  The force estimation problem in quantum metrology with an arbitrary
non-Markovian Gaussian bath is considered. No assumptions are made on the bath
spectrum and coupling strength with the probe. Considering the natural global
unitary evolution of both bath and probe and assuming initial global Gaussian
states we are able to solve the main issues of any quantum metrological
problem: the best achievable precision determined by the quantum Fisher
information, the best initial state and the best measurement. Studying the
short time behavior and comparing to regular Markovian dynamics we observe an
increase of quantum Fisher information. We emphasize that this phenomenon is
due to the ability to perform measurements below the correlation time of the
bath, activating non-Markovian effects. This brings huge consequences for the
sequential preparation-and- measurement scenario as the quantum Fisher
information becomes unbounded when the initial probe mean energy goes to
infinity, whereas its Markovian counterpart remains bounded by a constant. The
long time behavior shows the complexity and potential variety of non-Markovian
effects, somewhere between the exponential decay characteristic of Markovian
dynamics and the sinusoidal oscillations characteristic of resonant narrow
bands.
