We demonstrate continuous measurement and coherent control of the collective spin of an atomic ensemble undergoing Larmor precession in a high-finesse optical cavity. The coupling of the precessing spin to the cavity field yields phenomena similar to those observed in cavity optomechanics, including cavity amplification, damping, and optical spring shifts. These effects arise from autonomous optical feedback onto the atomic spin dynamics, conditioned by the cavity spectrum. We use this feedback to stabilize the spin in either its high- or low-energy state, where, in equilibrium with measurement backaction heating, it achieves a steady-state temperature, indicated by an asymmetry between the Stokes and the anti-Stokes scattering rates. For sufficiently large Larmor frequency, such feedback stabilizes the spin ensemble in a nearly pure quantum state, in spite of continuous measurement by the cavity field.

Exploring highly connected networks of qubits is invaluable for implementing various quantum algorithms and simulations. All-to-all connectivity allows for entangling qubits with reduced gate depth. On trapped ion quantum processors, the shared motional degree of freedom is routinely used to simultaneously entangle over a dozen qubits with high fidelity in the Mølmer-Sørensen gate. Here, we demonstrate the first implementation of a Mølmer-Sørensen-like interaction in circuit quantum electrodynamics through assisted by a shared photonic mode and Rabi driven qubits, which relaxes restrictions on qubit frequencies during fabrication and supports scalability. We achieve a two-qubit entangled state with maximum state fidelity or 97% in 310 ns and a three-qubit GHZ state with fidelity 90.5% in 217 ns, and a four-qubit GHZ state with 66% fidelity. The four-qubit gate is limited by shared resonator losses and the spread of qubit-resonator couplings, which must be addressed for high fidelity operations. Exploring highly connected networks of qubits is invaluable for implementing various quantum algorithms and simulations as it allows for entangling qubits with reduced circuit depth. Here, we demonstrate a multi-qubit STAR (Sideband Tone Assisted Rabi driven) gate. Our scheme is inspired by the ion qubit Mølmer-Sørensen gate and is mediated by a shared photonic mode and Rabi-driven superconducting qubits, which relaxes restrictions on qubit frequencies during fabrication and supports scalability. We achieve a two-qubit gate with maximum state fidelity of 95% in 310 ns, a three-qubit gate with state fidelity 90.5% in 217 ns, and a four-qubit gate with state fidelity 66% in 200 ns. Furthermore, we develop a model of the gate that show the four-qubit gate is limited by shared resonator losses and the spread of qubit-resonator couplings, which must be addressed to reach high-fidelity operations.