Statistics of dressed modes in a thermal state

Abstract

By a Wigner-function calculation, we evaluate the trace of a certain Gaussian operator arising in the theory of a boson system subject to both finite temperature and (weak) interaction. Thereby we rederive (and generalize) a recent result by Kocharovsky, Kocharovsky, and Scully [Phys. Rev. A 61, 053606 (2000)] in a way that is technically much simpler. One step uses a special case of the response of Wigner functions to linear transformations, and we demonstrate the general case by simple means. As an application we extract the counting statistics for each mode of the Bose gas. Key words: Gaussian operator, Wigner function, cold Bose gas. In a recent paper on dilute boson gases [1], Kocharovsky, Kocharovsky, and Scully faced the problem of evaluating the trace of a product of two 2-mode Gaussian operators. They solved it by exploiting the algebra of 2-mode squeezing operators in a remarkable tour de force occupying the appendix of [1]. The main objective of the present Brief Report is to put on record an alternative derivation that is simpler and more direct. In a few words, the essence of our calculation is this: We recall (i) that the trace of a product of two operators can be obtained as a phase space integral of the product of the respective Wigner functions [2]; (ii) that the Wigner functions of Gaussian operator functions are Gaussian functions of the phase space variables; and (iii) that integrals of Gaussians are immediate. The trace of interest is T(u, v; ǫ) = tr{G(u, v)ρǫ} = 〈G(u, v)〉 ǫ