Universität Stuttgart
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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 Fluctuations and correlations of quantum heat engines(2020) Denzler, Tobias; Lutz, Eric (Prof. Dr.)In this work we study the effect of quantum and thermal fluctuations on the statistics of quantum heat engine performance parameters, like efficiency and power. We begin by deriving an explicit solution for the characteristic function of the heat distribution of a thermal quantum harmonic oscillator. We then derive a general framework based on the standard two-point-measurement scheme to compute the efficiency distribution of a quantum Otto cycle. We analyze the generic properties of this distribution for scale-invariant driving Hamiltonians which describe a large class of single-particle, many-body, and nonlinear systems. We find that the efficiency is deterministic and that its mean is equal to the macroscopic efficiency for adiabatic driving. We continue our research by studying the efficiency large deviation function of two exemplary quantum heat engines, the harmonic oscillator and the two-level Otto cycles. While the efficiency statistics follow the ’universal’ theory of Verley et al. [Nature Commun. 5, 4721 (2014)] for nonadiabatic driving, we find that the latter framework does not apply in the adiabatic regime. We can relate this unusual property to the perfect anticorrelation between work output and heat input that suppresses thermal as well as quantum fluctuations. We then probe our findings in an experimental NMR setup using spin-1/2 systems and find them to agree rather well with our theoretical predictions. Afterward, we move on to the finite-time quantum Carnot cycle and investigate its power fluctuations. In particular, we consider how level degeneracy and level number, two commonly found properties in quantum systems, influence the relative work fluctuations. We find that their optimal performance may surpass those of nondegenerate two-level engines or harmonic oscillator motors. Our results highlight that these parameters can be employed to realize high-performance, high-stability cyclic quantum heat engines.Item Open Access Theory of yellow and green excitons in cuprous oxide with emphasis on correction terms and external fields(2022) Rommel, Patric; Main, Jörg (Prof. Dr.)Cuprous oxide has played a central role in the history of exciton physics, being the semiconductor where excitons were first experimentally discovered. Excitons formed from an electron in its lowest conduction band and a hole from its the highest valence band belong to the yellow exciton series. Recently, optical absorption experiments have followed this series up to principal quantum number n = 25 [T. Kazimierczuk et al., Nature 514, 343 (2014)]. This opens up possibilities for novel applications using the particular attributes of highly excited Rydberg system, for example in quantum information processing. For this, the properties of the excitons have to be understood thoroughly. In this thesis, we aim to advance the theoretical knowledge of the yellow and green exciton series in cuprous oxide. We use numerical simulation and analytical methods to investigate in detail the exchange splitting of the S states, the fine structure splitting of the D excitons, spectra in external magnetic fields in Faraday and Voigt configuration, second harmonic generation in forbidden directions, and autoionizing spectra in external electric and parallel magnetic and electric fields. For the latter, we apply the complex-coordinate-rotation method, which we then further use to calculate the green exciton resonances lying in the yellow continuum. We present absorption spectra for transitions from the crystal ground state and for interseries transitions from the yellow to the green series.Item Open Access Phase-space resolved decay rates of driven systems near the transition state(2020) Feldmaier, Matthias; Main, Jörg (Prof. Dr.)Die Bewegung einzelner Atome oder Moleküle bei chemischen Reaktionen lässt sich in vielen Fällen durch klassische Mechanik auf einer Born-Oppenheimer Potentialfläche beschreiben. Hierbei sind die Reaktanten oft durch eine Rang-1 Barriere von den Produkten getrennt. Eine solche Barriere ist durch einen instabilen Freiheitsgrad, die Reaktionskoordinate und eine gegebene Anzahl an stabilen Freiheitsgraden, die orthogonalen Moden, charakterisiert. Eine reagierende Trajektorie wird die Barriere meist in der Sattelregion, d. h. in einer direkten Umgebung des Sattels, überqueren. Diese Region fungiert als Flaschenhals für die Reaktion. Im Rahmen der Theorie der Übergangszustände (engl. transition state theory, TST) können Reaktionsraten über den Fluss reaktiver Trajektorien durch eine nur einmal durchstoßene Trennfläche (engl. dividing surface, DS) berechnet werden. Eine solche Trennfläche ist an der normal hyperbolischen invarianten Mannigfaltigkeit (NHIM) des Sattels verankert und trennt das System in Reaktanten und Produkte. Die NHIM ist dabei ein spezieller Unterraum des vollen Phasenraums und enthält Trajektorien, welche für alle Zeiten an die Sattelregion gebunden sind. Da diese Trajektorien somit weder zur Reaktanten- noch zur Produktseite gehören, bildet die NHIM einen Übergangszustand (engl. transition state, TS) der Reaktion. Für getriebene Systeme ist dieser zeitabhängig. In dieser Arbeit werden anhand eines zweidimensionalen, getriebenen Modellsystems mehrere Methoden zur Berechnung von NHIM und DS im Phasenraum vorgestellt. Basierend auf der Dynamik in einer direkten Umgebung der NHIM werden außerdem verschiedene Ansätze zur Berechnung des zugehörigen Zerfalls der Reaktantenpopulation nahe des TS diskutiert. Anschließend werden die vorgestellten Methoden auf ein realistischeres chemisches Modell angewandt, der getriebenen LiCN <-> LiNC Isomerisationsreaktion. Ein wichtiges Resultat hierbei ist, dass das externe Treiben dieses Systems einen großen Einfluss hat, sowohl auf die Dynamik von Trajektorien in der NHIM, als auch auf den zugehörigen Zerfall der Reaktantenpopulation nahe des TS.Item Open Access Nonequilibrium steady-state physics with quantum master equations(2021) Konopik, Michael; Lutz, Eric (Prof. Dr.)Item Open Access Symmetries and symmetrisation in quantum and electromagnetic multi-mode systems for balancing gain and loss(2021) Dizdarevic, Daniel; Main, Jörg (Apl. Prof. Dr.)Losses usually are an undesirable effect in physics. However, in combination with gain, novel and unexpected features occur. This is because gain and loss can effectively be described via an imaginary potential, which renders a Hamiltonian non-Hermitian. Although there are similarities to standard quantum mechanics, non-Hermitian quantum mechanics exhibits unique