Reaching the Limits of Ground-State Metrology with Many-Body Probes

Abstract

We investigate the physical requirements to reach the ultimate limits of ground-state metrology -- i.e., the estimation of an unknown parameter $\theta$ of an $N$-body Hamiltonian via measurements on its ground state -- using many-body probes featuring either short-range or long-range interactions. We characterize the conditions to saturate two fundamental limits of the quantum Fisher information $F_{\theta }$: the static ground-state bound $F_{\theta }\lesssim N^2/{\Delta }^2$, where $\Delta$ is the spectral gap, and the dynamical Heisenberg limit $F_{\theta }\lesssim N^2{\tau }^2$, where $\tau$ is the total protocol duration. To saturate the static limit, we argue that short-range interacting systems require a spectral gap that closes with $N$. At second-order quantum critical points this leads to a universal condition on the critical exponents, $\nu (d+z)=1$, which can be approached arbitrarily closely, as we show for the XXZ chain; cat-like ground states at first-order transitions provide an alternative route to static optimality. In contrast, we find that all-to-all interacting models can achieve optimality even in gapped regimes. Furthermore, reaching the dynamical bound is tightly related to the ground-state preparation time; we demonstrate that its optimal, inverse-gap profile can be obtained via local adiabatic protocols. By analyzing paradigmatic systems -- the transverse-field Ising chain, the XXZ chain, a squeezing Hamiltonian, and the two-mode Bose-Hubbard model -- we identify specific many-body probes capable of approaching both limits. Overall, our results provide a theoretical basis for the design of (local) many-body ground-state sensors operating at the ultimate limits of quantum precision.

20 + 16 pages, 8 figures, 4 + 3 tables