Section: New Results
System theory for quantum and quantum-like systems
Distributed source identification for wave equations: an observer-based approach
Participants : Marianne Chapouly, Mazyar Mirrahimi.
We consider the problem of identifying an unknown distributed source term for the wave equation inside a bounded domain. Assuming Dirichlet boundary conditions for the system, the output is given by the Neumann condition on a part of the boundary. It is a well-known result (applying observability inequality) that as soon as the observed part of the boundary satisfies the geometric control condition, the source term is identifiable. Furthermore, the minimal identifiability time corresponds to the minimal observability time. Here, we are rather interested in the practical problem of proposing an efficient inversion algorithm allowing us to identify the source in the case of minimal observation time. We propose, once again, to apply back-and-forth observer techniques to do this. However, as we are dealing with an infinite dimensional system, we have to deal with some additive problems such as the precomactness of the trajectories (in appropriate functional spaces) to ensure the convergence of the estimator. This work has led to a submitted conference publication [68] and a journal paper in preparation.
Inverse scattering for soft fault diagnosis in electric transmission lines
Participants : Michel Sorine, Qinghua Zhang.
Reflectometry is a technology studied in various domains to gather information about properties of media where waves are propagated and reflected. In electric engineering, today's advanced reflectometry methods provide an efficient solution for the diagnosis of electric transmission line hard faults (open and short circuits), but they are much less efficient for soft faults (spatially smooth variations of characteristic impedance). Studies on the relationship between the inverse scattering transform and the reflectometry technology for soft fault diagnosis have been started more than a quarter of a century ago [110] , but no real application of such methods has been reported, to our knowledge. In this work, we attempt to fill an important gap between inverse scattering transform and the practical reflectometry technology: it clarifies the relationship between the reflection coefficient measured with reflectometry instruments and the mathematical object of the same name defined in the inverse scattering theory, by reconciling finite length transmission lines with the inverse scattering transform defined on the infinite interval . The feasibility of applications of the inverse scattering transform to soft fault diagnosis is then studied by numerical simulation of lossless transmission lines affected by soft faults, and by the solution of the inverse scattering problem effectively retrieving smoothly varying characteristic impedance profiles from reflection coefficients. These results have been reported in [77] .
Modeling of electric transmission networks and multiconductor lines
Participants : Leila Djaziri, Mohamed Oumri, Michel Sorine, Qinghua Zhang.
The increasing number of electric transmission lines in modern engineering systems is amplifying the importance of the reliability of electric connections. In this context, two ANR projects, 0-DEFECT and INSCAN, have been started in 2009, both aiming at developing fault diagnosis techniques for transmission lines, with the former focusing on transmission networks, and the latter on multiconductor lines. See Sections 7.5 and 7.6 for more details about these two projects.
In order to prepare the studies on extensions of the inverse scattering transform (see also Section 6.2.2 ) to the cases of transmission networks and multiconductor lines, reduced modeling of such systems in terms of distributed RLCG parameters has been studied in two master training projects. The purpose has been to generalize the well known telegrapher's equations for a single transmission line to the considered more complex cases, and also to derive the corresponding Zakharov-Shabat equations, which are directly related to the inverse scattering transform. Numerical simulation of transmission networks has also been studied.
Some inverse scattering problems on star-shaped graphs
Participants : Filippo Visco Comandini, Mazyar Mirrahimi, Michel Sorine.
We consider some inverse scattering problems for Schrödinger operators over star-shaped graphs motivated by applications to the fault location of lossless electrical networks. We restrict ourselves to the case of minimal experimental setup consisting in measuring, at most, two reflection coefficients when an infinite homogeneous (potential-less) branch is added to the central node. First, by studying the asymptotic behavior of only one reflection coefficient in the high-frequency limit, we prove the identifiablity of the geometry of this star-shaped graph: the number of edges and their lengths. Next, we study the potential identification problem by inverse scattering, noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings (potentials with bounded H1 -norms). The main result states that, under some assumptions on the geometry of the graph, The measurement of two reflection coefficients, associated to two different sets of boundary conditions at the extremities of the tree, determines uniquely the potentials; it can be seen as a generalization of the theorem of the two boundary spectra on an interval. This work has led to a conference publication [57] and a submitted journal paper [76] .
Observer-based parameter estimation for quantum systems
Participants : Ashley Donovan, Zaki Leghtas, Mazyar Mirrahimi, Pierre Rouchon.
We had recently proposed an observer-based Hamiltonian identification algorithm for quantum systems [28] . The later paper provided a method to estimate the dipole moment matrix of a quantum system requiring the measurement of the populations on all states, which could be experimentally difficult to achieve. We propose here an extension to a 3-level quantum system, having access to the population of the ground state only. By a more adapted choice of the control field, we will show that a continuous measurement of this observable, alone, is enough to identify the field coupling parameters (dipole moment). This work has led to a conference publication [50] .
Also, through the two months stay of Ashley Donovan, we have considered the problem of the initial state reconstruction for quantum systems via continuous weak measurement. The decoherence due to the measurement induces a lost of information after a certain time-horizon; i.e. after a certain time interval the state of the system gets very near to a completely mixed state and therefore the measurement output is completely invaded by noise. This, we are interested in inversion algorithms applying the measurement output over a short time interval. Here, our approach consists in applying a recently developed back-and-forth nudging algorithm (developed by J. Blum and D. Auroux). This algorithm is based on the iterative application of two observers, one for the main system (over the time interval [0, T] ) and one for the same system in the time-reversed direction. As soon as the observer gains are well-defined so that the whole back-and-forth iteration leads to a contraction for the error dynamics, this iterative algorithm will provide a reliable estimator for the initial state of the system. This is a work in progress and must lead to publications in near future.
Quantum feedback by discrete quantum non-demolition measurements: towards on-demand generation of photon-number states
Participants : Hadis Amini, Mazyar Mirrahimi, Pierre Rouchon.
This work is done in collaboration with Michel Brune, Igor Dotsenko, Serge Haroche, Jean-Michel Raimond (Laboratoire Kastler-Brossel, ENS Paris).
We propose a quantum feedback scheme for the preparation and protection of photon number states of light trapped in a high-Q microwave cavity. A quantum non-demolition measurement of the cavity field provides information on the photon number distribution. The feedback loop is closed by injecting into the cavity a coherent pulse adjusted to increase the probability of the target photon number. In the ideal case (perfect cavity and measures), we present the feedback scheme and its detailed convergence proof through stochastic Lyapunov techniques based on super-martingales and other probabilistic arguments. The efficiency and reliability of the closed-loop state stabilization is assessed by quantum Monte-Carlo simulations. In realistic situations, we provide a quantum filter taking into account the cavity decay and the imperfections in the measurement process. Quantum Monte-Carlo simulations performed with experimental parameters illustrate convergence and protection of the photon number states. This work has led to one conference publication [52] and a journal publication [10] . This feedback algorithm is currently being integrated into the cavity QED experimental setup at Laboratoire Kastler-Brossel (ENS Paris).
