Chaos regularization of quantum tunneling rates

Quantum tunneling rates through a barrier separating two-dimensional, symmetric, double-well potentials are shown to depend on the classical dynamics of the billiard trajectories in each well and, hence, on the shape of the wells. For shapes that lead to regular (integrable) classical dynamics the tunneling rates fluctuate greatly with eigenenergies of the states sometimes by over two orders of magnitude. Contrarily, shapes that lead to completely chaotic trajectories lead to tunneling rates whose fluctuations are greatly reduced, a phenomenon we call regularization of tunneling rates. We show that a random-plane-wave theory of tunneling accounts for the mean tunneling rates and the small fluctuation variances for the chaotic systems.

Figure 1

Tunneling rates for integrable and chaotic wells and their sliding averages. Rates for (a) a two-dimensional integrable well, (b) a two-dimensional chaotic well, and (c) their sliding averages. Insets show the double-well and barrier configuration (not to scale) and some typical eigenstates. For the rectangular well (a) low tunneling and high tunneling rate states are shown. The sliding average window width in $E$ used for (c) is 4k. Parameters [see Fig. 2a for definitions] used for both (a) and (b) are Δ$=$0.1, $L=$ 2.4, $V_b=$ 1000, and $A=$ 4.8.

Figure 2

Well coordinates and tunneling rates. The coordinates for the theory are shown in (a). The sliding averages of the splittings are shown in (b) for the theory (red) and for the data from calculations of the concave wells (black) and the rectangular wells (blue). The standard deviations of the tunneling rates are shown in (c) for the theory (red), bowtie wells (black), and rectangular wells (blue).

Figure 3

Tunneling rates for various two-dimensional wells. Tunneling rates for (a) circular, (b) stadium, and (c) Sinai-shaped systems. Insets show the double-well and barrier configuration (not to scale) and some typical eigenstates. Other chaotic well shapes (butterfly and various concave-sided wells) also show the same regularization of tunneling rates. Roughly speaking, as the systems become more chaotic, the tunneling rate variation falls drastically.