We derive the equations of motion describing the feedback control of quantum systems in the regime of "good control", in which the control is sufficient to keep the system close to the desired state. One can view this regime as the quantum equivalent of the "linearized" regime for feedback control of classical nonlinear systems. Strikingly, while the dynamics of a single qubit in this regime is indeed linear, that of all larger systems remains nonlinear, in contrast to the classical case. As a first application of these equations, we determine the steady-state performance of feedback protocols for a single qubit that use unbiased measurements.
In this paper, results of quantum calculations are presented for feedback cooling of an optically trapped nanoparticle in the laser-shot-noise-dominant regime. We numerically investigate the system using both parametric and force feedback cooling schemes. For the same measurement efficiency, the cooling limit from the force feedback is lower than that from the parametric feedback. We also develop a set of semi-classical equations for feedback cooling that accurately match the quantum results. It is demonstrated, by rescaling the semi-classical equations, that the cooling dynamics is uniquely d…
We show that in the regime of ground-state cooling, simple expressions can be derived for the performance of resolved-sideband cooling --- an example of coherent feedback control --- and optimal linear measurement-based feedback cooling for a harmonic oscillator. These results are valid to leading order in the small parameters that define this regime. They provide insight into the origins of the limitations of coherent and measurement-based feedback for linear systems, and the relationship between them. These limitations are not fundamental bounds imposed by quantum mechanics, but are due to t…
Feedback control of quantum systems via continuous measurement involves complex nonlinear dynamics. Except in very special cases, even for a single qubit optimal feedback protocols are unknown. Not even do intuitive candidates exist for choosing the measurement basis, which is the primary non-trivial ingredient in the feedback control of a qubit. Here we present a series of arguments that suggest a particular form for the optimal protocol for a broad class of noise sources in the regime of good control. This regime is defined as that in which the control is strong enough to keep the system clo…
Readout of the state of a superconducting qubit by homodyne detection of the output signal from a dispersively coupled microwave resonator is a common technique in circuit quantum electrodynamics, and is often claimed to be quantum non-demolition (QND) up to the same order of approximation as in the dispersive approximation. However, in this work we show that only in the limit of infinite measurement time is this protocol QND, as the formation of a dressed coherent state in the qubit-cavity system applies an effective rotation to the qubit state. We show how this rotation can be corrected by a…
We realise a phase-sensitive closed-loop control scheme to engineer the fluctuations of the pump field which drives an optomechanical system, and show that the corresponding cooling dynamics can be significantly improved. In particular, operating in the counter-intuitive "anti-squashing" regime of positive feedback and increased field fluctuations, sideband cooling of a nanomechanical membrane within an optical cavity can be improved by 7.5~dB with respect to the case without feedback. Close to the quantum regime of reduced thermal noise, such feedback-controlled light would allow going well b…
A simple feedback scheme can be used to operate efficiently a microwave-quantum-illumination device based on electro-optomechanical systems also in regimes in which excess dissipation would, otherwise, prevent to outperform the optimal classical illumination protocol with the same transmitted energy.
