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Item Open Access Measurements of effective quasiparticle recombination times and of densities of electronic states at the Fermi level in superconducting Al- and Pb-films(1979) Epperlein, Peter W.; Eisenmenger, WolfgangTemperature-dependent quasiparticle recombination lifetimes tau exp (T) and densities N 0 of electronic states at the Fermi level have been measured from time decay experiments of excess quasiparticle concentrations in evaporated, superconducting Al- and Pb-tunnel junctions. Current pulses were used to inject excess, nonthermal quasiparticles in a single junction acting simultaneously as generator and detector. The experimental lifetimes in "unperturbed" Al show satisfactory agreement with calculations based on the 2Delta-phonon trapping lifetime model. Tau exp decreases with increasing perturbations of the Al film structure by oxygen background evaporation. In Pb the measured times indicate 2Delta-phonon volume losses. The densities N 0 in Pb-films and "unperturbed" as well as oxygen-perturbed Al-films differ by less than 5% from the corresponding bulk material data. Therefore, in trying to explain the enhancement of the transition temperature from 1.23 K to 1.85 K in perturbed, granular Al-films a change of N 0 can be ruled out.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 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 Selbstorganisation zwischen Mannigfaltigkeiten euklidischer und nichteuklidischer Geometrie durch Kooperation und Kompetition(2006) Güßmann, Martin; Wunner, Günter (Prof. Dr.)Gegenstand dieser Dissertation ist ein grundlegendes Problem biologischer Musterbildung. Im Laufe der Ontogenese von Wirbeltieren entstehen wohlgeordnete neuronale Verbindungen zwischen der Retina des Auges und dem Tectum, einer für die optische Informationsverarbeitung wesentlichen Struktur des Mittelhirns. Während die von den retinalen Ganglienzellen auf dem Tectum geknüpften synaptischen Kontakte anfangs noch ungeordnet und zufällig verteilt sind, kommt es im weiteren Verlauf der Ontogenese zur Generierung einer retinotopen Ordnung, d.h. benachbarte Zellen der Retina sind mit benachbarten Zellen des Tectum verbunden. Schon vor über zwanzig Jahren lieferten Häussler und von der Malsburg eine detaillierte analytische Behandlung für den Modellfall, dass die beiden Zellschichten als diskrete lineare Ketten mit jeweils derselben Anzahl von Zellen vorliegen. Die Ausbildung einer retinotopen Ordnung wurde dabei als das Resultat des Wechselspiels aus Kooperation und Konkurrenz zwischen den einzelnen synaptischen Kontakten betrachtet. Ziel dieser Dissertation ist die Verallgemeinerung dieses Modells auf Zellschichten beliebiger Geometrie und Dimension, um zu einer der biologischen Realität adäquateren Beschreibung zu gelangen. Dazu werden zunächst die zugrundeliegenden nichtlinearen Häussler-Gleichungen, welche die Dynamik der Verbindungsgewichte zwischen Retina und Tectum bestimmen, auf kontinuierliche Mannigfaltigkeiten beliebiger Geometrie und Dimension erweitert. Daran schließt sich eine ausführliche synergetische Systemanalyse dieser verallgemeinerten Häussler-Gleichungen an. Die sich daraus ergebenden generischen Ordnungsparametergleichungen stellen ein wesentliches neues Resultat dieser Arbeit dar. Sie dienen als Ausgangspunkt für die Analyse der Emergenz selbstorganisierter retinotoper Verbindungen in Zellschichten verschiedener Geometrien. Zunächst werden als einfachstes Beispiel zwei eindimensionale Mannigfaltigkeiten in Form von Saiten mit periodischen Randbedingungen betrachtet. Es wird der Nachweis erbracht, dass die Saite alle wesentlichen Eigenschaften der diskreten linearen Kette aufweist. Als eine erste Anwendung zweidimensionaler Mannigfaltigkeiten werden Ebenen mit periodischen Randbedingungen analysiert. Dabei zeigt sich, dass dieses System nicht in trivialer Weise in zwei Dimensionen entkoppelt. Ausgehend von einer eingehenden Untersuchung der Ordnungsparametergleichungen werden Bedingungen formuliert, bei denen die Superposition zweier Moden einen Zustand mit ausgeprägtem retinotopem Charakter liefert. Schließlich werden als ein Beispiel nichteuklidischer Mannigfaltigkeiten sphärische Geometrien