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Quantum Electronics and Laser Science Conference

Classical-wave analogs of quantum state measurement

K.F. Lee, Frank Reil, John E. Thomas

Published Nov 13, 2002• 1 citation
Publisher

Abstract

Summary form only given. It is well-known that in the paraxial approximation, the transverse mode of an optical field of frequency /spl omega/ obeys an equation which is formally identical in structure to the Schrodinger wave equation of quantum mechanics: i /spl part//spl epsiv/(x,y,z)//spl part/z = /spl nabla//sup 2//sub tr//spl epsiv/(x,y,z)/2k - 2/spl pi//spl kappa//spl Delta//spl chi/(x,y)/spl epsiv/(x,y,z). We employ a local oscillator (LO) comprising a phase-locked super-position of a large collimated Gaussian beam and a small focused Gaussian beam to directly measure the real and imaginary parts of the correlation function.

DOI
10.1109/qels.2001.961856

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Summary form only given. It is well-known that in the paraxial approximation, the transverse mode of an optical field of frequency /spl omega/ obeys an equation which is formally identical in structure to the Schrodinger wave equation of quantum mechanics: i /spl part//spl epsiv/(x,y,z)//spl part/z = /spl nabla//sup 2//sub tr//spl epsiv/(x,y,z)/2k - 2/spl pi//spl kappa//spl Delta//spl chi/(x,y)/spl epsiv/(x,y,z). We employ a local oscillator (LO) comprising a phase-locked super-position of a large collimated Gaussian beam and a small focused Gaussian beam to directly measure the real and imaginary parts of the correlation function.
One of the most delicate problems in quantum optics is the proper description of the phase of a single-mode radiation field. One has first to devise an experimental scheme for phase measurement, thus giving an operational definition of phase, and afterwards search for the proper quantum-mechanical description of the experimental procedure. Several such schemes which differ distinctly in their experimental setup have already been proposed and even partly realized. In this paper, I will describe them in some detail and present the outlines of their theoretical analysis.>
Summary form only given. In our experiments ultracold rubidium atoms are loaded into an optical standing wave. The amplitude of the standing wave is then modulated, creating a time dependent potential given by V(x,t)=/spl kappa/(1-2/spl epsiv/sin(2/spl pi/f/sub m/t))cos(2kx) where /spl kappa/ is the strength of the potential, /spl epsiv/ is the amplitude of modulation and f/sub m/ is the frequency of modulation. The dynamics of atoms in such a potential is chaotic except for small islands of stability which exist for certain values of /spl kappa/, /spl epsiv/ and f/sub m/. These islands are phase space resonances and rotate in phase space at the modulation frequency. By viewing the dynamics of the interaction stroboscopically, at intervals of the modulation frequency it is possible to observe these resonances in the final atomic velocity distribution.
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We have already shown that the coherent states are very close to the classical state of the well-defined amplitude and phase. A question not answered yet is: Does there exist an analogous quantum mechanical observable for the phase that is given by some Hermitian operator? We know from classical optics that phase of an EM field is an observable quantity. Hence, in quantum physics it should be associated with a Hermitian operator. In order to set up a quantitative description of the phase operator, we will introduce some of the ideas presented in the literature which led to the formulation of the Hermitian phase operator.
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