Undoing measurement-induced dephasing in circuit QED

We analyze the backaction of homodyne detection and photodetection on superconducting qubits in circuit quantum electrodynamics. Although both measurement schemes give rise to backaction in the form of stochastic phase rotations, which leads to dephasing, we show that this can be perfectly undone provided that the measurement signal is fully accounted for. This result improves on an earlier one [Phys. Rev. A 82, 012329 (2010)], showing that the method suggested can be made to realize a perfect two-qubit parity measurement. We propose a benchmarking experiment on a single qubit to demonstrate the method using homodyne detection. By analyzing the limited measurement efficiency of the detector and bandwidth of the amplifier, we show that the parameter values necessary to see the effect are within the limits of existing technology.

Figure 1

(a) Purity as a function of waiting time $t_{\mathrm{off}}$. After each measurement, a phase correction conditioned on the measurement trace was applied as described in the text. The parameters are given by ${\varepsilon }_d=\kappa$, $t_{\mathrm{meas}}=5/\kappa$, and $\chi =3\kappa$. The initial state is $|\psi \rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$. Inset: Purity as a function of measurement efficiency for the case of no feedback [dotted (red) curve], feedback without accounting for all photons [dashed (blue) curve], and feedback with full accounting [solid (green) curve]. Here, $t_{\mathrm{off}}=3/\kappa$. (b) Time evolution of the cavity field ${\alpha }_e$ [solid (green) curve] and ${\alpha }_g$ [dashed (red) curve]. After the measurement signal is turned off, the field starts to spiral back towards the origin. At certain points in time (arrows), ${\alpha }_e={\alpha }_g$, for which the qubit state is pure.

Figure 2

Setup used to model the finite bandwidth of the detector. The second cavity acts as a bandpass filter with bandwidth $2{\kappa }_b$, being perfectly transparent to signals at the measurement frequency but becoming increasingly more reflective as the frequency of the incoming signal becomes more detuned from the measurement frequency.

Figure 3

The dotted (green) curve shows the purity of the qubit density matrix after measurement, with phase correction after $t=t_{\mathrm{meas}}+t_{\mathrm{off}}$, as a function of the amplifier bandwidth. The solid (blue) line is the purity without any phase correction. Parameters are as in Fig. 1 with $t_{\mathrm{off}}=10/\kappa$. Inset: Purity vs measurement efficiency for $\text{BW}=10\kappa$ with [dotted (red) curve] and without [solid (blue) line] phase correction.