Dynamical Decoupling: Spin Echo vs CPMG

Refocusing Sequence

Noise Environment

Static disorder Δ (T2*-type) 1.0

Fluctuating noise σ (fast bath) 0.6

Sequence duration T 10.0

Live Readout

t = 0.00 / 10.00

Coherence |M(t)|

1.000

π-pulses fired

0 / 0

Sequence

Free Induction

Cycle

How it works

Each of N ≈ 240 "isochromats" is a copy of the same qubit that started in an equal superposition (a 90° pulse puts it on the equator of the Bloch sphere). Each one sees a slightly different local field, so its phase drifts at its own rate:

dφᵢ/dt = s(t) · [ δᵢ + ξᵢ(t) ]
δᵢ  ~ N(0, Δ²)      static, per-isochromat detuning ("inhomogeneous broadening")
ξᵢ(t)  Ornstein–Uhlenbeck  fast noise, correlation time τc, strength σ
s(t) = ±1, flips sign at every applied π-pulse

The ensemble coherence — the observable transverse magnetization — is the phase-averaged sum:

M(t) = | (1/N) Σᵢ exp(i·φᵢ(t)) |

With no pulse (free induction decay) the spread of static offsets δᵢ dephases the ensemble irreversibly on a timescale T2* = 1/Δ — this is the same mechanism that limits raw qubit coherence and NMR/MRI T2* contrast.

A Hahn echo — one π-pulse at T/2 — flips the sign of every isochromat's accumulated phase, so the *static* part of the dephasing exactly re-converges at t = T (a sharp echo). But it cannot track noise ξᵢ(t) that fluctuates faster than the T/2 spacing, so some coherence is lost to the fast bath.

CPMG spaces several π-pulses evenly across T. Each interval is shorter, so the fast noise looks more "static" within it and gets refocused too — this is dynamical decoupling: adding pulses filters out higher noise frequencies, at the cost of more control hardware. This is the real technique superconducting and trapped-ion qubits use to extend coherence time beyond T2*.

π-PULSE

Isochromats (individual phases) Ensemble average M(t)

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