Barrier Tunneling — Question 10

Question 10

Coherent multiple reflections can make two weakly transmitting barriers fully transparent

Two identical lossless barriers each have single-barrier transmission probability $\tau=.100$. In a coherent model their total transmitted amplitude is $t_{\mathrm{pair}}=\tau e^{i\chi}/[1-(1-\tau)e^{i\phi}]$, where $\phi$ is the round-trip phase including reflection phases and $\chi$ is an irrelevant overall phase. Exterior velocities are equal.

1. Derive the transmission probability and evaluate it at resonance $\phi=2\pi n$.

2. Find the antiresonant transmission at $\phi=(2n+1)\pi$ and compare with the incoherent multiple-reflection result.

Question 10 - Solution

Coherent multiple reflections can make two weakly transmitting barriers fully transparent

(a) Method and calculation

$$\boxed{T(\phi)=\frac{\tau^2}{1+(1-\tau)^2-2(1-\tau)\cos\phi}}.$$$$\phi=2\pi n:\quad\boxed{T_{\mathrm{res}}=\tau^2/[1-(1-\tau)]^2=1}.$$

(b) Interpretation and further calculation

$$\phi=(2n+1)\pi:\quad\boxed{T_{\mathrm{anti}}=\frac{\tau^2}{(2-\tau)^2}=.00277008}.$$$$T_{\mathrm{inc}}=\tau^2\sum_{j=0}^{\infty}(1-\tau)^{2j}=\boxed{\frac{\tau}{2-\tau}=.0526316}.$$

Physical check

The incoherent calculation adds path probabilities, while the coherent model adds amplitudes. Unity transmission requires the appropriate phase and symmetric lossless barriers; it is not predicted by multiplying two isolated transmission probabilities.