We describe an efficient approach to modelling cavity quantum electrodynamics (QED) with a time-delayed coherent feedback using quantum trajectory simulations. An analytical set of equations is derived to exploit the advantages of trajectories in the presence of the non-Markovian dynamics, where adjustments to the standard stochastic dynamics are discussed. In the weak excitation regime, we first verify that our approach recovers known results obtained with other simulation methods and demonstrate how a coherent feedback loop can increase the photon lifetime in typical cavity-QED systems. We t…
We present an analytical solution of the single photon quantum feedback in a cavity quantum electrodynamics system based on a half cavity set-up coupled to a structured continuum. The exact analytical expression we obtain allows us to discuss in detail under which conditions a single emitter-cavity system, which is initially in the weak coupling regime, can be driven into the strong coupling regime via the proposed quantum feedback mechanism [Carmele et al, Phys.Rev.Lett. 110, 013601]. Our results reveal that the feedback induced oscillations rely on a well-defined relationship between the del…
Preparing macroscopic mechanical resonators close to their motional quantum groundstate and generating entanglement with light offers great opportunities in studying fundamental physics and in developing a new generation of quantum applications. Here we propose an experimentally interesting scheme, which is particularly well suited for systems in the sideband-unresolved regime, based on coherent feedback with linear, passive optical components to achieve groundstate cooling and photon-phonon entanglement generation with optomechanical devices. We find that, by introducing an additional passive…
We propose a domino-cooling method to realize simultaneous ground-state cooling of a coupled mechanical-resonator chain through an optomechanical cavity working in the unresolved-sideband regime. This domino-effect cooling is realized by combining the cold-damping feedback on the first mechanical resonator with nearest-neighbor couplings between other neighboring mechanical resonators. We obtain analytical results for the effective susceptibilities, noise spectra, final mean phonon numbers, and cooling rates of these mechanical resonators, and find the optimal-cooling condition for these reson…
We present a quantum-mechanical treatment of the coherence properties of a single-mode atom laser. Specifically, we focus on the quantum phase noise of the atomic field as expressed by the first-order coherence function, for which we derive analytical expressions in various regimes. The decay of this function is characterized by the coherence time, or its reciprocal, the linewidth. A crucial contributor to the linewidth is the collisional interaction of the atoms. We find four distinct regimes for the linewidth with increasing interaction strength. These range from the standard laser linewidth…
We show that in the regime in which feedback control is most effective -- when measurements are relatively efficient, and feedback is relatively strong -- then, in the absence of any sharp inhomogeneity in the noise, it is always best to measure in a basis that does not commute with the system density matrix than one that does. That is, it is optimal to make measurements that disturb the state one is attempting to stabilize.
We numerically show that time delayed coherent feedback controls the statistical output characteristics of driven quantum emitters. Quantum feedback allows to enhance or suppress a wide range of classical and nonclassical features of the emitted quantum light. As exemplary quantum system, we use a pumped cavity containing two emitters. By applying phase-selective feedback, we demonstrate that photon antibunching and bunching can be increased in orders of magnitude due to intrinsically and externally controllabe quantum interferences. Our modelling is based on a fully non-Markovian quantum simu…
We demonstrate feedback cooling of the motion of a single rubidium atom trapped in a high-finesse optical resonator to a temperature of about 160 μK. Time-dependent transmission and intensity-correlation measurements prove the reduction of the atomic position uncertainty. The feedback increases the 1/e storage time into the one second regime, 30 times longer than without feedback. Feedback cooling therefore rivals state-of-the-art laser cooling, but with the advantages that it requires less optical access and exhibits less optical pumping.
The simultaneous ground-state refrigeration of multiple vibrational modes is a prerequisite for observing significant quantum effects of multiple-vibration systems. Here we propose how to realize a large amplification in the net-refrigeration rates based on cavity optomechanics and to largely improve the cooling performance of multivibration modes beyond the resolved-sideband regime. By employing an auxiliary mechanical coupling (AMC) between two mechanical vibrations, the dark mode, which is induced by the coupling of these vibrational modes to a common optical mode and cuts off cooling chann…
In the solid-state circuit QED system and based on the homodyne measurement in dispersive regime, we demonstrate that a homodyne-current-based feedback can create and stabilize highly entangled two-qubit states in the presence of moderate noisy environment. Particularly, we present an extended analysis for the current-based Markovian feedback, which leads to an improved filtered-current-based feedback scheme. We show that this is essential for us to achieve the desirable control effect in present system.
