(Practical, experiment-ready procedures to test the InterTop predictions: oscillatory visibility, synthetic phase, nonlocal imprint, geometric loop phases, and anisotropic decay.)
The protocols below are written so they can be implemented with (i) entangled photon / SPDC + cavity (or atomic-slit) platforms, and (ii) 2D superfluid platforms (thin He film, polariton condensate, cold-atom 2D gas). Each protocol lists required hardware controls, measurement sequence, statistics & sensitivity targets, control checks, and model-comparison analysis. Follow these exactly to produce data that cleanly discriminate InterTop from standard decoherence.
1. Twin-photon interferometer protocol (nonlocal holonomy test)
Goal: detect (a) oscillatory visibility V()V(\kappa) vs cavity leakage/coupling, (b) deterministic synthetic phase ()\Phi(\kappa), and (c) nonlocal phase imprint eff\beta_{\rm eff} in coincidence fringes when only one photon interacts with the cavity.
Setup
SPDC entangled photon source producing polarization- or time-bin-entangled pairs (A, B) with high brightness.
Photon A: routed through the cavity/atomic-slit apparatus where cavity parameter \kappa (and optionally g,g,\Delta) is controllable. Photon A's path contains a scanning interferometer (or spatial scanning detector) to record fringes when needed.
Photon B: remote reference arm with scanning detector for single and coincidence fringes; does not interact with cavity hardware.
High-efficiency single-photon detectors (SNSPDs preferred), time-tagging electronics, and coincidence processing. Detector efficiency \eta and dark counts bb characterized prior.
Fast control of \kappa (electro-optic or piezo control) and ability to perform slow adiabatic parameter loops in (,g,)(\kappa,g,\Delta) plane.
Measurement sequence (per \kappa point)
a. Parameter grid: choose MM points {i}\{\kappa_i\} spanning bright dark regime; typical M=15M=15--25. Include fine sampling near expected nodes if any pilot data suggest periodicity.
b. Per point acquisition: record coincidence fringe C(x;i)C(x;\kappa_i) by scanning detector coordinate xx (or interferometric phase) across one or more fringe periods. For each xx collect counts for time tintt_{\rm int} to accumulate NN coincidences per point (target N103N\gtrsim 10^3--10410^4 for robust phase/visibility fits). Simultaneously record singles on A and B.
c. Fit model per point: fit C(x;i)=C0[1+Vcoin(i)cos(kx+coin(i))]C(x;\kappa_i)=C_0[1+V_{\rm coin}(\kappa_i)\cos(kx+\Phi_{\rm coin}(\kappa_i))] with Poisson maximum-likelihood; extract V^coin,^coin\hat V_{\rm coin},\hat\Phi_{\rm coin} and uncertainties V,\sigma_{V},\sigma_{\Phi}. Also fit singles SB(x)S_B(x) to check single-arm phase stability B\Phi_B.
d. Phase extraction: build {i,^coin(i)}\{\kappa_i,\hat\Phi_{\rm coin}(\kappa_i)\} and {i,V^coin(i)}\{\kappa_i,\hat V_{\rm coin}(\kappa_i)\} arrays.
Tests & analysis
Test A (synthetic phase): fit ^coin()=0+eff\hat\Phi_{\rm coin}(\kappa)=\Phi_0+\beta_{\rm eff}\kappa. Compute ^eff\hat\beta_{\rm eff} and ^\sigma_{\hat\beta}. Reject null eff=0\beta_{\rm eff}=0 at chosen significance (5 for a strong claim).
Test B (visibility oscillation): fit full InterTop model
VIT()=V0e12(02+(0))cos(+0)V_{\rm IT}(\kappa)=V_0 e^{-\frac12(\sigma_0^2+\gamma(\kappa-\kappa_0))}\,|\cos(\beta\kappa+\phi_0)|
and competing decoherence forms Vdec()=V0exp[()]V_{\rm dec}(\kappa)=V_0'\exp[-\Gamma(\kappa)] (try linear and stretched-exponential \Gamma). Compare by AIC/BIC and residual periodogram (search for spectral power at frequency \beta).
Test C (nonlocality control): check that singles on B show no systematic B()\Phi_B(\kappa) drift while coincidences do. If coincidence phase shifts occur without single-arm shifts --- strong evidence against classical path/optical artifacts.
Test D (loop/adiabatic): perform closed loops in (,g)(\kappa,g), measure loop\Delta\Phi_{\rm loop} via forward/backward subtraction (see Sec. IV.E). Check traversal-speed invariance within adiabatic window.
Statistical thresholds: require ^/^5|\hat\beta|/\sigma_{\hat\beta} \ge 5 and AIC > 10 favoring InterTop, plus reproducibility over 3 experimental runs.
Controls to exclude artifacts
Classical phase insertion: imprint known classical phase on optical path; verify measurement pipeline.
Switch-off entanglement: repeat with classical coherent light to exclude classical cavity-induced phase artifacts.
Detector/FPGA cross-checks: vary detection gating and count rates to exclude pileup/dead-time artifacts.
Thermal/mechanical monitoring: log temperature/vibration to rule out correlated drift.
Sensitivity & resource estimates
Per-point phase uncertainty (NV2)1/2\sigma_\Phi \approx (N V^2)^{-1/2}. For V0.5V\sim0.5, N=104N=10^4 0.014\sigma_\Phi\sim 0.014 rad. For M=15M=15 points over span 1\Delta\kappa\sim 1 unit, ^/(Mrms)\sigma_{\hat\beta}\sim\sigma_\Phi/(\sqrt{M}\,\Delta\kappa_{\rm rms}) 0.004\sim 0.004 rad/unit. Detectable \beta 0.02 rad/unit at 5.
2. Superfluid interferometer protocol (2D film / polariton or cold-atom)
Goal: measure V(vs)V(v_s) and (vs)\Phi(v_s) vs superflow vsv_s, detect oscillatory visibility, resonant dips from vortex inertia, slip-driven time scaling, geometry dependence, and geometric loop phases.
