Adiabatic corrections for velocity-gauge simulations of electron dynamics in periodic potentials

We show how to significantly reduce the number of energy bands required to model the interaction of light with crystalline solids in the velocity gauge. We achieve this by deriving analytical corrections to the electric current density. These corrections depend only on band energies, the matrix elements of the momentum operator, and the macroscopic vector potential. Thus, the corrections can be evaluated independently from modeling the interaction with light. In addition to improving the convergence of velocity-gauge calculations, our analytical approach overcomes the long-standing problem of divergences in expressions for linear and nonlinear susceptibilities.Comment: Submitted to Computer Physics Communication

Strong-field Phenomena in Periodic Systems

The advent of visible-infrared laser pulses carrying a substantial fraction of their energy in a single field oscillation cycle has opened a new era in the experimental investigation of ultrafast processes in semiconductors and dielectrics (bulk as well as nanostructured), motivated by the quest for the ultimate frontiers of electron-based signal metrology and processing. Exploring ways to approach those frontiers requires insight into the physics underlying the interaction of strong high-frequency (optical) fields with electrons moving in periodic potentials. This Colloquium aims at providing this insight. Introduction to the foundations of strong-field phenomena defines and compares regimes of field--matter interaction in periodic systems, including (perfect) crystals as well as optical and semiconductor superlattices, followed by a review of recent experimental advances in the study of strong-field dynamics in crystals and nanostructures. Avenues toward measuring and controlling electronic processes up to petahertz frequencies are discussed

Ultrafast coherent electron dynamics in solids

Due to recent developments in high-field laser systems, intense sub-cycle pulses can be generated on a routine basis in laser laboratories around the world. The electric fields originating from such few-femtosecond laser pulses can be on the same order of magnitude as the internal electric fields in bulk, crystalline solids. Due to the short duration of the pulses the laser fluence can remain below the damage threshold of the material. This paves the way for exploring strong-field effects in solids in a non-destructive regime experimentally, and hence motivates theoretical investigations in this field. This thesis is about numerical studies of strong-field effects in insulators and semiconductors. In particular, calculations are performed at a quantum mechanical level in order to examine the importance of quantum coherence in light-matter interactions in the strong-field regime. The dynamics of electrons in one-dimensional spatially periodic potentials excited by laser pulses was simulated. Upon introducing phenomenological decoherence into the dynamical equations, it was found that the optical responses calculated from geometric phases of mixed quantum system were in excellent agreement with conventional approaches for evaluating the optically induced current and polarization response. The excellent agreement even extended to highly non-linear, strong-field regimes, and motivated the development of a numerical method to simulate open quantum mechanical systems governed by spatially periodic Hamiltonians subject to perturbations with broken translation symmetry. Density functional theory was also employed to obtain wave functions from first principles for a number of materials, for which time-resolved optical responses were calculated. Field-induced intraband motion was found to modify the interband transitions significantly at high field strengths for transitions that would otherwise be resonant at low field strengths. For semiconducting materials like GaAs, where the transition elements are strongly peaked at the centre of the Brillouin zone, a step-like excitation mechanism was revealed at field strengths on the order of 0.5 V/Å. Similar ab initio methods were used to model the optical Faraday effect in the insulating, wide band gap material Al2O3 for few-cycle pulses. The magnitude of the effect was predicted using non-perturbative methods. Time-dependent calculations confirmed that a near-instantaneous response is to be expected.Es wurde die Dynamik von Elektronen in Festkörpern, die durch intensive, Subzykluslaserpulse erregt werden numerisch untersucht. Die Berechnungen wurden auf der quantenmechanischen Ebene und in verschiedenen, unabhängigen elektromagnetischen Eichungen ausgeführt. Zuerst wurde die Dynamik der Elektronen in eindimensionalen periodischen Potentialen berechnet um die Gültigket von neuen numerischen Verfahren zu bestätigen. Eines dieser Verfahren ermöglicht Simulationen von räumlich periodischen, gemischten Quantensystemen mit Hamilton-Operatoren mit gebrochener Translationssymmetrie. Durch Anwendung der Dichtefunktionaltheorie wurden Wellenfunktionen für Halbleiter und Insulatoren hergeleitet. Danach konnt der zeitliche Verlauf des optisch induzierten Strom nach ersten Prinzipien bestimmt werden. Die Bedeutung von intraband Bewegungen für Elektronen im halbleitenden Material GaAs wurde ebenfalls untersucht. Bei Erregung mit resonanten Pulsen konnte ein stufenförmiger Anregungsmechanismus beobachtet werden. Ähnliche Methoden wurden verwendet, um die Größe des optischen Faraday-Effektes in einem Insulator mit einer Bandlücke, grösser der Fotonenergie beider Pulse, zu bestimmen. Diese Berechnungen deuten darauf hin, dass ultraschnelle Kontrolle der optisch induzierten Chiralität möglich ist

