Gradual partial-collapse theory for ideal nondemolition measurements of qubits in circuit QED

The conventional method of qubit measurements in circuit QED is employing the dispersive regime of qubit-cavity coupling, which results in an approximated scheme of quantum nondemolition (QND) readout. However, this scheme breaks down owing to the Purcell effect in the case of strong coupling and/or strong measurement drive. To remove the drawbacks of the dispersive readout, a recent proposal by virtue of longitudinal coupling suggests a new scheme to realize fast, high-fidelity and ideal QND readout of qubit state. In the present work, following dispersive readout, we construct the gradual partial-collapse theory for this new measurement scheme, in terms of both the quantum trajectory equation and quantum Bayesian approach. The longitudinal coupling provides as well a convenient method of cavity reset. In combination with the reset procedure, the established theory is expected to be useful for such as measurement-based feedback control and many other quantum applications associated with partial-collapse weak measurements.