  We experimentally and theoretically study the peak fraction of a
Bose-Einstein condensate loaded into a cubic optical lattice as the lattice
potential depth and entropy per particle are varied. This system is
well-described by the superfluid regime of the Bose-Hubbard model, which allows
for comparison with mean-field theories and exact quantum Monte Carlo (QMC)
simulations. Despite correcting for systematic discrepancies between condensate
fraction and peak fraction, we discover that the experiment consistently shows
the presence of a condensate at temperatures higher than the critical
temperature predicted by QMC simulations. This metastability suggests that
turning on the lattice potential is non-adiabatic. To confirm this behavior, we
compute the timescales for relaxation in this system, and find that
equilibration times are comparable with the known heating rates. The similarity
of these timescales implies that turning on the lattice potential adiabatically
may be impossible. Our results point to the urgent need for a better
theoretical and experimental understanding of the timescales for relaxation and
adiabaticity in strongly interacting quantum gases, and the importance of
model-independent probes of thermometry in optical lattices.
