Manipulating Andreev and Majorana bound states with microwaves

We study the interplay between Andreev (Majorana) bound states that form at the boundary of a (topological) superconductor and a train of microwave pulses. We find that the extra dynamical phase coming from the pulses can shift the phase of the Andreev reflection, resulting in the appearance of dynamical Andreev states. As an application we study the presence of the zero-bias peak in the differential conductance of a normal-topological superconductor junction—the simplest, yet somehow ambiguous, experimental signature for Majorana states. Adding microwave radiation to the measuring electrodes acts as a probe of the Andreev nature of the zero-bias peak.

Figure 1

(a) Sketch of the setup: blue is normal conductor, red is insulating barrier, and yellow is superconductor. (b) Sketch of the different trajectories for electrons (full lines) and holes (dashed). (c) Different applied voltage pulse trains as a function of time. From top to bottom: upright pulses, alternated pulses, and sine pulses. (d) Numerical computation of the zero-bias differential conductance as a function of time for different pulse trains (with vertical offset for clarity.)

Figure 2

Sketch of the accumulated phase along the paths through the interferometer. The paths have been flattened out for clarity. The clocks show the total phase accumulated (bottom) and the phase contributions from the pulses (blue arrows) and Andreev reflection (red arrows).

Figure 3

dc conductance in the presence of a train of voltage pulses with small intensity $\overline{n}=0.1$ (blue squares) and large intensity $\overline{n}=0.5$ (red circles) in the resonant situation $T=4{\tau }_F$. Dashed line: no voltage pulse. Symbols: numerical simulations. Lines: analytical model. Yellow arrows: evolution with increasing $\overline{n}$. The figure above each plot shows which type of voltage pulse train and system (regular $\text{S}$ or topological superconductor $\text{TS}$) was used. For the regular system we used $\alpha =0, \mathrm{\Delta }=0.075, E_F=2.0, E_z=0$, and $V_t=3.0$ (giving a barrier transmission of ${\left|d\right|}^2=0.3$). For the topological system the parameters used were $\alpha =1.0, \mathrm{\Delta }=0.5, E_F=1.0, E_z=2.0$, and $V_t=5.0$ (giving a barrier transmission of ${\left|d\right|}^2=0.17$).

Figure 4

Topological junction. Conductance in the presence of a sinusoidal perturbation as a function of dc bias, $V_b$, and driving frequency, $\omega$, for $\overline{n}=0.5$. The color scales are given in units of $e^2/h$. (a) Long junction. Negative (positive) $V_b$ shows the results from microscopic numerical [analytical with (7)] calculations. Some of the principle contributions for different pairs $(p,q)$ [(8)] are marked. ${\left|d\right|}^2=0.17$. (b) Shorter junction, obtained from numerical calculations including static disorder (quasiballistic regime) and at a finite temperature of 20 mK (for details see the Supplemental Material).