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arXiv:quant-ph/0303180 · Journal of Optics B Quantum and Semiclassical Optics

Continuous variable polarization entanglement, experiment and analysis

Warwick P. Bowen, Nicolas Treps, Roman Schnabel, Timothy C. Ralph, Ping Koy Lam

Published Aug 1, 2003• 15 citations
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Abstract

We generate and characterize continuous variable polarization entanglement between two optical beams. We first produce quadrature entanglement, and by performing local operations we transform it into a polarization basis. We extend two entanglement criteria, the inseparability criteria proposed by Duan et al (2000 Phys. Rev. Lett. 84 2722) and the Einstein–Podolsky–Rosen (EPR) paradox criteria proposed by Reid and Drummond (1988 Phys. Rev. Lett. 60 2731), to Stokes operators; and use them to characterize the entanglement. Our results for the EPR paradox criteria are visualized in terms of uncertainty balls on the Poincaré sphere. We demonstrate theoretically that using two quadrature entangled pairs it is possible to entangle three orthogonal Stokes operators between a pair of beams, although with a bound √3 times more stringent than for the quadrature entanglement.

DOI
10.1088/1464-4266/5/4/352

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We report on the generation of non-separable beams produced via the interaction of a linearly polarized beam with a cloud of cold caesium atoms placed in an optical cavity. We convert the squeezing of the two linear polarization modes into quadrature entanglement and show how to find the best entanglement generated in a two-mode system using the inseparability criterion for continuous variables (Duan et al 2000 Phys. Rev. Lett. 84 2722). We verify this method experimentally with a direct measurement of the inseparability using two homodyne detectors. We then map this entanglement into a polarization basis and achieve polarization entanglement .
Continuous variable polarization states have attracted much attention due to their apparent applicability to quantum information networks. Polarization entanglement was presented by transformation of the well-known quadrature entanglement onto the polarization basis. The inseparability criteria as well as the EPR paradox criteria for Stokes operators were also demonstrated. In this paper the generation of such highly quantum correlated states using the interference of two fiber-based polarization squeezing sources is presented. This squeezing can be efficiently created based on the nonlinear effect experienced by ultra-short pulses in fused silica fibers. Two linearly orthogonal polarized modes with identical amplitudes propagate through a polarization maintaining fiber, they experience the same nonlinearity and the phase of the interference between them is controlled to achieve circularly polarized light at the output.
We present an experimental demonstration of both quadrature and polarization entanglement generated via the interaction between a coherent linearly polarized field and cold atoms in a high finesse optical cavity. The nonlinear atom-field interaction produces two squeezed modes with orthogonal polarizations which are used to generate a pair of nonseparable beams, the entanglement of which is demonstrated by checking the inseparability criterion for continuous variables recently derived by Duan et al. [Phys. Rev. Lett. 84, 2722 (2000)]] and calculating the entanglement of formation [Phys. Rev. Lett. 91, 107901 (2003)]].
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.