We consider the many-body system of neutrinos interacting with each other through neutral current weak force. Emerging many-body effects in such a system could play important roles in some astrophysical sites such as the core collapse supernovae. In the literature this many-body system is usually treated within the mean field approximation which is an effective one-body description based on omitting entangled neutrino states. In this paper, we consider the original many-body system in an effective two flavor mixing scenario under the single angle approximation and present a solution without using the mean field approximation. Our solution is formulated around a special class of many-body eigenstates which do not undergo any level crossings as the neutrino self-interaction rate decreases while the neutrinos radiate from the supernova. In particular, an initial state which consists of electron neutrinos and antineutrinos of an orthogonal flavor can be entirely decomposed in terms of those eigenstates. Assuming that the conditions are perfectly adiabatic so that the evolution of these eigenstates follow their variation with the interaction rate, we show that this initial state develops a spectral split at exactly the same energy predicted by the mean field formulation.
We show that the spectral split of a neutrino ensemble which initially
consists of electron type neutrinos, is analogous to the BCS-BEC crossover
already observed in ultra cold atomic gas experiments. Such a neutrino ensemble
mimics the deleptonization burst of a core collapse supernova. Although these
two phenomena belong to very different domains of physics, the propagation of
neutrinos from highly interacting inner regions of the supernova to the vacuum
is reminiscent of the evolution of Cooper pairs between weak and strong
interaction regimes during the crossover. The Hamiltonians and the
corresponding many-body states undergo very similar transformations if one
replaces the pair quasispin of the latter with the neutrino isospin of the
former.Comment: 9 pages, 5 figure
We consider the entanglement of neutrinos evolving adiabatically under the effect of vacuum oscillations and self interactions through decreasing neutrino density similar to a core collapse supernova. For an initial state which consists only of electron type neutrinos, we analytically calculate the asymptotic value of the entropy of entanglement between the lowest energy neutrino and the rest of the system as a function of the number of neutrinos. We find that, as we increase neutrino number the entanglement entropy grows at first. But after going through a maximum, it approaches to zero in the limit of infinite number of neutrinos. We find that the number of neutrinos for which the entropy maximizes depends on the mixing angle.
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