Symmetries and transport in site-dependent driven quantum lattices

We explore the quantum dynamics of particles in a spatiotemporally driven lattice. A powerful numerical scheme is developed which provides us with the Floquet modes and thus enables a stroboscopic propagation of arbitrary initial states. A detailed symmetry analysis represents the cornerstone for an intricate manipulation of the Floquet spectrum. Specifically, we show how exact crossings can be converted into avoided ones, while the widths of these resulting avoided crossings can be engineered by adjusting parameters of the local driving. Asymptotic currents are shown to be controllable over a certain parameter range.

Figure 1

Snapshot of a spatiotemporally driven lattice consisting of Gaussian barriers of height $V_0$ with lattice spacing $L$. Three different driving laws $d_s\left(t\right)$ for $s=1,2,3$ are periodically repeated. The shaded barriers indicate the equidistant equilibrium positions of the barriers.

Figure 2

Absolute value of matrix elements $U_{\mu \nu }(T,0)$ of the time evolution operator for $\kappa =0$. The phases ${\delta }_s$ of the three barriers within one unit cell are $(0,0,0)$ in (a) and $(0,\pi ,0)$ in (b). The remaining parameters are $L=10,V_0=1.0,\omega =1.0,A=1.0$, and $\Delta =0.5$.

Figure 3

Husimi distributions in arbitrary units for $\sigma =0.5$ of a FBM at zero quasimomentum in a lattice with $n_p=3$ for four different settings of the barrier phases $({\delta }_1,{\delta }_2,{\delta }_3)$: (a) $(0,0,0)$, (b) $(0,\pi /2,0)$, (c) $(0,\pi ,0)$, and (d) $(0,\pi ,\pi /4)$. The barrier equilibrium positions are at $x_1=-10$, $x_2=0$, and $x_3=10$. The remaining parameters are as in Fig. 2.

Figure 4

Extracts of the Floquet spectra for different setups containing three barriers per unit cell. The three phases of the barrier driving laws are (a) $(0,2\pi /3,{\delta }_3)$, (b) $(0,{\delta }_2,0)$, and (c) $(0,0,0)$. In (c) the spectra for two different global driving amplitudes are shown. The remaining parameters are as in Fig. 2.

Figure 5

(a) Asymptotic quantum current $J\left(t_0\right)$ for three different configurations of the barrier phases $(0,2\pi /3,{\delta }_3)$. (b) Time-averaged quantum current as a function of the third barrier phase ${\delta }_3$. The lines are guides for the eye. The remaining parameters are as in Fig. 2.