  We study the transition from hadronic matter to a mixed phase of quarks and
hadrons at high baryon and isospin densities reached in heavy ion collisions.
We focus our attention on the role played by the nucleon symmetry energy at
high density.In this respect the inclusion of a scalar isovector meson, the
\delta-coupling, in the Hadron Lagrangian appears rather important. We study in
detail the formation of a drop of quark matter in the mixed phase, and we
discuss the effects on the quark drop nucleation probability of the finite size
and finite time duration of the high density region. We find that, if the
parameters of quark models are fixed so that the existence of quark stars is
allowed, then the density at which a mixed phase starts forming drops
dramatically in the range Z/A \sim 0.3--0.4. This opens the possibility to
verify the Witten-Bodmer hypothesis on absolute stability of quark matter using
ground-based experiments in which neutron-rich nuclei are employed. These
experiments can also provide rather stringent constraints on the Equation of
State (EoS) to be used for describing the pre-Supernova gravitational collapse.
Consistent simulations of neutron rich heavy ion collisions are performed in
order to show that even at relatively low energies, in the few AGeV range, the
system can enter such unstable mixed phase. Some precursor observables are
suggested, in particular a ``neutron trapping'' effect.
