning the phase-matching conditions in the SPDC process. We achieve this by tuning the orientation angle of the nonlinear crystal generating the entangled photons.
Starting from the standard Hamiltonian describing the optical non-linear process of spontaneous parametric down-conversion, we theoretically show that the generated entangled photon-pairs carry non-negligible orbital angular momentum in the degrees of freedom of relative movement in the type-II cases due to spatial symmetry breaking. We also show that the orbital angular momentum carried by photon-pairs in these degrees of freedom escapes detection in the traditional measurement scheme, which demands development of new techniques for further experimental investigations.
Identifying conservation laws is central to every subfield of physics, as they illuminate the underlying symmetries and fundamental principles. A prime example can be found in quantum optics: the conservation of orbital angular momentum (OAM) during spontaneous parametric down-conversion (SPDC) enables the generation of a photon pair with entangled OAM. In this Letter, we report on the observation of OAM conservation in SPDC pumped on the single-photon level by a preceding SPDC process. We implement this cascaded down-conversion scheme in free space, without waveguide confinement, and thereby set the stage for experiments on the direct generation of multiphoton high-dimensional entanglement using all degrees of freedom of light.