Section: Scientific Foundations

System theory for quantum and quantum-like systems

Quantization of waves propagation in transmission-line networks & Inverse scattering

Linear stationary waves. Our main example of classical system that is interesting to see as a quantum-like system is the Telegrapher Equation, a model of transmission lines, possibly connected into a network. This is the standard model for electrical networks, where $V$ and $I$ are the voltage and intensity functions of $z$ and $k$, the position and frequency and $R\left(z\right)$, $L\left(z\right)$, $C\left(z\right)$, $G\left(z\right)$ are the characteristics of the line:

Since the work of Noordergraaf [101] , this model is also used for hemodynamic networks with $V$ and $I$ respectively the blood pressure and flow in vessels considered as 1D media, and with $R=\frac{8\pi \eta }{S^2}$, $L=\frac{\rho }{S}$, $C=\frac{3S(r+h)}{E(2r+h)}$ where $\rho$ and $\eta$ are the density and viscosity of the blood ; $r$, $h$ and $E$ are the inner radius, thickness and Young modulus of the vessel. $S=\pi r^2$. The conductivity $G$ is a small constant for blood flow.

Monitoring such networks is leading us to consider the following inverse problem: get information on the functions $R$, $L$, $C$, $G$ from the reflection coefficient $ℛ\left(k\right)$ (ratio of reflected over direct waves) measured in some location by Time or Frequency Domain Reflectometry.

To study this problem it is convenient to use a Liouville transform, setting $x\left(z\right)={\int }_0^z\sqrt{L\left(z^{\text{'}}\right)C\left(z^{\text{'}}\right)}d{z}^{\text{'}}$, to introduce auxiliary functions ${\displaystyle q^{\pm }\left(x\right)=\frac{1}{4}\frac{d}{x}\left(ln\frac{L\left(x\right)}{C\left(x\right)}\right)\pm \frac{1}{2}\left(\frac{R\left(x\right)}{L\left(x\right)}-\frac{G\left(x\right)}{C\left(x\right)}\right)}$ and ${\displaystyle q_p\left(x\right)=\frac{1}{2}\left(\frac{R\left(x\right)}{L\left(x\right)}+\frac{G\left(x\right)}{C\left(x\right)}\right)}$, so that (2 ) becomes a Zakharov-Shabat system [89] that reduces to a Schrödinger equation in the lossless case ($R=G=0$):

and ${\displaystyle I(x,k)=\frac{1}{\sqrt{2}}{\left[\frac{C\left(x\right)}{L\left(x\right)}\right]}^{\frac{1}{4}}(v_1(x,k)+v_2(x,k))}$, ${\displaystyle V(x,k)=-\frac{1}{\sqrt{2}}{\left[\frac{L\left(x\right)}{C\left(x\right)}\right]}^{\frac{1}{4}}(v_1(x,k)-v_2(x,k))}$.

Our inverse problem becomes now an inverse scattering problem for a Zakharov-Shabat (or Schrödinger) equation: find the potentials $q^{\pm }$ and $q_p$ corresponding to $ℛ$. This classical problem of mathematical physics can be solved using e.g. the Gelfand-Levitan-Marchenko method.

Nonlinear traveling waves. In some recent publications  [92] , [91] , we use scattering theory to analyze a measured Arterial Blood Pressure (ABP) signal. Following a suggestion made in  [103] , a Korteweg-de Vries equation (KdV) is used as a physical model of the arterial flow during the pulse transit time. The signal analysis is based on the use of the Lax formalism: the iso-spectral property of the KdV flow allows to associate a constant spectrum to the non stationary signal. Let the non-dimensionalized KdV equation be

In the Lax formalism, $y$ is associated to a Lax pair: a Schrödinger operator, ${\displaystyle L\left(y\right)=-\frac{{\partial }^2}{\partial x^2}+y}$ and an anti-Hermitian operator ${\displaystyle M\left(y\right)=-4\frac{{\partial }^3}{\partial x^3}+3y\frac{\partial }{\partial x}+3\frac{\partial }{\partial x}y}$. The signal $y$ is playing here the role of the potential of $L\left(y\right)$ and is given by an operator equation equivalent to (4 ):

Scattering and inverse scattering transforms can be used to analyze $y$ in term of the spectrum of $L\left(y\right)$ and conversely. The “bound states” of $L\left(y\right)$ are of particular interest: if $L\left(y\right)$ is solution of (5 ) and $L\left(y\right(t\left)\right)$ has only bound states (no continuous spectrum), then this property is true at each time and $y$ is a soliton of KdV. For example the arterial pulse pressure is close to a soliton [86] , [14] .

