Feedback loops are central to most classical control procedures. A controller compares the signal measured by a sensor (system output) with the target value or set-point. It then adjusts an actuator (system input) to stabilize the signal around the target value. Generalizing this scheme to stabilize a micro-system's quantum state relies on quantum feedback, which must overcome a fundamental difficulty: the sensor measurements cause a random back-action on the system. An optimal compromise uses weak measurements, providing partial information with minimal perturbation. The controller should include the effect of this perturbation in the computation of the actuator's operation, which brings the incrementally perturbed state closer to the target. Although some aspects of this scenario have been experimentally demonstrated for the control of quantum or classical micro-system variables, continuous feedback loop operations that permanently stabilize quantum systems around a target state have not yet been realized. Here we have implemented such a real-time stabilizing quantum feedback scheme following a method inspired by ref. 13. It prepares on demand photon number states (Fock states) of a microwave field in a superconducting cavity, and subsequently reverses the effects of decoherence-induced field quantum jumps. The sensor is a beam of atoms crossing the cavity, which repeatedly performs weak quantum non-demolition measurements of the photon number. The controller is implemented in a real-time computer commanding the actuator, which injects adjusted small classical fields into the cavity between measurements. The microwave field is a quantum oscillator usable as a quantum memory or as a quantum bus swapping information between atoms. Our experiment demonstrates that active control can generate non-classical states of this oscillator and combat their decoherence, and is a significant step towards the implementation of complex quantum information operations.

Fock states with photon numbers n up to 7 are prepared on demand in a microwave superconducting cavity by a quantum feedback procedure which reverses decoherence-induced quantum jumps. Circular Rydberg atoms are used as quantum non-demolition sensors or as single photon emitter/absorber actuators. The quantum nature of these actuators matches the correction of singlephoton quantum jumps due to relaxation. The flexibility of this method is suited to the generation of arbitrary sequences of Fock states.PACS numbers: 42.50. Pq, 42.50.Dv, 03.67.Pp The preparation of non-classical field states and their protection against decoherence is an important aspect of quantum physics and of its application to information science. Among the proposed methods, including error correction [1], decoherence-free subspaces [2] and reservoir engineering [3], quantum feedback [4-6] is particularly promising. Its principle is to drive the quantum system towards a target state by the repeated action of a sensor-controller-actuator loop. The sensor performs quantum measurements and provides information to the controller. Taking into account the back-action of the measurement, the controller then estimates the system's state and programs the actuator to drive the system as close as possible to the target. A given feedback algorithm can in principle protect a wide class of target states. The operating point can be changed at any time and the system driven through a programmed trajectory in its Hilbert space.Photon number (Fock) states are appealing targets for quantum feedback operation. They combine a theoretical and intuitive simplicity (|n is an eigenstate of the field Hamiltonian with n quanta) with intrinsically nonclassical features (their Wigner functions take negative values for n ≥ 1). When coupled to an environment, Fock states lose rapidly their non-classicality with a time constant T n = T c /n, where T c is the lifetime of the field mean energy [7]. Hence, large n Fock states, whose decay time scale is much shorter than T c , are important tools for the exploration of decoherence at the quantum/classical boundary [8].The most intuitive algorithm for quantum feedback generation and protection of Fock states combines single photon emitters/absorbers together with a Quantum Non-Demolition (QND) photon counting sensor. Starting from vacuum, repeated photon emissions make the field climb the ladder of Fock states until the sensor recognizes that the target is reached. When an environmentinduced single-photon quantum jump occurs, the sensor detects it and the controller triggers the action of either an emitter or absorber to correct for it. FIG. 1. Scheme of the experimental set-up.Resonant actuators and QND sensors have already been used separately to prepare Fock states in a time short compared to T n . Repeated photon emission by resonant actuators has been achieved in cavity- [9] and circuit-QED experiments [10]. QND photon counting by repeated atomic sensor measurements has collapsed a coherent field into a Fock state with ra...