Using Spontaneous Emission of a Qubit as a Resource for Feedback Control

Persistent control of a transmon qubit is performed by a feedback protocol based on continuous heterodyne measurement of its fluorescence. By driving the qubit and cavity with microwave signals whose amplitudes depend linearly on the instantaneous values of the quadratures of the measured fluorescence field, we show that it is possible to stabilize permanently the qubit in any targeted state. Using a Josephson mixer as a phase-preserving amplifier, it was possible to reach a total measurement efficiency $\eta =35\%$, leading to a maximum of 59% of excitation and 44% of coherence for the stabilized states. The experiment demonstrates multiple-input multiple-output analog Markovian feedback in the quantum regime.

Figure 1

Feedback scheme and its implementation. (a) An arbitrary state of a transmon qubit is stabilized by feedback control based on a continuous measurement of its fluorescence field quadratures $V_I(t)$ and $V_Q(t)$. The control Hamiltonian $H_c=\hslash [u(t){\sigma }_x+v(t){\sigma }_y+w(t){\sigma }_z]$ depends linearly on $V_I(t)$ and $V_Q(t)$ and is designed to compensate in real time for the deviations of the qubit from its target state. The controller can be split into three physically distinct boxes. The Drift box adds a constant drive $(\overline{u},\overline{v})$ for static precompensation. The Rabi box adds a drive term $(\delta u,\delta v)$ proportional to the fluorescence field quadratures rotated by an angle $\alpha$. The FM box modulates the qubit frequency leading to a term $w$ proportional to the quadrature $V_{\beta }=V_I\cos \beta +V_Q\sin \beta$ of the fluorescence field. (b),(c) Scheme of the physical realizations of the Rabi and FM boxes (see the text).

Figure 2

Stabilization of the excited state. Targeting the excited state $|{\mathrm{\Psi }}_{0,0}⟩=|e⟩$, the parameters $G_{\mathrm{FM}}$, $\overline{u}$, and $\overline{v}$ are set to zero according to Eq. (2) (only the “Rabi box” of Fig. 1 is on). Measurements (symbols) and simulations (lines) represent $⟨{\sigma }_z⟩$ as a function of gain $G_R$ for fixed rotation angle $\alpha =\pi /2$ (left panel) and as a function of $\alpha$ for fixed $G_R/G_R^{\mathrm{opt}}=0.35$, 1, 11.4 (right panel). The case where the JPC amplifier is turned on (off) is shown in blue (gray). The qubit is measured after a feedback sequence of $30\mu \mathrm{s}$ ensuring that it is in a steady state. Statistical uncertainties are indicated by error bars in the left panel and by symbol size in the right panel. All the parameters entering the simulations are measured independently.

Figure 3

Stabilization of any state. (a) Measured (symbols) and simulated (continuous lines) values of $⟨{\sigma }_x⟩$ (orange) and $⟨{\sigma }_y⟩$ (green) as a function of the phase $\beta$ when targeting the state $|{\mathrm{\Psi }}_{\pi /2,\pi /2}⟩=(|e⟩+i|g⟩)/\sqrt{2}$. According to Eq. (2), one sets $G_R=G_R^{\mathrm{opt}}/2$, $\alpha =\pi /2$, $\overline{u}={\gamma }_1/8$, and $\overline{v}=0$. The gain $G_{\mathrm{FM}}$ is experimentally set to its optimal value $G_{\mathrm{FM}}^{\mathrm{opt}}$ giving the maximum $⟨{\sigma }_x{⟩}^2+⟨{\sigma }_y{⟩}^2$. The targeted state is most closely reached for $\beta =-10°$ (red) due to a slight nonlinearity in the FM box response [21]. As in Fig. 2, a comparison with the simulations serves as a calibration of the experimental offset on $\beta$. Statistical uncertainties are indicated by error bars (by symbol size in the other panels). (b) Measured (symbols) and simulated (continuous lines) values of $⟨{\sigma }_x⟩$, $⟨{\sigma }_y⟩$, and $⟨{\sigma }_z⟩$ (respectively, orange, green, and blue) as a function of the polar angle $\theta$ of targeted state $|{\mathrm{\Psi }}_{\theta ,\pi /2}⟩$. (c) Projection of the Bloch sphere on the $yz$ plane. Red dots represent the measured stabilized states of (b). The corresponding simulated states are shown as a red line. A purple dot indicates the stabilized states when aiming for $\theta =2\pi /3$.

Figure 4

Transient dynamics under feedback. Measured (symbols) and simulated (continuous lines) $⟨{\sigma }_x⟩$, $⟨{\sigma }_y⟩$, and $⟨{\sigma }_z⟩$ (respectively, orange, green, and blue) as a function of the duration $t$ of the feedback control. At $t=0$, the qubit is in thermal equilibrium, and all control parameters in Fig. 1 are set to 0. The upper panel corresponds to the target state $|{\mathrm{\Psi }}_{0,0}⟩=|e⟩$ [$G_R=G_R^{\mathrm{opt}}$, $\alpha =\pi /2$ as for the purple in Fig. 2(left)]. The lower panel corresponds to the target state $|{\mathrm{\Psi }}_{\pi /2,\pi /2}⟩=(|e⟩+i|g⟩)/\sqrt{2}$ [$\beta =-10°$ as for the red in Fig. 3]. Error bars are statistical uncertainties. Simulations are performed with the same model as in Figs. 2 and 3.