Dynamics of ion Coulomb crystals

The field of quantum simulations has achieved a remarkable success through the development of highly controllable and accessible quantum platforms, which pro- vide insights into the microscopic properties of complex large-scale systems that are otherwise difficult to analyze. Many of the platforms utilized in this pursuit are derived from the field of atomic, molecular, and optical physics. One particularly popular candidate is provided by trapped ions, whose vibrational and electronic degrees of freedom can be effectively combined through laser pulses to engineer desired model Hamiltonians or quantum circuits. Trapped ions constitute as well the basis for modern atomic clocks, the most precise frequency standards currently available. They find further applications in metrology, geodesy, and fundamental physics experiments. In this Thesis, we investigate the dynamics of vibrational modes in trapped ion crystals, utilizing them as a versatile platform to explore various many-body phenomena. We first focus on the expansion dynamics of local excitations and on heat transport within ion crystals hosting structural defects that undergo a sliding- to-pinned transition. We observe a significant reduction in conductivity when the crystal symmetry is spontaneously broken during the transition, and show that resonances between crystal eigenmodes lead to distinct softening signatures associated with energy localization. We then delve into the effects of thermal and quantum fluctuations on the vibrational modes of ion crystals near two distinct structural transitions. We observe the emergence of a prolonged symmetric phase stabilized by thermal and quantum fluctuations, and develop effective theories that reduce the degrees of freedom to the modes that drive the transitions. Finally, we discuss how to engineer spin-orbit coupling and on-site interaction energies for vibrational quantum excitations using two different external driving schemes. While the simulation of spin models with ions typically involves the use of two electronic states, we propose interpreting the two local oscillation modes in an ion crystal as a pseudospin. We show how using Floquet engineering ideas allows for spin flips in Coulomb-induced vibron hopping, resulting in a non-trivial coupling between spatial motion and spin evolution, that results in a markedly non-Abelian dynamics. Subsequently, we explore the simulation of Hubbard models in trapped ions by coupling the vibrational Fock states to an internal level system. Our findings include the observation of bound states in the strong interaction limit of the resulting Jaynes-Cummings-Hubbard model. By investigating these topics, we aim to contribute to the understanding of vibrational dynamics in trapped ion crystals, and shed light on their potential for simulating condensed matter systems, offering insights into phenomena that are otherwise challenging to explore.DFG/Sonderforschungsbereich 1227 DQ-mat/274200144/E

coordinative gait disorder in migraine disease

Klinische Studie mit Migräne-Patienten und gesunden Vergleichsprobanden zur Analyse des Gangbildes bezüglich der KoordinationGait analysis with migraine patients to find coordination disorder

On the relationship between stochastic turnpike and dissipativity notions

In this paper, we introduce and study different dissipativity notions and different turnpike properties for discrete-time stochastic nonlinear optimal control problems. The proposed stochastic dissipativity notions extend the classic notion of Jan C. Willems to $L^r$ random variables and to probability measures. Our stochastic turnpike properties range from a formulation for random variables via turnpike phenomena in probability and in probability measures to the turnpike property for the moments. Moreover, we investigate how different metrics (such as Wasserstein or L\'evy-Prokhorov) can be leveraged in the analysis. Our results are built upon stationarity concepts in distribution and in random variables and on the formulation of the stochastic optimal control problem as a finite-horizon Markov decision process. We investigate how the proposed dissipativity notions connect to the various stochastic turnpike properties and we work out the link between these two different forms of dissipativity