  Interaction of atoms with twisted light is the subject of intense
experimental and theoretical investigation. In almost all studies, the atom is
viewed as a localized probe of the twisted light field. However, as argued in
this paper, conceptually novel effects will arise if light-atom interaction is
studied in the double-twisted regime with delocalized atoms, that is, either
via twisted light absorption by atom vortex beam, or via two-twisted-photon
spectroscopy of atoms in a non-vortex but delocalized state. Even for
monochromatic twisted photons and for an infinitely narrow line, absorption
will occur over a finite range of detuning. Inside this range, a rapidly
varying absorption probability is predicted, revealing interference fringes
induced by two distinct paths leading to the same final state. The number,
location, height and contrast of these fringes can give additional information
on the excitation process which would not be accessible in usual spectroscopic
settings. Visibility of the predicted effects will be enhanced at the future
Gamma factory thanks to the large momenta of ions.
