Stimulated by the experimental realization of spin-dependent tunneling via gradient magnetic field [Phys. Rev. Lett. 111, 225301 (2013); Phys. Rev. Lett. 111, 185301 (2013)], we investigate dynamics of Bloch oscillations and Landau-Zener tunneling of
single spin-half particles in a periodic potential under the influence of a spin-dependent constant force. In analogy to the Wannier-Stark system, we call our system as the Wannier-Zeeman system. If there is no coupling between the two spin states, the system can be described by two crossing Wannier-Stark ladders with opposite tilts. The spatial crossing between two Wannier-Stark ladders becomes a spatial anti-crossing if the two spin states are coupled by external fields. For a wave-packet away from the spatial anti-crossing, due to the spin-dependent constant force, it will undergo spatial Landau-Zener transitions assisted by the intrinsic intra-band Bloch oscillations, which we call the Bloch-Landau-Zener dynamics. If the inter-spin coupling is sufficiently strong, the system undergoes adiabatic Bloch-Landau-Zener dynamics, in which the spin dynamics follows the local dressed states. Otherwise, for non-strong inter-spin couplings, the system undergoes non-adiabatic Bloch-Landau-Zener dynamics.
We investigate the energy spectrum for hybrid mechanical systems described by non-parity-symmetric quantum Rabi models. A set of analytical solutions in terms of the confluent Heun functions and their analytical energy spectrum are obtained. The anal
ytical energy spectrum includes regular and exceptional parts, which are both confirmed by direct numerical simulation. The regular part is determined by the zeros of the Wronskian for a pair of analytical solutions. The exceptional part is relevant to the isolated exact solutions and its energy eigenvalues are obtained by analyzing the truncation conditions for the confluent Heun functions. By analyzing the energy eigenvalues for exceptional points, we obtain the analytical conditions for the energy-level-crossings, which correspond to two-fold energy degeneracy.
