Cavity-enabled real-time observation of individual atomic collisions

Using the strong dispersive coupling to a high-cooperativity cavity, we demonstrate fast and non-destructive number-resolved detection of atoms in optical tweezers. We observe individual atom-atom collisions, quantum state jumps, and atom loss events with a time resolution of $100\mu$s through continuous measurement of cavity transmission. Using adaptive feedback control in combination with the non-destructive measurements, we further prepare a single atom with $92(2)\mathrm{\%}$ probability.