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Physical Review A

Berry phase and nonstationarity of a quantum state

J. Y. Zeng, Yian Lei

Published Jun 1, 1995• 10 citations
Publisher

Abstract

The intimate connection between the Berry phase and the nonstationarity of a quantum state is discussed. For a quantum system with time-independent Hamiltonian $H$, there may exist stationary states. It is shown that for a stationary state, the Berry phase always vanishes, and only for a nonstationary state after a cyclic evolution might the Berry phase appear. For time-dependent $H$, there exists no exactly stationary state, hence the Berry phase may appear.

DOI
10.1103/physreva.51.4415

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The intimate connection between the Berry phase and the nonstationarity of a quantum state is discussed. For a quantum system with time-independent Hamiltonian HHH, there may exist stationary states. It is shown that for a stationary state, the Berry phase always vanishes, and only for a nonstationary state after a cyclic evolution might the Berry phase appear. For time-dependent HHH, there exists no exactly stationary state, hence the Berry phase may appear.
Shows that (non-adiabatic) Berry phases in time-independent systems arise for fundamentally different reasons than those for time-dependent systems. For time-independent Hamiltonians the Berry phases are seen to only depend on the chosen cyclic initial state, the Hamiltonian merely providing the appropriate period. The author also discusses two time-dependent examples where the time dependence arises from different sources, namely a two-level atom in an intense laser and a harmonic oscillator with a periodic forcing term.
Berry pointed out that the Aharonov-Bohm phase could be considered as a Berry phase.In his argument,he gave up the adiabatic approximation,and the nonstationarity of quantum states was not concerned.In this paper we re-investigate Berry’s proof,and find that his final statement is correct,and in his argument,the adiabatic approximation should be used.In the AB effect,a moving electron is described as a moving wave packet (nonstationary),and the appearance of AB phase originates from the requirement that the evolution of the wave packet must satisfy the Schrdinger equation.
The evolution of a quantum mechanical system under a nonadiabatic external perturbation at a time interval (0,T) is considered. It is shown that all the cyclic states of the system, mod Psi (T))=e i phi mod Psi (0)), are determined by the eigenvalues in (quasi-energies) and associated eigenvectors mod phi in (t)) of the Schrodinger operator S q (t)=H(t)-i( delta / delta t) acting on some Hilbert space. The set of linearly independent cyclic (quasi-energy) states possess some properties similar to the properties of the stationary states of a closed system. The Berry phases of the states associated with eigenvectors of the discrete spectrum of S q (t), which are single-valued functions of omega =2 pi /T, are supplied by the partial derivatives of the corresponding eigenvalues (quasi-energies) with respect to omega . The approach developed is illustrated by several applications to time-dependent systems: the system under an adiabatic perturbation, the forced harmonic oscillator, and the two-level system. Even in the case of a system with a time-independent Hamiltonian there exist nonstationary quasi-energy states that have nontrivial Berry phases ( beta /2 pi not an integer). The criterion of existence of such states is formulated in terms of the energies of the system, and the corresponding expression for the Berry phases is obtained. Some examples of nonstationary cyclic states of closed systems, including the coherent states of a force-free oscillator and the Wannier states of electrons in the parabolic band, are considered.