Incoherent time evolution on a grid of Landau-Zener anticrossings

The mixing of Rydberg manifolds, induced by a ramped electric field F(t)=Ḟt, is modeled by interactions among two intersecting groups of parallel energy levels. The levels form a grid, each node of which is treated as an isolated two-level Landau-Zener anticrossing, characterized by probabilities D=exp(-2πγ) and A=1-D for diabatic and adiabatic transitions, respectively. The model assumes a core with one nonvanishing quantum defect ${\mathrm{\mu }}_0$ and with extremal Stark-level slopes, so that γ=‖${\mathrm{\mu }}_0$ ${\mathrm{\Vert }}^2$/3Ḟ${\mathit{n}}^{10}$ for all anticrossings, where the manifolds have principal quantum numbers n and n+1. Interference effects are ignored, which allows analytical treatment of the system’s evolution using path statistics or recursion relations. An initially populated edge state leads to a single-humped distribution of level populations. Analytical expressions are found for the probability distribution as a function of time, as well as its average and standard deviation. Limiting forms of the distribution at large times and in the diabatic (D→1) and adiabatic (D→0) limits are also given. These features are contrasted with those of a random walk. The relevance of the model to selective-field ionization is discussed.