Adiabatic approximation in the density matrix approach: non-degenerate systems

We study the adiabatic limit in the density matrix approach for a quantum system coupled to a weakly dissipative medium. The energy spectrum of the quantum model is supposed to be non-degenerate. In the absence of dissipation, the geometric phases for periodic Hamiltonians obtained previously by Berry are recovered in the present approach. We determine the necessary condition satisfied by the coefficients of the linear expansion of the non-unitary part of the Liouvillian in order that the imaginary phases acquired by the elements of the density matrix, due to dissipative effects, be geometric. The results derived are model independent. We apply them to spin model coupled to reservoir at thermodynamic equilibrium.