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Journal of Physics A Mathematical and Theoretical

Algebraic construction of coherent states for nonlinear quantum deformed systems

A N F Aleixo, A. B. Balantekin

Published Apr 3, 2012• 7 citations
Publisher

Abstract

A nonlinear deformed algebraic theory for quantum confined systems is constructed using a supersymmetric and shape-invariant approach. In the framework of this model, we present a mathematical Barut–Girardello procedure which leads to nonlinear deformed coherent states by the successive application of a deformed raising operator on the ground state of these systems. As an application, we use a simple exponential form for the nonlinearity function to study in detail the mathematical and physical properties of such states and their evolution in time.

DOI
10.1088/1751-8113/45/16/165302

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A nonlinear deformed algebraic theory for quantum confined systems is constructed using a supersymmetric and shape-invariant approach. In the framework of this model, we present a mathematical Barut–Girardello procedure which leads to nonlinear deformed coherent states by the successive application of a deformed raising operator on the ground state of these systems. As an application, we use a simple exponential form for the nonlinearity function to study in detail the mathematical and physical properties of such states and their evolution in time.
In this paper, within a parasupersymmetric and quantum deformed formalism, we introduce a class of bound-state problems which represents the coupling of a three-level atom with a two-dimensional potential system. We consider second-order parasupersymmetric quantum-mechanical models and a nonlinear deformed algebraic formulation for shape-invariant potential systems to study the quantum dynamics of physical observable, such as atomic level occupation, level transition probabilities, entropy and entanglement, in terms of time and deformation intensity. An application is given for a couple of shape-invariant potentials widely used to model quantum confined systems in several fields of physics, assuming a simple exponential form for the nonlinear deformation function.
Using an expanded algebraic formalism with the inclusion of inverse operators, we construct raised and decreased coherent states for a set of exactly solvable quantum confined systems. We assume in this procedure both the ladder-operator and the displacement-operator methods, showing the equivalence between the two approaches. For each coherent state defined, we present its expansion in the Hilbert eigenstate space Hes, eigenvalue equation, overcompleteness relation, as well as other intrinsic properties. Whenever possible, we present an interpretation based on nonlinear deformation models for these new forms of coherent states. We evaluate the relevance of the new coherent states in quantum entanglement and squeezing by taking, as an example, the case of a coupled system.
Generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show that this generalized formalism is able (a) to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system, (b) to reproduce, as particular cases, results already known for shape-invariant systems (such as standard harmonic oscillator and Pöschl–Teller potentials as well as quantum deformed harmonic oscillator models) and (c) point to a formalism that provides a unified description of the different kind of coherent states for quantum systems, deformed or not deformed.
In a previous paper (2007 J. Phys. A: Math. Theor. 40 11105), we constructed a class of coherent states for a polynomially deformed su (2) algebra. In this paper, we first prepare the discrete representations of the nonlinearly deformed su (1, 1) algebra. Then we extend the previous procedure to construct a discrete class of coherent states for a polynomial su (1, 1) algebra which contains the Barut–Girardello set and the Perelomov set of the SU (1, 1) coherent states as special cases. We also construct coherent states for the cubic algebra related to the conditionally solvable radial oscillator problem.
The generalized purely squeezed states for primary shape-invariant potentials systems, quantum deformed by different models, are constructed by the ladder-operator method within an algebraic approach based on supersymmetric quantum mechanics. The characteristic properties of these states as well as their quantum statistical properties and squeezing effects for generalized quadrature observables are studied and analyzed in terms of the quantum deformation parameter q. An application is given for a quantum deformed Pöschl–Teller potential system, and numerical results are presented and discussed in detail.
We present a general unified approach for finding the coherent states (CSs) of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the noncompact cases, to obtain the annihilation operator eigenstates, by finding the canonical conjugates of these operators. Generalized CSs, in the Perelomov sense, also follow from this construction. This allows us to explicitly construct CSs associated with various quantum optical systems.