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Journal of Physics A Mathematical and General

Generalized observables in polarization optics

Tiberiu S. Tudor

Published Aug 27, 2003• 30 citations
Publisher

Abstract

The operatorial theory of optical polarization devices can provide some of the clearest examples in which von Neumann's model of orthogonal measurement is violated. This paper analyses the global and spectral properties of some two- and three-layer optical polarizers of non-orthogonal kind by calculating the dyadic expressions of their operators in a Dirac-algebraic form. These constitute some of the most expressive examples of non-Hermitian operators corresponding to generalized observables and it is expected that the theory of generalized quantum measurement will take advantage of the simplicity of their physical realization.

DOI
10.1088/0305-4470/36/36/309

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The operatorial theory of optical polarization devices can provide some of the clearest examples in which von Neumann's model of orthogonal measurement is violated. This paper analyses the global and spectral properties of some two- and three-layer optical polarizers of non-orthogonal kind by calculating the dyadic expressions of their operators in a Dirac-algebraic form. These constitute some of the most expressive examples of non-Hermitian operators corresponding to generalized observables and it is expected that the theory of generalized quantum measurement will take advantage of the simplicity of their physical realization.
This paper constitutes an application of the polarization optics in the problem of quantum measurement. The non-Hermitian operators of the nonorthogonal multilayer optical polarizers represent observables in the sense of the generalized quantum theory of measurement. The intimate spectral structure of these polarizers can be disclosed in the frame of skew-angular vector bases and biorthonormal vector systems. We show that these polarizers correspond to skew projectors; their operators are "generated" by skew projectors in the sense of the spectral theorem of linear operators theory. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors.
Extracting information from a quantum system inevitably requires the performance of a measurement, and it is no surprise that the theory of measurement plays a central role in our subject. The physical nature of the measurement process remains one of the great philosophical problems in the formulation of quantum theory. Fortunately, however, it is sufficient for us to take a pragmatic view by asking what measurements are possible and how the theory describes them, without addressing the physical mechanism of the measurement process. This is the approach we shall adopt. We shall find that it leads us to a powerful and general description of both the probabilities associated with measurement outcomes and the manner in which the observation transforms the quantum state of the measured system. The simplest form of measurement was given a mathematical formulation by von Neumann, and we shall refer to measurements of this type as von Neumann measurements or projective measurements. It is this description of measurements that is usually introduced in elementary quantum theory courses. We start with an observable quantity A represented by a Hermitian operator Â, the eigenvalues of which are the possible results of the measurement of A. The relationship between the operator, its eigenstates {|λnñ}, and its (real) eigenvalues {λn} is expressed by the eigenvalue equation . . . Â |λn_ = λn|λn_. (4.1) . . .
In our paper we consider the non-classical behaviour of both the Hermitian (observable) Stokes parameters of light and the phase difference of two modes that describe the quantum polarization states of optical field. To characterize the degree of polarization of light we introduce a new quantity taking into account the quantum properties of different quantum states of two orthogonally polarized modes. The problem of determination of the phase difference in two modes of optical field for the quantum polarization states of light is discussed. To describe in general such a quantum field we introduce two pairs of the phase operators: the phase angles for the Stokes parameters of light in a three-dimensional picture of the Poincaré sphere. We also consider a special type of the eight-port polarization interferometer (polarimeter) for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phase operators and their fluctuations as well. Using an anisotropic (spatioperiodic) Kerr-like nonlinear medium associated with the polarization interferometer we could generate and also observe the polarization-squeezed phase states of light. The fluctuations in the phase difference between two orthogonally polarized modes for these non-classical states are less than the fluctuations for light in coherent state.
The state projection associated with the familiar (von Neumann) measurement of an observable does not describe the most general means of interrogating a quantum system. Two problems requiring a more general description are the discrimination between non–orthogonal quantum states and the simultaneous measurement of incompatible observables. We discuss both of these problems within the context of measurements of optical polarization. We illustrate the technique of field measurement by projection synthesis by describing a possible means for determining the optical phase probability distribution.
The spectral analysis of a non-Hermitian unbounded operator appearing in quantum physics is our main concern. The properties of such an operator are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operator and its adjoint can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.