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Optica

Optical spin–orbit interaction in spontaneous parametric downconversion

Yunkun Wu, Yutao Tang, Zixian Hu, Lan‐Tian Feng, Guang‐Can Guo, Xi‐Feng Ren, Guixin Li

Published Mar 31, 2023• 18 citations
Publisher

Abstract

Optical spin–orbit interaction (SOI), which can be used to simultaneously control the spin and orbital angular momentum of light, is important for both classical and quantum information applications. In linear and nonlinear optics, the SOI of light has been extensively explored in both artificial structures and conventional optical crystals. However, optical SOI in quantum nonlinear optical processes, such as spontaneous parametric downconversion (SPDC), has not been studied before. Here, we experimentally demonstrate that optical SOI in the SPDC process can be realized through a nonlinear crystal with threefold rotational symmetry. Two-photon quantum states with controlled angular momentum can be generated through the symmetry selection rules in nonlinear optics and the SOI of the pump wave. The proposed methodology may facilitate the generation and control of spin and orbital angular momentum of entangled photons.

DOI
10.1364/optica.487001

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Optical spin–orbit interaction (SOI), which can be used to simultaneously control the spin and orbital angular momentum of light, is important for both classical and quantum information applications. In linear and nonlinear optics, the SOI of light has been extensively explored in both artificial structures and conventional optical crystals. However, optical SOI in quantum nonlinear optical processes, such as spontaneous parametric downconversion (SPDC), has not been studied before. Here, we experimentally demonstrate that optical SOI in the SPDC process can be realized through a nonlinear crystal with threefold rotational symmetry. Two-photon quantum states with controlled angular momentum can be generated through the symmetry selection rules in nonlinear optics and the SOI of the pump wave. The proposed methodology may facilitate the generation and control of spin and orbital angular momentum of entangled photons.
Spontaneous parametric downconversion (SPDC) in quantum optics is an invaluable resource for the realization of high-dimensional qudits with spatial modes of light. One of the main open challenges is how to directly generate a desirable qudit state in the SPDC process. This problem can be addressed through advanced computational learning methods; however, due to difficulties in modeling the SPDC process by a fully differentiable algorithm, progress has been limited. Here, we overcome these limitations and introduce a physically constrained and differentiable model, validated against experimental results for shaped pump beams and structured crystals, capable of learning the relevant interaction parameters in the process. We avoid any restrictions induced by the stochastic nature of our physical model and integrate the dynamic equations governing the evolution under the SPDC Hamiltonian. We solve the inverse problem of designing a nonlinear quantum optical system that achieves the desired quantum state of downconverted photon pairs. The desired states are defined using either the second-order correlations between different spatial modes or by specifying the required density matrix. By learning nonlinear photonic crystal structures as well as different pump shapes, we successfully show how to generate maximally entangled states. Furthermore, we simulate all-optical coherent control over the generated quantum state by actively changing the profile of the pump beam. Our work can be useful for applications such as novel designs of high-dimensional quantum key distribution and quantum information processing protocols. In addition, our method can be readily applied for controlling other degrees of freedom of light in the SPDC process, such as spectral and temporal properties, and may even be used in condensed-matter systems having a similar interaction Hamiltonian.
Twisted light, or light with orbital angular momentum (OAM), plays an emerging role in both classical and quantum science, with important applications in areas as diverse as biophotonics, micromachines, spintronics, or quantum information. It offers fascinating opportunities for exploring new fundamental ideas in physics, as well as for being used as a tool for practical applications. One important point is to determine how to generate single photons, and two-photon states, with an appropriate OAM content. Here we describe the paraxial orbital angular momentum of entangled photon pairs generated by spontaneous parametric down-conversion (SPDC) in different non-collinear geometries. These geometries introduce a variety of new features. In particular, we find the OAM 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. 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.
We present a full vectorial analysis of the parametric downconversion process using a pump Bessel beam and a photon description based on Bessel wavefunctions. We find that, under the considered geometry, optical angular momentum is conserved but the crystal induces a coupling between the optical spin angular momentum (SAM) and the optical orbital angular momentum (OAM) ruled by the nonlinear properties of the crystal via the second order susceptibility tensor ?ijk(2). This leads to nontrivial correlations between the expected values of the SAM and OAM of the resulting two-photon states. Distributions of orbital angular momentum of the downconverted photons are evaluated in detail for a specific set-up.
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.
Abstract Spontaneous parametric down-conversion (SPDC) is the most widely used process for generating photon pairs entangled in various degrees of freedom such as polarization, time-energy, position-transverse momentum, and angle-orbital angular momentum (OAM). In SPDC, a pump photon interacts with a non-linear optical crystal and splits into two entangled photons called the signal and the idler photons. The SPDC process has been studied extensively in the last few decades for various pump and crystal configurations, and the entangled photon pairs produced by SPDC have been used in numerous experimental studies on quantum entanglement and entanglement-based real-world quantum-information applications. In this tutorial article, we present a thorough study of phase matching in BBO crystals for spontaneous parametric down-conversion and thereby also investigate the generation of entangled photons in such crystals. First, we present a theoretical derivation of two-photon wavefunction produced by SPDC in the frequency and transverse momentum bases. We then discuss in detail the effects due to various crystal and pump parameters including the length of the crystal, the angle between the optic axis and the pump propagation direction, the pump incidence angle on the crystal surface, the refraction at the crystal surfaces, and the pump propagation direction inside the crystal. These effects are extremely relevant in experimental situations. We then present our numerical and experimental results in order to illustrate how various experimental parameters affect the phase matching and thus the generation of entangled photons. Finally, using the two-photon wavefunction in the transverse wave-vector basis, we show how to derive the two-photon wavefunction in the OAM basis and thereby calculate the two-photon angular Schmidt spectrum. We expect this article to be useful for researchers working in various capacities with entangled photons generated by SPDC in BBO crystals.