Reversing Quantum Trajectories with Analog Feedback

Abstract

We demonstrate the active suppression of transmon qubit dephasing induced by dispersive measurement, using parametric amplification and analog feedback. By real-time processing of the homodyne record, the feedback controller reverts the stochastic quantum phase kick imparted by the measurement on the qubit. The feedback operation matches a model of quantum trajectories with a measurement efficiencyη ≈ 0.5, consistent with the result obtained by postselection. We overcome the bandwidth limitations of the amplification chain by numerically optimizing the signal processing in the feedback loop and provide a theoretical model explaining the optimization result. DOI: 10.1103/PhysRevLett.112.080501 PACS numbers: 03.67.Lx, 42.50.Dv, 42.50.Pq, 85.25.-j In a quantum measurement, information gain is accompanied by backaction, altering superposition states of the observed system [1]. Tunable strength measurements have been devised to balance the tradeoff between information gain and backaction. These can be realized, for example, by controlling the interaction of the observed qubit with an ancillary qubit, followed by strong measurement of the ancilla [2][3][4]. Depending on the choice of ancilla measurement basis, the observed qubit either acquires a stochastic phase kick, or is partially projected towards one of the basis states, in a direction that is determined by the measurement result. Similarly, a cavity mode can serve as an ancilla, with the measurement basis set by the detected field quadrature [5], and a continuous range of measurement results and associated kickbacks [5,6].For an efficient measurement [1], the correlation between the stochastic evolution of the system, also known as quantum trajectory, and the measurement record of the ancilla can be exploited to undo any unwanted backaction [7,8] [18]. Analog feedback, instead, is required to counteract the continuous range of measurement kickbacks in a qubit-cavity system. A first implementation of analog feedback relied on continuous monitoring of a driven qubit to stabilize Rabi oscillations [19].In this Letter, we demonstrate the real-time reversal of measurement-induced qubit dephasing in cQED, using phase-sensitive parametric amplification [20] and analog feedback control, as proposed in Ref. [21]. The recovery of coherence by feedback is quantitatively consistent with a measurement efficiencyη ≈ 0.5 for the homodyne detection chain, closely matching the result obtained by open-loop postselection. Furthermore, we demonstrate a numerical procedure that finds the optimal weight function for the homodyne signal integration, circumventing the inefficiency arising from the finite detection bandwidth.We study measurement-induced dephasing of a transmon qubit (transition frequency ω Q =2π ¼ 5.430 GHz) coupled to the fundamental mode of a 3D cavity (frequency f r ¼ 6.5433 GHz, linewidth κ=2π ¼ 1.4 MHz). The qubitcavity Hamiltonian in the presence of a measurement drive at frequency f m and valid in the dispersive regime of our experiment is [22] in a frame ro...

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