Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We present a quantum theory of feedback in which the homodyne photocurrent alters the dynamics of the source cavity. To the nonlinear stochastic (Ito) evolution of the conditioned system state we add a feedback term linear in the instantaneous stochastic (Stratonovich) photocurrent. Averaging over the photocurrent gives a feedback master equation which has the desired driftlike term, plus a diffusionlike term. We apply the model to phase locking a regularly pumped laser, and show that under ideal conditions the noise spectra of the output light exhibit perfect squeezing on resonance. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We present a formulation of feedback in quantum systems in which the best estimates of the dynamical variables are obtained continuously from the measurement record, and fed back to control the system. We apply this method to the problem of cooling and confining a single quantum degree of freedom, and compare it to current schemes in which the measurement signal is fed back directly in the manner usually considered in existing treatments of quantum feedback. Direct feedback may be combined with feedback by estimation, and the resulting combination, performed on a linear system, is closely analogous to classical LQG control theory with residual feedback. <s> BIB002 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We solve exactly the non-Markovian dynamics of a cavity mode in the presence of a feedback loop based on homodyne measurements, in the case of a nonzero feedback delay time. With an appropriate choice of the feedback parameters, this scheme is able to significantly increase the decoherence time of the cavity mode, even for delay times not much smaller than the decoherence time itself. <s> BIB003 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> In the conventional picture of quantum feedback control, sensors perform measurements on the system, a classical controller processes the results of the measurements, and actuators supply semiclassical potentials to alter the behavior of the quantum system. In this picture, the sensors tend to destroy coherence in the process of making measurements, and although the controller can use the actuators to act coherently on the quantum system, it is processing and feeding back classical information. This paper proposes an alternative method for quantum feedback control, in which the sensors, controller, and actuators are quantum systems that interact coherently with the system to be controlled. In this picture, the controller gets, processes, and feeds back quantum information. Controllers that operate using such quantum feedback loops can perform tasks such as entanglement transfer that are not possible using classical feedback. Necessary and sufficient conditions are presented for Hamiltonian quantum systems to be controllable and observable using both classical and quantum feedback. <s> BIB004 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> Unit-efficiency homodyne detection of the resonance fluorescence of a two-level atom collapses the quantum state of the atom to a stochastically moving point on the Bloch sphere. Recently, Hofmann, Mahler, and Hess [Phys. Rev. A 57, 4877 (1998)] showed that by making part of the coherent driving proportional to the homodyne photocurrent one can stabilize the state to any point on the bottom-half of the sphere. Here we reanalyze their proposal using the technique of stochastic master equations, allowing their results to be generalized in two ways. First, we show that any point on the upper- or lower-half, but not the equator, of the sphere may be stabilized. Second, we consider nonunit-efficiency detection, and quantify the effectiveness of the feedback by calculating the maximal purity obtainable in any particular direction in Bloch space. <s> BIB005 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We describe a protocol for continuously protecting unknown quantum states from decoherence that incorporates design principles from both quantum error correction and quantum feedback control. Our protocol uses continuous measurements and Hamiltonian operations, which are weaker control tools than are typically assumed for quantum error correction. We develop a cost function appropriate for unknown quantum states and use it to optimize our state-estimate feedback. Using Monte Carlo simulations, we study our protocol for the three-qubit bit-flip code in detail and demonstrate that it can improve the fidelity of quantum states beyond what is achievable using quantum error correction when the time between quantum error-correction cycles is limited. <s> BIB006 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time tau in the feedback circuit. Because a two-level atom is a non-linear optical system, an analytical solution is not possible. However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. We find the effect of the time delay to be qualitatively similar to chat of inefficient detection. The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. In the case where one tries to stabilize the atom in the excited state, an approximate analytical solution to the quantum trajectory is possible. The result, that the purity (P = 2Tr[rho (2)] - 1) of the average state is given by P = 1 - 4y tau (where gamma is the spontaneous emission rate) is found to agree very well with the numerical results. (C) 2001 Elsevier Science B.V. All rights reserved. <s> BIB007 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We compare two different approaches to the control of the dynamics of a continuously monitored open quantum system. The first is Markovian feedback, as introduced in quantum optics by Wiseman and Milburn [Phys. Rev. Lett. 70, 548 (1993)]. The second is feedback based on an estimate of the system state, developed recently by Doherty and Jacobs [Phys. Rev. A 60, 2700 (1999)]. Here we choose to call it, for brevity, Bayesian feedback. For systems with nonlinear dynamics, we expect these two methods of feedback control to give markedly different results. The simplest possible nonlinear system is a driven and damped two-level atom, so we choose this as our model system. The monitoring is taken to be homodyne detection of the atomic fluorescence, and the control is by modulating the driving. The aim of the feedback in both cases is to stabilize the internal state of the atom as close as possible to an arbitrarily chosen pure state, in the presence of inefficient detection and other forms of decoherence. Our results (obtained without recourse to stochastic simulations) prove that Bayesian feedback is never inferior, and is usually superior, to Markovian feedback. However, it would be far more difficult to implement than Markovian feedback and it loses its superiority when