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Optica

Inverse design of spontaneous parametric downconversion for generation of high-dimensional qudits

Eyal Rozenberg, Aviv Karnieli, Ofir Yesharim, Joshua Foley-Comer, Sivan Trajtenberg‐Mills, Daniel Freedman, Alex M. Bronstein, Ady Arie

Published Apr 29, 2022• 33 citations
Publisher

Abstract

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.

DOI
10.1364/optica.451115

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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.
Spontaneous parametric down-conversion 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 that takes into account all interaction effects, 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 every interaction parameter 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 down-converted 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 volume holograms 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 the spectral and temporal properties, and may even be used in condensed-matter systems having a similar interaction Hamiltonian.
We propose a novel, physically-constrained and differentiable approach for the generation of D-dimensional qudit states via spontaneous parametric down-conversion (SPDC) in quantum optics. We circumvent any limitations imposed by the inherently stochastic nature of the physical process and incorporate a set of stochastic dynamical equations governing its evolution under the SPDC Hamiltonian. We demonstrate the effectiveness of our model through the design of structured nonlinear photonic crystals (NLPCs) and shaped pump beams; and show, theoretically and experimentally, how to generate maximally entangled states in the spatial degree of freedom. The learning of NLPC structures offers a promising new avenue for shaping and controlling arbitrary quantum states and enables all-optical coherent control of the generated states. We believe that this approach can readily be extended from bulky crystals to thin Metasurfaces and potentially applied to other quantum systems sharing a similar Hamiltonian structures, such as superfluids and superconductors.
Spontaneous parametric down conversion (SPDC) is one of the most versatile techniques for the generation of correlated photon pair, whose quantum state is essential for photon entanglement that underpins many quantum applications like secure communication, quantum metrology and lithography and quantum imaging [1]. Inverse design of effective SPDC has been achieved by structured crystals with shaped pump beams [2]. Another popular platform for SPDC is nanofabricated structures with sub-wavelength thickness called metasurfaces, as they could achieve drastic enhancement of nonlinear light-matter interactions [3]. There remains the possibility to use inverse design to optimize metasurfaces for SPDC.
Entangled qudits, the high-dimensional entangled states, play an important role in the study of quantum information. How to prepare entangled qudits in an efficient and easy-to-operate manner is still a challenge in quantum technology. Here, we demonstrate a method to engineer frequency entangled qudits in a spontaneous parametric downconversion process. The proposal employs an angle-dependent phase-matching condition in a nonlinear crystal, which forms a classical-quantum mapping between the spatial (pump) and spectral (biphotons) degrees of freedom. In particular, the pump profile is separated into several bins in the spatial domain, and thus shapes the down-converted biphotons into discrete frequency modes in the joint spectral space. Our approach provides a feasible and efficient method to prepare a high-dimensional frequency entangled state. As an experimental demonstration, we generate a three-dimensional entangled state by using a homemade variable slit mask.
Abstract Introducing structure into photon pair generation via spontaneous parametric down‐conversion (SPDC) is shown to be useful for controlling the output state and exploiting new degrees of freedom for quantum technologies. This paper presents a new method for simulating first‐ and second‐order correlations of the down‐converted photons in the presence of structured pump beams and shaped nonlinear photonic crystals. This method is nonperturbative, and thus accounts for high‐order effects, and can be made very efficient using parallel computing. Experimental results of photodetection and coincidence rates in complex spatial configurations are recovered quantitatively by this method. These include SPDC in 2D nonlinear photonic crystals, as well as with structured light beams such as Laguerre Gaussian and Hermite Gaussian beams. This simulation method reveals conservation rules for the down‐converted signal and idler beams that depend on the nonlinear crystal modulation pattern and the pump shape. This scheme can facilitate the design of nonlinear crystals and pumping conditions for generating non‐classical light with pre‐defined properties.
Spatially entangled light has proven to be a critical resource in high-dimensional quantum information applications such as quantum cryptography, quantum sensing, and quantum computing. However, progress in the generation of arbitrary spatially entangled states for these protocols via spontaneous parametric down-conversion has been hindered due to the complex relationship between the nonlinear coupling and the generated quantum state. Here, we employ an inverse-design algorithm for the generation of non-trivial maximally entangled bi-photon qubit and qutrit states in the azimuthal index via spontaneous parametric down-conversion. These states are obtained by shaping the pump spatial distribution as a superposition of Laguerre-Gaussian radial modes. The generated bi-photon states are experimentally characterized through quantum state tomography, with their entanglement verified by breaking generalized Bell inequalities. Finally, we performed a proof-of-principle experiment demonstrating the potential utility of the inverse-designed states in entanglement-based quantum key distribution protocols, measuring quantum error rates of 3.95% and 7.03% for the generated qubit and qutrit states, respectively. Our results validate the inverse-design model in a free-space optical setup and pave the way for high-dimensional spatial-mode entangled state design and generation for quantum network applications.