Josephson response of a conventional and a noncentrosymmetric superconductor coupled via a double quantum dot

We consider transport through a Josephson junction consisting of a conventional $s$-wave superconductor coupled via a double quantum dot to a noncentrosymmetric superconductor with both singlet and triplet pairing. We calculate the Andreev bound state energies and the associated Josephson current. We demonstrate that the current-phase relation is a sensitive probe of the singlet-triplet ratio in the noncentrosymmetric superconductor. In particular, in the presence of an inhomogeneous magnetic field the system exhibits a $\phi$-junction behavior.

Figure 1

Schematic model of our setup. A double quantum dot (green) subject to an inhomogeneous magnetic field is tunnel coupled to a conventional $s$-wave superconductor (yellow) on the left as well as to a noncentrosymmetric superconductor (light blue) on the right.

Figure 2

Phase-dependent eigenenergies of the double quantum dot. Solid lines are the result of numerically diagonalizing Eq. (14); dotted lines correspond to the analytical approximation Eqs. (17). Parameters are $\delta =-0.2U$, ${\mathbf{B}}_1=-{\mathbf{B}}_2=0.25U{\mathbf{e}}_z$, ${\mathrm{\Gamma }}_{SL}=U$, $|{\mathrm{\Gamma }}_{T^0}|=0.5U$, and ${\mathrm{\Phi }}_{T^0}=\pi /2$.

Figure 3

Phase-dependent eigenenergies of the double quantum dot. Solid lines are the result of numerically diagonalizing Eq. (14); dotted lines correspond to the analytical approximation Eqs. (18). Parameters are $\delta =-0.2U$, ${\mathbf{B}}_1=0.25U{\mathbf{e}}_x{\mathbf{B}}_2=0.25U{\mathbf{e}}_y$, ${\mathrm{\Gamma }}_{SL}=U$, $|{\mathrm{\Gamma }}_{T^0}|=0.5U$, and ${\mathrm{\Phi }}_{T^0}=\pi /2$.

Figure 4

Phase-dependent eigenenergies of the double quantum dot. Solid lines are the result of numerically diagonalizing Eq. (14); dotted lines correspond to the analytical approximation Eqs. (17). Parameters are $\delta =-0.2U$, ${\mathbf{B}}_1={\mathbf{B}}_2=0$, ${\mathrm{\Gamma }}_{SL}=U$, ${\mathrm{\Gamma }}_{SR}=0.5U$, $|{\mathrm{\Gamma }}_{T^0}|=0.7U$, and ${\mathrm{\Phi }}_{T^0}=\pi /2$.

Figure 5

Phase-dependent eigenenergies of the double quantum dot. Solid lines are the result of numerically diagonalizing Eq. (14); dotted lines correspond to the analytical approximation Eqs. (17). Parameters are $\delta =-0.2U$, ${\mathbf{B}}_1=-{\mathbf{B}}_2=0.25U{\mathbf{e}}_z$, ${\mathrm{\Gamma }}_{SL}=U$, ${\mathrm{\Gamma }}_{SR}=0.5U$, $|{\mathrm{\Gamma }}_{T^0}|=0.7U$, and ${\mathrm{\Phi }}_{T^0}=\pi /2$.

Figure 6

Current-phase relation for different temperatures. Parameters are chosen as in Fig. 5.

Figure 7

Ratio between the maximal and minimal critical currents as a function of the triplet-singlet ratio ${\mathrm{\Gamma }}_{T^0}/{\mathrm{\Gamma }}_{SR}$ for different temperatures. Parameters are $\delta =-0.4U$, ${\mathbf{B}}_1=-{\mathbf{B}}_2=0.5U{\mathbf{e}}_z$, ${\mathrm{\Gamma }}_{SL}=U$, ${\mathrm{\Gamma }}_{SR}=0.4U$, and ${\mathrm{\Phi }}_{T^0}=\pi /2$.