Estimation PrOblems for Quantum & Quantumlike systems – EPOQ2

This research project has for goal to address a class of inverse problems rising from either the emerging application domain of ``quantum engineering' or from the classical applications where a natural quantization lead to quantum-like systems. Recent theoretical and experimental achievements have shown that the quantum dynamics can be studied within the framework of estimation and control theory, but give rise to unusual models that have not been completely explored yet. The main objective of this project is to provide a rather new point of view and pathway to study these multi-disciplinary subjects. This will be done by emphasizing on the control theoretical viewpoint (which has been rather absent in the previous efforts) and through a tight collaboration between researchers with various specialties in control engineering, mathematical analysis, theoretical and experimental physics and chemistry. Here, the problems and the applications in study are classified through 3 main themes. 1. Open-loop Hamiltonian identification and quantum chemical synthesis of molecular systems: Laboratory spectroscopic, dynamics, and kinetics data are known to be rich in information about the underlying forces between atoms and molecules. Knowledge of these forces is fundamental for all physical-chemical processes. Until now there has been no systematic means to extract the desired information from the laboratory data. The majority of previous efforts by chemists, physicists and numerical analysts have been concentrated around the optimization and statistical techniques. The main issues in most of the developed algorithms concern the existence of numerous local minima's, robustness with respect to laboratory noises, and the heavy cost of computations. In a rather new approach for this context, we propose to apply adaptive methods and nonlinear observers to address such estimation problems. Such strategy should improve the inversion quality from the viewpoints of robustness and computation cost. 2. Closed-loop parameter estimation and applications in quantum information and metrology: Our research on this subject regards two estimation problems of close natures. The first one with application in quantum information deals with the closed-loop generation of entangled states and the second one with application in metrology concerns the development of real-time feedbacks for high precision spectroscopy. Reliable production of entangled states for multi-qubit systems remains one of the main obstacles to achieve a robust processing of quantum information. As a sequel to our previous theoretical work on the closed-loop stabilization of such entangled states, we are here interested in an application of adaptive filtering techniques to improve the robustness margins with respect to laboratory uncertainties. The second application concerns the real-time synchronization of a laser frequency with an unknown atomic transition frequency. We are interested in achieving optical frequency standards based on the use of single ions isolated in an ion trap. Our approach to this synchronization problem is based on the adaptation of extremum-seeking algorithm to the stochastic dynamics of a single ion. 3. Estimation problems for quantum and quantized systems on networks: This part of the project concerns a class of inverse problems on the Schrödinger-type operators defined on networks with applications going from the nano-scale transmission networks to macroscopic cabling or hemodynamic networks. Here, we are particularly interested in observation of the electric signal propagation in transmission lines and of the blood pressure waves in the hemodynamic network. In the nano-scale transmission networks, Schrödinger-type equations appear naturally on the network in study. In the macroscopic scale, the first system directly, and the second one through a linear approximation of the 1D blood flow, give rise to the so-called ``telegrapher model''. This telegrapher model, after quantization through plane waves and an appropriate Liouville transformation, leads to a Schrödinger-type equation. Two particular classes of inverse problems with applications in fault-detection/diagnostics of the transmission networks and the signal processing of the blood pressure will be investigated. The first class concerns the topology (connectivities) and the geometry (lengths) of the network's metric graph. The second class corresponds to the analytic properties of the Schrödinger-type operators on the network: through the interpretation of the potentials in the Schrödinger type operators, we obtain information on the fine properties of the transmission lines or the blood pathways. This, in particular, would be helpful for the case of partial breakdowns or arterial tree characterization. Our approach here is to propose an adaptation of the scattering analysis and the inverse scattering techniques to the cases of these particular applications.