Inverse scattering as a generalized Fourier transform. For “pulse-shaped” signals $y$, meaning that $y\in L^1(ℝ;(1+{\left|x\right|}^2\left)dx\right)$, the squared eigenfunctions of $L\left(y\right)$ and their space derivatives are a basis in $L^1(ℝ;dx)$ (see e.g. [99] ) and we use this property to analyze signals. Remark that the Fourier transform corresponds to using the basis associated with $L\left(0\right)$. The expression of a signal $y$ in its associated basis is of particular interest. For a positive signal (as e.g. the arterial pressure), it is convenient to use $L(-y)$ as $-y$ is like a multi-well potential, and the Inverse scattering transform formula becomes:

where ${\psi }_n$ and $f(k,.)$ are solutions of $L(-y)f=k^{2}f$ with $k=i{\kappa }_n$, ${\kappa }_n>0$, for ${\psi }_n$ (bound states) and $k>0$ for $f(k,.)$ (Jost solutions). The discrete part of this expression is easy to compute and provides useful informations on $y$ in applications. The case $ℛ=0$ ($-y$ is a reflectionless potential) is then of particular interest as $2N$ parameters are sufficient to represent the signal. We investigate in particular approximation of pulse-shaped signals by such potentials corresponding to N-solitons.

Identification & control of quantum systems

Interesting applications for quantum control have motivated seminal studies in such wide-ranging fields as chemistry, metrology, optical networking and computer science. In chemistry, the ability of coherent light to manipulate molecular systems at the quantum scale has been demonstrated both theoretically and experimentally  [98] . In computer science, first generations of quantum logical gates (restrictive in fidelity) has been constructed using trapped ions controlled by laser fields (see e.g. the “Quantum Optics and Spectroscopy Group, Univ. Innsbruck”). All these advances and demands for more faithful algorithms for manipulating the quantum particles are driving the theoretical and experimental research towards the development of new control techniques adapted to these particular systems. A very restrictive property, particular to the quantum systems, is due to the destructive behavior of the measurement concept. One can not measure a quantum system without interfering and perturbing the system in a non-negligible manner.

Quantum decoherence (environmentally induced dissipations) is the main obstacle for improving the existing algorithms  [88] . Two approaches can be considered for this aim: first, to consider more resistant systems with respect to this quantum decoherence and developing faithful methods to manipulate the system in the time constants where the decoherence can not show up (in particular one can not consider the back-action of the measurement tool on the system); second, to consider dissipative models where the decoherence is also included and to develop control designs that best confronts the dissipative effects.

In the first direction, we consider the Schrödinger equation where $\Psi (t,x)$, $-\frac{1}{2}\Delta$, $V$, $\mu$ and $u\left(t\right)$ respectively represent the wavefunction, the kinetic energy operator, the internal potential, the dipole moment and the laser amplitude (control field):

While the finite dimensional approximations ($\Psi \left(t\right)\in {ℂ}^N$) have been very well studied (see e.g. the works by H. Rabitz, G. Turinici, ...), the infinite dimensional case ($\Psi (t,.)\in L^2({ℝ}^N;ℂ)$) remains fairly open. Some partial results on the controllability and the control strategies for such kind of systems in particular test cases have already been provided  [79] , [80] , [94] . As a first direction, in collaboration with K. Beauchard (CNRS, ENS Cachan) et J-M Coron (Paris-sud), we aim to extend the existing ideas to more general and interesting cases. We will consider in particular, the extension of the Lyapunov-based techniques developed in  [95] , [81] , [94] . Some technical problems, like the pre-compactness of the trajectories in relevant functional spaces, seem to be the main obstacles in this direction.

In the second direction, one needs to consider dissipative models taking the decoherence phenomena into account. Such models can be presented in the density operator language. In fact, to the Schrödinger equation (7 ), one can associate an equation in the density operator language where $\rho =\Psi {\Psi }^{*}$ represents the projection operator on the wavefunction $\Psi$ ($[A,B]=AB-BA$ is the commutator of the operators $A$ and $B$):

Whenever, we consider a quantum system in its environment with the quantum jumps induced by the vacuum fluctuations, we need to add the dissipative effect due to these stochastic jumps. Note that at this level, one also can consider a measurement tool as a part of the environment. The outputs being partial and not giving complete information about the state of the system (Heisenberg uncertainty principle), we consider a so-called quantum filtering equation in order to model the conditional evolution of the system. Whenever the measurement tool composes the only (or the only non-negligible) source of decoherence, this filter equation admits the following form:

where $L$ is the so-called Lindblad operator associated to the measurement, $0<\eta \le 1$ is the detector's efficiency and where the Wiener process $W_t$ corresponds to the system output $Y_t$ via the relation $d{W}_t=d{Y}_t-\text{Tr}\left[(L+L^{*}){\rho }_t\right]dt$. This filter equation, initially introduced by Belavkin  [82] , is the quantum analogous of a Kushner-Stratonovic equation. In collaboration with H. Mabuchi and his co-workers (Physics department, Caltech), we would like to investigate the derivation and the stochastic control of such filtering equations for different settings coming from different experiments  [96] .

Finally, as a dual to the control problem, physicists and chemists are also interested in the parameter identification for these quantum systems. Observing different physical observables for different choices of the input $u\left(t\right)$, they hope to derive more precise information about the unknown parameters of the system being parts of the internal Hamiltonian or the dipole moment. In collaboration with C. Le Bris (Ecole des ponts and Inria), G. Turinici (Paris Dauphine and Inria), P. Rouchon (Ecole des Mines) and H. Rabitz (Chemistry department, Princeton), we would like to propose new methods coming from the systems theory and well-adapted to this particular context. A first theoretical identifiability result has been proposed  [93] . Moreover, a first observer-based identification algorithm is under study.