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Physical Review A

Pancharatnam phase for ordinary and generalized squeezed states

Istok P. Mendaš

Published Feb 1, 1997• 9 citations
Publisher

Abstract

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 ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ 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.

DOI
10.1103/physreva.55.1514

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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.
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 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.
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.
We derive a formula of the nonadiabatic noncyclic Pancharatnam phase for a mesoscopic circuit with coupled inductors and capacitors. It shows that, because of coupling effect, the circuit is in squeezed quantum state initially, and the time evolution of Pancharatnam phase exhibits an oscillation in a complex way. Especially we find that when the capacity of the coupled capacitors is larger than that of other ones in the circuit, with the variation of time Pancharatnam phase becomes nearly periodic square-wave, which perhaps can provide a new approach for the realization of quantum logic gate.
The generalized time-dependent harmonic oscillator is studied. Though several approaches to the solution of this model have been available, yet a new approach is presented here, which is very suitable for the study of cyclic solutions and geometric phases. In this approach, finding the time evolution operator for the Schrödinger equation is reduced to solving an ordinary differential equation for a c-number vector which moves on a hyperboloid in a three-dimensional space. Cyclic solutions do not exist for all time intervals. A necessary and sufficient condition for the existence of cyclic solutions is given. There may exist some particular time interval in which all solutions with definite parity, or even all solutions are cyclic. Criteria for the appearance of such cases are given. The known relation that the nonadiabatic geometric phase for a cyclic solution is proportional to the classical Hannay angle is reestablished. However, this is valid only for special cyclic solutions. For more general ones, the nonadiabatic geometric phase may contain an extra term. Several cases with relatively simple Hamiltonians are solved and discussed in detail. Cyclic solutions exist in most cases. The pattern of the motion, say, finite or infinite, cannot be simply determined by the nature of the Hamiltonian (elliptic or hyperbolic, etc.). For a Hamiltonian with a definite nature, the motion can change from one pattern to another, that is, some kind of phase transition may occur, if some parameter in the Hamiltonian goes through some critical value.