Propagation effects in the target

In any target of finite thickness, the dynamics of the induced magnetic transition dipole moments will vary as a function of position in the target, since they are driven not only by the externally applied field, but also by the field scattered by the upstream dipoles. To determine these propagation effects, we treat the target as a medium of two-level atoms, and calculate the propagation of the SCU pulse through the target using the Maxwell−Bloch equations in the slowly varying envelope approximation^40,41

where Ω(x,t)=−2dℰ(x,t)/ħ is the Rabi frequency, with ℰ the slowly varying amplitude of the propagating field, d the magnetic dipole moment, L the target length, and ρ_eg(x, t) the density matrix element corresponding to the coherence between the ground state |g⟩ and the excited state |e⟩ induced by the propagating field. It follows from the nuclear dynamics governed by the equations of motion for the density operator ρˆ

Results of this analysis are shown in Extended Data Figs. ​Figs.22 and ​and33 for the parameters relevant to our experiment. Extended Data Fig. ​Fig.22 shows that there are indeed propagation effects, that is, the dipole dynamics depends on the position in the target because of the light scattered by the upstream nuclei. Nevertheless, the coherent control acts similarly everywhere inside the target: In the enhanced emission case, the excitation due to the first pulse is always rapidly driven back to the ground state by the second pulse. In the enhanced excitation case, the excitation due to the first pulse is always increased by the second pulse. To illustrate this feature in more detail, Extended Data Fig. ​Fig.33 compares the dipole dynamics at the target entry (x = 0), in the middle of the target (x = L/2), and at the end of the target (x = L). At all positions in the target, the two coherent control cases are clearly visible. Finally, the results shown as dashed lines in Fig. ​Fig.33 are obtained by averaging the spatially resolved dipole dynamics in Extended Data Fig. ​Fig.22 over the sample length.