This paper is concerned with the analysis of linear quantum optical networks. It provides a systematic approach to the construction a model for a given quantum network in terms of a system of quantum stochastic differential equations. This corresponds to a classical state space model. The linear quantum optical networks under consideration consist of interconnections between optical cavities, optical squeezers, and beamsplitters. These models can then be used in the design of quantum feedback control systems for these networks.
This paper is concerned with the analysis of linear quantum optical networks. It provides a systematic approach to the construction a model for a given quantum network in terms of a system of quantum stochastic differential equations. This corresponds to a classical state space model. The linear quantum optical networks under consideration consist of interconnections between optical cavities, optical squeezers, and beamsplitters. These models can then be used in the design of quantum feedback control systems for these networks.
This paper surveys some recent results on the theory of quantum linear systems and presents them within a unified framework.Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs).Such systems commonly arise in the area of quantum optics and related disciplines.Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems.With advances in quantum technology, the feedback control of such quantum systems is generating new challenges in the field of control theory.Potential applications of such quantum feedback control systems include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits.A recently emerging approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system.This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation.However, the design of coherent quantum feedback controllers remains a major challenge.This paper discusses recent results concerning the synthesis of H-infinity optimal controllers for linear quantum systems in the coherent control case.An important issue which arises both in the modelling of linear quantum systems and in the synthesis of linear coherent quantum controllers is the issue of physical realizability.This issue relates to the property of whether a given set of QSDEs corresponds to a physical quantum system satisfying the laws of quantum mechanics.The paper will cover recent results relating the question of physical realizability to notions occurring in linear systems theory such as lossless bounded real systems and dual J-J unitary systems.
In the last decade, there has been a growing interest in the integration of photonic components into VLSI circuits that in the long term may drive the need for systematic mixed electronic-photonic circuit design methods. On the other hand, theoretical and experimental investigations of coherent-feedback quantum control, the feedback control of a quantum system with another quantum system, have led to the development of a network synthesis theory for linear quantum stochastic models that are commonly employed in (linear) quantum optics, modelling such devices as optical cavities and optical parametric amplifiers. This paper makes theoretical contributions towards connecting these two independent developments, and anticipates potential future interest in systematic design methods for mixed linear electronic-photonic circuits, by establishing a network synthesis theory for linear stochastic systems with mixed quantum and classical degrees of freedom. It is shown how a physically realizable mixed quantum-classical linear stochastic system can be realized as a circuit composed of a feedback interconnection of a fully quantum linear subsystem, that can be implemented by quantum optical devices, and a classical linear sub-system, that can be implemented with standard electrical and electronic devices, together with appropriate interfaces that convert quantum signals to classical signals, and vice-versa. Two feedback architectures are proposed, and a decomposition lemma is derived that shows the structure of linear transformations of bosonic quantum signals into classical signals.
This paper surveys some recent results on the feedback control of quantum linear systems and the robustness properties of these systems. Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs). These systems can also be described in terms of a Hamiltonian operator H and a coupling operator L, which in the case of quantum linear systems have a specific quadratic and linear form respectively. Such systems commonly arise in the area of quantum optics and related disciplines. Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems. An important approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system. This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation. The paper discusses recent results concerning the synthesis of coherent quantum controllers such as in coherent quantum H∞ control. Another important issue in the design of quantum feedback controllers is the robustness of the closed loop quantum system to perturbations in the quantum plant dynamics arising from perturbations in the system Hamiltonian H and the system coupling operator L. The paper discusses recent robust stability results for such perturbed quantum systems which take the form of a quantum small gain theorem and a quantum Popov stability criterion. These results are usefu
This paper is concerned with the state space modelling of closed optical QED feedback control system in interacting Fock space which is shown to be a generalization of Langevin stochastic model in boson Fock space. This work includes the design of quantum networks of the feedback systems of QED cavity using two-mode beam splitter device at the input to the cavity and a single mode fractional beam splitter device at the output of the cavity in interacting mode and their stability analysis using Nyquist criterion.
The present article is concerned with the study of quantum electrodynamics (QED) in interacting Fock space and a generalised approach of modelling quantum feedback systems of the optical QED cavities. The dynamical behaviour and the state space representation in stochastic field of the optical QED system provide the basis for developing quantum feedback QED control system. This study is also motivated by the design of quantum networks of any number of optical QED baths connected in parallel. Nyquist stability of the feedback control system in the interacting Fock space using two-mode beam splitter device at the input and a single-mode fractional beam splitter device at the output of the cavity has been analysed. The degree of stability of the quantum feedback control system of the cavity QED is studied by measuring gain margin and phase margin of the system in terms of parameters of the beam splitter.
This thesis makes some theoretical contributions towards mixed quantum feedback network synthesis, quantum optical realization of classical linear stochastic systems and quantum feedback control designs. A mixed quantum-classical feedback network is an interconnected system consisting of a quantum system and a classical system connected by interfaces that convert quantum signals to classical signal (using homodyne detectors), and vice versa (using electro-optic modulators). In the area of mixed quantum-classical feedback networks, we present a network synthesis theory, which provides a natural framework for analysis and design for mixed linear systems. Physical realizability conditions are derived for linear stochastic differential equations to ensure that mixed systems can correspond to physical systems. The mixed network synthesis theory developed based on physical realizability conditions shows that how a classical of mixed quantum-classical systems described by linear stochastic differential equations can be built as a interconnection of linear quantum systems and linear classical systems using quantum optical devices as well as electrical and electric devices. However, an important practical problem for the implementation of mixed quantum-classical systems is the relatively slow speed of classical parts implemented with standard electrical and electronic devices, since a mixed system will not work correctly unless the electronic processing of classical devices is fast enough. Therefore, another interesting work is to show how classical linear stochastic systems build using electrical and electric devices can be physically implemented using quantum optical components. A complete procedure is proposed for a stable quantum linear stochastic system realizing a given stable classical linear stochastic system. The thesis explains how it may be possible to realize certain measurement feedback loops fully at the quantum level. In the area of quantum feedback control design, two numerical procedures based on extended linear matrix inequality (LMI) approach are proposed to design a coherent quantum controller in this thesis. The extended synthesis linear matrix inequalities are, in addition to new analysis tools, less conservative in comparison to the conventional counterparts since the optimization variables related to the system parameters in extended LMIs are independent of the symmetric Lyapunov matrix. These features may be useful in the optimal design of quantum optical networks. Time delays are frequently encountered in linear quantum feedback control systems such as long transmission lines between quantum plants and linear controllers, which may have an effect on the performance of closed-loop plant controller systems. Therefore, this thesis investigates the problem of linear quantum measurement-based feedback control systems subject to feedback-loop time delay described by linear stochastic differential equations. Several numerical procedures are proposed to design classical controllers that make quantum measurement-based feedback control systems with time delay stable and also guarantee that their desired control performance specifications are satisfied.