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Physical Review A

Justifying the thin-crystal approximation in spontaneous parametric down-conversion for collinear phase matching

Baghdasar Baghdasaryan, Fabian Steinlechner, S. Fritzsche

Published Jun 15, 2021• 16 citations
Publisher

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 ${w}_{p}$ and ${\ensuremath{\lambda}}_{p}$ are the beam waist and wavelength of the pump beam, respectively, and $L$ 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.

DOI
10.1103/physreva.103.063508

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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 w‾p=wp/\ensuremathλpL{\overline{w}}_{p}={w}_{p}/\sqrt{{\ensuremath{\lambda}}_{p}\phantom{\rule{0.16em}{0ex}}L}wp​=wp​/\ensuremathλp​L​, where wp{w}_{p}wp​ and \ensuremathλp{\ensuremath{\lambda}}_{p}\ensuremathλp​ 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.
Optimizing the brightness of a spontaneous parametric down-conversion (SPDC) source is an important task for many quantum information applications. We investigate the optimal focusing conditions to maximize the number of photons produced in an SPDC process and coupled with single-mode fibers. We provide a general expression for the two-photon wave function, generalizing previous known results, by considering collinear and noncollinear emission. We present analytical expressions for our results in the thin-crystal limit and clarify the relation between different focusing conditions already existing in the literature. Differently from what was previously reported, we show that the optimal ratio between the pump waist and the generated photons waist depends on the emission angle: It is 1/21/\sqrt{2}1/2​ for collinear degenerate emission and approaches 1/21/21/2 for larger collection angles. The role of spectral filters is also analyzed. We support and enrich our discussion with numerical simulations, performed for type-I SPDC in a $\ensuremath{\beta}$ barium borate crystal. For this type of emission, we also investigate the role of the transverse walk-off outside the thin-crystal regime.
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
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.
The process of spontaneous parametric down-conversion (SPDC) inXXX(2)-materials, in which a pump (P) photon of wavelength λpcan split into a pair of entangled signal (S) and idler (I) photons of lower energy, is an instrumental tool in optical quantum technologies. Recently, there has been a growing interest in pair generation in sub-wavelength-thin nonlinear crystals [1], [2], especially because they are not restricted by the phase-matching condition and exhibit SPDC in a very wide spectral and angular range. This also motivates their use in quantum imaging applications, for pushing the resolution of such schemes to the limit [3], [4]. Despite its seemingly simple appearance, there has not yet been a comprehensive theoretical analysis of photon-pair generation in a thin unstructured nonlinear slab. Specifically, a proper theory should embed the Fabry-Pérot type of interferences that the signal and idler photons experience in such a nonlinear etalon during SPDC. In our work, we develop the theory for description of SPDC in thin nonlinear etalons and calculate far-field properties of the generated pairs. We use the Green's function (GF) quantization approach for description of pairgeneration [5], combined with analytical expressions for the GF of multilayer systems [6], to develop our formalism. Because of the generality of the GF method, this formalism can describe any thickness of the slab, ultra-thin or thick, and far-field radiation in the non-paraxial regime. Fig. 1 (a) schematic of spdc in a nonlinear etalon.ϵ1,ϵ2\epsilon_{1},\epsilon_{2}ϵ1​,ϵ2​andϵ3\epsilon_{3}ϵ3​are linear permittivities of the mediums. Medium 2 with thicknesshhhhas a second-orderχ(2)\chi^{(2)}χ(2)nonlinearity. (b-e) Joint detection probabilities of different detection configurations and etalon thicknesses.
Photon sources based on spontaneous parametric down-conversion (SPDC) are central to experimental quantum optics and quantum technologies. Their performance is commonly quantified by three metrics: pair-collection probability, heralding efficiency, and spectral purity. In bulk-crystal SPDC, these metrics are known to be mutually constrained, yet the physical origin of the resulting trade-offs is often obscured. We show that these trade-offs originate from the frequency-dependent population of discrete spatial modes in the SPDC emission. By performing a Laguerre-Gauss mode decomposition at each frequency component, we show how spectral-spatial non-separability impacts collection probability, heralding efficiency, and purity. We apply this framework to two widely used quasi-phase-matching configurations: collinear degenerate type-0 and type-II SPDC in periodically poled bulk crystals, and quantify how different phase-matching functions shape the spectral-spatial mode structure. In particular, for type-II SPDC we compare standard periodically poled and aperiodically poled Gaussian phase matching. We experimentally validate some of our theoretical results using spatial- and spectral-projection measurements. This spectral-spatial mode analysis provides a quantitative and predictive framework for understanding and engineering bulk-crystal photon sources, enabling systematic multi-parameter optimization beyond qualitative design guidelines.
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 increase or decrease. In this work, we consider variable positions of focal planes and investigate how shifts of these focal planes influence the spatial and temporal properties of photon pairs. We present SPDC arrangements, in which the knowledge of the exact position of the focal planes is essential, as well as scenarios, where focal plane displacements do not contribute significantly to experimental outcomes. These findings are of particular interest for achieving higher efficiency in SPDC experiments.