Characteristic times for resonant tunneling in one dimension

It is shown that the properties of the propagator for transmission through an arbitrary one-dimensional potential lead in a natural way to two characteristic times ${\tau }^0$ and ${\tau }^L$ for decay, respectively, through the end points of the system, in terms of which the lifetime $\tau$ may be written as $\frac{2{\tau }^0{\tau }^L}{({\tau }^0+{\tau }^L)}$. We obtain the traversal times $t_{0L}$ and $t_{L0\mathrm{}}$ for scattering, respectively, from the left and right ends of the system. In terms of these characteristic times, it is found that, in general, for asymmetric or random potentials, the rate is given by $\frac{t_{0L}}{t_{L0\mathrm{}}}=\frac{(1-R^{\frac{1}{2}})}{(1+R^{\frac{1}{2}})}$, where $R$ stands for the reflection coefficient evaluated at resonance energy. A comparison with experiment shows that $t_{0L}$ may be even an order of magnitude different from $t_{L0\mathrm{}}$.