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Physical Review A

Enhanced entanglement from Ince-Gaussian pump beams in spontaneous parametric down-conversion

Baghdasar Baghdasaryan, S. Fritzsche

Published Nov 11, 2020• 29 citations
Publisher

Abstract

Spontaneous parametric down-conversion (SPDC) has been a reliable process for the generation of entangled photon pairs. In this process, a nonlinear quadratic crystal is pumped by a laser field in order to convert (high-energy) photons into correlated photon pairs whose efficient control plays an essential role in various applications of quantum information processing. In particular, the amount of entanglement has been successfully controlled by adjusting the spatial structure of the incident pump field. Here, we theoretically analyze how the entanglement of the down-converted two-photon state can be further enhanced by using Ince-Gaussian beams with well-defined ellipticity $\ensuremath{\epsilon}$, i.e., solutions of the paraxial wave equation in elliptical coordinates. These spatially structured beams are quite universal as they include both the Laguerre-Gaussian beams for $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ as well as the Hermite-Gaussian beams for $\ensuremath{\epsilon}\ensuremath{\rightarrow}\ensuremath{\infty}$. We demonstrate that the entanglement of the generated photon pairs in SPDC can be maximized by a proper choice of $\ensuremath{\epsilon}$ and that such an enhanced entanglement can be observed experimentally in terms of the Schmidt number.

DOI
10.1103/physreva.102.052412

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

Eyal Rozenberg, Aviv Karnieli, Ofir Yesharim +5 more

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.

