Transitional dynamics of a phase qubit-resonator system: requirements for fast readout of a phase qubit

We examine the non-stationary evolution of a coupled qubit-transmission line-resonator system coupled to an external drive and the resonator environment. By solving the equation for a non-stationary resonator field, we determined the requirements for a single-shot non-destructive dispersive measurement of the phase qubit state. Reliable isolation of the qubit from the "electromagnetic environment" is necessary for a dispersive readout and can be achieved if the whole system interacts with the external fields only through the resonator that is weakly coupled to the qubit. A set of inequalities involving resonator-qubit detuning and coupling parameter, the resonator leakage and the measurement time, together with the requirement of multi-photon outgoing flux is derived. It is shown, in particular, that a decrease of the measurement time requires an increase of the resonator leakage. This increase results in reducing the quality factor and decreasing the resolution of the resonator eigenfrequencies corresponding to different qubit states. The consistent character of the derived inequalities for two sets of experimental parameters is discussed. The obtained results will be useful for optimal design of experimental setups, parameters, and measurement protocols.