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We describe the decoherence process induced on a two-level quantum system indirect interaction with a non-equilibrium environment. The non-equilibriumfeature is represented by a non-stationary random function corresponding to thefluctuating transition frequency between two quantum states coupled to thesurroundings. In this framework, we compute the decoherence factors which havea characteristic "dip" related to the initial phases of the bath modes. Wetherefore study different types of environments, namely ohmic and supra-ohmic.These environments present different decoherence time-scales than the thermalenvironment we used to study. As a consequence, we compute analytically andnumerically the non-unitary geometric phase for the qubit in a quasi-cyclicevolution under the presence of these particular non-equilibrium environments.We show in which cases decoherence effects could, in principle, be controlledin order to perform a measurement of the geometric phase using standardprocedures.

Nonunitary geometric phases: a qubit coupled to an environment with random noise