Single fermion manipulation via superconducting phase differences in multiterminal Josephson junctions

We show how the superconducting phase difference in a Josephson junction may be used to split the Kramers degeneracy of its energy levels and to remove all the properties associated with time-reversal symmetry. The superconducting phase difference is known to be ineffective in two-terminal short Josephson junctions, where irrespective of the junction structure the induced Kramers degeneracy splitting is suppressed and the ground state fermion parity must stay even, so that a protected zero-energy Andreev level crossing may never appear. Our main result is that these limitations can be completely avoided by using multiterminal Josephson junctions. There the Kramers degeneracy breaking becomes comparable to the superconducting gap, and applying phase differences may cause the change of the ground state fermion parity from even to odd. We prove that the necessary condition for the appearance of a fermion parity switch is the presence of a “discrete vortex” in the junction: the situation when the phases of the superconducting leads wind by $2\pi$. Our approach offers strategies for creation of Majorana bound states as well as spin manipulation. Our proposal can be implemented using any low density, high spin-orbit material such as InAs quantum wells, and can be detected using standard tools.

Figure 1

(Top left) A superconducting ring (gray) allows one to concentrate the effect of a magnetic flux $\Phi$ on the small area of a Josephson junction (red). (Bottom left) The junction has subgap Andreev levels whose energy $\epsilon$ depends on the phase difference $2e\Phi /\hslash =\varphi$. Each level is doubly degenerate since in a short junction a finite phase difference does not induce a splitting of the Kramers degeneracy. (Top right) As explained in this work, Kramers degeneracy can be efficiently removed in a three-terminal junction, even in the absence of an external magnetic field. (Bottom right) Andreev spectrum for $2e{\Phi }_1/\hslash =-2e{\Phi }_2/\hslash =\varphi$. Both the splitting of Kramers degeneracy and Andreev level crossings at zero energy (marked by red circles) appear in the spectrum.

Figure 2

Three-terminal Josephson junction geometry. The scattering region (red) is a piece of a disordered two-dimensional material with spin-orbit coupling. It is connected to three superconducting leads (gray). In the normal state, the scattering region has a scattering matrix $s$. At energies smaller than the superconducting gap $\Delta$, modes leaving the scattering region are reflected at the interface with the superconductor by Andreev reflection processes (black arrows), described by a scattering matrix $r_A$.

Figure 3

Geometrical illustration of Eq. (19) in the case of three leads. The sum of the scalar products $\frac{1}{2}\langle \Psi |r_A|\Psi \rangle$ and $\frac{1}{2}\langle {\Psi }^{\prime }|r_A|{\Psi }^{\prime }\rangle$ must lie within the triangle on the complex plane whose vertices are the eigenvalues $i{e}^{i{\varphi }_1},i{e}^{i{\varphi }_2},i{e}^{i{\varphi }_3}$ of $r_A$. In the left panel, these phases do not surround the origin and the lowest allowed energy (in units of $\Delta$) is the minimum distance between the polygon and the origin [Eq. (20)]. In the right panel, the phases surround the origin, a discrete vortex is present in the junction and zero-energy solutions are allowed.

Figure 4

(Left) Phase dependence of the Andreev levels of a Rashba dot with $\mu =1/4ma$ for ${\varphi }_2={\varphi }_1$ (top) and ${\varphi }_2=2\pi -{\varphi }_1$ (bottom). Kramers degeneracy is present in the top panel (since one of the phase differences is zero), but not in the bottom panel. (Right) Energy difference $\delta \epsilon$ between the two lowest Andreev levels in a Rashba dot averaged over ${10}^2$ values of $\mu \in [0,1/2ma]$ for a fixed disorder configuration.

Figure 5

(Left) Examples of the minimum energy $\epsilon$ of an Andreev bound state as a function of $({\varphi }_1,{\varphi }_2)$. The first two examples are calculated using random scattering matrices, with and without with zero-energy crossings. The positions of the crossings are found numerically using the method of Appendix pp1, and are marked in blue. They form closed curves encircling domains of odd ground state fermion parity. The third example is for a QSH dot in the nontrivial phase, so that the fermion parity switch appears almost exactly at the boundary of the allowed zone. (Right) Ground state fermion parity $\langle P\rangle$ averaged over ${10}^4$ random matrices of size $n=6$, showing that fermion parity may only be odd only if the discrete vortex condition (21) is fulfilled.

Figure 6

(Top) Density of states $\rho \left(\epsilon \right)$ of a Rashba dot, computed along the diagonal ${\varphi }_2=2\pi -{\varphi }_1$ and averaged over $10{}^3$ values of $\mu \in [0,1/2ma]$ for a single disorder realization. Spin-orbit coupling $\alpha$ and disorder strength $u$ are the same as in Fig. 4. The dotted line shows the lower bound on the Andreev state energy (20). (Bottom) Density of states obtained from ${10}^6$ random scattering matrices with 10 modes per lead, computed for the three different values of $({\varphi }_1,{\varphi }_2)$ shown in the inset: in the gapped region [red, $(3\pi /4,\pi /4)$], in the presence of a discrete vortex [blue, $(4\pi /3,2\pi /3)$], and at the time-reversal invariant point [green, $(\pi ,\pi )$]. The black dashed lines are fits of Eqs. (24) and (25), with a single free parameter $\delta$.

Figure 7

Spectral properties of a three-terminal junction made in a Rashba dot and with finite $\Delta =0.01/2ma$, showing the effect of an increased size of the junction. All other parameters are as in Fig. 4. (Top) Energy difference $\delta \epsilon$ between the two lowest Andreev levels, averaged over 10 values of $\mu \in [0,1/2ma]$. (Bottom) Density of states of the junction, obtained by averaging over 200 values of $\mu \in [0,1/2ma]$, for a single disorder configuration and a fixed value of $\mu$ in the three arms of the junction. Black dots are the lower bound (20), which is valid in the limit $\Delta /E_T\to 0$.