Detecting topological superconductivity with ${\phi }_0$ Josephson junctions

The recent experimental discovery of ${\phi }_0$ Josephson junctions by Szombati et al. [Nat. Phys. 12, 568 (2016)], characterized by a finite phase offset in the supercurrent, requires the same ingredients as topological superconductors, which suggests a profound connection between these two distinct phenomena. Here, we show that a quantum dot ${\phi }_0$ Josephson junction can serve as a qualitative indicator for topological superconductivity: microscopically, we find that the phase shift in a junction of $s$-wave superconductors is due to the spin-orbit induced mixing of singly occupied states on the quantum dot, while for a topological superconductor junction it is due to singlet-triplet mixing. Because of this important difference, when the spin-orbit vector of the quantum dot and the external Zeeman field are orthogonal, the $s$-wave superconductors form a $\pi$ Josephson junction, while the topological superconductors have a finite offset ${\phi }_0$ by which topological superconductivity can be distinguished from conventional superconductivity. Our prediction can be immediately tested in nanowire systems currently used for Majorana fermion experiments and thus offers a realistic approach for detecting topological bound states.

Figure 1

Setups for ${\phi }_0\mathrm{JJs}$. (a) Two $s$-wave SCs (red) are tunnel coupled via a QD (yellow) with two orbitals $a$ and $b$. The QD is subject to an external magnetic field $\mathit{B}$ at some relative angle $\theta$ to its SOI ${\mathbf{\Omega }}_{\mathit{\text{D}}}$. (b) Same visual encodings. The SCs are replaced by two TSs (blue). The QD now couples to the two inner MBS (crosses) ${\mathrm{\Gamma }}_{1,2}$ of the TSs. (c) Spectrum of the bare QD as a function of $B$ for the double occupancy sector. Red bands contribute to our effective description; green bands do not. We have chosen $\delta =1$ meV, $g=40, U=0.9$ meV, and $U_{ab}=0.6$ meV, ${\mathrm{\Omega }}_{\text{D}}=0.1$ meV, so that $B^{\left(2\right)}=302$ mT. (d) Same as (c) but for the single occupancy sector with $B^{\left(1\right)}=432$ mT.

Figure 2

(a) Phase shift ${\phi }_{\nu }^0\left(\theta \right)$ (left panel) and Josephson current $I_{\nu }\left(\theta \right)$ at ${\phi }_{\nu }=0$ (right panel) for $\lambda =4$ and $\theta \in [{\theta }_c,\pi -{\theta }_c]$ with ${\theta }_c=0.3$. System parameters are chosen as in Fig. 1 with $B=B^{\left(2\right)}, V_g=-0.80$ meV, $t_{1a}=t_{2b}=0.01$ meV, $t_{1b}=0.05$ meV, and $t_{2a}=0.04$ meV. Compared to the SC JJ the phase shift (Josephson current at ${\phi }_{\text{S}}=0$) is nonzero for the TS JJ. (b) Experimental proposal. Left panel: SQUID geometry of a nanowire QD JJ and a reference junction without a QD connected in parallel. The current-phase relation of the Josephson current through the QD JJ is measured with respect to the reference junction by tuning the flux $f$ through the SQUID. Right panel: top view (upper panel) and side view (lower panel) of the QD JJ. Local gates are applied to define the tunnel coupled QD (yellow) as a short segment in a nanowire (gray) which is proximity-coupled to an $s$-wave SC (red). Local gates are also used to orient the dot SOI ${\mathbf{\Omega }}_{\text{D}}$ (green) and wire SOI ${\mathbf{\Omega }}_{\text{W}}$ (orange), respectively. To measure ${\phi }_{\nu }^0\left(\theta \right)$ and $I_{\nu }\left(\theta \right)$, a magnetic field $\mathbf{B}$ is rotated in the plane parallel to the SC film.

