We consider the estimation of an unknown parameter 𝜃 via a many-body probe. The probe is initially prepared in a product state and many-body time-independent interactions enhance its 𝜃 sensitivity during the dynamics and/or in the steady state. We present bounds on the quantum Fisher information, and corresponding optimal interacting Hamiltonians, for two paradigmatic scenarios for encoding 𝜃 : (i) via unitary Hamiltonian dynamics (dynamical metrology), and (ii) in the Gibbs and diagonal ensembles (time-averaged dephased state), two ubiquitous steady states of many-body open dynamics. We then move to the specific problem of estimating the strength of a magnetic field via interacting spins and derive two-body interacting Hamiltonians that can approach the fundamental precision bounds. In this case, we additionally analyze the transient regime leading to the steady states and characterize trade-offs between equilibration times and measurement precision. Overall, our results provide a comprehensive picture of the potential of many-body control in quantum sensing.
