An atomic Ramsey interferometer (auxiliary cavities R₁ and R₂) sandwiches the superconducting Fabry–Perot cavity C resonant at 51 GHz and cooled to 0.8 K (the mean number of blackbody photons is 0.05). The pulsed classical source S′ induces π/2 pulses resonant with the |g  |e transition in R₁ and R₂ (with relative phase φ_r) on the velocity-selected (250 m s⁻¹) Rydberg atom qubits (purple circles) prepared by laser excitation (blue arrow) from a rubidium atomic beam (green arrow) in B. The field-ionization detector D measures the qubits in the e/g basis with a 35% detection efficiency and an error rate of a few per cent (Supplementary Methods). The actuator S feeds cavity C by diffraction on the mirror edges. The controller K (a CPU-based ADwin Pro-II system) collects information from D to determine the real translation amplitude α applied by S. It sets the S-pulse duration through a PIN diode switch A (63-μs pulse for |α| = 0.1) as well as a 180° phase-shifter Φ controlling the sign of α.

Two feedback runs lasting 164 ms (2,000 loop iterations) stabilizing |n_t = 2 (left column) and |n_t = 3 (right column). The phase-shift per photon, φ₀ = 0.256π, allows controller K to discriminate n values between 0 and 7. For n_t = 2, the Ramsey phase is φ_r = −0.44 rad, corresponding to nearly equal e and g detection probabilities when n = 2. For n_t = 3, two Ramsey phases φ_r,1 = −0.44 rad and φ_r,2 = −1.24 rad are alternatively used, corresponding to equal e and g probabilities when n = 2 and n = 3, respectively. a, Sequences of qubit detection outcomes. The detection results are shown as blue downward bars for g and red upward bars for e. Two-atom detections in the same state appear as double-length bars. b, Estimated distance between the target and the actual state. c, Applied α-corrections (shown on a log scale as sgn(α)log|α|). d, Photon number probabilities estimated by K: P(n = n_t) is in green, P(n < n_t) in red, P(n > n_t) in blue. e, Field density operators ρ in the Fock-state basis estimated by K at four different times marked by arrows.

Plots a, b, c and d correspond to the target photon number states n_t = 1, 2, 3 and 4, respectively. The red histograms correspond to about 3,900 trajectories stopped when P(n_t) has reached the threshold value (0.8) for three successive iterations. These histograms describe the field at the time when the controller K has certified the ‘success’ of the quantum feedback procedure. The blue histograms correspond to 4,000 trajectories stopped at a fixed time (164 ms) and describe the feedback procedure steady-state. These histograms are reconstructed by a method independent of the feedback estimator. After interrupting the feedback, we record ten additional QND qubit samples (~2 detected atoms) with a Ramsey interferometer phase φ_r chosen in sequence among 4 values (φ_r = 1.17, 0.36, −0.44 and −1.24 rad). From these additional qubit detections, we reconstruct the final P_QND(n) distribution for each ensemble of trajectories by a maximum likelihood algorithm. Statistical error of the reconstructed P_QND(n) for different target states is about 0.01–0.02 for n = n_t and n_t ± 1, and it is significantly smaller than 0.01 for other photon numbers (see Supplementary Methods). The green histograms give the initial coherent state photon number distributions (a similar reconstruction was performed with a fixed time stop immediately after initial field injection).

a, Time evolution of the fraction of individual field trajectories, C_fr(t), that have reached the fidelity threshold (0.8) while converging towards |n_t = 3 in quantum feedback sequences (smooth line) and in passive QND ‘trials’ (stepped line). Statistics performed over 4,000 and 2,131 trajectories, respectively. The same Ramsey phase settings as in Fig. 2 have been used for both feedback and QND sequences. b, Recovery from a quantum jump: the lower plot shows probabilities (n,t) estimated by K and averaged over 2,561 trajectories, following the preparation at t = 0 of the Fock state |n = 2 by a QND measurement of an initial coherent state (colour code shown at right for the different values of n). The Ramsey phase settings are the same as in Fig. 2 for n_t = 3.The initial field density matrix of the field estimation algorithm is diagonal and corresponds to the red histogram in Fig. 3c. The experiment thus simulates the reaction of the quantum feedback procedure to a |3  |2 quantum jump occurring at t = 0, after the field has converged to the target. The upper plot in b shows the variation of the average modulus of the injection amplitude . Initially zero, grows rapidly to a maximum while the quantum jump is reversed. The controller finally quiets and returns to its average steady-state value.