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arXiv:quant-ph/0407184 · Physical Review A

Quantum polarization properties of two-mode energy eigenstates

Anita Sehat, Jonas Söderholm, Gunnar G. Bjork, P.B. Espinoza, A. B. Klimov, Luis Lorenzo Sánchez-Soto

Published Mar 30, 2005• 43 citations
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Abstract

We show that any pure, two-mode, $N$-photon state with $N$ odd or equal to two can be transformed into an orthogonal state using only linear optics. According to a recently suggested definition of polarization degree, this implies that all such states are fully polarized. This is also found to be true for any pure, two-mode, energy eigenstate belonging to a two-dimensional SU(2) orbit. Complete two- and three-photon bases whose basis states are related by only phase shifts or geometrical rotations are also derived.

DOI
10.1103/physreva.71.033818

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We show that the quantum Stokes parameters aptly describe the nonclassical two-polarization-mode Fock states and the transformation of their polarization states. In this description, the two-polarization-mode Fock states are the simultaneous eigenstates of the first and second Stokes parameters. Transformations of the polarization states, which are caused, for example, by optically active media, are described by the rotations in the Stokes vector space, which are given by unitary operators generated from the Hermitian Stokes parameters. The SU(2) symmetry of the Stokes parameters provides two independent degrees of freedom of polarimetric measurements that can exceed the standard quantum limit.
We examine whether the Stokes parameters of a two-mode electromagnetic field results from the superposition of the spins of the photons it contains. To this end we express any n-photon state as the result of the action on the vacuum of n creation operators generating photons which can have may different polarization states in general. We find that the macroscopic polarization holds as sum of the single-photon Stokes parameters only for the SU(2) orbits of photon-number states. The states that lack this property are entangled in every basis of independent field modes, so this is a class of entanglement beyond the reach of SU(2) transformations.
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We recall the framework of a consistent quantum description of polarization of light. Accordingly, the degree of polarization of a two-mode state $\mathrm{\ensuremath{\rho}\ifmmode \hat{}\else \^{}\fi{}}$ of the quantum radiation field can be defined as a distance of a related state ${\mathrm{\ensuremath{\rho}\ifmmode \hat{}\else \^{}\fi{}}}_{b}$ to the convex set of all SU(2)-invariant two-mode states. We explore a distance-type polarization measure in terms of the quantum Chernoff bound and derive its explicit expression. A comparison between the Chernoff and Bures degrees of polarization leads to interesting conclusions for some particular states chosen as illustrative examples.