Backtracking quantum trajectories with analog feedback

Circuit quantum electrodynamics offers a nearly ideal platform for the fundamental study of continuous quantum measurement. A nondemolition measurement of a superconducting qubit can be performed via homodyne detection of microwave transmission through a dispersively coupled cavity. By boosting the homodyne signal with a nearly noiseless phase-sensitive parametric amplifier, we experimentally show that a form of measurement backaction, consisting of stochastic quantum phase kicks on the measured qubit, is highly correlated with the fluctuations in the continuous homodyne record. We demonstrate a real-time analog feedback scheme that counteracts these phase kicks and thereby reduces measurement-induced dephasing. We develop a numerical optimization technique to overcome the bandwidth limitations of the amplification chain and provide a theoretical model for the optimization result. A quantum efficiency of 50\% is extracted for the complete analog feedback loop. Finally, we discuss the integration of this analog feedback technique to improve performance in our recent demonstration [1] of entanglement by dispersive parity measurement. $^{*}$equal contribution. [1] D. Rist\`e {\it et al.}, Nature 502, 350 (2013).