Figure 1

(a) The minimum overlap with the MI state, or localization, reached in a 7-site system with $U=8$, during 10 periods of driving. For $\omega =20$ (dashed line) the localization peaks at $K/\omega =2.4,5.5$—the zeros of ${\mathcal{J}}_0(K/\omega )$. When $\omega$ is reduced to $\omega =8$ (the first photon resonance, solid line), the peaks become extremely narrow and are centered on the zeros of ${\mathcal{J}}_1(K/\omega )$. The diamonds mark the points $K/\omega =3.5$ and 3.8 (see below). (b) Time evolution of the $\omega =8$ case for three system sizes, 7, 9, and 11 sites. For $K/\omega =3.5$, away from the resonance, the overlap with the initial state, $P(t)$, rapidly drops to zero. For $K/\omega =3.8$, at the peak of the resonance, the decay is much slower, indicating that the driving field preserves the MI state.Reuse & Permissions

Figure 2

(a) Quasienergy spectrum of a 2-site system, with $U=8$ and $\omega =20$. Only two of the three quasienergies are plotted (the remaining one oscillates weakly about zero), which make a series of close approaches to each other as $K$ is increased. Solid lines in (a) and (c) (red online) denote the perturbative solutions, which agree well with the exact results (black circles). (b) At the points of close approach, the tunneling is suppressed and the localization peaks. For all field strengths the tunneling is suppressed with respect to the undriven system, and the localization is thus enhanced. (c) At lower frequencies the behavior of the quasienergies changes dramatically. At $\omega =8$ the system is at the first photon resonance ($U=\omega$), and the behavior of the quasienergies is described extremely well by the perturbative solutions $ϵ=0,\pm 2{\mathcal{J}}_1(K/\omega )$. (d) As before, the localization is peaked at the points of quasienergy crossing, but in contrast to (b) the peaks are extremely sharp.Reuse & Permissions

Figure 3

Localization produced in a 7-site system ($U=8$) for two forms of periodic driving: (a) sinusoidal, (b) square wave. Both waveforms produce excellent localization at the first photon resonance, $\omega =U$, shown by solid black lines. At the second photon resonance ($2\omega =U$), shown by dashed lines (red online), the localization produced by the sinusoidal driving is considerably smaller, but the square wave still produces high peaks.Reuse & Permissions

Figure 4

The localization produced in a 5-site system with $U=8$ as a function of the frequency $\omega$ of the square wave driving field and its amplitude $K$. When $n\omega =U$ the localization is almost zero [the centers of the horizontal bands (blue online)] except at sharply defined peaks (red background online). Between the bands localization is good. The $n=1$, 2, 3, and 4 resonances are marked on the right. The upper-left triangle (blue online) displays poor localization and little structure, corresponding to the nonperturbative regime.Reuse & Permissions