Universität Stuttgart
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Item Open Access Übergangsraten eines getriebenen Spinsystems unter Berücksichtigung von Relaxation(2021) Maihöfer, MichaelDer Magnetismus hat die Menschen schon lange fasziniert. Obwohl viele Aspekte des Magnetismus geklärt sind, ist dieser in der Festkörperphysik auch heute noch ein offenes und aktives Forschungsgebiet. Das liegt nicht zuletzt daran, dass der Magnetismus ein kollektives Phänomen sehr vieler miteinander interagierender Teile bildet, deren magnetische Eigenschaften sich oft von denen der zugrundeliegenden Atome unterscheiden. Ein Trend der Forschung in diesem Gebiet ist es dabei, die Dimensionen des Festkörpers bis auf die Größenordnung von wenigen Atomen schrumpfen zu lassen und die dabei auftretenden magnetischen Eigenschaften von niedrigdimensionalen Festkörpern zu untersuchen. Diese Bemühungen waren auch sehr fruchtbar, und es wurde, um ein Beispiel zu nennen, der Giant Magnetoresistance Effect entdeckt, was seinen Entdeckern Albert Fert und Peter Grünberg 2007 den Nobelpreis in Physik einbrachte. Der Effekt bezeichnet das Auftreten eines magnetfeldrichtungsabhängigen elektrischen Widerstands in einem aus sich abwechselnden ferromagnetischen und nicht-magnetischen Dünnschichten bestehenden Material. Der Trend der Größenreduktion setzte sich fort, sodass nun auch die lateralen Dimensionen unterhalb der Größenordnung der charakteristischen Längenskalen, wie z.B. der Größe der magnetischen Domänen, gebracht wurden. Damit war das Feld des Mikromagnetismus (engl. micromagnetics) geboren. In gewisser Hinsicht vereinfacht dies die Beschreibung des Systems: Einerseits ist das System nun klein genug, sodass magnetische Domänen relevant werden, andererseits ist es groß genug, dass eine quantenmechanische Beschreibung noch nicht zwingend vonnöten ist. Oftmals reicht daher eine semiklassische Beschreibung des Makrospins über die bereits im Jahre 1955 phänomenologisch aufgestellte Landau-Lifshitz-Gilbert (LLG) Gleichung aus. Neuere Experimente legen allerdings nahe, dass im Bereich von Pikosekunden Abweichungen von Voraussagen der LLG-Gleichung auftreten und diese durch einen zusätzlichen Relaxationsterm ergänzt werden muss. Die in diesem Gebiet gewonnenen Erkenntnisse sind für viele technische Anwendungen relevant. Für die Entwicklung von magnetischen bzw. magnetooptischen Speichern ist die Erhöhung der Speicherdichte und der Lese- und Schreibgeschwindigkeit durch ein besseres Verständnis der magnetischen Anordnung und der Magnetisierungsumkehr, relevant. Ferner besteht die Hoffnung der Spintronics (abgeleitet aus den englischen Wörtern spin und electronics) die Informationsverarbeitung nicht mehr – wie in der Elektronik – durch elektrische Ladungen oder Ströme zu realisieren, sondern durch die Ausrichtung des magnetischen Moments der Elektronen. Demnach ist die Untersuchung der Umklappprozesse der Magnetisierung von zentralem Interesse. Ziel der vorliegenden Arbeit ist es die Rate zu bestimmen, mit der solche Umklappprozesse in einem Zweischichtenmodell stattfinden. Dies wird mithilfe der Methoden der Transition State Theory untersucht. Dieselbe Fragestellung wurde für die LLG-Gleichung bereits bearbeitet. Im Vergleich dazu wird nun in dieser Arbeit die um den Relaxationsterm erweiterte LLG-Gleichung herangezogen. Im Gegensatz zur LLG-Gleichung, die eine Differentialgleichung erster Ordnung ist, erlaubt die erweiterte LLG-Gleichung als Differentialgleichung zweiter Ordnung eine reichere Dynamik des Spinsystems. Die Transition State Theory (TST) wurde ursprünglich in der Chemie zur Bestimmung von Übergangsraten von chemischen Reaktionen entwickelt. Die grundlegende Idee der Transition State Theory ist dabei, den Ablauf einer chemischen Reaktion als eine klassische Trajektorie zwischen einem Ausgangs- und einem Endzustand zu beschreiben. Dabei muss diese Bahn eine Potentialhürde überwinden, die der Aktivierungsenergie der chemischen Reaktion entspricht. Die wesentliche Dynamik findet in der Nähe des Sattelpunktes statt, also der energiegünstigsten Stelle der Potentialhürde. Diese lokale Dynamik ist dann auch für die Übergangsrate zwischen den beiden Zuständen wesentlich und wird im Rahmen dieser Arbeit für das getriebene Spinsystem näher untersucht. Die Methoden der TST können auch die Landau-Lifshitz-Gilbert Gleichung bzw. der erweiterten LLG-Gleichung mit Relaxation hergestellt werden. Diese Bewegungsgleichung, zusammen mit einem effektiven Magnetfeld, welche eine bevorzugte Achse sowie eine Potentialbarriere darstellt, beschreibt Übergänge der Magnetisierung.Item Open Access Semiclassical quantization for the states