  Motivated by its role as a central pillar of current theories of dynamics of
spin ice in and out of equilibrium, we study the single-ion dynamics of the
magnetic rare earth ions in their local environments, subject to the effective
fields set up by the magnetic moments they interact with. This effective field
has a transverse component with respect to the local easy-axis of the crystal
electric field, which can induce quantum tunnelling. We go beyond the
projective spin-1/2 picture and use instead the full crystal-field Hamiltonian.
We find that the Kramers vs non-Kramers nature, as well as the symmetries of
the crystal-field Hamiltonian, result in different perturbative behaviour at
small fields ($\lesssim 1$ T), with transverse field effects being more
pronounced in Ho$_{2}$Ti$_{2}$O$_{7}$ than in Dy$_{2}$Ti$_{2}$O$_{7}$.
Remarkably, the energy splitting range we find is consistent with time scales
extracted from experiments. We also present a study of the static magnetic
response which highlights the anisotropy of the system in the form of an
off-diagonal $g$ tensor and we investigate the effects of thermal fluctuations
in the temperature regime of relevance to experiments. We show that there is a
narrow yet accessible window of experimental parameters where the anisotropic
response can be observed.
