Fig. 4: Time-reversed echo experiments.

a, The protocol of time-reversed echo, in which a sensing rotation towards the local amplifying direction (purple pulse, simulated by explicit microwave driving) is sandwiched between forward (t₊) and backward (t₋) evolutions under the TAT Hamiltonian. The reversal is achieved by a π/2 pulse around the Y-axis, which switches \({\widehat{\sigma }}^{x}\) with \({\widehat{\sigma }}^{z}\) and thereby transforms the Hamiltonian from \(\widehat{{\boldsymbol{\sigma }}}\cdot \widehat{{\boldsymbol{\sigma }}}+\lambda ({\widehat{\sigma }}^{z}{\widehat{\sigma }}^{z}-{\widehat{\sigma }}^{x}{\widehat{\sigma }}^{x})\) to \(\widehat{{\boldsymbol{\sigma }}}\cdot \widehat{{\boldsymbol{\sigma }}}-\lambda ({\widehat{\sigma }}^{z}{\widehat{\sigma }}^{z}-{\widehat{\sigma }}^{x}{\widehat{\sigma }}^{x})\). b, Measurement of the Y revival under the time reversal, without the sensing step (δθ = 0). The black curve is the reference decay without time reversal, and each of the other curves corresponds to dynamics reversed at a different t₊, indicated by the leftmost point on the curve (for example, for the brown curve, t₊ = 4.3 μs, as indicated by the brown horizontal arrows). c, Signal amplification measurement (in units of Bloch sphere radius) sweeping both t₊ and t₋, with sensing angle δθ = 15°. Each curve represents a different t₊, similar to b. The downward blue arrow indicates the point with maximum amplification, relative to the non-interacting reference at t₊ = t₋ = 0. The distance is averaged between the two amplifying pairs (Fig. 3b). d, Data in c replotted to compare the amplification without echo (black), with symmetric echo (brown) and with asymmetric echo (green). e, Mirror reflection (\({\mathcal{R}}\)) of the TAT dynamics, which reverses the direction of semi-classical flow and exchanges the sensing (\(\widehat{S}\)) and measurement (\(\widehat{M}\)) operators, represented here by their Pauli basis decompositions. This mirror reflection, when combined with time reversal (not drawn), which restores the original flow direction, forms a fundamental symmetry of the system. The directions of \(\widehat{S}\) and \(\widehat{M}\) are chosen as their local amplifying directions (that is, the TAT flow leads to amplification for sensing rotation around \(\widehat{S}\) and projective measurement along \(\widehat{M}\)). f, The symmetry in e maps the sensing operator \(\widehat{S}({t}_{+})\) to \(\widehat{M}(-{t}_{+})\) and the measurement operator \(\widehat{M}({t}_{+}-{t}_{-})\) to \(\widehat{S}({t}_{-}-{t}_{+})\), transforming the original asymmetric echo (top) to an equivalent protocol (bottom) with a modified forward evolution time \({t}_{+}^{{\prime} }={t}_{-}-{t}_{+}\), ensuring equal amplitude of amplification between them. Errors in the figure represent 1 s.d. accounting statistical uncertainties.