Experiment 06 · Planetary Grid · Kuramoto Network

Twelve oscillators.
One coupling constant.

Integrating dθᵢ/dt = ωᵢ + (K/N) Σⱼ sin(θⱼ − θᵢ) in real time. Below Kc ≈ 2.71 rad/s, the twelve nodes drift apart. Above Kc, they snap into phase — and the order parameter r climbs to 1.

EXPLAIN LIKE I'M 10
· everyday-language one-paragraph version
Twelve metronomes on the same table, each starting at its own pace. Turn the 'coupling' knob low and they stay disordered. Turn it up and watch them snap into step together. The exact moment they snap is what this page measures — it is the math behind how a network of qubits starts to behave like one bigger thing.
WHAT YOU ARE LOOKING AT
Twelve oscillators with slightly different natural frequencies, all coupled to each other. Turn up the coupling strength and watch them lock into phase together.
WHY THIS MATTERS FOR QUANTUM
A multi-qubit quantum computer is a network of coupled oscillators. The unresolved engineering question is how strongly to couple them. Too weak: they don't talk. Too strong: they noise each other into decoherence. Kuramoto's 1975 equations — the math behind this page — describe the exact moment a population spontaneously synchronises. The critical coupling K꜀ is the minimum needed for that lock; below it nothing happens, above it locking is automatic.
WHAT TO WATCH
The order parameter r. Below K꜀ it hovers near 0 (chaos). Above K꜀ it snaps near 1 (synchrony). The transition is sharp — that sharpness is the prediction.
Phase circle · N = 12
K = 2.00 · Kc = 2.71
r = 0.350
locked: 1/12
Order parameter r(t) · last 400 samples
Method note

The twelve intrinsic frequencies ωᵢ are drawn from a Gaussian centred at the active target with σ = 18 Hz. The mean-field phase transition occurs at Kc = σ √(8/π) ≈ 2.71. r is the magnitude of the centroid of the unit-circle phase distribution; locked = nodes within ±20° of the mean phase. Time-step dt = 0.015, integrator: forward Euler.

Coupling strength K · 2.00
0Kc = 2.718

The classical Kuramoto phase transition: increase K past the critical value and the population spontaneously synchronises. This is the cleanest known model of resonance-as-coupling, replacing the heuristic illustration that lived here before.

Simulation speed · 1.00×
0.1×1×3×

Scales the integration step. 1× corresponds to dt = 0.015; the synchronisation transition resolves in ≈1 s of real time at this setting.

Order parameter r
0.350
Locked nodes
1/12
Coupling regime
Sub-critical
Quantum mode · 4-qubit Lindblad realisation

One simulation, two transitions.

The classical network above syncs at Kc. The same K applied to four coupled qubits sometimes squeezes first — ξ² drops below 1 before r reaches 1. When that happens, the network is entangled but not yet phase-locked.

Live state · K = 2.00
r (classical)
0.071
phase coherence on xy-plane
|⟨J⟩|
0.141
max = 2.0 (CSS)
ξ² (Wineland)
147.942
< 1 ⇒ entangled
K_c · classical
1.60
mean-field sync threshold
Inverted scale of ξ² · the bar fills when entanglement is detected
▽ Above the classical limit · no entanglement detected
Local dephasing γ · 0.020
Each qubit dephases independently with this rate. Higher γ pushes the entanglement transition K_e to higher coupling.
r and ξ² over time
Orange · classical order parameter r ∈ [0, 1]. Cyan · ξ² spin-squeezing (lower is more entangled; dashed line at 1.0 is the classical limit).
K-sweep · steady state at t=8
Each point: take CSS at t=0, evolve under the chosen K and γ to t=8, plot r (orange) and ξ² (cyan). K_c is the classical threshold; the cyan curve crossing ξ²=1 marks the entanglement threshold K_e. When K_e < K_c there is a window of entangled but unsynchronised states — the regime mean-field theory hides.
Scope of the quantum panel
  • 1. N is just 4 here, not 12 — the full 16×16 density matrix evolves under exact Lindblad without unraveling, which keeps the result clean but caps the system size.
  • 2. Hamiltonian is XY exchange (σx σx + σy σy), not the full Kuramoto sin(θ_j − θ_i). XY is the quantum cousin — same K_c in mean-field, but it lives in the entire Hilbert space rather than on a torus.
  • 3. The spin-squeezing ξ² is a sufficient (not necessary) entanglement witness. ξ² > 1 doesn't mean the state is separable — only that this particular witness can't see the entanglement. ξ² < 1 is a strict guarantee.
  • 4. The K-sweep is computed at a fixed horizon t=8. At very small γ and very high K, the dynamics oscillate without settling — the snapshot under-represents the true asymptotic squeezing.
What this adds to the classical panel

The classical Kuramoto network above this panel tells you when the population phase-locks. This panel tells you when it entangles — a strictly stronger statement. In any regime where the cyan curve sits below 1 but r is still small, the system is doing something quantum that the classical model cannot see. If you want to design a multi-qubit processor, the K-window you care about is not where r=1; it is where ξ²<1 while still resolving distinct individual qubit phases. This panel exposes that window directly.