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Journal of Physics A Mathematical and General

Nonlinear shifts of q-Bose operators and q-coherent states

Alexei S. Zhedanov

Published Oct 7, 1991• 17 citations
Publisher

Abstract

The analogue of shift for q-Bose operators is found. In contrast with the case of ordinary Bose operators this shift is a nonlinear transformation: unitary operators of shift do not form a group. Two dual systems of q-coherent states associated with these shift operators are obtained. The matrix elements of the shift operator in canonical basis are expressed via the Charlier q-polynomials.

DOI
10.1088/0305-4470/24/19/001

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The analogue of shift for q-Bose operators is found. In contrast with the case of ordinary Bose operators this shift is a nonlinear transformation: unitary operators of shift do not form a group. Two dual systems of q-coherent states associated with these shift operators are obtained. The matrix elements of the shift operator in canonical basis are expressed via the Charlier q-polynomials.
In this paper, we considered a q -superoscillator model and constructed a transformation matrix which transforms the usual algebra of q -boson and q -fermion operators of the q -superoscillator into a nonlinear shift automorphism algebra. The q -supercoherent states for the q -superoscillator model is introduced by utilizing the transformation matrix. They are compared with the coherent states defined as eigenstates of the annihilation operators. Matrix elements of the transformation matrix on the Fock space bases are evaluated as well.
We start from the viewpoint of quantum canonical transformations, analogues of the usual classical canonical transformations. We show that in the quantum case, such transformations lead naturally to the definition of some important quantum optics states, including the vacuum, coherent states, squeezed vacuum and squeezed states. We then go on to consider quons, operators satisfying deformed versions of the canonical commutation relations. Extending the idea of a canonical transformation to deformed quantum commutators gives quon analogues of these states, quon states. We finally show that this approach provides us with defining relations for some quantum groups.
The single mode q-Bose field for -1<or=q<1 is shown to be a bounded operator with a continuous spectrum lying in the interval between +or-(2/(1-q)) 1 2/, thus becoming unbounded in the Bose limit (q to 1). The generalized eigenstates are determined in terms of q-Hermite polynomials whose properties are exploited to obtain orthonormality relations. It is observed that these are connected to important results in combinatorics. It is shown that the eigenstates are products of two q-exponentials of the q-creation operator acting on the Fock vacuum, in contrast to ordinary coherent states which involve only a single q-exponential of the creation operator. A method of representing operators in normal-ordered form based on this representation is described. The spectral resolution is employed to compute the asymptotic temporal behaviour of the particle number expectation value in states produced by coupling the q-Bose field to a source. It is shown that the limit does not commute with the limit q to 1. Specifically it is shown that for O<or=q<1 the asymptotic growth is linear in time in contrast to the quadratic growth for q=1.
We obtain the time-dependent correlation function describing the evolution of a single spin excitation state in a linear spin chain with isotropic nearest-neighbour XY coupling, where the Hamiltonian is related to the Jacobi matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case, an arbitrary element of the correlation function is expressed in a simple closed form. Its asymptotic limit corresponds to the Jacobi matrix of the Charlier polynomial, and may be understood as a unitary evolution resulting from a Heisenberg group element. Correlation functions for Hamiltonians corresponding to Jacobi matrices for the Hahn, dual Hahn and Racah polynomials are also studied. For the Hahn polynomials we obtain the general correlation function, some of its special cases and the limit related to the Meixner polynomials, where the su (1, 1) algebra describes the underlying symmetry. For the cases of dual Hahn and Racah polynomials, the general expressions of the correlation functions contain summations which are not of hypergeometric type. Simplifications, however, occur in special cases.