Direct Cavity Detection of Majorana Pairs

No experiment could directly test the particle-antiparticle duality of Majorana fermions, so far. However, this property represents a necessary ingredient towards the realization of topological quantum computing schemes. Here, we show how to complete this task by using microwave techniques. The direct coupling between a pair of overlapping Majorana bound states and the electric field from a microwave cavity is extremely difficult to detect due to the self-adjoint character of Majorana fermions which forbids direct energy exchanges with the cavity. We show theoretically how this problem can be circumvented by using photoassisted tunneling to fermionic reservoirs. The absence of a direct microwave transition inside the Majorana pair in spite of the light-Majorana coupling would represent a smoking gun for the Majorana self-adjoint character.

Figure 1

(a) Tight-binding model of our nanocircuit with $N_c=6$ sites (green dots). The consecutive sites are coupled by a tunnel-hopping constant $t$ and spin-orbit terms $\mathrm{\Lambda }$. All sites are tunnel coupled to the superconductor $S$ with a rate ${\mathrm{\Gamma }}_S$. The extremal sites are tunnel coupled with a rate ${\mathrm{\Gamma }}_N$ to normal metal contacts $N_{L(R)}$ with a bias voltage $V$. All sites are coupled to the microwave cavity with a constant $g$. (b) DOS $\nu (ϵ)$ at the ends of the chain or nanowire, versus $\omega$ and $E_z$, for cases $A$ (long nanowire with negligible coupling to $N_{L(R)}$) and $B$ (short nanowire and coupling to $N_{L(R)}$ and $S$ similar at zero energy). (c) Energetic scheme of the nanocircuit placed in the microwave cavity. The DOS in $S$ depends on the superconducting gap $\mathrm{\Delta }$ and the broadening parameter ${\mathrm{\Gamma }}_b$. The cavity microwave transmission $b_t/b_{\text{in}}$ or reflection $b_r/b_{\text{in}}$ is measured.

Figure 2

(a) $\mathrm{Im}[\chi ({\omega }_0)]$ versus ${\omega }_0$ and $E_z$ for case $A$ with $V=0$. (b) Scheme of the main features appearing in panel (a). (c) $\mathrm{Im}[\chi ({\omega }_0)]$ versus $E_z$ for constant values of ${\omega }_0$ corresponding to the symbols in panel (a). (d) Processes contributing to the features of panel (b) and forbidden processes $P$ and $Q$.

Figure 3

(a) $\mathrm{Im}[\chi ({\omega }_0)]$ versus ${\omega }_0$ and $E_z$ for case $B$ and $V=0$. (b) Corresponding $E_z-\mu$ map for ${\omega }_0=0.15\mathrm{\Delta }$. The pink circle corresponds to feature ① of Fig. 2, while the features with the yellow circles are specific to the short nanowire case.

Figure 4

(a) $\mathrm{Im}[\chi ({\omega }_0)]$ versus ${\omega }_0$ and $E_z$ for case $B$ and $eV=0.32\mathrm{\Delta }$. (b) Corresponding $\mu -E_z$ map of $\mathrm{Im}[\chi ({\omega }_0)]$ for ${\omega }_0=0.15\mathrm{\Delta }$.