  Sideband cooling is a popular method for cooling atoms to the ground state of
an optical trap. Applying the same method to molecules requires a number of
challenges to be overcome. Strong tensor Stark shifts in molecules cause the
optical trapping potential, and corresponding trap frequency, to depend
strongly on rotational, hyperfine and Zeeman state. Consequently, transition
frequencies depend on the motional quantum number and there are additional
heating mechanisms, either of which can be fatal for an effective sideband
cooling scheme. We develop the theory of sideband cooling in state-dependent
potentials, and derive an expression for the heating due to photon scattering.
We calculate the ac Stark shifts of molecular states in the presence of a
magnetic field, and for any polarization. We show that the complexity of
sideband cooling can be greatly reduced by applying a large magnetic field to
eliminate electron- and nuclear-spin degrees of freedom from the problem. We
consider how large the magnetic field needs to be, show that heating can be
managed sufficiently well, and present a simple recipe for cooling to the
ground state of motion.
