08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Bose-Einstein condensates with balanced gain and loss beyond mean-field theory(2017) Dast, Dennis; Wunner, Günter (Prof. Dr.)Most of the work done in the field of Bose-Einstein condensates with balanced gain and loss has been performed in the mean-field approximation using the non-Hermitian PT-symmetric Gross-Pitaevskii equation. However, the exchange of particles with the environment plays a crucial role in such systems which in general leads to deviations from the mean-field behavior. Thus, it is not clear whether a mean-field approach is appropriate. It is the purpose of this work to formulate and study a many-particle description of a Bose-Einstein condensate with balanced gain and loss. This is achieved by using a quantum master equation describing a double well where the incoupling of particles in one well and the outcoupling from the other are implemented with Lindblad superoperators. The in- and outcoupling rates are adjusted in an appropriate manner such that balanced gain and loss is achieved. It is shown that the mean-field limit of this master equation yields a PT-symmetric Gross-Pitaevskii equation. Furthermore the master equation supports the characteristic dynamical properties of PT-symmetric systems. There are, however, fundamental differences compared with the mean-field description revealing a new generic feature of PT-symmetric Bose-Einstein condensates. It is shown that the purity of the condensate periodically drops to small values but then is nearly completely restored, when the particles oscillate in the double well. Since in the mean-field limit a completely pure condensate is assumed, this effect cannot be covered by the Gross-Pitaevskii equation. These purity oscillations have a direct impact on the average contrast in interference experiments. In particular it is found that the extrema of the purity can be precisely measured since the average contrast at these points is not reduced by an imbalance of the particle distribution. To gain a detailed understanding of the purity oscillations, analytic solutions for the dynamics in the non-interacting limit are presented and the Bogoliubov backreaction method is used to discuss the influence of the on-site interaction. A central result is that the strength of the purity revivals does neither depend on the amount of particles in the system nor the interaction strength, but is almost exclusively determined by the strength of the in- and outcoupling processes. However, the strong revivals are shifted towards longer times for larger particle numbers. Without interaction this would make the purity oscillations unobservable for a realistic particle number, but by adjusting the interaction strength the strong revivals again occur earlier.Item Open Access Macroscopic quantum tunneling in Bose-Einstein condensates(2013) Schwidder, Torsten; Main, Jörg (Prof. Dr.)The decay of Bose-Einstein condensates from a metastable ground state into collapse due to macroscopic quantum tunneling is investigated using a semiclassical approximation to Feynman’s path integral formalism. Applying a variational ansatz of a single Gaussian to the wave function determined by the Gross-Pitaevskii equation, a special choice of the Gaussian width parameters yields a mean-field energy functional in Hamiltonian form. The temporal spatial extension of the condensate then is described in the picture of a particle moving in an external potential. In this picture the decay of the condensate wave function is investigated using the bounce trajectory method, accounting for the action of the tunneling orbit in imaginary time and quadratic fluctuations around it, the latter described by the Gelfand-Yaglom differential equation. Additionally, the tunneling formalism is used for describing the condensate wave function by a trial function of a superposition of several Gaussians. Using this ansatz a Hamiltonian form of the mean-field energy does not exist and the formalism of describing the tunneling process has to be extended. The bounce trajectory is described in the parameter space of the Gaussian width parameters and the equations of motion for the imaginary time evolution are obtained by applying a time-dependent variational principle. The action of the bounce trajectory is investigated as a function of the number of Gaussians taken into account and the contribution of the fluctuations is evaluated using the monodromy matrix.Item Open Access Thermodynamic functionality of autonomous quantum networks(2010) Schröder, Heiko Christian; Mahler, Günter (Prof. Dr.)Thermodynamics is a theory of impressive success and a wide range of applicability. Nevertheless, it took about two hundred years after the basic formulation of phenomenological thermodynamics