Mapping atomic trapping in an optical superlattice onto the libration of a planar rotor in electric fields

Abstract

We show that two seemingly unrelated problems -- the trapping of an atom in a one-dimensional optical superlattice (OSL) formed by the interference of optical lattices whose spatial periods differ by a factor of two, and the libration of a polar polarizable planar rotor (PR) in combined electric and optical fields -- have isomorphic Hamiltonians. Since the OSL gives rise to a periodic potential that acts on atomic translation via the AC Stark effect, it is possible to establish a map between the translations of atoms in this system and the rotations of the PR due to the coupling of the rotor's permanent and induced electric dipole moments with the external fields. The latter system belongs to the class of conditionally quasi-exactly solvable (C-QES) problems in quantum mechanics and shows intriguing spectral properties, such as avoided and genuine crossings, studied in details in our previous works [our works]. We make use of both the spectral characteristics and the quasi-exact solvability to treat ultracold atoms in an optical superlattice as a semifinite-gap system. The band structure of this system follows from the eigenenergies and their genuine and avoided crossings obtained as solutions of the Whittaker--Hill equation. Furthermore, the mapping makes it possible to establish correspondence between concepts developed for the two eigenproblems individually, such as localization on the one hand and orientation/alignment on the other. This correspondence may pave the way to unraveling the dynamics of the OSL system in analytic form.