ac Josephson transport through interacting quantum dots

We investigate the ac Josephson current through a quantum dot with strong Coulomb interaction attached to two superconducting and one normal lead. To this end, we perform a perturbation expansion in the tunneling couplings within a diagrammatic real-time technique. The ac Josephson current is connected to the reduced density matrix elements that describe superconducting correlations induced on the quantum dot via proximity effect. We analyze the dependence of the ac signal on the level position of the quantum dot, the charging energy, and the applied bias voltages.

Figure 1

A sketch of the device under investigation: A quantum dot with strong Coulomb interaction is weakly tunnel coupled to one normal and two superconducting leads.

Figure 2

Amplitude of the ac Josephson current in units of $2e{\Gamma }_S/\hslash$ as a function of dot level position $\varepsilon /U$. The temperature is $k_{B}T=U/10$, the bias voltage to the normal lead is ${\mu }_N=0$, and the bias voltages ${\mu }_{S1}$ and ${\mu }_{S2}$ applied to the superconductors are, respectively, given by (a) $-3U/2$ and $3U/2$ and (b) $-U/4$ and $U/4$. The solid curves are for ${\Gamma }_N={\Gamma }_S/10=U/100$ and the dashed lines for ${\Gamma }_N={\Gamma }_S=U/10$. The resonance conditions are indicated by vertical lines.

Figure 3

Amplitude of the ac Josephson current in units of $2e{\Gamma }_S/\hslash$ as a function of dot level position $\varepsilon /U$. The temperature is $k_{B}T=U/10$, the bias voltage to the normal lead is ${\mu }_N=0$, and the bias voltages ${\mu }_{S1}$ and ${\mu }_{S2}$ applied to the super-conductors are, respectively, given by (a) $-U$ and $3U/2$, (b) $-U/4$ and $3U/2$, and (c) $-U/4$ and $U/3$. The solid curves are for ${\Gamma }_N={\Gamma }_S/10=U/100$ and the dashed lines for ${\Gamma }_N={\Gamma }_S=U/10$. The resonance conditions are indicated by vertical lines.

Figure 4

Phase of the ac Josephson current in units of $\pi$ as a function of dot level position $\varepsilon /U$. The parameters are the same as in Figs. 2a and 2b.