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Physical Review A

Spontaneous parametric down-conversion: Revisiting the parameters of transverse entanglement outside the near zone

Mikhail Vladimirovich Fedorov, S. S. Mernova, K. V. Sliporod

Published May 13, 2022• 2 citations
Publisher

Abstract

The reduced density matrix ${\ensuremath{\rho}}_{r}(x,{x}^{\ensuremath{'}})$ of the transverse biphoton state beyond the near zone is found to be real on and only on diagonals in the plane of the photon's transverse coordinates $(x,{x}^{\ensuremath{'}})$. This remarkable feature is found to occur at any distances of photon propagation with diffraction spreading completely taken into account. The functions of the reduced density matrix at two orthogonal diagonals, ${\ensuremath{\rho}}_{r}(x,x)$ and ${\ensuremath{\rho}}_{r}(x,\ensuremath{-}x)$, are considered as the only two complementary single-particle distributions generated by the reduced density matrix. The ratio of widths of these two distributions is interpreted as the new entanglement parameter ${R}_{\mathrm{diag}}$ which is found to be very close to and completely compatible with the Schmidt entanglement parameter $K$.

DOI
10.1103/physreva.105.053709

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The reduced density matrix \ensuremathρr(x,x\ensuremath′){\ensuremath{\rho}}_{r}(x,{x}^{\ensuremath{'}})\ensuremathρr​(x,x\ensuremath′) of the transverse biphoton state beyond the near zone is found to be real on and only on diagonals in the plane of the photon's transverse coordinates (x,x\ensuremath′)(x,{x}^{\ensuremath{'}})(x,x\ensuremath′). This remarkable feature is found to occur at any distances of photon propagation with diffraction spreading completely taken into account. The functions of the reduced density matrix at two orthogonal diagonals, \ensuremathρr(x,x){\ensuremath{\rho}}_{r}(x,x)\ensuremathρr​(x,x) and \ensuremathρr(x,\ensuremath−x){\ensuremath{\rho}}_{r}(x,\ensuremath{-}x)\ensuremathρr​(x,\ensuremath−x), are considered as the only two complementary single-particle distributions generated by the reduced density matrix. The ratio of widths of these two distributions is interpreted as the new entanglement parameter Rdiag{R}_{\mathrm{diag}}Rdiag​ which is found to be very close to and completely compatible with the Schmidt entanglement parameter KKK.
As known, the degree of entanglement of biphoton states with respect to the transverse components of photon wave vectors (momenta) or coordinates can be characterized either by the parameter K associated with the Schmidt decompositions, or by the parameter R, defined as the ratio of the widths of the unconditional and conditional single-particle distributions of biphoton states. As entanglement is a fundamental characteristics of a state, it must be independent of a choice of its different representations, i.e. its entanglement parameters must be identical in the coordinate and momentum representations. Likewise, entanglement itself and its characterization parameters must remain constant along the photon propagation length. It's known, however, that only the parameter K obeys these requirements, whereas the parameter R in the coordinate representation deviates from K beyond the near zone and, thus, becomes inapplicable as a measure of the degree of entablement. But, as shown below, there is a way of saving the parameter R by means of modifying its definition in such a way that the modified parameter R turns out to be as good as the parameter K both in the momentum and coordinate representations, as well as in the near and far zones and everywhere between them. We will discuss also possible ways of measuring this newly defined parameter R experimentally.
We measure the transverse entanglement of photon pairs on their propagation from the near to the far field of spontaneous parametric down-conversion (SPDC). The Fedorov ratio, depending on the widths of conditional and unconditional intensity measurements, is shown to be only able to characterize entanglement in the near and far field zones of the source. Therefore we also follow a different approach. By evaluating the first-order coherence of a subsystem of the state we can quantify its entanglement. Unlike previous measurements, which determine the Fedorov ratio via intensity correlations, our setup is sensitive to both phase and modulus of the biphoton state and thus always grants experimental access to the full transverse entanglement of the SPDC state. It is shown theoretically that this scheme represents a direct measurement of the Schmidt number.
A method for generating two-photon wave packets (biphotons) with a small width of the spatial intensity correlation function (correlation radius) has been considered. We propose to prepare a two-photon wave packet inhomogeneously broadened in the angle by means of spontaneous parametric down-conversion in an inhomogeneous crystal. Such biphotons are not diffraction limited; i.e., their correlation radius is much larger than the inverse width of the spatial spectrum. However, the correlation radius of such a biphoton decreases as the biphoton propagates in free space, and transverse compression occurs at a certain distance from the crystal; i.e., the biphoton becomes diffraction limited.
Abstract As a tutorial to the spatial aspects of spontaneous parametric downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.
As a tutorial to the spatial aspects of Spontaneous Parametric Downconversion (SPDC), we present a detailed first-principles derivation of the transverse correlation width of photon pairs in degenerate collinear SPDC. This width defines the size of a biphoton birth zone, the region where the signal and idler photons are likely to be found when conditioning on the position of the destroyed pump photon. Along the way, we discuss the quantum-optical calculation of the amplitude for the SPDC process, as well as its simplified form for nearly collinear degenerate phase matching. Following this, we show how this biphoton amplitude can be approximated with a Double-Gaussian wavefunction, and give a brief discussion of the measurement statistics (and subsequent convenience) of such Double-Gaussian wavefunctions. Next, we use this approximation to get a simplified estimation of the transverse correlation width, and compare it to more accurate calculations as well as experimental results. We then conclude with a discussion of the concept of a biphoton birth zone, using it to develop intuition for the tradeoff between the first-order spatial coherence and bipohoton correlations in SPDC.
We investigate spectral and temporal features and entanglement of biphoton wave packets formed in spontaneous parametric down-conversion with a pulsed laser pump. The degree of entanglement is characterized by the experimentally measurable parameter RRR defined as the ratio of the coincidence and single-particle spectral widths. In the frequency representation, this parameter is found as a function of the pump-pulse duration $\ensuremath{\tau}$. The function $R(\ensuremath{\tau})$ is shown to have a minimum and even in the minimum, at rather natural conditions, the degree of entanglement is found to be very high (Rmin=73)({R}_{\mathrm{min}}=73)(Rmin​=73). The Schmidt number KKK is found analytically for both short and long pump pulses and interpolated for arbitrary pulse durations. All functional dependences of RRR and KKK are found to be identical and numerical coefficients are found to be rather close. Two-time temporal wave function of a biphoton state is investigated in detail, and a rather significant difference between the cases of short and long pump pulses is found to occur. In the case of long pulses, the temporal parameter Rt{R}_{t}Rt​ (defined as the ratio of durations of the single-particle and coincidence signals) is shown to be very close to the Schmidt number KKK.