The time-dependent Schrödinger equation (TDSE) models the quantum nature of molecular processes.  Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields. Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrödinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices. As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem. / eSSENCE

Quantum optimal control theory is applied to find optimal pulses for controlling the motion of an ion and a molecule for two different applications. Those optimal pulses enable the control of the dynamics of the system by driving the atom or the molecule from an initial state to desired states.The evolution equations obtained by means of the quantum optimal control theory are resolved iteratively using a monotonic convergent algorithm. A number of simulation parameters are varied in order to get the optimal pulses including the duration of the pulses, the time step of the time grid, a penalty factor that limits the maximal intensity of the fields, and a guess pulse which is used to start the optimal control.The optimal pulses obtained for each application are analyzed by Fourier transform, and also by looking at the time evolution of the populations that they generate in the system.The first application is the preparation of specific vibrational states of acetylene that are usually not reachable from the ground state, and that would remain unpopulated by usual spectroscopy. Relevant state energies and transition dipole moments are extracted from the experimental literature and especially from the global acetylene Hamiltonian conferring an uncommon precision to the control simulation. The control starts from the ground state. The target states belongs to the polyad Ns=1, Nr=5 of acetylene which includes two vibrational dark states and one vibrational bright state. First, the simulation is performed with the Schrödinger equation and in a second step, with the Liouville--von Neumann equation, as mixed states are prepared. Indeed, the control starts from a Boltzmann distribution of population in the rotational levels of the vibrational ground state chosen in order to simulate an experimental condition. But the distribution is truncated to limit the computational effort. One of the dark states appears to be a potential target for a realistic experimental investigation because the average population of the Rabi oscillation remains high and decoherence is expected to be weak. The optimal pulses obtained have a high fidelity, have a spectrum with well-resolved peak frequencies, and their experimental feasibility seems achievable within the current abilities of experimental laboratories.The second application is to propose an experimental realization of a microscopic physical device able to simulate quantum dynamics. The idea is to use the motional states of a Cd^+ ion trapped in an anharmonic potential to realize a quantum dynamics simulator of a single-particle Schrödinger equation. In this way, the motional states store the information and the optimal pulse manipulates this information to realize operations. In the present case, the simulated dynamics was the propagation of a wave packet in a harmonic potential. Starting from an initial quantum state, the pulse acts on the system to modify the motional states of the ion in such a way that the final superposition of motional states corresponds to the results of the dynamics. This simulation is performed with the Liouville--von Neumann equation and also with the Lindblad equation as dissipation is included to test the robustness of the pulse against perturbations of the potential. The optimal pulses that are obtained have a high fidelity which shows that the ion trap system has correctly realized the quantum dynamics simulation. The optimal pulses are valid for any initial condition if the potential of the simulation or the mass of the propagated wave packet is unchanged. / La théorie du contrôle optimal quantique est utilisée pour trouver des impulsions optimales permettant de contrôler la dynamique d'un atome et d'une molécule les menant d'un état initial à un état final. Les équations d'évolution obtenues grâce au contrôle optimal limitent l'intensité maximale de l'impulsion et sont résolues itérativement grâce à l'algorithme de Zhu--Rabitz. Le contrôle optimal est utilisé pour réaliser deux objectifs. Le premier est la préparation d'états vibrationnels de l'acétylène qui sont généralement inaccessibles par transition au départ de l'état vibrationnel fondamental. Ces états, appelés états sombres, sont les états cibles de la simulation. Ils appartiennent à la polyade Ns=1, Nr=5 de l'acétylène qui en contient deux ainsi qu'un état, dit brillant, qui lui est accessible depuis l'état fondamental. Les énergies des états du système et les moments de transitions dipolaires sont déterminés à partir d'un Hamiltonien très précis qui confère une précision inhabituelle à la simulation. Un des états sombres apparaît être un candidat potentiel pour une réalisation expérimentale car la population moyenne de cet état reste élevée après l'application de l'impulsion.Les niveaux rotationnels des états vibrationnels sont également pris en compte.Les impulsions optimales obtenues ont une fidélité élevée et leur spectre en fréquence présente des pics résolus.Le deuxième objectif est de proposer la réalisation expérimentale d'un dispositif microscopique capable de simuler une dynamique quantique. Ce travail montre qu'on peut utiliser les états de mouvement d'un ion de Cd^+ piégé dans un potentiel anharmonique pour réaliser la propagation d'un paquet d'onde dans un potentiel harmonique. Ce dispositif stocke l'information de la dynamique simulée grâce aux états de mouvements et l'impulsion optimale manipule l'information pour réaliser les propagations. En effet, démarrant d'un état quantique initial, l'impulsion agit sur le système en modifiant les états de mouvements de l'ion de telle sorte que la superposition finale des états de mouvements corresponde aux résultats de la dynamique. De la dissipation est incluse pour tester la robustesse de l'impulsion face à des perturbations du potentiel anharmonique. Les impulsions optimales obtenues ont une fidélité élevée ce qui montre que le système a correctement réalisé la simulation de dynamique quantique. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished