MSW Effect Revisted ====================================== .. admonition:: Pauli Matrices and Rotations :class: note Given a rotation .. math:: U = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin\theta & \cos \theta \end{pmatrix}, its effect on Pauli matrices are .. math:: U^\dagger \sigma_3 U &=\cos 2\theta \sigma_3 + \sin 2\theta \sigma_1 \\ U^\dagger \sigma_1 U & = -\sin 2\theta \sigma_3 + \cos 2\theta \sigma_1. Flavor Basis ------------------------- .. admonition:: Vacuum Oscillations :class: note Vacuum oscillations is already a Rabi oscillation at resonance with oscillation width :math:`\omega_v \sin 2\theta_v`. Neutrino oscillation in matter has a Hamiltonian in flavor basis .. math:: H^{(f)} = \left(- \frac{1}{2} \omega_v \cos 2\theta_v +\frac{1}{2}\lambda(x) \right)\sigma_3 + \frac{1}{2} \omega_v \sin 2\theta_v \sigma_1. The Schroding equation is .. math:: i \partial_x \Psi^{(f)} = H^{(f)} \Psi^{(f)}. To make connections to Rabi oscillations, we would like to remove the changing :math:`\sigma_3` terms, using a transformation .. math:: T = \begin{pmatrix} e^{-i \eta (x)} & 0 \\ 0 & e^{i \eta (x)} \end{pmatrix}, which transform the flavor basis to another basis .. math:: \begin{pmatrix} \psi_e \\ \psi_x \end{pmatrix} = \begin{pmatrix} e^{-i \eta (x)} & 0 \\ 0 & e^{i \eta (x)} \end{pmatrix} \begin{pmatrix} \psi_{a} \\ \psi_{b} \end{pmatrix}. The Schrodinger equation can be written into this new basis .. math:: i \partial_x (T \Psi^{(r)}) = H^{(f)} T\Psi^{(r)}, which is simplified to .. math:: i \partial_x \Psi^{(r)} = H^{(r)} \Psi^{(r)}, where .. math:: H^{(r)} = - \frac{1}{2}\omega_v \cos 2\theta_v \sigma_3 + \frac{1}{2} \omega_v \sin 2\theta_v \begin{pmatrix} 0 & e^{2i\eta(x)} \\ e^{-2i\eta(x)} & 0 \\ \end{pmatrix}, in which we remove the varying component of :math:`\sigma_3` elements using .. math:: \frac{d}{dx}\eta(x) = \frac{\lambda(x)}{2}. The final Hamiltonian would have some form .. math:: H^{(r)} = - \frac{1}{2}\omega_v \cos 2\theta_v \sigma_3 + \frac{1}{2} \omega_v \sin 2\theta_v \begin{pmatrix} 0 & e^{i\int_0^x \lambda(\tau)d\tau + 2i\eta(0)} \\ e^{-i\int_0^x \lambda(\tau)d\tau - 2i\eta(0)} & 0 \\ \end{pmatrix}, where :math:`\eta(0)` is chosen to conter the constant terms from the integral. 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 :math:`\exp{\left(i\int_0^x \lambda(\tau)d\tau\right)}` into many summations of :math:`\exp{\left( i a x + b \right)}` would be enough for an Rabi oscillation interpretation. Let's discuss the constant matter profile, :math:`\lambda(x) = \lambda_0`. Thus we have .. math:: \eta(x) = \frac{1}{2} \lambda_0 x. The Hamiltonian becomes .. math:: H^{(r)} = - \frac{1}{2}\omega_v \cos 2\theta_v \sigma_3 + \frac{1}{2} \omega_v \sin 2\theta_v \begin{pmatrix} 0 & e^{i\lambda_0 x} \\ e^{-i\lambda_0 x} & 0 \\ \end{pmatrix}, which is exactly a Rabi oscillation. The resonance condition is .. math:: \lambda_0 = \omega_v \cos 2\theta_v. Instanteneous Matter Basis ------------------------------------------------ Neutrino oscillation in matter has a Hamiltonian in flavor basis .. math:: H^{(f)} = \left(- \frac{1}{2} \omega_v \cos 2\theta_v +\frac{1}{2}\lambda(x) \right)\sigma_3 + \frac{1}{2} \omega_v \sin 2\theta_v \sigma_1. The Schroding equation is .. math:: i \partial_x \Psi^{(f)} = H^{(f)} \Psi^{(f)}, which can be transformed to instantaneous matter basis by applying a rotation :math:`U`, .. math:: i \partial_x \left( U\Psi^{(m)} \right)= H^{(f)} U\Psi^{(m)}, where .. math:: U = \begin{pmatrix} \cos \theta_m & \sin \theta_m \\ -\sin\theta_m & \cos \theta_m \end{pmatrix}. With a little algebra, we can write the system into .. math:: i \partial _x \Psi^{(m)} = H^{(m)}\Psi^{(m)} .. math:: H^{(m)} = U^\dagger H^{(f)} U - i U^\dagger \partial_x U. By setting the off-diagonal elements of the first term :math:`U^\dagger H^{(f)} U` to zero, we can derive the relation .. math:: \tan 2\theta_m = \frac{\sin 2\theta_v}{\cos 2\theta_v - \lambda/\omega_v}. Furthermore, we derive the term .. math:: i U^\dagger \partial_x U = - \dot\theta_m \sigma_2. We can calculate :math:`\dot\theta_m` by taking the derivative of :math:`\tan 2\theta_m`, .. math:: \frac{d}{dx} \tan 2\theta_m = \frac{2}{\cos^2 2\theta_m} \dot\theta_m, so that .. math:: \dot\theta_m &= \frac{1}{2} \cos^2 (2\theta_m) \frac{d}{dx} \tan 2\theta_m \\ & = \frac{1}{2} \frac{(\cos 2\theta_v - \lambda/\omega_v)^2}{ (\lambda/\omega_v)^2 + 1 - 2\lambda \cos 2\theta_v /\omega_v } \frac{d}{dx} \frac{\sin 2\theta_v}{\cos 2\theta_v - \lambda/\omega_v} \\ & = \frac{1}{2} \frac{(\cos 2\theta_v - \lambda/\omega_v)^2}{ (\lambda/\omega_v)^2 + 1 - 2\lambda \cos 2\theta_v /\omega_v } \frac{\sin 2\theta_v}{(\cos 2\theta_v - \lambda/\omega_v)^2} \frac{1}{\omega)v} \frac{d}{dx} \lambda(x) \\ & = \frac{1}{2} \sin 2\theta_m \frac{1}{\omega_m} \frac{d}{dx} \lambda(x).