Neutrino has long been regarded as the mysterious particle that is perhaps massless and travels at very close to the speed of light. Yet recent experiments, including the famous KamLAND experiment, has shown to us that neutrinos are very likely not massless and oscillate between three different flavors during their existence.

How were we able to deduce these facts from experimental results? The essential argument here revolves around the Mixing Matrix and one of its important parameters, θ₁₃.

Neutrinos are created and destroyed as weak eigenstates, yet as the neutrinos travel in space, they evolve according to their mass eigenstates, this process effectively mixes up the weak eigenstates of the neutrinos and allow them to turn into eachother (after measurement) as they travel.

Let us first get an idea of how the mass eigenstate evolves. Here is a mass eigenstate , a, of a neutrino at time t=0, in the rest frame (a, b, c denotes the three mass eigenstates).

As time passes it evolves according to the Schrödinger's equation:

And at time τ_a, the mass eigenstate becomes:

As the neutrinos are created, they are created in the weak eigentstates, which can be written as linear combinations of the mass eigenstates, after the neutrino of weak state α moves a distance of L, in the lab frame (α, β, γ denotes the three weak eigenstates):

Now, we are able to find the probability of measuring an oscillation of the initial neutrino weak eigenstate at the a destance L, by the following formula:

What determines the mixing probability is then the mixing matrix elements for this ocillation U_{κ={α, β, γ},k={a, b, c}}, and his mixing matrix can be specified by three mixing angles, θ₃₂, θ₁₂, θ_13.

In nuclear detector experiments, anti-neutrino are produced, by measuring the amount of neutrino oscillation of this particle, we will be able to determine the mixing angle θ₁₃, which is the key to CP violation in the neutrino sector.