Three-Level Landau-Zener Dynamics

We compute Landau-Zener probabilities for 3-level systems with a linear sweep of the uncoupled energy levels of the 3times3 Hamiltonian H(t). Two symmetry classes of Hamiltonians are studied: For H(t) in su(2) (expressible as a linear combination of the three spin 1 matrices), an analytic solution to the problem is obtained in terms of the parabolic cylinder D functions. For H(t) in su(3) (expressible as a linear combination of the eight Gell-Mann matrices), numerical solutions are obtained. In the adiabatic regime, full population transfer is obtained asymptotically at large time, but at intermediate times, all three levels are populated and Stückelberg oscillations are typically manifest. 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 Stückelberg oscillations are suppressed and that the temporal decay law of the population probabilities is not a simple exponential.