  We study the coherent dynamic interaction of a time-dependent spin-splitting
field with the homogeneous superconducting order parameter $\Delta(t)$ mediated
by spin-orbit coupling using the time-dependent Bogoliubov-de Gennes theory. In
the first part of the work we show that linear response of the superconductor
is strongly affected by the Zeeman field and spin-flip processes, giving rise
to multiple resonant frequencies of the superconducting Higgs modes. These
modes can be excited either by a quench, or by an additional non-stationary
component of the spin-splitting field, which couples linearly to the Higgs
modes. In the second part, we analyze the nonadiabatic dynamics of
quasiparticle states arising from the intersection of spectral branches from
different spin subbands, which can be provoked by a linearly growing Zeeman
field. We provide insights into the dependence of the order parameter
$\Delta(t)$ on this field and interference effects caused by tunneling of
states at the avoided crossing points. We also show that since the nonadiabatic
tunneling is related to spin-flip processes, the quasiparticle gas experiences
a dynamic magnetization that contributes to its spin susceptibility.
