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The European Physical Journal D

Spatial and temporal characteristics of spontaneous parametric down-conversion with varying focal planes of interacting beams

Richard Bernecker, Baghdasar Baghdasaryan, S. Fritzsche

Published Sep 1, 2023• 4 citations
Publisher

Abstract

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

DOI
10.1140/epjd/s10053-023-00754-x

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Spatial and temporal characteristics of spontaneous parametric down-conversion with varying focal planes of interacting beams

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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.

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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.

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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 ${\overline{w}}_{p}={w}_{p}/\sqrt{{\ensuremath{\lambda}}_{p}\phantom{\rule{0.16em}{0ex}}L}$, 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.

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Entangled photon pairs play a crucial role in emerging quantum technologies, acting as tamper-proof padlocks in secure quantum communication, ultra-precise probes in quantum metrology and high-fidelity information carriers in photonic quantum computing. Amongst the many technological possibilities for generating photon pairs, sources based on spontaneous parametric down-conversion (SPDC) in second-order nonlinear crystals are still the workhorse tool in quantum optics laboratories, and more recently even in long-distance quantum communication with satellites. SPDC is a widely used method to create entangled photon pairs. The number of photon pairs that can be detected in a particular choice of collection modes, typically being optical single-mode fibers, depends critically on the spatial characteristics of the multi-mode SPDC emission as well as the particular beam waist of the collection modes. While several studies have already addressed the issue of optimal fiber coupling of SPDC photons in theory and experiment, the results of these studies arrive at different conclusions. Here, we present the results of a comprehensive experimental study on the optimal collection of photon pairs into single-mode optical fibers. Our approach is based on quasi-phase-matched type II SPDC from a periodically poled KTiOPO4 (ppKTP) crystal. We discuss the influence of pump and collection focal parameters on the spectral brightness and heralding efficiency, as well as practical issues of alignment tolerances into an optical single mode fiber. Further, Using two-photon interference (Hong-Ou-Mandel interference), we assess the spectral bandwidth of the photon pairs for variable crystal lengths. The results are in good agreement with our theoretical model, thus providing the recipe of building ultra bright and stable entangled photon sources.

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Photon modes have an important role in characterizing the quantum sources of light. Proper coupling of various photon modes obtained in spontaneous parametric down conversion (SPDC) process in optical fibers is essential to generate an effective source of entangled photons. The two main pre-detection factors affecting the biphoton mode coupling in SPDC are the pump beam focusing parameter and the crystal thickness. We present the numerical and experimental results on the effect of pump focusing on conditional down-converted photon modes for a Type-I BBO crystal. We experimentally verify that biphoton coupling efficiency decreases asymptotically with pump beam focusing parameter. We attribute this behaviour to (a) the asymmetry in the spatial distribution of down-converted photons with the pump beam focusing parameter and (b) the ellipticity of biphoton modes introduced due to the focusing of the pump beam. We also show experimentally the ellipticity as well as quantify the asymmetry with the pump focusing parameter. These results may be useful in selecting optimum conditions in the down-conversion process for generating efficient sources of entangled photons for quantum information applications.

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Optimizing the coupling efficiency of spontaneous parametric down-conversion photon pairs into single-mode fibers

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Both correlated-mode coupling efficiency and photon pair flux (brightness) of spontaneous parametric down-conversion (SPDC) coupled into single-mode fibers are key parameters for various quantum information science applications. We present an experimental investigation on the optimization of the correlated-mode coupling efficiency and the brightness of frequency-degenerated SPDC photon pairs into single-mode optical fibers. Using photon pairs generated from a type-I phase-matched bulk $\ensuremath{\beta}$-barium borate crystal pumped by a continuous-wave laser, we carefully evaluated both the correlated-mode coupling efficiency and the brightness of SPDC light coupled to a single-mode optical fiber, while continuously changing the focusing of the pump and collection modes, experimenting with Rayleigh lengths from 3 to 280 (for the pump) and 0.2 to 20 (for the collection) times the crystal length. The measured data suggest that the correlated-mode coupling efficiency can be close to unity for any pump focus, however, the optimal Rayleigh length of the collection mode depends on the pump focus parameter, and the brightness can be increased by tightening the pump focus, provided that the Rayleigh length of the collection mode is precisely adjusted. A maximal correlated-mode coupling efficiency of $99.0\ifmmode\pm\else\textpm\fi{}1.3%$ was experimentally achieved. These results will be useful for photonic quantum science and technology using SPDC sources with high heralding efficiencies.

