Resonantly Tunable Majorana Polariton in a Microwave Cavity

We study the spectrum of a one-dimensional Kitaev chain placed in a microwave cavity. In the off-resonant regime, the frequency shift of the cavity can be used to identify the topological phase transition of the coupled system. In the resonant regime, the topology of the system is sensitive to the presence of photons in the microwave cavity and, moreover, for a large number of photons (classical limit), the physics becomes similar to that of periodically driven systems (Floquet insulators). We also analyze numerically a finite chain and show the existence of a degenerate subspace in the presence of the cavity that can be interpreted as a Majorana polariton.

Figure 1

A sketch of the system. In blue (light-gray) is the superconducting microwave cavity of length $L$, which supports quantized electromagnetic modes. The Majorana chain ${\gamma }_{1,2}^1\dots {\gamma }_N^{1,2}$ is depicted inside the cavity with the relevant parameters renormalized due to the coupling to the cavity: Chemical potential $\mu \to \mu +\delta \widehat{\mu }$ and hopping amplitude $t\to t+\delta \widehat{t}$ (with the hats denoting fluctuating quantum contributions).

Figure 2

The dimensionless functions $\Gamma (x)$ quantifying the microwave cavity frequency shift for an off-resonant cavity, given in Eq. (4) (inset), and ${\Gamma }_2(x)\equiv d^2\Gamma (x)/d{x}^2$ as a function of the parameter $x\equiv |\mu |/t$. There is a strong signature of the transition point $x=1$ (i.e., $|\mu |=t$) in ${\Gamma }_2$, which becomes singular. The function $\Gamma (x)$ instead is continuous and monotonic, decreasing with increasing the chemical potential $\mu$, the frequency shift being larger in the topological nontrivial phase.

Figure 3

Main: The energy splitting of the two lowest-energy states, $\Delta E_{\mathrm{maj}}$, for zero ($C=0$) and one excitation ($C=1$), as a function of the number of sites $N$. For these plots, we used $\mu =0.1$, $\alpha =0.1$, $\beta =0$, and $\omega =1$, all in units of $t$. Inset: The effective gap from the Majorana states to the continuum, $\Delta E_{\mathrm{gap}}$, for $C=1$, as a function of $N$ (using here $\alpha =0.2$ to exaggerate trends).