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Acta Physica Sinica

Transform relations between squeezed coherent state representation and quantum phase space distribution functions

Liang Xiu-Dong, Tai Yun-Jiao, Cheng Jian-Min, Zhai Long-Hua, Ye‐Jun Xu

Published Jan 1, 2015• 0 citations

Abstract

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.

DOI: 10.7498/aps.64.024207

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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 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.

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.

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

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.