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Chinese Physics

Wigner function of coherent state ofNcomponents

YE Yong-hua, Gao-Jian Zeng

Published Jun 1, 2007• 3 citations
Publisher

Abstract

In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms: the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius |β| = | α | with a separate angle of 2π/N , and the second term is composed of ½ N ( N– 1) Gaussian-cosine surfaces evenly centred in a circular region of radius |β| < | α | . Here, α is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.

DOI
10.1088/1009-1963/16/6/010

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Journal of Southwest Jiaotong University•Jan 1, 2009•0 citations
In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms: the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius |β| = | α | with a separate angle of 2π/N , and the second term is composed of ½ N ( N– 1) Gaussian-cosine surfaces evenly centred in a circular region of radius |β| < | α | . Here, α is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.
A measure of non-classicality of even and odd coherent states is studied. We first calculate the Wigner functions of the even and odd coherent states, which consists of two terms: the positive-definite Gaussian term and the wave term with negativity, and then calculate the integrated value ∊ max of the wave term of the Wigner functions of the even and odd coherent states in their area with negativity, and use ∊ max to measure non-classicality of the even and odd coherent states. For the even and odd coherent states with certain photon count, it is very convenient for us to use ∊ max to measure their non-classicality. The methods of our definition and calculation for ∊ max have theoretical reference value.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
The Wigner quasiprobability, along with some of its essentialproperties, is introduced and discussed in two versions, first covering real canonical variables such as W(q,p) and second a pair of complex conjugate coordinates such as W(α,α*). The reconstruction of the density operator ϱ of states is also given. Building upon the Susskind–Glogower concept of quantum phase operators, further aspects of phase operator algebras in the quantum optics of a harmonic oscillator are discussed in relation to the realization of the su(1,1) Lie algebra. Coherent phase states |ε⟩ are introduced in analogy to the common coherent states |α⟩ in two ways, as both eigenstates of certain operators and as states generated from a ground state |0⟩ by operators of the Lie group SU(1,1). The limiting transition to the non-normalizable Fritz London phase states |eiφ⟩ on the unit circle and an (over)-completeness relation for the coherent phase states are derived. The Wigner quasiprobability W(q,p) for the coherent phase states is calculated and graphically represented. From the Wigner quasiprobability, a phase distribution W(φ) is calculated by integrating over the radius, and its uncertainty is defined and presented. The Hilbert–Schmidt distance is discussed as a measure of the non-classicality of states, where most of our with Viktor Dodonov work was carried out.
We consider a set of operators = ( 1 ,..., N ) with diagonal representatives P ( n ) in the space of generalized coherent states | n ⟩: = ∫dµ( n ) P ( n )| n ⟩⟨ n |. We regularize the coherent-state path integral as a limit of a sequence of averages ⟨ ⟩ L over polygonal paths with L vertices n 1,..., L . The distribution of the path centroid = (1/ L )∑ l = 1 L P ( n l ) tends to the Wigner function W ( x ), the joint distribution for the operators: W ( x ) = lim L →∞ ⟨δ N ( x - )⟩ L . This result is proved in the case where the Hamiltonian commutes with . The Wigner function is non-positive if the dominant paths with path centroid in a certain region have Berry phases close to odd multiples of π. For finite L the path centroid distribution is a Wigner function convolved with a Gaussian of variance inversely proportional to L . The results are illustrated by numerical calculations of the spin Wigner function from SU (2) coherent states. The relevance to the quantum Monte Carlo sign problem is also discussed.
Wigner functions for the eigenstates of arbitrary power of annihilation operators were reconstructed using their expressions in Fock presentations.The distribution of the reconstructed Wigner functions in phase spaces was analyzed,based on which the nonclassical properties of these eigenstates were discussed.The results show that,the distribution of the Wigner functions for these eigenstates depend on the eigenvalues of the annihilation operators;the eignestates of the annihilation operators of the first power(i.e.,coherent states) are quasiclassical(their Wigner functions are always non-negative),and those of higher powers obviously exhibit nonclassical properties(their Wigner functions may become negative).