Each of the 60 qubits in the cloud starts on the Bloch-sphere equator (a π/2 pulse has already been 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. With no compensation the ensemble fans out and the net transverse magnetization decays as free-induction decay (T2*):
|ψ_i(t)⟩ = (|0⟩ + e^(iδ_i·t)|1⟩)/√2
M_xy(t) = |⟨e^(iδ·t)⟩| ≈ exp(-(Δt)²/2) (Gaussian FID envelope)
A π-pulse about the X-axis (spin echo) instantaneously flips each qubit's accumulated phase sign without changing δi. Because every spin then un-winds the phase it built up before the pulse, all static (slowly varying) detuning cancels exactly at the echo time — this is the Hahn echo:
φ_i(t) = δ_i · ∫₀ᵗ s(t′) dt′, s(t′) = (-1)^(pulses before t′)
Hahn echo (1 pulse at τ): φ_i(2τ) = δ_i·τ - δ_i·τ = 0 → full refocus
Firing N π-pulses at τ, 3τ, 5τ, … (2N-1)τ — the Carr-Purcell-Meiboom-Gill, or CPMG, sequence — refocuses the ensemble again at every even multiple of τ, which is exactly the dynamical-decoupling trick real quantum processors use to fight dephasing noise between gates: more pulses track a faster-fluctuating noise field and hold coherence closer to 1 for longer, at the cost of more control hardware overhead.
- Noise strength — spread Δ of the random per-qubit detuning; higher spread dephases the cloud 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.
- The bright arrow is the ensemble-average Bloch vector; its length is the live coherence readout.