  Finite systems of bosons and/or fermions described by the Hubbard model can
be realized using ultracold atoms confined in optical lattices. The ground
states of these systems often exhibit a coexistence of compressible superfluid
and incompressible Mott insulating regimes. We analyze such systems by studying
the out-of-equilibrium dynamics following a weak sudden quench of the trapping
potential. In particular, we show how the temporal variance of the site
occupations reveals the location of spatial boundaries between compressible and
incompressible regions. The feasibility of this approach is demonstrated for
several models using numerical simulations. We first consider integrable
systems, hard-core bosons (spinless fermions) confined by a harmonic potential,
where space separated Mott and superfluid phases coexist. Then, we analyze a
nonintegrable system, a $J-V-V'$ model with coexisting charge density wave and
superfluid phases. We find that the temporal variance of the site occupations
is a more effective measure than other standard indicators of phase boundaries
such as a local compressibility. Based on these examples, we argue that
analyzing temporal fluctuations is a valuable experimental tool for exploring
phase boundaries in trapped atom systems.