of cuprous oxide in consideration of the band structure(2021) Marquardt, MichaelExcitons are atom-like states in semiconductors like cuprous oxide formed by an electron and a positively charged hole. They are created by exciting an electron from the valence band into the conduction band where the electron forms a bound hydrogen-like state with the hole remaining in the valence band. In this thesis we will focus on excitons of the yellow series which have excitation energies corresponding to wavelengths of about 590 nm. Excitons in cuprous oxide have been studied intensively in experiments and quantum mechanical calculations. Those investigations showed that there are similarities to the hydrogen atom but also deviations caused by the band structure of the crystal. For the hydrogen atom it was possible to connect the quantum mechanical energy spectrum to classical Keplerian orbits in the Bohr-Sommerfeld model. The question arises whether this is possible for excitons in cuprous oxide as well. Semiclassical trace formulas relate fluctuations of the density of states to classical periodic orbits where the frequencies are related to the action or period of the periodic orbits while the amplitude is related to stability properties of the orbits. In this thesis we want to apply semiclassical theories for the calculation and interpretation of exciton spectra. In order to take the band structure of cuprous oxide into account in classical calculations we treat the quasispin and hole spin degrees of freedom with an adiabatic approach. Thereby, we assume the spin dynamics to be much faster than the classical motion and calculate the spin-dependent part of the Hamiltonian quantum mechanically while the exciton dynamics is treated classically. Cuprous oxide has a cubic Oh symmetry. Therefore, it has distinct symmetry planes in which two-dimensional classical exciton orbits occur. In order to simplify the problem we limit ourselves to orbits in the plane orthogonal to the [001] axis. For investigating the classical exciton dynamics we show a Poincaré surface of section and search for periodic orbits in the plane. Furthermore, we calculate the action, period and stability properties of these orbits and use them for semiclassical calculations.Item Open Access Application of machine learning to find exceptional points(2023) Egenlauf, PatrickIn open quantum systems, resonances can occur. These are quasi-bound states which can decay. By introducing a complex scaling, e.g. according to Reinhardt, and thus non-Hermitian operators, the complex energy eigenvalues of the resonances can be calculated. Here, the real part represents their energy, while the imaginary part unveils their lifetime. Resonances can degenerate, where a special case is the so-called exceptional point (EP) at which not only the eigenvalues but also the eigenvectors degenerate. Thus, the two resonances coalesce at the EP. An isolated EP can be described by a two-dimensional matrix model. A property of such an EP is that the two associated eigenvalues exchange their positions after one adiabatic orbit in parameter space around the EP. In 2007 the existence of these EPs was proven for the hydrogen atom in electric and magnetic fields by Cartarius. Due to limitations especially in magnetic field strengths, EPs in the hydrogen atom are not experimentally accessible. In 2014, a remarkable discovery by Kazimierczuk et al. revealed a mesmerizing hydrogen-like spectrum within cuprous oxide. This revelation stemmed from the resemblance between an exciton, a quasi-particle in a semiconductor consisting of electron and hole, and their atomic counterpart, the hydrogen atom. However, the fact that the excitons are environed by cuprous oxide necessitated consideration of the band structure to precisely describe the observed spectrum. This discovery kindled excitement as it provided a rare opportunity to bridge the realms of experimental and theoretical physics, inviting an enthralling dialogue between theory and experiment. For cuprous oxide the field strengths to observe EPs of resonances with small quantum numbers are much lower compared to the field strengths for the hydrogen atom, which is why it is favorable to find EPs in this system. This was already done for a hydrogen-like model, but to obtain experimentally comparable results the above mentioned band structure terms need to be considered. However, this increases the computational cost drastically for each diagonalization of the Hamiltonian due to its complexity. The existing methods to find EPs are based on a Taylor expansion around the EP. Due to the computational expensive diagonalizations of the