Quantum Tunneling in Spacetime — Interactive Simulation

1 · Wave-Packet Laboratory LIVE SOLVER

A Gaussian wave packet is propagated with the split-operator (Strang) spectral method on a 1024-point grid — unitary, spectrally accurate, with absorbing boundaries. Units: ħ = m = 1.

|ψ(x)|² probability density potential V(x) mean energy ⟨E⟩

Scenario

Barrier height V₀

Barrier width a

Packet momentum k₀

Sim speed

Energy ⟨E⟩

–

Transmitted

–

Reflected

–

Norm

–

Regime

–

What to try

Deep tunneling: set V₀ = 2.0, a = 5, k₀ = 1.2 (E ≈ 0.72 < V₀). A few percent of the probability leaks through a classically forbidden wall.
Exponential suppression: nudge the width up and watch transmission collapse — T ~ e^−2κa.
Above-barrier reflection: set E > V₀ (raise k₀). Classically T would be 1; the wave still partially reflects.
Resonant tunneling: choose the double barrier and sweep k₀ slowly — at quasi-bound-state energies, transmission spikes even though each barrier alone is nearly opaque. This is how resonant-tunneling diodes work.

2025 Nobel Prize. Tunneling is not only microscopic: Clarke, Devoret and Martinis showed in 1985 that the collective phase of a Josephson junction — the coordinated motion of billions of Cooper pairs — tunnels out of its metastable well. That discovery of macroscopic quantum tunneling earned the 2025 Nobel Prize in Physics and is the foundation of superconducting quantum computers.

ψ(t+Δt) = e^{−iV Δt/2} · 𝓕⁻¹ e^{−i k²Δt/2} 𝓕 · e^{−iV Δt/2} ψ(t) + 𝒪(Δt³)

2 · Transmission Explorer EXACT T(E)

The exact analytic transmission probability through a square barrier, as a function of incident energy. Toggle the log scale to see the brutal exponential suppression below the barrier — and the Ramsauer–Townsend-like oscillations above it.

Barrier height V₀

Barrier width a

E < V₀ : T = [ 1 + V₀² sinh²(κa) / 4E(V₀−E) ]⁻¹ , κ = √(2(V₀−E)) | E > V₀ : T = [ 1 + V₀² sin²(k′a) / 4E(E−V₀) ]⁻¹

3 · Tunneling-Time Observatory UNIQUE INSTRUMENT

The open question of the field, turned into an instrument you operate. A spin-½ wave packet tunnels while a weak magnetic field, confined to the barrier, makes its spin precess — the spin is the clock (Baz'–Rybachenko–Büttiker Larmor clock). The solver propagates a two-component spinor whose components see Zeeman-split barriers V₀ ∓ ω/2; the transmitted spin orientation is then converted into the two Larmor times. To our knowledge, no other public web tool runs this experiment live.

|ψ|² (both spin components) barrier with Zeeman splitting transmitted-spin dial (top right)

Barrier height V₀

Barrier width a

Packet momentum k₀

Larmor frequency ω

τ_y measured

–

τ_y analytic

–

τ_z measured

–

τ_z analytic

–

phase time τ_φ

–

Büttiker–Landauer

–

Transmission

–

How to operate it

Protocol: the spin starts along +x. Inside the barrier it precesses about the field (z); transmission also rotates it toward z because the spin-up component sees a slightly lower barrier. The transmitted-spin readout is taken at peak transmitted flux: τ_y = −⟨σ_y⟩/ω (precession clock) and τ_z = ⟨σ_z⟩/ω (back-action clock).
Weak-measurement limit: shrink ω toward 0.01 and the measured times converge to the analytic weak values τ_y = −ħ ∂(arg t)/∂V₀, τ_z = −ħ ∂(ln|t|)/∂V₀ (Büttiker 1983; Steinberg 1995). Crank ω up and watch back-action distort the clock — the measurement disturbing the measured.
Opaque-barrier check: for κa ≫ 1, τ_z → τ_BL = ma/ħκ while τ_y and τ_φ saturate — the Hartman effect, live, in the curves below.

τ_φ phase time τ_y Larmor precession τ_z back-action τ_BL = a/κ free flight a/v clock times vs barrier width, at the V₀ and E set above

Why this is the frontier. These are precisely the quantities the attoclock community has fought over for a decade. The 2024–2025 reanalyses (arXiv:2402.14431, arXiv:2503.07859) conclude the attoclock reads a non-local, phase-time-like quantity while Larmor-type clocks read a local, weak-value time — both visible above as different curves derived from the same transmission amplitude. This module reproduces, in a browser, the conceptual experiment that distinction is based on.