mathematical features like bi-orthogonal and self-orthogonal states. Such systems can be used to describe open quantum systems efficiently; though, the overall probability is not conserved in general. However, by balancing gain and loss, stable stationary states with intriguing properties can be realised. Balanced gain and loss occurs in combination with anti-unitary symmetries, which are related to time reversal. The simplest and most powerful symmetry in this regard is PT symmetry, which acted as the driving force behind the development of non-Hermitian quantum mechanics in the last two decades. Researchers produced some astounding results involving PT symmetry, like unidirectionally invisible structures and optimal robust wireless power transfer. Due to the generality of the PT operator, PT symmetry is applicable to almost any physical system, though, it is broken even for small perturbations. In the absence of symmetries, balanced gain and loss can still be achieved by means of symmetrisation or semi-symmetrisation, which are introduced in this thesis. Symmetrised non-Hermitian systems show similar features as symmetric ones, but they allow for a broader range of applications. Symmetrisation allows for the description of physical multi-well potentials with gain and loss. Yet, the lack of obvious symmetries or recognisable patterns makes symmetrised systems hard to understand intuitively. The relations between symmetries and symmetrisation are discussed in detail and both concepts are explicitly applied to one-dimensional multi-mode quantum systems, for which a simple matrix model is used as an example. Analytical symmetrised solutions are derived and it is explicitly demonstrated how symmetrisation can be used to systematically find two-mode systems with a stable stationary ground state. Further, it is shown that models with just two modes are only semi-symmetrisable, whereas they can be perfectly PT-symmetric. Semi-symmetrisation is also applied to multi-mode systems for the realisation of multi-mode chains and to spatially extended Gaussian multi-well potentials. Gaussian potentials can be used in experimental realisations with Bose-Einstein condensates involving non-linear contact interactions; these can be used to realise a self-stabilising mechanism of stationary states, thus making the system robust with respect to small perturbations. By deriving a mathematically equivalent model for inductively coupled electric resonant circuits, the concepts of symmetries and symmetrisation can be transferred from the quantum realm to the classical field of electrodynamics. While this provides a simple and, in particular, accessible platform for experiments, the possibility of applications for wireless power transfer are also discussed briefly, which concludes this thesis.Item Open Access Advances in transition-state theory and applications to driven systems(2023) Reiff, Johannes; Main, Jörg (Prof. Dr.)Chemical reactions are often described via the motion of an effective particle on a Born–Oppenheimer potential-energy surface. In this picture, trajectories typically turn from reactants to products when crossing a rank-1 saddle on the energy surface. The geometric properties of this bottleneck and its associated transition state play an important role for the dynamics of activated trajectories, i. e., trajectories that cross near threshold energy. Transition state theory is a well-established framework that can be used to analyze the dynamics near the rank-1 saddle. It focuses particularly on the determination of rates via the flux through the transition state, which has been investigated since the early 20th century and continues to be of relevance today. In this work, we focus mainly on the geometrical formulation of transition state theory. This description formalizes the distinction between reactants and products by defining a dividing surface in phase space based on the hyperbolic dynamics near the saddle. Specifically, a formally exact dividing surface can be constructed by anchoring it to the saddle's normally hyperbolic invariant manifold. This invariant manifold and its associated stable and unstable manifolds determine the fate of activated trajectories, and so they are of great interest in the field of chemical reaction kinetics. This dissertation is concerned with the development and application of numerical methods in the framework of transition state theory. We address the emergent dynamics of time-dependent chemical and physical model systems under periodic external driving of the transition barrier. In particular, we focus on the structure of the normally hyperbolic invariant manifold, its associated decay rates, and whether these rates can somehow be connected to Kramers's notion of escape rates. The range of models we investigate includes two simple but prototypical test cases with one and two driven saddles as well as the LiCN → LiNC isomerization reaction. We further show how transition state theory can be applied to celestial-mechanics, where it can be used to optimize orbits of satellites with respect to fuel consumption while accounting for time-dependent perturbations from the moon. The systems are mostly treated deterministically, but we also make use of the (generalized) Langevin equation when examining the absolute LiCN isomerization rates. In this context, we ask the fundamental question of how to define a rate, which is especially important at high temperatures.Item Open Access Quantum cooling : thermodynamics and information(2023) Soldati, Rodolfo R.; Lutz, Eric (Prof. Dr.)The theory of cooling is an important corner of thermodynamics, underlying many modern technological applications. As the field of quantum thermodynamics advances, refrigeration techniques must keep pace to fuel the innovations of quantum technologies. We study quantum cooling from its foundations to laboratory implementations within the specific paradigm of heat bath algorithmic cooling. Our study includes a detail modeling of experimental imperfections and establishes the fundamental cooling limits of the model, consolidating the algorithm as a viable quantum refrigeration method. Next, by developing the notion of virtual qubits, we demonstrate a cooling-boost protocol fueled by quantum coherences which is robust to experimental implementations. Aiming at aiding in the progress of refrigeration technologies, we conclude by studying the zero temperature equilibrium properties of a many-body system that can accommodate an autonomous quantum absorption refrigerator, and calculate its entanglement and critical properties, two features that, like quantum coherences, promise to improve the performance of quantum coolers.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.