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Journal of Modern Optics

Quantum correlation coeffi cient for position and momentum

Richard A. Campos

Published Jul 1, 1999• 16 citations
Publisher

Abstract

We study the general statistical measures of covariance and correlation coefficient for noncommuting quantum observables as exemplified by position and momentum operators in quantum optics. High values of the position-momentum quantum correlation coefficient are achievable with two-photon coherent states and Schrödinger cat states of the electromagnetic field for parametric values which renounce quadrature squeezing. We discuss quantum-optical paradigms to detect these measures.

DOI
10.1080/09500349908231335

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We study the general statistical measures of covariance and correlation coefficient for noncommuting quantum observables as exemplified by position and momentum operators in quantum optics. High values of the position-momentum quantum correlation coefficient are achievable with two-photon coherent states and Schrödinger cat states of the electromagnetic field for parametric values which renounce quadrature squeezing. We discuss quantum-optical paradigms to detect these measures.
Coherent and Squeezed States are widely discussed in literature, but there exist an other class of states (Covariance), minimizing more general inequality [Schrödinger, 1930] then Heisenberg’s one. To examine this relation and classify the Covariance, Squeezed and Coherent States, we consider the quantum optical problem of the propagation of electromagnetic waves in an anisotropic waveguide. The squeezed parameter, taking place in both — squeezed and covariance states, is expressed in explicit form through the main parameters of degenerated parametric amplification in an anisotropic waveguide. The expressions for the three independent quantum fluctuations are derived. The presence of nonlinearity is essential for the Covariance that appears in the Schrödinger Uncertainty Relation. Proposal for experimental realization of Covariance States in anisotropic optical waveguides is done.
Introduction We have now prepared the way for a quantum treatment of problems that are of interest in optics. We have examined some basic properties of the quantized field and have explored the coherent-state representation and some of its virtues. Until now our discussions have been essentially formal; we have not concerned ourselves with what is actually measured in the laboratory, nor with questions concerning the interactions of the quantized field with other systems. In the following sections we shall turn our attention to some of the quantum field theoretic quantities that are of particular interest in quantum optics and are of significance in the measurement of the field. Our discussions in this chapter will still be somewhat formal, in that we shall not yet come to grips with the problem of the interaction of the field with the apparatus. That problem will be tackled in Chapter 14. For the moment we shall continue to oversimplify and treat the field as a free field. However, within the idealized context of the free field we shall examine some of the questions encountered in measurements in quantum optics. We shall see that normally ordered correlations play a dominant role and examine some of their properties. Anti-normally ordered correlations will be seen to arise in much less common and less useful measurements of the field. We shall also examine the sense in which a photon may be said to be localized in a measurement in which a photoelectric emission is registered.
The two-mode even and odd coherent Schrödinger cat states are introduced for the generalized stationary and non-stationary harmonic oscillator. The expected values of positions and momenta and their dispersion matrices are calculated for the Schrödinger cat states. The quadrature squeezing and correlation phenomenon are studied and the Wigner and Husimi functions are constructed explicitly for these states. The photon statistics of the two-mode even and odd coherent states of the generalized oscillator is described in terms of the photon distribution function, which is expressed in terms of Hermite polynomials. The photon number, means and dispersions show oscillations which are characteristics of squeezing and correlation phenomenon.
It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrödinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.