We study theoretically dynamics of a driven-dissipative qubit-resonator system. Specifically, a transmon qubit is coupled to a transmission-line resonator; this system is considered to be probed via a resonator, by means of either continuous or pulsed measurements. Analytical results obtained in the semiclassical approximation are compared with calculations in the semi-quantum theory as well as with the previous experiments. We demonstrate that the temperature dependence of the resonator frequency shift can be used for the system thermometry and that the dynamics, displaying pinched-hysteretic curve, can be useful for realization of memory devices, the quantum memcapacitors.
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