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arXiv:2112.05934

SPDCinv: Inverse Quantum-Optical Design of High-Dimensional Qudits

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

Published Dec 11, 2021• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2112.05934

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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.
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.
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.
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.
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.
Nonlinear holography shapes the amplitude and phase of generated new harmonics using nonlinear processes. Classical nonlinear holography influenced many fields in optics, from information storage, demultiplexing of spatial information, and all-optical control of accelerating beams. Here, we extend the concept of nonlinear holography to the quantum regime. We directly shape the spatial quantum correlations of entangled photon pairs in two-dimensional patterned nonlinear photonic crystals using spontaneous parametric down conversion, without any pump shaping. The generated signal-idler pair obeys a parity conservation law that is governed by the nonlinear crystal. Furthermore, the quantum states exhibit quantum correlations and violate the Clauser-Horne-Shimony-Holt inequality, thus enabling entanglement-based quantum key distribution. Our demonstration paves the way for controllable on-chip quantum optics schemes using the high-dimensional spatial degree of freedom.
In the process of Spontaneous Parametric Down Conversion (SPDC), a pump photon spontaneously splits inside a quadratic nonlinear crystal to signal and idler photons. By spatially modulating the nonlinear coefficient, we form a bulk nonlinear metamaterial that can shape the first and second order correlations of the down-converted signal and idler photons in either the spatial domain or the spectral domains. These degrees of freedom span a multi-dimensional Hilbert space, which is beneficial for quantum information applications. Recently, we have designed and fabricated, using electric field poling, bulk nonlinear metamaterials in KTiOPO4, for generating spatially entangled signal-idler pairs. This includes a bi-photon Bell state in the Hermite-Gauss basis, or a state with 3 dominant pairs of coincidences, approximating a bi-photon qutrit. As for the spectral domain, we generated a variety of quantum states: high purity frequency uncorrelated states, frequency-bin Bell states and frequency entangled bi-photon qudit states. By increasing the pump power, we reach the regime of bright squeezed vacuum sources, enabling the generation of a square cluster state in the frequency domain.