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arXiv:quant-ph/9905101 · Physical Review A

Geometric phases for generalized squeezed coherent states

S. Seshadri, S. Lakshmibala, V. Balakrishnan

Published Feb 1, 1997• 21 citations
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Abstract

A simple technique is used to obtain a general formula for the Berry phase (and the corresponding Hannay angle) for an arbitrary Hamiltonian with an equally spaced spectrum and appropriate ladder operators connecting the eigenstates. The formalism is first applied to a general deformation of the oscillator involving both squeezing and displacement. Earlier results are shown to emerge as special cases. The analysis is then extended to multiphoton squeezed coherent states and the corresponding anholonomies deduced.

DOI
10.1103/physreva.55.869

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A simple technique is used to obtain a general formula for the Berry phase (and the corresponding Hannay angle) for an arbitrary Hamiltonian with an equally spaced spectrum and appropriate ladder operators connecting the eigenstates. The formalism is first applied to a general deformation of the oscillator involving both squeezing and displacement. Earlier results are shown to emerge as special cases. The analysis is then extended to multiphoton squeezed coherent states and the corresponding anholonomies deduced.
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of squeezed state. It is shown that the cyclic and quasicyclic squeezed states correspond to the periodic and quasiperiodic solutions of an effective Hamiltonian defined on an extended phase space, respectively. The geometric phase of the cyclic squeezed state is found to be a phase-space area swept out by a periodic orbit. Furthermore, a class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Their geometric phases are found to be independent of $\ensuremath{\Elzxh},$ and equal to $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle.
Motivated by some recent remarks on the ``removability'' of Berry's phase [Ph. De Sousa Gerbert, Ann. Phys. (N.Y.) 189, 155 (1989)] or its classical counterpart, the Hannay angle, we present here a fresh analysis of Berry's-phase calculations for some coherent states of general theoretical interest. Our reasoning, though semiclassical, lends itself easily for comparison with the exact results. Unlike spin-coherent states, both the (ordinary) harmonic-oscillator and squeezed coherent states are contained in Hamiltonians for which Berry's phase can be removed globally by a canonical transformation.
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.
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.
The geometric phases of the cyclic states of a generalized harmonic oscillator with nonadiabatic time-periodic parameters are discussed in the framework of a squeezed state. A class of cyclic states is expressed as a superposition of an infinite number of squeezed states. Then their geometric phases are obtained explicitly and found to be $\ensuremath{-}(n+1/2)$ times the classical nonadiabatic Hannay angle. It is shown that the analysis based on the squeezed state approach provides a clear picture of the geometric meaning of the quantal phase.
A group-theoretic treatment is given of the new phase found by Berry in the adiabatic evolution of a quantum-mechanical system in a finite-dimensional Hilbert space. It is shown how the Berry phases for the various eigenstates of the Hamiltonian are obtained from a set of angles associated with a group element. For the special case of a two-level system there is just one such angle which corresponds to the holonomy transformation associated with parallel transport around a closed curve on a sphere.