• quantum Fisher information
• quantum phase estimation
• 1D
• short-range effective interaction
• Fermi gas
• entanglement
• multimode cavity

This work is aimed to explore the possibility of realizing the
concept of many-body entanglement in quantum gases platforms and applying it
to phase estimations for metrological use in atomic interferometry experiments.
In particular, we consider the Hamiltonian of an effective two-level system of
ultracold interacting fermionic atoms in one dimension (1D), as it can be re-
alized in optical cavities after adiabatic elimination of the photon degrees of
freedom in conditions of large detuning from the atomic transition. The result-
ing effective interaction among the fermions mediated by the cavity photons,
can be tuned in strength after acting on the photon-atom coupling and on the
detuning, and in range by coupling the atoms to a number of cavity modes with
the same energy: the larger is the number, the shorter is the effective interac-
tion range. Such a platform has been realized in the group of Benjamin Lev
in Stanford, whom we have activated an on-purpose collaboration with. We
model the system with a Fermi-Hubbard Hamiltonian with hopping t and with
on-site U and nearest-neighbor J interactions, which we then solve by means of
Density-Matrix Renormalization Group (DMRG) methods in collaboration
with Davide Rossini (Physics Department, University of Pisa) and Jonathan
Keeling (St. Andrews University, Edinburgh, UK). We determine the quantum
phase diagram of the interacting fermion fluid while varying the two param-
eters U/t and J/t, finding a quite rich behavior including spin-density waves,
ferromagnetic-like and superfluid-like (both in 1D sense) ground states, and a
peculiar phase where particles clusterize. We quantitatively determine the de-
gree of the system metrological usability by calculating the Quantum Fisher
Information across the whole phase diagram, and find the cluster phase to be
especially convenient.