4 · False Vacuum Decay SPACETIME

Now scale tunneling up to the universe. If our vacuum is only a local minimum of some field potential, quantum tunneling nucleates bubbles of true vacuum (Coleman 1977). Each bubble wall accelerates toward the speed of light, converting everything it touches. Below: a stochastic visual analogue — nucleation events appear at a rate Γ ∝ e^−B, then grow and percolate.

Bounce action B (barrier)

True vacuum fraction

0.0%

Bubbles nucleated

From chalkboard to laboratory

False vacuum decay is no longer purely theoretical speculation about doomsday. Since 2024 it has been emulated in ferromagnetic-superfluid cold atoms (Zenesini et al., Nat. Phys. 2024), cold-atom lattice gauge theories, Rydberg atom arrays (arXiv:2512.04637), and a 5,564-qubit D-Wave annealer where true-vacuum bubbles formed, moved and merged in real time (2025). A key methodological advance — suppressing spurious edge nucleation with boundary "trenches" (arXiv:2504.02829) — has made bulk, cosmologically-relevant nucleation observable.

Hawking radiation as tunneling

The horizon of a black hole is itself a tunneling barrier. In the Parikh–Wilczek picture (2000), a quantum tunnels outward across the horizon while the hole shrinks to conserve energy; the emission rate Γ ∝ e^{−8πωM(1−ω/2M)} reproduces the Hawking temperature plus non-thermal corrections that carry correlations — an active line of argument (2025) that black-hole evaporation can be unitary. One mathematical object — the imaginary part of an action — connects the wave packet in module 1 to the evaporation of a black hole.

5 · How Long Does Tunneling Take?

The transmitted packet in module 1 emerges suspiciously early. Hartman showed in 1962 that the group delay saturates with barrier width — naively implying superluminal traversal. It doesn't: the transmitted pulse is a reshaped, attenuated replica of the incident packet's leading edge, and no signal outruns light. But the question "what does a clock strapped to the particle read?" stayed open for a century.

Modern attoclock experiments use the rotating field of an elliptically polarized laser as a clock hand while an electron tunnel-ionizes. After a decade of contradictory claims, 2024–2025 analyses (arXiv:2402.14431, arXiv:2503.07859) indicate the attoclock measures a non-local time akin to the Wigner phase time, while Larmor-type local clocks — interpreted through Steinberg's weak values — give a finite, position-resolved time density inside the barrier. Different clocks, different answers, one consistent quantum mechanics. Module 3 above puts both clock families in your hands: run the Larmor experiment, then compare its reading to the phase time computed from the same amplitude.

6 · When Relativity Joins In

Everything above is non-relativistic. Make the particle a Dirac fermion and tunneling changes character: for a sufficiently high step (V₀ − E > 2mc²) the wavefunction inside the barrier oscillates instead of decaying — the Klein paradox. For massless Dirac particles at normal incidence, transmission is exactly 1 through any electrostatic barrier; chirality forbids backscattering. Graphene's carriers obey precisely this equation, which is why Klein tunneling was first seen not at CERN but in carbon sheets (Katsnelson, Novoselov & Geim 2006; Young & Kim 2009). Recent proposals reach new Klein regimes by modulating barriers in time as well as space (arXiv:2510.21154) — and in 1+1D, a curved spacetime metric can be mapped into a local phase of the flat-space Dirac field, making tabletop analogue gravity with tunneling experiments possible.

The simulator in module 1 is honest about its limits: it solves the Schrödinger equation. Where its predictions would fail — Klein zone, pair creation, Zitterbewegung — is exactly where doc 03-relativistic-tunneling.md in this repository picks up.

7 · Selected References

Coleman, Fate of the false vacuum, Phys. Rev. D 15, 2929 (1977); Coleman & De Luccia, Phys. Rev. D 21, 3305 (1980).
Parikh & Wilczek, Hawking radiation as tunneling, Phys. Rev. Lett. 85, 5042 (2000).
Hartman, J. Appl. Phys. 33, 3427 (1962); Büttiker & Landauer, Phys. Rev. Lett. 49, 1739 (1982).
arXiv:2402.14431 — universal behavior of tunneling time in attoclock measurements (2024).
arXiv:2503.07859 — In Search of Lost Tunneling Time (2025).
arXiv:2504.02829 — cold-atom simulators of vacuum decay, edge-nucleation control (2025).
arXiv:2512.04637 — false vacuum decay in a Rydberg atom array.
Sci. Rep. 7, 40346 — mapping curved spacetimes into Dirac spinors.
Katsnelson, Novoselov & Geim, Nat. Phys. 2, 620 (2006) — Klein tunneling in graphene.
2025 Nobel Prize in Physics — Clarke, Devoret & Martinis, macroscopic quantum tunneling.

Full annotated bibliography in docs/references.md of this repository.