Superconducting circuits provide a new platform to study nonstationary cavity QED phenomena. An example of such a phenomenon is a dynamical Lamb effect which is a parametric excitation of an atom due to the nonadiabatic modulation of its Lamb shift. This effect has been initially introduced for a natural atom in a varying cavity, while we suggested its realization in a superconducting qubit-cavity system with dynamically tunable coupling. In the present paper, we study the interplay between the dynamical Lamb effect and the energy dissipation, which is unavoidable in realistic systems. We find that despite of naive expectations this interplay can lead to unexpected dynamical regimes. One of the most striking results is that photon generation from vacuum can be strongly enhanced due to the qubit relaxation, which opens a new channel for such a process. We also show that dissipation in the cavity can increase the qubit excited state population. Our results can be used for the experimental observation and investigation of the dynamical Lamb effect and accompanying quantum effects.
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