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Chinese Physics B

Wave function for the squeezed atomic coherent state in entangled state representation and some of its applications

Ji‐Suo Wang, Meng Xiang-Guo, Liang Bao-Long

Published Jan 1, 2010• 14 citations
Publisher

Abstract

Based on the Einstein, Podolsky, and Rosen (EPR) entangled state representation, this paper introduces the wave function for the squeezed atomic coherent state (SACS), which turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2 j. As important applications of the wave function, the Wigner function of the SACS and its marginal distribution are obtained and the eigenproblems of some Hamiltonians for the generalized angular momentum system are solved.

DOI
10.1088/1674-1056/19/1/014207

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Based on the Einstein, Podolsky, and Rosen (EPR) entangled state representation, this paper introduces the wave function for the squeezed atomic coherent state (SACS), which turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2 j. As important applications of the wave function, the Wigner function of the SACS and its marginal distribution are obtained and the eigenproblems of some Hamiltonians for the generalized angular momentum system are solved.
In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.
For a tripartite entangled system, based on the newly constructed tripartite Einstein–Podolsky–Rosen entangled state, we find the entangled state representation of the Wigner operator. The corresponding Wigner function then leads us to obtain the physical marginal distributions. We emphasize that, for an entangled-particle system, the physical meaning of the Wigner distribution function should lie in that its marginal distributions give the probability of finding the particles in an entangled way.
The general theory of quantum angular momentum is used to derive the unique Wigner distribution function for arbitrary angular-momentum states. We give the explicit distribution for atomic angular-momentum Dicke states, coherent states, and squeezed states that correspond to a collection of N two-level atoms. These Wigner functions W(\ensuremath{\theta},cphi) are represented as a pseudoprobability distribution in spherical phase space with spherical coordinates \ensuremath{\theta} and cphi on the surface of a sphere of radius \ensuremath{\Elzxh} \ensuremath{\surd}j(j+1) where j is the total angular-momentum eigenvalue.
We demonstrate the Einstein-Podolsky-Rosen (EPR) correlations in the Gaussian trial wavefunctional widely used in the Schrödinger representation of the self-interacting complex scalar fields (CSF). This demonstration is based on the explicit eigenstates ||ξ⟩, which possess the precise EPR entanglement and constitute a complete and orthonormal representation, of CSF ϕ and ϕ † in the Fock space. To show the entanglement of the Gaussian trial wavefunctional, we evaluate in the ⟨ξ||-representation its `entangled' Wigner functional resembling the Wigner function of the two-mode squeezed vacuum state, whose quantum entanglement is well known.