Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Large deviations for rare realizations of dynamical systems

A central problem in uncertainty quantification is how to characterize the impact that our incomplete knowledge about models has on the predictions we make from them. This question naturally lends itself to a probabilistic formulation, by making the unknown model parameters random with given statistics. In the following this approach is used in concert with tools from large deviation theory (LDT) and optimal control to estimate the probability that some observables in a dynamical system go above a large threshold after some time, given the prior statistical information about the system’s parameters and/or its initial conditions. It is established under which conditions the extreme events occur in a predictable way, as the minimizer of the LDT action functional, i.e. the instanton. In the first physical application, the appearance of rogue waves in a long-crested deep sea is investigated. First, the leading order equations are derived for the wave statistics in the framework of wave turbulence (WT), showing that the theory cannot go beyond Gaussianity, although it remains the main tool to understand the energetic transfers. It is shown how by applying our LDT method one can use the incomplete information contained in the spectrum (with the Gaussian statistics of WT) as prior and supplement this information with the governing nonlinear dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height. The rogue wave precursors in these pockets are wave patterns of regular height but with a very specific shape that is identified explicitly, thereby potentially allowing for early detection. Finally, the first experimental evidence of hydrodynamic instantons is presented using data collected in a long wave flume, elevating the instanton description to the role of a unifying theory of extreme water waves. Other applications of the method are illustrated: To the nonlinear Schrödinger equation with random initial conditions, relevant to fiber optics and integrable turbulence, and to a rod with random elasticity pulled by a time-dependent force. The latter represents an interesting nonequilibrium statistical mechanics setup with a strongly out-of-equilibrium transient (absence of local thermodynamic equilibrium) and a small number of degrees of freedom (small system), showing how the LDT method can be exploited to solve optimal-protocol problems

Time-dependent coupled-cluster for ultrafast spectroscopy

The ultimate reason for chemical reactivity is the electronic motion, occurring at an attosecond timescale. Until the last century, it was impossible to observe it directly, as the shortest available laser pulses had duration in the order of femtoseconds. Recent technological advances lead to sub-femtosecond laser pulses, making possible real-time observation and control of electron dynamics.My Ph.D. thesis aims to develop and implement a model for the interaction between ultrashort laser pulses and molecules. This is interesting as an extension of the theory and the computational tools available, to design experiments at laser facilities, and to predict and interpret their outcomes.The theoretical framework that we have chosen is the time-dependent coupled-cluster (TDCC) theory. We have implemented our code in the eT program, which represents the first released implementation of a TDCC method.After validating our procedures by comparison with the literature, we used our code to calculate the electronic response to a pump-probe sequence of laser pulses. We performed convergence tests of parameters on the LiH. Then, we observed and interpreted the effect of the delay between pump and probe pulses on the LiF transient absorption spectrum.We extended this implementation to a time-dependent equation-of-motion coupled-cluster (TD-EOM-CC) approach with the use of a reduced basis calculated with an asymmetric band Lanczos algorithm, and within the core-valence separation (CVS) approximation. This converged to the same spectral features as the TDCC but with much lower computational times, as we showed for LiF. We observed the limits of CVS approximation: for the LiH molecule, several peaks were not correctly retrieved. Finally, we modeled the transient absorption for the glycine molecule, which is a good candidate for experimental investigations.We also modeled the electronic impulsive stimulated Raman scattering (ISXRS) population transfer induced by an ultrashort laser pulse through the TD-EOM-CC model for Ne, CO, pyrrole, and p-aminophenol and visualized through a movie the real-time evolution of the electronic density of p-aminophenol.The significance of this work lies in the development of theoretical and computational tools to be used in attochemistry: one groundbreaking application can be the direct control of electrons, which would have a big impact on many research fields, like medicine, biology, and material science

Rank-adaptive structure-preserving reduced basis methods for Hamiltonian systems

This work proposes an adaptive structure-preserving model order reduction method for finite-dimensional parametrized Hamiltonian systems modeling non-dissipative phenomena. To overcome the slowly decaying Kolmogorov width typical of transport problems, the full model is approximated on local reduced spaces that are adapted in time using dynamical low-rank approximation techniques. The reduced dynamics is prescribed by approximating the symplectic projection of the Hamiltonian vector field in the tangent space to the local reduced space. This ensures that the canonical symplectic structure of the Hamiltonian dynamics is preserved during the reduction. In addition, accurate approximations with low-rank reduced solutions are obtained by allowing the dimension of the reduced space to change during the time evolution. Whenever the quality of the reduced solution, assessed via an error indicator, is not satisfactory, the reduced basis is augmented in the parameter direction that is worst approximated by the current basis. Extensive numerical tests involving wave interactions, nonlinear transport problems, and the Vlasov equation demonstrate the superior stability properties and considerable runtime speedups of the proposed method as compared to global and traditional reduced basis approaches

Electron-nuclear correlation, singly-excited Rydberg states and electron emission asymmetry in multiphoton ionization of H2

In this thesis multiphoton ionization of molecular hydrogen is investigated by using 50 fs laser pulses with 400 nm central wavelength and a Cold Target Recoil Ion Momentum Spectrometer (COLTRIMS), known as Reaction Microscope (ReMi). It was found that singly excited Rydberg states play a dominant role in the bound ionization process. In order to examine the importance of these Rydberg states for the dissociative ionization, the electron nuclear correlation and electron emission asymmetry was studied experimentally and theoretically. The kinetic energy distribution of the ions is simulated by numerically solving the time-dependent Schrödinger equation, whereas the electron localization asymmetry is modeled with a semiclassical theory. The presented findings indicate that dissociation via the H2+ 1sσg state is much less pronounced than commonly believed. Singly-excited Rydberg states are found to play the most important role in multiphoton bound and dissociative ionization of molecular hydrogen with 400 nm photons. The second part of the thesis reports about a pump-probe measurement using a pulse shaper setup in 4f geometry. To our knowledge this is the first report about combining a pulse shaper with a ReMi. The experimental data is compared to a former pump-probe measurement that uses a Mach-Zehnder interferometer to confirm the correct operation of the pulse shaper

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Potential field theory and its applications to classical mechanical problems

Advances in many scientific fields are expected to come from work in nanotechnology. Engineering at nano-scales presents novel problems that classical mechanics cannot solve. Many engineers are uncomfortable designing at this level because classical or continuum mechanics does not apply and quantum mechanics is said to apply in a tangible way. There are unique opportunities to contribute to the design, controls, and analysis of systems that are particularly suited to mechanical engineering. Within the derivations of classical mechanics are assumptions that limit its use to bulk engineering. These assumptions are examined to determine what principles can be extended to smaller scales. To allow engineers to do their job at these scales, it is necessary to understand strength and how changing scales affects the strength of material this leads directly to sets of variables necessary for engineering at any scale. Potential field theory is an old method that is experiencing a resurgence of interest. Potential fields are used to study quantum mechanics at the atomic scale, crack and dislocation mobility at the micro-scale, and even bulk analysis. It encompasses many problems that can be formulated using partial differential equations. These series solutions are well suited for computerized numerical approximation. Because of recent advances in computational abilities, potential field theory deserves a fresh look as a candidate for multiscale modeling and as the math that binds each level together