Quantum feedback experiments stabilizing Fock states of light in a cavity

The ubiquitous decoherence phenomenon is responsible for the lack of quantum superpositions at the macroscopic scale. It is increasingly difficult to isolate a quantum system from its environment when its size increases. Making use of the weird quantum properties of mesoscopic quantum states thus requires efficient means to combat decoherence. One option is real-time quantum feedback. It features the components of a conventional feedback: measurement of the system's state (sensor), analysis (controller), and feedback action (actuator) aiming to the target state. The random back-action of the measurements by the sensor makes quantum feedback much more difficult than its classical counterpart. Mesoscopic photon number (Fock) states feature a decoherence rate proportional to the photon number. They are thus a simple example of a fragile quantum resource. We demonstrated recently two quantum feedback schemes continuously stabilizing Fock state of a microwave field in a high-quality cavity. Sensitive atoms, crossing the field one at a time, are used as quantum nondemolition (QND) probes of its photon number. The feedback actuator is either a classical source [Sayrin et al., Nature (London) 477, 73 (2011)] or individual resonant atoms emitting or absorbing one photon each [Zhou et al., Phys. Rev. Lett. 108, 243602 (2012)]. Both schemes detect the quantum jumps of the photon number and efficiently correct their adverse effects, preparing and preserving a fragile quantum resource. We present an in-depth analysis of our methods, which sheds light onto the fundamental difficulties encountered in quantum feedback, in particular concerning the state estimation algorithm, and onto the ways to circumvent them. These results open the way to informationally optimal QND measurements or to the stabilization of mesoscopic field state superpositions. More generally, they can be cast in a variety of contexts to loosen the tight constraints set by decoherence in quantum metrology and quantum information processing.

Figure 1

General scheme of the cavity QED experiment. $\mathsf{C}$: high-$Q$ microwave cavity; $\mathsf{V}$: voltage supply to control the electric field in $\mathsf{C}$; ${\mathsf{R}}_{\mathsf{1}}$ and ${\mathsf{R}}_{\mathsf{2}}$: low-$Q$ Ramsey cavities; ${\mathsf{S}}^{\prime }$: microwave source. Toroids represent circular Rydberg atoms flying from the excitation zone $\mathsf{B}$ towards the ionization detector $\mathsf{D}$.

Figure 2

Experimental Ramsey fringes. Probability ${\pi }_g({\varphi }_{\mathrm{r}},0)$ for detecting the atom in state $g$ as a function of the Ramsey interferometer phase ${\varphi }_{\mathrm{r}}$ for an empty cavity. The dots are experimental. The solid line is a sine fit, used to measure the fringe contrast $c$ and offset ${\pi }_o$. In the paper, all presented Ramsey fringes are recorded by scanning the source frequency ${\omega }_{\mathrm{r}}$ while keeping the potential $V_{\mathrm{r}}$ constant.

Figure 3

Scheme of the quantum feedback experiment with a classical actuator (coherent microwave injection). $\mathsf{K}$: feedback controller; $\mathsf{S}$: microwave source coupled to $\mathsf{C}$; $\mathsf{A}$ and $\Phi$: amplitude and phase controls for microwave injection.

Figure 4

Coefficients $d_{n_{\mathrm{t}}}\left(n\right)$ defining the distance $d({\rho }_{\mathrm{t}},\rho )$ to four target Fock states ${\rho }_{\mathrm{t}}=|n_{\mathrm{t}}\rangle \left.〈n_{\mathrm{t}}\right|$ with $n_{\mathrm{t}}=1$ to 4.

Figure 5

Single sequences of the feedback experiment with $n_{\mathrm{t}}=1$ (left panel) and $n_{\mathrm{t}}=4$ (right panel). The frames present as a function of time $t$ the detected sensor states (upwards bars for $e$, downwards bars for $g$), the distance $d$ to $n_{\mathrm{t}}$, the feedback injection amplitude $\alpha$ chosen by $\mathsf{K}$ (in pseudologarithmic scale), and the photon-number distribution $p\left(n\right)$ inferred by $\mathsf{K}$ [color (grayscale)] together with its mean value $\overline{n}$ (solid black line).

