Frozen Gaussian Approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

International audienceThe paper is devoted to develop efficient domain decomposition methods for the linear Schrödinger equation beyond the semiclassical regime, which does not carry a small enough rescaled Planck constant for asymptotic methods (e.g. geometric optics) to produce a good accuracy, but which is too computationally expensive if direct methods (e.g. finite difference) are applied. This belongs to the category of computing middle-frequency wave propagation, where neither asymptotic nor direct methods can be directly used with both efficiency and accuracy. Motivated by recent works of the authors on absorbing boundary conditions [X. Antoine et al, J. Comput. Phys., 277 (2014), 268–304] and [X. Yang and J. Zhang, SIAM J. Numer. Anal., 52 (2014), 808–831], we introduce Semiclassical Schwarz Waveform Relaxation methods (SSWR), which are seamless integrations of semiclassical approximation to Schwarz Waveform Relaxation methods. Two versions are proposed respectively based on Herman-Kluk propagation and geometric optics, and we prove the convergence and provide numerical evidence of efficiency and accuracy of these methods

Many Body Quantum Chaos

This editorial remembers Shmuel Fishman, one of the founding fathers of the research field "quantum chaos", and puts into context his contributions to the scientific community with respect to the twelve papers that form the special issue

Critical role of quantum dynamical effects in the Raman spectroscopy of liquid water

Understanding the Raman spectroscopy at the atomistic level is important for the elucidation of dynamical processes in liquid water. Because the polarizability (or its time derivative) is often a highly nonlinear function of coordinates or/and momenta, we employ the linearized semiclassical initial value representation for quantum dynamical simulations of liquid water (and heavy water) under ambient conditions based on an ab initio based, flexible, polarizable model (the POLI2VS force field). It is shown that quantum dynamical effects play a critical role in reproducing the peaks in the intermediate region between the librational and bending bands, those between the bending and stretching bands, and the double-peak in the stretching band in the experimental isotropic Raman spectrum. In contrast, quantum dynamical effects are important but less decisive in the anisotropic Raman spectrum. By selectively freezing either the intramolecular O-H stretching or H-O-H bending mode, we demonstrate that the peak in the intermediate region (2000-2400 cm-1) of the isotropic Raman spectrum arises from the interplay of the stretching and bending motions while a substantial part of the peak in the same intermediate region of the anisotropic Raman spectrum may be attributed to the combined motion of the bending and librational modes

Electron dynamics in complex time and complex space

This thesis investigates the dynamics of electrons ionized by strong low frequency laser fields, from a semiclassical perspective, developing a trajectory-based formalism to describe the interactions of the outgoing electron with the remaining ion. Trajectory models for photoionization generally arise in the regime known as optical tunnelling, where the atom is subjected to a strong, slow field, which tilts the potential landscape around the ion, forming a potential energy barrier that electrons can then tunnel through. There are multiple approaches that enable the description of the ionized electron, but they are generally limited or models derived by analogy, and the status of the trajectories is unclear. This thesis analyses this trajectory language in the context of the Analytical R-Matrix theory of photoionization, deriving a trajectory model from the fundamentals, and showing that this requires both the time and the position of the trajectory to be complex. I analyse this complex component of the position and I show that it requires careful handling: of the potentials where it appears, and of the paths in the complex plane that the trajectory is taken through. In this connection, I show that the Coulomb potential of the ion induces branch cuts in the complex time plane that the integration path needs to avoid, and I show how to navigate these branch cuts. I then use this formalism to uncover a kinematic mechanism for the recently discovered (Near-)Zero Energy Structures of above-threshold ionization. In addition, I analyse the generation of high-order harmonics of the driving laser that are emitted when the photoelectron recollides with the ion, using a pair of counter-rotating circularly polarized pulses to drive the emission, both in the context of the conservation of spin angular momentum and as a probe of the long-wavelength breakdown of the dipole approximation.Open Acces

