High-multipole excitations of hydrogen-like atoms by twisted photons near a phase singularity

We calculate transition amplitudes and cross sections for excitation of hydrogen-like atoms by the twisted photon states, or photon states with angular momentum projection on the direction of propagation exceeding (h) over bar. If the target atom is located at distances of the order of atomic size near the phase singularity in the vortex center, the transition rates into the states with orbital angular momentum l(f) > 1 become comparable with the rates for electric dipole transitions. It is shown that when the transition rates are normalized to the local photon flux, the resulting cross sections for l(f) > 1 are singular near the optical vortex center. The relation to the 'quantum core' concept introduced by Berry and Dennis is discussed.