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Europhysics Letters (EPL)

The Exact Solution and Geometric Phase for the SU (1, 1) System

Jing‐Bo Xu, Tiezheng Qian, Xiao-Chun Gao

Published May 15, 1991• 5 citations
Publisher

Abstract

The exact solution and time evolution operator for the SC7(1, 1) system are obtained by making use of the Lewis-Riesenfeld theory. The time-evolution operator is used to derive the Aharonov-Anandan phase. The noncyclic evolution is then discussed with the help of the Aharonov-Anandan phase. Finally, the Berry phase is calculated by taking the adiabatic limit. © 1991 IOP Publishing Ltd.

DOI
10.1209/0295-5075/15/2/002

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中国科学通报：英文版•Jan 1, 1992•0 citations
The exact solution and time evolution operator for the SC7(1, 1) system are obtained by making use of the Lewis-Riesenfeld theory. The time-evolution operator is used to derive the Aharonov-Anandan phase. The noncyclic evolution is then discussed with the help of the Aharonov-Anandan phase. Finally, the Berry phase is calculated by taking the adiabatic limit. © 1991 IOP Publishing Ltd.
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.
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.
We study the dynamics of a quantum system with a time-dependent Hamiltonian which is given by a linear combination of SU(1,1) generators and coupled with a heat bath. An effective Hamiltonian of this dissipative system is derived. With the help of a time-dependent gauge transformation, we obtain the exact solutions of the time-dependent Schrödinger equation, from which the time-evolution operator and the non-adiabatic and adiabatic Berry phases, which depend on time, are calculated explicitly. In the weak dissipation limit, an additional term besides the original Berry phase is found. The additional phase does not have a geometrical meaning due to the dissipation.
Using the time-dependent unitary transformation instead of the invariant operator, the solutions of SU(1, 1) and SU(2) time-dependent quantum systems are obtained. It is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the Hamiltonian and the other correspond to the Floquet operator which gives the cyclic states and their associated phases over one period of the evolution. The non-adiabatic Berry's (or Aharonov-Anandan) phases are determined totally by such a unitary transformation.