Each of the 60 qubits starts in phase (a π/2 pulse already applied) and precesses at its own static detuning δi, drawn from a Gaussian of width Δ set by the noise slider — inhomogeneous broadening from a fluctuating local field. The phase disk (top-left) is the equatorial plane of the Bloch sphere viewed from directly above: with no compensation the dots fan out and the bright ensemble-average arrow shrinks (T2* free-induction decay):
φ_i(t) = δ_i · ∫₀ᵗ s(t′) dt′, s(t′) = (-1)^(pulses before t′)
M_xy(t) = |⟨e^(iφ)⟩| (≈ Gaussian FID envelope with no pulses)
A π-pulse instantaneously flips the sign of every qubit's ongoing phase accumulation without touching δi. Because each spin then un-winds exactly what it wound up before the flip, static detuning cancels completely at 2τ — the Hahn echo — and firing N pulses at τ, 3τ, 5τ, …, (2N-1)τ (Carr–Purcell–Meiboom–Gill, CPMG) refocuses the ensemble again at every even multiple of τ up to 2Nτ, verified numerically below:
N=4, τ=1s → signIntegral(t)=0 exactly at t = 2,4,6,8s
(confirmed by standalone script; the sign-integral shared by every qubit,
independent of δ_i, is what returns to zero — not an approximation)
The coherence strip-chart (top-right) plots |Mxy(t)| as it scrolls, with a dashed line at every fired pulse — you can watch the decay curve dip and snap back at each echo. The phase-trajectory diagram (bottom) tracks φi(t) for 12 sampled qubits directly — the classic textbook spin-echo picture of individually fanning lines pinched back to zero at each refocusing time.
- Noise strength — spread Δ of the random per-qubit detuning; higher spread fans the disk out faster between pulses.
- τ — time between successive π-pulses (and to the first pulse).
- π-pulses (N) — 0 gives plain free-induction decay; N≥1 applies Hahn echo (N=1) or CPMG (N>1) refocusing.
- Drag inside the phase disk to rotate the viewing frame — a fixed reference-angle offset, exactly like orbiting a camera around the original 3D Bloch sphere.