Setup options (choose per platform)
Probe interferometer: a Mach--Zehnder or two-path interferometer for phonons/polaritons/matter-wave probes whose arms traverse distinct regions of the 2D film and form a closed loop sensitive to vortex circulation.
Flow control: ability to set and vary superflow speed vsv_s and anisotropy aa (trap shaping).
Vortex imaging: in-situ imaging (e.g., phase-contrast, tracer particles, or high-resolution density imaging for cold atoms) to record vortex nucleation events and motion.
Fast acquisition: fringe-tracking hardware to record phase jumps and telegraph noise.
Measurement sequence
a. Parameter grid: choose MM values of vsv_s spanning low (no nucleation) to high (frequent tunneling); sample anisotropy values as needed. Typical M=20M=20.
b. Per point acquisition: for each vsv_s, record repeated fringe scans to obtain V^(vs)\hat V(v_s) and ^(vs)\hat\Phi(v_s), with integration times TT chosen to probe both short-time behavior (resolve individual slips) and long-time ensemble averages. Acquire several TT-values (e.g., T=0.01,0.1,1,5T=0.01,0.1,1,5 s) to test slip-scaling.
c. Simultaneous vortex monitoring: record images (or time series) to detect vortex events and extract (vs)\Gamma(v_s), vortex positions, and (if possible) dynamical response to small drives (to extract mv,,Km_v,\eta,K).
d. Closed-loop protocols: choose two control parameters (e.g., vsv_s and trap anisotropy aa) and perform forward/backward loops in parameter space; measure geo\phi_{\rm geo} per Sec. IV.E.
Tests & analysis
Visibility model fits: fit measured V(vs;T)V(v_s;T) to the master law (IV.D Eq. D1)
V(vs;T)=V0e12Varmot(vs)12(vs)Tslip2cos((vs)).V(v_s;T)=V_0 e^{-\tfrac12\mathrm{Var}_{\rm mot}(v_s)-\tfrac12\Gamma(v_s)T\sigma_{\rm slip}^2}\;|\cos(\overline{\Phi}(v_s))|.
Constrain Varmot\mathrm{Var}_{\rm mot} using independently measured mv(vs),0(vs),S()m_v(v_s),\omega_0(v_s),S_\xi(\omega) where available (use Eq. (4)).
Resonance test: identify dips in V(vs)V(v_s) correlated with 0(vs)\omega_0(v_s) crossing bath spectral peaks (from S()S_\xi(\omega)). Correlation coefficient > 0.8 between predicted resonance location and visibility dip is strong evidence for inertia-mediated holonomy variance.
Slip-scaling test: for slip-dominated regime, plot 2ln(V/V0)-2\ln(V/V_0) vs TT at fixed vsv_s. Expect linear relation with slope (vs)slip2\Gamma(v_s)\sigma_{\rm slip}^2. Verify slope matches direct vortex-counting (vs)\Gamma(v_s)\times measured 2\langle\delta\phi^2\rangle from imaging geometry.
Oscillation detection: after removing envelope via estimated variance contributions, compute residual carrier cos((vs))|\cos(\overline{\Phi}(v_s))| and fit (vs)=0+vs\overline{\Phi}(v_s)=\Phi_0+\beta v_s to extract \beta. Use Fourier analysis of residuals to find frequency \beta.
Loop-phase extraction: use forward/backward subtraction to obtain ^geo\hat\phi_{\rm geo}; vary area and confirm linear small-area scaling per Eq. (6).
Geometry dependence: repeat interferometry with different loop areas/shapes to map G|\mathbf{G}| dependence.
Controls
No-flow baseline: measure at vs=0v_s=0 to characterize technical noise floor and extract V0V_0.
Engineered noise injection: add controlled phonon drive to modify S()S_\xi(\omega) and verify predicted changes in Varmot\mathrm{Var}_{\rm mot}.
Imaging cross-check: correlate individual phase slips with vortex crossing times to establish causal link.
Single-run/ensemble checks: perform many independent runs to test non-Gaussianity: compute kurtosis & skewness of phase distributions; Nonzero kurtosis indicates discrete-slip dominance.
Sensitivity & resource estimates
Required per-point counts depend on probe modality: for photonic polariton probes, N104N\sim10^4 detected photons per point gives 0.02\sigma_\Phi\lesssim0.02 rad. For matter-wave or phonon probes, SNR differs --- aim for phase precision 0.05\sigma_\Phi\lesssim 0.05 rad.
To resolve 0.5\beta\sim 0.5 rad/(mm/s) over vs1\Delta v_s\sim 1 mm/s with M=20M=20, need ^0.1\sigma_{\hat\beta}\lesssim0.1 rad/(mm/s) typical requirements similar to twin-photon case: total effective counts Ntot104N_{\rm tot}\sim 10^4--10510^5 across grid.
3. Joint model-comparison and decision procedure (applies to both platforms)
Model set:
H0H_0 (decay-only): Vdec()=V0exp[()]V_{\rm dec}(\lambda)=V_0\exp[-\Gamma(\lambda)], \Phi constant or random with variance predicted by decoherence.
H1H_1 (InterTop holonomy): VIT()=V0e12Var()cos(+0)V_{\rm IT}(\lambda)=V_0 e^{-\tfrac12\mathrm{Var}(\lambda)}|\cos(\beta\lambda+\phi_0)|, ()=0+\Phi(\lambda)=\Phi_0+\beta\lambda (or with loop-generalization).
Fitting & statistics:
a. Fit both models by MLE accounting for Poisson photon statistics (or Gaussian counts if large). Use joint fits to amplitude and phase data when available.
b. Compute AIC, BIC, and likelihood-ratio test statistic =2(lnL1lnL0)\Lambda=2(\ln\mathcal{L}_{1}-\ln\mathcal{L}_{0}). For nested models use Wilks' theorem (if applicable) to assess p-value; otherwise use parametric bootstrap to obtain empirical null distribution.
c. Examine residuals of H0H_0: significant periodic structure (via periodogram F-test) at frequency \beta strongly contradicts H0H_0.
d. Decision thresholds: require (a) ^/^5\hat\beta/\sigma_{\hat\beta}\ge5, (b) AIC>10 favoring H1H_1, (c) nonlocality test passed (for twin-photon), or geometry/loop tests passed (for superfluid).
Reporting: always report best-fit parameters with 95% CI, AIC/BIC, p-values, and raw residuals. Include model-predicted curves overlaid on data and periodograms.
4. Control checklist to rule out classical artifacts (must be performed)
Demonstrate that single-arm or classical-beam fringes do not show the same phase dependence as the coincidence (twin-photon) or that single-arm phase tracks differ.
Calibrate cavity-induced dispersive optical path changes independently (measure with classical probe) and subtract known contributions.
Vary detector positions; confirm phase shifts are global (fringe translation) not local artifact.
Turn off entanglement or switch off holonomy engineering (e.g., open cavity) to show disappearance of \beta-like signals.
Conduct environmental-noise correlation analysis (temperature, RF pick-up) to exclude common-mode technical phase drifts.
5. Data products to archive & publish
