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Chinese Physics

Time-dependent Landau system and non-adiabatic Berry phase in two dimensions

Jing Hui, Jiansheng Wu

Published Jul 1, 2000• 8 citations
Publisher

Abstract

By applying the time-independent unitary transformation, the time-dependent Landau system is transformed into a product of a time-independent Landau system's Hamiltonian and a factor only depending on time, which can be solved exactly. Both the invariant operator and the eigenstate are obtained. In the periodical time-dependent case, the non-adiabatic Berry's phase is also presented.

DOI
10.1088/1009-1963/9/7/001

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Journal of Physics A Mathematical and General•Oct 7, 1990•26 citations
By applying the time-independent unitary transformation, the time-dependent Landau system is transformed into a product of a time-independent Landau system's Hamiltonian and a factor only depending on time, which can be solved exactly. Both the invariant operator and the eigenstate are obtained. In the periodical time-dependent case, the non-adiabatic Berry's phase is also presented.
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.
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.
A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.