Figure 1

Schematic representation of the driving-induced transitions for driving frequencies $\omega =1/4$, 1/2, 1, and 2. See the text for the meaning of the various symbols. In each case the direct, first-order (solid arrows) and indirect, second-order (dashed arrows) transition amplitudes interfere to produce a ratchet current. For clarity, in the graph for $\omega =1/2$ the first dashed arrow of the various second-order processes has been omitted.Reuse & Permissions

Figure 2

Comparison of the time-dependent current, in units of $\hslash$, predicted by the effective three-level model [pale (green) dotted line] and the exact numerical results [dark (blue) solid line]. Parameters are $\omega =1$ and $\alpha =\beta =0.2$. The three-level model predicts Rabi oscillations which fit the exact results extremely well for weak driving ($K=0.1$). For strong driving ($K=0.5$) the exact results show additional small, high-frequency oscillations, but the main behavior is still reasonably well described by the effective model.Reuse & Permissions

Figure 3

Time-averaged current, in units of $\hslash$, averaged over 4000 driving periods for a weakly driven system ($K=0.05$, asymmetry parameters $\alpha =\beta =0.2$), plotted as a function of the driving frequency. Four peaks appear, at driving frequencies of $\omega =0.25$, 0.5, 1, and 2, in agreement with the resonant condition (18). Inset: Magnified view, showing the existence of subresonances at commensurate fractions of the resonant frequencies. We also note the reduction of the $\omega =0.5$ resonance when the time average is evaluated continuously instead of stroboscopically.Reuse & Permissions

Figure 4

Comparison of the analytical results with numerically exact data for the four principal resonances, $\omega =0.25$, 0.5, 1, and 2 (see Table ). As before, currents are measured in units of $\hslash$. The values for $\omega =1/2$ are actually negative, so for convenience we plot their absolute value. For all the curves the asymmetry parameters were set to be $\alpha =\beta =0.2$. We can see that for weak driving strengths the agreement is excellent; the effective three-level model produces quantitatively accurate results. For higher driving strengths the model diverges from the exact results, as expected for a perturbative result.Reuse & Permissions

Figure 5

Rabi frequencies, ${\Omega }_R$, predicted by the effective three-level model [see Eq. (44) and Table ] are shown by lines, to compare with data extracted from numerically exact simulations. As in Fig. 4, we see excellent agreement between analytics and numerics.Reuse & Permissions

Figure 6

The stroboscopic current is evaluated at discrete times $t_n=t_0+nT$. Here we show the dependence of the stroboscopic time average on the initial time sampling point $t_0$. The current is measured in units of $\hslash$. For $\omega =0.25$, 1, and 2 the variation is weak, but for $\omega =0.5$ the dependence is strong and contributes to the low value of the continuous current average for this value of the driving frequency. The number of driving periods used to compute the stroboscopic average is, from top to bottom, 400, 400, 600, and 300. This amounts to about 10 complete Rabi periods ($2\pi /{\Omega }_R$) in the first two cases and to one Rabi period in the last two.Reuse & Permissions