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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Accurate relative-phase and time-delay maps all over the emission cone of hyperentangled photon source

Salem F. Hegazy, Jala M. El-Azab, Yehia A. Badr, Salah S. A. Obayya

Published Apr 27, 2016• 1 citation
Publisher

Abstract

High flux of hyperentangled photons entails collecting the two-photon emission over relatively wide extent in frequency and transverse space within which the photon pairs are simultaneously entangled in multiple degrees of freedom. In this paper, we present a numerical approach to determining the spatial-spectral relative-phase and time-delay maps of hyperentangled photons all over the spontaneous parametric down conversion (SPDC) emission cone. We consider the hyperentangled-photons produced by superimposing noncollinear SPDC emissions of two crossed and coherentlypumped nonlinear crystals. We adopt a vectorial representation for all parameters of concern. This enables us to study special settings such as the self-compensation via oblique pump incidence. While rigorous quantum treatment of SPDC emission requires Gaussian state representation, in low-gain regime (like the case of the study), it is well approximated to the first order to superposition of vacuum and two-photon states. The relative phase and time-delay maps are then calculated between the two-photon wavepackets created along symmetrical locations of the crystals. Assuming monochromatic plane-wave pump field, the mutual signal-idler relations like energy conservation and transversemomentum conservation define well one of the two-photon with reference to its conjugate. The weaker conservation of longitudinal momentum (due to relatively thin crystals) allows two-photon emission directions coplanar with the pump beam while spreading around the perfect phase-matching direction. While prior works often adopt first-order approximation, it is shown that the relative-phase map is a very well approximated to a quadratic function in the polar angle of the two-photon emission while negligibly varying with the azimuthal angle.

