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Physical Review A

Modal spectrum in spontaneous parametric down-conversion with noncollinear phase matching

Yingwen Zhang, Filippus Stefanus Roux

Published Jun 5, 2014• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.89.063802

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Mar 5, 2013•0 citations
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.
Considering the quantum state produced in type I spontaneous parametric down-conversion with collinear, degenerate signal and idler beams, and a Gaussian pump, we show that the azimuthal Schmidt number in the Laguerre-Gaussian (LG) basis increases when the radial indices of the LG modes detected in the signal and idler beams are different. These observations are confirmed by the good agreement between theoretical and experimental results. The theoretical results are obtained by deriving expressions for the probability amplitude to detect LG modes with any combination of azimuthal and radial indices in a down-converted photonic quantum state.
The radial degrees of freedom of the biphoton states that are produced in spontaneous parametric down-conversion (SPDC) in the Laguerre-Gaussian (LG) basis are investigated, theoretically and experimentally. We calculated the theoretical azimuthal Schmidt numbers for different combinations of radial indices and found that a larger azimuthal Schmidt number is obtained for higher radial indices of the signal and idler beams. Moreover, larger azimuthal Schmidt numbers are also obtained when the difference between the two radial indices increases. Comparing these theoretical predictions with the azimuthal Schmidt numbers obtained from experimentally measurements, we found good agreement. Experimentally we demonstrated that it is possible to obtain a threefold increase in the azimuthal Schmidt number while maintaining a reasonable coincidence count rate by using LG modes with slightly larger radial indices.
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.
The Bessel-Gaussian modal spectrum, generated in spontaneous parametric down-conversion of a Gaussian pump beam, is considered. This is done by first deriving a general expression for the true probability of detecting specific transverse spatial modes for the pairs of photons generated in a spontaneous parametric down-conversion process. These expressions are applied for Bessel-Gaussian modes in type I phase matching with collinear, degenerate down-converted beams. The result shows that a broad orbital angular momentum spectrum can be obtained for Bessel-Gaussian modes under certain conditions.
The OAM or spiral bandwidth indicates the dimensionality of an entangled state that is produced by the spontaneous parametric down-conversion process. Normally this bandwidth is determined by modulating the signal and idler beams with helical phase functions with opposite azimuthal indices on the spatial light modulators in the signal and idler beams, respectively. We added an additional binary Bessel function to the helical phase, thereby specifying the radial dependence of the mode to be Bessel-Gaussian (BG) modes. This comes down to a post selection process, which is known to have the ability to increase entanglement. The result is a modification to the shape of the OAM spectrum, which leads to a higher dimensionality for the quantum states. We perform analytical calculations to show that the bandwidths obtained by measuring in the BG modal basis are larger than those for the LG modes. These theoretical predictions are confirmed by experimental measurements of the bandwidths for LG modes and for BG modes with different transverse scales.