-0.5 -0.25 0 0.25 0.5 -0.4 -0.3 -0.2 Quasienergy 2 [4] T. Salger, et al. , Science 326 , 1241 (2009). [5] D H. Dunlap and V.M. Kenkre, Phys. Rev. B 34 , 3625 (1986). FIG. 1: Quasienergy spectrum of the ratchet system for time- symmetric driving ( K = 2 . 4, ω = 1, β = 0). The major- ity of the quasienergies show little dependence on the spa- tial asymmetry α , but the narrow avoided crossings (with a gap of Δ ϵ ≃ 0 . 0014) at α = ± 0 . 32 (highlighted by red circles) give rise to long-lived transient currents, with a dura- tion much longer than typical experimental observation times, even though the temporal symmetry of the driving is not bro- k en. FIG. 2: Decay of the continuously averaged current 〈 I ( t ) 〉 for the strongly driven ratchet system, K = 2 . 4, ω = 1, for different asymmetry parameters α and β . A typical experi- mental observation time, indicated by the vertical green line, is taken from the recent work of the Bonn group [4]. For β = 0 . 2 (black solid line) time-symmetry of the driving is explicitly broken and 〈 I ( t ) 〉 approaches a non-zero asymp- totic value. When β = 0, however, the time-symmetry of the driving is not broken and asymptotically the ratchet current must decay to zero. For α = 0 . 32 (red dashed line) the aver- age current decays extremely slowly, due to a narrow avoided crossing in the quasienergy spectrum (see Fig. 1). We also plot the stroboscopically averaged value of the current (green dot-dashed line) for these driving parameters, which asymp- totically approaches a constant value. For other values of α (blue dash-dot-dot line) the current decays more rapidly, but nonetheless remains significant over timescales much longer than those used in experiment. 100 200 300 Time / T 0.01 0.1 Time averaged current  stroboscopic Bonn experiment