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Item Open Access Classical and semiclassical approaches to excitons in cuprous oxide(2024) Ertl, Jan; Main, Jörg (Prof. Dr.)When an electron is excited from the valence into the conduction band it leaves behind a positively charged hole in the valence band to which it can couple through the Coulomb interaction. Bound states of electrons and holes, the excitons, are the solid state analogue of the hydrogen atom. As such they follow a Rydberg series. T. Kazimierczuk et al. [Nature 514, 343 (2014)] were able to show the existence of Rydberg excitons in cuprous oxide up to principle quantum number n=25. These states then have extensions in the µm range and thus lie in a region where the correspondence principle is applicable and quantum mechanics turns into classical mechanics. A more precise study of experimental spectra reveals significant deviations from a purely hydrogen-like behavior. These deviations can be traced to the complex valence band structure of cuprous oxide which inherits the cubic symmetry of the system. A theoretical description of the band structure introduces new degrees of freedom, i.e., a quasispin I=1 describing the three-fold degenerate valence band. Due to the coupling of quasispin and hole spin the valence band splits resulting in a yellow exciton series and two green exciton series with light and heavy holes. In this thesis we provide a semiclassical interpretation for excitons in cuprous oxide beyond the hydrogen-like model. To this end we introduce an adiabatic approach diagonalizing the band structure part of the Hamiltonian in a basis for quasi- and hole spin. This leads to a description via energy surfaces in momentum space, which correspond to the different exciton series. Classical dynamics can be calculated by choosing the energy surface of the series under interest and integrating Hamilton's equations of motion. Due to the energy surfaces the symmetry is drastically reduced compared to the hydrogen-like problem now allowing for the existence of fully three-dimensional orbits as well as the possibility of chaotic dynamics. For the yellow exciton series we find mostly regular phase space regions with quasi-periodic motion on near-integrable tori and small chaotic phase space regions. To demonstrate the existence of classical exciton orbits in the quantum spectra we show that the quantum mechanical recurrence spectra exhibit peaks, which, by application of semiclassical theories and a scaling transformation, can be directly related to classical periodic exciton orbits. An analysis of the energy dependence reveals that the dynamics deviations' from a purely hydrogen-like behavior increase with decreasing energy. Starting from the full Hamiltonian we develop a spherical model from which we are able to derive the quantum defects of the yellow exciton series using a semiclassical torus quantization. A comparison with quantum mechanical calculations show good agreement with our semiclassical results, thus allowing to identify individual quantum states by a corresponding classical exciton orbit in analogy to Bohr's atomic model. Finally, we provide a comparison of yellow exciton series with the two distinct green exciton series. The phase space is analyzed by application of Poincaré surfaces of section and Lagrangian descriptors. In addition, we investigate the Lyapunov stability of individual orbits. The analysis reveals the existence of a classically chaotic exciton dynamics for both yellow and green excitons, however, the chaotic regions are more pronounced for the green than for the yellow excitons. Excitons in cuprous oxide thus provide an example of a two-particle system with chaos even without the application of external fields.Item Open Access Bound states in the continuum in cuprous oxide quantum wells(2024) Aslanidis, AngelosExcitons were first introduced in the 1930s by J. A. Frenkel, as the quanta of the excitation of an electron in a semiconductor. When an electron in a semiconductor is excited, it leaves behind a positively charged electron hole. The electron and hole, bound by the Coulomb force, form a quasi-particle known as an exciton. This exciton, made up of both a negative and a positive charge, can be thought of as the solid-state analog of a hydrogen atom. The first experimental observation of excitons was made by Gross and Karryev in Cu2O in 1952. This thesis specifically explores Wannier-Mott excitons. Due to different bandgaps of the adjoint materials, the exciton can be considered trapped in a quantum potential well. Moreover, the higher quantum-confinement subbands couple with the continuum of the lower ones, resulting in resonance states above the scattering threshold. Under certain circumstances, some resonance states appear to have an infinite lifetime, which means these states are bound. These so-called bound states in the continuum (BIC) are the main subject of this thesis. They will be further investigated by approximating an exciton trapped in a cuprous oxide quantum well through quantum defect theory (QDT) and comparing it with numerically precise calculations based on a large B-spline basis. First, there will be a theoretical introduction to excitons in general. Further, the QDT will be explained, and the method of approximating the system will be detailed. After that, the method of approximating the wavefunctions in a B-spline basis together with the complex-coordinate-rotation method will be explained. Lastly, the results of both methods will be compared and discussed.