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Communications in Theoretical Physics

Wigner Distribution Function and Husimi Function of a Kind of Squeezed Coherent State

Fan Hong‐Yi, Liu Shu-Guang

Published Mar 1, 2007• 12 citations
Publisher

Abstract

We find a new κ-parameter squeezed coherent state | p , q ⟩ κ representation, which possesses well-behaved features, i.e., its Wigner function's marginal distribution in the `` q -direction'' and in the `` p -direction'' is the Gaussian form exp {−κ( q '− q ) 2 }, and exp {( p '− p ) 2 /κ}, respectively. Based on this, the Husimi function of | p , q ⟩ κ is also obtained, which is a Gaussian broaden version of the Wigner function. The | p , q ⟩ κ state provides a good representative space for studying various properties of the Husimi operator.

DOI
10.1088/0253-6102/47/3/009

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We find a new κ-parameter squeezed coherent state | p , q ⟩ κ representation, which possesses well-behaved features, i.e., its Wigner function's marginal distribution in the q -direction'' and in the p -direction'' is the Gaussian form exp {−κ( q '− q ) 2 }, and exp {( p '− p ) 2 /κ}, respectively. Based on this, the Husimi function of | p , q ⟩ κ is also obtained, which is a Gaussian broaden version of the Wigner function. The | p , q ⟩ κ state provides a good representative space for studying various properties of the Husimi operator.
我们发现一个新κ-参数挤压了协调州的| p ， q > _κ 表示，它拥有行为优良特征，即，它的 Wigner 功能“ s 边缘的分发在”q方向“并且在里面”p方向“是 Gaussian 形式 exp {-κ(q''- q )~ 2 }，并且 exp {( p ”-p)~2/κ},分别地。基于这， | p 的 Husimi 功能， q > _ κ也被获得，它是 Gaussian 拓宽 Wigner 功能的版本。| p， q > _ κ状态为学习 Husimi 操作员的各种各样的性质提供一个好代表性的空格。
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.
In this paper, the so-called Husimi operator Δ h ( q , p ; κ), which is introduced by smoothing out the Wigner operator Δ w ( q , p ) by averaging over the "coarse graining" function exp[-κ( q '- q ) 2 - ( p ' - p ) 2 /κ], is now regarded as a Weyl correspondence connecting the Husimi operator Δ h ( q , p ; κ) with its classical correspondence, since the integration kernel is just the Wigner operator. In this way we can easily identify | p , q ;κ⟩ such that Δ h ( q , p ;κ) = | p , q ;κ⟩ ⟨ p , q ;κ|, where | p , q ; κ⟩ is a new kind of squeezed coherent states. The entangled Husimi operator is also treated in this way. Thus a simple way to find the Husimi operator is presented.
We find that a kind of atomic coherent state, formed as exp[ξJ + – ξ*J − ] |00〉, when the SU(2) generators J± are taken as Fan's form, J + = (1/2)(a 1 – a 2 † )(a 1 † – a 2 ), J − = (1/2)(a † 1 + a 2 ) (a 1 + a 2 † ), and J 0 = (1/2)(a 1 † a 2 † – a 1 a 2 ), is simultaneously a two-mode squeezed state. We analyse this squeezed state's physical properites, such as the cross-correlation function, the Wigner function, and its marginal distribution as well as the Husimi function.