obvious simplifying approximations are made. It is thus not clear which form of feedback would be better in the face of inevitable experimental imperfections. <s> BIB008 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We have studied theoretically the basic operation of a quantum feedback loop designed to maintain a desired phase of quantum coherent oscillations in a single solid-state qubit. The degree of oscillations synchronization with external harmonic signal is calculated as a function of feedback strength, taking into account available bandwidth and coupling to environment. The feedback can efficiently suppress the dephasing of oscillations if the qubit coupling to the detector is stronger than the coupling to the environment. <s> BIB009 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We show that quantum feedback control can be used as a quantum error correction process for errors induced by weak continuous measurement. In particular, when the error model is restricted to one, perfectly measured, error channel per physical qubit, quantum feedback can act to perfectly protect a stabilizer codespace. Using the stabilizer formalism we derive an explicit scheme, involving feedback and an additional constant Hamiltonian, to protect art (n-1)-qubit logical state encoded in n physical qubits. This works for both Poisson (jump) and white-noise (diffusion) measurement processes. As an example, we show that detected-spontaneous emission error correction with a driving Hamiltonian can greatly reduce the amount of redundancy required to protect a state from that which has been previously postulated. <s> BIB010 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We describe a scheme for quantum-error correction that employs feedback and weak measurement rather than the standard tools of projective measurement and fast controlled unitary gates. The advantage of this scheme over previous protocols [for example, Ahn et al. Phys. Rev. A 65, 042301 (2001)], is that it requires little side processing while remaining robust to measurement inefficiency, and is therefore considerably more practical. We evaluate the performance of our scheme by simulating the correction of bit flips. We also consider implementation in a solid-state quantum-computation architecture and estimate the maximal error rate that could be corrected with current technology. <s> BIB011 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> The goal of this article is to provide a largely self-contained introduction to the modelling of controlled quantum systems under continuous observation, and to the design of feedback controls that prepare particular quantum states. We describe a bottom-up approach, where a field-theoretic model is subjected to statistical inference and is ultimately controlled. As an example, the formalism is applied to a highly idealized interaction of an atomic ensemble with an optical field. Our aim is to provide a unified outline for the modelling, from first principles, of realistic experiments in quantum control. <s> BIB012 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained. <s> BIB013 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as"coherent feedback control.'' It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones. <s> BIB014 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator. <s> BIB015 </s> Quantum control theory and applications: A survey <s> Feedback stabilization/control <s> I present an experimental realization of a coherent-feedback control system that was recently proposed for testing basic principles of linear quantum stochastic control theory [M. R. James, H. I. Nurdin and I. R. Petersen, to appear in IEEE Transactions on Automatic Control (2008), arXiv:quant-ph/0703150v2]. For a dynamical plant consisting of an optical ring-resonator, I demonstrate ~ 7 dB broadband disturbance rejection of injected laser signals via all-optical feedback with a tailored dynamic compensator. Comparison of the results with a transfer function model pinpoints critical parameters that determine the coherent-feedback control system's performance. <s> BIB016
Based on the use of feedback information from measurements or estimation, one can design a feedback controller to control or stabilize a quantum system. In Markovian quantum feedback, the feedback Hamiltonian is commonly selected as a linear function of the feedback signal. For example, the feedback Hamiltonian in Markovian feedback control via homodyne detection may be selected as H f b (t) = F (t)I(t), where F (t) is a Hermitian operator and I(t) is a current from the measurement BIB001 , BIB008 , BIB005 . Then the feedback Hamiltonian will enter the system Hamiltonian to alter the system evolution. Markovian quantum feedback has been widely applied in the stabilization of arbitrary pure states of two-level systems BIB005 , noiseless subspace generation BIB013 , continuous quantum error correction BIB010 , etc. In Bayesian quantum feedback, the feedback Hamiltonian is a general function of the measurement record BIB002 , BIB009 , which is used to control the system dynamics. Bayesian feedback is usually superior to Markovian feedback since it uses more information. However, it is more difficult to implement Bayesian feedback than Markovian feedback due to the existence of the estimation step BIB008 . Bayesian feedback has also been applied in the preparation of quantum states BIB009 and quantum error correction BIB006 , BIB011 . If the feedback delay cannot be ignored, the feedback Hamiltonian must include a delay parameter BIB003 which in some cases has a qualitatively similar effect to that of unprecise measurement BIB007 . Both Markovian feedback and Bayesian feedback use information from measurement (or estimation) and the feedback controller is usually a classical controller. Another different feedback paradigm is coherent quantum feedback, where a quantum controller (i.e., the controller is another quantum system) directly affects the system dynamics to be controlled BIB004 , BIB016 , BIB014 . In quantum feedback control, the filtering SME is an important dynamical equation for the design of the feedback control system. For example, for a quantum system described by an SME, one may use stochastic Lyapunov techniques to design a feedback control law for asymptotically stabilizing the quantum system to an objective state. For the model BIB012 , if S = {ρ ∈ C N ×N : ρ = ρ † , trρ = 1, ρ ≥ 0}, we have the following result BIB015 : Theorem 10 Consider the system (16) evolving in the set S. Let the final state ρ f = v f v † f , where v f is an eigenstate |φ m (1 ≤ m ≤ N ), and let γ > 0. Consider the following control law: B through the boundary tr(ρρ f ) = γ, and u t = 1 otherwise. Then there exists a γ > 0 such that u t globally stabilizes (16) around ρ f and Eρ t → ρ f as t → ∞.