The Floquet-Boltzmann equation

Periodically driven quantum systems can be used to realize quantum pumps, ratchets, artificial gauge fields and novel topological states of matter. Starting from the Keldysh approach, we develop a formalism, the Floquet-Boltzmann equation, to describe the dynamics and the scattering of quasiparticles in such systems. The theory builds on a separation of time-scales. Rapid, periodic oscillations occurring on a time scale $T_0=2 \pi/\Omega$, are treated using the Floquet formalism and quasiparticles are defined as eigenstates of a non-interacting Floquet Hamiltonian. The dynamics on much longer time scales, however, is modelled by a Boltzmann equation which describes the semiclassical dynamics of the Floquet-quasiparticles and their scattering processes. As the energy is conserved only modulo $\hbar \Omega$, the interacting system heats up in the long-time limit. As a first application of this approach, we compute the heating rate for a cold-atom system, where a periodical shaking of the lattice was used to realize the Haldane model.Comment: 12 pages + 3 pages of appendix, 13 figure

Directed motion of doublons and holes in periodically driven Mott insulators

Periodically driven systems can lead to a directed motion of particles. We investigate this ratchet effect for a bosonic Mott insulator where both a staggered hopping and a staggered local potential vary periodically in time. If driving frequencies are smaller than the interaction strength and the density of excitations is small, one obtains effectively a one-particle quantum ratchet describing the motion of doubly occupied sites (doublons) and empty sites (holes). Such a simple quantum machine can be used to manipulate the excitations of the Mott insulator. For suitably chosen parameters, for example, holes and doublons move in opposite direction. To investigate whether the periodic driving can be used to move particles "uphill", i.e., against an external force, we study the influence of a linear potential $- g x$. For long times, transport is only possible when the driving frequency $\omega$ and the external force $g$ are commensurate, $n_0 g = m_0 \omega$, with $\frac{n_0}{2},m_0 \in \mathbb{Z}$.Comment: 11 pages, 9 figure

Real-space imaging of a topological protected edge state with ultracold atoms in an amplitude-chirped optical lattice

Topological states of matter, as quantum Hall systems or topological insulators, cannot be distinguished from ordinary matter by local measurements in the bulk of the material. Instead, global measurements are required, revealing topological invariants as the Chern number. At the heart of topological materials are topologically protected edge states that occur at the intersection between regions of different topological order. Ultracold atomic gases in optical lattices are promising new platforms for topological states of matter, though the observation of edge states has so far been restricted in these systems to the state space imposed by the internal atomic structure. Here we report on the observation of an edge state between two topological distinct phases of an atomic physics system in real space using optical microscopy. An interface between two spatial regions of different topological order is realized in a one-dimensional optical lattice of spatially chirped amplitude. To reach this, a magnetic field gradient causes a spatial variation of the Raman detuning in an atomic rubidium three- level system and a corresponding spatial variation of the coupling between momentum eigenstates. This novel experimental technique realizes a cold atom system described by a Dirac equation with an inhomogeneous mass term closely related to the SSH-model. The observed edge state is characterized by measuring the overlap to various initial states, revealing that this topological state has singlet nature in contrast to the other system eigenstates, which occur pairwise. We also determine the size of the energy gap to the adjacent eigenstate doublet. Our findings hold prospects for the spectroscopy of surface states in topological matter and for the quantum simulation of interacting Dirac systems

Propagation and spectral properties of quantum walks in electric fields

We study one-dimensional quantum walks in a homogeneous electric field. The field is given by a phase which depends linearly on position and is applied after each step. The long time propagation properties of this system, such as revivals, ballistic expansion and Anderson localization, depend very sensitively on the value of the electric field $\Phi$, e.g., on whether $\Phi/(2\pi)$ is rational or irrational. We relate these properties to the continued fraction expansion of the field. When the field is given only with finite accuracy, the beginning of the expansion allows analogous conclusions about the behavior on finite time scales.Comment: 7 pages, 4 figure

Periodically driven many-body quantum systems : Quantum Ratchets, Topological States and the Floquet-Boltzmann Equation

