Fractional Dynamics of Open Quantum Systems

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We can describe an open quantum system starting from a closed Hamiltonian system if the open system is a part of the closed system (Weiss, 1993). However situations can arise where it is difficult or impossible to find a Hamiltonian system comprising the given quantum system. As a result, the theory of open and non-Hamiltonian quantum systems can be considered as a fundamental generalization (Kossakowski, 1972; Davies, 1976; Ingarden and Kossakowski, 1975; Tarasov, 2005, 2008b) of the quantum Hamiltonian mechanics. The quantum operations that describe dynamics of open systems can be considered as real completely positive trace-preserving superoperators on the operator space. These superoperators form a completely positive semigroup. The infinitesimal generator of this semigroup is completely dissipative (Kossakowski, 1972; Davies, 1976; Ingarden and Kossakowski, 1975; Tarasov, 2008b). Fractional power of operators (Balakrishnan, 1960; Komatsu, 1966; Berens et al., 1968; Yosida, 1995; Martinez and Sanz, 2000) and superoperators (Tarasov, 2008b, 2009a) can be used as a possible approach to describe fractional dynamics of open quantum systems. We consider superoperators that are fractional powers of completely dissipative superoperators (Tarasov, 2009a). We prove that the suggested superoperators are infinitesimal generators of completely positive semigroups for fractional quantum dynamics. The quantum Markovian equation, which includes an explicit form of completely dissipative superoperator, is the most general type of Markovian master equation describing non-unitary evolution of the density operator that is trace-preserving and completely positive for any initial condition.