Quantum Bayesian approach to circuit QED measurement with moderate bandwidth

We consider continuous quantum measurement of a superconducting qubit in the circuit QED setup with a moderate bandwidth of the measurement resonator, i.e., when the “bad-cavity” limit is not applicable. The goal is a simple description of the quantum evolution due to measurement, i.e., the measurement backaction. Extending the quantum Bayesian approach previously developed for the bad-cavity regime, we show that the evolution equations remain the same, but now they should be applied to the entangled qubit-resonator state, instead of the qubit state alone. The derivation uses only elementary quantum mechanics and basic properties of coherent states, thus being accessible to nonexperts.

Figure 1

Schematic of the circuit QED setup. Microwave field of frequency ${\omega }_{\mathrm{d}}$ is transmitted through (or reflected from) the resonator, whose frequency slightly changes, ${\omega }_{\mathrm{r}}\pm \chi$, depending on the qubit state. After amplification, the microwave is sent to the IQ mixer, which produces two quadrature signals: $I\left(t\right)$ and $Q\left(t\right)$. In the case of a phase-sensitive amplifier we define $I\left(t\right)$ as the signal corresponding to the amplified quadrature, while for a phase-preserving amplifier we define $I\left(t\right)$ as the quadrature carrying information about the qubit state.

Figure 2

Comparison between (a) transmission and (b) reflection configurations. The incoming drive field $A_{\mathrm{d}}$ is mostly reflected, but its small part enters the resonator, contributing to the field change as $\dot{\alpha }=\sqrt{{\kappa }_{\mathrm{in}}}{A}_{\mathrm{d}}$ in the transmission case and $\dot{\alpha }=\sqrt{{\kappa }_{\mathrm{out}}}{A}_{\mathrm{d}}$ in the reflection case. For the resonator field, ${\left|\alpha \right|}^2$ is the average number of photons, while for the propagating fields, ${\left|F\right|}^2$ and $|A_{\mathrm{d}}{|}^2$ are the average numbers of photons per second.

Figure 3

Illustration of the history-tail idea. Since the qubit does not evolve by itself, we can think in terms of superposition [Eq. (46)] of two evolutions of the resonator state $\left[\right|{\alpha }_0\left(t\right)\rangle$ and $|{\alpha }_1\left(t\right)\rangle ]$, which leave the “history record” in the form of propagating field, leaked from the resonator. We call this propagating field a history tail. The quantum Bayesian formalism developed in this paper is based on measuring small pieces of the history tail in the textbook way. The corresponding state collapse leads to change of the coefficients in the superposition, which is the evolution due to measurement.

Figure 4

Illustration of the effect of vacuum noise. The vacuum noise $v\left(t\right)$ incident from the output side affects the resonator state $\alpha \left(t\right)$ via the coupling ${\kappa }_{\mathrm{out}}$. Therefore, $v\left(t\right)$ contributes to the outgoing field $F$ twice: due to direct reflection and due to the field leaking from the resonator later. We mostly consider the case ${\kappa }_{\mathrm{out}}\approx \kappa ,{\kappa }_{\mathrm{in}}\ll \kappa$, so that we can neglect the effect of the vacuum noise $v_{\mathrm{add}}\left(t\right)$ incident from the input side, which adds to the drive field $A_{\mathrm{d}}=-i\varepsilon /\sqrt{{\kappa }_{\mathrm{in}}}$. (The outgoing field from the input port is not important and not shown.)