Entanglement in three-flavor collective neutrino oscillations

Extreme conditions present in the interiors of the core-collapse supernovae make neutrino-neutrino interactions not only feasible but dominant in specific regions, leading to the nonlinear evolution of the neutrino flavor. Results obtained when such collective neutrino oscillations are treated in the mean-field approximation deviate from the results using the many-body picture because of the ignored quantum correlations. We present the first three-flavor many-body calculations of the collective neutrino oscillations. The entanglement is quantified in terms of the entanglement entropy and the components of the polarization vector. We propose a qualitative measure of entanglement in terms of flavor-lepton number conserved quantities. We find that in the cases considered in the present work, the entanglement can be underestimated in the two-flavor approximation. The dependence of the entanglement on mass ordering is also investigated. We also explore the mixing of mass eigenstates in different mass orderings.

Figure 1

The comparison of the probability of finding the $q=2$ frequency mode neutrino in ${\nu }_1$ mass eigenstate ($P_{{\nu }_1}$) in the three-flavor and the two-flavor case. Different mixing angles ${\theta }_{12}$ and ${\theta }_{13}$ are considered in the two-flavor case. The results are for $N=5$ neutrinos with initial state $|\psi ⟩=|{\nu }_e{\nu }_e{\nu }_e{\nu }_e{\nu }_e⟩$ in NO.

Figure 2

Left panels: the temporal evolution of $P_3$ (top), $P_8$ (middle) and $S$ (bottom) for $N=5$ neutrinos with initial state $|\psi ⟩=|{\nu }_e{\nu }_e{\nu }_e{\nu }_e{\nu }_e⟩$ in NO [left] and IO [right]. Right panels: the temporal evolution of $P_{{\nu }_1}$ (top), $P_{{\nu }_2}$ (middle) and $P_{{\nu }_3}$ (bottom) for $N=5$ neutrinos with initial state $|\psi ⟩=|{\nu }_e{\nu }_e{\nu }_e{\nu }_e{\nu }_e⟩$ in NO [left] and IO [right].

Figure 3

Left panels: the temporal evolution of $P_3$ (top), $P_8$ (middle) and $S$ (bottom) for $N=3$ neutrinos with initial state $|\psi ⟩=|{\nu }_e{\nu }_{\mu }{\nu }_{\tau }⟩$ in NO (left) and IO (right). Right panels: the temporal evolution of $P_{{\nu }_1}$ (top), $P_{{\nu }_2}$ (middle) and $P_{{\nu }_3}$ (bottom) for $N=3$ neutrinos with initial state $|\psi ⟩=|{\nu }_e{\nu }_{\mu }{\nu }_{\tau }⟩$ in NO (left) and IO (right). Red solid, green dash-dotted and blue dashed lines represent the first, second and third neutrino, respectively.

Figure 4

The temporal evolution of $P_3$ (top), $P_8$ (middle) and $S$ (bottom) for $N=5$ neutrinos with initial state $|\psi ⟩=|{\nu }_e{\nu }_e{\nu }_{\mu }{\nu }_{\mu }{\nu }_{\tau }⟩$ in NO (left) and IO (right). Right panels: The temporal evolution $P_{{\nu }_1}$ (top), $P_{{\nu }_2}$ (middle) and $P_{{\nu }_3}$ (bottom) for $N=5$ neutrinos with initial state $|\psi ⟩=|{\nu }_e{\nu }_e{\nu }_{\mu }{\nu }_{\mu }{\nu }_{\tau }⟩$ in NO (left) and IO (right).

Figure 5

The asymptotic $P_3$ and $P_8$ values in the ${\widehat{e}}_3\text{−}{\widehat{e}}_8$ plane for $N=3$ (left panel) and $N=5$ (right panel) neutrinos with pure and mixed initial states in both (NO) and (IO). The points are labeled by the neutrino frequency mode, i.e., $q$.