  Selection of "magic" trapping conditions with ultracold atoms or molecules,
where pairs of internal states experience identical trapping potentials, brings
substantial benefits to precision measurements and quantum computing schemes.
Working at such conditions could ensure that detrimental effects of inevitable
inhomogeneities across an ultracold sample are significantly reduced. However,
this aspect of confinement remains unexplored for ultracold polar molecules.
Here, we present means to control the AC Stark shift of rotational states of
ultracold polar molecules, when subjected to both trapping laser light and an
external electric field. We show that both the strength and relative
orientation of the two fields influence the trapping potential. In particular,
we predict "magic electric field strengths" and a "magic angle", where the
Stark shift is independent of the DC external field and rotational states of
the molecule.
