← SCPN-Quantum-Control

Phase synchronisation

A crowd of oscillators, each ticking at its own rate, coupled so that each feels the average of the rest. Below a critical coupling they stay a blur; above it they snap into step. That transition — the Kuramoto model — is the physical idea the whole platform simulates and controls. Drive it yourself below; the panel runs the library's exact mean-field kernel.

The model in one equation

Each oscillator $i$ has a phase $\theta_i$ and its own natural frequency $\omega_i$. It advances at its own rate plus a pull toward the population's collective phase, scaled by the coupling $K$. The collective phase is captured by the complex order parameter $R\,e^{i\psi}$ — the average of every oscillator's unit phasor. Its magnitude $R$ runs from $0$ (a uniform blur) to $1$ (perfect lock).

$$\dot\theta_i \;=\; \omega_i \;+\; K\,R\,\sin(\psi-\theta_i),\qquad R\,e^{i\psi} \;=\; \frac{1}{N}\sum_{j=1}^{N} e^{i\theta_j}$$

The mean-field kernel writes the coupling in the algebraically identical form $K\big(\bar{s}\cos\theta_i-\bar{c}\sin\theta_i\big)$ with $\bar{c}=\langle\cos\theta\rangle,\ \bar{s}=\langle\sin\theta\rangle$, so a single pass over the population drives every oscillator. It is integrated with fixed-step fourth-order Runge–Kutta — the same kernel the platform pins its browser panel to.

Live — mean-field Kuramoto, RK4
order parameter R: — coupling K: — state: —

The left wheel shows each oscillator as a dot on the unit circle; the arrow is the order-parameter vector $R\,e^{i\psi}$. The right trace is $R$ over time. Start with $K$ low and the dots scatter around the ring — $R$ hugs zero. Raise $K$ past the point where the pull overcomes the frequency spread and a cluster forms: the arrow lengthens and $R$ climbs toward one. Widen the frequency spread and you need more coupling to lock — the competition that sets the critical point.

Deeper: why the browser panel is checked to a tolerance, not bit-for-bit
This panel runs the same mean-field derivative, the same fixed-step RK4 step, and the same order-parameter definition as the library's Python reference and its Rust WASM kernel. The three are pinned to each other — but across languages the elementary $\sin$ and $\cos$ can differ in the last bit, so the reference deliberately checks the panel within a tolerance rather than demanding bit-identical output. Verified here against the Python reference over four trajectories (incoherent, partial, locked, and zero-coupling), the largest divergence in $R$ was under $2\times10^{-14}$ — roundoff, far inside the reference's own tolerance.

Why it sits under everything here

The Kuramoto network is the workload the platform maps onto quantum and analog hardware. The same coupling problem compiles to a Trotterised circuit for a gate machine, to a native Hamiltonian for neutral-atom, circuit-QED and photonic backends, and to this classical mean-field kernel for a fast, honest baseline. Synchronisation is the quantity every one of those routes is trying to reproduce and steer.

RouteWhat carries the dynamicsWhere it runs
Classical mean-fieldthis RK4 kernel over $\theta_i$browser / CPU baseline
Trotterised circuitXY Hamiltonian, gate decompositiongate-based QPU
Native analogRydberg / exchange / beam-splitter couplersneutral-atom, circuit-QED, photonic
Networked classicalper-pair coupling matrix $K_{nm}$CPU, arbitrary topology
The science Algorithms Hardware results Back to overview

Evidence boundary: this is the classical mean-field simulator, run live in your browser from the library's exact kernel — a baseline and a teaching tool, not a hardware result. It makes no quantum-advantage claim; the hardware observations live on the results pages, with their own raw counts and statistics. Frequencies and initial phases are generated on the page; the physics you see is the integrator, which is checked against the Python reference to better than $2\times10^{-14}$.