About the general criteria for correct phenomenological relaxation operators of quantum systems

We define three general conditions that the relaxation operator in the von Neumann equation for the density matrix of an open quantum system should satisfy. The first one provides the correct relation between the macroscopic current and polarization in a quantum medium, the second is defined by thermodynamic arguments and the third one is imposed by gauge invariance of the solution. The corrected phenomenological relaxation operator is proposed for any quantum system with a discrete number of basis functions, as well as for a quantum system described by the density matrix in coordinate representation. The dissipative three-dimensional oscillator in an arbitrary oriented magnetic field is considered in detail. We have shown that the minimally modified relaxation operator in comparison with the standard relaxation operator, corresponding to tau-approximation, eventually leads to simpler and noncontradictory equations for the dynamics of averaged values of physical quantities.