analysiert, was insbesondere auch durch die reale Form von Retina und Tectum motiviert ist. Ein wesentliches Ergebnis der nichtlinearen Analyse im Fall der Kugel besteht in der Erkenntnis, dass stationäre Lösungen der Ordnungsparametergleichungen existieren, die einer perfekten 1-1-Retinotopie entsprechen.Item Open Access Hartree-Fock-Roothaan-Rechnungen für Vielelektronen-Atome in Neutronenstern-Magnetfeldern(2007) Engel, Dirk; Wunner, Günter (Prof. Dr.)Die Interpretation thermischer Neutronensternspektren erfordert umfangreiche Datenbestände atomarer Dipolübergänge in intensiven Magnetfeldern. Zu diesem Zweck wird in der vorliegenden Arbeit die neue HFFER-Methode zur schnellen Berechnung der Wellenfunktionen, Energien und Oszillatorstärken von Atomen und Ionen mittlerer Kernladungszahl Z in Neutronenstern-Magnetfeldern B>10^7 T entwickelt. Das gekoppelte System der Hartree-Fock-Gleichungen wird selbstkonsistent mit longitudinalen Wellenfunktionen und transversalen Amplituden der Landau-Zustände bis n=7 gelöst. Bei dem präsentierten Ab-initio-Verfahren werden die transversalen Landau-Amplituden durch Lösung der Hartree-Fock-Roothaan-Gleichungen für jedes Elektron berechnet. Die longitudinalen Wellenfunktionen ergeben sich aus dem System der eindimensionalen Hartree-Fock-Gleichungen, die mithilfe der Finiten-Elemente-Methode in einer geeigneten B-Spline-Basis gelöst werden. Alle Algorithmen lassen sich sehr effizient parallelisiert auf einem Computercluster implementieren. Typischerweise benötigt die Berechnung eines Eisen-Grundzustands mit N=26 Elektronen auf p=N=26 Cluster-Prozessoren eine Laufzeit von unter 500 Sekunden. Die Arbeit präsentiert numerische Rechnungen für Grundzustände und verschiedene angeregte Zustände von Atomen und Ionen mit Kernladungszahlen Z=2,...,26 und N=2,...,26 Elektronen. Soweit möglich, werden die Ergebnisse einerseits mit früheren adiabatischen Rechnungen und andererseits mit neueren Quanten-Monte-Carlo-Simulationen verglichen.Item Open Access Exceptional points in atomic spectra and Bose-Einstein condensates(2008) Cartarius, Holger; Main, Jörg (Prof. Dr.)Exceptional points are a special type of degeneracy which can appear for the resonances of parameter-dependent quantum spectra described by non-Hermitian Hamiltonians. They represent positions in the parameter space at which two or even more resonances pass through a branch point singularity. At the critical parameter values, the energies, the widths, and the wave functions describing the resonances are identical. The branching eigenstates show a geometric phase for a parameter space loop around the branch point. In this thesis exceptional points are investigated in two important quantum systems. The first system is the hydrogen atom in crossed external electric and magnetic fields. It is a representative for the class of atoms in static external fields, which are as fundamental quantum system accessible both with experimental and theoretical methods and are ideally suited to study the influence of exceptional points. The resonance spectra of the hydrogen atom are numerically calculated with the complex rotation method. A procedure to systematically search for exceptional points is elaborated and the existence of exceptional points is proven. The influence of the branch point singularities on the resonance energies, the wave functions, and the photoionization cross section is analyzed. In addition, a possibility for the observation of exceptional points in an experiment with atoms is proposed. The investigation of the resonances in spectra of the hydrogen atom in this thesis furthermore reveals structures which can provide an insight into the ionization mechanism. The ionization mechanism of the hydrogen atom in crossed electric and magnetic fields has been investigated, e.g., by application of the transition state theory. Here, calculations are performed which give clear evidence for an important influence of the classical transition state in the quantum spectrum. A second class of quantum systems in which exceptional points appear are the stationary states of Bose-Einstein condensates. They are described by the nonlinear Gross-Pitaevskii equation and it is known that by a variation of the system's parameters