Chiral Majorana fermions resonance exchange moudulated by quantum dot coupling strength

Abstract

We study the resonance exchanges of two chiral Majorana fermions in two distinct systems theoretically in this work: one is an isolated Majorana zero mode interacting with complexes formed by two chiral Majorana fermions and a Majorana zero mode, and the other involves isolated quantum dots that are coupled to a system composed of Majorana fermions and a quantum dot. Our research results reveal that both of these coupled systems can facilitate the effective transmissions of the two chiral Majorana fermions as <inline-formula><tex-math id="M1">\begin{document}$ {\gamma _1} \to - {\gamma _2} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M1.png"/></alternatives></inline-formula>and <inline-formula><tex-math id="M2">\begin{document}$ {\gamma _2} \to - {\gamma _1} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M2.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M2.png"/></alternatives></inline-formula>, and the resonant tunneling effects in the two systems are equivalent. Therefore, quantum dots can replace Majorana zero modes to achieve resonant tunneling. In order to observe the resonance exchange of two chiral Majorana fermions with the two quantum dots, a circuit based on anomalous quantum Hall insulator proximity-coupled with s-wave superconductor is proposed as shown in figure. The numerical results indicate that the resonant exchange of chiral Majorana fermions can be modulated by the coupling strength between the two quantum dots, and it is particularly noteworthy that the tunneling process is independent of the superconducting phase. If one of the chiral Majorana fermions undergoes resonance coupling with another quantum dot or Majorana zero mode, an additional negative sign is obtained, leading to <inline-formula><tex-math id="M3">\begin{document}$ - {\gamma _2} \to {\gamma _1} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M3.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M3.png"/></alternatives></inline-formula>. After experiencing two resonance exchange processes, the final result is <inline-formula><tex-math id="M4">\begin{document}$ {\gamma _1} \to {\gamma _2} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M4.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M4.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ {\gamma _2} \to - {\gamma _1} $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M5.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="13-20240739_M5.png"/></alternatives></inline-formula>, which implies the realization of non-Abelian braiding operations. Our conclusion is that the modulation of coupling strength between two quantum dots can be used to achieve the switch of Majorana fermions braiding-like operation, which is independent of superconducting phase. Therefore, the designed scheme provides a new way for adjusting the braiding-like operation of Majorana fermions. These findings may have potential applications in the realization of topological quantum computers.