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Journal of Mathematical Physics

The close connection between the definition and construction of raised and decreased coherent states with the inverse generators in the algebraic description of quantum confined systems

A N F Aleixo, A. B. Balantekin

Published May 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.5037599

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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.
We consider quantum confined systems represented by supersymmetric quantum mechanics where the Hamiltonians are products of two operators. We introduce inverses of these operators as well as pair operators, which are products of the original operators and their inverses. We show that the eigenstates of these pair operators are purely squeezed states, and using them we construct raised and decreased extensions of the squeezed states. We apply this procedure both to the ladder-operator and to the displacement-operator methods, showing the equivalence between those the two approaches. Finally, we show how the raised and decreased squeezed states can be dynamically produced by coupling the Hamiltonian with a two-level atom.
Using an algebraic approach, we construct purely coherent and squeezed states for a set of exactly solvable quantum confined systems assuming a generalization of both the ladder-operator and the displacement-operator methods, showing the equivalence between the two procedures. We obtain the analogue of the displacement and the squeeze operators and study their basic relations and their connections with the properties of the purely coherent and squeezed states that they produce. In a second approach, we construct generalized squeezed coherent and intelligent states for these systems considering the successive action of the displacement and the squeeze operators (in the first case) and the combined action of these operators (in the second case). We apply the generalized approach for shape-invariant systems with potential parameters related by a translation and by a scaling and show that the presence of the generalizing functionals of the potential parameters in the formalism opens a large number of possibilities in the construction of these special quantum states.
We introduce, via their action on the eigenstate basis, the inverse or the one-sided inverse of the algebraic generators for a class of important potentials used to model quantum confined systems in several fields of physics. We obtain the complete algebraic formulation of the systems with the inclusion of the inverse generators. We establish the relations of this complete algebraic approach with the nonlinear quantum deformation approach and the generalization of the phase operators’ definition for these systems. We outline extensions for all the results obtained in this study for the case of the two-parameter quantum deformed systems which preserves the algebraic nature of the undeformed systems.
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.
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.
Using the nonlinear coherent states method , a formalism for the construction of the coherent states associated with ‘ inverse bosonic operators ’ and their dual family has been proposed. Generalizing the approach, the ‘inverse of f -deformed ladder operators’ corresponding to the nonlinear coherent states in the context of quantum optics and the associated coherent states have been introduced. Finally, after applying the proposal to a few known physical systems, particular nonclassical features as sub-Poissonian statistics and the squeezing of the quadratures of the radiation field corresponding to the introduced states have been investigated.