  We analytically study the effect of gravitational and harmonic forces on
ultra-cold atoms with synthetic spin-orbit coupling (SOC). In particular, we
focus on the recently observed transitions between internal states induced by
acceleration of the external modes. Our description corresponds to a
generalized version of the Landau-Zener (LZ) model: the dimensionality is
enlarged to combine the quantum treatment of the external variables with the
internal-state characterization; additionally, atomic-interaction effects are
considered. The emergence of the basic model is analytically traced. Namely, by
using a sequence of unitary transformations and a subsequent reduction to the
spin space, the SOC Hamiltonian, with the gravitational potential incorporated,
is exactly converted into the primary LZ scenario. Moreover, the transitions
induced by harmonic acceleration are approximately cast into the framework of
the basic LZ model through a complete analytical procedure. We evaluate how the
validity of our picture depends on the system preparation and on the magnitude
of atomic-interaction effects. The identification of the regime of
applicability and the rigorous characterization of the parameters of the
effective model provide elements to control the transitions.
