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European Journal of Physics

Optical field-strength polarization of two-mode single-photon states

Jesús Liñares, María C. Nistal, David Barral, Vicente Moreno

Published Jul 9, 2010• 10 citations
Publisher

Abstract

We present a quantum analysis of two-mode single-photon states based on the probability distributions of the optical field strength (or position quadrature) in order to describe their quantum polarization characteristics, where polarization is understood as a significative confinement of the optical field-strength values on determined regions of the two-mode optical field-strength plane. We will show that the mentioned probability distributions along with the values of quantum Stokes parameters allow us to characterize the polarization of a two-mode single-photon state, in an analogous way to the classical case, and to distinguish conceptually between mixture and partially polarized quantum states; in this way, we propose a simple definition of the quantum polarization degree based on the recent concept of distance measure to an unpolarized distribution, which gives rise to a depolarization degree equivalent to an overlapping between the probability distribution of the quantum state and a non-polarized two-mode Gaussian distribution. The work is particularly intended to university physics teachers and graduate students as well as to physicists and specialists concerned with the issue of optical polarization.

DOI
10.1088/0143-0807/31/5/001

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We present a quantum analysis of two-mode single-photon states based on the probability distributions of the optical field strength (or position quadrature) in order to describe their quantum polarization characteristics, where polarization is understood as a significative confinement of the optical field-strength values on determined regions of the two-mode optical field-strength plane. We will show that the mentioned probability distributions along with the values of quantum Stokes parameters allow us to characterize the polarization of a two-mode single-photon state, in an analogous way to the classical case, and to distinguish conceptually between mixture and partially polarized quantum states; in this way, we propose a simple definition of the quantum polarization degree based on the recent concept of distance measure to an unpolarized distribution, which gives rise to a depolarization degree equivalent to an overlapping between the probability distribution of the quantum state and a non-polarized two-mode Gaussian distribution. The work is particularly intended to university physics teachers and graduate students as well as to physicists and specialists concerned with the issue of optical polarization.
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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.
A quantum analysis of the generalized polarization properties of multimode non-stationary states based on their optical field-strength probability distributions is presented. The quantum generalized polarization is understood as a significant confinement of the probability distribution along certain regions of a multidimensional optical field-strength space. The analysis is addressed to quantum states generated in multimode linear and nonlinear waveguiding (integrated) photonic devices, such as multimode waveguiding directional couplers and waveguiding parametric amplifiers, whose modes fulfill a spatial modal orthogonality. In particular, the generalized polarization degree of coherent, squeezed and Schrödinger’s cat states is analyzed.
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.
Quantum polarization is an important property that has been extensively used for quantum communication purposes. In this work, the mixture of an unpolarized two-mode state and a two-mode pure state is analysed. This mixture is controlled by a single parameter, namely the degree of purity. The quantum polarization of the two-mode mixed state is discussed using the quantum Stokes parameters and the quantum degree of polarization. A relation between the degree of purity and the quantum degree of polarization is established. The mixture is used to model coherent light amplified by an erbium-doped fibre amplifier, which focuses the loss of polarization due to the unpolarized light resulting from spontaneous emission. Finally, the use of such a mixed state in the quantum key distribution using multiphoton pulses is analysed.
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.