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Journal of Mathematical Physics

A consistent bridge for the construction of generalized nonclassical states for confined systems through the inclusion of inverse generators in the algebraic formalism

A N F Aleixo, A. B. Balantekin

Published Jan 1, 2019• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.5037792

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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 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.
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.
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 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.
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.
Among the three methods of defining the squeezed states of the harmonic oscillator, only two, i.e. the minimum uncertainty and ladder-operator methods were extended to investigate the squeezed states of the general symmetry systems. The only exception is the displacement-operator formalism. In this letter we propose a displacement-operator approach to the squeezed states for general systems. We show that the ladder-operator squeezed states for general symmetry systems can be obtained by applying an exponential displacement-operator on the ``vacuum'' state. The general formalism is presented and it is illustrated with su(1,1) as an explicit example. As a byproduct, the squeezed states of the q-deformed oscillator are obtained and, in particular, expressed in terms of the {\\it usual exponential} displacement-operator acting the ``vacuum'' state.