Quantum versus classical dynamics in a driven barrier: The role of kinematic effects

We study the dynamics of the classical and quantum mechanical scattering of a wave packet from an oscillating barrier. Our main focus is on the dependence of the transmission coefficient on the initial energy of the wave packet for a wide range of oscillation frequencies. The behavior of the quantum transmission coefficient is affected by tunneling phenomena, resonances, and kinematic effects emanating from the time dependence of the potential. We show that when kinematic effects dominate (mainly in intermediate frequencies), classical mechanics provides very good approximation of quantum results. In that frequency region, the classical and quantum transmission coefficients are in optimal agreement. Moreover, the transmission threshold (i.e., the energy above which the transmission coefficient becomes larger than a specific small threshold value) is found to exhibit a minimum. We also consider the form of the transmitted wave packet and we find that for low values of the frequency the incoming classical and quantum wave packet can be split into a train of well-separated coherent pulses, a phenomenon that admits purely classical kinematic interpretation.

Figure 1

The oscillating barrier at its equilibrium position (solid line) and its extremal positions (dashed line). The boundaries of the interaction region are shown with the dotted line.

Figure 2

Classical (open circles) and quantum (solid circles) transmission coefficient as a function of the energy of the incoming wave packet for three frequency values, namely (a) $\omega =3\times {10}^{-4}$ a.u., (b) $\omega =3\times {10}^{-3}$ a.u., and (c) $\omega =3\times {10}^{-2}$ a.u. The classical (dotted line) and quantum (dashed line) transmission coefficients for the static barrier are also shown.

Figure 3

Contour plots of the (a) classical and (b) quantum transmission coefficients as a function of the energy $E$ of the in-coming wave packet and of the frequency of the oscillation $\omega$. The height of the potential barrier is shown as a solid vertical line. The horizontal dashed lines correspond to frequencies ${\log }_{10}\omega =-2.5$ and ${\log }_{10}\omega =-3.5$ (see text).

Figure 4

Contour plot of the difference between the classical and the quantum transmission coefficient $\Delta T=T_C-T_Q$ as a function of the energy $E$ of the incoming wave packet and of the frequency of the oscillation $\omega$. The height of the potential barrier is shown as a solid vertical line. The horizontal dashed lines correspond to frequencies ${\log }_{10}\omega =-2.5$ and ${\log }_{10}\omega =-3.5$ (see text).

Figure 5

The difference between the classical and quantum transmission coefficients $\Delta T=T_C-T_Q$ as a function of the energy $E$ of the incoming wave packet for $\omega =3\times {10}^{-3}$ a.u.

Figure 6

Schematic representation on the $x-t$ plane of an orbit exhibiting five collisions with the oscillating barrier for $\omega =3\times {10}^{-3}$ a.u. and $E=0.02667$ a.u. The orbit is represented as joined linear segments and the boundaries of the barrier as sinusoidal curves. The points of impact are denoted with black dots.

Figure 7

$\Delta \rho ={\rho }_{>2,t}-{\rho }_{>2,r}$ (solid line) and $\Delta T=T_C-T_Q$ (dashed line) as a function of the energy of the incoming wave packet for $\omega =3\times {10}^{-3}$ a.u.

Figure 8

Classical (open circles) and quantum (solid circles) transmission thresholds $E_T$ as a function of the oscillation frequency.

Figure 9

Representation on the $x-t$ plane of a part of a wave packet (bundle of initially straight lines) with $E=2\times {10}^{-3}$ a.u. interacting with a laterally harmonically oscillating barrier (region between two sinusoidal curves) oscillating with frequency (a) $\omega =3\times {10}^{-4}$ a.u., (b) $\omega =3\times {10}^{-3}$ a.u., and (c) $\omega =3\times {10}^{-2}$ a.u. In (b) the phases of the barrier for which transmission occurs are indicated with thick solid lines.

Figure 10

The classical (solid line) and quantum (dotted line) time-dependent transmission coefficient for $\omega =6\times {10}^{-4}$ a.u. and (a) $E=0.03$ a.u. and (b) $E=0.013$ a.u.

Figure 11

Snapshots of the time evolution of the (a) classical and (b) quantum probability distribution of the transmitted wave packet for $\omega =6\times {10}^{-4}$ a.u., $E=0.013$ a.u., and four values of time, namely $t=53600$ a.u., $t=120300$ a.u., $190400$ a.u., and $t=257100$ a.u.

Figure 12

(a) Schematic representation on the $x-t$ plane of a part of a wave packet (bundle of initially straight lines) interacting with a laterally harmonically oscillating barrier (region between two sinusoidal curves). The right boundary of the interaction region is shown with a dashed line. (b) Schematic representation of the regions A, B, C: the orbits in region A are transmitted whereas the orbits in regions B and C are reflected (see text).