Boundary Potential Method for Describing Electron Teleportation in an Interferometer with a Topological Superconductor

Abstract

One-dimensional topological superconductors accommodate a pair of Majorana zero modes at their ends. In an interferometer containing such a topological superconductor, electron transport is significantly affected by the Majorana zero modes constituting a non-local state localized near both ends of the superconductor. When the number of electrons $\mathcal{N}$ in the superconductor is constrained by a charging effect, the resonant tunneling through the non-local state is expected to result in unusual transport properties. This resonant tunneling, called electron teleportation, is not easy to describe because there is no simple method to handle the constraint on $\mathcal{N}$. Here, we propose a boundary potential method based on scattering theory for calculating the conductance of the interferometer under a given constraint on $\mathcal{N}$. This method enables us to calculate the conductance taking account of relevant charging energy and details of the system.

Figures (4)

• Figure 1: Interferometer consists of a topological superconductor wire of $N$ sites, which is connected to ground by a capacitor, and an infinitely long normal metal lead. The left end (i.e., the 1st site) and the right end (i.e., the $N$th site) of the superconductor are connected to the 1st and $M$th sites of the normal metal lead, respectively, where $t_{1}$ ($t_{2}$) represents the transfer integral between the $1$st ($N$th) site in the superconductor and the $1$st ($M$th) site in the normal metal lead. A magnetic flux $\Phi$ pierces the loop formed by the superconductor and the normal metal lead.
• Figure 2: Schematics of scattering processes that transfer an electron at the $M$th site to an electron at the 1st site when the ground state is ${\rm G}_{2l}$. Only the processes involving the non-local state with $E_{1}$ are shown. The process (a) is included in $\bigl[V_{1 M}^{e\sigma, e\sigma'}(E)\bigr]_{+}^{\mathcal{E}}$ whereas the process (b) accompanying the splitting and recombination of a Cooper pair is included in $\bigl[V_{1 M}^{e\sigma, e\sigma'}(E)\bigr]_{-}^{\mathcal{E}}$.
• Figure 3: (Color online) Dimensionless non-local conductance $g(V)$ of the interferometer in the absence of the charging effect at $eV/E_{g} = 0$ and $0.01$.
• Figure 4: (Color online) Dimensionless non-local conductance $g(V)$ in the presence of the charging effect with $\delta U/E_{g} =$ (a) $0.0002$, (b) $0.002$, (c) $0.02$, and (d) $0.02$. In (a)--(c), solid lines (magenta) represent $g(0)$ in the even ground state ${\rm G}_{2l}$, and dashed lines (blue) represent $g(0)$ in the odd ground state ${\rm G}_{2l\pm 1}$. In (d), the solid line (magenta) and the dashed line (blue) represent $g(V)$ at $eV/E_{g} = 0.018$ under the constrains ${\rm G}_{2l} \leftrightarrow {\rm E}_{2l + 1}$ and ${\rm G}_{2l-1} \leftrightarrow {\rm E}_{2l}$, respectively.