Dynamics of a Rydberg hydrogen atom near a topologically insulating surface

We investigate the classical dynamics of a Rydberg hydrogen atom near the surface of a planar topological insulator. The system is described by a Hamiltonian consisting of the free-hydrogen part and the hydrogen-surface potential. The latter includes the interactions between the electron and both image electric charges and image magnetic monopoles. Owing to the axial symmetry, the $z$ component of angular momentum $l_z$ is conserved. Here we consider the $l_z=0$ case. The structure of the phase space is explored extensively by means of numerical techniques and Poincar'{e} surfaces of section for the recently discovered topological insulator TlBiSe${}_2$. The phase space of the system is separated into regions of vibrational and rotational motion. We show that vibrational-rotational-vibrational type transitions can be tuned with the topological magnetoelectric polarizability.