Figure 6

Absolute values of the cavity field density matrix elements estimated by the controller during the realization of the feedback experiment with $n_{\mathrm{t}}=4$ presented in Fig. 5 (right panel). (a) Initial coherent state ($t=0$ ms) with the average photon number $n_{\mathrm{t}}$. (b)–(d) Estimated state at 15.6 ms (high population in $\left|n_{\mathrm{t}}\right.〉$ after initial feedback convergence), at 48.4 ms (after a sudden quantum jump to $\left|n_{\mathrm{t}}-1\right.〉$), and at 51.5 ms (during the correction phase), respectively.

Figure 7

Establishment of the steady-state regime. The probabilities $\overline{p}(n_{\mathrm{t}},t)$ (squares), $\overline{p}(n<n_{\mathrm{t}},t)$ (triangles), and $\overline{p}(n>n_{\mathrm{t}},t)$ (circles) are averaged over 4000 trajectories for each $n_{\mathrm{t}}$ value. Panels (a) to (d) correspond to $n_{\mathrm{t}}=1$ to 4, respectively. Solid lines are a fit with the sum of two exponential functions.

Figure 8

Time evolution of the normalized standard deviation $\overline{\sigma }\left(t\right)$ averaged over 4000 trajectories for each $n_{\mathrm{t}}$ value. Panels (a) to (d) correspond to $n_{\mathrm{t}}=1$ to 4, respectively. Solid lines are fits to an exponential decay.

Figure 9

Injection amplitudes chosen during feedback operation with $\left|n_{\mathrm{t}}=4\right.〉$. (a) Histogram of nonzero values of $\alpha$. (b) Absolute value of $\alpha$ averaged over 4000 trajectories of 164 ms duration as a function of the current probability of $n_{\mathrm{t}}$.

Figure 10

Photon-number distribution $p_{\mathrm{QND}}\left(n\right)$, measured by an independent QND process, for the target photon numbers $n_{\mathrm{t}}$ from 1 to 4 prepared by means of the coherent actuator feedback. (a) Reference initial coherent field with $n_{\mathrm{t}}$ photons on the average. (b) Field reconstructed in the steady-state mode after interrupting the feedback loop at 164 ms. (c) Converged field when $\mathsf{K}$ estimates that $p\left(n_{\mathrm{t}}\right)>0.8$.

Figure 11

Probability $F_{\mathrm{conv}}(n_{\mathrm{t}},t)$ for $p\left(n_{\mathrm{t}}\right)$ to reach a threshold level of 0.8 (smooth line). Panels (a) to (d) correspond to $n_{\mathrm{t}}$ from 1 to 4. Each curve is determined from 4000 trajectories. The horizontal lines depict the 0.63 level defining the convergence time. The staircase line presents a passive trial-and-error QND method for Fock state preparation.

Figure 12

Optimization of the QND measurement duration of the trial-and-error Fock state preparation for $n_{\mathrm{t}}=2$.

Figure 13

Scheme of the quantum feedback experiment with emitting and absorbing resonant atoms used as quantum actuators. (a) Experimental setup. The controller $\mathsf{K}$ sets the durations of the microwave pulses in ${\mathsf{R}}_{\mathsf{1}}$ and ${\mathsf{R}}_{\mathsf{2}}$ and the potential $\mathsf{V}$ applied to one of the mirrors of $\mathsf{C}$. Light (magenta) and dark (blue) toroids represent sensor (dispersive) and actuator (resonant) samples. Their positions with respect to the Ramsey zones, cavity, and detectors are to scale. They correspond to the end of the $k$th feedback loop after the detection of atom $k$. (b) Schematic control sequence for the realization of the three atomic sample types for an atom crossing the cavity center at time $t_0$. The rectangles represent the Ramsey pulses. The solid line represents the time dependence of the potential $\mathsf{V}$, which is switched between two values $V_{\mathrm{dis}}$ (lower) and $V_{\mathrm{res}}$ (upper), corresponding to dispersive and resonant atom-field interaction, respectively.

