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Journal of Physics A Mathematical and General

Floquet theory and the non-adiabatic Berry phase

D J Moore

Published Jul 7, 1990• 45 citations
Publisher

Abstract

An efficient calculational algorithm is provided for the operator decomposition approach to non-adiabatic Berry phases for systems with periodic Hamiltonians.
DOI
10.1088/0305-4470/23/13/006

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The present paper analyses the relation between the theory of the time-dependent wave operator and the Berry phase concept. It is proved that the wave operator approach is consistent with the non-adiabatic (Aharonov–Anandan) Berry phase, given that the wave operator and the parallel transport commute. It is then demonstrated that the non-Abelian Aharonov–Anandan phase can be calculated by working inside a reduced active space in the framework of wave operator theory. Finally an adiabatic transport formula is derived in the wave operator context and the influence of this effective Hamiltonian theory on the Berry phase is analysed. The theoretical results concerning the non-adiabatic Berry phase are confirmed numerically by considering a photodissociation process in the framework of the generalized Floquet theory.
It is well known that Hermitian operators have real eigenvalues while non-Hermitian ones may have complex eigenvalues. Recently, numerical and analytical results indicated that the spectra of many non-Hermitians Hamiltonians HHH are indeed real if they are invariant under the combined action of self-adjoint parity PPP and time reversal T.T.T. The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the PTPTPT-symmetric Hamiltonians follow from their pseudo-Hermiticity. It is possible to explain these observations by the concept of pseudo-Hermitian operators and to formulate completeness and orthonormality relations. Most of the effort has been devoted to study time-independent non-Hermitian systems. In this paper, we study the exactly solvable time-dependent periodic pseudo-Hermitian Hamiltonians. The method introduced, to make the reality of eigenvalues and phases, is based on a Floquet decomposition of the evolution operator UH(t)=ZH(t)exp(iMHt){U}^{H}(t)={Z}^{H}(t)exp(i{M}^{H}t)UH(t)=ZH(t)exp(iMHt) associated with the periodic pseudo-hermitian Hamiltonian H(t)=H(t+T)H(t)=H(t+T)H(t)=H(t+T). One of the results found in this paper concerns a calculation of Berry's phase for periodic, but not necessarily adiabatic, pseudo-Hermitian Hamiltonians. A two-level pseudo-Hermitian system is discussed as an illustrative example.
In the system of several interacting spins, geometric phases have been researched intensively. However, the studies are mainly focused on the adiabatic case (Berry phase), so it is necessary for us to study the non-adiabatic counterpart (Aharonov and Anandan phase). In this paper, we analyze both the non-degenerate and degenerate geometric phase of Lipkin—Meskov—Glick type model, which has many application in Bose—Einstein condensates and entanglement theory. Furthermore, in order to calculate degenerate geometric phases, the Floquet theorem and decomposition of operator are generalized. And the general formula is achieved.