Dynamics of the nonequilibrium spin-boson model: A benchmark of master equations and their validity

In recent years, there has been tremendous focus on identifying whether effective descriptions of open quantum systems such as master equations can accurately describe the dynamics of open quantum systems. One particular question is whether they provide the correct steady state in the long time limit. The transient regime is also of interest. Description of evolution by various master equations---some of them being not completely positive---is benchmarked against exact solutions [see, e.g., R. Hartmann and W. T. Strunz, Phys. Rev. A 101, 012103 (2020)]. An important property of true evolution is its non-Markovian features, which are not captured by the simplest completely positive master equations. In this paper we consider a non-Markovian yet completely positive evolution (known as refined weak coupling or cumulant equation) for the spin-boson model with an overdamped Drude-Lorentz spectral density and arbitrary coupling. We benchmark it against a numerically exact solution, as well as against other master equations, for different coupling strengths and temperatures. We find the cumulant to be a better description in the weak-coupling regime where it is supposed to be valid. For the examples considered it shows superiority at moderate and strong couplings in the low-temperature regime for all examples considered. In the high-temperature regime, however, its advantage vanishes. This indicates that the cumulant equation is a good candidate for simulations at weak to moderate coupling and low temperature. Our calculations are greatly facilitated due to our concise formulation of the cumulant equation by means of representation of the density matrix in the SU(N) basis.