Figure 1

(a) Schematic of the four-terminal device studied in this work. A double layer is coupled to four superconducting electrodes. The system is phase biased and a Josephson current can flow without dissipation. The bilayer is in an exciton-condensate phase. (b) Assuming translational invariance along the contacts, the system can be considered as one dimensional. In a mean-field approximation, the superconducting and exciton ordered phases can be described through space-dependent order parameters $\Gamma \left(x\right)$ and $\Delta \left(x\right)$, respectively. The order parameter $\Gamma$ is different from zero only in a region of length $L$ sandwiched between the electrodes. To allow for a current-carrying state, the phase of the exciton condensate must be space dependent. The four superconducting electrodes have different phases ${\varphi }_i$. For simplicity, we assume that the amplitude of superconducting order parameter is equal to $\Delta$ in all electrodes.Reuse & Permissions

Figure 2

One-dimensional linearized bands for the two layers. For a Fermi-level location in the figure, top- (bottom-) layer states near the right Fermi point are right movers (left movers), whereas states near the the left Fermi point are left movers (right movers).Reuse & Permissions

Figure 3

Four-particle coherent Andreev reflection: this illustration is a schematic for the process responsible for the conversion of supercurrent from the two superconductors into an excitonic counterflow supercurrent. An artificial normal (N) region in which both superconducting and EC pairing potentials are zero has been inserted between the superconductors (S) and the exciton condensate (EC). At the superconductor normal interface, an electron impinging from the normal region is perfectly reflected as a hole (arrow C) or, vice versa, a hole is reflected as an electron (arrow A). Current is conserved by emission and absorption of a Cooper pair. This is the normal Andreev reflection process. At the right side, a similar but distinct process takes place. The exciton pairing potential has the effect of reflecting electrons and holes from one layer to the other (arrows B and D). Since real interlayer tunneling is absent, the interlayer reflection is always accompanied by the emission of an exciton (electron-hole pair). Two excitons are emitted, one for the process B and one for D. The wave functions for the complete path ABCD can be matched only at a precise energy, which depends on the various phases at play and is the energy of the excitonic Andreev bound state (EABS) mentioned in the text. The process illustrated converts Cooper pairs in the two superconductors into two excitons.Reuse & Permissions

Figure 4

In the main panel, we show the density of states of the system $\rho (\varepsilon ,\Delta ,|\Gamma \left|\right)$ (in units of $1/E_{\mathrm{T}}$) as a function of energy $\varepsilon$ (in units of $E_{\mathrm{T}}$) as computed numerically from Eq. (35) for a random choice of the phases ${\varphi }_i,\gamma$, and $qL$ and for $\Delta =5E_{\mathrm{T}}$ and $\left|\Gamma \right|=100E_{\mathrm{T}}$. Lower inset: the contribution due to the presence of the superconducting electrodes ${\rho }_{\text{BS}}\left(\varepsilon \right)$ as a function of $\varepsilon$. Upper inset: the contribution due to the EC ${\rho }_{\text{EC}}\left(\varepsilon \right)$ as a function of $\varepsilon$. These two contributions can be considered as independent in the strong-coupling limit $\left|\Gamma \right|\gg \Delta ,\hslash v_{\text{F}}/L$.Reuse & Permissions

Figure 5

The Josephson current as a function of ${\varphi }_T-{\varphi }_B$ normalized in two different ways. In the upper panel, the current is normalized with respect to the critical current, while in the lower panel, it is expressed in units of $e{v}_{\text{F}}/L$.Reuse & Permissions

Figure 6

A four-terminal device studied, which realizes a protected qubit. The connections between pairs of terminals $A,A^{\prime }$ and $B,B^{\prime }$ ensure that the phase differences ${\varphi }_T=\varphi \left(B^{\prime }\right)-\varphi \left(A\right)$ and ${\varphi }_B=\varphi \left(A^{\prime }\right)-\varphi \left(B\right)$ are opposite: ${\varphi }_B=-{\varphi }_T$. This results in a doubly periodic free-energy dependence on the phase difference $\phi =\varphi \left(B\right)-\varphi \left(A\right)$ and formation of two degenerate states that can be used for quantum computation.Reuse & Permissions