Electron Momentum Distributions from Strong-Field-Induced Ionization of Atoms and Molecules

High-intensity femtosecond laser pulses in the visible or infrared range can induce electron emission. This single-ionization process may be interpreted as a sequence of (nonadiabatic) tunnel ionization and subsequent acceleration of the electron by the external oscillating field in the presence of the electrostatic force between electron and parent ion. Based on the analysis of photoelectron momentum distributions from the numerical solution of the time-dependent Schrödinger equation, this thesis theoretically studies a variety of phenomena taking place in atoms as well as in molecules in strong fields. The underlying physical mechanisms are revealed by simplified models which take the nonperturbative character of the ionization process into account. The simulation results for several settings are directly compared to measurements, offering the possibility to benchmark state-of-the-art theory and experiment against each other. One example of this is an investigation of the nonadiabatic strong-field ionization of atomic hydrogen in an attoclock setting. More generally, the deflection of the photoelectrons is analyzed in different attoclock configurations to explore the initial conditions of electrons at the tunnel exit—the position where the electron appears after tunneling. When a molecule is ionized, its orbital structure influences the liberated electron wave packet. The orbital imprint on the momentum-space phase of the wave packet, which encodes spatial information, is demonstrated and an interferometric approach to access these phases is evaluated. A characterization of the freed wave packet is crucial as it influences subsequent processes. Such secondary processes are induced when the electron is driven back to the parent ion and scatters off. Similar to focusing of light by a lens, the Coulomb attraction forces scattered electron wave packets through focal points, causing a shift of their phase. Due to the interference of outgoing waves, these phases become visible in electron momentum distributions. For a faithful description, these focal-point effects must be included in a prefactor of the exponentiated action in semiclassical models. Furthermore, the control of electron scattering dynamics is demonstrated for low-energy electrons close to the continuum threshold by means of near-single-cycle terahertz pulses. The temporally-localized preparation of the electron wave packet by a femtosecond laser pulse at a well-defined time within the terahertz field enables a switching between different regimes of dynamics, ranging from recollision-free acceleration to extensive scattering phenomena. In contrast to most studies in the electric dipole approximation that consider only the temporal evolution of the external electric field, various beyond-dipole effects in strong-field ionization are explored in the present work. The microscopic mechanisms of nondipole modifications are thoroughly analyzed. There, the effects of the spatially-varying electric field and of the magnetic field as well as their fingerprints on the geometry of the momentum distributions are identified. Furthermore, the subcycle time resolution of the light-induced momentum transfer in an attoclock-like setup is explored theoretically. Electron recollisions entirely change the observed nondipole effects and render the observations sensitive to the electronic target structure. The high-order above-threshold ionization caused by large-angle scattering is investigated both for exemplary atoms and for diatomic molecules through examination of nondipole shifts of the lateral momentum distribution. The phases of the electron wave packets are also altered by beyond-dipole effects. It is shown that this results in a displacement of ring-link structures known as above-threshold ionization rings that are caused by intercycle interference. In addition, the holographic structures arising from the subcycle interference of scattered and nonscattered electrons are modified

Nonadiabatic effects in the dynamics of atoms confined in a cylindric time-orbiting-potential magnetic trap

In a time-orbiting-potential magnetic trap the neutral atoms are confined by means of an inhomogeneous magnetic field superimposed to an uniform rotating one. We perform an analytic study of the atomic motion by taking into account the nonadiabatic effects arising from the spin dynamics about the local magnetic field. Geometric-like magnetic-fields determined by the Berry's phase appear within the quantum description. The application of a variational procedure on the original quantum equation leads to a set of dynamical evolution equations for the quantum average value of the position operator and of the spin variables. Within this approximation we derive the quantum-mechanical ground state configuration matching the classical adiabatic solution and perform some numerical simulations.Comment: 12 pages, 4 figure

