We consider quantum systems with a Hamiltonian containing a weak perturbation, i.e., $\mathit{H}={\mathit{H}}_{\mathit{0}}+\mathit{\lambda }·\tilde{\mathit{H}}, \mathit{\lambda }=\{{\lambda }_1,{\lambda }_2,...,\}, \tilde{\mathit{H}} =\{H_1,H_2,...,\}, \left|\mathit{\lambda }\right|\ll 1$, and address situations where $\tilde{\mathit{H}}$ is known but the values of the couplings $\mathit{\lambda }$ are unknown and should be determined by performing measurements on the system. We consider two scenarios: in the first one we assume that measurements are performed on a given stationary state of the system, e.g., the ground state, whereas in the second one an initial state is prepared and then measured after evolution. In both cases, we look for the optimal measurements to estimate the couplings and evaluate the ultimate limits to precision. In particular, we derive general results for one and two couplings and analyze in detail some specific qubit models. Our results indicate that dynamical estimation schemes may provide enhanced precision upon a suitable choice of the initial preparation and the interaction time.

• Received 13 October 2023
• Accepted 11 March 2024

DOI:https://doi.org/10.1103/PhysRevA.109.032427