2305.01150_Quantum computing neutrino physics
Flavor of neutrinos important for matter interaction
Thus understand flavor evolution of neutrinos
Many neutrinos with high density will undergo dynamics in flavor space due to $\nu-\nu$ interactions
Neutrinos decouple (?) interactions between hadrons and $\mu$ and $\tau$ neutrinos are supressed
# Neutrino flavor oscillation
If neutrinos have mass and mass eigenstates are not equal to falvor eigenstates (PMNS matrix analog to CKM matrix) flavor oscillations happen
Misalignment of states can be modified in medium (Through interactions with the particles I guess) MSW-effect
Neutrinos scatter with electrons in matter, similar effect to photons leading to refratctive index $n$ of materials
Scattering gives “potential term” which shifts mass eigenstates, which changes neutrino oscillation (?)
In extreme environments like core collapse super novae the background medium can consist of neutrinos semselfs (instead of electons)
“Effective potential” due to weak neutral current scattering between neutrinos
In low energy EFT given by an effective hamiltonian: $$\mathcal{H}=\frac{G_F}{\sqrt{2}}\sum_{\alpha,\beta}(\bar{\nu}_\alpha\gamma^\mu\nu_\alpha)(\bar{\nu}_\beta\gamma_\mu\nu_\beta)$$
Gives rise to off diagonal potential in flavor basis (?) - Electron interactions dont do that
This couples all neutrinos to each other (?) Flavor history of neutrino $i$ will influence flavor evolution of neutrino $j$ and vicer versa.
Connection to quantum information:
Pairwise interaction between neutrinos, similar to spin-spin coupling of quantum many body systems
Hilber space for $N$ neutrinos of $3^N$ (for $3$ flavors)
To simplify problem: Assumption: Neglect multi particle interactions: Model as single particle interaction with “Mean field” from other particles
With this assumption hilber space is $3N$
# Neutrino Hamiltonian
Usually multiple neutrinos modeled as interacting plane waves in a box
Limitations: Does not include finite size of wave package
Hamiltonian for $n_f$ flavors in terms of $SU(n_f)$ generators, e.g. for $n_f = 2$: $$J^+_p = a^\dagger_1(p)a_2(p)$$$$J^-_p = a^\dagger_2(p)a_1(p)$$ $$J^z_p = \frac{1}{2}\left(a^\dagger_1(p)a_1(p)-a^\dagger_2(p)a_2(p)\right)$$
The $J$s are the $SU(2)$ generators, the $a$s are fermionic creation and annihilation operators
Fermionic operators are in mass basis, meaning $a_1$ for neutrino with $m_1$ and $a_2$ accordingly
Write operators in terms of pauli matrices (?) $\vec{J}_p=\vec{\sigma}_p/2$.
Hamiltonian with vacuum mixing and $\nu-\nu$ interactions can be written as $$\mathcal{H}=\sum_p\omega_p\vec{B}\cdot\vec{J}_p + \frac{\sqrt{2}G_F}{V}\sum_{p,q}(1-\vec{e}_p\cdot \vec{e}_q)\vec{J}_p\cdot\vec{J}_q,$$ either in mass basis $\ket{\nu_1}, \ket{\nu_2}$ or flavor basis $\ket{\nu_e}, \ket{\nu_x}$.
The $\vec{B}$ is the background field “Point along the mass basis??” In mass basis $\vec{B}=(0,0,1)$ in flavor basis $\vec{B}=(\sin2\Theta,0,-\cos2\Theta)$
The $\omega_p$ are the vacuum neutrino oscillation frequencies for momentum p
$V$ is the quantization volume (?)
In mean field approximation replace the interaction term $\vec{J}_p\cdot\vec{J}_q$ with approximate one particle operator: $\vec{J}_p\cdot\vec{J}_q\approx \vec{J}_p\cdot\braket{\vec{J}_q}+\braket{\vec{J}_p}\cdot\vec{J}_q - \braket{\vec{J}_p}\cdot\braket{\vec{J}_q}$
This leads to individual neutrinos remaining uncorrelated!
# Quantum dynamics
# Quantifying entanglement
- Entanglement Quantum entropies to measure amount of entanglement
- Divide system into two sub systems $A$, e.g. single neutrino and rest system $A^c$
- Take partial trace to trace out rest system $\rho_A = \text{Tr}_{A^c}[\rho]$
- In case of the single neutrino subsystem one would have $$\rho_A = \sum_{i_1,…,i_{A-1},i_{A+1},…,i_N}\braket{\nu_{i_1},…,i_{A-1},i_{A+1},…,i_N|\rho|\nu_{i_1},…,i_{A-1},i_{A+1},…,i_N}$$
- The entanglement entropy is the Von Neumann entropy of the subsystem $$S_A = -\text{Tr}[\rho_A\log\rho_A]$$