the ground state and a second stationary solution are born together in a tangent bifurcation. It is pointed out in this thesis that the mean field energies, the chemical potentials, and the wave functions show at the point of bifurcation the behavior of an exceptional point. The results allow for the extension of the concept of exceptional points to nonlinear quantum systems. Two types of condensates are investigated for this purpose. Bose-Einstein condensates with a laser-induced gravity-like 1/r interaction exhibit analytic solutions which directly prove the existence of exceptional points. The results obtained in this system are used to identify and describe exceptional points in the Bose-Einstein condensation of dipolar gases which is of high experimental interest and has already been realized.Item Open Access Semiklassische Quantisierung chaotischer Billardsysteme mit C4v-Symmetrie(2001) Bücheler, SteffenDie Arbeit beschäftigt sich mit einem System mit C4v-Symmetrie - dem Hyperbelbillard. Die grundlegenden Ideen der Bahnsuche und die Eigenschaften des Systems werden in Kapitel 2 besprochen. Sie bilden die Voraussetzungen für die semiklassische Quantisierung. Kapitel 3 führt in die semiklassische Theorie ein und schließt mit einigen quantenmechanischen Betrachtungen. Das "Pade-Verfahren", angewandt in Kapitel 4, ist das erste semiklassische Verfahren, das die Energieeigenwerte ermitteln soll. Die Konvergenzbetrachtungen spielen dabei eine besondere Rolle. In Kapitel 5 wird ein weiteres Verfahren zur semiklassischen Quantisierung besprochen, die "harmonische Inversion", die in Kapitel 6 für die Anwendung auf kreuzkorrelierte Signale erweitert wird. Schließlich werden die Ergebnisse aller Verfahren in Kapitel 7 zusammengefaßt.Item Open Access Wave packet dynamics in atomic systems and Bose-Einstein condensates(2008) Fabcic, Tomaz; Main, Jörg (Prof. Dr.)The wave packet dynamics in atomic systems and in Bose-Einstein condensates is investigated by means of a time-dependent variational principle. The wave packets are assumed to be parametrized by a set of time-dependent parameters. The time evolution of the parameters of the coupled Gaussian wave packets can be calculated from a set of ordinary differential equations, as obtained from the time-dependent variational principle. Unfortunately, the set of equations is ill-behaved in most practical applications, depending on the number of propagated Gaussian wave packets, and methods for regularization are needed. A general method for regularization based on applying adequate nonholonomic inequality constraints to the evolution of the parameters, keeping the equations of motion well-behaved is presented. The power of the method is demonstrated for a non-integrable system with two degrees of freedom. The Gaussian wave packet (GWP) method is applied to the three-dimensional hydrogen atom. The regularization based on the introduction of Kustaanheimo-Stiefel coordinates and a fictitious time is performed. The regularization implies a transition from the three-dimensional physical position space to a four-dimensional parameter space, spanned by the Kustaanheimo-Stiefel coordinates. The regularization is accompanied by a restriction on physically allowed wave functions. The Coulomb potential is transformed to a harmonic potential and GWPs are the exact solutions, provided they fulfill the restriction. The effect of the restriction on the four-dimensional GWP is discussed and it is shown that the GWPs can satisfy the restriction if the Gaussian parameter space is reduced in a certain way. The exact analytic evolution of the restricted GWP is presented, and the expansion of a localized initial wave function in the restricted Gaussian basis set and its analytic propagation in the fictitious time are shown. Symmetry subspaces with conserved magnetic quantum number m and with conserved angular momentum quantum numbers l,m are treated separately. The method is also applied to the non-integrable H atom in in a homogeneous magnetic field and in perpendicular external electric and magnetic fields. The evolution of the wave packets is determined by the constrained time-dependent variational principle. The numerical results are compared to numerically exact values