We present a broad summary of research involving the application of quantum feedback control techniques to optical setups, from the early enhancement of optical amplitude squeezing to the recent stabilisation of photon number states in a microwave cavity, dwelling mostly on the latest experimental advances. Feedback control of quantum optical continuous variables, quantum nondemolition memories, feedback cooling, quantum state control, adaptive quantum measurements, and coherent feedback strategies will all be touched upon in our discussion.

Quantum control is a broad field of study, engaging the engineering, mathematics, and physical sciences communities in an effort to analyse, design, and experimentally demonstrate techniques whereby the dynamics of physical systems operating at the quantum regime is steered towards desired aims by external, time-dependent manipulation [

Due to the high degree of coherent control, to the wide availability of well-established experimental techniques, and to the relatively low technical noise and decoherence enjoyed by optical setups, the quantum optics community has been in a position to pioneer most of the quantum control techniques developed so far and is still definitely at the forefront of such research. In particular, quantum optics allows for fast and relatively efficient detections in the quantum regime, for manipulations by control fields on time scales much shorter than the system’s typical dynamical time scales, as well as for efficient input-output interfaces (as for travelling modes impinging on optical cavities). These advantages make quantum optical systems particularly well suited for the implementation of closed-loop (“feedback”) control techniques, where some (classical or quantum) information is extracted from the system and used to condition the control operations. Feedback control is, in a sense, the next step in quantum control techniques, with the remarkable possibility of stabilising specific target states in the face of decoherence and noise. Also, feedback techniques can be applied to cool a number of diverse quantum degrees of freedom, and thus to help bringing them into the quantum regime. Over the last decade, several quantum optical demonstrations of feedback techniques have been achieved, climaxing with the recent, spectacular stabilisation of photon number states in a cavity QED setup [

Our paper will follow a combined historical-contextual order. We shall start with the introduction of some basic terminology and notions concerning quantum feedback; next, we will move on to consider the measurement-based feedback control of quantum continuous variables in general, and then specialise our treatment to the important cases of quantum memories for continuous variables and feedback cooling; we will hence hit the deep quantum regime by considering the feedback control of highly coherent systems with few quantum excitations in cavity QED and linear optics setups; we will then review advances in adaptive measurement techniques in the context of optical quantum metrology; our last stop will be to consider the notion and experimental achievements in the subarea of coherent feedback; finally, we shall include some cursory outlook on research in the area. Note that we will emphasise primarily experimental achievements with quantum optical systems and, although partial reference to the accompanying theoretical literature will be provided, no attempt will be made at a comprehensive coverage of the vast general theoretical literature on quantum feedback control. For that, the reader may refer to textbooks with a broader scope [

We introduce here the fundamental notions required for a basic understanding of feedback control in quantum systems. We will limit ourselves to defining and briefly sketching some terminology and typical issues encountered. A detailed, systematic treatment may be found in [

Ideally, a closed quantum system can be prepared in a pure state vector

Assuming, for simplicity, that the global system-environment state

The conditional evolution, being dependent on the outcome of quantum measurements, involves a fundamental probabilistic element. In order to describe such a dynamics, one has hence to define appropriate stochastic increments (if the monitoring is continuous) or “quantum jumps” [

This probabilistic element in the dynamics, whereby the quantum trajectories (i.e., the pure conditional states) evolve continuously but not differentiably in time, is the expression of the fundamental probabilistic nature of quantum states and is referred to as the “back-action” noise induced by the measurement process. In this context, especially in relation to quantum control, it is relevant to introduce the notion of “strength” of a quantum measurement. The strongest possible class of measurements is represented by projective measurements, where the operators ^{1}

After the monitoring, measurement-based feedback loops are typically closed by actuators that apply coherent quantum manipulations on the system where some parameter depends on the measurement outcome at the previous step. Ideally, such actuators act on time-scales which are fast with respect to the system dynamics (and so do the detectors).

In this paper, we will refer to

Depiction of measurement-based (a) and coherent (b) feedback loops. In (a), the output system, after having interacted with the system, is measured, and the classical information contained in the measurement outcome is then used to affect the evolution of the quantum system (typically by modifying the Hamiltonian); in (b), the output quantum state is coherently manipulated (typically by a unitary operation), and then fed back into the quantum system as an input state through a coherent coupling.

Measurement-based feedback loop

Coherent feedback loop

In quantum optics, the light quadratures (the quantum counterparts of electric and magnetic fields of electromagnetic waves) may be treated as canonical quantum operators

It is in fact a metrological application of light fields, namely, the interferometric detection of gravitational waves, that first sparkled the quest for the production of nonclassical, squeezed states [^{2}

Such early all-optical developments, and more broadly the general interest in quantum optics and in the generation of squeezed light, motivated the establishment of a fully quantum theory of continuously monitored systems and feedback control, mainly tied to quantum optical settings and to measurement processes accessible to light fields (essentially homodyne and heterodyne detections, and their derivations). This eventually led to a theory that can be expressed in terms of quantum Langevin equations in the Heisenberg picture or “general-dyne” unravellings and stochastic Schrödinger equations in the Schrödinger picture [

The interest in the production of squeezed light has not dwindled since the eighties. Quite on the contrary, it has been further strengthened by the clarification, in the context of continuous variable quantum information, of the relationship between squeezing, Einstein-Podolski-Rosen correlations, and quantum entanglement [

There are, however, two areas, quantum nondemolition atomic quantum memories and cooling of quantum mechanical oscillators, where the measurement-based feedback control of continuous variables has found favourable ground for experimental implementation.

