Classical Reconstruction of the PMNS Matrix Using a Mechanical Neutrino Oscillator

TL;DR

This paper demonstrates a classical mechanical analog of three-flavor neutrino oscillations by constructing a system of three coupled pendulums. It reconstructs a mechanical mixing matrix from measured amplitudes and beat frequencies, and derives a scaling relation that maps mechanical evolution time to the neutrino oscillation parameter L/E, clarifying that the analogy is phase-based rather than a dynamical replica. The results show qualitative agreement with the PMNS structure, notably accurate reproduction of θ_{12} and θ_{13}, and a beat-frequency hierarchy that mirrors the neutrino mass-squared differences. The work offers an accessible pedagogical visualization of flavor mixing and highlights both the utility and the limitations of classical analogs for quantum phenomena.

Abstract

Neutrino oscillations arise from quantum interference between neutrino mass eigenstates and are governed by the PMNS matrix. Although this is an intrinsically quantum phenomenon, its mathematical structure is analogous to systems of coupled classical oscillators. In this work, a three--pendulum system connected by springs is constructed as a classical analog of three--flavor neutrino oscillations. Measurements of amplitude transfer, normal--mode structure, and beat frequencies are used to extract a mechanical mixing matrix, which is compared with the structure of the PMNS matrix under the assumption of zero CP violation. A scaling relation linking mechanical time evolution to the neutrino \(L/E\) behavior is derived, clarifying the scope and limitations of the analogy. The experiment demonstrates how abstract concepts of neutrino mixing can be visualized using simple and accessible classical systems, offering both pedagogical value and a qualitative understanding of flavor oscillation dynamics.

Figures (13)

• Figure 1: Schematic diagram of the three–pendulum coupled system.
• Figure 2: Photograph of the actual experimental apparatus used to construct the mechanical PMNS analog.
• Figure 3: Beat envelope for the coupled motion between Pendulum 1 and Pendulum 2, showing the characteristic slow modulation of energy transfer in the mechanical analog.
• Figure 4: Simulated time-series evolution of the angular displacement of all three pendulums using the experimentally measured normal-mode frequencies.
• Figure 5: Normalized amplitudes for each pendulum under the three different initial excitations. Each column corresponds to a normal mode analog in the mechanical PMNS matrix.