PHY.10:5.3 - Refocus a hidden physical difference

A prepared ensemble has a transverse phase signal. Consider two idealized accounts of its free decay. In account S, each member has a fixed frequency offset delta, drawn from the Lorentzian density

p(delta)=Gamma/[pi (delta²+Gamma²)], with Gamma>0.

Each phase advances by delta t. Averaging over the ensemble gives M_S(t)=exp(-Gamma t) for t≥0. The individual offsets remain fixed even though the mean signal decays.

In account D, the phase instead has independent Gaussian increments with variance 2 Gamma dt over an interval dt. This Markov dephasing gives the same free signal, M_D(t)=exp(-Gamma t). The two accounts agree on this free-decay observation.

Apply a refocusing rotation at time tau and read the signal at 2 tau. In the ideal pulse comparison, the sign of phase accumulation is reversed for the second interval. Under S, each accumulated phase becomes delta tau-delta tau=0; the ensemble signal returns to 1. Under D, the two intervals have independent phase increments. Subtracting them leaves variance 4 Gamma tau, so the signal remains exp(-2 Gamma tau).

For Gamma=10 s⁻¹ and tau=0.1 s, the predicted refocused signals are 1 and approximately 0.135. The intervention exposes a difference hidden by the equal free decays.

A physical pulse has finite duration, range and accuracy. Its response over the occupied frequency range, other relaxation during the sequence and readout error must be included where they can change this separation. For illustration, if their combined effect on each predicted normalized signal is bounded by 0.05, the predicted indication intervals around 1 and 0.135 remain disjoint. This bound is a condition of that proposed implementation, not supplied by the ideal calculation.

If the pulse cannot refocus a consequential part of the ensemble, a small return can have that cause as well as irreversible dephasing. Change the pulse or reference comparison, retain its bounded effect, or leave the interpretation conditional. A partial echo can also motivate an account with both static variation and changing noise. The performed comparison then guides which hidden dynamics to retain.