From Dynamical to Steady-State Many-Body Metrology: Precision Limits and Their Attainability with Two-Body Interactions

We consider the estimation of an unknown parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>θ</a:mi></a:math> via a many-body probe. The probe is initially prepared in a product state and many-body time-independent interactions enhance its <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>θ</c:mi></c:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>θ</e:mi></e:math>: (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.