# Measuring the phase in neutrino oscillations

Hi,

When we talk about neutrino oscillations, the discussion is always about the phase that a particular flavour eigenstate picks up with time. The phase is usually partly geometric and partly dynamic. I have a question about this.

How do we measure the phase experimentally? What is the basic idea behind the setup that allows us to compare on beam of neutrinos which have evolved with a geometric and dynamic phase with a beam of "unevolved" neutrinos? This really puzzles me since most experiments involve straight paths i.e. nowhere does the neutrino beam travel in a circle or something to come back and recombine or something like that. I can see that if we bring them together, then of course, we will get an interference pattern but how do we do this?

Thank you.

## Answers and Replies

Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
The phase is relative to the point of origin, and it is measured by looking at the different composition of flavor eigenstates in the beam. It has nothing to do with an "interference pattern".

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.

Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
Can you write down the equation for neutrino oscillations? If so, can you identify the thing that's the phase?

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 $<v_{\mu}(0)|v_{e}(t)>$. 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 $|<v_{\mu}(0)|v_{e}(t)>|^{2}$ 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

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Bump! Any help on this?

I was also thinking, is it possible to have two seperate coherent neutrino sources? This would be one way of getting an interference pattern between unevolved neutrinos and those neutrinos that have evolved as they travel from the first source to the second.

Thank you.

Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
You're kind of making a mess here. "Phase" is nothing magical - it's the offset you have in a function like cos(x). If you have an oscillation that can be expressed as Acos(something) + Bsin(something else), it's rewritable as Ccos(something + phase).

Nothing more, nothing less.