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Zeitschrift für Naturforschung A

Sudarshan-Glauber P, Q, and Wigner W Distributions in Correlated-Spontaneous-Emission Lasers

Marlan O. Scully, Shi-Yao Zhu

Published Feb 1, 1997• 0 citations
Publisher

Abstract

Abstract A comparative study of Sudarshan-Glauber P, Q and Wigner distributions in correlated-spontaneous-emission lasers has been made. Immediately giving the information of squeezing by exam-ining the diffusion coefficients is the advantage of the Sudarshan-Glauber P representation.
DOI
10.1515/zna-1997-1-229

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Much attention has been given in the past to two classes of squeezed states: the squeezed vacuum and coherent states. Here we study the effects of squeezing on number states and on thermal field states. The statistical properties of these various squeezed states are discussed using the second-order correlation functions. The quasiprobabilities of the Wigner, Q, and positive P representations are calculated and compared for the squeezed states. The Glauber P representation for the squeezed thermal state explicitly shows the limit of its applicability. The photon number distributions of the squeezed number and squeezed thermal states are extensively discussed and new interference effects in phase space are shown to lead to highly structured number distributions.
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Phase properties of squeezed number states and squeezed thermal states are studied. Exact analytical formulae for phase distributions based on different phase approaches are derived and illustrated graphically. It is shown that the phase quasiprobability distribution P (W) ( theta ) associated with the Wigner function does not depend on the photon number of the initial number state and has the same form for both kinds of squeezed states under consideration while the Pegg-Barnett phase distribution and the phase quasiprobability distributions associated with the Q function and the Glauber-Sudarshan P function, approach P (W) ( theta ) in the limit of highly excited states.
It is shown that, to any quasiprobability distribution corresponding to a given density operator, one can associate a class of positive distributions in an extended space. In situations where the quasiprobability distributions are singular, these distributions are shown to provide regularized versions thereof. The positive distribution introduced by Drummond and Gardiner (1981) in the context of the Glauber-Sudarshan P-representation is shown to arise as a special case. One can thus associate a positive distribution to the Wigner function such that its moments give the averages of Weyl-ordered operators with respect to the given density operator. It is also found that one can associate to the Glauber-Sudarshan P-function, a distribution in the extended space which, though not positive, interestingly, involves the Wigner function (1932). A measurement scheme which directly yields these positive distributions is presented.