Neutrinos · D07 · A changing quantum compass

Neutrino Matter Resonance

Send one coherent neutrino state through a changing electron density. Follow its path on the Bloch sphere, compare vacuum evolution, and discover when slow conversion breaks down.

Electron flavor here $\text{—}$

Upper matter branch $\text{—}$

Vacuum phase average after exit $\text{—}$

PREPARE A DENSITY CROSSING

Turn the compass. Follow the state.

Density profileEntrance stateAntineutrino: reverse the matter potentialEnergy

E

10\,\mathrm{MeV}

Profile length

L

10000\,\mathrm{km}

Entrance electron density proxy

\rho Y_e

100\,\frac{\mathrm{g}}{\mathrm{cm}^{3}}

Vacuum mixing angle

\theta

33.4\,{}^\circ

Mass splitting

Mass-squared splitting scale

\Delta m^2

7.5\,\times10^{-5}\,\mathrm{eV}^{2}

Settings prepare the next calculation. Scrubbing, camera views and replay reuse the current numerical trajectory.

Compare Slow with Sudden: the density range is identical. What changes is how far the state has to follow the rotating axis.

Two active flavors, prescribed one-dimensional density and coherent forward scattering. This is a quantum-state diagram, not a particle orbit or detector record. No absorption, three-flavor corrections, solar production averaging, Earth structure, interactions or instrument response. Numerical midpoint evolution is unitary per step; the refinement difference is a diagnostic, not a rigorous error bound. Sources and conventions below.

Coherent electron flavor probability

Compare vacuum

Mint: numerical matter evolution. Dashed: the same entrance state propagated in vacuum. Gold marker: resonance crossing, when present. This is a calculation, not repeated measurements of a travelling particle.

Density and the resonance condition

The horizontal axis is distance through an illustrative one-dimensional profile. It is not solar radius or a measured Earth model.

Instantaneous branch populations

Gold: upper branch population. Rose: local mixing strength. Constant branch population indicates following in an instantaneous basis; it need not imply constant flavor.

After leaving the medium

Instantly set the potential to zero at exit, project onto vacuum mass states, then average away their relative phase. This average is a separate operation; it is not the coherent endpoint probability.

Computed vacuum phase average: $0$
Adiabatic reference: slow, vacuum exit: $0$

Check the calculation

Maximum normalization drift: $0$
Maximum step refinement difference: $0$

Exact unitary exponentials at interval midpoints. Compare 4,800 with 9,600 steps at 1,201 common positions. The difference diagnoses convergence; it is not an error guarantee. CSV retains all 9,601 positions. Dense chart columns show the sampled minimum and maximum within each pixel.

Conventions, model boundaries & sources

Electron flavor is the north pole; the other active flavor is the south pole. Bloch coordinates encode amplitudes, not real-space directions. Gold is the normalized trace-free Hamiltonian axis. The two-flavor splitting is positive. Antineutrinos reverse the charged-current potential; this real two-flavor model has no CP phase. Entrance eigenstates are recomputed when the medium changes. Vacuum comparisons always use the identical prepared entrance state.

Density means mass density times electrons per nucleon. The decreasing profile follows a shifted exponential over four decades and reaches zero exactly at exit. Solar-like means the shape and illustrative parameters, not a solar density fit. Constant density is abruptly removed at exit; its adiabatic vacuum-exit reference is inapplicable. No absorption or detector sampling is modeled.

Read the MSW story →Return to vacuum propagation →

CONTINUE EXPLORING

From flavor to evidence

Propagation predicts a flavor probability. Which parts of the experiment turn an interaction into a usable record?

Start with the essentials

Focus question: Can an almost electron-like entrance eigenstate emerge almost entirely as the other flavor without absorption?
One-sentence intuition: Slow variation can preserve an instantaneous branch population while changing its flavor composition. Resonance alone does not guarantee conversion.

Core mathematical model

Trace-free evolution

i\hbar c\,\frac{d\psi}{dx}=\left[\frac{\Delta m^2}{4E}\begin{pmatrix}-\cos 2\theta&\sin 2\theta\\\sin 2\theta&\cos 2\theta\end{pmatrix}+\frac{V}{2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}\right]\psi

The flavor basis is electron, then other active. A common trace phase is removed. Distance uses kilometres, energy eV internally and the positive mass-squared splitting eV squared.

The electron potential

V=\sqrt{2}G_F N_e\simeq 7.6324\times10^{-14}\,\mathrm{eV}\,\frac{\rho Y_e}{\mathrm{g\,cm^{-3}}}

Density is mass density times electrons per nucleon. The potential changes sign for antineutrinos. Common active-flavor neutral-current terms do not change this two-flavor evolution.

Resonance is maximal local mixing

2EV=\Delta m^2\cos 2\theta,\qquad \sin^2 2\theta_m=\frac{(\Delta m^2\sin 2\theta)^2}{(\Delta m^2\cos 2\theta-2EV)^2+(\Delta m^2\sin 2\theta)^2}

For positive splitting and angles below 45 degrees, the positive-density resonance occurs for neutrinos. The gold axis is then transverse to the flavor axis. At zero mixing the exact level crossing has no off-diagonal conversion.

Averaging after exit

\overline{P}_e=p_1\cos^2\theta+p_2\sin^2\theta,\qquad p_1+p_2=1

Project the exiting state onto vacuum mass states and discard their relative phase. This is not the coherent electron probability at the last position. The separate adiabatic reference assumes slow following all the way to a vacuum exit.

Common difficulties

A Bloch sphere is not a neutrino orbit

Typical misconceptionThe neutrino physically spirals around a sphere inside the Sun.

Better mental modelThe sphere encodes a normalized two-component quantum state. Its coordinates are amplitude coherences and flavor imbalance, not spatial coordinates.

Resonance is not absorption

Typical misconceptionElectron neutrinos disappear because matter blocks them.

Better mental modelThis Hermitian model conserves total active probability. The medium changes coherent evolution; no absorption is included.

Run the experiment

01
Watch the compass turn
Use Solar-like gradient and replay. Scrub across the gold resonance marker.
What to observe: The gold evolution axis rotates as density decreases. The state remains normalized while its electron probability changes.
02
Give the state time to follow
Compare Slow crossing and Sudden crossing. Both prepare the same upper entrance eigenstate and the same density range.
What to observe: A longer length makes the same density change slower per kilometre. The slow case nearly preserves the upper branch; the sudden case does not.
03
Hold the density fixed
Choose Constant resonance. Inspect the probability and branch curves.
What to observe: The local mixing is almost maximal; coherent flavor oscillations persist. The upper branch population is constant because the Hamiltonian is constant. The abrupt exit invalidates the slow-exit reference.
04
Reverse the potential
Return to Solar-like gradient, select Antineutrino, then Propagate.
What to observe: For the positive splitting and angle used here, the positive-density resonance disappears. This two-flavor model has no CP phase; a matter-induced difference is not CP discovery.
05
Test a numerical claim
Inspect normalization and refinement diagnostics, then export the entire trajectory.
What to observe: Per-step unitarity protects normalization; density discretization can still change the trajectory. The step comparison diagnoses one numerical approximation, not all omitted physics.