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Frontiers in Optics

Direct Measurement of Transverse Mode Entanglement in Two-Photon States

H. Di Lorenzo Pires, Carlos Henrique Monken, Martin P. van Exter

Published Jan 1, 2009• 12 citations
Publisher

Abstract

We introduce and implement a new method to measure the Schmidt number of spatially entangled two-photon states by exploiting a connection between the Schmidt decomposition and the coherent-mode decomposition in classical coherence theory.
DOI
10.1364/fio.2009.jma6

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Optics Letters•Jul 6, 2021•8 citations
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.
A complete characterization of entanglement in a two-qutrit state generated using the transverse spatial correlations of two parametric down-converted photons is discussed. We verify entanglement for a particular case via entanglement witness operators which are decomposed into a sum of local observables. Experimentally, these operators are accomplished by using a spatial light modulator and a polarizing beam splitter. The quantification of entanglement is computed by the negativity. We also realize an experiment for the verification of the Schmidt number of quantum correlated photons. This measure of entanglement counts the number of global superpositions of product states which are needed to describe a compound quantum system. For unraveling a certain value of this measure, a family of entanglement and Schmidt number tests is constructed.
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.