Physical Review A•Oct 28, 2022•8 citations
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) 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.
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.
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.
Entangled photon pairs play a crucial role in emerging quantum technologies, acting as tamper-proof padlocks in secure quantum communication, ultra-precise probes in quantum metrology and high-fidelity information carriers in photonic quantum computing. Amongst the many technological possibilities for generating photon pairs, sources based on spontaneous parametric down-conversion (SPDC) in second-order nonlinear crystals are still the workhorse tool in quantum optics laboratories, and more recently even in long-distance quantum communication with satellites. SPDC is a widely used method to create entangled photon pairs. The number of photon pairs that can be detected in a particular choice of collection modes, typically being optical single-mode fibers, depends critically on the spatial characteristics of the multi-mode SPDC emission as well as the particular beam waist of the collection modes. While several studies have already addressed the issue of optimal fiber coupling of SPDC photons in theory and experiment, the results of these studies arrive at different conclusions. Here, we present the results of a comprehensive experimental study on the optimal collection of photon pairs into single-mode optical fibers. Our approach is based on quasi-phase-matched type II SPDC from a periodically poled KTiOPO4 (ppKTP) crystal. We discuss the influence of pump and collection focal parameters on the spectral brightness and heralding efficiency, as well as practical issues of alignment tolerances into an optical single mode fiber. Further, Using two-photon interference (Hong-Ou-Mandel interference), we assess the spectral bandwidth of the photon pairs for variable crystal lengths. The results are in good agreement with our theoretical model, thus providing the recipe of building ultra bright and stable entangled photon sources.
Photon modes have an important role in characterizing the quantum sources of light. Proper coupling of various photon modes obtained in spontaneous parametric down conversion (SPDC) process in optical fibers is essential to generate an effective source of entangled photons. The two main pre-detection factors affecting the biphoton mode coupling in SPDC are the pump beam focusing parameter and the crystal thickness. We present the numerical and experimental results on the effect of pump focusing on conditional down-converted photon modes for a Type-I BBO crystal. We experimentally verify that biphoton coupling efficiency decreases asymptotically with pump beam focusing parameter. We attribute this behaviour to (a) the asymmetry in the spatial distribution of down-converted photons with the pump beam focusing parameter and (b) the ellipticity of biphoton modes introduced due to the focusing of the pump beam. We also show experimentally the ellipticity as well as quantify the asymmetry with the pump focusing parameter. These results may be useful in selecting optimum conditions in the down-conversion process for generating efficient sources of entangled photons for quantum information applications.
Both correlated-mode coupling efficiency and photon pair flux (brightness) of spontaneous parametric down-conversion (SPDC) coupled into single-mode fibers are key parameters for various quantum information science applications. We present an experimental investigation on the optimization of the correlated-mode coupling efficiency and the brightness of frequency-degenerated SPDC photon pairs into single-mode optical fibers. Using photon pairs generated from a type-I phase-matched bulk $\ensuremath{\beta}$-barium borate crystal pumped by a continuous-wave laser, we carefully evaluated both the correlated-mode coupling efficiency and the brightness of SPDC light coupled to a single-mode optical fiber, while continuously changing the focusing of the pump and collection modes, experimenting with Rayleigh lengths from 3 to 280 (for the pump) and 0.2 to 20 (for the collection) times the crystal length. The measured data suggest that the correlated-mode coupling efficiency can be close to unity for any pump focus, however, the optimal Rayleigh length of the collection mode depends on the pump focus parameter, and the brightness can be increased by tightening the pump focus, provided that the Rayleigh length of the collection mode is precisely adjusted. A maximal correlated-mode coupling efficiency of $99.0\ifmmode\pm\else\textpm\fi{}1.3%$ was experimentally achieved. These results will be useful for photonic quantum science and technology using SPDC sources with high heralding efficiencies.