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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Probability density function for representing quantum states of polarized optical field in the basis of linearly polarized photons

Łukasz Michalik, Andrzej W. Domański

Published Apr 21, 2012• 0 citations
Publisher

Abstract

In this paper authors discuss the inverse problem for the density operator describing a bi-modal quantum mixed states of polarized optical field with the reduced probability density function. It is reduced because we assume that photons phase is known - it is represented by the Dirac delta function in the probability distribution. We ask for example if it is possible to represent an elliptically polarized plane wave in the basis of linearly polarized photons (or photons with any other arbitrary chosen phase). Our goal is to define a reversible integral transformation in order to represent the reduced probability density function by the density operator describing a mixed state and to analyze the uniqueness of the solution. This problem is similar to calculating Glauber-Sudarshan function when representing a quantum mixed state in the coherent states basis. However the integral transformation that we search is not that easy to define. It is based on convolution and cross-correlation operations. The operator that generates this transformation is defined using the Stokes operators.

DOI
10.1117/12.922207

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In this paper authors discuss the inverse problem for the density operator describing a bi-modal quantum mixed states of polarized optical field with the reduced probability density function. It is reduced because we assume that photons phase is known - it is represented by the Dirac delta function in the probability distribution. We ask for example if it is possible to represent an elliptically polarized plane wave in the basis of linearly polarized photons (or photons with any other arbitrary chosen phase). Our goal is to define a reversible integral transformation in order to represent the reduced probability density function by the density operator describing a mixed state and to analyze the uniqueness of the solution. This problem is similar to calculating Glauber-Sudarshan function when representing a quantum mixed state in the coherent states basis. However the integral transformation that we search is not that easy to define. It is based on convolution and cross-correlation operations. The operator that generates this transformation is defined using the Stokes operators.
In this paper authors present a novel idea for reconstructing quantum states of polarized optical field expressed in the one-photon basis. The quantum state reconstruction is a measurement of a quasidistribution function P that allows to obtain a density operator for the analyzed state. The most commonly used approach for a quantum state reconstruction is based on the so called quantum tomography. The marginal probability distributions for the Wigner function are measured for different angles of projection and then they are put together into the two-dimensional distribution by the inverse Radon transformation. In our approach the analyzed field is combined with the reference field which is well defined, it can be for example a photon-beam containing linearly polarized photons only. The proposed thought experiment assumes that we measure the degree of linear polarization of this combined beam using a rotating polarizer. We obtain the searched distribution P by changing the parameters of the reference beam – its intensity and state of polarization. This series of measurements allows to build the system of equations The values of the function P at various points P(θ i, ϕ j) are the searched variables of this system. Accuracy of that solution depends on how many measurements are done.
Abstract In this paper the authors discuss an alternative way for reconstructing one-photon mixed states of a partially polarized optical field. The task is to represent the probability density distribution describing these kind of states with the Stokes parameters which also characterize the effective state of polarization. These parameters can be measured by means of the degree of polarization with an experimental setup containing a rotating linear polarizer and a circular polarizer. A thought experiment is presented which assumes that the measurement is undertaken on an analyzed beam coupled with a reference beam containing photons polarized in a well-known way. The method discussed in the paper is an alternative for the most commonly used quantum tomography approach.
In this paper the authors present the idea for reconstructing one-photon states. Reconstructing a quantum state means measuring the probability distribution P that allows one to write the density operator for the analysed state. The most commonly known approach for the quantum reconstruction is the quantum tomography. Our alternative method assumes that the analysed field is coupled with the reference field which is described by the parameters settled during a measurement. In the proposed gedankenexperiment the degree of linear polarisation (DOLP) of this combined beam is measured using a rotating linear polariser. We state that it is possible to obtain the P-function by changing the parameters of reference beams and by preparing the series of DOLP measurements. This series of data leads to the system of equations. The values of the P-function at chosen points are the unknowns of this system. This article focuses on the numerical algorithm for solving these equations.
In the paper a quantum description of an effective state of polarization for an electromagnetic field containing variously polarized photons is presented. The discussion of the problem includes the quantum character of a single photon, comparison with a classical field, matrix description of photon polarization and physical interpretation of the representation of a density operator that represents photons in a statistical ensemble. Full text: PDF References: R.J. Glauber, Quantum Theory of Optical Coherence (Wiley-VCH 2007). C.C. Gerry, P.L. Knight, Introductory Quantum Optics (Cambrige Univeristy Press 2005) R.J. Glauber, New Techn. Ideas Quant. Measur. Theory 480, 336 (1986). L. Michalik, J. Jasinski, P.L. Makowski, A.W. Domanski,Problem of degree of polarization for photons, Proc. SPIE 7745, 77450K (2010). [CrossRef] T. Tudor, A. Gheondea, Pauli algebraic forms of normal and nonnormal operators, Jour. Opt. Soc. Am. A 24, 204 (2007). [CrossRef]
In this paper we discuss the reconstruction process of one-photon mixed states of partially polarised light. To solve this issue, we obtain the Stokes parameters by means of the degree of polarisation. The density operator describing the examined state is represented with these parameters. In the proposed two-beams method the degree of polarisation is measured on the analysed beam combined with reference beams containing photons with a settled state of polarisation. Coupling these beams allows one to obtain the Stokes parameters from the intensity contrast behind the rotating polariser. We discuss the influence of the quantum uncertainty on this technique of one-photon states reconstruction and we consider it for three aspects – the possibility of reducing the number of reference beams that are needed, the optimal state of polarisation of reference beams and the accuracy of the reconstruction method.