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Journal of Physics B Atomic Molecular and Optical Physics

Generalized Husimi operator as the rotated and squeezed coherent state projector and its application

Ye‐Jun Xu, Wang Xian-cai, Gui Chuan-you, Zheng-fang Zhu

Published Jan 29, 2013• 0 citations
Publisher

Abstract

We propose a so-called generalized Husimi operator ?gh(x, p, ?, ?) corresponding to the generalized Husimi function and prove ?gh(x, p, ?, ?) = |x, p, ?, ???x, p, ?, ?|, i.e. ?gh(x, p, ?, ?) is actually a pure state projector. Further, |x, p, ?, ?? turns out to be a rotated and squeezed coherent state . As an application, we propose a measured scheme on the generalized Husimi function in the context of a single trapped ion.

DOI
10.1088/0953-4075/46/4/045501

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We propose a so-called generalized Husimi operator ?gh(x, p, ?, ?) corresponding to the generalized Husimi function and prove ?gh(x, p, ?, ?) = |x, p, ?, ???x, p, ?, ?|, i.e. ?gh(x, p, ?, ?) is actually a pure state projector. Further, |x, p, ?, ?? turns out to be a rotated and squeezed coherent state . As an application, we propose a measured scheme on the generalized Husimi function in the context of a single trapped ion.
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.
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.
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.