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arXiv:1205.1979 · Physical Review A

Entanglement witnesses and measures for bright squeezed vacuum

Magdalena Stobińska, Falk Töppel, Pavel Sekatski, Maria V. Chekhova

Published Aug 22, 2012• 29 citations
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Abstract

Quantum entanglement is a fascinating phenomenon, especially if it is observed at the macroscopic scale. Importantly, macroscopic quantum correlations can be revealed only by accurate measurement outcomes and strategies. Here, we formulate feasible entanglement witnesses for bright squeezed vacuum in the form of the macroscopically populated polarization triplet Bell states. Their testing involves efficient photodetection and the measurement of the Stokes operators' variances. We also calculate the measures of entanglement for these states such as the Schmidt number and the logarithmic negativity. Our results show that the bright squeezed vacuum degree of polarization entanglement scales as the mean photon number squared. We analyze the applicability of an operational analog of the Schmidt number.

DOI
10.1103/physreva.86.022323

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Quantum entanglement is a fascinating phenomenon, especially if it is observed at the macroscopic scale. Importantly, macroscopic quantum correlations can be revealed only by accurate measurement outcomes and strategies. Here, we formulate feasible entanglement witnesses for bright squeezed vacuum in the form of the macroscopically populated polarization triplet Bell states. Their testing involves efficient photodetection and the measurement of the Stokes operators' variances. We also calculate the measures of entanglement for these states such as the Schmidt number and the logarithmic negativity. Our results show that the bright squeezed vacuum degree of polarization entanglement scales as the mean photon number squared. We analyze the applicability of an operational analog of the Schmidt number.
Photon-number correlation measurements are performed on bright squeezed vacuum states using a standard Bell-test setup, and quantum correlations are observed for conjugate polarization-frequency modes. We further test the entanglement witnesses for these states and demonstrate the violation of the separability criteria, which infers that all of the macroscopic Bell states, containing typically 106{10}^{6}106 photons per pulse, are polarization entangled. The study also reveals the symmetry of macroscopic Bell states with respect to local polarization transformations.
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.
Normalized quantum Stokes operators introduced in Phys. Rev. A 95, 042113 (2017) enable one to better observe non-classical correlations of entangled states of optical fields with undefined photon numbers.For a given run of an experiment the new quantum Stokes operators are defined by the differences of the measured intensities (or photon numbers) at the exits of a polarizer divided by their sum.It is this ratio that is to be averaged, and not the numerator and the denominator separately, as it is in the conventional approach.The new approach allows to construct more robust entanglement indicators against photon-loss noise, which can detect entangled optical states in situations in which witnesses using standard Stokes operators fail.Here we show an extension of this approach beyond phenomena linked with polarization.We discuss EPR-like experiments involving correlations produced by optical beams in a multi-mode bright squeezed vacuum state.EPR-inspired entanglement conditions for all prime numbers of modes are presented.The conditions are much more resistant to noise due to photon loss than similar ones which employ standard Glauber-like intensity, correlations.
A concept of polarization entanglement for continuous variables is introduced. For this purpose the Stokes-parameter operators and the associated Poincar\'e sphere, which describe the quantum-optical polarization properties of light, are defined and their basic properties are reviewed. The general features of the Stokes operators are illustrated by evaluation of their means and variances for a range of simple polarization states. Some of the examples show polarization squeezing, in which the variances of one or more Stokes parameters are smaller than the coherent-state value. The main object of the paper is the application of these concepts to bright squeezed light. It is shown that a light beam formed by interference of two orthogonally polarized quadrature-squeezed beams exhibits squeezing in some of the Stokes parameters. Passage of such a primary polarization-squeezed beam through suitable optical components generates a pair of polarization-entangled light beams with the nature of a two-mode squeezed state. Implementation of these schemes using the double-fiber Sagnac interferometer provides an efficient method for the generation of bright nonclassical polarization states. The important advantage of these nonclassical polarization states for quantum communication is the possibility of experimentally determining all of the relevant conjugate variables of both squeezed and entangled fields using only linear optical elements followed by direct detection.
In this work we focus our attention on the correlation properties of a new class of bright nonclassical states of light, which are called the macroscopic Bell states [1,2]. These states are of interest in quantum optics due to their rarely combined two properties: brightness and quantumness. Recently we have investigated the polarization properties of these states, namely we have shown that the macroscopic singlet state is absolutely unpolarized and free of polarization noise [2], while the triplet states reveal the effect of hidden polarization [3]. Now we experimentally prove that both singlet and triplet macroscopic Bell states are entangled.