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International Journal of Modern Physics B

COHERENT STATES FOR THE FINITE su(2)-OSCILLATOR MODEL

Luis Edgar Vicent

Published May 20, 2006• 2 citations
Publisher

Abstract

A finite oscillator is a system that has a discrete and finite number of spatial positions and energy levels accessible. The su(2)-oscillator model takes the generators of the angular momentum su(2) algebra and, giving to each one a definite physical interpretation, provides a consistent mathematical description of a finite oscillator. This work searches an example of identification of generalized coherent states, within the collection of states and wavefunctions of the finite su(2) oscillator model.

DOI
10.1142/s0217979206034418

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A finite oscillator is a system that has a discrete and finite number of spatial positions and energy levels accessible. The su(2)-oscillator model takes the generators of the angular momentum su(2) algebra and, giving to each one a definite physical interpretation, provides a consistent mathematical description of a finite oscillator. This work searches an example of identification of generalized coherent states, within the collection of states and wavefunctions of the finite su(2) oscillator model.
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The one-dimensional finite oscillator model satisfies that under commutator brackets, all operators close into the (dynamical) Lie algebra su(2); therefore the eigenvalues of position, momentum and energy operators are discrete and the eigenvalue of a compact u(l) central operator fixes the dimension of the su(2) representation. In this work we review the one-dimensional finite oscillator model with u(2) as the dynamical algebra where the wave functions, which involve Kravchuk polynomials, are discrete functions of angular momentum theory.
Based on the coherence property under the Fourier transform, we introduce two types of coherent states into the finite discrete oscillator model of s u (2) algebra, and show that they are stable under the fractional Fourier–Kravchuk transform. Using the set of discrete coherent states of this model as biorthonormal bases, we propose two discrete transforms. The first transform maps f ∈ C N functions to the unit circle, and the second transform maps the same functions to Bargmann space; both transforms have analytic inverses. These transformations establish a connection between Fourier and Bargmann expansions, respectively, with the basis of Kravchuk symmetric functions. They also provide a framework for the study of finite-dimensional systems in a periodic or complex continuous space. We give some examples of applications to qudit systems.
A finite two-dimensional oscillator is built as the direct product of two finite one-dimensional oscillators, using the dynamical Lie algebra su(2)x⊕su(2)y. The position space in this model is a square grid of points. While the ordinary `continuous' two-dimensional quantum oscillator has a symmetry algebra u(2), the symmetry algebra of the finite model is only u(1)x⊕u(1)y, because it lacks rotations in the position (and momentum) plane. We show how to `import' an SO(2) group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid. We thus propose a finite analogue for fractional U(2) Fourier transforms.
Abstract From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called su(2) oscillator model in dimension 2j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the su(2) oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions. 1