We consider the time-optimal control by magnetic fields of a spin 1/2 particle in a dissipative environment. This system is used as an illustrative example to show the role of singular extremals in the control of quantum systems. We analyze a simple case where the control law is explicitly determined. We experimentally implement the optimal control using techniques of nuclear magnetic resonance. To our knowledge, this is the first experimental demonstration of singular extremals in quantum systems with bounded control amplitudes.

We analyze the simultaneous time-optimal control of two-spin systems. The two non coupled spins which differ in the value of their chemical offsets are controlled by the same magnetic fields. Using an appropriate rotating frame, we restrict the study to the case of opposite shifts. We then show that the optimal solution of the inversion problem in a rotating frame is composed of a pulse sequence of maximum intensity and is similar to the optimal solution for inverting only one spin by using a nonresonant control field in the laboratory frame. An example is implemented experimentally using techniques of Nuclear Magnetic Resonance.

Magnetic Resonance Imaging has become nowadays an indispensable tool with applications ranging from medicine to material science. However, so far the physical limits of the maximum achievable experimental contrast were unknown. We introduce an approach based on principles of optimal control theory to explore these physical limits, providing a benchmark for numerically optimized robust pulse sequences which can take into account experimental imperfections. This approach is demonstrated experimentally using a model system of two spatially separated liquids corresponding to blood in its oxygenated and deoxygenated forms. Since its discovery in the forties, Nuclear Magnetic Resonance (NMR) has become a powerful tool 1,2 to study the state of matter in a variety of domains extending from biology and chemistry 3 to solid-state physics and quantum computing 4,5 . The power of NMR techniques is maybe best illustrated by medical imaging 6 , where it is possible e.g. to produce a three-dimensional picture of the human brain. NMR spectroscopy and Magnetic Resonance Imaging (MRI) involve the manipulation of nuclear spins via their interaction with magnetic fields. All experiments in liquid phase can be described in a first approach as follows. A sample is held in a strong and uniform longitudinal magnetic field denoted B 0 . The magnetization of the sample is then manipulated by a particular sequence of transverse radio-frequency magnetic pulses B 1 in order to prepare the system in a particular state. The analysis of the radio-frequency signal that is subsequently emitted by the nuclear spins leads to information about the structure of the molecule and its spatial position. One deduces from this simple description that the crucial point of this process is the initial preparation of the sample, i.e. to design a corresponding pulse sequence to reach this particular state with maximum efficiency. The maximum achievable efficiency can be determined for the transfer between well defined initial and target states 7 if relaxation effects can be neglected. In imaging applications, where relaxation forms the basis for contrast, a very large number of different strategies have been proposed and implemented so far with the rapid improvement of NMR and MRI technology 2,6 . However, there was no general approach to provide the maximum possible performance and the majority of these pulse sequences have been built on the basis of intuitive and qualitative reasonings or on inversion methods such as the Shinnar-Le Roux algorithm 8 . Note that this latter can be applied only in the case where there is no relaxation effect and radio-frequency inhomogeneity.A completely different point of view emerges if this problem is approached from an optimal control perspective. Optimal control theory was created in its modern version at the end of the 1950s with the Pontryagin Maximum Principle (PMP) [9][10][11] . Developed originally for problems in space mechanics, optimal control has become a key tool in a large spectrum of applications including eng...

We consider the time-optimal control of an ensemble of uncoupled spin 1/2 particles in the presence of relaxation and radiation damping effects, whose dynamics is governed by nonlinear equations generalizing the standard linear Bloch equations. For a single spin, the optimal control strategy can be fully characterized analytically. However, in order to take into account the inhomogeneity of the static magnetic field, an ensemble of isochromats at different frequencies must be considered. For this case, numerically optimized pulse sequences are computed and the dynamics under the corresponding optimal field is experimentally demonstrated using nuclear magnetic resonance techniques.

Here we study the optimum efficiency of the excitation of maximum quantum (MaxQ) coherence using analytical and numerical methods based on optimal control theory. The theoretical limit of the achievable MaxQ amplitude and the minimum time to achieve this limit are explored for a set of model systems consisting of up to five coupled spins. In addition to arbitrary pulse shapes, two simple pulse sequence families of practical interest are considered in the optimizations. Compared to conventional approaches, substantial gains were found both in terms of the achieved MaxQ amplitude and in pulse sequence durations. For a model system, theoretically predicted gains of a factor of three compared to the conventional pulse sequence were experimentally demonstrated. Motivated by the numerical results, also two novel analytical transfer schemes were found: Compared to conventional approaches based on non-selective pulses and delays, double-quantum coherence in two-spin systems can be created twice as fast using isotropic mixing and hard spin-selective pulses. Also it is proved that in a chain of three weakly coupled spins with the same coupling constants, triple-quantum coherence can be created in a time-optimal fashion using so-called geodesic pulses.