Figure 3

Tunneling sequences (up to Hermitian conjugation) of the SC JJ for contributions $\propto cos{\phi }_{\text{S}}$. We use the basis $|n_{1\mathbf{k}\Uparrow },n_{1\mathbf{k}\Downarrow },n_{a\uparrow },n_{a\downarrow },n_{b\uparrow },n_{b\downarrow },n_{2\mathbf{q}\Uparrow },n_{2\mathbf{q}\Downarrow }\rangle ={\left({\gamma }_{1\mathbf{k}\Uparrow }^{†}\right)}^{n_{1\mathbf{k}\Uparrow }}{\left({\gamma }_{1\mathbf{k}\Downarrow }^{†}\right)}^{n_{1\mathbf{k}\Downarrow }}{\left(d_{a\uparrow }^{†}\right)}^{n_{a\uparrow }}{\left(d_{a\downarrow }^{†}\right)}^{n_{a\downarrow }}{\left(d_{b\uparrow }^{†}\right)}^{n_{b\uparrow }}{\left(d_{b\downarrow }^{†}\right)}^{n_{b\downarrow }}{\left({\gamma }_{2\mathbf{q}\Uparrow }^{†}\right)}^{n_{2\mathbf{q}\Uparrow }}{\left({\gamma }_{2\mathbf{q}\Downarrow }^{†}\right)}^{n_{2\mathbf{q}\Downarrow }}|0_1,0_{\text{D}},0_2\rangle$. Filled (empty) dots are used to visually represent a filled (empty) level.

Figure 4

Same as Fig. 3 but for contributions $\propto sin{\phi }_{\text{S}}$ to the effective Hamiltonian of the SC JJ.

Figure 5

Tunneling sequences of the TS JJ for $\theta =\pi /2$. We use the basis $|n_1,n_{a\uparrow },n_{a\downarrow },n_{b\uparrow },n_{b\downarrow },n_2\rangle ={\left(C_1^{†}\right)}^{n_1}{\left(d_{a\uparrow }^{†}\right)}^{n_{a\uparrow }}{\left(d_{a\downarrow }^{†}\right)}^{n_{a\downarrow }}{\left(d_{b\uparrow }^{†}\right)}^{n_{b\uparrow }}{\left(d_{b\downarrow }^{†}\right)}^{n_{b\downarrow }}{\left(C_2^{†}\right)}^{n_2}|0_1,0_{\text{D}},0_2\rangle$. Filled (empty) dots are used to visually represent a filled (an empty) level. (a) Tunneling sequences that give contributions $\propto cos\left({\phi }_{\text{TS}}\right)$. (b) Tunneling sequences that give contributions $\propto sin\left({\phi }_{\text{TS}}\right)$.

Figure 6

Phase shift ${\phi }_{\text{TS}}^0$ as a function of the magnitude of the external magnetic field $B$ at $\theta =\pi /2$ for $\lambda =1,4$. For $\lambda =2,3$ the phase shift is independent of $B$ and given by ${\phi }_{\text{TS}}^0=\pi /2$. For the SC JJ we do not observe a phase shift when $\theta =\pi /2, {\phi }_{\text{S}}^0=0$. We see that the phase shift is peaked at $B=B^{\left(2\right)}$ when the singlet triplet mixing is maximal and it saturates at $\pi /2$ when $B\gg B^{\left(2\right)}$. Note however that our perturbative approach is not valid when $B\ll B^{\left(2\right)}$, because additional energy levels would have to be taken into account.

Figure 7

Estimate of the critical angle ${\theta }_c$ when $\lambda =4$ by analyzing the conditions for the weak coupling limit as a function of $\theta$. The system parameters are chosen as in the main text and the Appendixes. In the left panel we plot $\pi {\nu }_F{t}^2/(\mathrm{\Omega }sin\theta )$ (red dashed) and $\pi {\nu }_F{t}^2/|E_4^{\left(1\right)}-E_{-}^{\left(2\right)}|$ (red solid). In the right panel we plot $t/(\mathrm{\Omega }sin\theta )$ (blue dashed) and $t/|E_4^{\left(1\right)}-E_{-}^{\left(2\right)}|$ (blue solid). We find that ${\theta }_c=0.3$. This choice of critical angle also works for $\lambda =1,2,3$.

Figure 8

Magnitude of the critical current $|I_{\nu }^c\left(\theta \right)|$ for different choices of $\lambda$. The system parameters are chosen as in the main text and the Appendixes.

Figure 9

Phase shift ${\phi }_{\nu }^0\left(\theta \right)$ (top row) and Josephson current $I_{\nu }\left(\theta \right)$ at ${\phi }_{\text{S}}=0$ (bottom row) for $\lambda =1,2,3$. The system parameters are chosen as in the main text. The jumps in the Josephson current $I_{\text{TS}}\left(\theta \right)$ correspond to a change of the ground state of the junction.