The manipulation of quantum materials by light constitutes a new emergent field in condensed matter systems. Strong light-matter coupling and ultrafast optics are powerful tools for inducing new quantum phases in strongly correlated systems, with one particularly fascinating perspective being the modification of superconductivity with light. However, the microscopic behaviour in condensed matter systems often remains challenging to observe directly. Ultracold quantum gases provide a highly microscopically controllable platform for quantum simulation of complex many-body phenomena in quantum materials coupled to light.

This thesis investigates the quantum simulation of strongly interacting fermions coupled to a high-finesse optical cavity, focusing on light-induced density waves. We begin by reporting the observation of optical bistability in the on-axis driven atom-cavity system, which arises from the Fermi gas acting as a Kerr medium, where the intensity-dependent refractive index is caused by a collective atomic density wave influenced by the strong short-range correlations.

By utilising the tunable short-range interactions via a magnetic Feshbach resonance and engineering long-range photon-mediated interactions in the cavity quantum electrodynamics setup through transverse pumping, we explore the emergence of density-wave order in a two-component Fermi gas, providing new insights into the many-body physics of strongly correlated systems with strong long-range interactions. This enables the study of both steady-state and dynamical properties of density-wave formation. By monitoring the photons leaking from the cavity, we investigate the onset of density-wave order as the long-range interaction strength is adiabatically ramped across the critical point and observe the variations of the phase boundary throughout the BEC-BCS crossover. Moreover, we examine the divergence of the density-wave response near the critical point.

Following a quench of long-range interactions, the cavity electrodynamics setup allows for real-time observation of density-wave ordering dynamics, revealing an exponential growth of the order parameter. The early-time collective dynamics is analogous to an inverted harmonic oscillator, where the imaginary oscillation frequency sets the growth rate of the unstable polaritonic mode. We also explore the critical slow down during finite-speed ramps of long-range interactions.

Finally, we investigate light-matter interaction between charge-density-wave order, pair-density-wave order and the cavity field due to the strong dispersive coupling of atoms and pairs to light. This is achieved by transversely pumping the atom-cavity system near a strongly coupled photoassociation transition. The two distinct light-matter interaction channels, along with the mutual coupling between atoms and pairs, result in frustration or intertwining depending on the sign and magnitude of the pair-cavity coupling, which is controlled by tuning the cavity resonance frequency relative to the molecular line. We observe the asymmetric onset of density-wave order via the superradiant cavity light field in the the vicinity of the photoassociation line at unitarity and examine the pair contribution for various molecular detunings.

These results provide valuable insights into light-induced quantum phases in strongly interacting fermions, contributing to the broader understanding of cavity quantum materials.