Reversing Quantum Trajectories with Analog Feedback

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 efficiency $\tilde{\eta }\approx 0.5$, 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.

Figure 1

Measurement-induced dephasing and analog feedback scheme. (a) Diagram of the key elements of the experimental setup. Qubit measurement and control drives are coupled to the input port of an asymmetrically coupled 3D cavity (${\kappa }_{\mathrm{in}}/{\kappa }_{\text{out}}\approx 1/30$). The signal emitted at the output port is added to the pump tone, which biases the JPA to a voltage gain $G=16$ and a bandwidth ${\kappa }_{\mathrm{JPA}}/2\pi =5.7\mathrm{MHz}$ (Fig. S7). The reflected, amplified signal [20, 23] is directed by a circulator to a semiconductor amplifier (HEMT) at 3 K. At room temperature, the signal is split into two arms, one for data acquisition and another feeding the FPGA-based feedback controller (see Fig. S1 for setup details). (b) Cavity spectroscopy for the qubit prepared in $|0⟩$ and $|1⟩$. Measurement pulses are applied at $f_m$ (green arrow). (c) Echo sequence, where in the second half a measurement pulse with amplitude ${\tilde{\epsilon }}_m$ is inserted to study its dephasing effect on the qubit. The second $\pi /2$ pulse is compiled into the tomographic rotation $R_{\mathrm{n⃗},\overline{\phi }}$, where $R_{\mathrm{n⃗}}$ is either $R_y(-\pi /2)$, $R_x(\pi /2)$ or $I$, and the axis is rotated by $\overline{\phi }$ around $z$ to cancel the deterministic phase shift. (d) Parametric plot of the averaged homodyne response $⟨V_Q⟩$ versus $⟨V_I⟩$ for measurement phase $\varphi =\pi /2$ and 0, respectively, for the qubit in $|0⟩$ (red) and $|1⟩$ (blue), with ${\tilde{\epsilon }}_m=0.4\mathrm{V}$. Dashed circle: signal corresponding to ${\overline{n}}_{\mathrm{ph}}=0.1$ intracavity average photon. (e) Qubit coherence $r_{\mathrm{ol}}$ as a function of ${\tilde{\epsilon }}_m$. The best-fit curve gives the lever arm ${\epsilon }_m/{\tilde{\epsilon }}_m=2\pi \times 1.2\mathrm{MHz}/\mathrm{V}$. For ${\tilde{\epsilon }}_m=0$, $r_{\mathrm{ol}}=r_{\text{off}}=0.79\pm 0.01$.

Figure 2

Conditional qubit tomography and cancellation of measurement-induced dephasing by analog feedback. (a) The measurement $M_Q$ is performed with a pulse at $f_m$ with amplitude ${\tilde{\epsilon }}_m=0.4\mathrm{V}$ and 500 ns length (dashed trace). The homodyne record $V_Q$ is acquired for a total duration of $1.25\mu \mathrm{s}$ from the start of the measurement pulse. Light (dark) trace: single (average) record. (b) Measurement scheme. (c) Conditional state tomography (left) and corresponding fraction of counts $C$ (right) in open-loop operation. Solid (dashed) curves: data (model with $\tilde{\eta }=0.50$). The tomography outcomes $M_I$ are binned on $V_{\text{int}}=\sum _{n}w[n]V_Q[n]$, where $V_Q$ is sampled every 10 ns. The weight function $w=w_{\text{opt}}$ is obtained by numerical optimization using the records $V_Q$ (see also Fig. 4. (d) Stochastic qubit phase $\delta \phi$ (dots) and absolute coherence $r$ (squares), binned on $V_{\text{int}}$, and model for $\delta \phi$ with $\tilde{\eta }=0.50$ (solid) and 1 (dashed line). In closed-loop operation [(e), corresponding to $c_{\mathrm{fb}}=-10$ in Fig. 3], $V_Q$ is fed to the feedback controller, which calculates $V_{\text{int}}$ using $w_{\text{opt}}$ and translates it into $\delta \phi$, setting the phase of $R_{\mathrm{n⃗},\phi }$. (f) Measured distribution of $\delta \phi$ (grey scale) produced by $M_Q$ and refocusing by analog feedback. This refocusing increases the unconditioned coherence from $r_{\mathrm{ol}}=0.40$ (black arrow) to $r_{\mathrm{cl}}=0.56$ (pink arrow). Dashed circle: maximum $r=r_{\text{off}}$ that would be obtained with $\tilde{\eta }=1$.

Figure 3

Extraction of measurement efficiency from the extent of coherence recovery. (a) Coherence versus feedback gain $c_{\mathrm{fb}}$ for ${\tilde{\epsilon }}_m=0.2–0.7\mathrm{V}$, with $w_{\text{opt}}$ optimized at ${\tilde{\epsilon }}_m=0.4\mathrm{V}$. Top left: average homodyne voltage $⟨V_Q⟩$ for the same range of ${\tilde{\epsilon }}_m$. The maximum coherence $r_{\mathrm{cl}}$ corresponds to the optimum feedback gain $c_{\text{opt}}$ (lower inset), directly proportional to ${\tilde{\epsilon }}_m$. The horizontal dashed line indicates the coherence $r_{\text{off}}$ for no measurement drive (${\tilde{\epsilon }}_m=0$). Error bars are the standard deviations of eight repetitions. (b) Contour plot of the measurement efficiency $\tilde{\eta }$, with curves at 0.1 steps. For each ${\tilde{\epsilon }}_m$, $r_{\mathrm{cl}}$ is obtained by a quadratic fit of $r$ around the maximum and $r_{\mathrm{ol}}$ is the measured average for $c_{\mathrm{fb}}=0$ in (a). The best fit of Eq. (2) (orange dashed line) to the data yields $\tilde{\eta }=0.49\pm 0.01$.

Figure 4

(a) Measurement efficiency as a function of JPA gain and bandwidth. The experiment in Fig. 2 is repeated for different pump powers (with the JPA resonance kept at $f_m$), setting the JPA voltage gain $G$ and bandwidth ${\kappa }_{\mathrm{JPA}}$ (triangles). For $G\lesssim 10$, the gain is insufficient to overcome the noise temperature of the HEMT (noise temperature $\sim 3\mathrm{K}$). For higher $G$, the infinite-bandwidth $w_{\infty }$ becomes suboptimal as $V_Q$ is low-pass filtered by the JPA when ${\kappa }_{\mathrm{JPA}}$ approaches $\kappa$ (dots). By numerically optimizing the weight function ($w_{\text{opt}}$), this filtering is undone and $\tilde{\eta }\approx 0.5$ is recovered (squares). For $G=1$ the pump is turned off and the JPA is intentionally detuned by $\sim 200\mathrm{MHz}$ from $f_m$. (b) Numerically optimized $w_{\text{opt}}$ for $G=2.5$ (full dots) and 23 (empty dots), model $w_{\infty }$ for infinite detector bandwidth (dashed line) and mode-matched $w_{\mathrm{mm}}$ (dot-dashed line) for ${\kappa }_{\mathrm{JPA}}/2\pi =3.9\mathrm{MHz}$ ($G=23$).