Fractional Generalization of Quantum Markovian Master Equation

TL;DR

This paper introduces a fractional generalization of the quantum Markovian master equation, using superoperators that model varying degrees of environmental influence on quantum systems, and solves it for a harmonic oscillator with linear friction.

Contribution

It develops a fractional extension of the quantum Markovian equation using superoperators, demonstrating their properties and solving the equation for specific quantum systems.

Findings

Superoperators are infinitesimal generators of completely positive semigroups.

The fractional power parameter models different environmental influences.

The solution describes a continuum from no environment to full influence.

Abstract

We prove 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 $0<\alpha<1$.