This article is cited in 24 papers

Fractional generalization of the quantum Markovian master equation

V. E. Tarasov

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University

Abstract: We propose a generalization of the quantum Markovian equation for observables. In this generalized equation, we use superoperators that are fractional powers of completely dissipative superoperators. We prove that the suggested superoperators are infinitesimal generators of completely positive semigroups and describe the properties of this semigroup. We solve the proposed fractional quantum Markovian equation for the harmonic oscillator with linear friction. A fractional power of the Markovian superoperator can be considered a parameter describing a measure of "screening" of the environment of the quantum system: the environmental influence on the system is absent for $\alpha=0$, the environment completely influences the system for $\alpha=1$, and we have a powerlike environmental influence for $0<\alpha<1$.

Keywords: fractional power of an operator, non-Hamiltonian quantum system, quantum Markovian equation, completely positive semigroup.

Received: 27.03.2008
Revised: 23.06.2008

DOI: 10.4213/tmf6310

 English version: Theoretical and Mathematical Physics, 2009, 158:2, 179–195

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