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Modern Physics Letters B

BERRY PHASE IN AN EFFECTIVE SU(1,1) SYSTEM

Yan-Hong Jin, Zhi-Jian Li, J.-Q. Liang

Published Sep 10, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984902004524

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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.
We study the time evolution of a class of exactly solvable time-dependent quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators with the help of the invariant Hermitian operator. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the invariant operator. The adiabatic and non-adiabatic Berry phases are calculated with the exact solutions. Moreover, we derive an explicit time-evolution operator which is used to investigate the time-dependent two-photon squeezing states and SU(2) squeezing states. The squeezing properties of the time-dependent SU(1,1) coherent states are also discussed.
We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.
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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.
For a quantum system with time dependent boundary conditions, it is shown that the ef-fective Hamiltonian can be constructed in a simple way without any geometric treatemnt. In this paper the effective Hamiltonian, constructed in the above way, of quantum particle in a spherical box of varying radius is used to calculate the Berry phase of the wave function. We found that the effective Hamiltonian and original one are formally related with a static gauge transformation, the quantum states of the two Hamiltonians differ by a Berry phase.