Dynamical trapping and chaotic scattering of the harmonically driven barrier

A detailed analysis of the classical nonlinear dynamics of a single driven square potential barrier with harmonically oscillating position is performed. The system exhibits dynamical trapping which is associated with the existence of a stable island in phase space. Due to the unstable periodic orbits of the KAM structure, the driven barrier is a chaotic scatterer and shows stickiness of scattering trajectories in the vicinity of the stable island. The transmission function of a suitably prepared ensemble yields results which are very similar to tunneling resonances in the quantum mechanical regime. However, the origin of these resonances is different in the classical regime.

Figure 1

The ac-driven potential barrier.

Figure 2

(a) A typical Poincaré section showing trapped particles. (b) Enlargement of island 4.

Figure 3

Motion of trapped particles. The black line represents the particle, the two gray lines represent the edges of the barrier. (a) is the trajectory associated with the central periodic orbit of the stable island, (b) is a typical trajectory of a quasiperiodic orbit. The numbers in (a) correspond to the island numbers in Fig. 2a.

Figure 4

Maximum barrier potential allowing for periodic orbits as a function of the barrier’s thickness. Only pairs of parameters below this curve lead to stable orbits.

Figure 5

Overview of the parameter dependence of the KAM island. The parameters are $l=0.1$ and (a) $V_0=0.895$, (b) $V_0=0.867$, (c) $V_0=0.86$, (d) $V_0=0.76$, (e) $V_0=0.7$, (f) $V_0=0.553$, (g) $V_0=0.5$, (h) $V_0=0.05$.

Figure 6

Phase-space volume of the stable island as a function of $V_0$ for different $l$.

Figure 7

Plot of the Poincaré section and the periodic orbits of period one to 21. Stable orbits are shown as dots, unstable orbits as crosses.

Figure 8

Stable (a) and unstable (b) manifolds of the UPOs of the system.

Figure 9

Velocity change $\mid v_{\mathrm{in}}\mid -\mid v_{\mathrm{out}}\mid$ of the particle as a function of velocity $v_{\mathrm{in}}$ and phase of the first collision ${\phi }_1$ in gray scale.

Figure 10

Typical trajectories for acceleration (a) and deceleration (b) at ($v_{\mathrm{in}}=0.4$, ${\phi }_1=2.3$) and ($v_{\mathrm{in}}=1.5$, ${\phi }_1=4.5$).

Figure 11

Number of collisions of the particle as a function of velocity $v_{\mathrm{in}}$ and phase ${\phi }_1$ represented as gray scale. The collision number is also shown in the plot.

Figure 12

Enlargement of part of Fig. 9 by a factor of ${10}^{13}$.

Figure 13

Logarithm of the dwell time of the particle as a function of velocity $v_{\mathrm{in}}$ and phase ${\phi }_1$ represented as gray scale.

Figure 14

The distribution of the sticking times, using a logarithmic adjusted bin size. By fitting a power law we found an exponent of $\gamma =-2.5$.

Figure 15

Singularities of the scattering function not associated with chaotic scattering. (a) is the trajectory of a whispering gallery orbit, (b) is a low velocity peak.

Figure 16

Transmission function for different driving frequencies of the classical system. (a) $\omega =0$, (b) $\omega =0.0003$, (c) $\omega =0.03$.

Figure 17

Transmission function of a single particle as a function of the initial velocity and (a) the phase of the first collision ${\phi }_1$, (b) the initial phase ${\phi }_0$.