Figure 14

Experimental vacuum Rabi oscillations. The probability ${\pi }_g$ to detect a resonant atom in state $g$ is measured as a function of the effective interaction time $t$. Initially, the atom-cavity system is prepared in a joint state $\left|e,0\right.〉$ (a) or $\left|g,1\right.〉$ (b). The dots are experimental, with statistical error bars. Solid lines are phenomenological fits to exponentially damped oscillating functions.

Figure 15

Rabi oscillations measured in Fock states from $n=1$ (a) to 6 (f). The initial atomic state is $\left|g\right.〉$ (the data for the initial level $\left|e\right.〉$ are not presented in this figure). The points are experimental with statistical error bars. Solid lines are fits to phenomenological damped oscillating functions taking into account the impurity of the prepared Fock states.

Figure 16

Frequency and damping rate of the fits to the experimental Rabi oscillations. The initial atom-cavity state is either $\left|g,n+1\right.〉$ (triangles) or $\left|e,n\right.〉$ (squares). Straight lines are linear fits.

Figure 17

Optimization by Monte Carlo simulations of the sequence partition into $N_{sn}$ sensor and $N_{ct}$ control samples. (a) Average population $\overline{p}\left(n_{\mathrm{t}}\right)$ in the target state $n_{\mathrm{t}}=3$ (2000 trajectories) as a function of $N_{sn}$ and $N_{ct}$ with $N_{sn}+N_{ct}⩽20$. (b) Convergence time ${\tau }_{\mathrm{conv}}\left(n_{\mathrm{t}}\right)$, in ms, for reaching $p\left(n_{\mathrm{t}}\right)\ge 0.8$ in $63\%$ of trajectories used in (a). Dark gray color indicates partitions for which the convergence time is larger than the simulated sequence duration of 145 ms.

Figure 18

Probability of a photon exchange between photon-number states and actuator atoms. (a) Emission probability of an emitter. Triangles and squares correspond to the effective interaction times $t_e^{2\pi }\left(n_{\mathrm{t}}\right)$ and $0.8t_e^{2\pi }\left(n_{\mathrm{t}}\right)$, respectively, with $n_{\mathrm{t}}=3$. (b) Absorption probability of an absorber. Triangles and squares correspond to the interaction times $t_g^{2\pi }\left(n_{\mathrm{t}}\right)$ and $1.2t_g^{2\pi }\left(n_{\mathrm{t}}\right)$, respectively.

Figure 19

Numerical optimization of the resonant effective interaction times. Average over 2000 sequences of the population in the target state $n_{\mathrm{t}}=3$ (a) and $n_{\mathrm{t}}=7$ (b) is calculated for each pair of values $t_{em}$ and $t_{ab}$ marked by black circles. The color (gray) gradient is interpolated to guide the eye.

Figure 20

Single sequences of the feedback experiment with $n_{\mathrm{t}}=1$ (left), $n_{\mathrm{t}}=4$ (center), and $n_{\mathrm{t}}=7$ (right). The frames present as a function of the total time $t$ the detected sensor states (upwards bars for $e$, downwards bars for $g$), the distance $d$ to $n_{\mathrm{t}}$, the actuators sent by $\mathsf{K}$ (red bars for emitters, blue bars for absorbers), and the photon-number distribution $p\left(n\right)$ inferred by $\mathsf{K}$ [color (grayscale)] together with its mean value $\overline{n}$ (solid black line).

