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Physical Review A

Exact integrability of the two-level system: Berry’s phase and nonadiabatic corrections

Nilanjana Datta, Gautam Kumar Ghosh, M.H. Engineer

Published Jul 1, 1989• 38 citations
Publisher

Abstract

The general time-dependent quantum two-level system is shown to admit an exact invariant. This leads to a classical formulation of the problem, which is then solved by standard techniques of Hamiltonian mechanics. Berry's phase and nonadiabatic corrections to it emerge as an asymptotic limit of the exact dynamics. A novel connection between Berry's phase and Hannay's angle is shown to exist in this case.

DOI
10.1103/physreva.40.526

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The general time-dependent quantum two-level system is shown to admit an exact invariant. This leads to a classical formulation of the problem, which is then solved by standard techniques of Hamiltonian mechanics. Berry's phase and nonadiabatic corrections to it emerge as an asymptotic limit of the exact dynamics. A novel connection between Berry's phase and Hannay's angle is shown to exist in this case.
In a recent paper [Y. C. Ge and M. S. Child, Phys. Rev. Lett. 78, 2507 (1997)], by using a Gaussian wave function, Ge and Child presented a nonadiabatic relation between the quantum Berry phase and the classical Hannay angle for the time-dependent harmonic oscillator. In this paper, we present a perspective for this relation without the use of a trial wave function. In particular, an exact explicit formula for the cyclic evolution over the period T in the parameter space of action invariant is obtained; the $\ensuremath{-}(n+1/2)$ relation between the quantum geometric angle and the Hannay angle is rigorously established.
The Lewis' invariant and exact solution for the driven generalized time-dependent harmonic oscillator is found by making use of the Lewis-Riesenfeld quantum theory. Then, the adiabatic asymptotic limit of the exact solution is discussed and the Berry's phase for this system is obtained. We then proceed to use the exact solution to construct the coherent state and calculate the corresponding exact classical phase angle. This phase angle can give the Hannay's angle in the adjabatic limit. The relation between the exact Lewis' phase and the corresponding classical phase angle is discussed.
Exact Schr\"odinger wave functions of a time-dependent harmonic oscillator are found in analytically closed forms for the eigenstates of the generalized invariant and the instantaneous Hamiltonian. The cyclic initial state (CIS) and corresponding nonadiabatic Berry phase are also found exactly for a $\ensuremath{\tau}$-periodic Hamiltonian. There may exist $N\ensuremath{\tau}$-periodic CISs and corresponding Berry phases, but the cases with unstable classical motions do not have CISs in which cases Berry phases do not exist.
A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.