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 α=0, the environment completely influences the system for α=1, and we have a powerlike environmental influence for 0<α<1.
Keywords:fractional power of an operator, non-Hamiltonian quantum system, quantum Markovian equation, completely positive semigroup.
Citation:
V. E. Tarasov, “Fractional generalization of the quantum Markovian master equation”, TMF, 158:2 (2009), 214–233; Theoret. and Math. Phys., 158:2 (2009), 179–195
This publication is cited in the following 23 articles:
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Tarasov V.E., “Non-Markovian Dynamics of Open Quantum System With Memory”, Ann. Phys., 434 (2021), 168667
Tarasov V.E., “Lattice Fractional Quantum Field Theory: Exact Differences Approach”, Mod. Phys. Lett. A, 36:14 (2021), 2140001
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Tarasov V.E., “Quantum Maps With Memory From Generalized Lindblad Equation”, Entropy, 23:5 (2021), 544
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Tarasov V.E., Tarasova V.V., “Time-Dependent Fractional Dynamics With Memory in Quantum and Economic Physics”, Ann. Phys., 383 (2017), 579–599
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Tarasov V.E., “Fractional Diffusion Equations for Open Quantum System”, Nonlinear Dyn., 71:4, SI (2013), 663–670
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Tarasov V.E., “Quantum Dissipation From Power-Law Memory”, Ann. Phys., 327:6 (2012), 1719–1729
Tarasov V.E., “The Fractional Oscillator as an Open System”, Cent. Eur. J. Phys., 10:2 (2012), 382–389
Sirin H., Buyukkilic F., Ertik H., Demirhan D., “The effect of time fractality on the transition coefficients: Historical Stern-Gerlach experiment revisited”, Chaos Solitons & Fractals, 44:1–3 (2011), 43–47