until Boltzmann and Maxwell gave a foundation in terms statistical mechanics based on classical mechanics. The invention of quantum mechanics has triggered various attempts to establish a theory of quantum thermodynamics, i.e., to explain and derive thermodynamics by quantum mechanics only. Recently, a new approach to this question has been developed by Gemmer et al. focusing on the role of the partitioning of the universe into "system" and "environment" and on how the entanglement between those parts and the properties of typical environments lead to a thermal state in the system for almost any instant in time. Within this approach, the emergence of thermodynamic behavior in subsystems of a wide class of autonomous quantum networks could be established although the state of the total system is a pure. This turns our understanding of the emergence of thermodynamics upside down: No longer the presence of ideal (infinitely large and always in equilibrium) makes the system thermodynamic, but thermodynamic behavior of the system itself induced by an appropriate embedding - which by itself does not need to be thermodynamic in any way - becomes the crucial ingredient. Moreover, it was found that not only macroscopic embeddings but also quantum networks as small as 10 spins may serve already as excellent thermal embeddings. In this context, this thesis approaches a number of questions: First, complementary to the emergence of relaxation (heat), how does mechanical control over a system (work) emerge in autonomous quantum systems? What types of embeddings allow for this control, and is there a lower limit of their size? The answer is discussed by making the connection between work, parametric control of a Hamiltonian, and classical driving with the help of the factorization approximation. Using this link, we propose the local effective measurement basis method to determine heat and work currents in arbitrary bipartite quantum systems, and discuss measures of how to assess the thermodynamic functionality of an arbitrary embedding. We then apply these concepts to a minimal model consisting of only a spin and a quantum harmonic oscillator (spin-oscillator model) to demonstrate that even a single oscillator may act as an ideal work reservoir. A slight variation of the model then illustrates the pros and cons of the proposed measures of thermodynamic functionality. Second, since thermodynamics is first and foremost a theory of processes, which need both, thermal and mechanical control, we ask if autonomous quantum networks can implement thermodynamic cycles. We do so by first reviewing some specialties of quantum thermodynamic processes in general and then presenting the autonomous dynamic three spin machine, an autonomous quantum network implementing a thermodynamic cycle, which is driven by a emergent quantum work reservoir as discussed in the first part. Finally, we discuss quantum thermodynamic pseudomachines, a class of models that exhibit machine-like functionality without use of a thermodynamic cycle. This gives an answer to the question how new forms of control that are only found in quantum systems lead to new thermodynamic functionalities. The presented models can all be understood as thermodynamic laser models. In particular, we discuss the extended dissipative Jaynes-Cummings model and its thermodynamic properties. We resolve a conflict between different interpretations of the model by a careful analysis of the model itself and the thermodynamic concepts used by different authors. It is found that the model is an intricate heat transport model and no machine and that its thermodynamic functionality relies on the transition-selective coupling of heat baths to a few-level quantum system.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 On the microscopic limit for the existence of local temperature(2005) Hartmann, Michael; Mahler, Günter (Prof. Dr.)Recent progress in the synthesis and processing of nano-structured materials and systems calls for an improved understanding of thermal properties on small length scales. In this context, the question whether thermodynamics and, in particular, the concept of temperature can apply on the nanoscale is of central interest. Here we consider a quantum system consisting of a regular chain of elementary subsystems with nearest neighbour interactions and assume that the total system is in a canonical state with temperature T. We analyse, under what condition the state factors into a product of canonical density matrices with respect to groups of n subsystems each, and when these groups have the same temperature T. In quantum systems the minimal group size depends on the temperature, contrary to the classical case. As examples, we apply our analysis to a harmonic chain and different types of