Nonlinear extension of the quantum dynamical semigroup

Jakub Rembieliński and Paweł Caban

Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Lodz, Pomorska 149/153, 90-236 Łódź, Poland

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In this paper we consider deterministic nonlinear time evolutions satisfying so called convex quasi-linearity condition. Such evolutions preserve the equivalence of ensembles and therefore are free from problems with signaling. We show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasi-linear operations also possesses the semigroup property. Next we generalize the Gorini-Kossakowski-Sudarshan-Lindblad type equation for the considered evolution. As examples we discuss the general qubit evolution in our model as well as an extension of the Jaynes-Cummings model. We apply our formalism to spin density matrix of a charged particle moving in the electromagnetic field as well as to flavor evolution of solar neutrinos.

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