ac Josephson Effect in Finite-Length Nanowire Junctions with Majorana Modes

It has been predicted that superconducting junctions made with topological nanowires hosting Majorana bound states (MBS) exhibit an anomalous $4\pi$-periodic Josephson effect. Finding an experimental setup with these unconventional properties poses, however, a serious challenge: for finite-length wires, the equilibrium supercurrents are always $2\pi$ periodic as anticrossings of states with the same fermionic parity are possible. We show, however, that the anomaly survives in the transient regime of the ac Josephson effect. Transients are, moreover, protected against decay by quasiparticle poisoning as a consequence of the quantum Zeno effect, which fixes the parity of Majorana qubits. The resulting long-lived ac Josephson transients may be effectively used to detect MBS.

Figure 1

(a) Schematics: A nanowire of length $L=L_{S^{\prime }}+L_N+L_{S^{\prime }}$ in contact with two $s$-wave superconductors (with gap ${\Delta }_S$), develops a proximity-induced SC gap ${\Delta }_{S^{\prime }}<{\Delta }_S$ and four Majorana modes ${\gamma }_{1,2,3,4}$. The transparency of the junction can be controlled by a gate voltage $V_g$. (b) Energy of the four lowest many-body states in the TS phase, as a function of SC phase difference $\varphi$ across $N$. They correspond to even (solid) and odd (dashed) fillings of the two lowest ABS, see Fig. 2b. Note the avoided crossing of size ${\delta }_{\pi }$ at $\varphi =\pi$ and the detachment gap ${\delta }_0$ at $\varphi =0$.

Figure 2

Andreev bound states (ABS) for a SS’NS’S junction with normal conductance $G=0.72G_0$ (with $G_0=2e^2/h$), an induced gap ${\Delta }_{S^{\prime }}=218\mu \mathrm{eV}$, and wire’s length of $4\mu \mathrm{m}$ (where $l_{so}=216\mathrm{nm}$ for a InSb wire). In the topologically trivial phase (a), $\mathcal{B}=0.36\mathrm{meV}=0.5{\mathcal{B}}_c$, whereas in the nontrivial phase (b), $\mathcal{B}=1.1\mathrm{meV}=1.5{\mathcal{B}}_c$. The dashed line denotes the wire effective gap ${\Delta }_{\mathrm{eff}}=130\mu \mathrm{eV}$ that separates localized ABS from the quasicontinuum.

Figure 3

Top: diagrams of the different dynamical regimes of the ac Josephson current as a function of junction bias $V$ and dimensionless conductance $G/G_0$, for (a) the topologically trivial and (b) nontrivial phases of a $8.7\mathrm{\text{−}}\mu \mathrm{m}$ long InSb wire similar to that of Fig. 2. Typical current profiles are presented in (c). Conventional adiabatic (blue, top) and multiple Andreev reflection (MAR) (green, bottom) regimes are both $2\pi$ periodic. A wide region of beating $4\pi \mathrm{\text{-periodic}}$ ac Josephson current (red, middle) appears in the nonadiabatic TS phase and is absent in the trivial case. Each regime has a distinct frequency transform (d). The inset shows the beat splitting $\pm \delta \omega$ of the anomalous peak at ${\omega }_J/2$ as a function of the bias $V$, both for the even (solid) and the odd sectors (dashed). Panel (e) presents a typical dissipative decay from the $4\pi$ regime into the stationary MAR regime, which develops a dc current component (peak at $\omega =0$).

Figure 4

Transient time scales as $t_T\equiv \frac{{\hslash }^2{\Gamma }_0}{(2\mathrm{eV}{)}^2}\tau$, with dimensionless function $\tau (G/G_0)$ plotted in (a). The transient $4\pi$ regime [red shaded region, (b)] becomes longer as the relaxation rates to the quasicontinuum ${\Gamma }_0$ increase (Quantum Zeno effect). The two main panels in (b) show the ac Josephson current vs. time ($2\mathrm{eV}=0.03{\Delta }_{S^{\prime }}$) and two values of ${\Gamma }_0$ highlighted in the inset, which shows the transient time $t_T\propto {\Gamma }_0$.