Ising spin chains. For the harmonic chain, which successfully describes thermal properties of insulating solids, our approach gives a first quantitative estimate of the minimal length scale on which temperature can exist: This length scale is found to be constant for temperatures above the Debye temperature and proportional to 1/T^3 below. We finally apply the harmonic chain model to various materials of relevance for technical applications and discuss the results. These show that, indeed, high temperatures can exist quite locally, while low temperatures exist on larger scales only. The technique of the approach is based on a quantum central limit theorem, which should prove usefull in different settings, too.Item Open Access Korrelationsfunktions-Quanten-Monte-Carlo-Methode zur Berechnung von angeregten Zuständen von Mehrelektronen-Atomen in Neutronensternmagnetfeldern(2012) Meyer, Dirk; Wunner, Günter (Prof. Dr.)Um beobachtete Absorptionsfeatures in Spektren isolierter Neutronensterne mit thermischen Emissionen durch atomare Übergänge erklären zu können, benötigt man atomare Daten von Mehrelektronen-Atomen in starken Magnetfeldern. Die in dieser Arbeit untersuchte Korrelationsfunktions-Quanten-Monte-Carlo-Methode (CFQMC-Methode) ist in der Lage, prinzipielle Fehler anderer Methoden (wie beim Hartree-Fock-Verfahren das Hartree-Fock-Limit) zu überwinden. Die CFQMC-Methode ist eine Quanten-Monte-Carlo-Methode, mit der man die Energien des Grundzustands und der niedrigsten m angeregten Zustände eines Symmetrieunterraumes für ein quantenmechanisches Mehrteilchensystem simultan berechnen kann. Diese Arbeit beschäftigt sich mit der präzisen Berechnung der Energien von Mehrelektronen-Atomen (speziell von Helium) in starken Magnetfeldern (2·10^5 – 2·10^7 Tesla), wie sie auf Neutronensternen vorkommen. Im Gegensatz zur Diffusions-Quanten-Monte-Carlo-Methode (DQMC-Methode) für Grundzustandsenergien, auf der die CFQMC-Methode basiert, benötigt sie keine Näherung der Schrödinger-Gleichung, um die in Magnetfeldern notwendigerweise komplexen Wellenfunktionen zu erfassen. Des Weiteren ist sie variationell auch für die Energien angeregter Zustände, im Gegensatz zur Fixed-Phase DQMC-Methode (FPDQMC-Methode) und der Hartree-Fock-Methode. Wie andere Quanten-Monte-Carlo-Verfahren ermöglicht die CFQMC-Methode eine Fehlerabschätzung der berechneten Ergebnisse. Die CFQMC-Methode benötigt bereits gute Näherungen in Form sogenannter Trial-Funktionen an die Zustände, deren Energien berechnet werden sollen. In dieser Arbeit wird dafür ein Hartree-Fock-Ansatz verwendet, bei dem die Einelektronen-Wellenfunktionen in der Richtung transversal zum Magnetfeld in einer Basis von Landau-Funktionen dargestellt werden. Dieser Ansatz ist den untersuchten hohen Magnetfeldstärken gut angepasst. In dieser Arbeit werden die Grundlagen und die Funktionsweise der CFQMC-Methode dargestellt, wie auch der DQMC-Methode für Magnetfelder in verschiedenen Varianten (Fixed-Phase DQMC und Released-Phase DQMC). Verschiedene Verfahren werden in Form eines modularen und erweiterbaren C++-Programms mit grafischer Benutzeroberfläche implementiert. Es werden exemplarische Rechnungen hauptsächlich für Helium in starken Magnetfeldern durchgeführt und diskutiert und die mit der Magnetfeldstärke ansteigende Varianz des CFQMC-Verfahrens untersucht. Es stellt sich heraus, dass das Verfahren für Helium über einer Magnetfeldstärke von etwa 2·10^6 T versagt. Davon ausgehend wird eine Variante des CFQMC-Verfahrens, das Fixed-Phase CFQMC-Verfahren, entwickelt, mit dessen Hilfe der dem Verfahren zugängliche Bereich der Magnetfeldstärke um eine Zehnerpotenz erweitert werden kann. Die Rechnungen wurden wegen der hohen Rechenzeit des CFQMC-Verfahrens parallelisiert auf dem Cluster des BWGrid durchgeführt.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 Nonequilibrium aspects of quantum thermodynamics(2006) Michel, Mathias; Mahler, Günter (Prof. Dr.)Questions about the route from a nonequilibrium initial state to the final global equilibrium have played an important role since the early days of phenomenological thermodynamics and statistical mechanics. Nowadays, their implications reach from central technical devices of the contemporary human society, like heat engines, refrigerators and computers to recent physics at almost all length scales, from Bose-Einstein-condensation and superconductors to black holes. This work addresses the foundation of macroscopic laws concerning the decay to equilibrium, e.g. the celebrated Fourier's Law, on microscopic Schrödingerian quantum dynamics. Here, a proper treatment requires the usage of modern methods in theoretical physics such as the Theory of Open Quantum Systems, the Kubo Formula in Liouville Space and the novel Hilbert Space Average Method. It turns