HahnEcho

HahnEcho, or Hahn echo, is a pulse-echo technique used in nuclear magnetic resonance (NMR) and electron spin resonance (ESR) to refocus dephased spins and recover signal that has been lost to inhomogeneous broadening. It was introduced by Erwin Hahn in 1950 and stands as a foundational method for measuring transverse coherence in spin systems.

The standard Hahn echo sequence begins with a 90-degree pulse that creates transverse magnetization. After a delay tau, a 180-degree pulse inverts the spin vectors, causing their phase evolution due to static field inhomogeneities to reverse. At a time 2tau after the initial pulse, the spins rephase and an echo is observed. The echo amplitude reflects the surviving transverse magnetization and is influenced by relaxation processes, pulse imperfections, and dynamic dephasing mechanisms.

This simple two-pulse scheme gave rise to many developments, including the Carr-Purcell (CP) sequence and the Carr-Purcell-Meiboom-Gill (CPMG) sequence, which employ trains of 180-degree pulses to extend coherence times and mitigate pulse errors. Hahn echoes are widely used to measure the transverse relaxation time T2, characterize material properties, and study molecular dynamics in solid-state and liquid-state NMR, as well as in ESR experiments. In quantum information science, the Hahn echo concept underpins basic coherence measurements and forms a basis for more complex dynamical decoupling techniques used to preserve quantum states in noisy environments.