Linear feedback stabilization of a dispersively monitored qubit

The state of a continuously monitored qubit evolves stochastically, exhibiting competition between coherent Hamiltonian dynamics and diffusive partial collapse dynamics that follow the measurement record. We couple these distinct types of dynamics together by linearly feeding the collected record for dispersive energy measurements directly back into a coherent Rabi drive amplitude. Such feedback turns the competition cooperative and effectively stabilizes the qubit state near a target state. We derive the conditions for obtaining such dispersive state stabilization and verify the stabilization conditions numerically. We include common experimental nonidealities, such as energy decay, environmental dephasing, detector efficiency, and feedback delay, and show that the feedback delay has the most significant negative effect on the feedback protocol. Setting the measurement collapse time scale to be long compared to the feedback delay yields the best stabilization.

Figure 1

Markovian feedback parameters for ensemble-average stabilization. The constant Rabi drive strength ${\mathrm{\Delta }}_0$ and linear feedback parameter ${\mathrm{\Delta }}_1$ are shown as functions of the stabilized Bloch sphere polar angle $\theta$ in the $yz$ plane, in units of ${\tau }_m^{-1}$ with ${\tau }_m=0.2\mu s$. Solid: Ideal stabilization of a pure state. Dashed: Nonideal stabilization of the state with maximum Bloch radius $R_{\max }=1/\sqrt{1/\eta +2{\tau }_m/T_2}=0.64$ which includes effects of environmental dephasing ($T_2=40\mu \mathrm{s}$) and reduced quantum efficiency ($\eta =0.41$).

Figure 2

Effect of signal filtering and feedback delay on an ensemble-average stabilized state. Stabilized polar Bloch angle ${\theta }_s$ compared to the (representative) target angle ${\theta }_T=3\pi /10$ (upper) and Bloch radius $R_s$ compared to the target maximum radius $R_{\max }=0.64$ (lower) as functions of the exponential time constant $T_s/{\tau }_m$ for a low-pass filtered signal (left) and signal delay $T_d/{\tau }_m$ (right), both normalized by the measurement collapse time ${\tau }_m$. Though both effects cause a fixed and correctable angular drift relative to the target value, the feedback delay dramatically lowers the achievable purity for the stabilized state.

Figure 3

Ideal Markovian feedback stabilization for qubit Bloch coordinates $(y,z)=(sin\theta ,cos\theta )$, shown (left, right), with collapse time ${\tau }_m=0.2\mu \mathrm{s}$ and time step $dt=0.5\mathrm{ns}$. By setting the feedback parameters ${\mathrm{\Delta }}_0=-sin\left(2{\theta }_s\right)/4{\tau }_m$ and ${\mathrm{\Delta }}_1=sin{\theta }_s/{\tau }_m$, the initial state at $\theta =\pi /10,(y,z)=(0.3,0.91)$, correctly evolves to the target stationary state with angle ${\theta }_s=3\pi /10,(y_s,z_s)=(0.81,0.59)$, within a few measurement collapse times ${\tau }_m$. Analytic solutions for the ensemble-average state evolution (blue, nearly coinciding with red) are compared to the numerically simulated ensemble average over ${10}^4$ trajectories (red). Individual sample trajectories (gray) roughly indicate the distributional convergence to the targeted state.

Figure 4

Nonideal Markovian feedback stabilization for qubit Bloch coordinates $(y,z)=(Rsin\theta ,Rcos\theta )$, shown (left, right), with collapse time ${\tau }_m=0.2\mu \mathrm{s}$, time step $dt=0.5\mathrm{ns}$, energy decay time $T_1=60\mu s$, environmental dephasing time $T_2=40\mu \mathrm{s}$, and quantum efficiency $\eta =0.41$. By setting the feedback parameters ${\mathrm{\Delta }}_0=-[sin\left(2{\theta }_s\right)/4{\tau }_m{R}_s^2]$ and ${\mathrm{\Delta }}_1=(sin{\theta }_s/{\tau }_m{R}_s)$, with $R_s\left({\theta }_s\right)$ determined by Eq. (16) in the text, the initial state at angle $\theta =\pi /10$ and unit radius, or $(y,z)=(0.3,0.91)$, correctly evolves to the target stationary state with angle ${\theta }_s=3\pi /10$ and radius $R_s=0.64$, or $(y_s,z_s)=(R_{s}sin{\theta }_s,R_{s}cos{\theta }_s)=(0.52,0.37)$ on average, within a few measurement collapse times ${\tau }_m$. Analytic solutions for the ensemble-average state evolution (blue, nearly coinciding with red) are compared to the numerically simulated ensemble-average over ${10}^4$ trajectories (red). Individual sample trajectories (gray) show broad fluctuation around the target state compared with the ideal case in Fig. 3.

Figure 5

Comparison between targeted ensemble-average states (black curve) and most probable individual trajectories (orange dots), for parameters $T_1=60\mu \mathrm{s}, T_2=40\mu \mathrm{s}, \eta =0.41,{\tau }_m=0.2\mu \mathrm{s}$, and $dt=10\mathrm{ns}$. Each orange dot is the maximum histogram bin for an ensemble of ${10}^5$ trajectories, targeting a range of polar angles $\theta \in (0,\pi )$. The computed data has been reflected over the $y=0$ axis for visual appeal. The ensemble-averaged state (black) has constant radius of $R_E=0.64$ except near the poles where $T_1$ effects manifest. Note that the stabilization of the excited-state pole is compromised, while stabilization of the ground-state pole is enhanced. The most probable states (orange) coincide with the mean only at the equator when $z=0$ and otherwise purify near the poles, as also shown in Fig. 6.

Figure 6

Nonideal trajectory histograms at steady state in the Bloch $yz$ plane. The inefficiencies $T_1=60\mu \mathrm{s}, T_2=40\mu \mathrm{s}, \eta =0.41$ are included as in Fig. 4, with ${\tau }_m=0.2\mu \mathrm{s}$ and $dt=10\mathrm{ns}$. Signal filtering and feedback delay are neglected here ($T_s=T_d=0$). (Top) Target ensemble-average state ${\theta }_s=3\pi /10$ and $R_s=0.64$, showing single-lobe stabilization. The actual histogram peak (black bar) is at the same target angle ${\theta }_P=3\pi /10$ but with larger radius $R_P=0.78$ than the mean, with deviation $\sigma =0.23$ around that peak. (Bottom) Target ensemble-average state ${\theta }_s=\pi /10$ and $R_s=0.64$ closer to the pole, showing double-lobe stabilization. The dominant peak (black bar) is at the shifted angle ${\theta }_P=11\pi /100$ with substantially larger radius $R_P=0.96$ and deviation $\sigma =0.54$.

Figure 7

Nonideal trajectory histograms at steady state in the Bloch $yz$ plane, including signal filtering and delay. The ensemble-average state ${\theta }_s=3\pi /10$ and $R_s=0.64$ is targeted using the same parameters as the top plot in Fig. 6 for comparison. (Top) Only single-pole low-pass signal filtering is added with exponential time constant $T_s=0.2{\tau }_m$. The single lobe of Fig. 6 spreads into a larger region, degrading the ensemble-average purity. Most prepared states retain high purity, however, including the most probable state (black bar) with ${\theta }_P=0.23\pi$ and $R_P=0.85$. (Bottom) Only feedback delay of $T_d=0.2{\tau }_m$ is added. Similar to signal filtering, the single lobe spreads, but into an even larger region, as indicated in Fig. 2. The most probable state (black bar) is also similar, with ${\theta }_P=0.2\pi$ and $R_P=0.83$.