Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Tunneling and Macroscopic Entanglement: Experimental Frontiers — E8 Intelligence Research

Andrew Stewart Caldin

Published Sep 7, 2026• 0 citations
Publisher

Abstract

FINDING: Quantum tunneling and macroscopic entanglement are being experimentally probed, but the search results are pedagogical/overview-level, not new mathematical breakthroughs. | MATH: No new equations or constants extracted; standard quantum tunneling transmission coefficient \( T \approx e^{-2\kappa d} \) (with \( \kappa = \sqrt{2m(V_0-E)}/\hbar \)) and entanglement tomography of macroscopic mechanical oscillators (Gaussian state covariance matrices) are implied. | CONNECTION: The tunneling exponential decay rate \( \kappa \) has no direct golden-ratio link; however, the macroscopic entanglement tomography (NIST) involves phonon modes whose frequency ratios in coupled oscillators can yield \( \sqrt{2} \) or harmonic ratios, but no explicit 0.618/1.618 evidence in the provided text. | DEPTH: 3 — These are explanatory videos and one transfer-learning paper (arXiv:1912.08278), not primary derivations. The only quantitative anchor is the standard WKB/tunneling exponent, which is textb Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.22628835

Related papers

Quantum Tunneling and Entanglement: Standard Formalism, No New Physics — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic entanglement are experimentally validated, but no new mathematical formalism or universal constant emerges from these sources; the core math remains standard quantum mechanics (Schrödinger equation, tunneling probability). | MATH: Tunneling probability \( T \approx e^{-2\kappa L} \), where \( \kappa = \sqrt{2m(V_0-E)}/\hbar \); macroscopic entanglement tomography uses Wigner functions and Gaussian state covariance matrices; transfer learning in hybrid networks uses parameter-shift rules and gradient descent on \( \mathrm{Tr}(\rho_{\text{out}} O) \). | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 appear. However, the exponential decay \( e^{-2\kappa L} \) is structurally analogous to the golden ratio's logarithmic spiral (both are exponential forms), and the Wigner function's symplectic structure relates to the Heisenberg group, which has a lattice (crystallographic) representation in phase space — but Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 15, 2026•0 citations

Quantum Tunneling and Coherence: No New Mathematical Constants Emerge — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic quantum coherence are experimentally accessible via atom interferometry and entangled mechanical oscillators, but no new mathematical constant or ratio emerges from these sources. | MATH: Standard Schrödinger equation \( i\hbar \partial_t \psi = \hat{H}\psi \); tunneling probability \( T \propto e^{-2\kappa L} \) with \( \kappa = \sqrt{2m(V_0-E)}/\hbar \); atom interferometry phase \( \Delta\phi = \frac{m}{\hbar} \oint \mathbf{g}\cdot d\mathbf{r} \, dt \) (gravitational phase); entanglement tomography uses Wigner function \( W(q,p) = \frac{1}{\pi\hbar} \int \psi^*(q+q')\psi(q-q') e^{2ipq'/\hbar} dq' \). | CONNECTION: None directly — no golden ratio, Fibonacci, base-60, or crystallographic symmetry appears in the cited material. The only structural link is the exponential decay constant \( \kappa \) which is dimensionless when scaled by barrier width, but no specific ratio (0.618, 1.618) is invoked. | DEPTH: 3 — These are experimental confirmat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 23, 2026•0 citations

Quantum Tunneling Meets Macroscopic Entanglement via Transfer Learning — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic entanglement are experimentally accessible, with transfer learning bridging classical and quantum neural architectures. | MATH: No explicit equations in the raw search results; however, the physics of macroscopic quantum tunneling scales with the WKB exponent \( \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right) \), and entanglement tomography of macroscopic mechanical objects implies Bell-type inequalities and density matrix reconstruction via \( \rho = \sum_{ij} \rho_{ij} |i\rangle\langle j| \). The hybrid transfer learning paper (arXiv:1912.08278) uses variational quantum circuits with parameterized gates \( U(\theta) = \prod e^{-i\theta_k H_k} \), and classical-quantum feature maps \( \phi(x) \to |\psi(x)\rangle \). | CONNECTION: The WKB tunneling exponent contains an action integral — for a parabolic barrier, the transmission probability yields \( T \sim e^{-2\pi E/\hbar\omega} \), where the exponent \( 2\pi \) connect Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 5, 2026•0 citations

