Abstract: Linear quantum measurements with independent particles are bounded by the Standard Quantum Limit (SQL). However, the SQL can be overcome if it is possible to generate entanglement between many particles. Such entanglement can be created via the interaction of atoms with light in an optical resonator. For an optical transition clock, we have demonstrated a 4dB improvement of the clock precision by optically induced entanglement.

However, the metrological gain from entanglement is often limited by the final state readout rather than the state preparation, especially for more complex entangled many-body states. An alternative is to use a time-reversal protocol to amplify small displacement of the entangled state. We implement such a time-reversal protocol through a controlled sign change in the many-body Hamiltonian of atomic spins coupled to an optical cavity. With this approach, we demonstrate an improvement of 12 dB beyond the SQL. We also use the time-reversed Hamiltonian to experimentally investigate the relation between quantum information scrambling, out-of-time-order correlators and metrological gain.