This thesis reports on quantum simulation experiments conducted on strongly correlated fermionic atoms coupled to light in an optical cavity. To this day, the quantum-mechanical description of strongly correlated many-body systems remains a considerable theoretical challenge. Quantum simulation experiments with ultracold atoms contribute to the understanding of such systems through controlled studies of complex, engineered many-body states. Advances in measurement and control techniques continuously drive the development of this field, enabling the creation, detection, and manipulation of novel states of matter.

The experiments presented in this thesis build upon the prior observation of cavity-induced density-wave ordering in a gas of strongly interacting 6Li atoms. In particular, we report on an extension of the cavity quantum electrodynamics setup through the implementation of a high-resolution imaging system. The imaging setup enables the in situ observation of the density waves through resonant absorption imaging within a single experimental realization. Using this technique, we investigate ordering both following adiabatic preparation and after a quench of the cavity-induced long-range interaction. Our observations support the expectation of a density modulation with a single spatial frequency and phase that extends across the entire atomic cloud. We further present a correlation study between atomic and photonic observables, confirming their correspondence and thereby directly testing the theoretical foundation of cavity-mediated interactions. By imaging density waves in the vicinity of the first higher-order transverse cavity mode, we demonstrate the capability of the imaging setup to reveal local order beyond global observables.

Additionally, this thesis presents two studies of density-wave ordering based on measurements of the cavity photon leakage. The first study focuses on the dynamics of order formation following an instantaneous quench. We observe an exponential growth of the order parameter, with a growth rate predominantly governed by the quench strength, in agreement with a theoretical description based on a random phase approximation.
In the second study, we realize a system of three mutually coupled orders, enabled by the simultaneous coupling of light to atoms and atom pairs via a photoassociation resonance. In combination with the strong correlations present in the gas, this gives rise to three coupled order parameters: the cavity photon field, a density wave, and a pair-density wave. The mutual competition and enhancement between these orders result in a measurable shift of the density-wave order phase boundary.

We conclude the thesis by reporting on recent experimental advances in the implementation of a homogeneous trapping potential, leading to a uniform atomic density along two spatial directions. This represents work towards an ideal starting point for a range of exciting future experiments, including spectroscopic measurements, local potential patterning, and correlation studies in long-range interacting Fermi gases, leveraging the numerical aperture of the imaging system.