Three-level Landau-Zener dynamics

We compute Landau-Zener probabilities for three-level systems with a linear sweep of the uncoupled energy levels of the 3 x 3 Hamiltonian matrix H(t). Two symmetry classes of Hamiltonians are studied: For H(t) is an element of su(2) (expressible as a linear combination of the three spin-1 matrices), an analytic solution to the dynamical problem is obtained in terms of the parabolic cylinder D functions. For H(t)is an element of su(3) (expressible as a linear combination of the eight Gell-Mann matrices), numerical solutions are calculated. In the adiabatic regime, full population transfer is obtained asymptotically at large times, but at intermediate times, all three levels are populated, and Stuckelberg oscillations can manifest from the occurrence of two avoided crossings. For the open system, (wherein interaction with a reservoir occurs), we numerically solve a Markovian quantum master equation for the density matrix with Lindblad operators that models interaction with isotropic white Gaussian noise. We find that Stuckelberg oscillations are suppressed and that the decoherence cannot be modeled in terms of a simple exponential.