Optica•Apr 29, 2022•33 citations
Spontaneous parametric down-conversion (SPDC) has been a reliable process for the generation of entangled photon pairs. In this process, a nonlinear quadratic crystal is pumped by a laser field in order to convert (high-energy) photons into correlated photon pairs whose efficient control plays an essential role in various applications of quantum information processing. In particular, the amount of entanglement has been successfully controlled by adjusting the spatial structure of the incident pump field. Here, we theoretically analyze how the entanglement of the down-converted two-photon state can be further enhanced by using Ince-Gaussian beams with well-defined ellipticity \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, i.e., solutions of the paraxial wave equation in elliptical coordinates. These spatially structured beams are quite universal as they include both the Laguerre-Gaussian beams for \ensuremathϵ\ensuremath→0\ensuremath{\epsilon}\ensuremath{\rightarrow}0\ensuremathϵ\ensuremath→0 as well as the Hermite-Gaussian beams for \ensuremathϵ\ensuremath→\ensuremath∞\ensuremath{\epsilon}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathϵ\ensuremath→\ensuremath∞. We demonstrate that the entanglement of the generated photon pairs in SPDC can be maximized by a proper choice of \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ and that such an enhanced entanglement can be observed experimentally in terms of the Schmidt number.
High-dimensional spatial entanglement generated via spontaneous parametric down-conversion (SPDC) provides a powerful resource for quantum information processing, yet its mode structure becomes increasingly complex when the pump occupies a higher-order Laguerre--Gaussian (LG) mode. Here, we develop an algebraic framework for characterizing biphoton spatial-mode entanglement by factorizing LG-pumped SPDC into a spatial-mode beam-splitter transformation followed by spatial two-mode squeezing. By choosing the biphoton LG basis radius as the geometric mean of the pump-beam radius and the crystal-induced correlation width, we obtain a natural modal basis in which the two operations can be treated separately. When the pump-beam radius matches the correlation width, spatial squeezing vanishes, and the two circular-mode components of the pump are independently conserved across the signal and idler photons, leading to conservation of both orbital angular momentum and the total spatial mode number. We further derive an analytical expression for the Schmidt number for arbitrary LG pump modes and squeezing strengths, revealing how the radial and azimuthal pump indices govern the dimensionality of the spatial entanglement. This algebraic characterization provides a unified description of the mode structure, conservation laws, and entanglement dimensionality of higher-order-LG-pumped SPDC.
We demonstrate experimental generation of spatially entangled photon pairs by spontaneous parametric down-conversion (SPDC) using a partial spatially coherent pump beam. By varying the spatial coherence of the pump, we show its influence on the down-converted photon's spatial correlations and on its degree of entanglement, in excellent agreement with theory. We then exploit this property to produce pairs of photons with a specific degree of entanglement by tailoring of the pump coherence length. This work thus unravels the fundamental transfer of coherence occurring in SPDC processes, and provides a simple experimental scheme to generate photon pairs with a well-defined degree of spatial entanglement, which may be useful for quantum communication and information processing.
Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based on photonic entanglement. The efficiency of photon pair generation is often characterized by means of a $\mathrm{sinc}(L\mathrm{\ensuremath{\Delta}}k/2)$ function, where LLL is the length of the nonlinear medium and $\mathrm{\ensuremath{\Delta}}k$ is the phase mismatch between the pump and down-converted fields. In theoretical investigations, the sinc\mathrm{sinc}sinc behavior of the phase mismatch has often been approximated by a Gaussian function $exp(\ensuremath{-}\ensuremath{\alpha}{x}^{2})$ in order to derive analytical expressions for the SPDC process. Different values have been chosen in the literature for the optimization factor $\ensuremath{\alpha}$, for instance, by comparing the widths of sinc\mathrm{sinc}sinc and Gaussian functions or the momentum of down-converted photons. As a consequence, different values of $\ensuremath{\alpha}$ provide different theoretical predictions for the same setup. Therefore an informed and unique choice of this parameter is necessary. In this paper, we present a choice of $\ensuremath{\alpha}$ which maximizes the validity of the Gaussian approximation. Moreover, we also discuss the so-called super-Gaussian and cosine-Gaussian approximations as practical alternatives with improved predictive power for experiments.
Abstract Spontaneous parametric down-conversion (SPDC) is a widely used process to prepare entangled photon pairs. In SPDC, a second-order nonlinear crystal is pumped by a coherent laser beam to generate photon pairs. The photon pairs are usually detected by single-mode fibers (SMF), where only photons in a Gaussian mode can be collected. The collection modes possess typical Gaussian parameters, namely a beam waist and a focal plane position. The collection efficiency of photons highly depends on the choice of both parameters. The exact focal plane position of the pump beam relative to those of the detection modes is difficult to determine in a real experiment. Usually, theoretical and experimental studies assume that the focal plane positions of the pump and the generated beams are positioned in the center of the crystal. The displacement of beam focal planes can lead to deviations from expected results and the coupling efficiency into SMF can decrease. In this study, we theoretically examine variable positions of focal planes in the Laguerre–Gaussian basis, a popular experimental modal decomposition of the spatial biphoton state. We explore how the choice of focal plane positions affects the spatial and temporal properties and the purity of the photon pairs. We present SPDC setups where precise knowledge of the focal plane position is essential and scenarios where focal plane displacements have negligible impact on experimental outcomes. Graphical Abstract
Spatially engineered photons from spontaneous parametric down-conversion (SPDC) are a valuable tool for studying and applying photonic entanglement. An advantage of SPDC is that simple expressions for the two-photon state can be obtained using justified approximations. In particular, the thin-crystal approximation has often been invoked in the engineering of high-dimensional entangled states. Knowledge of the conditions under which the thin-crystal approximation remains valid is essential for the realization of experimental setups. We provide a quantitative guideline on the validity of the thin-crystal approximation in calculating the two-photon spatial state. In particular, we show that the applicability of this regime is related to the focusing parameter ${\overline{w}}_{p}={w}_{p}/\sqrt{{\ensuremath{\lambda}}_{p}\phantom{\rule{0.16em}{0ex}}L}$, where wp{w}_{p}wp​ and ${\ensuremath{\lambda}}_{p}$ are the beam waist and wavelength of the pump beam, respectively, and LLL is the length of the nonlinear crystal. Additionally, the validity of the thin-crystal regime is investigated concerning the size of a subspace in the Laguerre Gaussian basis, into which the two-photon state can be projected in a given experiment.
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.