Unphysical time evolution in Markovian and non-Markovian Caldeira-Leggett master equations

We investigate a non-Markovian master equation of the Caldeira-Leggett model obtained in the weak coupling limit up to the second order expansion. The evolution of the non-Markovian master equation is unphysical when the stationary solution is unphysical, and we prove that the solution always remains physical for small times. Moreover, we are able to find even stronger anomalous behavior, when the trace of the solution is diverging. We also provide results for the Markovian master equation and show that positivity violations occur for various types of initial conditions even when the stationary solution is physical. In our study we use Gaussian initial states to simplify the problem and employ a sufficient and necessary condition to detect unphysical behavior. Based on our numerical experiments we conclude that the non-Markovian master equation is superior to the Markovian one.