Journées bisontines sur le contrôle quantique : systèmes d’EDPs et applications à l’IRM
|Organisateur et contact : Nabile Boussaid|
Dates : 9 Mars 2015-> 11 Mars 2015
Lieu : Laboratoire de mathématiques, Salle 316B, Bâtiment Métrologie, UFR S&T, Besançon
- Nina Amini : Interpolation approach to Hamiltonian-varying quantum systems and the adiabatic theorem
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this talk, we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure could be taken as the error of adiabatic approximation. We prove under certain conditions, this error can be precisely estimated for an arbitrarily given interpolating function. It is easy to deduce from the error estimation, whether the quantitative conditions of the adiabatic theorem are consistent for a particular variation of Hamiltonian. We find that the conventional quantitative conditions do not necessarily imply a small error, even for a small variation of the system Hamiltonian being less than a threshold value.
- Ugo Boscain : Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems
We study the controllability of a closed control-affine quantum system driven by two or more external fields. We provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the Hamiltonian in dependence of the controls seen as parameters. Such spectral condition is structurally stable in the case of three controls or in the case of two controls when the Hamiltonian is real. The spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite-dimensional case. In the
finite-dimensional case it implies that the system is Lie-bracket generating when lifted to the group of unitary transformations, and in particular that it is exactly controllable. Hence, Lie algebraic conditions are deduced from purely spectral properties. We conclude the article by proving that approximate and exact controllability are equivalent properties for general finite-dimensional quantum systems.
- Nicolas Boulant : Controlling temperature in parallel transmission and in magnetic resonance imaging at ultra-high field
Magnetic Resonance Imaging (MRI) at ultra-high field (UHF) improves the ignal-to-noise ratio and thus allows reaching in principle higher image spatial resolution. The consequent shortening of the radio-frequency (RF) wavelength on the other hand leads to severe inhomogeneities of the control field in organs such as the human brain. To mitigate these inhomogeneities, adiabatic and composite pulses have been used with success at lower fields but these techniques are limited at UHF as they require high energy deposits in the tissues. Parallel transmission (pTx), which consists in placing around the subject several RF transmitters that can be controlled independently (amplitude and phase), has been a very promising technology to solve this problem. Safety management however becomes more complex as there is an infinite number of possible interference scenarios between the different fields, each one yielding a different energy deposit distribution. Yet, taking over-conservative safety margins would clearly prevent from exploiting the full potential of the powerful MRI scanners. Within that context, and considering that temperature is the most relevant safety metric, the design of the different (interfering) control fields of a pTx system and under explicit temperature rise constraint is reported, the temperature being predicted by the Pennes’ bio-heat (partial differential equation) model.
- Nabile Boussaïd : Regular propagators of bilinear quantum systems
During this talk, I will present results on the eistence of solutions of abstract bilinear systems with Radon controls. The analysis of the regularity of the solutions in space, time and with repect to the control will give us different extensions of the famous negative result of Ball, Marsden and Slemrod.
Time permitting, I will also consider question linked to the minimal energy for the control as well as the minimal time.
- Francesca Carlotta Chittaro : Stabilisability of a family of systems with two levels
- Michel Duprez : Partial controllability of parabolic systems
- Paolo Mason : Control of the bilinear Schrödinger equation with three inputs via adiabatic methods
In this talk I will present a method to control the Schrödinger equation with three entries based on the adiabatic approximation and exploiting passages through conical intersections. I will also show that conical intersections are structurally stable and that generically each double eigenvalue of the Hamiltonian corresponds to a conical intersection in finite dimension or in infinite dimensional physically significant cases.
- Mario Sigalotti : Some remarks on zero-time controllability of finite-dimensional closed quantum systems
We start by giving a proof of the equivalence between approximate and exact controllability for closed finite-dimensional quantum systems. The proof works independenty of the type of dependence of the Hamiltonians on the control parameters. We then focus on affine-control systems, obtaining a characterization of zero-time controllability. Ongoing work on orbit foliation will also be discussed.
- Dominique Sugny : Robust optimal control for an ensemble of spins
We show on several examples from Nuclear Magnetic Resonance or Magnetic Resonance Imaging how to optimaly control an inhomogneous ensemble of spins. Some analytic and numeric solutions are presented présentées.
- Léo Van Damme : Analytic solutions for a spin ensemble control