# 17.2. Decoherence¶

Neutrinos flavor oscillations is, in principle, related to decoherence due to wave packet speration dueing their propagation.

1. What is the typical decoherence length?

Reference [Kersten2016]

Smaller than 100km for $$\Delta m_{13}$$

Of order 1000km for $$\Delta m_{12}$$

1. How to describe decoherence effect phenomenologically?

Reference [Akhmedov2014]

$i \frac{d}{dt} \rho = [H, \rho] -i \frac{1}{L_{\mathrm{coherence}}} (1-\hat D) \rho,$

where $$\hat D$$ is the projection operator that projects out the diagonal elements.

Suppose we have only the decoherence effect, and

$\begin{split}\rho = \begin{pmatrix} a & c\\ c^* & b \end{pmatrix},\end{split}$

the equation describes a damping of the coherences (off diagonal elements),

$\begin{split}i\frac{d}{dt} \begin{pmatrix} a & c\\ c^* & b \end{pmatrix} = -i \frac{1}{L_{\mathrm{coherence}}}\begin{pmatrix} 0 & c\\ c^* & 0 \end{pmatrix}.\end{split}$

So we have damping of $$c$$,

$\frac{d}{dt} c = - \frac{1}{L_{\mathrm{coherence}}}c,$

which is solved

$c = \exp \left( -\frac{t}{L_{\mathrm{coherence}}} \right) .$
1. Think of decoherence in flavor isospin picture.

Density matrix and flavor isospin are related to each other

$\rho = \frac{1}{2} ( 1 + \boldsymbol{\sigma} \cdot \mathbf P ).$

Coherence elements (off diagonal elements) in the density matrix are related to $$P_1, P_2$$.

However, the Hamiltonian forces the flavor isospin to precess.

The equation of motion is

$\frac{d}{dt}(s_k) + (\delta_{k1}s_1 + \delta_{k2}s_2) = (\mathbf s \times \mathbf H)_k.$
1. Kinetic spread of wave packet corresponds to an energy spread of the wave packet.

Reference [Kersten2016]

The energy spread for supernova neutrinos can be as large as 1MeV.

## 17.2.1. Refs & Notes¶

 [Akhmedov2014] Akhmedov, E., Kopp, J., & Lindner, M. (2014). Decoherence by wave packet separation and collective neutrino oscillations
 [Kersten2016] (1, 2) Kersten, J., & Smirnov, A. Y. (2016). Decoherence and oscillations of supernova neutrinos. European Physical Journal C, 76(6).

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