Controlling and manipulating complex many-body quantum systems will be a key ingredient for the development of next-generation technologies. While the realisation of a universal quantum machine is still out of reach, in recent years experimental systems of ultracold atoms have already evolved into a vivid field of research for quantum simulation. Crucially, such systems even allow for the successful quantum engineering of targeted many-body systems by means of coherent periodic driving. The essential properties of these Floquet systems encompass two main aspects: fast driving facilitates the simulation of effective static systems, and interactions lead to unique heating effects as energy is only conserved modulo the driving frequency. Within this thesis we theoretically study both of these aspects in respective model systems. In part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott insulator in a designed one-dimensional Floquet system. Here, periodic driving in combination with breaking all mirror symmetries of the system can induce directed motion of particles. In the limit of small excitation densities, the effectively non-interacting quantum ratchet determines the motion of holes and doublons in the Mott insulator and can in fact be used to manipulate the dynamics of such. This little quantum machine can also be used to drive particles against an external force, where transport is possible but requires the fulfilment of a commensurability condition for long times. In part II, we discuss the role of interactions for periodically driven systems by means of a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, we develop this semiclassical formalism based on a clear separation of time scales. The result is a description of the dynamics and the scattering of Floquet quasiparticles in such systems. Here, the property of discrete energy violation is naturally encoded in our formalism predicting the heating of interacting Floquet systems to infinite temperatures in the long-time limit. As a first application of this approach, we investigate a cold atom setup realising the Haldane model by means of periodic shaking. While homogeneous systems heat up globally, a confining potential evokes thermoelectric transport effects resulting from spatially dependent heating characteristics. Moreover, we show that the interplay of intrinsic heating, macroscopic diffusion and non-trivial topological properties of the Haldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous particle transport as the result of developing temperature gradients. In part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing a theoretical framework for a possible simulation of a topological edge state in a one-dimensional optical lattice. In this case, the one-dimensional Dirac equation with spatially varying mass is important, which captures the topological properties of a corresponding system of the BDI symmetry class. We analytically discuss such system and investigate the role of mean-field interaction effects. We also identify the emergence of dynamical instabilities in a realisation with bosonic atoms

Electric quantum walks with individual atoms

We report on the experimental realization of electric quantum walks, which mimic the effect of an electric field on a charged particle in a lattice. Starting from a textbook implementation of discrete-time quantum walks, we introduce an extra operation in each step to implement the effect of the field. The recorded dynamics of such a quantum particle exhibits features closely related to Bloch oscillations and interband tunneling. In particular, we explore the regime of strong fields, demonstrating contrasting quantum behaviors: quantum resonances vs. dynamical localization depending on whether the accumulated Bloch phase is a rational or irrational fraction of 2\pi.Comment: 5 pages, 4 figure

Mit Lachen die Wahrheit zu sagen Entwicklungsgeschichte und Autorintention des Barockromans Simplicissimus

The bloody epoch of the Thirty Years\u27 War and the wide spread decline of German culture and social life is the major topic in Simplicissimus, the first High German novel written by Hans Jacob Christoph von Grimmelshausen in 1668. The purpose of this thesis is to summarize the development of this novel and to find Grimmelshausen\u27s intention in writing his book which was immensely popular upon its publication, but disappeared into obscurity during the German classical period. Since its rediscovery in the middle of the 19th century, scholars have puzzled over the satirical meaning of Simplicissimus as well as the identity of its author. By concentrating on the sources available to Grimmelshausen, researchers have proposed a variety of interpretations of the text. Some see the book as Germany\u27s best example of the Spanish picaresque novel, others include it in the genre of Bildungsroman , whose classic example is Goethe\u27s Wilhelm Meister. Many readers marvel over the satirical and witty language of the novel, while others admire its sophisticated structures. It is evident that scholars have yet to reach a consensus on a comprehensive interpretation of Simplicissimus. Valuable insight could be gained from discovering Grimmelshausen\u27s intentions. By searching the text carefully for statements made by the author about his opinion on social customs, war, family, religion etc., a composite picture of Grimmelshausen\u27s world view was constructed. By combining these assumed attitudes and opinions with his personal development and the social and historical influences of his time, the conclusion emerged that the author wrote his novel out of an ethical didactic urge to confront his readers with the ills of their time. The economic, political and religious changes of the 17th century uprooted the established medieval structures of European life, causing millions of deaths and devastation in large parts of Germany. Grimmelshausen saw the deterioration of life directly related to the changes in the beliefs and behaviors of his contemporaries and felt compelled to alarm his fellow men. A second motivational force for his literary endeavors was the necessity to supplement his small income as tavern owner, although he certainly could not have anticipated the astonishing success of his book. The confusing array of literary forms in Simplicissimus is the result of the author\u27s concern with his message which subordinated the adherence to a specific genre, but nevertheless produced a masterpiece of German literature

Remediation of Contaminated Sites — Case Histories

In this paper, two major remediation projects located in the German Ruhr District will be introduced. The key to an effective land re-utilization plan is a harmonized management of ground investigation, risk assessment and clean-up strategies. Geotechnical and hydrogeological techniques, as well as economical risks involved in the recycling of industrial wasteland will be discussed. Criteria which led to the acceptance of certain remediation techniques will be reflected and their validity will be examined. It will be analyzed whether the projected performance objectives have been met and the demands of both, the investors and the environmental protection agencies have been satisfied. Special attention will be paid to the cost-effectiveness of the remediation strategies proposed