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Journal of Physics A Mathematical and Theoretical

New formulation of nonlinear vector coherent states off-deformed spin–orbit Hamiltonians

Joseph Ben Geloun, Mahouton Norbert Hounkonnou

Published Aug 1, 2007• 14 citations
Publisher

Abstract

We provide, with a new formulation of vector, coherent states for nonlinear spin–orbit Hamiltonian models in terms of the matrix eigenvalue problem for generalized annihilation operators. Nonlinear quaternion vector coherent states are also discussed.
DOI
10.1088/1751-8113/40/33/f01

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Through canonical and f deformation quantizations, classes of nonlinear spin-orbit interaction models are considered. These generalized models are relevant in condensed matter physics and quantum optics in domains of nonlinear spin-Hall effect or of the multiphoton Jaynes-Cummings [Proc. IEEE 51, 89 (1963)] model, for instance. A spectral decomposition follows the exact diagonalization of the Hamiltonian and leads to the definition of ladder operators acting on the associated Hilbert space. Nonlinear vector coherent states are then constructed from a general lowering operator action. Explicit solutions of their related moment problems in ordinary and f-deformed theories are displayed, thus providing new classes of coherent states for such generalized spin-orbit models.
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In this work, we construct nonlinear coherent states for Hamiltonians with linear and quadratic terms in the number operator by the generalization of two definitions: as eigenstates of a deformed annihilation operator and as those states obtained by the application of a deformed displacement operator on the vacuum state. We evaluate their temporal dependence, analyze its dispersion relations and some of their statistical properties for two model Hamiltonians, one supporting a finite number of bound states and the other supporting an infinite number of bound states.
In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.
As a substantial generalization of the technique for constructing canonical and the related nonlinear and q -deformed coherent states, we present here a method for constructing vector coherent states (VCS) in the same spirit. These VCS may have a finite or an infinite number of components. The resulting formalism, which involves an assumption on the existence of a resolution of the identity, is broad enough to include all the definitions of coherent states existing in the current literature, subject to this restriction. As examples, we first apply the technique to construct VCS using the Plancherel isometry for groups and VCS associated with Clifford algebras, in particular quaternions. As physical examples, we discuss VCS for a quantum optical model and finally apply the general technique to build VCS over certain matrix domains.
A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable zzz. In the present scheme, the variable zzz is replaced by a matrix valued function over appropriate domains. As particular examples, we analyze the quaternionic extensions of the the canonical coherent states and the Gilmore-Perelomov and Barut-Girardello coherent states arising from representations of SU(1,1).