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

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