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Modern Physics Letters A

ATOMIC COHERENT STATES AS THE EIGENSTATES OF A TWO-DIMENSIONAL ANISOTROPIC HARMONIC OSCILLATOR IN A UNIFORM MAGNETIC FIELD

Xiang‐Guo Meng, Ji‐Suo Wang, Hong-Yi Fan

Published Dec 14, 2009• 3 citations
Publisher

Abstract

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.

DOI
10.1142/s021773230903120x

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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.
We review how to rely on the quantum entanglement idea of Einstein–Podolsky–Rosen and the developed Dirac's symbolic method to set up two kinds of entangled state representations for describing the motion and states of an electron in uniform magnetic field. The entangled states can be employed for conveniently expressing Landau wave function and Laughlin wave function with a fresh look. We analyze the entanglement involved in electron's coordinates (or momenta) eigenstates, and in the angular momentum-orbit radius entangled state. Various applications of these two representations, such as in developing angular momentum theory, squeezing mechanism, Wigner function and tomography theory for this system are presented. Thus the present review systematically summarizes a distinct approach for tackling this physical system.
We reveal that the two-variable Hermite function h m , n , which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator's wave functions. The Schmidt decomposition of h m , n is derived. It also turns out that h m , n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S ( r ) H m , n (μ a 1 † ,μ a 2 † )|00⟩, which is the minimum uncertainty state for sum squeezing, in ⟨η| representation is calculated.