Periodic pseudo-Hermitian Hamiltonian: Nonadiabatic geometric phase

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 H are indeed real if they are invariant under the combined action of self-adjoint parity P and time reversal T. The concept of a pseudo-Hermitian operator showed that the remarkable spectral properties of the PT-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 U-H (t) = Z(H) (t) exp(iM(H)t) associated with the periodic pseudo-hermitian Hamiltonian 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.