Geometric phase decomposition for exactly solvable cases with interaction

Abstract

In the paper a problem of geometric phase decomposition for general evolutions in the Hilbert space is addressed. The decomposition of total phase into dynamical and geometrical parts employs a state representation in the noninteraction picture. The noninteraction picture is introduced with the help of decomposition of the system evolution operator into two parts, the one pertaining to free evolution and the other to interaction. The procedure requires the problem to be an exactly solvable case of the Schrödinger equation. The most common class of such problems includes dipole Hamiltonians, for which the evolution operator can be decomposed into a combination of unitary operators. Geometric phase decomposition in the noninteraction picture can be applied to general noncyclic evolutions, but for the cyclic states it reduces to the Floquet decomposition. Defined this way, geometric phase possesses characteristic features of geometric phase for the free system, but extends some stationary properties to temporal dependences. As a case, the time-dependent relationship between geometric phase and the nonstationarity of a quantum state is illustrated by an example of a spin-½ in a rotating magnetic field. It is shown that geometric phase reaches a maximum when the spin state becomes completely nonstationary.