Population dynamics and phase effects in periodic level crossings

We present an analytic study of the population dynamics of a two-state system interacting with an external field and subjected to periodic level crossings. We apply an evolution matrix approach to calculate the excited-state population at the crossings (the nodes) and at the antinodes. The results are expressed in terms of only two parameters: the transition probability p for a quarter period from a crossing to an antinode, and the transition probability P for a half period between two successive crossings. We find that the values of the excited-state population at the antinodes can form global (gross) structures. We show that these structures and the population dynamics as a whole are very sensitive to the initial phase φ of the frequency-modulated field, particularly in the limits φ=0 (cosine modulation) and φ=π/2 (sine modulation). We calculate the parameters p and P by using two analytic approaches: one based on the original Landau-Zener model, and the other based on the finite Landau-Zener model. Both approaches unexpectedly lead to the same results. The notion of the global structures and the relevant parametrization in terms of p and P allow us to find various distinctive cases of population dynamics, such as population swapping, completely periodic evolution, superpositional trapping, and stepwise evolution.