Measuring the phase in neutrino oscillations

by McLaren Rulez
Tags: measuring, neutrino, oscillations, phase
 P: 261 Thank you for replying, Vanadium 50. In Griffiths' Quantum Mechanics, he mentions that a beam of particles (all in the same state $\psi$) can be split into two and one of the two acquires a relative phase to the other, then the beam, when recombined is of the form $\Psi = \frac{1}{2}(\psi + e^{i\phi}\psi)$. Now, we see that $|\Psi|^{2}= |\psi|^{2}cos^{2}(\frac{\phi}{2})$ which is an interference pattern. I assumed this was what happened with neutrinos also. I don't quite understand how we can deduce the phase by looking at the different compositions of flavour eigenstates in the beam. For instance, if we start with $|\nu_{e}(0)>$ and it acquires a phase $e^{i\phi}$ after time t, then we have $e^{i\phi}|\nu_{e}(t)>$. Now, if we try to take any measurement $|<\nu_{\alpha}(0)|e^{i\phi}|\nu_{e}(t)>|^{2}$, where $\alpha = \mu, e$ or $\tau$ the phase will cancel away. So how can we measure it? Thank you for your help.
 P: 261 I'm not sure what exactly the equation you have in mind is. Basically, I consider a flavour eigenstate to start with, say the electron neutrino. The case considered here is a two neutrino mixing case. I express it at t=0 as follows $|v_{e}(0)>= cos\theta |v_{1}>+sin\theta |v_{2}>$ After time t, it evolves into $|v_{e}(t)>= e^{-i\omega_{1} t}cos\theta |v_{1}>+e^{-i\omega_{2} t}sin\theta |v_{2}>$ Suppose we consider a time $T = \frac{2\pi}{\omega_{2} - \omega_{1}}$. Then, we see that we can express the previous relation as $|v_{e}(t)>= e^{i\phi}|v_{e}(0)>$, where $\phi=\frac{-2\pi \omega_{1}}{\omega_{2} - \omega_{1}}$ $\phi$ is my phase, the one I want to measure. After any time which is an integer multiple of $T$ we will get this form for the relation i.e. $|v_{e}(T)>= e^{in\phi}|v_{e}(0)>$ The previous analysis can be generalized to arbitrary t. Consider the inner product of this state with a muon flavour eigenstate . This is a complex number, so let's call it $re^{i\phi}$. Again, the $\phi$ is my phase. Let's say I want to see what is the probability of finding a muon neutrino at time t if I measure it. Then that probability is given by $||^{2}$ which is $r^2$. I saw all this derivation in a paper which then proceeded to work out the $r$ and $\phi$ for each case (electron-electron, electron-muon, etc.) It turns out that both the $r$ and $\phi$ are functions of the mixing angle $\theta$ and the author proposed that by measuring the phase, one could work out the mixing angle. In that paper, the author considers a slightly different case; he seperates the dynamic and geometric phase and the $\phi$ he uses is actualy the geometric phase. Nonetheless, my basic question is how to measure any type of phase for neutrinos. The reference is http://prd.aps.org/abstract/PRD/v63/i5/e053003