Semiclassical initial value representation for complex dynamics

Semiclassical initial value representations (SC-IVRs) are popular methods for an approximate description of the quantum dynamics of atomic and molecular systems. A very efficient special case is the propagator by Herman and Kluk, which will be the basis for the investigations in this work. It consists of a phase space integration over initial conditions of classical trajectories which are guiding Gaussian wavepackets. A complex phase factor in the integrand allows for interference between different trajectories, which leads to soft quantum effects being naturally included in the description. The underlying classical trajectories allow for an approximate description of the dynamics of large quantum systems that are inaccessible for a full quantum propagation. Moreover, they also provide an intuitive understanding of quantum phenomena in terms of classical dynamics. The main focus of this work is on further approximations to Herman-Kluk propagation whose applicability to complex dynamics is limited by the number of trajectories that are needed for numerical convergence of the phase space integration. The central idea for these approximations is the semiclassical hybrid formalism which utilizes the costly Herman-Kluk propagator only for a small number of system degrees of freedom (DOFs). The remaining environmental DOFs are treated on the level of Heller's thawed Gaussian wavepacket dynamics, a single trajectory method which is exact only for at most harmonic potentials. If the environmental DOFs are weakly coupled and therefore close to their potential minimum, this level of accuracy is sufficient to account for their effect on the system. Thus, the hybrid approximation efficiently combines accuracy and low numerical cost. As a central theoretical result, we apply this hybrid idea to a time-averaging scheme to arrive at a method for the calculation of vibrational spectra of molecules that is both accurate and efficient. This time-averaged hybrid propagation is then used to study the vibrational dynamics of an iodine-like Morse oscillator bilinearly coupled to a Caldeira-Leggett bath of harmonic oscillators. We first validate the method by comparing it to full quantum and Herman-Kluk propagation for appropriately sized environments. After having established its accuracy, we include more bath DOFs to investigate the influence of the Caldeira-Leggett counter term on the shift of the vibrational levels of the Morse oscillator. As a result, we find out that a redshift, which is observed experimentally for, e.g., iodine in a rare gas matrix, occurs only if the counter term is not included in the Hamiltonian. We then move away from the model bath and on to a realistic, experimentally relevant environment consisting of krypton atoms. We put the iodine molecule into a cluster of 17 krypton atoms and investigate the loss of coherence of the iodine vibration upon coupling to just a few normal coordinates of the bath. These modes with the same symmetry as the iodine vibration turn out to be sufficient to reproduce the expected qualitative dependence on bath temperature and initial state of the iodine molecule. With these few normal modes, a full quantum calculation yields values for coherence loss rates that are close to experimental results. Furthermore, a comparison to semiclassical calculations with more bath modes included confirms the importance of the few highly symmetric normal coordinates. Then, we apply the time-averaged hybrid formalism once more to calculate the vibrational spectrum of the iodine molecule in this now anharmonic krypton environment. Using a krypton matrix instead of a cluster geometry, we find the correct qualitative and also quite good quantitative agreement for the shift of the iodine potential. Finally, we will investigate a more fundamental question, namely, if SC-IVRs contain the spin effects due to the Pauli exclusion principle. To this end, we apply a number of SC-IVRs to the scattering of two electrons with initial states corresponding to either parallel or antiparallel spin. We compare the outcome to full quantum results and find that the difference is resolved by those methods that comprise multiple interfering trajectories

