  We propose and investigate a scheme for engineering a synthetic thermal bath
for a bosonic quantum gas in a one-dimensional optical lattice based on
Markovian feedback control. The performance of our scheme is quantified by the
fidelity between the steady state of the system and the effective thermal
state. For double-well and triple-well systems with non-interacting particles,
the steady state is found to be an exact thermal state, which is attributed to
the fact that the transfer rates between all pairs of coupled eigenstates
satisfy detailed balance condition. The scenario changes when there are more
lattice sites, where the detailed balance condition does not hold any more, but
remains an accurate approximation. Remarkably, our scheme performs very well at
low and high temperature regimes, with the fidelity close to one. The
performance at the intermediate temperature regime (where a crossover into a
Bose condensed regime occurs) is slightly worse, and the fidelity shows a
gentle decrease with increasing system size. We also discuss the interacting
cases. In contrast to the non-interacting cases, the scheme is found to perform
better at a higher temperature. Another difference is that the minimal
temperature that can be engineered is nonzero and increases with the
interaction strength.
