Collective dynamics of driven-dissipative atomic systems in optical cavities

This thesis reports the theoretical description of spatio-temporal pattern forming in atomic ensembles interacting with photons. The atoms are confined in a cavity, are driven by one or several external lasers, and scatter or emit light into the resonator. Ordered structures emerge in parameter ranges where the emitted photons constructively interfere. The formation of spatial patterns can be understood as a consequence of long-range interactions between the atoms mediated by the multiple-scattered cavity photons. In this thesis we develop mean-field models to characterize the out-of-equilibrium dynamics of the atoms including the onset, the transient, and the stationary state of the ordered structures. Here, we identify the conditions for which the emerging dynamical phases exhibit features of spatial self-organization and synchronization. We point out that these features can be measured by observing the light at the cavity output. Finally, we argue that the statistical mechanics of this driven-dissipative systems provides key insights into the dynamics of long-range interacting systems, which is to date largely unexplored.Diese Arbeit behandelt die theoretische Beschreibung von raumzeitlichen Strukturen in mit Licht wechselwirkenden atomaren Ensembles. Die Atome befinden sich in einem Resonator, werden von einem oder mehreren externen Lasern getrieben und emittieren Photonen in den Resonator. Geordnete Strukturen entstehen für Parameterbereiche, in denen die emittierten Photonen konstruktiv interferieren. Das Ausbilden von räumlichen Strukturen kann als Folge einer langreichweitigen Wechselwirkung zwischen den Atomen verstanden werden, welche durch die mehrfach gestreuten Photonen vermittelt wird. In dieser Doktorarbeit entwickeln wir Molekularfeldmodelle um die Nichtgleichgewichtsdynamik der Atome, unter Einbezug des Einsetzen, der vorübergehenden Dynamik und dem stationären Zustand der geordneten Strukturen, zu charakterisieren. Dabei identifizieren wir die Bedingungen für welche die ausbildenden dynamischen Phasen Eigenschaften von räumlicher Selbstorganisation und Synchronisation zeigen. Wir weisen darauf hin, dass diese Eigenschaften durch das von dem Resonator abgegebene Licht gemessen werden können. Schließlich argumentieren wir, dass die statistische Mechanik dieser getriebenen dissipativen Systeme Schlüsselerkenntnisse über die Dynamik von langreichweitig wechselwirkenden Systemen bereitstellt, welche bis heute weitgehend unerforscht sind

An Inverse Problem for a Nonlinear Stochastic Differential Equation Modeling Price Dynamics

Diffusion processes are widely used for mathematical modeling in finance e.g. in modeling foreign exchange rates. This paper presents a non-linear stochastic continuous time model that captures the main characteristics of price dynamics. The generalized mean reversion process discloses various features of observed price movements such as multi-modality of the distributions, multiple equilibria, and regime switching. The attractors depend substantially on the economic environment. The model reveals a significant connection between exchange rates and its fundamentals. Furthermore, it is consistent with traditional flexible exchange rate models. Stochastic differential equations describing diffusion processes are directly linked to the forward Kolmogorov equation. In order to calibrate the models, efficient algorithms identifying the system parameters are in demand. Taking into account nonlinear effects in volatility and drift and dependence on observed economic data, which are not directly modeled, one obtains problems which cannot be treated by standard numerical methods. The coefficients are rapidly oscillatory and strong instabilities may arise. To handle these problems we develop numerical methods, which are used to simulate the nonlinear dynamics of exchange rates depending on economic data