  The observation of significant dipolar effects in gases of ultra-cold polar
molecules typically demands a strong external electric field to polarize the
molecules. We show that even in the absence of a significant polarization,
dipolar effects may play a crucial role in the physics of polar molecules in
bilayers, provided that the molecules in each layer are initially prepared in a
different rotational state. Then, inter-layer dipolar interactions result in a
non-local swap of the rotational state between molecules in different layers,
even for weak applied electric fields. The inter-layer scattering due to the
dipole-dipole interaction leads to a non-trivial dependence of the swapping
rate on density, temperature, inter-layer spacing, and population imbalance.
For reactive molecules like KRb, chemical recombination immediately follows a
non-local swap and dominates the losses even for temperatures well above
quantum degeneracy, and could be hence observed under current experimental
conditions.
