Spin Echo

Dynamical decoupling and refocusing

Pulse Sequence

Echo Time 2τ: 2.0 ms Frequency Spread Δω: 5.0 rad/ms Number of Spins: 50

Decoherence

T₂ (dephasing): 10.0 ms

Enable T₂ decay🔑 Unlock

Visualization

Speed: 1.0× Show individual spins Show total magnetization

Spin Echo Physics

The Hahn Spin Echo refocuses dephasing from inhomogeneous fields.

Pulse Sequence:

90° pulse: rotates spins into x-y plane
Free evolution τ: spins dephase (fan out)
180° π-pulse: flips spins (reverses evolution)
Free evolution τ: spins rephase (echo!)

Key Physics:

Each spin precesses at ω₀ + δω
After time τ: φᵢ = δωᵢ × τ
π-pulse flips: φᵢ → -φᵢ
At time 2τ: total phase = 0 (refocused!)
Echo amplitude decays as exp(-2τ/T₂)

Applications:

NMR spectroscopy and MRI
Dynamical decoupling for quantum computing
Quantum memory protection
Removing inhomogeneous broadening