Conduction noise and motional narrowing of the nuclear magnetic resonance line in sliding spin-density waves

We report on joint conduction noise and ${}_{}{}^{13}{}_{}{}^{}\mathrm{C}$ NMR measurements in the sliding spin-density wave (SDW) state of (TMTSF${)}_2$ ${\mathrm{PF}}_6$, where TMTSF is tetramethyltetraselenafulvalene. The noise distribution is narrow and scales linearly with current. Narrowing of the NMR line was observed simultaneously. The comparison of these two sets of data shows that the sliding motion is highly coherent and that the conduction noise frequency is equal to the winding rate of the SDW. In the depinned volume, a substantial fraction of the conduction electrons participate in the sliding mode. Consequences on the origin of conduction noise and on other transport measurements are discussed.