Exact analytical solutions to the master equation of quantum Brownian motion for a general environment

We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value problems for integro-differential equations which led to incorrect master equation coefficients and impacts on the description of nonlocal dissipation effects in all earlier derivations. Furthermore, we provide explicit, exact analytical results for the master equation coefficients and its solutions in a wide variety of cases, including ohmic, sub-ohmic and supra-ohmic environments with a finite cut-off.

Research highlights► We study the model of a quantum oscillator linearly coupled to a bath of oscillators. ► We derive the master equation and solutions for general spectra and temperatures. ► We generalize to cases with an external force and arbitrary number of oscillators. ► Other derivations have incorrect diffusion and force response for nonlocal damping. ► We give exact results for ohmic, sub-ohmic and supra-ohmic environments.