Critical current for an insulating regime of an underdamped current-biased topological Josephson junction

We study analytically an underdamped current-biased topological Josephson junction. First, we consider a simplified model at zero temperature, where the parity of the nonlocal fermionic state formed by Majorana bound states (MBSs) localized on the junction is fixed, and show that a transition from insulating to conducting state in this case is governed by single-quasiparticle tunneling rather than by Cooper pair tunneling, in contrast to a nontopological Josephson junction. This results in a significantly lower critical current for the transition from insulating to conducting state. We propose that if the length of the system is finite, the transition from insulating to conducting state occurs at exponentially higher bias current due to hybridization of the states with different parities as a result of the overlap of MBSs localized on the junction and at the edges of the topological nanowire forming the junction. Finally, we discuss how the appearance of MBSs can be established experimentally by measuring the critical current for an insulating regime at different values of the applied magnetic field.

Figure 1

Schematic representation of the system: (a) cross section of a semiconducting nanowire (SE, green) with layers of superconductor (S, blue) on two facets and magnetic field $B$ applied along the nanowire, (b) a Josephson junction in the nontopological state ($B<B_c$, only Cooper pairs can tunnel), and (c) a Josephson junction in the topological regime ($B>B_c$, with competition between Cooper-pair and single-quasiparticle tunneling).

Figure 2

The effective potential energy $V\left(\varphi \right)$ as a function of the phase difference $\varphi$; see Eq. (3). We schematically indicate the $4\pi$ tunneling between minima of an effective potential in the two limits: (a) $E_M\gg E_J$ and (b) $E_M\ll E_J$. In the latter limit, the potential also exhibits a set of local minima, shifted from the absolute minima by $E_M$.

Figure 3

Schematic of the equivalent electric circuit for an underdamped topological Josephson junction. The applied current $I$ is divided between the shunting impedance $Z$ (current $I_b$) and the topological junction, effectively represented by the capacitance $C$, Cooper-pair tunneling element $E_J$, and single-quasiparticle tunneling element $E_M$.

Figure 4

Schematic representation of the overlap of MBSs ${\gamma }_{1\left(2\right)}$ on the junction and ${\gamma }_{0\left(3\right)}$ on the nanowire edges with associated splittings ${\delta }_L$ and ${\delta }_R$ for the left and right parts of the wire, respectively. The lengths of the corresponding parts are given by $L_L$ and $L_R$.

Figure 5

Two lowest energy levels in a fixed phase regime (in the limit $E_M\gg E_J$). The ground-state energy (blue curve) can be seen as an effective potential $V_{\mathrm{eff}}$ [see Eq. (31)] in the adiabatic limit such that one can neglect Landau-Zener transitions. As a result, $2\pi$ phase slips are restored.

Figure 6

Schematic illustration of the critical current $I_c$ as function of magnetic field $B$. At $B=B_i>B_c$, the overlap between MBSs goes to zero, $\delta =0$. As a result, the critical current $I_c$ drops exponentially. The schematic plateaus of $I_c$ correspond to $2\pi$ periodicity, while the dips correspond to (mostly) $4\pi$ periodicity of the Josephson junction.