and show excellent agreement. Another class of systems where the variational Gaussian wave packet method yields good results are cold gases. In this thesis Bose-Einstein condensates are investigated, where in addition to the common short-range contact interaction two different long-range interactions, viz. a laser induced gravity-like 1/r-interaction or a magnetic dipole-dipole interaction are present. The dynamics as resulting from the time-dependent extended Gross-Pitaevskii equation for Bose-Einstein condensates with attractive 1/r-interaction is investigated with both the GWP method and numerically exact calculations. It is shown that these condensates exhibit signatures known from the nonlinear dynamics of autonomous Hamiltonian systems. The two stationary solutions created in a tangent bifurcation at a critical value of the scattering length are identified as elliptical and hyperbolical fixed points, corresponding to stable and unstable stationary states of the condensate. The stable stationary state is surrounded by elliptical islands, corresponding to condensates periodically oscillating in time, whereas condensate wave functions in the unstable region undergo a collapse within finite time. For negative scattering lengths below the tangent bifurcation no stationary solutions exist, i.e., the condensate is always unstable and collapses. The dynamics of condensates with inter-atomic magnetic dipole-dipole interaction is investigated in the mean field limit using a Gaussian trial function. The anisotropy of the magnetic dipole-dipole interaction breaks the spherical symmetry and for ordered dipoles an effectively two-dimensional system with cylindrical symmetry is obtained. Special attention is payed to the regularity of the dynamics and it is shown that a transition from regular to chaotic motion takes place with increasing energy where regions of regular and chaotic motion coexist. It is shown that stable modes exist at energies high above the saddle point energy, i.e. at energies where a collapse of the condensate is expected.Item Open Access A quantum approach to thermodynamics(2003) Gemmer, Jochen; Mahler, Günter (Prof. Dr.)Since the days of Boltzmann the foundation of thermodynamics has been subject to an ongoing discussion. Various problems like the precise definition of entropy, the origin of the second law, etc. have always been issues. For a long time the discussion has not been in the center of interest, for statistical physics work extremely well from a practical point of view. But now, as research deals with single small objects like molecules or atoms, the question arises whether or not thermodynamics are applicable to them, which leads back to the old question of the foundation of thermodynamics. We examine whether thermodynamics can be viewed as emerging from Schrödinger-type quantum dynamics. Therefore we partially transfer concepts of thermodynamical behavior in Gamma-space to the Hilbertspace of compound systems. Central roles play the local Von Neumann entropy of a system and the inevitability of entanglement with its environment, causing it to increase.Item Open Access Thermal and nonthermal properties of closed bipartite quantum systems(2007) Schmidt, Harry; Mahler, Günter (Prof. Dr.)We investigate a two-level system in contact with a larger quantum environment, often consisting of many two-level systems itself (spin environment) which may or may not interact. The total system is considered to be closed. The environment typically is in a canonical state with a given temperature initially. Depending on the precise spectral structure of the environment and the type of coupling between both systems, the smaller part may relax to a canonical state with the same temperature as the environment (i.e., thermal relaxation) or to some other quasiequilibrium state (nonthermal relaxation). The type of (quasi)equilibrium state can be related to the distribution of certain properties of the energy eigenvectors of the total system. We examine these distributions for several abstract and concrete (spin environment) Hamiltonian systems; the significant aspect of these distributions can be related to the relative strength of the local and interaction parts of the Hamiltonian.