Full Counting Statistics of Multiple Andreev Reflections

We derive the full distribution of transmitted particles through a superconducting point contact of arbitrary transparency under voltage bias. The charge transport is dominated by multiple Andreev reflections. The counting statistics is a multinomial distribution of processes, in which multiple charges $ne$ ($n=1,2,3,\dots$) are transferred through the contact. For zero temperature we obtain analytical expressions for the probabilities of the multiple Andreev reflections. The current, shot noise, and high current cumulants in a variety of situations can be obtained from our result.

Figure 1

Schematic representation of the MARs for BCS superconductors with gap $\Delta$. We have sketched the density of states of both electrodes. In the upper left panel we describe the process in which a single electron tunnels through the system overcoming the gap due to a voltage $eV\ge 2\Delta$. The other panels show MARs of order $n=2,3,4$. In these processes an incoming electron at energy $E$ undergoes at least $n-1$ Andreev reflections to finally reach an empty state at energy $E+\mathrm{n}\mathrm{e}\mathrm{V}$. In these MARs a charge $ne$ is transferred with a probability, which for low transparencies goes as $T^n$. At zero temperature they have a threshold voltage $eV=2\Delta /n$. The arrows pointing to the left in the energy trajectories indicate that a quasiparticle can be normal reflected. The lines at energies below $E$ and above $E+\mathrm{n}\mathrm{e}\mathrm{V}$ indicate that after a detour a quasiparticle can be backscattered to finally contribute to the MAR of order $n$.

Figure 2

Current contribution of processes $n=1,\dots ,10$, from right to left, as a function of voltage for BCS superconductors of gap $\Delta$. The different panels correspond to different transmissions. Notice the logarithmic scale in the upper panels.

Figure 3

(b),(c) Second and third cumulant at zero temperature for a contact between BCS superconductors. Both are normalized to the first cumulant (the average current) shown in panel (a). The transmissions are indicated in the plots.