Exponentially small quantum correction to conductance

When time-reversal symmetry is broken, the average conductance through a chaotic cavity, from an entrance lead with $N_1$ open channels to an exit lead with $N_2$ open channels, is given by $N_1N_2/M$, where $M=N_1+N_2$. We show that, when tunnel barriers of reflectivity $\gamma$ are placed on the leads, two correction terms appear in the average conductance, and that one of them is proportional to $\gamma^{M}$. Since $M\sim \hbar^{-1}$, this correction is exponentially small in the semiclassical limit. Surprisingly, we derive this term from a semiclassical approximation, generally expected to give only leading orders in powers of $\hbar$. Even though the theory is built perturbatively both in $\gamma$ and in $1/M$, the final result is exact.