Nonadiabatic transitions between adiabatic surfaces: Phase diffusion in superconducting atomic point contacts

Motivated by experiments with current-biased superconducting atomic point contacts the general problem of nonadiabatic transitions between adiabatic surfaces in presence of strong dissipation is studied. For a single-channel device the supercurrent is determined by the diffusive motion of the superconducting phase difference on two Andreev levels. These surfaces are uncoupled only in the adiabatic limit of low to moderate transmissions while for high transmissions curve crossings are important. Starting from a general master equation of the full density matrix an approximate time evolution equation for the populations on the adiabatic surfaces in the overdamped limit is derived from which the relevant observables can be obtained. Specific results for the case of atomic point contacts are in agreement with experimental observations that cannot be explained by conventional theory.

Figure 1

(Color online) Energy surfaces $E_{\pm }$ of the bound Andreev states for a single channel with transmission $\tau$ (solid) together with the diabatic surfaces $V_{1,2}$ (dotted). The minimal energy gap between the Andreev levels at the Landau-Zener point $\phi =\pi$ is indicated by the arrow. See text for details.

Figure 2

Equivalent circuit of the RCSJ model including a weak link.

Figure 3

(Color online) Transition rate according to Eq. (46) and scaled with the frequency ${\Omega }_0={\Delta }_0\left(\pi \right)/\hslash$ vs the transmission for various inverse temperatures ${\Delta }_S/k_{B}T=1$ (solid), 2 (dashed), and 5 (dotted). Other parameters are $E_C/{\Delta }_S=2$, $\gamma \hslash /{\Delta }_S=11$, and $I_{\text{dc}}{\phi }_0/{\Delta }_S=0.3$. See text for details.

Figure 4

(Color online) Steady-state populations for the $n_{-}$ [upper solid (blue) line] and the $n_{+}$ [lower solid (red) line] Andreev levels for a transmission $\tau =0.99$ according to Eq. (31). Also shown are the populations according to the dynamics on uncoupled surfaces (dashed lines) (Refs. 21, 22). Parameters are $k_{B}T/{\Delta }_S=0.8,{\phi }_0{I}_{\text{dc}}/{\Delta }_S=0.3$ and $\hslash \gamma /{\Delta }_S=11$.

Figure 5

(Color online) Average supercurrent vs average voltage in units of $R{I}_0$ with $I_0=\left(e{\Delta }_S/\hslash \right)\left(1-\sqrt{1-\tau }\right)$ being the critical current of the junction. Parameters are the same as in Fig. 4. Squares (red) denote the results for the coupled dynamics and circles (blue) for the standard dynamics on averaged surfaces (see text).

Figure 6

(Color online) Maximal supercurrent as a function of temperature. Parameters are chosen as in Fig. 4. Squares (red) depict the data for coupled dynamics and circles (blue) for the standard dynamics on averaged surfaces (see text).