A well-known approach to the storage and retrieval of the quantum state of a travelling light field employs the collective interaction of light with atomic ensembles contained in room temperature vapour cells [

Although the coherent interaction would already correlate the atomic ensemble with the light passing through it, the operation of these continuous variables memories is actually optimised by a measurement-based feedback loop. The formal description of this feedback scheme is rather simple and illustrates very well the action of homodyne feedback on quantum fields. It is therefore worthwhile to sketch it here (see [

Atomic QND memories, operating at room temperature according to the principles sketched above, surpassed the quantum threshold (beating any possible classical “measure and prepare” strategy [

The past ten years have seen a widespread and intense effort by the physics community to achieve ground state cooling of massive harmonic oscillators. Such a line of research, drawing its origins from gravitational wave detectors, holds considerable promise for applications in precision sensing [^{3}

Although similar techniques had long been mastered in the classical regime [

Simplified schematics of a cold damping feedback loop in optomechanics: the cavity output is monitored by phase-sensisitve homodyne detection at the frequency of a cavity mode which is optomechanically coupled to a material oscillator (in this particular depiction, a trapped bead in a levitating setup [

Analogous techniques, employing active measurement-based feedback control and often called “parametric cooling” (where the control parameter is generally different from case to case), were also recently applied to cool cantilevers [

With few exceptions, the experimental achievements covered so far, while genuinely “quantum” in that they are subject to measurement back-action and act on quantum variables, operate in a semiclassical regime, where the quantum fluctuations are small with respect to the average fields. We will now move on to demonstrations in the deep quantum regime, where fields and matter are controlled down to a few quanta.

Cavity QED systems, where high finesse cavity light, typically in the microwave region, is strongly coupled with flying or trapped atoms, is arguably one of the most controllable quantum systems and has hence an outstanding pedigree in the demonstration of quantum information and control primitives [^{4}

It should therefore come as no surprise that the first demonstration of deep quantum feedback control techniques was achieved with a cavity QED setup [

Further advances in feedback state control have made use of qubits (two-level quantum systems) encoded in the polarisation of travelling photons. These linear optical setups have been extremely successful since the early days of quantum information, enjoying low decoherence rates and ample possibilities for coherent manipulations on single qubits. They were in fact employed in the earliest demonstrations of quantum teleportation [^{5}

For the first experimental demonstration of real-time steady state control by measurement-based feedback, we have to turn back to a cavity QED setup, one with a remarkable track record of experimental achievements over the last 20 years [

This experiment demonstrated final fidelities of about 0.8 for Fock states

The feedback scheme of [

On the theoretical side, it is worth noting that schemes to achieve and preserve highly entangled steady states between internal atomic degrees of freedom, analogous to those discussed for continuous variables in Section

Alongside their potential for the control of quantum states and operations, feedback techniques have also found application in precision measurements and quantum metrology in optical setups. It has long been known that, because of the uncertainty relations, quantum mechanics impose bounds on the attainable precision of quantum measurements [

Besides the recourse to optimised nonclassical states, it has also long been known that the adoption of feedback techniques enhances the precision of quantum measurements. The notion of “adaptive” quantum measurement was first formulated in a quantum optical setting by considering the estimate of the phase of an optical mode [

With the notable exception of [^{6}

Adaptive quantum state estimation, where the objective is the reconstruction of a quantum state given a certain number of available copies, and whose efficiency was theoretically proven in [

Somewhat related to these developments are the demonstrations of state discrimination by adaptive measurements.^{7}

Measurement-based quantum feedback is based on the extraction of ^{8}

In quantum optics, coherent feedback control can be typically implemented by letting the output of a cavity undergo some controlled coherent evolution and then feeding it back as a cavity input. These situations can be described by the so-called input-output formalism [

Let us also mention that, over the past few years, considerable theoretical work has been devoted to the understanding of the open- and closed-loop coherent controllability of infinite dimensional bosonic systems, by

The coherent control and manipulation of quantum systems is growing more and more central to quantum optics and, more broadly, to the whole physics community [

A number of colleagues have helped the author, directly or indirectly (by collaboration on topics relevant to this paper), during the completion of this paper, among whom he would like to mention M. Barbieri, D. Vitali, F. Sciarrino, A. R. R. Carvalho, S. Mancini, and P. Barker. Special thanks go to M.G. Genoni, whose assistance at all stages of this endeavour was invaluable.

Although it must be noted that there are strategies, based on the so-called Zeno dynamics [

Let us remind the reader that the variances (“uncertainties”) of canonical quadratures are constrained by the Heisenberg uncertainty principle:

Such schemes, which are proving very successful, have been since understood to be equivalent to standard cavity cooling, in the sense that they use the cavity to preferentially scatter blue detuned photons, and hence extract energy from the harmonic oscillator in the process.

Arguably, the advantages of standard QED set-ups are offset by difficulties in terms of scalability, especially with respect to integrated optical components of recent developments in the linear optical arena [

It is also worth remarking that similar examples of single-shot “feedforward” manipulations are also employed in all forms of quantum teleportation, both with polarised photons [

A Kalman filter [

By “state discrimination,” one refers to the task of distinguishing between different non-orthogonal quantum states with limited available resources (e.g., number of copies of the state). Note that no single-shot measurement can deterministically distinguish between two non-orthogonal quantum states.

Let us remark that there exists no procedure capable of converting a single copy of an arbitrary quantum state into classical information.