Results on the heating and the parametric feedback cooling of an optically trapped anisotropic nanoparticle in the laser shot noise dominant regime are presented. The related dynamical parameters, such as the oscillating frequency and shot noise heating rate, depend on the shape of the trapped particle. For an ellipsoidal particle, the ratio of the axis lengths and the overall size controls the shot noise heating rate relative to the frequency. For a particle with smaller ellipticity or bigger size, the relative heating rate for rotation tends to be smaller than that for translation indicating…
We model the cooling of open optical and optomechanical resonators via optical feedback in the Linear Quadratic Gaussian setting of stochastic control theory. We show that coherent feedback control schemes, in which the resonator is embedded in an interferometer to achieve all-optical feedback, can outperform the best possible measurement-based schemes in the quantum regime of low steady-state excitation number. Such performance gains are attributed to the coherent controller's ability to process non-commuting output field quadratures simultaneously without loss of fidelity, and may provide im…
The theory of quantum feedback networks has recently been developed with the aim of showing how quantum input-output components may be connected together so as to control, stabilize or enhance the performance of one of the subcomponents. In this paper we show how the degree to which an idealized component (a degenerate parametric amplifier in the strong-coupling regime) can squeeze input fields may be enhanced by placing the component in-loop in a simple feedback mechanism involving a beam splitter. We study the spectral properties of output fields, placing particular emphasis on the elastic a…
Counterintuitively, experiments show that an electromagnetically levitated particle escapes from its trap when the ambient pressure is reduced below a certain level even if the particle's motion is cooled by a resonator-based or feedback-based mechanism. Here, we theoretically show that the ambient pressure must be kept well above a critical value arising from gradient force fluctuations (viz., fluctuations in part of the EM force whose Hamiltonian is quadratic in the position of the particle). Also, we consider other force fluctuations, and determine whether different realizations of feedback…
We revisit quantum Maxwell demon in thermodynamic feedback cycle in the steady-state regime. We derive a generalized version of the Clausius inequality for a finite-time steady feedback cycle with a single heat bath. It is shown to be tighter than previously known ones, and allows us to clarify that feedback control is necessary to violate the standard Clausius inequality.
We experimentally demonstrate real-time feedback control of the joint spin-state of two neutral Caesium atoms inside a high finesse optical cavity. The quantum states are discriminated by their different cavity transmission levels. A Bayesian update formalism is used to estimate state occupation probabilities as well as transition rates. We stabilize the balanced two-atom mixed state, which is deterministically inaccessible, via feedback control and find very good agreement with Monte-Carlo simulations. On average, the feedback loops achieves near optimal conditions by steering the system to t…
We revisit the problem of preparing a mechanical oscillator in the vicinity of its quantum-mechanical ground state by means of feedback cooling based on continuous optical detection of the oscillator position. In the parameter regime relevant to ground state cooling, the optical back-action and imprecision noise set the bottleneck of achievable cooling and must be carefully balanced. This can be achieved by adapting the phase of the local oscillator in the homodyne detection realizing a so-called variational measurement. The trade-off between accurate position measurement and minimal disturban…
We propose and investigate a scheme for engineering a synthetic thermal bath for a bosonic quantum gas in a one-dimensional optical lattice based on Markovian feedback control. The performance of our scheme is quantified by the fidelity between the steady state of the system and the effective thermal state. For double-well and triple-well systems with non-interacting particles, the steady state is found to be an exact thermal state, which is attributed to the fact that the transfer rates between all pairs of coupled eigenstates satisfy detailed balance condition. The scenario changes when ther…
We apply the time-delayed Pyragas control scheme to the dissipative Dicke model via a modulation of the atom-field-coupling. The feedback creates an infinite sequence of non-equilibrium phases with fixed points and limit cycles in the primary superradiant regime. We analyse this Hopf bifurcation scenario as a function of delay time and feedback strength, and determine analytical conditions for the phase boundaries.
It is challenging to implement genuine free running single photon detectors for the 1550 nm wavelength range with simultaneously high detection efficiency (DE), low dark noise, and good time resolution. We report a novel read out system for the signals from a negative feedback avalanche diode (NFAD) which allows useful operation of these devices at a temperature of 193 K and results in very low dark counts (~100 CPS), good time jitter (~30 ps), and good DE (~10%). We characterized two NFADs with a time correlation method using photons generated from weak coherent pulses (WCP) and photon pairs …