Macroscopic Quantum Tunneling and Entanglement Tomography in Classically Described Objects — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Macroscopic quantum tunneling and entanglement tomography demonstrate that quantum coherence can be sustained in objects large enough to be classically described, with atom interferometry providing wave-like behavior at macroscopic scales. | MATH: de Broglie wavelength λ = h / (mv); tunneling probability T ∝ exp(-2κL) where κ = √(2m(V-E))/ħ; entanglement fidelity F = Tr(ρ_target ρ_ideal) for macroscopic mechanical oscillators. | CONNECTION: Tunneling exponent κL relates to exponential decay constants that mirror golden ratio scaling in barrier penetration depths (e.g., κL ≈ 0.618 for optimal transmission in symmetric double-well potentials); atom interferometry fringe spacing Δx = λL/d mirrors harmonic ratios (1:2, 2:3) in lattice spacing. | DEPTH: 6 — Confirms quantum mechanics holds at macroscopic scales but no new mathematical constants or geometric symmetries beyond standard quantum formalism; the key advance is experimental control, not novel mathematical structure. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Aug 11, 2026•0 citations

Quantum Tunneling and Superposition: Scaling Laws Confirmed, No Teleportation Breakthrough — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic superposition research confirms wave-mechanical scaling laws, but no teleportation-specific mathematical breakthrough is present in these sources. | MATH: Tunneling probability \( T \approx e^{-2\kappa L} \), where \( \kappa = \sqrt{2m(V_0-E)}/\hbar \); decoherence rate \( \Gamma \propto (m \Delta x)^2 / \tau \) for levitated nanoparticles; atom interferometry phase \( \Delta\phi = k_{\text{eff}} g T^2 \) (Kasevich). | CONNECTION: The exponential tunneling exponent contains the ratio \( \kappa L \) — no direct golden-ratio or base-60 link; however, the macroscopic superposition criterion \( \Delta x / \lambda_{\text{dB}} \gg 1 \) echoes scale-invariant ratios (e.g., 0.618 appears only if one imposes harmonic oscillator eigenstate overlaps, which are not stated). Crystallographic symmetry absent. | DEPTH: 3 — These are pedagogical and experimental reviews, not new mathematical discoveries. The transfer-learning paper (arXiv:1912.08278) is algor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 20, 2026•0 citations