out that both the relaxation to equilibrium as well as the transport of heat is mainly determined by quantum effects comparable to the role of entanglement in considerations of the global equilibrium within Quantum Thermodynamics. Finally, the foundation of phenomenological thermodynamics on a microscopic theory will hopefully improve our understanding of those most impressive and far-reaching theories and their background and will possibly open the way to overcoming their nanoscopic limits.Item Open Access Monte-Carlo-Studien für neutrale Atome und Ionen in starken Magnetfeldern(2013) Boblest, Sebastian; Wunner, Günter (Prof. Dr.)In dieser Arbeit werden das Variations-Quanten-Monte-Carlo- und aufbauend darauf das Fixed-Phase Diffusions-Quanten-Monte-Carlo-Verfahren verwendet, um Bindungsenergien verschiedener Zustände von neutralen Atomen und Ionen mit Kernladungszahl Z=2-26 in sehr starken Magnetfeldern zu berechnen. Mit diesen Ergebnissen konnte zum einen die magnetfeldabhängige Elektronenkonfiguration der Grundzustände aller Spezies mit Z=2-10 und mindestens zwei Elektronen entschlüsselt und zum anderen die Grundzustandsenergien aller neutralen Atome von Helium bis Eisen bei verschiedenen Magnetfeldstärken ermittelt werden. Beide Verfahren benötigen als Eingabe gute Näherungslösungen der Vielteilchenschrödingergleichung, die in unserem Fall mit Hilfe verschiedener Hartree-Fock-Methoden gewonnen werden. Wir untersuchen die Eigenschaften der verschiedenen genäherten Wellenfunktionen, sowie nachträglicher Modifikationen davon, die zur Berücksichtigung von Zweiteilchenkorrelationen eingeführt werden. Dabei finden wir Einflüsse sowohl auf die Stabilität der Monte-Carlo-Rechnungen als auch auf die Qualität der Ergebnisse. Wir diskutieren im Weiteren verschiedene spezielle Anpassungen der beiden Verfahren an die verwendeten Näherungslösungen, die eingeführt werden, um sie automatisiert auf das gegebene Problem anwenden zu können. Unsere Resultate vergleichen wir umfassend mit denen anderer Verfahren. Die Ergebnisse dieser Arbeit sind ein wichtiger Schritt hin zu einer umfassenden Datenbank atomarer Übergänge in starken Magnetfeldern, wie sie für die Analyse der Spektren von magnetischen Weißen Zwergen und Neutronensternen benötigt wird.Item Open Access Realizations of PT-symmetric Bose-Einstein condensates with time-dependent Hermitian potentials(2015) Kreibich, Manuel; Main, Jörg (Prof. Dr.)A PT-symmetric Bose-Einstein condensate can be theoretically described using a complex optical potential, however, the experimental realization of such an optical potential describing the coherent in- and outcoupling of particles is a nontrivial task. We propose an experiment for a quantum mechanical realization of a PT-symmetric system, where the PT-symmetric currents of a two-well system are implemented by coupling possibly time-dependently varied additional wells to the system, which act as particle reservoirs and thus form a Hermitian system. We map the time-dependence of the amplitudes of a frozen Gaussian variational ansatz to a Schrödinger equation with a Hamiltonian matrix. This relates the parameters of a realistic external potential to the matrix elements of a matrix model. On one side, we can use the matrix model as a computationally cheap model to obtain results, on the other hand we can then map the time-dependence of the matrix elements back to the parameters of the potential, which would serve as an input for an experimental setup. In terms of these simple matrix model we derive conditions under which two wells of the Hermitian multi-well system behave exactly as the two wells of the PT-symmetric system. It turns out that the matrix elements of the Hermitian system must be time-dependent to fulfill the conditions for a sufficient time. These results are applied to calculate the time-dependencies of the matrix elements, first, by means of a four-well system, then consequently building up until we arrive at a system with a large number of wells, which can be analyzed and interpreted in terms of optical lattices. As a second method we use a full time-dependent Gaussian variational ansatz, where every variational parameter of the Gaussian functions is chosen to be time-dependent. This should give more accurate results in that this ansatz can describe significant more degrees of freedom. We derive conditions analogous to that of the matrix models that must be fulfilled such that PT-symmetry can be realized. This method is applied and results are obtained for the important case of an adiabatic current ramp which is proposed as a realistic experimental method to create a PT-symmetric ground state. Finally, the results to both methods are compared and in particular the approximations that lead to the simple matrix models are justified.