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arXiv:1202.3410 · Journal of Physics A Mathematical and Theoretical

Generalized squeezed–coherent states of the finite one-dimensional oscillator and matrix multi-orthogonality

Vincent X. Genest, Luc Vinet, Alexei S. Zhedanov

Published May 4, 2012• 3 citations

Abstract

A set of generalized squeezed–coherent states for the finite oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the generators. These matrix elements are given in the ( N + 1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3 × 3 matrix multi-orthogonal polynomials Q n ( k ) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed–coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2 × 2 matrix orthogonal polynomials and the squeezed–coherent states tend to those of the standard oscillator.

DOI: 10.1088/1751-8113/45/20/205207

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A set of generalized squeezed–coherent states for the finite oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the generators. These matrix elements are given in the ( N + 1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3 × 3 matrix multi-orthogonal polynomials Q n ( k ) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed–coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2 × 2 matrix orthogonal polynomials and the squeezed–coherent states tend to those of the standard oscillator.

The Pegg-Barnett phase-operator formalism utilizes a finite basis set to represent operators of the harmonic oscillator; this enables the phase to be represented by a Hermitian operator, but rests on taking the dimensionality of the basis set to infinity for observable quantities. Simultaneously, in their approach Pegg and Barnett consider quantum states of a harmonic oscillator which are normalized in the Fock space, i.e., the dimensionality of the basis set in which the states of the harmonic oscillator are defined is supposed to be infinite, while the phase operator is defined in the finite-dimensional basis set. In this paper we address the problem of a consistent definition of a coherent state within a finite state basis. We employ displacement operators to define such coherent states and numerically evaluate observables as a function of the size of the basis set. We investigate phase properties of these coherent states. We find that if the dimensionality of the state space is much larger than the mean occupation number of the coherent states, then the results obtained in the finite-dimensional basis are applicable in the case of a ordinary quantum-mechanical harmonic oscillator. These coherent states are minimum uncertainty states with respect to quadrature operators (i.e., the position and momentum operators) and do not exhibit quadrature squeezing. A weakly excited (compared with the dimensionality of the state space) coherent state in finite-dimensional basis is not strictly speaking a minimum uncertainty state with respect to the number and phase operators. We give definitions of amplitude and phase squeezing and show that weakly excited coherent states can be amplitude squeezed. In the high-intensity limit (again compared with the dimensionality of the state space) these states exhibit phase squeezing.

We explore squeezed coherent states of a three-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered X 1 -Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three-mode squeezed coherent states resolve the identity operator. We evaluate Mandel’s Q -parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Furthermore, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate the Wigner function of these three-mode squeezed coherent states and demonstrate the squeezing property explicitly.

We extend our approach to constmction of coherent states for oscillator-like systems connected with given system of orthogonal polynomials to the case of the classical orthogonal polynomials of a discrete variable. In this paper we consider oscillntmor-liksey stem associated with the Meixnct polynomials and compare associated Hamiltonian with the Hamiltonian of the relativistic linear oscillator obtained by the other point of view. We also construct generalized coherent states both Rarut-Girardello and Perelomov types €or this system.