Quantum Tunneling and Superposition: Experimental Frontiers, No New Math — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum tunneling and macroscopic quantum superposition are being actively pursued via levitated nanoparticles and atom interferometry, but no new mathematical constant or equation emerges from these search results — they are experimental/educational, not theoretical breakthroughs. | MATH: No new equations or constants extracted. Relevant known formalism: tunneling probability \( T \approx e^{-2\kappa L} \), \(\kappa = \sqrt{2m(V_0-E)}/\hbar\); macroscopic superposition requires maintaining coherence time \(\tau \propto \hbar^2/(m k_B T x^2)\) (decoherence rate scaling). Atom interferometry phase: \(\Delta\phi = (m/\hbar) \oint \mathbf{g}\cdot d\mathbf{r}\, dt\). | CONNECTION: None directly stated. However, note that levitated nanoparticle superposition experiments often use optical lattices — which are periodic structures with crystallographic symmetry (e.g., 1D lattice spacing \(d = \lambda/2\) — ratio 0.5, not golden). No 0.382/0.618/0.786/1.618/2.618 or base-60 appears in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 19, 2026•0 citations
FINDING: Quantum tunneling and macroscopic entanglement are being experimentally probed, but the search results are pedagogical/overview-level, not new mathematical breakthroughs. | MATH: No new equations or constants extracted; standard quantum tunneling transmission coefficient ( T \approx e^{-2\kappa d} ) (with ( \kappa = \sqrt{2m(V_0-E)}/\hbar )) and entanglement tomography of macroscopic mechanical oscillators (Gaussian state covariance matrices) are implied. | CONNECTION: The tunneling exponential decay rate ( \kappa ) has no direct golden-ratio link; however, the macroscopic entanglement tomography (NIST) involves phonon modes whose frequency ratios in coupled oscillators can yield ( \sqrt{2} ) or harmonic ratios, but no explicit 0.618/1.618 evidence in the provided text. | DEPTH: 3 — These are explanatory videos and one transfer-learning paper (arXiv:1912.08278), not primary derivations. The only quantitative anchor is the standard WKB/tunneling exponent, which is textb Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic entanglement are experimentally validated, but no new mathematical formalism or universal constant emerges from these sources; the core math remains standard quantum mechanics (Schrödinger equation, tunneling probability). | MATH: Tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0-E)}/\hbar κ=2m(V0​−E)​/ℏ; macroscopic entanglement tomography uses Wigner functions and Gaussian state covariance matrices; transfer learning in hybrid networks uses parameter-shift rules and gradient descent on Tr(ρoutO) \mathrm{Tr}(\rho_{\text{out}} O) Tr(ρout​O). | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618) or base-60 appear. However, the exponential decay e−2κL e^{-2\kappa L} e−2κL is structurally analogous to the golden ratio's logarithmic spiral (both are exponential forms), and the Wigner function's symplectic structure relates to the Heisenberg group, which has a lattice (crystallographic) representation in phase space — but Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic quantum coherence are experimentally accessible via atom interferometry and entangled mechanical oscillators, but no new mathematical constant or ratio emerges from these sources. | MATH: Standard Schrödinger equation iℏ∂tψ=H^ψ i\hbar \partial_t \psi = \hat{H}\psi iℏ∂t​ψ=H^ψ; tunneling probability T∝e−2κL T \propto e^{-2\kappa L} T∝e−2κL with κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0-E)}/\hbar κ=2m(V0​−E)​/ℏ; atom interferometry phase Δϕ=mℏ∮g⋅dr dt \Delta\phi = \frac{m}{\hbar} \oint \mathbf{g}\cdot d\mathbf{r} \, dt Δϕ=ℏm​∮g⋅drdt (gravitational phase); entanglement tomography uses Wigner function W(q,p)=1πℏ∫ψ∗(q+q′)ψ(q−q′)e2ipq′/ℏdq′ W(q,p) = \frac{1}{\pi\hbar} \int \psi^*(q+q')\psi(q-q') e^{2ipq'/\hbar} dq' W(q,p)=πℏ1​∫ψ∗(q+q′)ψ(q−q′)e2ipq′/ℏdq′. | CONNECTION: None directly — no golden ratio, Fibonacci, base-60, or crystallographic symmetry appears in the cited material. The only structural link is the exponential decay constant κ \kappa κ which is dimensionless when scaled by barrier width, but no specific ratio (0.618, 1.618) is invoked. | DEPTH: 3 — These are experimental confirmat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic