It is one of the challenges of modern physics to describe the effect of the (macroscopic) environment on the evolution of a quantum system. Its roots are in the perennial problem of the incompatibility of irreversible processes with the time-reversal symmetry of the system. The Lindblad equation approach is one of the ways to account for the environment-system interaction in a consistent form. In a subsequent development the ''square-root method'', originally devised by B. Reznik (Phys. Rev. Lett. 76 (1996) 1192), was used for the reduced density matrix evolution under effect of the Lindblad operators (A. Yahalom and R. Englman, Physica A 371 (2006) 368). In this formulation the nm element of the time (t) dependent density matrix is written in the form q nm Γ°tΓ ΒΌ 1 A P A aΒΌ1 c a n Γ°tΓc aΓ m Γ°tΓ. The so called ''square root factors'', the c(t)'s, are non-square matrices and are averaged over A systems (a) of the ensemble. This square-root description is exact.
The method is here applied to incorporate Lindblad-type processes into the evolution of a degenerate quantum system. Specifically, the effect of a dissipative environment on the topological or Berry phase (BP) is here studied for a system consisting of a spin-doubly degenerate orbital (E), belonging to the cubic G3 2 or C 8 representation, subject to time-periodically varying stress and magnetic fields. The environmental effects redistribute the component amplitude of the initial wave packet and, with a general precessional motion of the magnetic field, change the BP markedly. The changes in the BP are suppressed for the magnetic field precessing on a large circle, in accordance with previous results (D.M. Tong et al., Phys. Rev. Lett. 93 (2004) 080405).