Geometric nature of the environment-induced Berry phase and geometric dephasing

We investigate the geometric phase or Berry phase acquired by a spin half which is both subject to a slowly varying magnetic field and weakly coupled to a dissipative environment ( either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original Berry phase ( for an isolated system) is the flux of a monopole field through the loop traversed by the magnetic field, the environment-induced modification of the phase is the flux of a quadrupolelike field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the Berry phase, this geometric dephasing is gauge invariant for open paths of the magnetic field.