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Journal of Physics Conference Series

Degrees of polarization for a quantum field

Luis Lorenzo Sánchez-Soto, Jonas Söderholm, E. C. Yustas, A. B. Klimov, Gunnar G. Bjork

Published Apr 1, 2006• 12 citations
Publisher

Abstract

Unpolarized light is invariant with respect to any SU(2) polarization transformation. Since this fully characterizes the set of density matrices representing unpolarized states, we introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We discuss different candidates of distance, and show that they induce fundamentally different degrees of polarization.

DOI
10.1088/1742-6596/36/1/026

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Unpolarized light is invariant with respect to any SU(2) polarization transformation. Since this fully characterizes the set of density matrices representing unpolarized states, we introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We discuss different candidates of distance, and show that they induce fundamentally different degrees of polarization.
It is well established that unpolarized light is invariant with respect to any SU(2) polarization transformation. This requirement fully characterizes the set of density matrices representing unpolarized states. We introduce the degree of polarization of a quantum state as its distance to the set of unpolarized states. We use two different candidates of distance, namely the Hilbert-Schmidt and the Bures metric, showing that they induce fundamentally different degrees of polarization. We apply these notions to relevant field states and we demonstrate that they avoid some of the problems arising with the classical definition.
Quantum optics entails polarization properties that cannot be fully described by the classical Stokes parameters. In this work, we characterize the polarization of classical as well as quantum fields by means of a probability distribution on the Poincar\'e sphere. This serves to define the degree of polarization of a field state as the distance between the corresponding polarization distribution and the uniform distribution representing unpolarized light. We apply this definition to relevant quantum field states such as SU(2) coherent, squeezed, number, and phase states.
The degree of polarization of a quantum field can be defined as its distance to an appropriate set of states. When we take unpolarized states as this reference set, the states optimizing this degree for a fixed average number of photons N‾\overline{N}N present a fairly symmetric, parabolic photon statistic, with a variance scaling as N‾2{\overline{N}}^{2}N2. Although no standard optical process yields such a statistic, we show that, to an excellent approximation, a highly squeezed vacuum can be taken as maximally polarized. We also consider the distance of a field to the set of its SU(2) transformed, finding that certain linear superpositions of SU(2) coherent states make this degree to be unity.
We address a quantitative determination of the degree of polarization of type-II unpolarized light via the computation of the distance between the polarization distribution and the uniform distribution associated with fully unpolarized light (i.e., type-I unpolarized light or natural light). We determine the maximum degree of polarization for type-II unpolarized light and the states reaching it. We show that the degree of polarization can be arbitrarily large, approaching complete polarization for increasing mean photon numbers.
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.
A new consequent description of the polarization structure of light within the framework of quantum optics is given by using the polarization gauge SU(2) invariance of free electromagnetic fields and a related concept of the polarization (P) spin. Within this approach the quantum light may be thought as a mixture of the usual polarized photons and unpolarized biphotons. New classes of unpolarized light states generated by P-scalar biphotons are examined. Possible applications are briefly discussed.