Coherent time evolution on a grid of Landau-Zener anticrossings

A model of the time evolution of two interacting Rydberg manifolds of energy levels subject to a linearly ramped electric field is solved exactly in the Landau-Zener (LZ) approximation. Each manifold’s levels are treated as linear in time, parallel, equally spaced, and infinite in number. Their pairwise interactions produce a regular two-dimensional grid of isolated anticrossings. The time development of an initially populated state is then governed by two-level LZ transitions at avoided crossings and adiabatic evolution between them, parametrized by the LZ transition probability $D$ and a dynamical phase unit φ. The resulting probability distributions of levels are given analytically in the form of recursion relations, generating functions, integral representations involving $D$ and φ, and in certain limits by Bessel or Whittaker functions. Level populations are mapped out versus location on the grid for a range of cases. Interference effects lead to two principal types of probability distributions: a braiding adiabatic pattern with revivals for small $D$ and a diabatic pattern for $D\to 1$ in which only certain levels parallel to the initial one are appreciably populated. The sensitivity of the coherent evolution to φ is discussed, along with the relation of this model to others and to selective-field ionization.