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.
The standard method of „measuring“ quantum wavefunction is the technique of {it indirect} quantum state tomography. Owing to conceptual novelty and possible advantages,an alternative {\it direct} scheme was proposed and demonstrated recently in quantum optics system. In this work we present a study on the direct scheme of measuring qubit state in the circuit QED system, based on weak measurement and weak value concepts. To be applied to generic parameter conditions, our formulation and analysis are carried out for finite strength weak measurement, and in particular beyond the bad-cavity and weak-response limits. The proposed study is accessible to the present state-of-the-art circuit-QED experiments.
Developing efficient and reliable schemes for practical quantum measurements is of essential importance to quantum information science and quantum metrology. In this work, for the increasinglyimportant superconducting circuit-QED setup, we present a rigorous approach starting with the quantum trajectory equation (QTE) to establish an {\it exact} quantum Bayesian rule. For the „realistic“ back-action (no qubit state information gain), we obtain important correction factors for arbitrary setup parameters. For the „spooky“ information gain back-action, we establish new prior distribution knowledge for the Bayesian inference, which differ from the standard Gaussian distribution and ensure to give strictly the same results as that by numerically integrating the QTE. Compared to the QTE approach, while keeping the same accuracy, the obtained quantum Bayesian rule has much higher efficiency to compute the stochastic change of the measured state. The generic method of this work opens also a new way to construct exact quantum Bayesian rules for quantum measurement in other systems.
For continuous weak measurement of qubits, we obtain exact expressions for weak values (WVs) from the post-selection restricted average of measurement outputs, by using both the quantumtrajectory-equation (QTE) and quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the „very“ weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that our result incorporates non-perturbative correction which could be important when the AAV’s WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier’s noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful result for homodyne measurement in circuit-QED system, which allows for measuring the real and imaginary parts of the AAV’s WV by simply tuning the phase of the local oscillator. This advantage can be exploited as efficient state-tomography technique.
There exist two scenarios of quantum weak measurement theories. One is the well-known quantum trajectory theory which, in terms of continuous differential equation, has been broadlyapplied in quantum optics and quantum control problems. Another is the relatively newer quantum Bayesian approach, which has the advantage of being more efficient to infer the state of the measured quantum system merely based on certain integrated output of measurements. In this work, we aim to develop a quantum Bayesian rule for weak measurements of qubits in circuit quantum electrodynamics (QED). Starting with the optical quantum trajectory equation, our analysis pays particular attention to the nature of the cavity field under continuous quadrature monitoring. This allows our treatment unrestricted to the „bad-cavity“ and weak-response limits, thus making the obtained rule applicable to general setup parameters. With accuracy well proven numerically in this work, we expect this proposed approach to be useful for future circuit-QED measurement and control experiments.