Efficient multipole representation for matter-wave optics

Technical optics with matter waves requires a universal description of three-dimensional traps, lenses, and complex matter-wave fields. In analogy to the two-dimensional Zernike expansion in beam optics, we present a three-dimensional multipole expansion for Bose-condensed matter waves and optical devices. We characterize real magnetic chip traps, optical dipole traps, and the complex matter-wave field in terms of spherical harmonics and radial Stringari polynomials. We illustrate this procedure for typical harmonic model potentials as well as real magnetic and optical dipole traps. Eventually, we use the multipole expansion to characterize the aberrations of a ballistically interacting expanding Bose–Einstein condensate in (3 + 1) dimensions. In particular, we find deviations from the quadratic phase ansatz in the popular scaling approximation. The scheme is data efficient by representing millions of complex amplitudes of a field on a Cartesian grid in terms of a low order multipole expansion without precision loss. This universal multipole description of aberrations can be used to optimize matterwave optics setups, for example, in matter-wave interferometers.