Relativistic Real-Time Methods

Recent advances in laser technology enable to follow electronic motion at its natural time-scale with ultrafast pulses, leading the way towards atto- and femtosecond spectroscopic experiments of unprecedented resolution. Understanding of these laser-driven processes, which almost inevitably involve non-linear light-matter interactions and non-equilibrium electron dynamics, is challenging and requires a common effort of theory and experiment. Real-time electronic structure methods provide the most straightforward way to simulate experiments and to gain insights into non-equilibrium electronic processes. In this Chapter, we summarize the fundamental theory underlying the relativistic particle-field interaction Hamiltonian as well as equation-of-motion for exact-state wave function in terms of the one- and two-electron reduced density matrix. Further, we discuss the relativistic real-time electron dynamics mean-field methods with an emphasis on Density-Functional Theory and Gaussian basis, starting from the four-component (Dirac) picture and continue to the two-component (Pauli) picture, where we introduce various flavours of modern exact two-component (X2C) Hamiltonians for real-time electron dynamics. We also overview several numerical techniques for real-time propagation and signal processing in quantum electron dynamics. We close this Chapter by listing selected applications of real-time electron dynamics to frequency-resolved and time-resolved spectroscopies

Classical Strongly Coupled QGP I: The Model and Molecular Dynamics Simulations

We propose a model for the description of strongly interacting quarks and gluon quasiparticles at $T=(1-3)T_c$, as a classical and nonrelativistic colored Coulomb gas. The sign and strength of the inter-particle interactions are fixed by the scalar product of their classical {\it color vectors} subject to Wong's equations. The model displays a number of phases as the Coulomb coupling is increased ranging from a gas, to a liquid, to a crystal with antiferromagnetic-like color ordering. We analyze the model using Molecular Dynamics (MD) simulations and discuss the density-density correlator in real time. We extract pertinent decorrelation times, diffusion and viscosity constants for all phases. The classical results when extrapolated to the sQGP suggest that the phase is liquid-like, with a diffusion constant $D\approx 0.1/T$ and a bulk viscosity to entropy density ratio $\eta/s\approx 1/3$.Comment: 11 pages, 14 figure

Ultrafast coherent electron dynamics in solids

Due to recent developments in high-field laser systems, intense sub-cycle pulses can be generated on a routine basis in laser laboratories around the world. The electric fields originating from such few-femtosecond laser pulses can be on the same order of magnitude as the internal electric fields in bulk, crystalline solids. Due to the short duration of the pulses the laser fluence can remain below the damage threshold of the material. This paves the way for exploring strong-field effects in solids in a non-destructive regime experimentally, and hence motivates theoretical investigations in this field. This thesis is about numerical studies of strong-field effects in insulators and semiconductors. In particular, calculations are performed at a quantum mechanical level in order to examine the importance of quantum coherence in light-matter interactions in the strong-field regime. The dynamics of electrons in one-dimensional spatially periodic potentials excited by laser pulses was simulated. Upon introducing phenomenological decoherence into the dynamical equations, it was found that the optical responses calculated from geometric phases of mixed quantum system were in excellent agreement with conventional approaches for evaluating the optically induced current and polarization response. The excellent agreement even extended to highly non-linear, strong-field regimes, and motivated the development of a numerical method to simulate open quantum mechanical systems governed by spatially periodic Hamiltonians subject to perturbations with broken translation symmetry. Density functional theory was also employed to obtain wave functions from first principles for a number of materials, for which time-resolved optical responses were calculated. Field-induced intraband motion was found to modify the interband transitions significantly at high field strengths for transitions that would otherwise be resonant at low field strengths. For semiconducting materials like GaAs, where the transition elements are strongly peaked at the centre of the Brillouin zone, a step-like excitation mechanism was revealed at field strengths on the order of 0.5 V/Å. Similar ab initio methods were used to model the optical Faraday effect in the insulating, wide band gap material Al2O3 for few-cycle pulses. The magnitude of the effect was predicted using non-perturbative methods. Time-dependent calculations confirmed that a near-instantaneous response is to be expected.Es wurde die Dynamik von Elektronen in Festkörpern, die durch intensive, Subzykluslaserpulse erregt werden numerisch untersucht. Die Berechnungen wurden auf der quantenmechanischen Ebene und in verschiedenen, unabhängigen elektromagnetischen Eichungen ausgeführt. Zuerst wurde die Dynamik der Elektronen in eindimensionalen periodischen Potentialen berechnet um die Gültigket von neuen numerischen Verfahren zu bestätigen. Eines dieser Verfahren ermöglicht Simulationen von räumlich periodischen, gemischten Quantensystemen mit Hamilton-Operatoren mit gebrochener Translationssymmetrie. Durch Anwendung der Dichtefunktionaltheorie wurden Wellenfunktionen für Halbleiter und Insulatoren hergeleitet. Danach konnt der zeitliche Verlauf des optisch induzierten Strom nach ersten Prinzipien bestimmt werden. Die Bedeutung von intraband Bewegungen für Elektronen im halbleitenden Material GaAs wurde ebenfalls untersucht. Bei Erregung mit resonanten Pulsen konnte ein stufenförmiger Anregungsmechanismus beobachtet werden. Ähnliche Methoden wurden verwendet, um die Größe des optischen Faraday-Effektes in einem Insulator mit einer Bandlücke, grösser der Fotonenergie beider Pulse, zu bestimmen. Diese Berechnungen deuten darauf hin, dass ultraschnelle Kontrolle der optisch induzierten Chiralität möglich ist

