Links
Professor John GoughChair in Mathematics Specialist Subjects: Mathematical Physics; Open Quantum Systems; Quantum Filtering Control & Information; Classical and Quantum ProbabilityQuantum Open Systems & Quantum ControlJohn Gough has taken up the established chair in Mathematics at Aberystwyth and is setting up a research group in the area of quantum open systems and quantum control with Rolf Gohm. This involves applications of classical and quantum probability to model the measurement, estimation and control of quantum systems. Whilst mathematical rigorous approaches to open quantum systems date back to the 1970�s, the field of quantum control has emerged only in relatively recent times. Driving this is the fact that quantum control has now become experimentally realizable, and has also started to enter into mainstream engineering research. It deals with the problem of estimation and control of quantum mechanical systems, and significantly it is now developing in conjunction with laboratory-based experiments. The field is inherently interdisciplinary, bringing together expertise from mathematical, theoretical and experimental physics as well as from the mathematical theory for communication and control engineering. The open systems approach provides the mathematical framework for the formulation and analysis of fundamentals, in particular, it allows a convenient description of procedures, familiar from an engineering perspective, but that would be difficult to formulate within the traditional (closed) quantum theory. The picture below shows a setup that would be relatively straightforward to a control engineer, however, we wish to work within the quantum domain: the plant is a quantum mechanical system, a Boson input field (say light modeled as a quantum stochastic process) interacts with the system; the output field is measured by a detector and the recorded output (photo-current) is passed through a filter and then to an actuator. The measurement process must necessarily be non-demolition.
The familiar Schrödinger equation for the unitary U=U(t) describing a closed dynamical system, governed by a self-adjoint Hamiltonian H, is dU(t) = -iHU(t) dt. A generalization for open systems is given by the quantum stochastic differential equation (QSDE)
For the choice of S unitary, L arbitrary and H again self-adjoint, the process U determines a unitary adapted process. Taking A to be the input field and measuring output
where Recent research has focused on (Markov) approximation schemes to the unitary QSDEs: in joint work with Dr Ramon van Handel (CalTech), the correct form of the coefficients (S, L, H) occurring in adiabatic elimination problems was derived and shown to differ from those appearing in the standard physics literature. In collaboration with Slava Belavkin (Nottingham) the Hamiltonian-Jacobi-Bellman theory relating to optimal control of quantum systems. In collaboration with Matthew James and Masahiro Yanagisawa (ANU, Canberra), the general theory of quantum feedforward and feedback networks is being investigated: for example, the form of coefficients (S, L, H) for an input/output system with feedback is given below. This work extends the transfer function approach of Yanagisawa and Kimura to non-linear couplings, and to scattering
The future research within the group will focus on quantum feedback control and some of the projects to be addressed are the general theory of quantum feedback, stability of open dynamical systems, robustness, Lyapunov control, H-infinity control, dissipative analysis, geometric and algebraic methods, etc., all of which require special treatment in the quantum domain. |
BSc, MSc, PhD (NUI, Dublin)
Mathematical and Physical Sciences Tel: 01970 622755 Fax: 01970 622826 Email: jug@aber.ac.uk
|
, leads to the filtered state 
at time t satisfying the quantum filtering equation (stochastic Schrödinger equation)
, 
and B = B(t) is the stochastic process given by 
, B(0)=0. The process B has the law of a Wiener (Brownian motion) process and corresponds to the classical notion of an innovations process from classical communications theory. The filtered state is the conditional state based on the observations of the process Y up to time t (equivalently, continuous measurement of the observable 
of the plant) and is optimal in the least squares sense of estimation. The combination of the filter and actuator gives a controller that can then act on the plant. Filtering based on Poisson processes - counting quanta! - is also possible.
.
| Partner Institutions |
|
| Supported by |  |