Feedback is a fundamental and well-understood method of control used since ancient times; this thesis investigates coherent feedback control in a quantum Hall regime. Quantum Feedback requires a level of isolation from measurement in order to maintain the coherence of the electron wavefunction, unlike the classical variant of feedback where measurement is commonplace as information is easily duplicated. Here we build upon a Landauer Büttiker model for electrons as flying qubits in a closed feedback arrangement which uses scattering matrices to represent and connect the elements of the feedback device to better analyse the qualities that modify feedback effects in a variety of systems. Here we will also investigate the effects of finite bias at each stage of the thesis, with a focus on what effect it has on the function of the feedback system and the extent of the effect depending on how feedback is employed.
Feedback is a fundamental and well-understood method of control used since ancient times; this thesis investigates coherent feedback control in a quantum Hall regime. Quantum Feedback requires a level of isolation from measurement in order to maintain the coherence of the electron wavefunction, unlike the classical variant of feedback where measurement is commonplace as information is easily duplicated. Here we build upon a Landauer Büttiker model for electrons as flying qubits in a closed feedback arrangement which uses scattering matrices to represent and connect the elements of the feedback device to better analyse the qualities that modify feedback effects in a variety of systems. Here we will also investigate the effects of finite bias at each stage of the thesis, with a focus on what effect it has on the function of the feedback system and the extent of the effect depending on how feedback is employed.
Abstract We consider coherent feedback control of quantum transport and focus on the application of simple controllers and the effects of a finite bias voltage. We show that simple single-parameter controllers can give rise to a range of useful effects such as amplification of changes in plant transmission, increased resolution of energy filtration, and the detection of differences between otherwise indistinguishable plants. We explore how these effects are impacted by the phase-averaging effects associated with finite bias and identify important voltage scales for the maintenance of the functionalities achieved through feedback control.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.
Feedback control in quantum transport has been predicted to give rise to several interesting effects, among them quantum state stabilization and the realization of a mesoscopic Maxwell's daemon. These results were derived under the assumption that control operations on the system are affected instantaneously after the measurement of electronic jumps through it. In this contribution, I describe how to include a delay between detection and control operation in the master equation theory of feedback-controlled quantum transport. I investigate the consequences of delay for the state stabilization and Maxwell's daemon schemes. Furthermore, I describe how delay can be used as a tool to probe coherent oscillations of electrons within a transport system and how this formalism can be used to model finite detector bandwidth.
Abstract : During the three-year performance period we have carried out a program of theoretical research on the foundations and potential applications of coherent-feedback quantum control. We have focused on potential applications in quantum-enhanced meterology and information processing, and on hardware platforms of an optical or opto-mechanical nature. The studies we have been performing begin to build a picture of how coherent feedback can provide a kind of circuit/network theory for quantum engineering, enabling rigorous analysis and numerical simulation of advanced devices and systems. We have begun to establish analytical approaches for comparing the performance of quantum-enhanced feedback control systems with classical counterparts, leveraging canonical results from classical control theory together with advanced methods for numerical simulations of proposed quantum systems. All of our results point to the interest of continued future study in this area, and of future investments in the experimental realization of the kinds of systems we propose.
A stochastic dynamical system is a system composed of many interacting components which includes stochastic behavior. Feedback control theory is designed to use information from the current state of the system in order to temper the final state toward the desired outcome. In this dissertation, I perform studies on two choices of classic and quantum dynamical systems to cover vast variety of statistical tools and methods. For classical model, I perform a study on finite-size effects at the phase transition in the Nagel-Schreckenberg traffic model as an example of non-equilibrium many body stochastic dynamical system. For quantum mechanical model, I introduce an optimal measurement-based feedback control protocol for cooling a single qubit as an example of complex system with one cell and many degrees of freedom. I examine the Nagel-Schreckenberg traffic model for a variety of maximum speeds. I show that the low density limit can be described as a dilute gas of vehicles with a repulsive core. At the transition to jamming, I observe finite-size effects in a variety of quantities describing the flow and the density correlations, but only if the maximum speed Vmax is larger than a certain value. A finite-size scaling analysis of several order parameters shows universal behavior, with scaling exponents that depend on Vmax The jamming transition at large Vmax can be viewed as the nucleation of jams in a background of freely flowing vehicles. Feedback control of quantum systems via continuous measurements involves complex nonlinear dynamics. As a result, even for a single qubit the optimal measurement for feedback control is known only in very special cases. I show here that for a broad class of noise processes, a series of compelling arguments can be applied to greatly simplify the problem of steady-state preparation of the ground-state, while loosing little in the way of optimality. Using numerical optimization to solve this simplified control problem, I obtain for the first time a non-trivial feedback protocol valid for all feedback strengths in the regime of good control. The protocol can be described relatively simply, and contains a discontinuity as a function of feedback strength.
This dissertation focuses on two connected areas: quantum computation and quantum control. Two proposals to construct a quantum computer, using nuclear magnetic resonance (NMR) and superconductivity, are introduced. We give details about the modeling, qubit realization, one and two qubit gates and measurement in the language that mathematicians can understand and fill gaps in the original literatures. Two experimental examples using liquid NMR are also presented. Then we proceed to investigate an example of quantum control, that of a magnetometer using quantum feedback. Previous research has shown that feedback makes the measurement robust to an unknown parameter, the number of atoms involved, with the assumption that the feedback is noise free. To evaluate the effect of the feedback noise, we extend the original model by an input noise term. We then compute the steady state performance of the Kalman filter for both the closed-loop and open-loop cases and retrieve the estimation error variances. The results are compared and criteria for evaluating the effects of input noise are obtained. Computations and simulations show that the level of input noise affects the measurement by changing the region where closed loop feedback is beneficial.