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Journal of Sichuan Normal University

The Pancharatnam Phase of Generalized Time-dependent Oscillator

Hou Bang-pin

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

By using the time evolution operator for the generalized time-dependent oscillator, the quantum phase of Pancharatnam type and its dynamical and geometric part are obtained.

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Physical Review A•Feb 1, 1997•9 citations
The Pancharatnam phase for displaced number states of the harmonic oscillator is discussed. In particular, it is examined how a single quantum oscillator, driven by a suitable transient external force, evolves form the initial eigenstate u m (x) to the final, displaced number state modified with a suitable phase factor. The significance of this, usually neglected, phase factor for the solution of the relevant time-dependent Schrodinger equation and for the geometric phase accumulated in the wavefunction during the time evolution of the system is examined. The general expression for the geometric phase for a noncyclic evolution from an initial displaced number state at time t 1 to the final state at time t 2 is derived and two applications, that of delta and harmonic forcing, are worked out. The special case of cyclic evolution is subsequently discussed and, in particular, the conditions leading to such an evolution are derived. The relationship to the classical notion of cyclic evolution is also examined in some detail and it is demonstrated that the general expression for the Pancharatnam phase reduces to the corresponding Berry phase.
The Pancharatnam phase for ordinary and generalized squeezed states of the harmonic oscillator is discussed. In particular, the general expressions for the total and dynamical phases are obtained yielding, from their difference, the geometric phase for a nonadiabatic and noncyclic evolution from an initial generalized squeezed state at time t∣IH{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}t∣IH to the final state at a time t\ensuremath{''}. The special case of a two-photon interaction process, such as parametric amplification, is subsequently discussed. In particular, the conditions leading to cyclic evolution are examined and the corresponding Aharonov-Anandan phase is found from the general expression for the Pancharatnam phase.