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 Δmₐₜₘ². The neutrino mass eigenstates in matter may stay coherent between these multiple resonances, giving rise to oscillations in the survival probabilities of neutrino species. We provide an analytical approximation to these inevitable phase effects, that relates the density profile of the shock wave to the oscillation pattern. The phase effects are present only if the multiple resonances encountered by neutrinos are semiadiabatic, which typically happens for 10^-5sin²θ₁₃10^-3. The observability of these oscillations is severely limited by the inability of the detectors to reconstruct the neutrino energy faithfully. For typical shock wave profiles, the detection of these phase effects seems rather unlikely. However, if the effects are indeed identified in the νₑ spectra, they would establish inverted hierarchy and a nonzero value of θ₁₃.