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 an optical superlattice (OSL) and the libration of a planar rigid rotor in combined electric and optical fields (PR) – have isomorphic Hamiltonians. Formed by the interference of optical lattices whose spatial periods differ by a factor of two, OSL gives rise to a periodic potential that acts on atomic translation via the AC Stark effect. The PR system, also known as the generalized planar pendulum, is realized by subjecting a planar rigid rotor to combined orienting and aligning interactions due to the coupling of the rotor's permanent and induced electric dipole moments to the combined fields. The PR system belongs to the class of conditionally quasi-exactly solvable problems and exhibits intriguing spectral properties that have been established in our previous work Becker et al (2017 Eur. Phys. J. D 71 149). Herein, we make use of both the PR $\mapsto$ OSL mapping and the quasi-exact solvability of the PR system to treat ultracold atoms in an OSL as a semifinite-gap system. The band structure of this system follows from the eigenenergies and their genuine and avoided crossings obtained previously for the PR system as solutions of the Whittaker-Hill equation. These solutions characterize both the squeezing and the tunneling of atoms trapped in an OSL and pave the way to unraveling their dynamics in analytic form. Furthermore, the PR $\mapsto$ OSL 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.