Berry Phase and its Unitarily Invariant Generalization

Diana Monteoliva, H. J. Korsch, J A Nunez

A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.

Robert Olkiewicz

Donald H. Kobe

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.

Luiz Agostinho Ferreira, Abraham Hirsz Zimerman, J. F. Gomes

Mikhail Orionovich Katanaev

Barbara Sanborn, Sergeĭ K. Suslov, Luc Vinet

We present quadratic dynamical invariant and evaluate Berry's phase for the time-dependent Schroedinger equation with the most general variable quadratic Hamiltonian.