Braiding of Majorana Fermions in a Cavity

We study the dynamical process of braiding Majorana bound states (MBS) in the presence of the coupling to photons in a microwave cavity. We show theoretically that the $\pi /4$ phase associated with the braiding of MBS, as well as the parity of the ground state are imprinted into the photonic field of the cavity, which can be detected by dispersive readout techniques. These manifestations are purely dynamical, they occur in the absence of any splitting of the MBS that are exchanged, and they disappear in the static setups studied previously. Conversely, the cavity can affect the braiding phase, which in turn should allow for cavity controlled braiding.

Figure 1

Sketch of the system and the cavity-monitored Majorana braiding (exchange) process of the MBS ${\gamma }_1$ and ${\gamma }_2$. Left: The $Y$ junction with the end ${\gamma }_i$, $i=1$, 2, 3, and the middle ${\gamma }_0$ MBS, along with the couplings ${\mathrm{\Delta }}_i^{\mathrm{tot}}(t)={\mathrm{\Delta }}_i(t)+{\delta }_i(a^{†}+a)$ in the presence of the cavity field $\widehat{E}$ (green). Right: the braiding steps, with the dashed line $i$ corresponding to a splitting ${\mathrm{\Delta }}_i\gg {\mathrm{\Delta }}_j$, with $i\ne j$ (top), and the evolution of the vector ${\overrightarrow{n}}_{\mathrm{\Delta }}(t)$ in Eq. (1) on the Bloch sphere during the braiding process (bottom). The octant spanned by this vector connects to the Berry phase accumulated by the ground state wave function that pertains to the Majorana braiding statistics.

Figure 2

Dependence of the reflection coefficient $R_{\tau }({\omega }_0,\mathrm{\Omega })$ with respect to the braiding frequency $\mathrm{\Omega }$ for the two parities $\tau =\pm 1$. In the main (inset) plot we show $R_{\tau }(\mathrm{\Omega })$ with the Berry phase contribution ${\gamma }_B=\pi /4$ included (neglected) in the reflection coefficient. The red (full) and the blue (dashed) correspond to parity $\tau =1$ and $\tau =-1$, respectively. The position of the peak shift according to Eq. (15), for the braiding trajectory ${\gamma }_B=\pi /4$ (main), while they coincide at ${\gamma }_B=0$ (inset). We consider $\omega =1$, ${\mathrm{\Delta }}_0=1.1$, $\kappa =\mathrm{\Gamma }=0.002$, as well as ${\alpha }_1={\alpha }_2=0$ and ${\alpha }_3=0.01$ (we remind that ${\alpha }_i={\delta }_i/{\mathrm{\Delta }}_i$) with all energies expressed in terms of the cavity frequency ${\omega }_0$.

Figure 3

Reflection coefficient $R_{\tau }({\omega }_0,\mathrm{\Delta })$ (upper row) and the phase shift ${\xi }_{\tau }({\omega }_0,\mathrm{\Delta })$ (lower row) as a function of the detuning $\mathrm{\Delta }={\mathrm{\Delta }}_0-{\omega }_0$ for the two parities $\tau =\pm 1$. In the left (right) columns we show the plots for ${\gamma }_B=\pi /4$ included (neglected), respectively. The red (full) and the blue (dashed) correspond to parity $\tau =1$ and $\tau =-1$, respectively. The position of the peaks shift between the two parities is $\mathrm{\Omega }{\gamma }_B/\pi$. The values of the parameters are the same as in Fig. 2.