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arXiv:1007.1322 · Physical Review Letters

Entangling Different Degrees of Freedom by Quadrature Squeezing Cylindrically Polarized Modes

C. Gabriel, Andrea Aiello, W. Zhong, Tijmen Godfried Euser, Nicolas Y. Joly, Peter Banzer, Michael Förtsch, Dominique Elser, Ulrik L. Andersen, Christoph Marquardt, Philip St.J. Russell, Gerd Leuchs

Published Feb 11, 2011• 135 citations
arXivHTMLPDF

Abstract

Quantum systems such as, for example, photons, atoms, or Bose-Einstein condensates, prepared in complex states where entanglement between distinct degrees of freedom is present, may display several intriguing features. In this Letter we introduce the concept of such complex quantum states for intense beams of light by exploiting the properties of cylindrically polarized modes. We show that already in a classical picture the spatial and polarization field variables of these modes cannot be factorized. Theoretically it is proven that by quadrature squeezing cylindrically polarized modes one generates entanglement between these two different degrees of freedom. Experimentally we demonstrate amplitude squeezing of an azimuthally polarized mode by exploiting the nonlinear Kerr effect in a specially tailored photonic crystal fiber. These results display that such novel continuous-variable entangled systems can, in principle, be realized.

DOI
10.1103/physrevlett.106.060502

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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.
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.
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Squeezed states, special kinds of entangled states, are known as a useful resource for quantum metrology. In interferometric sensors, they allow us to overcome the ‘classical’ projection noise limit stemming from the independent nature of the individual photons or atoms within the interferometer. Motivated by the potential impact on metrology as well as by fundamental questions in the context of entanglement, a lot of theoretical and experimental effort has been made to study squeezed states. The first squeezed states useful for quantum-enhanced metrology have been proposed and generated in quantum optics, where the squeezed variables are the coherences of the light field. In this tutorial, we focus on spin squeezing in atomic systems. We give an introduction to its concepts and discuss its generation in Bose–Einstein condensates. We discuss in detail the experimental requirements necessary for the generation and direct detection of coherent spin squeezing. Two exemplary experiments demonstrating adiabatically prepared spin squeezing based on motional degrees of freedom and diabatically realized spin squeezing based on internal hyperfine degrees of freedom are discussed.
We explore in detail the possibility of intracavity generation of continuous-variable (CV) entangled states of light beams under mode phase-locked conditions. We show that such quantum states can be generated in a self-phase-locked nondegenerate optical parametric oscillator (NOPO) based on type II phase-matched down-conversion combined with a linear mixer of two orthogonally polarized modes of the subharmonics in a cavity. A quantum theory of this device, recently realized in the experiment, is developed for both subthreshold and above-threshold operational regimes. We show that the system providing high-level phase coherence between two generated modes, unlike the ordinary NOPO, also exhibits different types of quantum correlations between photon numbers and phases of these modes. We quantify the CV entanglement as two-mode squeezing and show that the maximal degree of the integral two-mode squeezing (which is 5050%50 relative to the level of vacuum fluctuations) is achieved at the pump field intensity close to the generation threshold of a self-phase-locked NOPO, provided that the constant of linear coupling between the two polarizations is much less than the mode detunings. The peculiarities of CV entanglement for the case of unitary, nondissipative dynamics of the system under consideration are also cleared up.