  We extend our analysis of the effects of the interplay of quantum phases and
nonlinearity to address saturation effects in small quantum systems.
  We find that initial phases dramatically control the dependence of
self-trapping on initial asymmetry of quasiparticle population and can compete
or act with nonlinearity as well as saturation effects. We find that there is a
minimum finite saturation value in order to obtain self-trapping that crucially
depends on the initial quasiparticle phases and present a detailed
phase-diagram in terms of the control parameters of the system: nonlinearity
and saturation.
