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Physical Review A

Unpolarized light: Classical and quantum states

J. Lehner, Ulf Leonhardt, Henryk Paul

Published Apr 1, 1996• 81 citations

Abstract

We give a general definition of unpolarized light that rests on invariance properties rather than specific values of second-order field correlations or, equivalently, the Stokes parameters. It turns out that to require invariance with respect to rotation around the propagation axis and symmetry with respect to left- and right-handed circular polarization is not sufficient to fully characterize completely unpolarized light, such as natural light. Actually, a third requirement demanding invariance with respect to phase retardation is needed. Depending on whether it is satisfied or not, we speak of type I or type II unpolarized light, and we specify the general form of both classical distribution functions and density operators consistent with the respective conditions. \textcopyright{} 1996 The American Physical Society.

DOI: 10.1103/physreva.53.2727

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We give a general definition of unpolarized light that rests on invariance properties rather than specific values of second-order field correlations or, equivalently, the Stokes parameters. It turns out that to require invariance with respect to rotation around the propagation axis and symmetry with respect to left- and right-handed circular polarization is not sufficient to fully characterize completely unpolarized light, such as natural light. Actually, a third requirement demanding invariance with respect to phase retardation is needed. Depending on whether it is satisfied or not, we speak of type I or type II unpolarized light, and we specify the general form of both classical distribution functions and density operators consistent with the respective conditions. \textcopyright{} 1996 The American Physical Society.

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.

We investigate the polarization properties of unpolarized light, first in the temporal domain realized by

1.55 \mu \mathrm{m}

ASE light emitted by an erbium-doped fiber amplifier and second in the spatial domain by exploiting the polarization dispersion of a Cornu depolarizer. We find novel Stokes parameter correlations and invariances for temporally unpolarized light and novel states on the Poincaré sphere for spatially unpolarized light. All these results give new fresh insight into the field of classical polarization with interesting promising quantum and sensing application perspectives.

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.

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 capitalize on a multipolar expansion of the polarization density matrix, in which multipoles appear as successive moments of the Stokes variables. When all the multipoles up to a given order

K

vanish, we can properly say that the state is

K\mathrm{th}

-order unpolarized, as it lacks of polarization information to that order. First-order unpolarized states coincide with the corresponding classical ones, whereas unpolarized to any order tally with the quantum notion of fully invariant states. In between these two extreme cases, there is a rich variety of situations that are explored here. The existence of hidden polarization emerges in a natural way in this context.