Josephson-Majorana cycle in topological single-electron hybrid transistors

Charge transport through a small topological superconducting island in contact with a normal and a superconducting electrode occurs through a cycle that involves coherent oscillations of Cooper pairs and tunneling in/out the normal electrode through a Majorana bound state, the Josephson-Majorana cycle. We illustrate this mechanism by studying the current-voltage characteristics of a superconductor-topological superconductor-normal metal single-electron transistor. At low bias and temperature the Josephson-Majorana cycle is the dominant mechanism for transport. We discuss a three-terminal configuration where the nonlocal character of the Majorana bound states is emergent.

Figure 1

The system consists of a superconducting island connected to two normal metals and to a superconductor. A quantum wire with strong spin-orbit interaction is deposited on top of the superconducting island. The whole system is immersed in an external magnetic field. By choosing properly the parameters the proximized wire enters a topological phase with two Majorana bound states localized at the two ends of the wire. Tunneling into the normal leads happens through the Majorana states while the coupling of the central island to the superconducting lead is due to the Josephson effect.

Figure 2

Josephson-Majorana cycle for $n_g\sim 1$. The states differing by one Cooper pair, $|0,0\rangle$ and $|1,0\rangle$, are coupled via the Josephson term. The coupling between $|0,1\rangle$ and $|0,0\rangle$ is given by the regular term in the tunneling Hamiltonian, the connection between $|0,1\rangle$ and $|1,0\rangle$ by the anomalous one.

Figure 3

Current-voltage characteristics for $k_{B}T=E_C/100$, $\Gamma =k_{B}T/10$, $E_J=E_C/5$. In the lowest order in the transmission the current is different from zero only in the region $n_g\sim 1$ where two states coupled by the Josephson matrix element are nearly degenerate. Outside the region of maximum current the threshold voltage $V^{-}$ is visible as a tiny line in the Coulomb blockade region.

Figure 4

Current-voltage characteristics and its dependence on the bias charge $n_V=CV/e$ and the gate charge $n_g=C{V}_g/e$. Parameters are: $k_{B}T=E_C/100$, $\Gamma =k_{B}T/10$, $E_J=\Gamma /\sqrt{3}$. The threshold voltage to observe a current is $n_V\simeq \pm 0.5$. The slight displacement of the resonance condition from $n_g=1$ is due to the Lamb shift from the normal leads.