  We discuss the rotational cooling of diatomic molecules in a Bose-Einstein
condensate (BEC) of ultra-cold atoms by emission of phonons with orbital
angular momentum. Despite the superfluidity of the BEC there is no frictionless
rotation for typical molecules since the dominant cooling occurs via emission
of particle-like phonons. Only for macro-dimers, whose size becomes comparable
or larger than the condensate healing length, a Landau-like, critical angular
momentum exists below which phonon emission is suppressed. We find that the
rotational relaxation of typical molecules is in general faster than the
cooling of the linear motion of impurities in a BEC. This also leads to a
finite lifetime of angulons, quasi-particles of rotating molecules coupled to
phonons with orbital angular-momentum. We analyze the dynamics of rotational
cooling for homo-nuclear diatomic molecules based on a quantum Boltzmann
equation including single- and two-phonon scattering and discuss the effect of
thermal phonons.
