Long-distance coherence of Majorana wires

Theoretically, a pair of Majorana bound states in a topological superconductor forms a single fermionic level even at large separations, implying that the parity information is stored nonlocally. The nonlocality leads to a long-distance coherence for electrons tunneling through a Coulomb-blockaded Majorana wire [L. Fu, Phys. Rev. Lett. 104, 056402 (2010)], an effect that can be observed, e.g., in an interferometer. Here, we examine theoretically the coherent electron transfer, taking into account that tunneling implies the long-distance transfer of charge, which is carried by one-dimensional plasmons. We show that the charge dynamics does not affect the coherence of the electron tunneling process in a topological superconductor consisting of a semiconductor wire proximitized by a single bulk superconductor. The coherence may be strongly suppressed, however, if the topological superconductivity derives from a semiconductor wire proximitized by a granular superconductor.

Figure 1

Interferometer setup used to measure the coherent tunneling through a Coulomb-blockaded topological superconductor. Each interferometer arm $j=1,2$ is modeled as an array of topological superconductor islands, connected via Josephson junctions. Majorana bound states exist at the ends of every island, but only the Majorana bound states ${\gamma }_{j,1}$ and ${\gamma }_{j,2N}$ at the far left and right of the array enter into the low-energy theory. The figure also schematically shows the capacitive coupling between islands and between each island and ground, as well as the flux $\mathrm{\Phi }$ through the interference loop.

Figure 2

Plasmon spectrum of a long ($N={10}^4$) Josephson junction array in the harmonic approximation, overlaid with the linear (acoustic) approximation at small momenta (solid blue line) and the plasmon energy at $k=N$ (dashed blue line). The parameters chosen are $E_{\mathrm{J}}C/e^2=10$ and $C_{\mathrm{g}}/C={10}^{-4}$. Inset: Plasmon spectrum of a much shorter ($N=100$) Josephson junction array for the same values of $E_{\mathrm{J}}$, $C$, and $C_{\mathrm{g}}$. The acoustic plasmon branch at small $k$ has almost disappeared.

Figure 3

Normalized coherent amplitude ${\mathcal{G}}_j^{\left(N\right)}/{\mathcal{G}}_j^{(1,\mathrm{eff})}$ as a function of the number of superconducting islands $N$. The insets focus on the exponential $N$ dependence in short systems. The data points show the amplitude ratio ${\mathcal{G}}_j^{\left(N\right)}/{\mathcal{G}}_j^{(1,\mathrm{eff})}$ for $E_{\mathrm{J}}C/e^2=5$, 10, and 15 and $C_{\mathrm{g}}/C={10}^{-4}$ (bottom to top data sets, left), for $E_{\mathrm{J}}C/e^2=5$, 10, and 15 and $E_{\mathrm{J}}{C}_{\mathrm{g}}/e^2=1\times {10}^{-3}$ (bottom to top, center), and $C/C_{\mathrm{g}}=5\times {10}^3$, ${10}^4$, $1.5\times {10}^4$ and $E_{\mathrm{J}}C/e^2=10$ (top to bottom, right). The dotted curves follow from the asymptotic expressions (15) and (17).