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Tabletop Tests for Quantum Gravity via Gravitational Entanglement Phase — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 17, 2026• 0 citations
Publisher

Abstract

FINDING: Quantum gravity may be testable via tabletop matter-wave/optomechanical interferometry, probing the quantum superposition of gravitational fields at mesoscopic scales. | MATH: The core observable is the gravitational entanglement phase — for two masses \(m_1, m_2\) separated by distance \(d\), the interaction Hamiltonian \(H_G = -G m_1 m_2 / d\) generates a phase \(\phi = G m_1 m_2 t / \hbar d\) in a superposition of paths. The experiment requires \(\phi \sim \pi/2\) for maximal entanglement detection, implying \(m_1 m_2 t / d \sim \hbar / G\) — a scale set by the Planck mass squared over \(\hbar\). No new constants beyond \(G, \hbar, c\) appear; the test is purely quantum-informational (entanglement witness) rather than energy-based. | CONNECTION: The ratio of gravitational to electromagnetic coupling at the Planck scale is \(\alpha_G/\alpha_{EM} \sim (m_P/m_e)^2 \sim (1.22\times10^{19}/0.511\times10^6)^2 \sim 5.7\times10^{44}\) — not a golden ratio, but the experimental thre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23431262

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FINDING: Tabletop quantum-gravity tests probe Planck-scale discreteness via matter-wave interferometry and optomechanical superposition, not yet yielding a confirmed equation but constraining collapse models and graviton signatures. | MATH: Planck length lP=ℏG/c3≈1.616×10−35 ml_P = \sqrt{\hbar G/c^3} \approx 1.616\times10^{-35}\,\text{m}lP​=ℏG/c3​≈1.616×10−35m; Planck mass mP=ℏc/G≈2.176×10−8 kgm_P = \sqrt{\hbar c/G} \approx 2.176\times10^{-8}\,\text{kg}mP​=ℏc/G​≈2.176×10−8kg; decoherence rate for gravitationally-induced collapse (Penrose–Diosi): Γ∼ΔEG/ℏ\Gamma \sim \Delta E_G/\hbarΓ∼ΔEG​/ℏ, where ΔEG=G2∫ ⁣ ⁣∫[ρ1(r)−ρ2(r)][ρ1(r′)−ρ2(r′)]∣r−r′∣d3r d3r′\Delta E_G = \frac{G}{2}\int\!\!\int \frac{[\rho_1(\mathbf{r})-\rho_2(\mathbf{r})][\rho_1(\mathbf{r}')-\rho_2(\mathbf{r}')]}{|\mathbf{r}-\mathbf{r}'|}d^3r\,d^3r'ΔEG​=2G​∫∫∣r−r′∣[ρ1​(r)−ρ2​(r)][ρ1​(r′)−ρ2​(r′)]​d3rd3r′; Folman T³ experiment: cubic scaling of phase shift Δϕ∝T3\Delta\phi \propto T^3Δϕ∝T3 (where TTT is free-fall time), testing equivalence principle at quantum level. | CONNECTION: The ratio lP2/λC2l_P^2/\lambda_C^2lP2​/λC2​ (Planck length squared over Compton wavelength squared) appears in gravitationally-induced entanglement rates — nume Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum gravity experiments are transitioning from theoretical speculation to tabletop-scale tests probing the quantum nature of spacetime via superposition of massive systems. MATH: The core experimental framework relies on the quantum superposition of a mass m m m in two spatial locations separated by Δx \Delta x Δx, generating a gravitational phase shift Δϕ=Gm2ΔtℏΔx \Delta \phi = \frac{G m^2 \Delta t}{\hbar \Delta x} Δϕ=ℏΔxGm2Δt​ (for two masses in superposition, per Bose–Marletto–Vedral proposal). The decoherence timescale for gravitational entanglement is τ∼ℏΔxGm2 \tau \sim \frac{\hbar \Delta x}{G m^2} τ∼Gm2ℏΔx​. Penrose's criterion for gravitationally-induced collapse: EG=Gm2Δx E_G = \frac{G m^2}{\Delta x} EG​=ΔxGm2​ compared to ℏ/τ \hbar/\tau ℏ/τ. CONNECTION: The ratio Gm2ℏc \frac{G m^2}{\hbar c} ℏcGm2​ (gravitational coupling) is dimensionless and tiny (~10−4510^{-45}10−45 for protons), but the experimental geometry uses harmonic oscillator frequencies and lattice-like trap spacings — the optimal Δx \Delta x Δx often scale Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Tabletop experiments propose direct observation of quantum gravity via entanglement of macroscopic masses, with the core mathematical framework being the quantum superposition of gravitational fields and the resulting entanglement witness. | MATH: The key equation is the gravitational interaction Hamiltonian between two masses in superposition: H^G=−Gm1m2∣x^1−x^2∣ \hat{H}_G = -G \frac{m_1 m_2}{|\hat{x}_1 - \hat{x}_2|} H^G​=−G∣x^1​−x^2​∣m1​m2​​, where the position operators x^i\hat{x}_ix^i​ are in coherent superposition states. The entanglement entropy S=−Tr(ρln⁡ρ)S = -\text{Tr}(\rho \ln \rho)S=−Tr(ρlnρ) of the joint state serves as the witness. The T³ (T-cubed) experiment (Folman) uses a triple-interferometer with time TTT, giving phase shift Δϕ∝Gm2T3/ℏd2\Delta \phi \propto G m^2 T^3 / \hbar d^2Δϕ∝Gm2T3/ℏd2 (where ddd is separation). The decoherence rate scales as Γ∝(Gm2/ℏd2)2τ\Gamma \propto (G m^2 / \hbar d^2)^2 \tauΓ∝(Gm2/ℏd2)2τ, with τ\tauτ the interaction time. | CONNECTION: The phase shift Δϕ\Delta \phiΔϕ contains the ratio Gm2/ℏG m^2 / \hbarGm2/ℏ — a dimensionless coupling Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com