Hamiltonian, these methods are inefficient or even not applicable. Hence, a new method is required to accurately and efficiently identify EPs in cuprous oxide. Inspired by the remarkable advances in machine learning, especially within the realm of physics, a novel method on the foundation of Gaussian process regression (GPR) is developed. As a prominent member of the supervised machine learning family, GPR serves as a powerful and innovative approach to predict the positions of EPs in cuprous oxide. The used data to train a GPR model is obtained by simulations. Hence, the error is only due to numerical inaccuracies, which can be neglected. Unlike neural networks, GPR offers the advantage of precisely passing through the provided training points, which is a key motivation for its utilization. Yet, the optimization of the searching process goes beyond the new method. An efficient algorithm is devised to enhance the search for EPs in cuprous oxide, which contributes to the discovery of promising EPs and thus enables a possible experimental verification of these data.Item Open Access Multi-fidelity Bayesian machine learning for global optimization(2022) Kuchelmeister, ManuelThe computational optimization and exploration of materials is a challenging task, due to the high dimensionality of the search space and the high cost of accurate quantum mechanical calculations. To reduce the number of costly calculations, the Bayesian Optimization Structure Search (BOSS) has been developed. BOSS combines sample-efficient active learning with Gaussian process regression. This work introduces several multi-fidelity approaches that can reduce the number of costly, accurate calculations even further by incorporating information from inexpensive but less accurate calculations. Using the intrinsic model of coregionalization, BOSS samples data from multiple atomistic calculations based on quantum chemistry (Gaussian16, using CCSD(T)), density-functional theory (FHI-aims, using a PBE-exchange correlation functional) and force fields (AMBER18). Multi-fidelity BOSS samples both, lower and higher-fidelity calculations, while maintaining CCSD(T) accuracy for the global minimum inference. We tested our new multi-fidelity approaches on a 4D alanine conformer search. There, multi-fidelity BOSS has reduced the computational cost, measured in CPU hours, by up to 90%. We found that the efficiency of the approaches depends mostly on the correlation and the computational cost difference between the fidelities. These tests serve as a benchmark for the great potential that multi-fidelity learning can have to reduce the cost of expensive structure-search problems.Item Open Access Quantum kernel methods and applications to differential equations(2024) Flórez Ablan, RobertoQuantum computers have the potential to surpass classical computers in specific tasks, promising advantages in many fields. Machine Learning (ML), a domain with significant societal impact, is a key area of interest for exploring the applications of quantum computing. Here, we investigate two research directions aimed at understanding how current quantum computers can be used to solve ML problems. First, we study Quantum Kernels (QKs). By calculating inner products between quantum states, QKs can be used to define similarity measures between points. QKs are a promising approach to Quantum Machine Learning (QML) but, in general, they have not been shown to outperform classical ML methods. A key reason for this is that QKs suffer from the exponential concentration problem. As the number of qubits increases, the kernel matrices become similar to the identity matrix, preventing generalization. One strategy to alleviate the exponential concentration problem is to rescale the data points that enter the quantum model. This technique is known as bandwidth tuning and has been shown to allow generalization in QKs. However, it has been numerically demonstrated that using this method results in QKs that cannot provide a quantum advantage over classical methods. In this thesis, we propose an explanation for this phenomenon. We show that due to the size of the rescaling factors, the QKs become similar to polynomial and RBF kernels, which are classically tractable. Second, we implemented a Differential Equation (DE) solver based on variational quantum methods. A Quantum Neural Network (QNN) or QK, is used to represent an ansatz for the solution of a DE. The DE information is included into a loss function, which is minimized using a classical optimizer. In the case of a QK, the optimized parameters are the coefficients of a linear combination of QKs evaluated at the data points. In the