Full Counting Statistics of Multiple Andreev Reflection

We derive the full counting statistics of charge transfer through a voltage biased superconducting junction. We find that, for measurement times much longer than the inverse Josephson frequency, the counting statistics describes a correlated transfer of quanta of multiple electron charges, each quantum associated with the transfer of a single quasiparticle. An expression for the counting statistics in terms of the quasiparticle scattering amplitudes is derived.

Figure 1

Left: A schematic picture of the junction. Right: Multiple Andreev reflection processes in the voltage biased superconducting junction. An electron-like quasiparticle injected at an energy $E_0$ can undergo multiple Andreev reflections before being emitted as an $\alpha$-type (e or h) quasiparticle at an energy $E_n=E_0+\mathrm{n}\mathrm{e}\mathrm{V}$. These scattering processes have the amplitudes $s_{n0}^{\alpha e}$, shown in the figure. Note that all quasiparticles injected from the left at energies $E_0+2\mathrm{n}\mathrm{e}\mathrm{V}$ and from the right at $E_0+(2n+1)eV$ scatter on the same “ladder” in energy space.

Figure 2

The probability $P_n=(1/2eV){\int }_{-\Delta -2eV}^{-\Delta }{P}_n(E_0)d{E}_0$ of $n$-electron scattering events, as a function of voltage for $n\in \{1,2,3,4\}$ in a fully transparent junction (left) and a tunnel junction (right).

Figure 3

The zero-frequency current noise (upper panel) and the third cumulant (lower panel) as a function of voltage for transparencies $D=0.1$, 0.5, 0.9, 0.99, and 1. All curves are for a single mode junction at temperatures $kT\ll \Delta$.

Figure 4

Upper panel: the Fano factor $S/I$ (solid line) and $C_3/I$ (dash-dotted line) in the tunnel limit $D=0.01$. Lower panel: the current (solid line), noise (dash-dotted line), and third cumulant (dashed line) for a short, diffusive superconducting junction.