Phase space analysis of quantum effects in strong field ionisation

This thesis explores quantum effects during strong field ionisation, with emphasis on both classical and quantum phase-space interpretational tools. Specifically, this involves investigating the presence of momentum gates during the enhanced ionisation of H₂⁺. These structures cycle through the momentum space without following the time-profile of the external field. By computing autocorrelation functions and Wigner quasiprobability distributions, we establish that momentum gates may occur for static driving fields, and even for no external field at all. Their primary cause is an interference-induced bridging mechanism that occurs if both wells in the molecule are populated. Their cyclic motion in momentum space has a non-classical evolution, as seen from the quantum Liouville equation. Additionally, we employ the quantum trajectory method to seek another criteria for non-classicality. Using an analytical method, we then compute the different eigenfrequencies governing the system in a field-free setting. This provides an in depth understanding that is applied to the time-dependent case. There, the frequency of the quantum bridge, intrinsic to the molecule, is higher than that of the external field. This leads the quasiprobability distribution to sometimes counter-intuitively flow in the direction opposed to the electric-field gradient. These ionisation mechanisms form an optimisation problem that can be controlled using the appropriate molecular targets, driving fields and coherent superposition of states. We investigate the impact of multiple parameters at once by employing machine learning dimensionality reduction techniques. This allows us to disentangle the different effects at play and establish a hierarchy of parameters for controlling ionisation. The features encountered are explained with phase-space arguments and optimal conditions are found for both static and time-dependent fields. The conclusions presented throughout this thesis can in the future be expanded towards multielectron systems, incorporating decoherence and multiple degrees of freedom

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

Dynamics and spectroscopy of strongly coupled electrons and nuclei

This thesis describes work on several research topics in which transport and spectroscopy are influenced by strong electron-nuclear or nuclear-nuclear interactions. First, I give a broad overview of the motivations and background for the main topics covered in this thesis. In the next section, I explore the applicability of perturbative quantum master equations to linear absorption and nonlinear two-dimensional and pump-probe spectroscopies. Next, I introduce a theory of charge transport in organic semiconductors that unifies two popular pictures: incoherent polaron hopping and transient localization due to dynamic disorder. In the next section, I investigate the impact of phonon anharmonicity on the charge transport dynamics of soft semiconductors. Finally, I present a new method of efficiently calculating anharmonic vibrational spectra from ab initio molecular potential energy surfaces

Evaluation and analysis of vibrationally resolved electronic spectra with ab initio semiclassical dynamics

Linear and nonlinear spectroscopy techniques are widely used to study numerous important chemical and physical processes. However, the interpretation of these experimental spectra often becomes very complicated because a particular spectrum constitutes a mere footprint of a host of various, possibly intertwined, effects. In this spirit, calculations in the time-dependent picture provide a useful tool for decoding such spectra. Nevertheless, the ultimate challenge is to devise a theoretical framework that could yield sufficient efficiency as well as accuracy to describe the molecular system of interest in a satisfactory way. The strategy is relatively straightforward for low dimensional systems that are directly tractable, e.g., with exact quantum dynamics performed on an equidistant grid. Despite the formidable overall exponential scaling, these calculations can be significantly accelerated by using higher-order split-operator propagation schemes. In general however, one is forced to seek an affordable balance between physical accuracy on one hand and computational efficiency on the other by employing, for instance, some of the techniques from the broad family of semiclassical methods based on classical trajectories. To this end, the thawed Gaussian approximation (TGA) is combined with an on-the-fly ab initio scheme (OTF-AI). The resulting OTF-AI-TGA algorithm is efficient enough to treat all vibrational degrees of freedom (DOFs) on an equal footing even in case of larger molecules such as pentathiophene (105 DOFs). Moreover, in sharp contrast to popular approaches based on global harmonic approximation, OTF-AI-TGA reproduces almost perfectly the absorption and photoelectron spectra of ammonia, i.e., spectra with strong dependence on large amplitude motions. In addition to the mere reproduction of experimental spectra, a novel systematic approach is introduced to assess the importance and the dynamical couplings of individual vibrational DOFs. This is in turn used to gain a deeper insight into the associated physical and chemical processes by attributing specific spectral features to the underlying molecular motion. Specifically, in the case of oligothiophenes, this approach was used to assign the dynamical interplay between quinoid and aromatic characters of individual rings to particular spectral patterns and, furthermore, to explain the changes in the vibrational line shape with an increasing number of rings. Furthermore, in systems that are too large to be treated with accurate quantum methods, efficient methods such as OTF-AI-TGA are expected to be useful as a preliminary tool for identification of the subspace of the important DOFs. On this subspace, one can then unleash some of the less efficient yet better-suited methods. In summary, OTF-AI-TGA combined with this novel analysis approach is intended to provide the first crucial step in a hierarchical computational protocol for studying large molecules such as dyes