Dynamical simulation of the injection of vortices into a Majorana edge mode

The chiral edge modes of a topological superconductor can transport fermionic quasiparticles with Abelian exchange statistics, but they can also transport non-Abelian anyons: edge vortices bound to a $\pi$-phase domain wall that propagates along the boundary. A pair of such edge vortices is injected by the application of an $h/2e$ flux bias over a Josephson junction. Existing descriptions of the injection process rely on the instantaneous scattering approximation of the adiabatic regime [Beenakker et al., Phys. Rev. Lett. 122, 146803 (2019)], where the internal dynamics of the Josephson junction is ignored. Here, we go beyond that approximation in a time-dependent many-body simulation of the injection process, followed by a braiding of mobile edge vortices with a pair of immobile Abrikosov vortices in the bulk of the superconductor. Our simulation sheds light on the properties of the Josephson junction needed for a successful implementation of a flying topological qubit.

Figure 1

Edge-vortex injector [10], consisting of a Josephson junction in a topological superconductor with copropagating chiral edge modes. An $h/2e$ flux increment injects a pair of edge vortices on opposite edges with a protected fermion parity. The corresponding phase domain wall is represented by green lines. The adiabatic description of the injection process assumes that the injection time $t_{\mathrm{inj}}=(2\pi {\xi }_{\mathrm{J}}/W){(\mathrm{d}\phi /\mathrm{d}t)}^{-1}$ is long compared to the propagation time $W/v$ along the junction. In this work we relax that assumption to simulate a device (Fig. 2) in which these dynamically injected edge vortices are braided with Abrikosov bulk vortices.

Figure 2

(a) Full braiding device: two injectors (as in Fig. 1) are used to produce pairs of edge vortices. The pair of edge vortices at the back exchanges parity with the bulk vortices upon overtaking a bulk vortex, which is detected by an $e/2$ charge measurement at the exit. (b) Dispersion of Majorana edge modes (magenta), calculated for an infinite strip of a topological superconductor ($\mathcal{N}=1$). (c) Lowest-energy levels in an infinite Josephson junction (described in Sec. 2) as a function of the superconducting phase. At $\varphi =\pi$ these modes become degenerate and correspond to chiral Majorana edge states propagating along the junction [24].

Figure 3

Time snapshots of a dynamical simulation of the full device during the injection and braiding protocol. (a) Bogoliubov quasiparticle density as defined in Eq. (4.3) and (b) current density as defined in (3.1). In this simulation $v{t}_{\mathrm{inj}}=1.5W\ll L$, so the edge vortices injected at the back and front junctions are well separated, creating two separate $e/2$ charge pulses upon fusion. An animated version can be found in [25].

Figure 4

(a) Simulated (red) and theoretical (gray) current density at the exit of the superconductor. A system without (with) vortices is represented by dashed (solid) lines. The pulse width $t_{\mathrm{inj}}\approx \tau /5.17$ is indicated. (b) Corresponding charge increase for different values of the interjunction separation $L$, with values of $L/v\tau$ shown on top of the curves. All simulations have $\tau =500a/v$ and $W=42a$.

Figure 5

Evolution of the parity operator expectation value in the initial ground state (a) without and (b) with vortices. In (b), the parity of the vortices is separated from the edges, and a parity switch is observed. Convergence of the curves as a function of $E_{\max }$ is shown in color.

Figure 6

Top: Net charge increase at the exit of the superconductors (a) without and (b) with vortices, with four geometrically induced path length differences between the edges $\delta x$ for $\tau =500a/v$. Bottom: Parity of the edge sector for the same data sets. The calculated parity is independent of $\delta x$. In this case the data sets overlap, making the different curves indistinguishable.

Figure 7

(a) and (b) Quasiparticle occupation of the energy levels (thick colored lines; a thick line signifies a strong occupation) above the ground state level ($E=0$), superimposed on the time-independent energy spectrum of $H\left(\phi \right)$ (thin gray lines). The color of the lines distinguishes between junction (blue) and edge (red) states. At fast injection in (b), the quasiparticle occupation in the junction levels $E_{\mu }\ge {\mathrm{\Delta }}_{\mathrm{J}}$ at final time is high. We have removed the vortex state.

Figure 8

Bogoliubov quasiparticle number inside the junction (red) and inside the edges (blue) as a function of time. (a) shows that the junction excitation fully escapes into the edges, while in (b), at short injection time, the junction contains residual quasiparticles. The bottom panels show the corresponding quasiparticle densities at two different times. The integration window used to calculate the quasiparticle number inside the junction is marked with a blue rectangle. An animated visualization can be found in [25].

Figure 9

(a) Quasiparticle number inside the junction as a function of time for two values of the injection time. (b) Residual quasiparticle number in the junction at some final time $t_f=500a/v$ as a function of the ratio $v{t}_{\mathrm{inj}}/W$. An exponential fit yields $\langle {\widehat{N}}_{\mathrm{junc}}\left(t_{\mathrm{f}}\right)\rangle =N_{0}exp(v{t}_{\mathrm{inj}}/W\beta )$, with $\beta =0.31$ and $N_0=0.5$.

Figure 10

Convergence of the charge at the exit using two methods. Left: With charge expressed in the local basis using Eq. (A8), where $E_{\max }$ is the maximum energy of the states in the sum over $\alpha$. Right: With charge expressed in the basis of eigenstates of $H\left(0\right)$ using Eq. (A13), where $E_{\max }$ is the maximum energy of the states in the sums over $\alpha , \mu$, and $\nu$.

Figure 11

Convergence of the parity sector ${\widehat{P}}^{\prime }$ at final time $t_{\mathrm{f}}$ as a function of the index $n_{\max }$, which counts the number of eigenstates included in the calculation. This is done for different values of $v{t}_{\mathrm{inj}}/W$, which are displayed on the curves. For energies above $E_{\max }={\mathrm{\Delta }}_{\mathrm{J}}$, the hybridized edge-junction states are necessary in the convergence of the operator.

Figure 12

Three snapshots of the braiding protocol for two values of $v{t}_{\mathrm{inj}}/W$. The top panel shows the snapshots in terms of the local excitation density, and the bottom panel shows them in terms of the local current density.

Figure 13

First row: Quasiparticle occupation of the energy levels (thick colored lines). The color of the lines distinguishes between junction (blue) and edge (red) states. Second row: Quasiparticle number in the junction (blue) and at the edges (red) and their sum at the final value (gray dashed line). Third row: Current at the exit of the superconductor. Fourth row: Net charge creation at the exit of the superconductor. All results are shown for different values of $v{t}_{\mathrm{inj}}/W$ from 2.3 (left) to 0.1 (right).