Nonequilibrium Fractional Josephson Effect

Josephson tunnel junctions exhibit a supercurrent typically proportional to the sine of the superconducting phase difference $\varphi$. In general, a term proportional to $\cos (\varphi )$ is also present, alongside microscopic electronic retardation effects. We show that voltage pulses sharply varying in time prompt a significant impact of the $\cos (\varphi )$ term. Its interplay with the $\sin (\varphi )$ term results in a nonequilibrium fractional Josephson effect (NFJE) $\sim \sin (\varphi /2)$ in the presence of bound states close to zero frequency. Our microscopic analysis reveals that the interference of nonequilibrium virtual quasiparticle excitations is responsible for this phenomenon. We also analyze this phenomenon for topological Josephson junctions with Majorana bound states. Remarkably, the NFJE is independent of the ground state fermion parity unlike its equilibrium counterpart.

Figure 1

The dominant microscopic process for the Josephson pair current. A Cooper pair breaks up into two excitations (circles), here involving TTBSs at frequency ${\omega }_0$, combining after tunneling. The un-biased density of states (DOS) ($t\ll 0$) is shown in red, while its biased counterpart ($t\gg 0$) is shown in green. The interference (shaded blue) of the IQPs tunneling before and after the voltage step causes the ${\omega }_J^{\prime }={\omega }_J/2\pm 2{\omega }_0$ oscillations.

Figure 2

(a) Normalized current response to a step bias $V(t)=V_0\mathrm{\Theta }(t)$, with $e{V}_0=(\pi /2)\mathrm{\Delta }$, $\zeta =150\mathrm{\Delta }$, $\mathrm{\Gamma }=0.1\mathrm{\Delta }$, and $I_0=(e{\mathcal{T}}^2/\hslash )(2{\mathrm{\Delta }}^2/{\hslash }^2{\zeta }^2)(\hslash /\mathrm{\Delta })=\pi \mathrm{\Delta }/(8e{R}_N)$. We consider TTBSs in both leads at ${\omega }_0=0.1\mathrm{\Delta }$ with $h_n=h_s=1$ [73]. Strong ${\omega }_J/2$ oscillations dominate initially, eventually decaying to SJE oscillations. (b) Short-time Fourier transform of the current in (a), showing the frequency components versus time, revealing the $\mathrm{NFJE}({\omega }_J/2)\to \mathrm{SJE}({\omega }_J)$ transition at time $\sim \hslash /\mathrm{\Gamma }$ (dashed line). The discussion section provides an estimate for it, which determines the remaining parameters.

Figure 3

Normalized current ($I/I_0$) response to a square wave bias with $e{V}_0=0.8\mathrm{\Delta }$ and duty-cycle $D=0.75$, resolved in Fourier domain. In (a) we assume BCS superconducting leads, and in (b) TTBSs at ${\omega }_0=0.1\mathrm{\Delta }$ in both leads [73] with $\mathrm{\Gamma }=0.1\mathrm{\Delta }$. For ${\omega }_d\gtrsim \mathrm{\Gamma }/\hslash$, the NFJE oscillations dominate each pulse cycle in the presence of TTBSs, whereas for ${\omega }_d\lesssim \mathrm{\Gamma }/\hslash$, they decay to reveal the static normal current through the TTBS. The stripes arise due to the nonlinear response to the periodic voltage drive, creating resonances. Some of the prominent ones are marked in green(red), corresponding to positive bias(both) section(s) of each square pulse being commensurate with the NFJE oscillation period.

Figure 4

Normalized current response to a small-amplitude step bias $V(t)=V_0\mathrm{\Theta }(t)$, with $e{V}_0=0.4\mathrm{\Delta }$, $\zeta =150\mathrm{\Delta }$, and $\mathrm{\Gamma }=0.1\mathrm{\Delta }$. In (a), we consider topologically superconducting (TS) leads based on Kitaev chains, and in (b) we consider TTBSs in both leads at ${\omega }_0=0.1\mathrm{\Delta }=\mathrm{\Gamma }$ with $h_n=h_s=1$ [73] ($e{V}_0=4\hslash {\omega }_0$). Unlike the TS junction in (a), there is no noticeable NFJE for the TTBSs in (b).