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An ultra-strong coupling regime takes place in a compound system when acoupling strength between the subsystems exceeds one tenth of the systemeigenfrequency. It transforms into a deep-strong coupling regime when thecoupling strength exceeds the system eigenfrequency. In these regimes, thereare difficulties with description of relaxation processes without explicitconsidering of environment degrees of freedom. To correctly evaluate therelaxation rates, it is necessary to consider the interaction of the systemwith its environment taking into account the counter-rotating wave anddiamagnetic terms. We develop a self-consistent theory for calculation of therelaxation rates in the systems, in which the coupling strength is of the orderof the system eigenfrequency. We demonstrate that the increase in the couplingstrength can lead to a significant decrease in the relaxation rates. Inparticular, we show that for frequency-independent density of states of theenvironment, the relaxation rates decrease exponentially with the increase inthe coupling strength. This fact can be used to suppress losses by tuning thestrength coupling and the environment states.

Self-consistent description of relaxation processes in systems with ultra- and deep-strong coupling