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

The multi Davydov-Ansatz: Apoptosis of moving Gaussian basis functions with applications to open quantum system dynamics

We utilize the multi Davydov-Ansatz, an Ansatz of the bosonic many-body wave function in terms of moving Gaussian basis functions, to illuminate several aspects of open quantum system dynamics and quantum many-body theory. By two artifices alongside the time-dependent variational principle we extract from this Ansatz, commonly considered ill-behaved and not converging, a highly stable and converging method. Its extremely favourable scaling of the numerical effort with the number of degrees of freedom facilitates exploration of the zero and non-zero temperature physics of both system and environment of open quantum systems in the strong coupling regime, even in cases where the system is laser-driven. The discovery that strongly coupling a system to an environment may, apart from the introduction of dissipation and decoherence also serve as a resource for the system has fuelled the research on strongly correlated open quantum systems. Although the advent of ultra powerful data processors enables advanced methods to tackle these systems, their explicit treatment without further assumptions remains an eminently challenging task. With the multi Davydov-Ansatz we numerically exactly calculate the dynamics of various open systems coupled strongly to an environment. In particular, we illuminate diverse aspects of laser-driven molecular dynamics in dissipative environments. Based on a rigorous investigation of the time-dependent variational principle for moving Gaussian basis functions, we systematically develop a linear algebra formulation of the system of equations of motion for the Ansatz parameters. On its basis we precisely isolate the origin of the issues related to the multi Davydov-Ansatz and solve the long-standing convergence problem of the method by a regularization termed apoptosis. We show exemplary for the ohmic and sub-ohmic Spin-Boson model that apoptosis renders the multi Davydov-Ansatz a highly stable method with an outstanding speed of convergence, suited to numerically exactly reproduce the dynamics of the model at surprisingly humble numerical effort even for strong coupling strengths. Furthermore, since they are not suited to efficiently reproduce Fock number states in many-body systems, we shed some light on possible extensions of the Gaussian basis functions in the multi Davydov-Ansatz in terms of displaced number states and in terms of squeezed states. In particular we argue that due to the emergence of an inappropriate number of equations of motion, there is no straightforward generalization of the multi Davydov-Ansatz by displaced number states. For the purpose of further optimization of the multi Davydov-Ansatz, we investigate in detail the impact on the numerical effort of different representations of an open system's environment. In particular, different frequency discretizations for given continuous spectral densities are examined with respect to the speed of convergence of the system dynamics to the continuum limit. We utilize a Windowed Fourier Transform as an a priori measure for the quality of the discretized representation of bath correlations. Furthermore, efficient representations of the environment for shifted initial conditions in general and non-zero temperature in particular are found systematically. As an alternative representation of an environment of mutually uncoupled harmonic oscillators, we investigate an environment represented in terms of a linear chain of effective modes. In this context we detail how to consistently reformulate the effective mode representation in second quantization, removing inadvertent double excitations introduced by the original formulation. We show that the alternative representation is beneficial in cases where the bath spectral density is highly structured, while for the ohmic and sub-ohmic spectral density of the Spin-Boson model it is of no advantage. Once we have identified the numerically most efficient representation of the environment, we apply the multi Davydov-Ansatz in order to illuminate several aspects of open quantum system dynamics whose investigation has previously remained occlusive. In particular, the access to the exact dynamics of the environmental degrees of freedom allows to shed light on the question for the channels through which energy can be interchanged between system and environment in the considered systems. Firstly, in a system-bath setup we survey the vibrational relaxation dynamics of deuterium dimers at a silicon surface. The investigation of the relaxation dynamics requires the quantum mechanical treatment of multiple system levels, which in turn prohibits a treatment of the environmental dynamics on a perturbative level. We demonstrate that the multi Davydov-Ansatz allows for a numerically exact calculation of the system dynamics with multiple system levels and a huge number of surface vibrations explicitly taken into account. Furthermore, due to the structure of the spectral density of the environment, the effective mode representation allows for this system to dramatically reduce the numerical effort. Secondly we shall investigate in detail the relaxation dynamics of an exciton in a one-dimensional molecular crystal. Since the strong coupling regime renders highly complicated the phonon dynamics, apoptosis turns out to be inevitably required in order to reliably converge the system dynamics. We show that the multi Davydov-Ansatz equipped with apoptosis allows for an extremely efficient calculation of the exciton and phonon dynamics, for both large hopping integrals and large molecular crystals. Furthermore we illuminate diverse aspects of laser-driven molecular dynamics in a dissipative environment. By restriction to two electronic energy levels we determine the channels through which system and environment interchange energy in the vicinity of an avoided crossing in a dissipative Landau-Zener model. In particular, we reveal that the final transition probability can be tuned by coupling to the environment for both diagonal and off-diagonal coupling. By appropriately adjusting the initial excitation of the system, the final transition probability is shown to converge to a fixed value for increasing coupling. Finally, we investigate in detail laser-induced population transfer by rapid adiabatic passage in a dissipative environment. By application of the multi Davydov-Ansatz it is shown for zero as well as for non-zero temperature that strongly coupling the system to an environment can serve as a resource for the population inversion. In particular, we shall examine how the coupling to the environment compensates for the decay channels in the system even if the laser pulse is only weakly chirped.:1. Introduction 2. Prerequisites 2.1. Harmonic oscillator basics 2.2. Canonical coherent states of the harmonic oscillator 2.3. Overcompleteness of CS and the Segal-Bargmann transformation 2.4. Density operator representation in terms of CS 2.5. Ideal squeezed states 2.6. Displaced number states 2.7. On the variational principle 3. Real time propagation with CS 3.1. Variational principle with CS 3.1.1. Gauge freedom in the vMCG Ansatz 3.1.2. Equations of motion for the vMCG Ansatz 3.2. Standard form of the linear system 3.3. Regularity of the coefficient matrix 3.3.1. Regularization in the case of vanishing coefficients 3.3.2. Apoptosis of CS 3.4. The route to Semiclassics 3.5. Variational principle with DNS and squeezed states 3.6. The multi Davydov-Ansatz 3.7. The multi Davydov-Ansatz at non-zero temperature 4. Open Quantum Systems 4.1. System-Bath Hamiltonian 4.2. The road to classical dissipation 4.3. The impact of apoptosis and regularization of the -matrix 4.3.1. Multi Davydov-Ansatz for the Quantum Rabi model 4.3.2. Multi Davydov-Ansatz and the Spin-Boson model 4.3.2.1. Spin-Boson model in the ohmic regime 4.3.2.2. Spin-Boson model in the sub-ohmic regime 4.4. The Windowed Fourier Transform 4.5. The sub-ohmic case and the problem of oversampling 4.5.1. On the polarized initial condition 4.5.2. On the treatment of non-zero temperature 4.6. The Effective Mode Representation 5. Applications 5.1. Vibrational relaxation dynamics at surfaces 5.2. Relaxation dynamics of the Holstein polaron 5.3. The dissipative Landau Zener Model 5.3.1. Coupling to a single environmental mode 5.3.2. Coupling to multiple environmental modes 5.4. Rapid Adiabatic Passage with a dissipative environment 6. Summary And Outlook List of abbreviations Appendix A. Closure relation of displaced number states B. Hamilton equations: classical vs. CCS for a Morse oscillator C. Equations of motion for the multi Davydov-Ansatz C.1. D2-Ansatz C.2. D1-Ansatz D. Details of implementation E. Calculation of the BCF F. Calculation of the polarized initial condition for = 0 Bibliography List of publication

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture