WKB approximation for multichannel barrier penetrability

Using the method of the local transmission matrix, we generalize the well-known WKB formula for barrier penetrability to multichannel systems. We compare the WKB penetrability with a solution of the coupled-channels equations, and show that the WKB formula works well at energies well below the lowest adiabatic barrier. We also discuss the eigenchannel approach to a multichannel tunneling, which may improve the performance of the WKB formula near and above the barrier.

Figure 1

The eigenpotentials for the three-channel problem obtained by diagonalizing the coupling matrix $\mathbf{W}\left(x\right)$ at each $x$ (the solid lines). The uncoupled barrier $V\left(x\right)$ is shown by the dotted line for a comparison.

Figure 2

The barrier penetrability for the uncoupled barrier $V\left(x\right)$ as a function of energy $E$ in linear (the upper panel) and the logarithmic (the lower panel) scales. The solid and the dashed lines are the exact solution and the WKB approximation, respectively.

Figure 3

Same as Fig. 2, but for the three-channel problem. The dotted line shows the penetrability in the no-coupling limit.

Figure 4

Comparison of the barrier penetrability calculated by several methods. The solid line shows the exact solution of the coupled-channel equations. The dashed line is obtained with the modified adiabatic method which takes the nonadiabatic correction into account, while the filled circles are results of the eigenchannel approximation. The upper and the lower panels show the penetrability in linear and logarithmic scales, respectively.