DOI
10.1117/12.2227679

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High flux of hyperentangled photons entails collecting the two-photon emission over relatively wide extent in frequency and transverse space within which the photon pairs are simultaneously entangled in multiple degrees of freedom. In this paper, we present a numerical approach to determining the spatial-spectral relative-phase and time-delay maps of hyperentangled photons all over the spontaneous parametric down conversion (SPDC) emission cone. We consider the hyperentangled-photons produced by superimposing noncollinear SPDC emissions of two crossed and coherentlypumped nonlinear crystals. We adopt a vectorial representation for all parameters of concern. This enables us to study special settings such as the self-compensation via oblique pump incidence. While rigorous quantum treatment of SPDC emission requires Gaussian state representation, in low-gain regime (like the case of the study), it is well approximated to the first order to superposition of vacuum and two-photon states. The relative phase and time-delay maps are then calculated between the two-photon wavepackets created along symmetrical locations of the crystals. Assuming monochromatic plane-wave pump field, the mutual signal-idler relations like energy conservation and transversemomentum conservation define well one of the two-photon with reference to its conjugate. The weaker conservation of longitudinal momentum (due to relatively thin crystals) allows two-photon emission directions coplanar with the pump beam while spreading around the perfect phase-matching direction. While prior works often adopt first-order approximation, it is shown that the relative-phase map is a very well approximated to a quadratic function in the polar angle of the two-photon emission while negligibly varying with the azimuthal angle.
Realizing high flux of hyperentangled photons requires collecting photon pairs simultaneously entangled in multiple degrees of freedom over relatively wide spectral and angular emission ranges. We consider the hyperentangled photons produced by superimposing noncollinear spontaneous parametric down conversion (SPDC) emissions of two crossed and coherently pumped nonlinear crystals. We present an approach for determining the directional-spectral relative-phase and time-delay maps of hyperentangled photons all over the SPDC emission cone. A vectorial representation is adopted for all parameters of concern. This enables us to examine unconventional arrangements such as the autocompensation of relative-phase and time-delay via oblique pump incidence. While prior works often adopt first-order approximation, it is shown that the actual directional relative-phase map is very well approximated by a quadratic function of the polar angle of the two-photon emission while negligibly varying with the azimuthal angle.
The two-photon state generated by overlapping the noncollinear spontaneous parametric down conversion (SPDC) emissions of two coherently pumped type-I crystals undergoes spatial and spectral decoherence precluding the capture of high pair counts with high state fidelity. Here we present a tunable compensation method for tackling the rising state decoherence in the two degrees of freedom in the case of crystals with negative birefringence. We determine the directional-spectral phase function of the two-photon state initially and after compensation, taking into account the finite spectral width of the pump beam. Subsequently, the optimal characteristics of the compensation crystals are computed by criteria of flat phase function in spectrum and space. The results of our model are compared with measurements of some recent experiments.
Abstract Spontaneous parametric down-conversion (SPDC) is the most widely used process for generating photon pairs entangled in various degrees of freedom such as polarization, time-energy, position-transverse momentum, and angle-orbital angular momentum (OAM). In SPDC, a pump photon interacts with a non-linear optical crystal and splits into two entangled photons called the signal and the idler photons. The SPDC process has been studied extensively in the last few decades for various pump and crystal configurations, and the entangled photon pairs produced by SPDC have been used in numerous experimental studies on quantum entanglement and entanglement-based real-world quantum-information applications. In this tutorial article, we present a thorough study of phase matching in BBO crystals for spontaneous parametric down-conversion and thereby also investigate the generation of entangled photons in such crystals. First, we present a theoretical derivation of two-photon wavefunction produced by SPDC in the frequency and transverse momentum bases. We then discuss in detail the effects due to various crystal and pump parameters including the length of the crystal, the angle between the optic axis and the pump propagation direction, the pump incidence angle on the crystal surface, the refraction at the crystal surfaces, and the pump propagation direction inside the crystal. These effects are extremely relevant in experimental situations. We then present our numerical and experimental results in order to illustrate how various experimental parameters affect the phase matching and thus the generation of entangled photons. Finally, using the two-photon wavefunction in the transverse wave-vector basis, we show how to derive the two-photon wavefunction in the OAM basis and thereby calculate the two-photon angular Schmidt spectrum. We expect this article to be useful for researchers working in various capacities with entangled photons generated by SPDC in BBO crystals.
The generation of high-fidelity polarization-entangled photon pairs, to date, has been demonstrated on specific spatial modes or over relatively narrow apertures. We put forward and demonstrate an experimental scheme to extend the temporal and spatial indistinguishability of polarization-entangled photons over wide emission angles, which can be applied to cover the whole spontaneous parametric downconversion (SPDC) cone. Over such a wide angular extent, while the time-delay map is almost flat which renders the conventional compensation via a birefringent element an appropriate approach, the relative-phase map—verified as a quadratic function—necessitates a tunable compensation paradigm. Here, to do so, we employ a phase-only two-dimensional spatial light modulator (2D SLM) loaded by the complementary of the relative phase map to equalize the phase variations for one third of the noncollinear SPDC emission. After eliminating the temporal and spatial distinguishability over the 2D SLM area, a 97% polarization visibility is verified for the entangled photon pairs scattered widely across the SPDC cone.
Hyperentanglement is a promising resource for achieving high capacity quantum communication. Here, we propose a compact scheme for the generation of path–frequency hyperentangled photon pairs via spontaneous parametric down-conversion (SPDC) processes, where six different paths and two different frequencies are covered. A two-dimensional periodical χ (2) nonlinear photonic crystal (NPC) is designed to satisfy type-I quasi-phase-matching conditions in the plane perpendicular to the incident pump beam, and a perfect phase match is achieved along the pump beam’s direction to ensure high conversion efficiency, with theoretically estimated photon flux up to 2.068 × 105 pairs⋅s−1⋅mm−2. We theoretically calculate the joint-spectral amplitude (JSA) of the generated photon pair and perform Schmidt decomposition on it, where the resulting entropy S of entanglement and effective Schmidt rank K reach 3.2789 and 6.4675, respectively. Our hyperentangled photon source scheme could provide new avenues for high-dimensional quantum communication and high-speed quantum information processing.
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.