Geometric Phases In Open Quantum Systems

The geometric (Berry) phase of a two-level system in a dissipative environment is analyzed by using the second-quantized formulation, which provides a unified and gauge-invariant treatment of adiabatic and nonadiabatic phases and is thus applicable to a quantitative analysis of transitional regions away from ideal adiabaticity. In view of the recent experimental observation of the Berry phase in a superconducting qubit, we illustrate our formulation for a concrete adiabatic case in the Ohmic dissipation. The correction to the total phase together with the geometry-dependent dephasing time is given in a transparent way. The behavior of the geometric phase away from ideal adiabaticity is also analyzed in some detail.Further we adopt the standard master equation method to consider the geometric phase under finite temperature. The correction of geometric phases arising from environment is analyzed. The dephasing time scale and the relaxation time scale are investigated. We find that all these quantities have an exponential dependence on temperature T and meanwhile they also have a close dependence on geometric parameterθ. Energy transports between spin and environment are also checked.