*3.3. Response to an ac Drive*

We finally address the full two-time character of the cross-correlation in Equation (5). Specifically, we consider the case of ac driving by introducing a monoharmonic driving term to the voltage

$$V(t) = V\_0 + A\cos(\Omega t)\tag{7}$$

where the static part is set by *V*0 = *tC*/2 and the amplitude of the ac driving is *A* = *tC*/2 with the driving frequency *Ω* = *tC*/10. To reduce the computational effort, we consider only the short nanoribbons in this case (*L* = 2 nm). In Figure 5, we show the propagation of the full two-time cross-correlation from the initial time *t*0. AGNR data are shown in Figure 5a–c and ZNGR data are shown in Figure 5d–f. In contrast to the previous steady state results, here we show the initial transient (up to 50 fs), which includes relaxation effects.

We observe that the ac driving does not change the overall picture of traversal time, i.e., the time it takes for the information to traverse through the nanoribbons can be clearly read off from the separation of peaks along the anti-diagonal. Compared to the long-time limit in Figures 2–4, the initial transient only shows some additional oscillations but the main features seen in the steady state data are still visible. The two-time correlations also show the effect of disorder; as in the dc case, the signal gets considerably disturbed for hopping disorder (cf. Figure 3) and for on-site disorder (cf. Figure 4), but in the latter case the signal destruction is less severe. In the disorder energy scale considered in Figure 5 (hopping and on-site disorder strength: *w*, *f* = *tC*/2), it is also observed that the first traversal event peaks are more clearly visible for the case of on-site disorder (Figure 5c,f) compared to the hopping disorder (Figure 5c,e). However, the coherence of subsequent traversal events without disorder (shown in the side-peaks of Figure 5a,d), is strongly suppressed by either kind of disorder. In calculations not shown here, we have checked that other types of ac driving (biharmonic drive, faster/slower modulation, lower/higher intensity) have no effect on the qualitative behaviour of traversal times.

**Figure 5.** Two-time current cross-correlations for fixed length (*L* = 2 nm) GNRs with ac driving. (**a**) AGNR without disorder, (**b**) AGNR with hopping disorder *w* = *tC*/2, (**c**) AGNR with on-site disorder *f* = *tC*/2, (**d**) ZGNR without disorder, (**e**) ZGNR with hopping disorder *w* = *tC*/2, (**f**) ZGNR with on-site disorder *f* = *tC*/2.
