We experimentally study the emergence of antiferromagnetic correlations between ultracold fermionic atoms in a two-dimensional optical lattice with decreasing temperature. We determine the uniform magnetic susceptibility of the two-dimensional Hubbard model from simultaneous measurements of the in situ density distribution of both spin components. At half filling and strong interactions our data approach the Heisenberg model of localized spins with antiferromagnetic correlations. Moreover, we observe a fast decay of magnetic correlations when doping the system away from half filling.

The high controllability of analogue quantum simulators using ultracold atoms in optical lattices pushes forward the frontiers in the experimental investigation of the fermionic Hubbard model. The bilayer Hubbard model is a step beyond the two-dimensional Hubbard model that extends the latter by incorporating a coupling between two two-dimensional Hubbard systems. This is also a step forward in the idea of analogue quantum simulate real materials such as the copper oxide high temperature superconductors which possess a coupled layer structure. This thesis is dedicated to the experimental implementation of an analogue quantum simulator for a bilayer Hubbard system with cold atoms in optical mono-and bi-chromatic lattices. The measurement of competing magnetic order in the bilayer Hubbard system is a result of the work during the course of this thesis. This measurement requires a high controllability of the system which goes along with a precise calibration and fundamental characterisation of the implemented bilayer Hubbard system. Here, the calibration of the interaction strength by means of a comparison between data and theoretical predictions becomes possible through one major outcome of this thesis. This is a method to compute interacting Wannier functions in an optical superlattice. A further major outcome of this thesis is the measurement of thermodynamics, density fluctuations and entropy in the bilayer system considered as a monolayer Hubbard system with reservoir. Further outcomes of this thesis for a fundamental characterisation of the bilayer Hubbard system are:1. The characterisation of the Hubbard band insulator.2. The computation of the potential map for an optical bi-chromatic superlattice. The calibration of the optical bi-chromatic superlattice by comparing experimentaldata to theoretical predictions. Here, the band projection position operator method to compute non-interacting Wannier functions in a superlattice was successfully implemented in this thesis. The latter are the starting point for the newly developed method to compute interacting Wannier functions.V Furthermore, I thank Prof. Dr. Reinhard Kienberger and Prof. Dr. Florian Schreck for their support during the Master's thesis. VII Bibliography 133List of Figures 1451 This is in contrast to the scattering force which light exerts on atoms when its frequency is close to the resonance of the atomic transition, i.e. ∆ ≈ 0. This scattering force saturates for strong intensity and decreases with 1/∆ 2 while the dipole force does not saturate and only decreases with 1/∆. 2 Here, ΩR is the Rabi frequency in the two level picture.

Dynamic magnetoelectric coupling in the improper ferroelectric Cu 1−x Zn x O (x = 0, x = 0.05) was investigated using terahertz time-domain spectroscopy to probe electromagnon and magnon modes. Zinc substitution was found to reduce the antiferromagnetic ordering temperature and widen the multiferroic phase, under the dual influences of spin dilution and a reduction in unit-cell volume. The impact of Zn substitution on lattice dynamics was elucidated by Raman and Fourier-transform spectroscopy, and shell-model calculations. Pronounced softenings of the A u phonons, active along the direction of ferroelectric polarization, occur in the multiferroic state of Cu 1−x Zn x O, and indicate strong spin-phonon coupling. The commensurate antiferromagnetic phase also exhibits spin-phonon coupling, as evidenced by a Raman-active zone-folded acoustic phonon, and spin dilution reduces the spin-phonon coupling coefficient. While the phonon and magnon modes broaden and shift as a result of alloy-induced disorder, the electromagnon is relatively insensitive to Zn substitution, increasing in energy without widening. The results demonstrate that electromagnons and dynamic magnetoelectric coupling can be maintained even in disordered spin systems.

We study the particle-hole symmetry in the Hubbard model using ultracold fermionic atoms in an optical lattice. We demonstrate the mapping between charge and spin degrees of freedom and, in particular, show the occurrence of a state with "incompressible" magnetisation for attractive interactions. Our results present a novel approach to quantum simulation by giving access to strongly-correlated phases of matter through an experimental mapping to easier detectable observables.

The mechanism of fermionic pairing is the key to understanding various phenomena such as hightemperature superconductivity and the pseudogap phase in cuprate materials. We study the pair correlations in the attractive Hubbard model using ultracold fermions in a two-dimensional optical lattice. By combining the fluctuation-dissipation theorem and the compressibility equation of state, we extract the interacting pair correlation functions and deduce a characteristic length scale of pairs as a function of interaction and density filling. At sufficiently low filling and weak on-site interaction, we observe that the pair correlations extend over a few lattice sites even at temperatures above the superfluid transition temperature.