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arXiv:quant-ph/0610032 · Physical Review A

Maximally polarized states for quantum light fields

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

Published Oct 15, 2007• 11 citations
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Abstract

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 $\overline{N}$ present a fairly symmetric, parabolic photon statistic, with a variance scaling as ${\overline{N}}^{2}$. 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.

DOI
10.1103/physreva.76.043820

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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.
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 SU(2) group is used in two different fields of quantum optics, the quantum polarization and quantum interferometry.Quantum degrees of polarization may be based on distances of a polarization state from the set of unpolarized states.The maximum polarization is achieved in the case where the state is pure and then the distribution of the photon-number sums is optimized.In quantum interferometry, the SU(2) intelligent states have also the property that the Fisher measure of information is equal to the inverse minimum detectable phase shift on the usual simplifying condition.Previously, the optimization of the Fisher information under a constraint was studied.Now, in the framework of constraint optimization, states similar to the SU(2) intelligent states are treated.
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.
We introduce a probability distribution for polarization of three-dimensional quantum light fields as a marginal of the quadrature QQQ function for a three-mode field by removing the variables irrelevant for polarization (total intensity and global phase). The probability distribution turns out to be determined by projection on SU(3) coherent states. We introduce a degree of polarization as the distance between the polarization distribution and the uniform distribution associated with completely unpolarized light. We study the relation between two- and three-dimensional polarization by considering field states with a component in the vacuum state. We apply this formalism to some relevant field states.
The classical definition of degree of polarization (DOP) is expressed in the quantum domain by replacing intensities through quantum mechanical average values of relevant number operators and is viewed as the first generalization of intensity. This definition assigns inaccurately the unpolarized status to some typical optical fields, e.g., amplitude-coherent phase-randomized and hidden-polarized light, which are not truly unpolarized light. The apparent paradoxical trait is circumvented by proposing a new definition of DOP in quantum optics through the second generalization of intensity. The correspondence of a new DOP to the usual DOP in quantum optics is established. It is seen that the two definitions disagree significantly for intense optical fields but coincide for weak light (thermal light) or for optical fields in which occupancy of photons in the orthogonal mode is very feeble. Our proposed definition of DOP, similar to other proposals in literature, reveals an interesting feature that states of polarization of optical quantum fields depend upon the average photons (intensity) present therein.