Minimal instances for ℤ₂ lattice gauge theories and spin pumps

The implementation and characterization of minimal instances of complex many-body systems have fundamental importance for the development of quantum simulators. This thesis reports on the implementation of two such minimal instances: a minimal instance for ℤ₂ lattice gauge theories coupled to matter and a minimal instance for spin pumping. ℤ₂ lattice gauge theories are characterized by a ℤ₂ gauge degree-of-freedom on every lattice link. Matter particles moving across these links pick up a phase depending on the traversed link's gauge field. To conserve the local gauge symmetry during this motion, also the gauge field itself changes. A corresponding minimal instance consists of a single matter particle on two lattice sites, which are connected by a link. The presented implementation of this minimal instance is based on two-component ultracold rubidium atoms in a periodically-driven optical double-well potential with a state-dependent energy offset between neighboring sites. In this implementation, one component represents the matter, while the other component represents the gauge field. The resulting effective Floquet Hamiltonian exhibits ℤ₂ symmetry for specific modulation parameters, if the periodic driving is resonant with the on-site interaction energy. The dynamics was investigated for different initial states and compared to a numerical analysis. This numerical analysis is in agreement with the measurements and reveals nontrivial dynamics as expected from a ℤ₂ lattice gauge theory coupled to matter. Furthermore, symmetry-breaking terms are identified and routes to overcome these limitations are discussed. Finally, a way to couple minimal instances to a one-dimensional extended system is proposed. The results constitute a first step towards quantum simulation of gauge theories and provide important insights for future studies using Floquet-based techniques. A quantum spin pump generates a spin transport in response to a cyclic and adiabatic variation of a one-dimensional Hamiltonian. The spins are transported in opposite directions and thereby exhibit a vanishing charge transport. In a spin-conserving system with homogeneously filled bands, the spin transport per pump cycle is quantized and deeply connected to the pump cycle's topology. Therefore, such a spin pump can be interpreted as a dynamical version of a two-dimensional time-reversal symmetric quantum spin Hall system. In the experiment, a minimal instance of such a spin pump was implemented with ultracold rubidium atoms in an optical double-well potential. In the limit of piecewise isolated instances throughout the pump cycle, where each instance shifts by one lattice site per half pump cycle, a one-dimensional spin pump is realized. To characterize the response of the pump, the spin separation and the occurring spin currents are measured. For the detection of the spin separation, the center-of-mass positions of both spin components are analyzed in in-situ absorption images. For the detection of spin currents, a novel detection method was developed, which connects the superexchange oscillations after a projection onto static double wells to the instantaneous spin current. This newly developed method can be applied to a general class of systems and in combination with single-site detection, it also allows for a local detection of instantaneous spin currents. The results demonstrate a corner stone for the implementation of a spin pump, which can be extended, e.g. by adding time-reversal-invariant spin--orbit interactions or by breaking time-reversal symmetry. This extension then results either in a system described by a nontrivial ℤ₂ invariant or a quantum spin Hall system described by a spin-Chern number.Die Implementierung und Charakterisierung minimaler Bausteine von komplexen Vielteilchensystemen