Reduced dynamics and the master equation of open quantum systems

An exact reduced density operator of a quantum system interacting with a bosonic thermal reservoir is derived by means of the simple algebraic method. The necessary and sufficient condition is found that the time-convolutionless master equation becomes exact up to the second order with respect to the system-reservoir interaction. The result is examined by means of the boson-detector model. The reduced dynamics of a quantum system interacting with a classical reservoir is also discussed.