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

High-frequency light-matter interaction in atoms and molecules

The field of attosecond science is a novel and fast-evolving research area that aims at unravelling the motion of particles in atoms, molecules, and solids. Therein, attochemistry thrives to understand, monitor, and one-day control the movement of electrons in molecules, which will open a new path to steer nuclear dynamics and photochemical reactions. In order to observe the motion of electrons, attosecond resolution and, thus, attosecond light pulses are needed. These attosecond pulses are inherently rooted in the high-frequency regime, ranging from XUV to soft and hard x-ray radiation. Depending on the energy, intensity, and aimed-at observable, different light-matter interactions can be studied. In this work, we tackle three different kinds of high-frequency light-matter interaction that originate in three different energy regimes and allow us to gain novel insight into the dynamics of molecules. In the XUV regime, the ionisation dynamics of correlated, multi-particle systems is studied together with few-cycle effects. In the soft x-ray regime, attosecond x-ray absorption is introduced as a novel tool to observe coupled electron and nuclear dynamics in a neutral molecule. In the hard x-ray regime, we focus on ultrafast, non-resonant x-ray scattering, which can be transformed into a future technology capable of observing electron dynamics. We are confident that this work will benefit the general understanding of high-frequency light-matter interaction in atoms and molecules, as well as aid and initiate new experiments in the field of attochemistry using XUV ionisation, x-ray absorption, and x-ray scattering

Simulation of charge transport in amorphous organic semiconductors

Viele Anwendungen der organischen Elektronik wie beispielsweise organische Leuchtdioden (OLEDs) oder organische Photovoltaik (OPV) basieren auf amorphen halbleitenden Molekülen. Die fast unendlichen Variationsmöglichkeiten organischer Materialien erschweren gezielte experimentelle Materialentwicklung. Bestehende Theorien für den Ladungstransport in amorphen Materialien basieren weitgehend auf empirischen Materialparametern und können so nicht zur prädiktiven Vorhersage von Eigenschaften neuer Materialien benutzt werden. In dieser Arbeit werden Modelle entwickelt, die dem Zweck dienen, den Zusammenhang zwischen mikroskopischen Moleküleigenschaften und deren makroskopischer Leitfähigkeit zu verstehen und somit die Ladungsträgermobilität neuer Materialien vorauszusagen. Zwei Aspekte stellen bei der Entwicklung von Modellen für den Ladungstransport in amorphen Materialien besondere Herausforderungen dar. Zum einen müssen Effekte auf vielen Größenskalen berücksichtigt werden, die von der elektronischen Struktur einzelner Moleküle in Subnanometerbereich bis hin zu Perkolationseffekten im Mikrometermaßstab reichen. Zum anderen hängt die Ladungsträgermobilität exponentiell von der Energieunordnung im amorphen Material ab. Diese wird von der Konformation einzelner Moleküle sowie von deren Wechselwirkung mit ihrer ungeordneten Umgebung bestimmt. Um die Effekte auf allen Größenskalen zu berücksichtigen, wird in dieser Arbeit ein Multiskalenmodell zur Simulation von Ladungstransport in organischen Halbleitern vorgestellt. Die Energieunordnung atomar aufgelöster Morphologien wird mithilfe der Quantum Patch Methode bestimmt, die die elektronische Struktur amorpher Moleküle selbstkonsistent bestimmt. Die mit diesem Modell berechnete Ladungsträgermobilität zeigt gute Übereinstimmung mit experimentellen Daten. Darüber hinaus erlaubt das Modell eine Zerlegung der Ladungsträgermobilität in Faktoren, die von einzelnen Moleküleigenschaften abhängen. Dies ermöglicht die Ableitung von Designkriterien für neue organische Moleküle. Mithilfe dieser Kriterien wurde die Elektronenmobilität eines bekannten Materials durch Änderung der chemischen Struktur gezielt erhöht. Bei dieser Modifikation wurden die Energielevels bewusst konstant gehalten, um optische Eigenschaften nicht zu verändern. Das somit gewonnene Material wurde synthetisiert und elektronisch charakterisiert. In Übereinstimmung mit den theoretischen Vorhersagen zeigt das Material eine um drei Größenordnungen erhöhte Elektronenmobilität. Dieses Beispiel demonstriert die Durchführbarkeit von in-silico Materialentwicklung