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 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 Dimensionierung gekoppelter Schwingkreise mit ausgeglichenem Gewinn und Verlust(2024) Rehwald, SimonDas Ziel dieser Arbeit ist ein System aus zwei gekoppelten elektromagnetischen Schwingkreisen mit ausgeglichenem Gewinn und Verlust zu realisieren. Dazu kann eine fundamentale Symmetrie der Physik, die PT-Symmetrie, betrachtet werden. Für die Realisierung von Systemen mit ausgeglichenem Gewinn und Verlust werden in dieser Arbeit experimentell realisierbare Schwingkreise mit intrinsischen Verlusten und einer äußeren Anregung betrachtet. Dafür wird der zur Beschreibung dieser gekoppelten Schwingkreise verwendete Moden-Formalismus von Grund auf hergeleitet. Dadurch können die effektiven Größen, durch welche die gekoppelten Schwingkreise und deren Symmetrie untersucht werden, auf die in den Schwingkreisen verwendeten Bauteile zurückgeführt werden. Neben der PT-Symmetrie wird zusätzlich noch die anti-PT-Symmetrie untersucht, bei der Gewinn und Verlust zwar auch symmetrisch, aber nicht ausgeglichen sind. Für die Dimensionierung verschiedener PT- und anti-PT-symmetrischer Systeme werden insgesamt drei verschiedene Kopplungsvarianten betrachtet und verglichen. Dabei werden jeweils möglichst einfache Schwingkreise betrachtet, damit die in dieser Arbeit theoretisch betrachteten Systeme auch einfach experimentell realisiert werden können.Item Open Access Nonequilibrium thermodynamics of quantum coherence beyond linear response(2024) Rodrigues, Franklin L. S.; Lutz, EricQuantum thermodynamics allows for the interconversion of quantum coherence and mechanical work. Quantum coherence is thus a potential physical resource for quantum machines. However, formulating a general nonequilibrium thermodynamics of quantum coherence has turned out to be challenging. In particular, precise conditions under which coherence is beneficial to or, on the contrary, detrimental for work extraction from a system have remained elusive. We here develop a generic dynamic-Bayesian-network approach to the far-from-equilibrium thermodynamics of coherence. We concretely derive generalized fluctuation relations and a maximum-work theorem that fully account for quantum coherence at all times, for both closed and open dynamics. We obtain criteria for successful coherence-to-work conversion, and identify a nonequilibrium regime where maximum work extraction is increased by quantum coherence for fast processes beyond linear response.Item Open Access Thermodynamics in the presence of initial coherences and correlations(2024) Rodrigues, Franklin L. S.; Lutz, Eric (Prof. Dr.)One of the main trends in modern technology is the constant miniaturization of electronic devices. In doing so, it is inevitable that quantum effects will eventually be an unavoidable feature in the planing and fabrication of new instruments; considerations about their power output and energy efficiency lie beyond the scope of classical physics. This regime demands a generalization of the laws of thermodynamics that accounts for the interconversion between classical and quantum resources. This thesis fills this gap by providing a general method to extend the second law of thermodynamics to account for genuine quantum features. These include state coherences consumed in the energy basis and quantum correlations between the system of interest and its environment. We show with our framework that the consumption of coherences is a necessary condition for improved work extraction. For it to be sufficient, we show that one must carefully consider the decoherence timescales. We proceed to investigate two general classes of systems in which quantum resources naturally occur. We begin by studying the thermal properties of generalized Gibbs ensembles, i.e., systems whose conserved quantities do not commute. Then, we explore open quantum systems where the steady state contains non-negligible correlations as consequence of strong interaction strengths with their enviroments. In both cases, we show under which conditions one can use quantum resources to improve the performance of thermal processes, paving the way to efficient design at the nanoscale.