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Chinese Physics Letters

From the Thermo Wigner Operator to the Thermo Husimi Operator in Thermo Field Dynamics

Xuefen Xu, Zhu Shi-Qun

Published Sep 1, 2010• 0 citations
Publisher

Abstract

Using the thermo entangled state representation and based on the thermo Wigner operator we first introduce the thermo Husimi operator, which turns out to be a projective operator of the thermo squeezed coherent state, therefore the positive definite property of Husimi distribution function can be insured. In addition, we find the use of the thermo Husimi operator in simplifying the form of master equations.

DOI
10.1088/0256-307x/27/9/090305

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Using the thermo entangled state representation and based on the thermo Wigner operator we first introduce the thermo Husimi operator, which turns out to be a projective operator of the thermo squeezed coherent state, therefore the positive definite property of Husimi distribution function can be insured. In addition, we find the use of the thermo Husimi operator in simplifying the form of master equations.
For the first time we introduce the Husimi operator Delta_h(gamma,varepsilon;kappa) for studying Husimi distribution in phase space(gamma,varepsilon) for electron's states in uniform magnetic field, where kappa is the Gaussian spatial width parameter. Using the Wigner operator in the entangled state |lambda> representation [Hong-Yi Fan, Phys. Lett. A 301 (2002)153; A 126 (1987) 145) we find that Delta_h(gamma,varepsilon;kappa) is just a pure squeezed coherent state density operator |gamma,varepsilon>_kappa kappa
A new operator representation, called squeezed coherent state representation, is introduced since Husimi operator has the form of squeezed coherent state. We fisrt introduce its specific integral expression. When κ = 1, this representation is reduced to the usual P function. As an example, we calculate the squeezed coherent state representation for thermal field to illustrate a difference between P function and the squeezed coherent state representation. Especially, in order to better apply this representation to quantum optics, we reveal the integral transformations between the squeezed coherent state representation, respectively, and the following three functions: Wigner function, Q function, and Husimi function.
We introduce a generalized positive-definite operator Δ g (q,p) by smoothing out the Wigner operator Δ w (q,p) and by averaging over the “coarse graining" function. The function is then regarded as the classical Weyl correspondence of the operator Δ g (q,p); in this way we can easily identify a quantum state |Φ〉 such that Δ g (q,p) = |Φ〉 〈Φ|, and |Φ〉 turns out to be a new kind of squeezed coherent state. Correspondingly, the generalized distribution function for any state |Ψ〉 is 〈Ψ|Δ g (q,p)|Ψ〉 = |〈Φ|Ψ〉| 2 , which is obviously positive-definite and is a generalization of the Husimi function.