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Journal of Physics A Mathematical and General

The Aharonov-Anandan phase for quasi-energy trajectory-coherent states

Andrey Yu. Trifonov, A A Yevseyevich

Published Oct 7, 1995• 6 citations
Publisher

Abstract

Quasi-energy spectral series ( epsilon nu (h(cross)), Psi ( epsilon nu )) which, in the limit h(cross) to 0, correspond to stable motions of a classical system along closed phase trajectories are built up in terms of a quasi-classical approximation for the Schrodinger equation with an arbitrary T-periodic h(cross) -1 (pseudo)differential Hamilton operator. Using the procedure of splitting the quantum-mechanical phase into dynamic and geometric components, the "geometric" contribution of the Aharonov-Anandan phase gamma epsilon ( nu ) to the quasi-energy spectrum is calculated. It is shown that the gamma epsilon ( nu ) phase, in the adiabatic approximation, coincides with the Berry phase that corresponds to a cyclic evolution of a stable rest-point of a classical system. Some examples are considered.

DOI
10.1088/0305-4470/28/19/019

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Quasi-energy spectral series ( epsilon nu (h(cross)), Psi ( epsilon nu )) which, in the limit h(cross) to 0, correspond to stable motions of a classical system along closed phase trajectories are built up in terms of a quasi-classical approximation for the Schrodinger equation with an arbitrary T-periodic h(cross) -1 (pseudo)differential Hamilton operator. Using the procedure of splitting the quantum-mechanical phase into dynamic and geometric components, the "geometric" contribution of the Aharonov-Anandan phase gamma epsilon ( nu ) to the quasi-energy spectrum is calculated. It is shown that the gamma epsilon ( nu ) phase, in the adiabatic approximation, coincides with the Berry phase that corresponds to a cyclic evolution of a stable rest-point of a classical system. Some examples are considered.
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.
We show that cyclic quantum evolution can be realized and the Aharonov-Anandan (AA) geometric phase can be determined for any spin-j system driven by periodic fields. Two methods are extended for the study of this problem: the generalized spin-coherent-state technique and the Floquet quasienergy approach. Using the former approach, we have developed a generalized Bloch-sphere model and presented a SU(2) Lie-group formulation of the AA geometric phase in the spin-coherent state. We show that the AA phase is equal to j times the solid angle enclosed by the trajectory traced out by the tip of a generalized Bloch vector. General analytic formulas are obtained for the Bloch vector trajectory and the AA geometric phase in terms of external physical parameters. In addition to these findings, we have also approached the same problem from an alternative but complementary point of view without recourse to the concept of coherent-state terminology. Here we first determine the Floquet quasienergy eigenvalues and eigenvectors for the spin-j system driven by periodic fields. This in turn allows the construction of the time-evolution propagator, the total wave function, and the AA geometric phase in a more general fashion.
If the time evolution of a quantum state leads back to the initial state, a geometric phase is accumulated that is known as the Berry phase for adiabatic evolution or as the Aharonov-Anandan (AA) phase for nonadiabatic evolution. We evaluate these geometric phases using Floquet theory for systems in time-dependent external fields with a focus on paths leading through a degeneracy of the eigenenergies. Contrary to expectations, the low-frequency limits of the two phases do not always coincide. This happens as the degeneracy leads to a slow convergence of the quantum states to adiabaticity, resulting in a nonzero finite or divergent contribution to the AA phase. Steering the system adiabatically through a degeneracy provides control over the geometric phase as it can cause a π shift of the Berry phase. On the other hand, we revisit an example of degeneracy crossing proposed by AA. We find that, at suitable driving frequencies, both geometric-phase definitions give the same result and the dynamical phase is zero due to the symmetry of time evolution about the point of degeneracy, providing an advantageous setup for manipulation of quantum states.
We construct an operator for the Aharonov-Anandan phase for time independent Hamiltonians. This operator is shown to generate the motion of cyclic quantum systems through an equation of evolution involving only geometric quantities, i.e. the distance between quantum states, the geometric phase and the total length of evolution. From this equation, we derive an operator for the Samuel and Bhandari phase (SB-phase) for non cyclic evolutions. Finally we show how the SB-phase can be used to construct an operator corresponding to a quantum clock which commutator with the Hamiltonian has a canonical expectation value.