Dynamical Decoupling: Spin Echo vs CPMG — 2D Phasor View

A qubit prepared in an equal superposition loses coherence when its transverse-plane phase drifts away from the reference clock — either because every member of an ensemble sees a slightly different static field (inhomogeneous broadening, T2*) or because the field itself fluctuates in time. Because the transverse-plane phase is a single angle, its natural home is a 2D phasor circle rather than a 3D sphere the trajectory never leaves the equator of. This simulator integrates ~220 real isochromats' phases under a chosen static disorder and a fluctuating Ornstein–Uhlenbeck noise bath, plots them live on that phasor plane, and — going further than a 3D rendering can — draws the ensemble coherence M(t) as an actual time-series curve next to an exact analytic Gaussian-decay overlay, so you can watch the simulation numerically converge to the closed-form prediction. Switching between Free Induction Decay, a Hahn echo, and 4- or 8-pulse CPMG sequences shows how inserting π-pulses reverses accumulated dephasing — the real dynamical-decoupling technique used to extend coherence time in quantum hardware.