Geometrical Helicity Multiplication in Nonlinear Phonon Transfer

This paper derives and tests a parameter-free geometrical law for helicity transfer in the nonlinear conversion of the doubly degenerate infrared-active Eᵤ phonon mode into the Raman-active Eg mode in Bi₂Se₃. The analysis uses the open polarization-resolved experimental data deposited by Minakova et al. for coherent phonon angular-momentum transfer. The componentwise quadratic crystal interaction reported for the source experiment is written in the circular basis as Qout,+∝ Qin,-\,2 and Qout,-∝ Qin,+\,2. Reducing this operator to normalized helicity produces the continuous transfer law Hₒᵤₜ=-2Hᵢₙ1+Hᵢₙ². The law predicts not only reversal of handedness but also an increase in the magnitude of helicity for every intermediate elliptical state satisfying 0<|Hᵢₙ|<1. In the transformed helicity coordinate χ=artanh(H), the nonlinear bounded relation becomes the exact multiplication law χₒᵤₜ=-2χᵢₙ. The same quadratic operator also gives the polarization-dependent output-intensity relation S0,out=C/2S0,in²(1+Hᵢₙ²), where C is one common experiment-dependent transfer coefficient. For equal input spectral intensity and otherwise unchanged coupling conditions, the law predicts the calibration-independent ratio S0,outᶜⁱʳᶜS0,outˡⁱⁿ=2. At the central Eᵤ and Eg mode bins, the helicity law reduces the summed squared error by a factor of $46.90$ relative to the simpler relation Hₒᵤₜ=-Hᵢₙ. The helicity-aware intensity law gives R²=0.9703, compared with R²=0.0546 for a helicity-independent quadratic-intensity model. Statewise transfer coefficients vary by approximately $4%$, and leave-one-polarization-state-out predictions remain within approximately $7%$ of the deposited output intensities. Robustness was tested across $81$ matched input-output spectral windows. The quadratic law outperformed simple sign reversal throughout the complete window grid, with improvement factors ranging from approximately $24$ to $147$. A freely fitted nonlinear order remained centered near the preregistered quadratic value $p=2$. Within the Quantum Measurement Units framework, the result identifies one geometrical transfer index connecting nonlinear order, mode freq multiplication, phase winding, continuous helicity transformation, and crystal pseudo-angular-momentum closure: freqₒᵤₜ2\,freqᵢₙ, φₒᵤₜ=-2φᵢₙ+φ₀, χₒᵤₜ=-2χᵢₙ, and lₒᵤₜ≡2lᵢₙ 3. The analysis distinguishes the previously reported quadratic crystal interaction from the new contribution established here: its reduction to a parameter-free continuous helicity map, the associated output-intensity invariant, and their direct validation against the deposited polarization-resolved measurements. The paper concludes with falsifiable predictions for dense ellipticity scans, matched circular and linear input states, direct phase-winding measurements, held-out polarization tests, and higher-order opposite-helicity processes. The general order-p helicity law is Hₒᵤₜ⁽ᵖ⁾=-[p\,artanh(Hᵢₙ)].