Figure 1

Josephson junction in a one-dimensional (1D) (proximity-induced) topological superconductor. The two Majorana states ${\eta }_1$, ${\eta }_2$ located at each side of the junction are hybridized (Josephson energy $E_J$), whereas the remaining two ${\eta }_3$, ${\eta }_4$ are spatially separated and weakly coupled (coupling energies ${\delta }_j$). The system is assumed to be current biased and shunted with resistance $R$ (as shown) or phase biased ($\phi$ fixed), and driven with a finite frequency ${\omega }_d$.Reuse & Permissions

Figure 2

Spectra and connections computed in a Rashba nanowire model. (a) Andreev spectrum (${\epsilon }_n>0$ shown for $n\le 18$). (b) Connections $M_{1,2}$ (black line) and $M_{1,-2}$ (light line) as a function of phase difference. Model connections from Eq. (3) (dotted lines) agree closely with the numerical results in the nanowire model. (c) Connections $M_{2,n}$, $2<n\le 18$ between a low-energy Andreev level and the continuum. The other low-energy level is weakly coupled to the continuum $|M_{1,n}|\ll |M_{2,n}|$. (d) Schematic many-body spectrum with Landau-Zener ($P_{\text{LZ}}$) and continuum transitions ($P_D$) indicated. Here, $|n_1{n}_2\rangle$ denote the occupation numbers of levels 1 and 2. Even- and odd-parity states are drawn in black and light lines, respectively.Reuse & Permissions

Figure 3

Time-averaged current-phase relation $\overline{I}\left({\varphi }_0\right)$ as a function of excitation frequency ${\omega }_d$, keeping ac amplitude fixed at $\hslash s_d{\omega }_d=0.1E_J$ and ${\Gamma }_D={10}^{-5}{E}_J$ (lighter color: larger current). We take $\Delta /E_J=1.5$ and ${\delta }_{+}/E_J={10}^{-2}$, ${\delta }_{-}/E_J={10}^{-3}$. The dashed line indicates the threshold $\hslash {\omega }_d=\Delta -{\epsilon }_2\left(\phi \right)$ for continuum transitions. The $n$-photon resonances $n\hslash {\omega }_d={\epsilon }_2\left(\phi \right)-{\epsilon }_1\left(\phi \right)$ between the Andreev bound states for $n=1,2$ are indicated. The frequency-dependent threshold at ${\varphi }_0+2s_d\approx \pi$ is due to sweeping past the point $\phi =\pi$ where the equilibrium current phase relation changes sign and LZ transitions occur. Inset: as main panel, up to a higher frequency ${\omega }_d=2\left|\Delta \right|$.Reuse & Permissions

Figure 4

Tuning the $4\pi$-periodic Josephson effect. (a) Relationship between the effective parameter $P_D$ and the microscopic model, for $\Delta =1.5E_J$ and $\overline{V}\gg {\Gamma }_{q,r}$, ${\delta }^2/E_J$. (b) Effective current-phase relation, as obtained from master equation (4) with $\hslash {\omega }_d/E_J=1.35$ (solid lines), and the analytical result (12) with $P_D$ from Fig. 4a (dashed lines). (c) The $4\pi$ Shapiro kink for $P_{\text{LZ}}=1$ and $P_D=0,0.25,0.5$ from Eq. (10). For $P_D=0$, the result is shown as a dotted line; the solid line indicates an exact result from Eq. (13) without the mean-field approximation. (d) Composite result from Fig. 4c and Eq. (12) for the Shapiro steps with parameters $R{I}_c/\left(\hslash {\omega }_r\right)=0.2$, $s_r\equiv R{I}_r/\left(\hslash {\omega }_r\right)=0.5$, $P_{\text{LZ}}=1$ and different $P_D$. The curves are offset horizontally for clarity.Reuse & Permissions

Figure 5

Time-averaged current-phase relation for a nanowire system. As in Fig. 3 in the main text, but for the nanowire system whose spectrum and connections are displayed in Fig. 2, corresponding to parameters given in Ref. 28. The assumed continuum inverse lifetime is ${\Gamma }_0/E_J={10}^{-3}$.Reuse & Permissions

Figure 6

Probability of continuum relaxation for a nanowire system. As in Fig. 4a in the main text, but for the nanowire system whose spectrum and connections displayed in Fig. 2, and shown as a function of $s_d$ with fixed $e\overline{V}=0.01E_J$. The offset from zero is due to a finite spontaneous quasiparticle poisoning and relaxation with rates ${\Gamma }_{r,q}/E_J={10}^{-4}$. The result is not sensitive to continuum lifetime ${\Gamma }_0^{-1}$.Reuse & Permissions