Phase diffusion and fractional Shapiro steps in superconducting quantum point contacts

We study the influence of classical phase diffusion on the fractional Shapiro steps in resistively shunted superconducting quantum point contacts. The problem is mapped onto a Smoluchowski equation with a time dependent potential. A numerical solution for the probability density of the phase difference between the leads gives access to the mean current and the mean voltage across the contact. Analytical solutions are derived in some limiting cases. We find that the effect of temperature is stronger on fractional than on integer steps, in accordance with preliminary experimental findings. We further extend the analysis to a more general environment including two resistances and a finite capacitance.

Figure 1

Equivalent circuit of the resistively and capacitively shunted junction (RCSJ) model in the presence of a microwave field $v_{\mathit{ac}}\cos \omega t$.

Figure 2

Andreev levels as a function of the phase difference for $\tau =0.9$ and 1.

Figure 3

Evolution of the adiabatic current-phase relation (10) with the transmission $\tau$ at zero temperature.

Figure 4

Lattice representation of the Fourier-space Smoluchowski equation. The amplitude of the microwave $v_{\mathit{ac}}$ couples the $n$ chains, while the harmonics of the current-phase relation couple the $k$ chains.

Figure 5

Mean current as a function of the mean voltage across the contact, for three different values of the transmission and the following parameters (expressed in reduced units as indicated in the text): $R={10}^{-4}$, $T={10}^{-2}$, $v_{\mathit{ac}}=4.4\times {10}^{-3}$, $\omega ∕2\pi =12.5\times {10}^{-3}$.

Figure 6

Zoom of Fig. 5.

Figure 7

Evolution of step 1 with temperature.

Figure 8

Evolution of step $1∕2$ with temperature. The decrease of the amplitude is clearly much stronger than for step 1.

Figure 9

Scaling of the size of Shapiro steps of order $n∕k$ with temperature for a contact with $\tau =0.99$ at fixed $k\alpha =2.5$ and $R={10}^{-3}$. The full line corresponds to a fit of the form $1∕(1+a{k}^{2}T)$.

Figure 10

Generalization of the RCSJ model.

Figure 11

Supercurrent peak for a contact with $\tau =0.9$ in the circuit of Fig. 10 with $R=0.001$, $r=R∕10$, $T=0.02$, and $e^2∕C=0.0001$, 0.0002, 0.0004, 0.0008, and 0.0016 from bottom to top. As can be observed the results converge to an asymptotic curve corresponding to the RSJ model.