Pauli Matrices and Rotations
Given a rotation
its effect on Pauli matrices are
Vacuum Oscillations
Vacuum oscillations is already a Rabi oscillation at resonance with oscillation width
Neutrino oscillation in matter has a Hamiltonian in flavor basis
The Schroding equation is
To make connections to Rabi oscillations, we would like to remove the changing
which transform the flavor basis to another basis
The Schrodinger equation can be written into this new basis
which is simplified to
where
in which we remove the varying component of
The final Hamiltonian would have some form
where
For arbitary matter profile, we could first apply Fourier expand the profile into trig function then use Jacobi-Anger expansion so that the system becomes a lot of Rabi oscillations.
Any transformations or expansions that decompose
Let’s discuss the constant matter profile,
The Hamiltonian becomes
which is exactly a Rabi oscillation. The resonance condition is
Neutrino oscillation in matter has a Hamiltonian in flavor basis
The Schroding equation is
which can be transformed to instantaneous matter basis by applying a rotation
where
With a little algebra, we can write the system into
By setting the off-diagonal elements of the first term
Furthermore, we derive the term
We can calculate
so that
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