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International Journal of Modern Physics B

THE CONSISTENCY OF THE ADIABATIC AND EXACT GEOMETRIC PHASES

Xinyan Jia, Weidong Li, J.-Q. Liang

Published Mar 30, 2008• 0 citations
Publisher

Abstract

With the help of the time-dependent gauge transformation, by which a set of time-invariant eigenstates of the Hamiltonian can be obtained, the geometric phase of the time-dependent system is obtained analytically. In the limit of the adiabatic condition, this phase exactly reduces to the adiabatic geometric phase. To illustrate the present theory, two solvable time-dependent systems are considered.

DOI
10.1142/s0217979208039095

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With the help of the time-dependent gauge transformation, by which a set of time-invariant eigenstates of the Hamiltonian can be obtained, the geometric phase of the time-dependent system is obtained analytically. In the limit of the adiabatic condition, this phase exactly reduces to the adiabatic geometric phase. To illustrate the present theory, two solvable time-dependent systems are considered.
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The present paper finds the complete set of exact solutions of the general time-dependent dynamical models for quantum decoherence, by making use of the Lewis–Riesenfeld invariant theory and the invariant-related unitary transformation formulation. Based on this, the general explicit expression for the decoherence factor is then obtained and the adiabatic classical limit of an illustrative example is discussed. The result (i.e., the adiabatic classical limit) obtained in this paper is consistent with what is obtained by other authors, and furthermore we obtain more general results concerning time-dependent nonadiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor. PACS Nos.: 03.65.Ge, 03.65.Bz
First we put forward an essential condition, for the existence of the non-zero adiabatic geometric phase of a quantum-mechanical Hamiltonian system, that the Hamiltonian operators at different times do not commute. Then it is shown that constraint =0 determines completely the phase relation of normalized eigenstate vectors │n' (t)> at different times. According to this property, we advance the sufficient and necessary condition for the existence of the non-zero adiabatic geometric phase, which is |n′(T)>≠|n′(0)> when =0. And also we derive a general time-integral formula for the adiabatic geometric phase along the line. Finally, as an application of it, we calculate the geometric phase of a spin 1/2 system once discussed by Solem and Biedenharn. It is pointed out that the problem encountered by them lies in the multi-valuedness of the eigenstate vector in parameter space.