Dynamics of open quantum systems

A tutorial introduction is presented for the calculation of the time dynamics for models of dissipative quantum mechanics where a small quantum system is coupled to noninteracting bosonic or fermionic reservoirs. We discuss the basics and applications of the real-time renormalization group method (RTRG), where the Fourier variable $E$ conjugate to the time variable $t$ is used as flow parameter. It is shown how a well-controlled perturbation theory can be set up in the coupling to the bath where secular and logarithmic terms are resummed to all order of perturbation theory. This is achieved by a set of perturbative RG equations for the effective Liouvillian and the effective vertices. Truncating the RG equations we show how the time dynamics can be obtained analytically for exponentially small, intermediate and exponentially large times. Basic applications are discussed for particle and potential energy exchange within the interacting resonant level model, spin exchange for the nonequilibrium Kondo model, and heat exchange within the ohmic spin boson model at zero bias.