Monodromy and the structure of the energy spectrum of hydrogen in near perpendicular electric and magnetic fields

Abstract

We examine the energy spectrum of hydrogen in weak near-perpendicular electric and magnetic fields using quantum computations and semiclassical analysis. The spectral structure is displayed in a lattice constructed by plotting the difference between total energy and first order energy versus first order energy, for all states of a given principal quantum number $n$. We have used classical perturbation theory to derive three approximate constants of the motion. Quantization of the resulting approximate actions produces a corresponding lattice in agreement with our quantum calculations. For some field arrangements, one of the actions is an intrinsically multivalued function of the constants of the motion. This produces defects in the lattices and signifies the presence of a classical phenomenon called monodromy. The presence of monodromy in the case of exactly perpendicular fields was previously predicted by Sadovski\'{\i} and Cushman. Our quantum calculations and independent semiclassical analysis confirm their prediction, and show that the presence of monodromy not only persists into near-perpendicular fields, but also assumes other manifestations over a wider range of field strength ratios. We find that in near-perpendicular fields the structure of the spectrum is divided into six distinct parameter regions, which we characterize by the presence and type of lattice defect.