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Communications in Theoretical Physics

Aharonov-Anandan Phase and Superadiabatic Approximaton

Jing‐Bo Xu, Dong-Hui Xu, Xiao-Chun Gao

Published Jun 1, 1993• 0 citations
Publisher

Abstract

In this paper, the exact evolving state for a two-level quantum system is found by making use of the Lewis-Riesenfeld invariant theory. In principle, the correction to an arbitrary order in the adiabatic approximation parameter can be obtained from this exact evolving state. To the -th order in , it is shown that the exact evolving state reduces to the superadiabatic basis introduced by Berry recently. The Aharonov-Anandan phase and its adiabatic limit (Berry phase for the system) are also calculated.

DOI
10.1088/0253-6102/19/4/391

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In this paper, the exact evolving state for a two-level quantum system is found by making use of the Lewis-Riesenfeld invariant theory. In principle, the correction to an arbitrary order in the adiabatic approximation parameter can be obtained from this exact evolving state. To the -th order in , it is shown that the exact evolving state reduces to the superadiabatic basis introduced by Berry recently. The Aharonov-Anandan phase and its adiabatic limit (Berry phase for the system) are also calculated.
As a non-adiabatic generalization of a recent work of Gerry, the manifestation of the Aharonov-Anandan phase in the degenerate parametric amplifier of quantum optics is discussed by making use of the Lewis-Riesenfeld quantum theory. We find the Lewis invariant for the system and study its evolution. The Aharonov-Anandan phase is then calculated and the connection between the Aharonov-Anandan phase difference and the average photon-number of the output field is obtained. Finally, the observable effect of the phase difference is investigated.
In this paper, the exact solution and time-evolution operator for the system with a su(1,1)?s{\mathrm{?}}_{\mathit{s}}?s​h(4) Hamiltonian are found by making use of Lewis-Riesenfeld quantum theory. The time-evolution operator is then used to derive the Aharonov-Anandan phase. Finally, the noncyclic evolution is discussed with the help of the Aharonov-Anandan phase and the Berry phase is calculated by taking the adiabatic limit.
We present the time evolution of the electronic two-level model in rotating wave approximation and calculate the Aharonov–Anandan phase of the vector states with cyclic evolution either within the period of an external monochromatic electric field or the period corresponding to the generalized Rabi frequency. The Aharonov–Anandan phase in the most general case is shown to be dependent on the initial vector state, unless the system evolves in the adiabatic regime; in the latter case, the Aharonov–Anandan phase recovers Berry's phase. In the quasi-resonant regime of the electronic two-level model, the different relations between Γ and the frequency ω of the electric field correspond to distinct regimes of the coupling between this external field and matter.
We introduce a perturbative approach to solving the time-dependent Schr\"odinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-1∕21∕21∕2 problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.