case of a QNN, the optimized parameters are the phases of the quantum gates. The QNN implementation was included into the open-source QML python library sQUlearn. A preliminary hyperparameter study was conducted for QKs. Based on our limited investigation, we conclude that QKs leveraging the fidelity between quantum states, known as Fidelity Quantum Kernels (FQKs), demonstrate superior performance compared to those employing a semi-classical approach, referred to as Projected Quantum Kernels (PQKs).Item Open Access Item Open Access Predicting instantaneous decay rates at the transition state using neural networks and investigating their usefulness for the reaction rates of arbitrary ensembles(2021) Mishra, PoojaIn this thesis we are going to discuss the TST (chapter 2 will discuss it). Briefly explaining the NHIM and the instantaneous decay rates. The main topic of discussion is the neural networks, as discussed in chapter 3 and their emerging usage in the market. We will build our own neural network to predict the time-dependent instantaneous decay rates of a chemical reaction from random samples at the position and velocity coordinates on the NHIM. Also we are going to discuss the prediction of neural networks taking coordinates of a trajectory as the initial conditions. And eventually we will investigate on the usefulness for the reaction rates of arbitrary ensembles by predicting the time-dependent instantaneous rates of trajectories near the NHIM.Item Open Access Wave functions and oscillator strengths in a two-band model for Rydberg excitons in cuprous oxide quantum wells(2024) Kühner, LeonRydberg physics is the study of systems involving highly excited states of atoms or molecules, known as Rydberg states. In these states, one or more electrons are far from the nucleus, giving the atom exaggerated properties such as large size, long lifetimes, and strong interactions with external fields and nearby particles. These unique features make Rydberg systems a valuable tool for exploring a range of phenomena in atomic physics, quantum optics, and condensed matter physics. They are particularly important for applications in quantum technologies, such as quantum simulation and computation and sensing. Another candidate for Rydberg physics are excitons. When an electron is excited from the valence band to the conduction band the electron in the conduction band and the positively charged hole in the valance band can form hydrogen-like states. Excitons in cuprous oxide, though with relatively low principal quantum numbers, have already been detected in the 1950s by Gross and Hayashi. In 2014 it was possible to measure exciton states with a principal quantum number up to n=25, since then the exciton Rydberg physics has attracted large attention. These states have radii in the range of microns. Rydberg excitons show a large variety of phenomena which do not occur in atomic physics, for example the structure of the valence band leads to a breaking of the spherical symmetry, the spin-orbit coupling leads to the occurrence of a green and yellow exciton series, and central-cell corrections have effects on even parity states. Other effects occur when Rydberg excitons are confined in quantum wells. Such effects have been experimentally observed in GaAs. Thin layers in cuprous oxide have already been produced. Therefore, the observation of excitons in cuprous oxide quantum wells is expected soon. Excitons in quantum wells allow one to investigate the dimensional crossover from three-dimensional systems with weak confinement to two-dimensional systems with strong confinement. For this system the energy spectra have already been computed and effects like overlapping Rydberg series and resonances have been discussed. The theoretical calculations have so far been restricted to the computation of eigenenergies in a hydrogen-like model ignoring the impact of the valence band. The aim of this thesis is to study the effects of Rydberg excitons which rely on the wave functions. Such effects are the behavior of wave functions from weak to strong confinement and the quenching behavior in these regions that are visualized in this thesis. Numerically the wave functions are expanded in a B-spline basis. Also resonances above the first scattering threshold as well as bound states in the continuum above this threshold are visualized. Further, wave functions that undergo an avoided crossing are investigated. Another aspect is the influence of electrostatic effects for exciton states in quantum wells. These lead to the appearance of surface excitons, which can be seen in the visualization of these states. Oscillator strengths are