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AIP conference proceedings

Covariance, Squeezed and Coherent States: Proposal for Experimental Realization of Covariance States

Andrey K. Angelow

Published Jan 1, 2007• 5 citations
Publisher

Abstract

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.

DOI
10.1063/1.2733158

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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.
Characteristic uncertainty relations and their related squeezed states are briefly reviewed and compared in accordance with the generalizations of three equivalent definitions of canonical coherent states. The standard SU(1, 1) coherent states are shown to be the unique states that minimize the Schrödinger uncertainty relation for every pair of the three generators and the Robertson relation for the three generators. The characteristic uncertainty inequalities are naturally extended to the case of several states. It is shown that these inequalities can be written in the equivalent complementary form.
Certain superposition (linear combination) of energy eigenfunctions of linear harmonic oscillators are called coherent states because they are of significance in optics for the representation of coherent light waves. A hallmark of coherent states is that the variances (squares of the uncertainties) of x and px are constant in time and further their product becomes the minimum value allowed by the Heisenberg uncertainty principle. On the other hand, some linear combinations of harmonic oscillator energy functions give rise to squeezed states . A curious property of squeezed states is that the variances of x and px oscillate in time 180o out of phase with one another with the frequency twice of the oscillator. The wave packet of a coherent state possesses a minimum uncertainty. 〈x〉 and 〈px〉 have the same oscillatory forms as in the classical case. The coherent states have also been called minimum uncertainty coherent states, the Schrödinger coherent states or the Glauber coherent states [1-6]. The squeezed states are sometimes referred to in the literature as two-photon coherent states and generalized coherent states.
This special issue offers a large selection of contributions dealing with recent advances in the theoretical and experimental aspects of quantum correlations and fluctuations. The issue results from a proposal to dedicate a special issue to the topics covered in the 6th International Conference on Squeezed States and Uncertainty Relations (ICSSUR '99) which took place in Naples, Italy on 24-29 May 1999. This proposal was received enthusiastically by the conference organisers, and the unexpectedly large number of contributions indicates the high level of interest in coherent effects in optics and related systems. The published papers encompass a wide range of topics: quantum measurements and noise, quantum interference and correlations, quantum communication and information, generation and characterization of non-classical states of light, squeezed states, and other quantum harmonic oscillators and quantum-like systems. With thanks to the Editor and the Editorial Office for all the work done, the guest editors express the hope that analogous thematic issues could appear in the near future.
Using only elementary quantum-mechanical concepts, the statistical properties of various harmonic oscillator states that are linear superpositions of its energy eigenfunctions are described. These superpositions include coherent states and squeezed (or two-photon coherent) states. The resulting Gaussian, minimum-uncertainty wave packets are shown to oscillate back and forth for both coherent and squeezed states, but with an oscillating ‘‘width’’ for the squeezed states. Also examined are the principles underlying the production of squeezed electromagnetic waves via parametric amplification or four-wave mixing, their measurement by homodyne detection, and the connection between squeezing and non-Poissonian counting statistics.
An introductory survey on the Schroedinger uncertainty relation and its minimization states is presented with minimal number of formulas and some historical points. The case of the two canonical observables, position and momentum, is discussed in greater detail: basic properties of the two subsets of minimization states (canonical squeezed and coherent states) are reviewed and compared. The case of two non-canonical observables is breafly outlined. Stanfard SU(1,1) and SU(2) group-related coherent states can be defined as states that minimize Schroedinger inequality for the three pairs of generators simultaneously. The symmetry of the Heisenberg and Schroedinger relations is also discussed, and two natural generalizations to the cases of several observables and several states are noted.