Phase effects in neutrino conversions during a supernova shock wave

Neutrinos escaping from a core collapse supernova a few seconds after bounce pass through the shock wave, where they may encounter one or more resonances corresponding to $\Delta m^2_{atm}$. The neutrino mass eigenstates in matter stay coherent between these multiple resonances, giving rise to oscillations in the survival probabilities of neutrino species. We point out these novel ``phase effects'', and provide an analytic approximation to them that relates the density profile of the shock wave to the oscillation pattern. The phase effects can be strong if the multiple resonances encountered by neutrinos are semi-adiabatic, which typically happens for $10^{-5} \lsim \sin^2 \theta_{13} \lsim 10^{-3}$. These effects would manifest themselves in oscillations in the observed neutrino spectra for certain neutrino mixing scenarios. The time dependent neutrino spectra from a future galactic supernova can then be used to identify the neutrino mixing scenario as well as reconstruct the density profile in the region around the reverse shock