investigated and rely on the behavior of the wave function. In our system the oscillator strengths are no longer translational invariant. Ultimately, this work provides a comprehensive exploration of Rydberg exciton wave functions, which could be instrumental in advancing the use of these systems in emerging quantum applications.Item Open Access Semiclassical analysis and interpretation of quantum mechanically computed Cu2O exciton spectra(2021) Schumacher, MoritzIn solid state physics the energy dispersion of the electrons is described by the band structure of the solid. When an electron is excited from a valence band into a conduction band it creates an unoccupied state in the valence band which is commonly known as a hole and can be treated as a positively charged quasi particle. Instead of considering all the interactions of the excited electron with the electrons remaining in the valence bands one can equivalently consider only the interaction between the excited electron and the hole. With the Coulomb attraction between those two they can form bound states which are called excitons. This suggests a simple description in analogy to the hydrogen atom which is a good approximation under certain conditions, for example a sufficiently large extension of the exciton such that the crystal background can be treated as a continuum. In this thesis we consider excitons in cuprous oxide which can be described as a hydrogen-like system in a first approximation. Depending on the valence bands and conduction bands involved one can distinguish between different exciton series which are named after the corresponding color of light needed for their excitation. The two series with the lowest excitation energies are therefore called yellow and green series. The yellow series with lowest excitation energy has been investigated intensely in experiments and a hydrogen-like exciton spectrum could be observed. However also deviations from the hydrogen-like behavior have been found which are visible as a fine-structure splitting in the spectrum. Theoretical investigations could attribute those deviations mainly to the complex band structure of cuprous oxide. For a more complete theoretical description of the yellow excitons a present coupling of the yellow and green series has to be taken into account. Therefore the valence bands involved in the green series have to be included. This can be achieved by the introduction of a quasi-spin which couples with the hole spin. The coupling strength of the two series is controlled by the spin-orbit coupling constant. Its value is given by the separation between the valence bands involved in the two series at the Gamma point. In 2014 highly excited yellow exciton states with a principal quantum number of up to n = 25 could be observe by T. Kazimierczuk et al. in experiments. For those large quantum numbers the correspondence principle becomes applicable and a classical or semiclassical treatment should be possible. In this thesis we want to investigate connections between the quantum mechanical spectrum and the associated classical dynamics of the yellow excitons. We include the complex valence band structure involved in the yellow and green series and therefore account for the deviations from a hydrogen-like behavior. Numerical calculation of the quantum mechanical exciton spectrum requires the diagonalization of a Hamiltonian using a large but truncated basis set. Although the agreement to experiments is very good, those calculations do not provide direct information about the associated classical exciton dynamics. For hydrogen-like systems classical orbits forming Kepler ellipses are connected to the Rydberg spectrum by the Bohr-Sommerfeld model. The classical phase space structure is not changing with energy as all bound states can be connected to classical elliptic Kepler orbits. This is not the case for excitons in cuprous oxide. The associated classical dynamics is different for every state in the spectrum as the ratio between the corresponding energy and the spin-orbit coupling constant varies. It is possible to avoid this energy dependence by scaling the coupling constant with the energy such that the ratio between energy and the resulting scaled coupling constant remains constant over the whole spectrum. This leads to a scaled quantum spectrum. For every bound classical dynamics characterized by a given energy there exists a corresponding scaled quantum spectrum. A connection between the scaled quantum mechanical exciton spectra and classical exciton dynamics is established by semiclassical trace formulas. They relate fluctuations of the quantum density of states to a