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arXiv:2403.04561

Einstein-Podolsky-Rosen correlations in spontaneous parametric down-conversion: Beyond the Gaussian approximation

Alexandre Gutenberg da Costa Moura, Carlos Henrique Monken

Published Mar 7, 2024• 0 citations
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Abstract

We present analytic expressions for the coincidence detection probability amplitudes of photon pairs generated by spontaneous parametric down-conversion in both momentum and position spaces, without making use of the Gaussian approximation, and taking into account the effects of birefringence in the nonlinear crystal. We also present experimental data supporting our theoretical predictions, using Einstein-Podolsky-Rosen correlations as benchmarks, for 8 different pump beam configurations.

DOI
10.48550/arxiv.2403.04561

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Physical Review A•Jun 1, 1996•126 citations
We present analytic expressions for the coincidence detection probability amplitudes of photon pairs generated by spontaneous parametric down-conversion in both momentum and position spaces, without making use of the Gaussian approximation, and taking into account the effects of birefringence in the nonlinear crystal. We also present experimental data supporting our theoretical predictions, using Einstein-Podolsky-Rosen correlations as benchmarks, for 8 different pump beam configurations.
Spontaneous parametric down-conversion is the primary source of position-correlated and momentum-anti-correlated photon pairs that form the canonical Einstein-Podolsky-Rosen (EPR) state. Their transverse spatial correlations are usually analyzed within the thin-crystal approximation, where the two-photon wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. In practice, however, birefringence-induced transverse walk-off breaks this factorization and couples these degrees of freedom. Here, we show that this coupling persists even for nominally thin crystals once the free-space propagation of the joint spatial intensity is taken into account. This sum-difference coordinate coupling leads to a distinctive tapering of the transverse correlations near the crystal image plane - an effect that standard factorized models cannot capture. Numerical simulations and experimental data clearly confirm this novel behavior. Our findings provide a more complete description of photon-pair generation in birefringent nonlinear media and clarify fundamental limits on spatially resolved quantum imaging and spatial-mode quantum information processing with EPR states.
Spontaneous parametric down-conversion is the primary source of position-correlated and momentum-anticorrelated photon pairs that form the canonical Einstein-Podolsky-Rosen (EPR) state. Their transverse spatial correlations are usually analyzed within the thin-crystal approximation, where the two-photon wavefunction is assumed to factorize into independent functions of the sum and difference coordinates. In practice, however, birefringence-induced transverse walk-off breaks this factorization and couples these degrees of freedom. Here, we show that this coupling persists even for nominally thin crystals once the free-space propagation of the joint spatial intensity is taken into account. This sum-difference coordinate coupling leads to a distinctive tapering of the transverse correlations near the crystal image plane-an effect that standard factorized models cannot capture. Numerical simulations and experimental data clearly confirm this novel behavior. Our findings provide a more complete description of photon-pair generation in birefringent nonlinear media and clarify fundamental limits on spatially resolved quantum imaging and spatial-mode quantum information processing with EPR states.
We describe the generation of correlated photon pairs by means of spontaneous parametric down-conversion of an optical pump in the form of a finite energy Airy beam. The optical system function, which contributes to the propagation of the down-converted beam before being registered by the detectors, is computed. The spectral function is utilized to calculate the biphoton amplitude for finding the coincidence count of the inbound Airy photons in both far-field and near-field configurations. We report the reconstruction of the finite energy Airy beam in the spatial correlation of the down-converted beams in near field scenario. In far field, the coincidence counts resembles the probability density of the biphoton in momentum space, revealing a direct mapping of the anti-correlation of the biphoton momentum. By examining the spatial Schmidt modes, we also demonstrate that longer crystals have tighter real-space correlations, but higher-dimensional angular correlations, whereas shorter crystals have fewer modes in momentum space and broader multimode correlations in position space.
Abstract As a tutorial, we examine the absolute brightness and number statistics of photon pairs generated in spontaneous parametric down-conversion (SPDC) from first principles. In doing so, we demonstrate how the diverse implementations of SPDC can be understood through a single common framework, and use this to derive straightforward formulas for the biphoton generation rate (pairs per second) in a variety of different circumstances. In particular, we consider the common cases of both collimated and focused Gaussian pump beams in a bulk nonlinear crystal, as well as in nonlinear waveguides and micro-ring resonators. Furthermore, we examine the number statistics of down-converted light using a non-perturbative approximation (the multi-mode squeezed vacuum), to provide quantitative formulas for the relative likelihood of multi-pair production events, and explore how the quantum state of the pump affects the subsequent statistics of the down-converted light. Following this, we consider the limits of the undepleted pump approximation, and conclude by performing experiments to test the effectiveness of our theoretical predictions for the biphoton generation rate in a variety of different sources.
Using a multidimensional Gaussian approximation of the wave function for the signal and idler light generated by spontaneous parametric down-conversion, we derive analytical expressions for the second-order coherence function and the fourth-order coherence function (which is proportional to the signal-idler photon coincidence rate). The magnitudes of these functions are expressed as products of Gaussian functions of the azimuthal angles, the polar angles, and the time delay. Their widths determine six parameters: the coherence angles and coherence time, and the entanglement angles and entanglement time. We show how these parameters are governed by the pump-beam waist, the pump spectral width, and the crystal length. We thereby derive relations analogous to the van Cittert--Zernike theorem and the Siegert relation for thermal light. We show that the normalized photon coincidence rate decreases sharply as the signal and idler apertures become mismatched or misaligned. We experimentally confirm this latter prediction by using parametrically down-converted light obtained from a LiIO3{\mathrm{LiIO}}_{3}LiIO3​ crystal pumped by Kr+{\mathrm{Kr}}^{+}Kr+-ion laser radiation at 413 nm. \textcopyright{}1996 The American Physical Society.