Geodesic
Fractional generalization of the~quantum Markovian master equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 158 (2009) no. 2, pp. 214-233

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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 $0\alpha1$.
Keywords: fractional power of an operator, non-Hamiltonian quantum system, completely positive semigroup.
Mots-clés : quantum Markovian equation 
@article{TMF_2009_158_2_a3,
     author = {V. E. Tarasov},
     title = {Fractional generalization of the~quantum {Markovian} master equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {214--233},
     publisher = {mathdoc},
     volume = {158},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2009_158_2_a3/}
}
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V. E. Tarasov. Fractional generalization of the~quantum Markovian master equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 158 (2009) no. 2, pp. 214-233. http://geodesic.mathdoc.fr/item/TMF_2009_158_2_a3/