superposition of oscillations with frequencies determined by the period or action of classical periodic orbits. Their amplitudes are related to stability properties of the orbits. We therefore apply a Fourier transform and a technique for high-resolution spectral analysis called harmonic inversion to numerically calculated scaled quantum exciton spectra. The resulting quantum recurrence spectra exhibit peaks at positions given by the action of classical periodic orbits of the associated classical dynamics. Their contribution to the quantum spectra is given by the amplitudes of the peaks. By appropriately including the band structure of cuprous oxide it is possible to treat the excitons classically. With the application of semiclassical theories a semiclassical recurrence spectrum can be obtained from parameters of numerically integrated classical periodic orbits. A comparison of the quantum and semiclassical recurrence spectra shows very good agreement. This allows us to obtain information about the periodic orbits (e.g. shape, action, stability, etc.) contributing to the scaled quantum spectra. This thesis thus provides a deeper insight into the classical exciton dynamics in cuprous oxide and the relation between the fine-structure splitting present in quantum spectra and the associated classical dynamics.Item Open Access Relation between ionized gas kinematics and Lyman-alpha observables in galaxies(2023) Schaible, Anna LenaIn the early universe, galaxies can be detected via their Lyman-alpha emission, which is redshifted in the optical wavelength range (1215:7 Å, 2p to 1s transition of hydrogen). However, not all galaxies in the early universe show Ly-alpha emission. The reasons for this are only partly understood. Neutral hydrogen has a high absorption cross-section for Ly-alpha photons. In galaxies, this leads to a spatial and spectral random walk of the Ly-alpha photons in the interstellar-medium (ISM), which increases the absorption probability by interstellar dust of the Ly-alpha photons and is a main reason that not all galaxies show Ly-alpha emission. The random walk of the photons leads to a diffusion spatially and spectrally, which makes it harder to detect the Ly-alpha radiation. The study of spatially resolved observations with samples of galaxies with and without Ly-alpha emission can give important insights in the prevailing ISM conditions for promoting Ly-alpha escape. Such studies can be performed well on a sample of nearby galaxies, which resemble galaxies from the early universe. For nearby galaxies multi-wavelength observations can be obtained, which allow a detailed study of the ISM conditions influencing the Ly-alpha escape. This thesis uses integral field spectroscopic data obtained from the Potsdam Multi Aperture Spectrophotometer at the Calar Alto 3.5 m telescope to investigate the kinematics of ionized gas in 42 nearby galaxies with young stellar populations and active star formation. We use the Balmer-alpha line (6562:8 Å, 3 to 2 transition) as a tracer for the intrinsic Ly-alpha radiation field in the galaxies. Additionally, we use photometric observations from the Hubble Space Telescope for the Lyman-alpha Reference Sample (LARS) and Extended Lyman-alpha Reference Sample (eLARS) galaxies to obtain the Ly-alpha observables. Turbulent kinematics may shift emitting and absorbing material out of resonance, increasing the likelihood of Ly-alpha escaping from galaxies. To test this hypothesis, we perform a global analysis of the kinematic properties of the LARS and eLARS sample, along with their Ly-alpha observables. We derive velocity fields and velocity dispersion maps from the H-alpha observations, and then we focus on the relation between integrated kinematic quantities and the Ly-alpha observables (luminosity, equivalent width and escape fraction). Prior to the analysis, we apply a newly introduced gradient method to correct our data for point spread function smearing. Our results from Kendall tau statistic tests between ionized gas kinematics and Ly-alpha observables support the hypothesis that galaxies dominated by turbulent kinematics, rather than ordered motions, favor the escape of Ly-alpha. Furthermore, we apply a multivariate linear regression method on the Ly-alpha observables luminosity, equivalent width and escape fraction to asses the importance of the integrated kinematic parameters. Again, we find that intrinsic velocity dispersion is an important parameter in affecting the emergence of Ly-alpha emission. We therefore suggest that dispersion dominated ionized gas kinematics may be a necessary, but not a sufficient, condition for facilitating Ly-alpha escape.