  In contrast to interferometry-based quantum sensing, where interparticle
interaction is detrimental, quantum many-body probes exploit such interactions
to achieve quantum-enhanced sensitivity. In most of the studied quantum
many-body probes, the interaction is considered to be short-ranged. Here, we
investigate the impact of long-range interaction at various filling factors on
the performance of Stark quantum probes for measuring a small gradient field.
These probes harness the ground state Stark localization phase transition which
happens at an infinitesimal gradient field as the system size increases. Our
results show that while super-Heisenberg precision is always achievable in all
ranges of interaction, the long-range interacting Stark probe reveals two
distinct behaviors. First, by algebraically increasing the range of
interaction, the localization power enhances and thus the sensitivity of the
probe decreases. Second, as the interaction range becomes close to a fully
connected graph its effective localization power disappears and thus the
sensitivity of the probe starts to enhance again. The super-Heisenberg
precision is achievable throughout the extended phase until the transition
point and remains valid even when the state preparation time is incorporated in
the resource analysis. As the probe enters the localized phase, the sensitivity
decreases and its performance becomes size-independent, following a universal
behavior. In addition, our analysis shows that lower filling factors lead to
better precision for measuring weak gradient fields.
