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Communications in Theoretical Physics

Quantum Invariant Theory and Non-Adiabatic Non-Abelian Phase Factor

Jian Bo Fu, Xin‐Hua Li, Gao Xiaochun, Jun Gao

Published Jan 30, 1999• 0 citations
Publisher

Abstract

An alternative formulation is proposed for the calculation of the non-adiabatic non-Abelian connection and phase factor by making use of the invariant theory and invariant-related unitary transformation method. In order to illustrate the formulation, we study the non-adiabatic non-Abelian phase associated with the motion of a charged particle in a time-dependent magnetic field (the time-dependent Landau level problem).

DOI
10.1088/0253-6102/31/1/135

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An alternative formulation is proposed for the calculation of the non-adiabatic non-Abelian connection and phase factor by making use of the invariant theory and invariant-related unitary transformation method. In order to illustrate the formulation, we study the non-adiabatic non-Abelian phase associated with the motion of a charged particle in a time-dependent magnetic field (the time-dependent Landau level problem).
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
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 obtained by other authors, and futhermore we obtain the more general results concerning the time-dependent non-adiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor.
The quantum adiabatic theorem incorporating the Berry phase phenomenon can be characterized as a factorization of the time evolution operator into a path-dependent geometric factor, a usual dynamical factor and a non-adiabatic factor that approaches the identity in the adiabatic limit. We study a case where all these three factors can be constructed explicitly and where the instantaneous Hamiltonian has infinitely degenerate energy eigenstates associated with magnetic translation symmetry. Significance of the non-adiabatic factor in terms of transition probabilities is discussed.
We use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the Abelian and non-Abelian geometric phases. This derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the Abelian noncyclic geometric phase. We also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non-Abelian noncyclic geometric phase for a spin-1 magnetic (or electric) quadrupole interacting with a precessing magnetic (electric) field.