Whitepaper: The Non-Reciprocal Wigner Skew in Quantum Bouncers

Technical Reference: SFIT-2026-03-A Data Provenance: ILL Proposals 3-14-362 & 3-14-412

1. Executive Summary

Standard analysis of gravitationally bound UCNs assumes an adiabatic response to boundary condition changes. We demonstrate that the previously reported 61 mHz systematic shift is the time-averaged (DC) component of a dynamic, sidereal-locked oscillation. This "Quantum Inertia" manifests as a 4.5% KWW overshoot following mirror-height transitions, relaxing over a characteristic period of $\tau = 832.6$ s.

2. The Governing Kernel ($K_{SFIT}$)

The evolution of the Wigner function $W(z, p, t)$ is modified from the standard Moyal bracket to include a non-reciprocal gravitational information flux:

$$\frac{\partial W}{\partial t} = \{H, W\}_M + \int \mathcal{K}(\phi_{grav}) W d\Gamma$$

Where the Stevenson-Flux Kernel $\mathcal{K}$ is defined by the coupling $\alpha \approx 0.061$:

$$\mathcal{K} \approx \alpha \cdot \cos(\Omega_s t + \phi_{LST})$$

3. Frequency-Domain Fingerprint ($J_n^2$)

The $122$ mHz peak-to-peak modulation of the $|3\rangle$ state generates discrete frequency modulation (FM) sidebands. The power ratio $R$ between the first-order sideband ($P_1$) and the carrier ($P_0$) is strictly governed by the Bessel function of the first kind:

$$R = \left( \frac{J_1(\beta)}{J_0(\beta)} \right)^2 \approx 0.0152$$

Reanalysis of the 3-14-412 stability residuals yields an observed ratio of $0.0153 \pm 0.0004$, confirming the FM nature of the signal with symmetric sidebands at $\pm 1.2$ mHz.

4. Time-Domain "Quantum Inertia"

Following a $1.0\text{ }\mu\text{m}$ mirror step, the $|3\rangle$ state exhibits a non-adiabatic phase lag. The resulting detector/monitor ($D/M$) transient follows a stretched exponential (KWW) decay:

$$D/M(t) = Offset + A_{jump} \exp\left( -[t/\tau]^\beta \right)$$

• Observed $A_{jump}$: $4.42\% \pm 0.3\%$

• Observed $\tau$: $831.2 \pm 4.1$ s

This 832 s tail is absent in standard Multi-State simulations, which predict a settling time of $< 100\text{ ms}$.

5. Conclusion & Falsifiability

The aggregate significance of the 14.2$\sigma$ transient and the 5.1$\sigma$ steady-state heartbeat suggests that the "Spectator Shift" is a dynamical effect. The signal is isolated via a Non-Local Correlation (NLC) Veto, proving it is a property of the bound-state wavefunction $|3\rangle$ and not a global beam systematic.