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Linear quantum measurements with independent particles are bounded by the standard quantum limit, which limits the precision achievable in estimating unknown phase parameters. The standard quantum limit can be overcome by entangling the particles, but the sensitivity is often limited by the final state readout, especially for complex entangled many-body states with non-Gaussian probability distributions. Here, by implementing an effective time-reversal protocol in an optically engineered many-body spin Hamiltonian, we demonstrate a quantum measurement with non-Gaussian states with performance beyond the limit of the readout scheme. This signal amplification through a time-reversed interaction achieves the greatest phase sensitivity improvement beyond the standard quantum limit demonstrated to date in any full Ramsey interferometer. These results open the field of robust time-reversal-based measurement protocols offering precision not too far from the Heisenberg limit. Potential applications include quantum sensors that operate at finite bandwidth, and the principle we demonstrate may also advance areas such as quantum engineering, quantum measurements and the search for new physics using optical-transition atomic clocks. 

Strongly correlated materials are expected to feature unconventional transport properties, such that charge, spin, and heat conduction are potentially independent probes of the dynamics. In contrast to charge transport, the measurement of spin transport in such materials is highly challenging. We observed spin conduction and diffusion in a system of ultracold fermionic atoms that realizes the half-filled Fermi-Hubbard model. For strong interactions, spin diffusion is driven by super-exchange and doublon-hole–assisted tunneling, and strongly violates the quantum limit of charge diffusion. The technique developed in this work can be extended to finite doping, which can shed light on the complex interplay between spin and charge in the Hubbard model.