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Optics Letters

Characterizing the spatial entanglement from laser modes analogous to quantum wave functions

M. X. Hsieh, X. L. Zheng, Y. T. Yu, H. C. Liang, K. F. Huang, Y. F. Chen

Published Jul 6, 2021• 8 citations
Publisher

Abstract

The Schmidt decomposition is exploited to study the spatial entanglement of laser transverse modes analogous to quantum Lissajous states. Based on the inverse Fourier transform, the stationary Lissajous state can be analytically derived as a coherent superposition of degenerate Hermite–Gaussian eigenmodes. With the derived stationary state, the Schmidt modes and the participation number N can be employed to evaluate the spatial localization and the quantum entanglement. The larger the participation number, the more localized is the stationary coherent state on the Lissajous figure. Moreover, the larger the participation number, the higher is the spatial entanglement.

DOI
10.1364/ol.434069

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The Schmidt decomposition is exploited to study the spatial entanglement of laser transverse modes analogous to quantum Lissajous states. Based on the inverse Fourier transform, the stationary Lissajous state can be analytically derived as a coherent superposition of degenerate Hermite–Gaussian eigenmodes. With the derived stationary state, the Schmidt modes and the participation number N can be employed to evaluate the spatial localization and the quantum entanglement. The larger the participation number, the more localized is the stationary coherent state on the Lissajous figure. Moreover, the larger the participation number, the higher is the spatial entanglement.
The propagation-dependent spatial entanglement for the structured laser beams generated by an arbitrary incident Hermite-Gaussian (HG) mode passing through an astigmatic mode converter (AMC) is theoretically explored. The structured output beams are analytically decomposed into the expansion of HG modes for any given rotation angle of the AMC. Based on the Schmidt decomposition, the propagation-dependent spatial entanglements of the structured output modes are quantified with the von Neumann entropy. To manifest the propagation-dependent entropy, the probability distribution of the expanded HG modes in the structured output beam is quantitatively analyzed.
We experimentally study the properties of coherent mode decomposition for the intensity correlation function of quasi-thermal light. We use the technique of spatial mode selection developed for studying the transverse entanglement of photon pairs, and show that it can be extended to characterize classical spatial correlations. Our results demonstrate the existence of a unique, for a given thermal source, basis of coherent modes, correlated in a way much resembling the Schmidt modes of spatially entangled photons.
We introduce and experimentally demonstrate a method to measure the Schmidt number of pure two-photon states entangled in transverse-mode structure. Our method is based on the connection between the Schmidt decomposition in quantum theory and the coherent-mode decomposition in classical coherence theory. We apply the method to two-photon states generated by spontaneous parametric down conversion and show that our results are in excellent agreement with numerical calculations based on the Schmidt decomposition.
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Quantum entanglement by a beam splitter (BS) is shown to be analogous to laser mode transformation by an astigmatic mode converter (AMC). Schmidt decomposition is used to characterize the entanglement by an AMC for generating orbital angular momentum and by a BS for creating quantum photon interference. The probability distributions of Schmidt decomposition are calculated to manifest the sameness and difference between AMC and BS in generating entanglement. Finally, the theoretical patterns of mode transformations by an AMC are confirmed with experimental results to validate the present analysis.