  We suggest and analyze a new scheme to adiabatically cool bosonic atoms to
picokelvin temperatures which should allow the observation of magnetic ordering
via superexchange in optical lattices. The starting point is a gapped phase
called the spin Mott phase where each site is occupied by one spin-up and one
spin-down atom. An adiabatic ramp leads to an xy-ferromagnetic phase. We show
that the combination of time-dependent density matrix renormalization group
methods with quantum trajectories can be used to fully address possible
experimental limitations due to decoherence, and demonstrate that the magnetic
correlations are robust for experimentally realizable ramp speeds. Using a
microscopic master equation treatment of light scattering in the many-particle
system, we test the robustness of adiabatic state preparation against
decoherence. Due to different ground-state symmetries, we also find a
metastable state with xy-ferromagnetic order if the ramp crosses to regimes
where the ground state is a z-ferromagnet. The bosonic spin Mott phase as the
initial gapped state for adiabatic cooling has many features in common with a
fermionic band insulator, but the use of bosons should enable experiments with
substantially lower initial entropies.
