Lamb Shift and the Vacuum Rabi Splitting in a Strongly Dissipative Environment.

We study the vacuum Rabi splitting of a qubit ultrastrongly coupled to a high- cavity mode and a radiation reservoir. Three methods are employed: a numerically exact variational approach with a multiple Davydov ansatz, the rotating-wave approximation (RWA), and the transformed RWA. Agreement between the variational results and the transformed RWA results is found in the regime of validity of the latter, where the RWA breaks down completely.