  We show that the quasi-adiabatic evolution of a system governed by the Dicke
Hamiltonian can be described in terms of a self-induced quantum many-body
metrological protocol. This effect relies on the sensitivity of the ground
state to a small symmetry-breaking perturbation at the quantum phase
transition, that leads to the collapse of the wavefunciton into one of two
possible ground states. The scaling of the final state properties with the
number of atoms and with the intensity of the symmetry breaking field, can be
interpreted in terms of the precession time of an effective quantum
metrological protocol. We show that our ideas can be tested with spin-phonon
interactions in trapped ion setups. Our work points to a classification of
quantum phase transitions in terms of the capability of many-body quantum
systems for parameter estimation.
