A Cavity-Microscope for Quantum Simulations with Locally-Controllable All-to-All Interactions

This thesis presents the development, construction, and benchmark of an experimental platform that combines cold fermionic 6Li atoms with locally controllable light-matter interactions. To enable local control, a new device, the cavity-microscope, was created. This device combines a high-finesse cavity with an on-axis high numerical aperture microscope. The cavity allows for strong interactions between photons and atoms, while the microscope enables high-resolution spatial engineering of the optical properties of the atoms. This is achieved by coupling the excited state of the effective two-level system to an auxiliary state with the help of a control laser beam that is focused to a small spot by the cavity-microscope.We provide an overview of our experimental apparatus, with a particular focus on the design and fabrication process for the cavity-microscope. This includes the vibration-damping platform of the cavity-microscope, which helps to reduce the mechanical resonances of the mounting structure. Additionally, we discuss the progress of our newly developed next-generation cavity-microscope, which simplifies the optical design and improves the performance of our current device. We provide a comprehensive description of the Hamiltonian that governs our laboratory system with a particular focus on the various measurement techniques that enable us to analyze the atom-cavity system. In particular, we demonstrate that the measurement of the cavity's dispersive shift can be used to infer in real-time different properties of the atomic cloud, such as atom count, temperature, and internal-state occupation.We demonstrate the capability of our cavity-microscope to manipulate the interactions between light and matter by inferring the 3D density profile of the trapped atom cloud using the scanning probe technique. In detail, the control is achieved by a combination of Floquet and wavefront engineering of the control beam. With a spatial light modulator, we correct optical aberrations of the cavity-microscope based on a Hartmann--Shack scheme directly on the atomic cloud. In our experiment, we realize an all-to-all interacting, disordered spin system by subjecting the atomic cloud in our cavity to controllable quasi-random light shifts. By spectroscopically probing the low-energy excitations of the system, we explore the competition of interactions with disorder and show disorder-induced breaking of the strong collective coupling.Finally, we investigate the essential requirements for achieving control over high-rank cavity-mediated all-to-all interaction in our system. This control is crucial for enabling accurate quantum simulations of quantum gravity using the holographic duality of the Sachdev--Ye--Kitaev (SYK) model. To validate our findings, we performed two separate numerical experiments that demonstrate the robustness of our results.