Josephson effect in topological superconducting rings coupled to a microwave cavity

We theoretically study a one-dimensional $p$-wave superconducting mesoscopic ring interrupted by a weak link and coupled inductively to a microwave cavity. We establish an input-output description for the cavity field in the presence of the ring, and identify the electronic contributions to the cavity response and their dependence on various parameters, such as the magnetic flux, chemical potential, and cavity frequency. We show that the cavity response is $4\pi$ periodic as a function of the magnetic flux in the topological region, stemming from the so-called fractional Josephson current carried by the Majorana fermions, while it is $2\pi$ periodic in the nontopological phase, consistent with the normal Josephson effect. We find a strong dependence of the signal on the cavity frequency, as well as on the parity of the ground state. Our model takes into account fully the interplay between the low-energy Majorana modes and the gapped bulks states, which we show is crucial for visualizing the evolution of the Josephson effect during the transition from the topological to the trivial phase.

Figure 1

Top: A sketch of the combined topological superconductor fluxed-biased by the cavity. This is probed in the input-output method, and the output signal ${\widehat{b}}_{\mathrm{out}}$ is to be compared with the input one ${\widehat{b}}_{\mathrm{in}}$. Bottom: (a) A zoom into the topological superconductor in the shape of a ring. The magnetic field of the cavity ${\widehat{B}}_{\mathrm{cav}}$ pierces the ring and gives rise to a fluctuating magnetic flux. A perpendicular, external magnetic field $B_{\mathrm{ext}}$ is also applied, so that the total flux felt by the electrons in the ring is ${\mathrm{\Phi }}_{\mathrm{tot}}={\mathrm{\Phi }}_{\mathrm{ext}}+{\widehat{\mathrm{\Phi }}}_{\mathrm{cav}}$. The ring is interrupted by a weak link, that pertains to the hopping strength $t^{\prime }\ll t$ $(t$ is the hopping strength within the ring; see the text). In the topological region, there are Majorana fermions on the left $\left({\gamma }_L\right)$ and on the right $\left({\gamma }_R\right)$ of the superconductor, and a Josephson current $I_J$ is flowing within the ring. (b) A possible implementation of the geometry in (a) based on a spin-orbit nanowire flux-biased in a SQUID geometry. Depicted in light red is a conventional superconducting loop that is interrupted so that there is the Josephson current flowing only through the wire.

Figure 2

The spectrum of the ring as a function of the flux $\mathrm{\Phi }$ in the nontopological (left) and topological (right) regions, respectively. In red are depicted the Majorana energy levels that show oscillations with a period ${\mathrm{\Phi }}_0$.

Figure 3

Left row (right row), from top to bottom: Dependence of the Josephson, nondiagonal real, and nondiagonal imaginary susceptibilities respectively, in the topological (nontopological) region on the magnetic flux $\mathrm{\Phi }/{\mathrm{\Phi }}_0$ for $f_M=0$. The red, green, and blue lines correspond to $\mu =-1.5$ $\left(\mu =-2.5\right)$, $\mu =-1$ $\left(\mu =-3\right)$, and $\mu =-0.5$ $\left(\mu =-3.5\right)$, respectively. The phase transition point is at $\mu =-2$. The parameters for the plots are $\mathrm{\Delta }=0.1,t^{\prime }=0.2,N=50$, and all energies are expressed in terms of $t=1$.

Figure 4

Dependence of the Josephson, nondiagonal real, and nondiagonal imaginary susceptibilities, respectively, on the magnetic flux $\mathrm{\Phi }$ and parity $f_M$. The dashed black (full red) curves correspond to parity $f_M=0$ $\left(f_M=1\right)$. The oscillations of the susceptibilities for the two parties are both shifted by ${\mathrm{\Phi }}_0$, as well as asymmetric in amplitudes (once superimposed). The parameters for the plots are $\mu =-0.5,\mathrm{\Delta }=0.1,t^{\prime }=0.2,N=50$, and all energies are expressed in terms of $t=1$.

Figure 5

Dependence of the real part of the nondiagonal susceptibility on the frequency $\omega$ for two values of the external flux, $\mathrm{\Phi }=0$ (left) and $\mathrm{\Phi }={\mathrm{\Phi }}_0/2$ (right), and for the two parity states $f_M=0$ (dashed black) and $f_M=1$ (full red) lines, respectively. The parameters for the plots are $\mu =-0.5,\mathrm{\Delta }=0.1,t^{\prime }=0.5,N=50$, and all energies are expressed in terms of $t=1$.