Extended Data Fig. 5: Dimer refocusing explanation of asymmetric echo.
From: Signal amplification in a solid-state sensor through asymmetric many-body echo
a, Heisenberg picture illustration of the asymmetric echo, where the sensing (purple star) and measurement (black box) operators happen at time t+ and t+ − t−, respectively. Their anti-matched phase accumulation at t+ = t−/2 explains the observed maximum amplification at such time (see Supplement and our complementary work by N.T.L. et al., in preparation). b, Physical interpretation of the susceptibility matrix. Black dots are the resulting states after sensing along X and Z, for non-interacting sensors. Orange (blue) arrows represent further twisting (decay) dynamics, as described by the diagonal (off-diagonal) components of the susceptibility matrix. c, Spin dimers (gray spins with blue arrow between them) naturally occur in a positionally-disordered spin ensemble. d, Energy spectrum of an isolated spin dimer, and the “Λ” (“ladder”) type processes that describe the diagonal (off-diagonal) components of the susceptibility matrix. The gray disks and the curvy lines connecting them represent the initial state \(| \pm Y\rangle \), which is a coherent superposition \(| {\phi }_{-}\rangle \pm i| {\psi }_{+}\rangle \). e, Analytically calculated amplification of an isolated dimer coupled by TAT Hamiltonian with coupling strength JD = 2π × 40 kHz, under the symmetric echo (brown) and asymmetric echo with t+ = t−/2 (green). f, Same as (e), after averaging over positional disorder. g, Plotting the amplification of ΔX and ΔZ separately for the dataset in main text Fig. 4(c), focusing on t+ = 1.73 μs case. The harmonic dimer spectra (inset) explains the common peaks at t− = 2t+. h, Same as (g) but measured under an XYZ Hamiltonian (see Supplement) that alters the dimer spectra (inset). The anharmonic spectra leads to a splitting of the peaks. The two colored dashed vertical lines represent the predicted peak locations based on the dimer spectra, which determines the refocusing conditions (see Supplement for discussions). Errors represent 1 s.d. accounting statistical uncertainties.