Figure 21

Evolution of the photon-number distribution $\overline{p}(n,t)$ averaged over 4000 feedback sequences with $n_{\mathrm{t}}=3$. The upper panel shows $\overline{p}(n=n_{\mathrm{t}},t)$ (green, thick curve), $\overline{p}(n<n_{\mathrm{t}},t)$ (blue, dotted curve), and $\overline{p}(n>n_{\mathrm{t}},t)$ (red, thin curve). The lower panel is a zoom into the region highlighted in the upper panel. $N_{sn}$ and $N_{ct}$ indicate the periodic division of the atomic sequence into sensor samples detecting the field decay [reduction of $p\left(n_{\mathrm{t}}\right)$] and control samples correcting for it [increase of $p\left(n_{\mathrm{t}}\right)$].

Figure 22

Probabilities of the decision made by $\mathsf{K}$ on the control samples (emitter: dashed red line; sensor: solid green line; absorber: dashed-dotted blue line) as a function of $(\overline{n}-n_{\mathrm{t}})$. Data extracted from 4000 realizations of the 140-ms-long experiment for each $n_{\mathrm{t}}$, from 1 to 7.

Figure 23

Probability $F_{\mathrm{conv}}(n_{\mathrm{t}},t)$ for $p\left(n_{\mathrm{t}}\right)$ to reach a 0.8 threshold level for $n_{\mathrm{t}}=1$ to 7. Each curve is determined from 4000 sequences. The horizontal line shows the 0.63 level defining the convergence time $t_{\mathrm{conv}}$.

Figure 24

Photon-number distribution prepared by means of the quantum actuator feedback for the target photon numbers $n_{\mathrm{t}}$ from 1 to 7. (a) Reference Poisson distribution with $n_{\mathrm{t}}$ photons on the average. (b) Photon-number distribution $p_{\mathrm{QND}}\left(n\right)$ measured by an independent QND process in the steady-state mode after interrupting the feedback loop at 140 ms. (c) Photon-number distribution $p_{\mathrm{QND}}\left(n\right)$ measured in the convergence mode when $\mathsf{K}$ estimates that $p\left(n_{\mathrm{t}}\right)>0.8$. For (b) and (c), $p_{\mathrm{QND}}\left(n\right)$ is measured from $n_0$ to $n_0+7$, with $n_0=0$ for $n_{\mathrm{t}}\le 5$ and $n_0=2$ for $n_{\mathrm{t}}>5$.

Figure 25

Calibration of the phase shift per photon ${\varphi }_0$. (a) Thick black line: histogram of the measured atomic coherence phase ${\phi }_{\mathrm{a}}$. It is fitted by a sum of seven Gaussian peaks with the same width, corresponding to photon numbers $n$ from zero to six (green thin lines for the individual peaks, red thin line for their sum). (b) Position of the peaks as a function of $n$ (dots) and second-order polynomial fit (line).

Figure 26

Measurement of the detector efficiency $\varepsilon$. Ramsey fringes ${\pi }_g({\varphi }_{\mathrm{r}},0)$ in the vacuum field (black with larger contrast) and Ramsey fringes ${\pi }_g({\varphi }_{\mathrm{r}},0d)$ conditioned on no atom being detected in the resonant sample (red with smaller contrast).

Figure 27

Control of the injection amplitude and phase.

Figure 28

Calibration of the injected amplitude. Dots: probability ${\pi }_e$ for detecting the probe atom in $\left|e\right.〉$ as a function of the injection duration $t_{\mathsf{S}}$. The Ramsey interferometer phase is ${\varphi }_{\mathrm{r}}=-{\varphi }_0/2$. Line: theoretical fit (A5), with $\tau$ as the only adjustable parameter.

Figure 29

Rabi oscillations of two atoms both prepared in state $\left|e\right.〉$ and simultaneously coupled to the cavity mode. The atoms are detected both in $\left|e\right.〉$ (a), both in $\left|g\right.〉$ (b), or one in $\left|e\right.〉$ and the other in $\left|g\right.〉$ (c). Solid red lines are theoretical fits with the phenomenological exponential damping.