The rotating-wave approximation: consistency and applicability from an open quantum system analysis

We provide an in-depth and thorough treatment of the validity of the rotating-wave approximation (RWA) in an open quantum system. We find that when it is introduced after tracing out the environment all timescales of the open system are correctly reproduced, but the details of the quantum state in general will not be. The RWA made before the trace is more problematic: it results in incorrect values for environmentally induced shifts to system frequencies, and the resulting theory has no Markovian limit. In either form, the RWA gives an inaccurate quantum state which makes it inappropriate for calculating entanglement dynamics or similar detailed properties of the state dynamics. We also emphasize the fact that even under the RWA the master equation for a combination of systems and external forces is not a simple combination of the master equations of the systems and forces. Such a combination may be tempting, because the RWA guarantees that a master equation so constructed will have a valid mathematical form; however, it will not accurately reflect the dynamics of the physical system. To obtain the correct master equation for the composite system a proper consideration of the non-Markovian dynamics is required.