Directed transport and localization in phase-modulated driven lattices

We explore the dynamics of noninteracting particles loaded into a phase-modulated one-dimensional lattice formed by laterally oscillating square barriers. Tuning the parameters of the driven unit cell of the lattice selected parts of the classical phase space can be manipulated in a controllable manner. We find superdiffusion in position space for all parameters regimes. A directed current of an ensemble of particles can be created through locally breaking the spatiotemporal symmetries of the time-driven potential. Magnitude and direction of the current are tunable. Several mechanisms for transient localization and trapping of particles in different wells of the driven unit cell are presented and analyzed.

Figure 1

Schematic illustration of the lattice of laterally oscillating barriers.

Figure 2

(Color online) Stroboscopic PSS of the lattice of uniformly oscillating square potentials. For a better illustration we have added in (a) the potential energy to the kinetic energy of the particles, in order to avoid the discontinuities in the “regular” PSS [see (b)].

Figure 3

(Color online) (a) Central periodic orbit $\left(x\approx 2.1,p\approx 1.4\right)$ for the resonance with winding number $w=2$ in the lattice of uniformly oscillating barriers. (b) Periodic two bounce orbit between the first and second barrier in the case of the specific phase period three $\left\{\dots ,0,\frac{\pi }{10},\frac{3\pi }{10},0,\frac{\pi }{10},\frac{3\pi }{10},\dots \right\}$.

Figure 4

(Color online) Magnification of the Poincaré surface of section in the region of the upper boundary of the chaotic sea. The dashed curve is the last stable KAM torus, i.e., this invariant curve delimits the chaotic sea.

Figure 5

(Color online) Poincaré surface of section of a single chaotic trajectory, which has been started in the upper region and stopped once it has crossed the $p=0$ axis. The diamonds and stars are periodic orbits corresponding to truncations of the continued fraction expansion of $w_1=\frac{2\gamma +1}{\gamma }$ and $w_2=\frac{29\gamma -1}{11\gamma }$, respectively.

Figure 6

(Color online) Flux through the chains of periodic orbits, belonging to convergents of $w_1$ and $w_2$, as a function of the level $j$. The black curve is ${\Phi }_{r_j/s_j}\sim C{\xi }^{-j}$ with $\xi \approx 4.339$.

Figure 7

(Color online) Stroboscopic PSS of the lattice with a linear phase gradient of period 3 (a), 5 (b) and (10). The length of the unit cell increases with the period of the gradient. The dashed curves are the FISCs.

Figure 8

(Color online) Magnification of the Poincaré surface of section in the upper region of the FISC for two different phase gradients. (a) is the equidistant gradient of period 3 and in (b) the phase of the “middle” barrier has been shifted by $\alpha =-0.03\pi /3$. The dashed curves are the FISCs.

Figure 9

PSS for a lattice with a gradient of period 3 and four different global potential heights. The values are $V_0=2.0$ (a), $V_0=13.0$ (b), $V_0=22.0$ (c), and $V_0=32.0$ (d).

Figure 10

(Color online) (a) shows the absolute value of the mean position averaged over an ensemble of ${10}^5$ particles, which has been started with small momentum $p\le 0.1$ in the chaotic sea as a function of time for various phase periods. For $n=6$ the evolution of the standard deviation has been plotted in the inset. Accordingly, (b) shows the mean velocity as a function of time.

Figure 11

(Color online) Cumulative distribution function $P\left(t\right)$ for the length of ballistic flights (phase period $n=6$). Fitting a power law to the distribution yields an exponent of $\mu =1.6$.

Figure 12

Transport velocity $v_{\text{mean}}$, as a function of the phase shift $\alpha =-\frac{0.1\pi }{3}\dots 0$ of the “middle” barrier. With a star we have marked the setup, which we have chosen in Sec. 3 to produce the PSS shown in Fig. 8b.

Figure 13

(Color online) Cumulative distribution function $P\left(t\right)$ of the dwell time for the three different wells with fixed potential height $V_0=4$ for setup (c). In the inset $P\left(t\right)$ belonging to the second well for different values of $V_0=4\dots 8$ is shown.