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

Quantum Annealing: An Overview

In this review, after providing the basic physical concept behind quantum annealing (or adiabatic quantum computation), we present an overview of some recent theoretical as well as experimental developments pointing to the issues which are still debated. With a brief discussion on the fundamental ideas of continuous and discontinuous quantum phase transitions, we discuss the Kibble-Zurek scaling of defect generation following a ramping of a quantum many body system across a quantum critical point. In the process, we discuss associated models, both pure and disordered, and shed light on implementations and some recent applications of the quantum annealing protocols. Furthermore, we discuss the effect of environmental coupling on quantum annealing. Some possible ways to speed up the annealing protocol in closed systems are elaborated upon: We especially focus on the recipes to avoid discontinuous quantum phase transitions occurring in some models where energy gaps vanish exponentially with the system size.Comment: Final version; in pres

On the rate of convergence of Schwarz waveform relaxation methods for the time-dependent Schrödinger equation

International audienceThis paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided

WKB approaches to restore time in quantum cosmology: predictions and shortcomings

In this review, we analyse different aspects concerning the possibility to separate a gravity-matter system into a part which lives close to a quasi-classical state and a “small” quantum subset. The considered approaches are all relying on a WKB expansion of the dynamics by an order parameter and the natural arena consists of the Bianchi universe minisuperspace. We first discuss how, limiting the WKB expansion to the first order of approximation, it is possible to recover for the quantum subsystem a Schrödinger equation, as written on the classical gravitational background. Then, after having tested the validity of the approximation scheme for the Bianchi I model, we give some applications for the quantum subsystem in the so-called “corner” configuration of the Bianchi IX model. We individualize the quantum variable in the small one of the two anisotropy degrees of freedom. The most surprising result is the possibility to obtain a non-singular Bianchi IX cosmology when the scenario is extrapolated backwards in time. In this respect, we provide some basic hints on the extension of this result to the generic cosmological solution. In the last part of the review, we consider the same scheme to the next order of approximation identifying the quantum subset as made of matter variables only. This way, we are considering the very fundamental problem of non-unitary morphology of the quantum gravity corrections to quantum field theory discussing some proposed reformulations. Instead of constructing the time dependence via that one of the classical gravitational variables on the label time as in previous works, we analyse a recent proposal to construct time by fixing a reference frame. This scheme can be reached both introducing the so-called “kinematical action”, as well as by the well-known Kuchar–Torre formulation. In both cases, the Schrödinger equation, amended for quantum gravity corrections, has the same morphology and we provide a cosmological implementation of the model, to elucidate its possible predictions

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

Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems

Over the last few years, extraordinary advances in experimental and theoretical tools have allowed us to monitor and control matter at short time and atomic scales with a high degree of precision. An appealing and challenging route toward engineering materials with tailored properties is to find ways to design or selectively manipulate materials, especially at the quantum level. To this end, having a state-of-the-art ab initio computer simulation tool that enables a reliable and accurate simulation of light-induced changes in the physical and chemical properties of complex systems is of utmost importance. The first principles real-space-based Octopus project was born with that idea in mind, i.e., to provide a unique framework that allows us to describe non-equilibrium phenomena in molecular complexes, low dimensional materials, and extended systems by accounting for electronic, ionic, and photon quantum mechanical effects within a generalized time-dependent density functional theory. This article aims to present the new features that have been implemented over the last few years, including technical developments related to performance and massive parallelism. We also describe the major theoretical developments to address ultrafast light-driven processes, such as the new theoretical framework of quantum electrodynamics density-functional formalism for the description of novel light-matter hybrid states. Those advances, and others being released soon as part of the Octopus package, will allow the scientific community to simulate and characterize spatial and time-resolved spectroscopies, ultrafast phenomena in molecules and materials, and new emergent states of matter (quantum electrodynamical-materials)