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

Schmidt number for X-entanglement of photon pairs

D. B. Horoshko, Giuseppe Patera, A. Gatti, Mikhail I. Kolobov

Published Apr 21, 2012• 0 citations
Publisher

Abstract

We calculate the Schmidt number for a two-dimensional model of the nonfactorable spatiotemporal wave-function of biphotons produced in type-I spontaneous parametric down-conversion with degenerate and collinear phase- matching taking into consideration a major part of the broad spectral and angular bandwidth of the down- converted light. We derive an analytical expression for the Schmidt number as a function of the filter bandwidth in the limit of spectrally narrow pump.

DOI
10.1117/12.922465

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We calculate the Schmidt number for a two-dimensional model of the nonfactorable spatiotemporal wave-function of biphotons produced in type-I spontaneous parametric down-conversion with degenerate and collinear phase- matching taking into consideration a major part of the broad spectral and angular bandwidth of the down- converted light. We derive an analytical expression for the Schmidt number as a function of the filter bandwidth in the limit of spectrally narrow pump.
In this work, the Schmidt number of the two-photon state generated by parametric down-conversion (PDC) is evaluated in the framework of a fully spatiotemporal model for PDC. A comparison with the results obtained in either purely spatial or purely temporal models shows that the degree of entanglement of the PDC state cannot be trivially reduced to the product of the Schmidt numbers obtained in models with lower dimensionality, unless the detected bandwidth is very narrow. This result is a consequence of the nonfactorability of the state in the spatial and temporal degrees of freedoms of twin photons. In the limit of a broad pump beam, we provide a geometrical interpretation of the Schmidt number as the ratio between the volume of the phase-matching region and of a correlation volume.
A simple analytic decomposition of the spatially entangled two-photon field allows us to generalize the earlier results of Law and Eberly [Phys. Rev. Lett. 92, 127903 (2004)] to more realistic experimental geometries of spontaneous parametric down-conversion. We quantify analytically how spatial filtering reduces the Schmidt number or dimensionality of the two-photon entanglement from a ``generated'' value to a generally much lower ``usable or detected'' value, for both collinear and noncollinear phase matching. We also discuss the intimate relation between the (two-photon) Schmidt number and the classical (one-photon) concept of etendue or geometric extent.
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.
We investigate the effect of the down-conversion angle between the signal and idler beams in spontaneous parametric down-conversion on the bandwidth of the modal spectrum (Schmidt number) of the down-converted quantum state. For this purpose, we consider both the full Schmidt number and the azimuthal Schmidt number in the Laguerre-Gaussian basis. For the full Schmidt number, we show that the approximation of the phase-matching function with a Gaussian function gives results that disagree significantly from those obtained with the more physical sinc-type phase-matching function. We found a drastic increase in both the Schmidt number and the azimuthal Schmidt number for small down-conversion angles, in agreement with recent experimental results.
High-dimensional entangled states have now been identified as excellent candidates towards the enhancement of the bandwidth of quantum systems. A variety of methods exist that aim to certify and set the bounds to entanglement. However, rapid and accurate approaches for precise quantification of the dimensions remain a challenge. Here, we report a facile, rapid and robust approach that quantifies a wide range of the spatial dimensions of high-dimensional entangled states using an interferometric technique using only two images. Our process works for a class of pure two-photon states, regardless of its separability and provides a fast and easy way to accurately measure the spatial Schmidt number which is a quantitative measure of dimensionality of entanglement. We apply our method to two-photon states generated by spontaneous parametric down-conversion (SPDC) and show that our results are in excellent agreement with numerical estimates. Since such estimates only exist for collinear phase matching in SPDC crystals, we also derive a noncollinear phase-matching condition applicable for type-2 crystals. In these nonlinear crystals, the commonly applied assumption of degenerate daughter photons no longer holds, leading to deviations from earlier theoretical predictions. Our interferometric technique provides excellent results for both type-0 and type-2 collinear phase-matching conditions, endorsing its wide applicability in quantum photonics. Published by the American Physical Society 2024