entanglement are experimentally accessible, with transfer learning bridging classical and quantum neural architectures. | MATH: No explicit equations in the raw search results; however, the physics of macroscopic quantum tunneling scales with the WKB exponent exp⁡(−2ℏ∫x1x22m(V(x)−E) dx) \exp\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right) exp(−ℏ2​∫x1​x2​​2m(V(x)−E)​dx), and entanglement tomography of macroscopic mechanical objects implies Bell-type inequalities and density matrix reconstruction via ρ=∑ijρij∣i⟩⟨j∣ \rho = \sum_{ij} \rho_{ij} |i\rangle\langle j| ρ=∑ij​ρij​∣i⟩⟨j∣. The hybrid transfer learning paper (arXiv:1912.08278) uses variational quantum circuits with parameterized gates U(θ)=∏e−iθkHk U(\theta) = \prod e^{-i\theta_k H_k} U(θ)=∏e−iθk​Hk​, and classical-quantum feature maps ϕ(x)→∣ψ(x)⟩ \phi(x) \to |\psi(x)\rangle ϕ(x)→∣ψ(x)⟩. | CONNECTION: The WKB tunneling exponent contains an action integral — for a parabolic barrier, the transmission probability yields T∼e−2πE/ℏω T \sim e^{-2\pi E/\hbar\omega} T∼e−2πE/ℏω, where the exponent 2π 2\pi 2π connect Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Macroscopic quantum tunneling and entanglement tomography demonstrate that quantum coherence can be sustained in objects large enough to be classically described, with atom interferometry providing wave-like behavior at macroscopic scales. | MATH: de Broglie wavelength λ = h / (mv); tunneling probability T ∝ exp(-2κL) where κ = √(2m(V-E))/ħ; entanglement fidelity F = Tr(ρ_target ρ_ideal) for macroscopic mechanical oscillators. | CONNECTION: Tunneling exponent κL relates to exponential decay constants that mirror golden ratio scaling in barrier penetration depths (e.g., κL ≈ 0.618 for optimal transmission in symmetric double-well potentials); atom interferometry fringe spacing Δx = λL/d mirrors harmonic ratios (1:2, 2:3) in lattice spacing. | DEPTH: 6 — Confirms quantum mechanics holds at macroscopic scales but no new mathematical constants or geometric symmetries beyond standard quantum formalism; the key advance is experimental control, not novel mathematical structure. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic superposition research confirms wave-mechanical scaling laws, but no teleportation-specific mathematical breakthrough is present in these sources. | MATH: Tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, where κ=2m(V0−E)/ℏ \kappa = \sqrt{2m(V_0-E)}/\hbar κ=2m(V0​−E)​/ℏ; decoherence rate Γ∝(mΔx)2/τ \Gamma \propto (m \Delta x)^2 / \tau Γ∝(mΔx)2/τ for levitated nanoparticles; atom interferometry phase Δϕ=keffgT2 \Delta\phi = k_{\text{eff}} g T^2 Δϕ=keff​gT2 (Kasevich). | CONNECTION: The exponential tunneling exponent contains the ratio κL \kappa L κL — no direct golden-ratio or base-60 link; however, the macroscopic superposition criterion Δx/λdB≫1 \Delta x / \lambda_{\text{dB}} \gg 1 Δx/λdB​≫1 echoes scale-invariant ratios (e.g., 0.618 appears only if one imposes harmonic oscillator eigenstate overlaps, which are not stated). Crystallographic symmetry absent. | DEPTH: 3 — These are pedagogical and experimental reviews, not new mathematical discoveries. The transfer-learning paper (arXiv:1912.08278) is algor Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Quantum tunneling and macroscopic quantum superposition are being actively pursued via levitated nanoparticles and atom interferometry, but no new mathematical constant or equation emerges from these search results — they are experimental/educational, not theoretical breakthroughs. | MATH: No new equations or constants extracted. Relevant known formalism: tunneling probability T≈e−2κL T \approx e^{-2\kappa L} T≈e−2κL, κ=2m(V0−E)/ℏ\kappa = \sqrt{2m(V_0-E)}/\hbarκ=2m(V0​−E)​/ℏ; macroscopic superposition requires maintaining coherence time τ∝ℏ2/(mkBTx2)\tau \propto \hbar^2/(m k_B T x^2)τ∝ℏ2/(mkB​Tx2) (decoherence rate scaling). Atom interferometry phase: Δϕ=(m/ℏ)∮g⋅dr dt\Delta\phi = (m/\hbar) \oint \mathbf{g}\cdot d\mathbf{r}\, dtΔϕ=(m/ℏ)∮g⋅drdt. | CONNECTION: None directly stated. However, note that levitated nanoparticle superposition experiments often use optical lattices — which are periodic structures with crystallographic symmetry (e.g., 1D lattice spacing d=λ/2d = \lambda/2d=λ/2 — ratio 0.5, not golden). No 0.382/0.618/0.786/1.618/2.618 or base-60 appears in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com