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#### Institute

- Department Physik (6) (remove)

This thesis examines the interaction of light with single dibenzanthanthrene (DBATT) dye molecules. DBATT serves as a model for a quantum-mechanical two-level system. By means of strong focusing of the incident light and cooling of the dye molecules to temperatures below 2 K, a particularly efficient light-matter interaction can be realized. This enables observation of the nonlinearity inherent to a two-level system, e.g., in the form of saturation of the fluorescence signal with light beams containing only a few photons per lifetime of the excited molecular state.
Various nonlinear phenomena arise when two light beams of different frequency are sent to a single molecule. These processes can be exploited to coherently manipulate the transmission of a beam focused onto a single molecule via a second light beam. The occurrent effects, i.e., the AC-Stark shift, stimulated Rayleigh scattering, and three-photon amplification, are detected in the transmitted signal. In addition, four-wave mixing and the dependence of the excited state population on the phase difference of the two incident beams are demonstrated by the use of measurements with subnanosecond time resolution. These results show the possible application of organic dye molecules in the field of quantum information processing where nonlinearities on the single photon and single emitter level are highly sought-after.
In this work, the experimental and theoretical principles of single molecule spectroscopy are discussed. Particular attention is placed on the investigation of the coherent light-matter interaction using transmission measurements. A significant change in this transmitted signal, via the scattering of a single molecule, requires a strong light-matter coupling. To quantify the efficiency of this interaction, the maximum possible coupling of a focused beam and a single emitter is discussed. The results presented herein show that the achieved coupling is typically 5% of the theoretical maximum. Thus, the interaction of a molecule with two light fields with different frequencies is investigated. The nonlinear effects that arise are described qualitatively within the dressed-atom model and quantitatively with a Fourier ansatz. The experimental techniques are explained in detail and the measurement results are presented and discussed.

The manipulation of liquid crystals by external potentials, such as electric or magnetic fields, is of great importance in many industrial and research applications. To gain insight into the impact of the different mechanisms in such a complex multi-particle system is a scientific challenge. Interactions between particles and external potential, particle-particle interactions, and, in case of colloidal systems, hydrodynamic interactions lead, in their interplay, to fascinating dynamical states and unusual diffusion behavior.
This work presents new insights into the dynamics of colloidal liquid crystals with computer simulations. For this purpose several model systems of increasing complexity are studied. The first model system is the most basic model system possible: a system of hard spherocylinders that only interact via excluded volume. In a next step Brownian motion, the random motion of colloidal particles in a fluid, is included via Langevin Dynamics. Finally, also the impact of hydrodynamic interactions between the particles is studied within a Lattice Boltzmann framework.

Triangulations, which can intuitively be described as a tessellation of space into simplicial building blocks, are structures that arise in various different branches of physics: They can be used for describing complicated and curved objects in a discretized way, e.g., in foams, gels or porous media, or for discretizing curved boundaries for fluid simulations or dissipative systems. Interpreting triangulations as (maximal planar) graphs makes it possible to use them in graph theory or statistical physics, e.g., as small-world networks, as networks of spins or in biological physics as actin networks. Since one can find an analogue of the Einstein-Hilbert action on triangulations, they can even be used for formulating theories of quantum gravity. Triangulations have also important applications in mathematics, especially in discrete topology.
Despite their wide occurrence in different branches of physics and mathematics, there are still some fundamental open questions about triangulations in general. It is a prior unknown how many triangulations there are for a given set of points or a given manifold, or even whether there are exponentially many triangulations or more, a question that relates to a well-defined behavior of certain quantum geometry models. Another major unknown question is whether elementary steps transforming triangulations into each other, which are used in computer simulations, are ergodic. Using triangulations as model for spacetime, it is not clear whether there is a meaningful continuum limit that can be identified with the usual and well-tested theory of general relativity.
Within this thesis some of these fundamental questions about triangulations are answered by the use of Markov chain Monte Carlo simulations, which are a probabilistic method for calculating statistical expectation values, or more generally a tool for calculating high-dimensional integrals. Additionally, some details about the Wang-Landau algorithm, which is the primary used numerical method in this thesis, will be examined in detail.

Understanding and predicting the behaviour of liquids at interfaces poses formidable scientific challenges and is highly relevant for the thriving fields of micro- and nanofluidics. External friction forces, such as in liquids slipping over solids, play a central role.
This work presents a well-founded extension of (generalised) fluctuating hydrodynamics to systems with slip boundaries or more general external friction forces. The theory naturally includes thermal fluctuations, which become important on small length scales. Moreover, they are fundamentally related to dissipative processes and reveal microscopic details of friction. A resulting fluctuating slip boundary condition is applied to derive stochastic thin film equations on slip substrates and to calculate the autocorrelation function of the tangential interaction force at a liquid-solid boundary.
In addition, a complementary approach for arbitrary small (classical) scales offers an alternative description. The formalism transfers notions from macroscopic hydrodynamics to the microscale, and external friction appears as a combination of static external forces and additional viscous dissipation.

A fluid of hard spherocylinders serves as a simple model system for
Liquid Crystals. Within classical density functional theory, the free energy
of anisotropic hard bodies can be written in terms of weighted densities solely depending on geometry and position of a single oriented particle.
We improve upon this Fundamental Measure Theory by providing the
exact low-density limit and proposing more suitable expressions
for the dense system. To compare the presented approaches, we
discuss the phase diagram of hard spherocylinders as well as
interfacial and elastic properties. The analytic results for some simple hard-body systems constrain the general form of the functional.
This work provides a sophisticated density functional, which allows
to deduce macroscopic phenomena from the geometry of the fluid‘s particles. It is straightforward to describe mixtures and confined
fluids or to extend the theory to soft interaction potentials.

Quantum Walks in Time
(2014)

Quantum walks are a fascinating concept. They reveal the
counterintuitive and complex dynamics imprinted on all particles underlying the laws of quantum mechanics. Amongst other phenomena, quantum walks show that atoms, electrons and photons propagate drastically faster than any classical particle due to their wavelike nature.
In this book we give an introduction into the world of quantum walks and discuss their applications in quantum simulations and quantum computing. We present three innovative, optical experiments, which expose the astonishing properties of particles on a quantum walk. With the so-called "Quantum Walks in Time"-architecture we reveal the complete spectrum of single particle dynamics—rangingfrom a fast ballistic spread up to a strict localization effect—and investigate experimentally challenging two-particle effects, as the creation of bound states.