Adiabatic quantum pumping of chiral Majorana fermions

We investigate adiabatic quantum pumping of chiral Majorana states in a system composed of two Mach-Zehnder type interferometers coupled via a quantum point contact. The pumped current is generated by periodic modulation of the phases accumulated by traveling around each interferometer. Using scattering matrix formalism we show that the pumped current reveals a definite signature of the chiral nature of the Majorana states involved in transport in this geometry. Furthermore, by tuning the coupling between the two interferometers the pump can operate in a regime where finite pumped current and zero conductance are expected.

Figure 1

Schematic of the Mach-Zehnder interferometer studied in this paper. Two superconducting islands, SC${}_1$ and SC${}_2$, are connected via a point contact (t). In the left (in) and right (out) lead, two chiral Dirac fermion modes, ${\varphi }^e$ and ${\varphi }^h$, propagate. The entire setup is placed on top of a 3D topological insulator. Majorana fermions ${\gamma }_l$ ($l=$ 1–4) are the mediating states in the central interferometer region. See the text for more details.

Figure 2

Contour plot of the pumped current $I_{p,R}/I_0$ [Eq. (8)] as a function of the phases ${\theta }_1$ and ${\theta }_2$ for $t=0.8$. (a) ${\theta }_2={\theta }_1$ line, (b) ${\theta }_2={\theta }_1-\pi$ line, (c) ${\theta }_1=0$ line, (d) ${\theta }_2=0$ line, and (e) ${\theta }_1=-\pi /2$ and ${\theta }_2=0$ point. The maximum (minimum) values $I_{p,R}/I_0=\pm 1.12$.

Figure 3

Pumped current [Eq. (8)] as a function of ${\theta }_1$ for different values of $t$ and for three values of $\delta {\theta }_{-}\equiv {\theta }_1-{\theta }_2$. The legend shows the ($t$,$\delta {\theta }_{-}$) values of each curve in each panel. (a) The top panel shows six curves for $(0.99,0)$, $(0.7,0)$, $(0.3,0)$, $(0.99,\pi )$, $(0.7,\pi )$, and $(0.3,\pi )$. (b) The bottom panel shows three curves for $(0.99,\pi /2)$, $(0.7,\pi /2)$, and $(0.3,\pi /2)$.

Figure 4

Contour plot of the conductance $G$ [Eq. (10)] in units of $e^2/h$ as a function of the phases ${\theta }_1$ and ${\theta }_2$ for $t=0.8$. (a) ${\theta }_2={\theta }_1$ line, (b) ${\theta }_2={\theta }_1-\pi$ line, (c) ${\theta }_1=0$ line, (d) ${\theta }_2=0$ line, and (e) ${\theta }_1=-\pi /2$ and ${\theta }_2=0$ point.

Figure 5

Conductance [Eq. (10)] as a function of ${\theta }_1$ for different values of $t$ and for three values of $\delta {\theta }_{-}={\theta }_1-{\theta }_2$. The legend shows the $(t,\delta {\theta }_{-})$ values of each curve in each panel. (a) The top panel shows six curves for $(0.99,0)$, $(0.7,0)$, $(0.3,0)$, $(0.99,\pi )$, $(0.7,\pi )$, and $(0.3,\pi )$. (b) The bottom panel shows three curves for $(0.99,\pi /2)$, $(0.7,\pi /2)$, and $(0.3,\pi /2)$.