Feedback-stabilization of an arbitrary pure state of a two-level atom

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.

Quantum-state protection in cavities

We show how an initially prepared quantum state of a radiation mode in a cavity can be preserved for a long time using a feedback scheme based on the injection of appropriately prepared atoms. We present a feedback scheme both for optical cavities, which can be continuously monitored by a photodetector, and for microwave cavities, which can be monitored only indirectly via the detection of atoms that have interacted with the cavity field. We also discuss the possibility of applying these methods for decoherence control in quantum information processing.

Frequency stabilised grating feedback laser diode for atom cooling applications

The free running linewidth of an external cavity grating feedback diode laser is on the order of a few megahertz and is limited by the mechanical and acoustic vibrations of the external cavity. Such frequency fluctuations can be removed by electronic feedback. We present a hybrid stabilisation technique that uses both a Fabry-Perot confocal cavity and an atomic resonance to achieve excellent short and long term frequency stability. The system has been shown to reduce the laser linewidth of an external cavity diode laser by an order of magnitude to 140 kHz, while limiting frequency excursions to 60 kHz relative to an absolute reference over periods of several hours. The scheme also presents a simple way to frequency offset two lasers many gigahertz apart which should find a use in atom cooling experiments, where hyperfine ground-state frequency separations are often required.

Controlling chaos in a Lorenz-like system using feedback

We demonstrate that the dynamics of an autonomous chaotic laser can be controlled to a periodic or steady state under self-synchronization. In general, past the chaos threshold the dependence of the laser output on feedback applied to the pump is submerged in the Lorenz-like chaotic pulsation. However there exist specific feedback delays that stabilize the chaos to periodic behavior or even steady state. The range of control depends critically on the feedback delay time and amplitude. Our experimental results are compared with the complex Lorenz equations which show good agreement.

Controlling the decoherence of a ''meter'' via stroboscopic feedback

We propose a simple modification of the experimental scheme employed by Brune rt ni. [Phys. Rev. Lett. 79, 4887 (1996)] for the generation and detection of a Schrodinger cat state, in which the decoherence of the cat state can be significantly slowed down using an appropriate feedback.

A framework for valuing derivative securities

This paper develops a general framework for valuing a wide range of derivative securities. Rather than focusing on the stochastic process of the underlying security and developing an instantaneously-riskless hedge portfolio, we focus on the terminal distribution of the underlying security. This enables the derivative security to be valued as the weighted sum of a number of component pieces. The component pieces are simply the different payoffs that the security generates in different states of the world, and they are weighted by the probability of the particular state of the world occurring. A full set of derivations is provided. To illustrate its use, the valuation framework is applied to plain-vanilla call and put options, as well as a range of derivatives including caps, floors, collars, supershares, and digital options.

Non-Markovian homodyne-mediated feedback on a two-level atom: a quantum trajectory treatment

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.

Bayesian feedback versus Markovian feedback in a two-level atom

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.

Practical scheme for error control using feedback

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 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.

Quantum trajectories for the realistic measurement of a solid-state charge qubit

We present a new model for the continuous measurement of a coupled quantum dot charge qubit. We model the effects of a realistic measurement, namely adding noise to, and filtering, the current through the detector. This is achieved by embedding the detector in an equivalent circuit for measurement. Our aim is to describe the evolution of the qubit state conditioned on the macroscopic output of the external circuit. We achieve this by generalizing a recently developed quantum trajectory theory for realistic photodetectors [P. Warszawski, H. M. Wiseman, and H. Mabuchi, Phys. Rev. A 65, 023802 (2002)] to treat solid-state detectors. This yields stochastic equations whose (numerical) solutions are the realistic quantum trajectories of the conditioned qubit state. We derive our general theory in the context of a low transparency quantum point contact. Areas of application for our theory and its relation to previous work are discussed.

Dynamical creation of entanglement by homodyne-mediated feedback

For two two-level atoms coupled to a single Bosonic mode that is driven and heavily damped, the steady state can be entangled by resonantly driving the system [S. Schneider and G. J. Milburn, Phys. Rev. A 65, 042107 (2002)]. We present a scheme to significantly increase the steady-state entanglement by using homodyne-mediated feedback, in which the Bosonic mode is that of an electromagnetic cavity, the output of which is measured and the resulting homodyne photocurrent is used to modulate the field driving the qubits. Such feedback can increase the nonlinear response to both the decoherence process of the two-qubit system and the coherent evolution of individual qubits. We present the properties of the entangled states using the SO(3) Q function.

Optimal unravellings for feedback control in linear quantum systems

For quantum systems with linear dynamics in phase space much of classical feedback control theory applies. However, there are some questions that are sensible only for the quantum case: Given a fixed interaction between the system and the environment what is the optimal measurement on the environment for a particular control problem? We show that for a broad class of optimal (state- based) control problems ( the stationary linear-quadratic-Gaussian class), this question is a semidefinite program. Moreover, the answer also applies to Markovian (current-based) feedback.

High-fidelity measurement and quantum feedback control in circuit QED

Circuit QED is a promising solid-state quantum computing architecture. It also has excellent potential as a platform for quantum control-especially quantum feedback control-experiments. However, the current scheme for measurement in circuit QED is low efficiency and has low signal-to-noise ratio for single-shot measurements. The low quality of this measurement makes the implementation of feedback difficult, and here we propose two schemes for measurement in circuit QED architectures that can significantly improve signal-to-noise ratio and potentially achieve quantum-limited measurement. Such measurements would enable the implementation of quantum feedback protocols and we illustrate this with a simple entanglement-stabilization scheme.

An unexpected role for visual feedback in vehicle steering control

Some motor tasks can be completed, quite literally, with our eyes shut. Most people can touch their nose without looking or reach for an object after only a brief glance at its location. This distinction leads to one of the defining questions of movement control: is information gleaned prior to starting the movement sufficient to complete the task (open loop), or is feedback about the progress of the movement required (closed loop)? One task that has commanded considerable interest in the literature over the years is that of steering a vehicle, in particular lane-correction and lane-changing tasks. Recent work has suggested that this type of task can proceed in a fundamentally open loop manner [1 and 2], with feedback mainly serving to correct minor, accumulating errors. This paper reevaluates the conclusions of these studies by conducting a new set of experiments in a driving simulator. We demonstrate that, in fact, drivers rely on regular visual feedback, even during the well-practiced steering task of lane changing. Without feedback, drivers fail to initiate the return phase of the maneuver, resulting in systematic errors in final heading. The results provide new insight into the control of vehicle heading, suggesting that drivers employ a simple policy of “turn and see,” with only limited understanding of the relationship between steering angle and vehicle heading.