hat eine grundlegende Bedeutung für die Entwicklung von Quantensimulatoren. In dieser Doktorarbeit werden die Implementierungen zweier minimaler Bausteine präsentiert: Ein minimaler Baustein für ℤ₂-Gittereichtheorien gekoppelt an Materie und ein minimaler Baustein für Spinpumpen. ℤ₂-Gittereichtheorien sind durch einen ℤ₂-Eichfreiheitsgrad auf jeder Gitterkante charakterisiert. Wenn Materieteilchen sich über diese Kanten bewegen, sammeln sie eine Phase ein, die vom Eichfeld der überquerten Kante abhängt. Um die lokale Eichsymmetrie auch während dieser Bewegung zu erhalten, ändert sich das Eichfeld selbst. Ein entspreche der minimaler Baustein besteht aus einem Materieteilchen auf zwei Gitterplätzen, die durch eine Kante verknüpft sind. Die hier vorgestellte Implementierung des minimalen Bausteins basiert auf ultrakalten Rubidiumatomen mit zwei Komponenten in einem periodisch getriebenen, optischen Doppeltopfpotential, welches einen komponentenabhängigen Energieunterschied zwischen benachbarten Gitterplätzen aufweist. In dieser Implementierung stellt eine Komponente die Materie und die andere Komponente das Eichfeld dar. Der entsprechende effektive Floquet-Hamiltonoperator ist ℤ₂-symmetrisch für spezifische Modulationsparameter, wenn die periodische Modulation resonant mit der Wechselwirkungsenergie auf einem Gitterplatz ist. Die Dynamik für verschiedene Anfangszustände wurde untersucht und mit einer numerischen Analyse verglichen. Diese numerische Analyse stimmt mit den Messungen überein und zeigt, dass die Messungen einer nicht-triviale Dynamik folgen wie es von einer ℤ₂-Gittereichtheorie gekoppelt an Materie erwartet wird. Zudem werden eichsymmetriebrechende Terme identifiziert und Wege aufgezeigt diese Beschränkungen zu umgehen. Schießlich wird eine Möglichkeit vorgeschlagen wie aus minimalen Bausteinen ein eindimensionales erweitertes System entstehen kann. Diese Ergebnisse stellen einen ersten Schritt zur Quantensimulation von Eichtheorien dar und ermöglichen wichtige Einblicke für zukünftige Studien mit Floquet-basierten Techniken. Eine Quantenspinpumpe generiert einen Spintransport als Antwort auf eine zyklische und adiabatische Änderung eines eindimensionalen Hamiltonoperators. Dabei werden die Spins in entgegengesetzte Richtungen transportiert ohne dass Ladungstransport auftritt. In einem System mit Spinerhaltung und homogen gefüllten Bändern ist der Spintransport pro Pumpzyklus quantisiert und mit der Topologie des Pumpzykluses verknüpft. Im Experiment wurde ein minimaler Baustein für solch eine Spinpumpe mit ultrakalten Rubidiumatomen in optischen Doppeltopfpotentialen implementiert. Im Fall von während des Pumpzykluses abschnittsweise isolierten Bausteinen, die sich jeden halben Pumpzyklus um einen Gitterplatz verschieben, wird eine eindimensionale Spinpumpe realisiert. Um das Verhalten der Pumpe zu charakterisieren, werden die Spinauftrennung und die auftretenden Spinströme gemessen. Die Spinauftrennung wird aus den Schwerpunktspositionen der beiden Spinkomponenten in in-situ Absorptionsbildern bestimmt. Für die Detektion der Spinströme wurde eine neue Methode entwickelt, die die Superaustauschoszillationen nach einer Projektion auf statische Doppeltöpfe mit dem unmittelbaren Spinstrom verbindet. Diese neu entwickelte Methode kann auf allgemeine Systeme angewendet werden und in Kombination mit Einzelplatzauflösung sogar unmittelbare Spinströme lokal bestimmen. Diese Ergebnisse legen den Grundstein für die Implementierung von Spinpumpen, die dann beispielsweise durch das Hinzufügen von zeitumkehrerhaltenden Spin--Orbit-Wechselwirkungen oder das Brechen der Zeitumkehrsymmetrie erweitert werden können. Dadurch entsteht dann entweder ein System mit einer mit nicht-trivialen \Ztwo{}-Erhaltungsgröße oder ein nicht-triviales Quanten-Spin-Hall-System, welches durch eine Spin-Chernzahl beschrieben wird