Berry phase with environment: classical versus quantum
  We discuss the concept of the Berry phase in a dissipative system. We show that one can identify a Berry phase in a weakly-dissipative system and find the respective correction to this quantity, induced by the environment. This correction is expressed in terms of the symmetrized noise power and is therefore insensitive to the nature of the noise representing the environment, namely whether it is classical or quantum mechanical. It is only the spectrum of the noise which counts. We analyze a model of a spin-half (qubit) anisotropically coupled to its environment and explicitly show the coincidence between the effect of a quantum environment and a classical one. 
Robert S. Whitney, Yuriy Makhlin, Alexander Shnirman, Yuval Gefen