Squeezing light with Majorana fermions

Coupling a semiconducting nanowire to a microwave cavity provides a powerful means to assess the presence or absence of isolated Majorana fermions in the nanowire. These exotic bound states can cause a significant cavity frequency shift but also a strong cavity nonlinearity leading, for instance, to light squeezing. The dependence of these effects on the nanowire gate voltages gives direct signatures of the unique properties of Majorana fermions, such as their self-adjoint character and their exponential confinement.

Figure 1

(a) Scheme our our setup. The microwave cavity is made from the various superconducting contacts in purple. The nanowire (yellow) is placed between the center and ground conductors of the cavity. It is tunnel contacted to a grounded superconducting contact (purple) and capacitively contacted to three gate electrodes with voltage $V_0$ (blue) and two gate electrodes with voltage $V_1$ (pink) used to impose chemical potentials ${\mu }_0$ and ${\mu }_1$ in different sections of the nanowire. A normal-metal contact (gray) with bias voltage $V$ is tunnel contacted to the nanowire to perform a conductance spectroscopy. (b) Chemical potential (left axis) and schematic quasiparticle density probability (right axis) in the nanowire versus coordinate $z$. Four MBSs appear at the boundaries between the topological ($\mu ={\mu }_1$) and nontopological ($\mu ={\mu }_0$) sections of the nanowire. In a finite length system, the MBS wave functions overlap.

Figure 2

(a) Conductance $G_N$ between the normal-metal contact and the ground versus the bias voltage $V$ and the chemical potential ${\mu }_1$. (b) Coupling energies $\varepsilon$ and $\tilde{\varepsilon }$ (full and dotted black lines), transverse coupling ${\gamma }_e^t$ (pink dashed line), dispersive shift $\chi$ of the cavity frequency (blue line with open circles), and Kerr nonlinearity $K$ (red line with full circles) versus ${\mu }_1$. In this figure, we have used $\Delta =500\mu \mathrm{eV}$, $E_z/\Delta =2$, ${\mu }_0/\Delta =1.9$, ${\alpha }_{\text{so}}\sim 8.{10}^4\text{m}{\text{s}}^{-1}$, $L_T=L_{\mathrm{NT}}=1000\mathrm{nm}$, ${\alpha }_c{V}_{\text{rms}}=4\mu \mathrm{V}$, ${\omega }_{\mathrm{cav}}/2\pi =8\mathrm{GHz}$, $\Gamma =2\mu \mathrm{eV}$, $T=10\mathrm{mK}$, and $G_k=e^2/h$. The topological transition is located well outside the ${\mu }_1$ range considered here since ${\mu }_c/\Delta =1.73$.

Figure 3

Schematic representation of the current flow in the nanocircuit depending on the value of ${\mu }_1$. Panel (a) corresponds to ${\mu }_0$ and ${\mu }_1$ close to ${\mu }_c$ so that $\varepsilon$ and $\tilde{\varepsilon }$ are finite and the four MBSs are coupled together. In this regime, the current flows from the superconducting contact (purple) to the normal-metal contact (gray) through the four Majorana bound states. Panel (b) corresponds to a value of ${\mu }_1$ far below ${\mu }_c$ so that the coupling $\varepsilon$ between MBS 1 and other MBSs vanishes. In this case, the current flows from the superconducting contact to the normal-metal contact through MBS 1 only.

Figure 4

Schematic representation of the coupling mechanism between MBSs and cavity photons. (a) When ${\mu }_0$ and ${\mu }_1$ are close to the critical potential ${\mu }_c$, cavity photons modify the coupling between all consecutive MBSs. This yields a coupling between the nanowire and the cavity with a transverse component in the MBSs even charge sector, represented here on a Bloch sphere. (b) When ${\mu }_1$ is far below ${\mu }_c$, only MBSs 2 and 3 remain coupled by cavity photons. In this case, one can only have a longitudinal coupling in the nanowire even charge sector, as a direct consequence of the self-adjoint character of Majorana fermion operators.

Figure 5

(a) Diagram of the cavity behavior depending on the steady power $P_{\mathrm{in}}$ applied to its input port and the nanowire potential ${\mu }_1$. For $P_{\mathrm{in}}>P_{\mathrm{in}}^{\mathrm{crit}}$, the behavior of the cavity becomes hysteretic. Inset: modulus $t_{\mathrm{cav}}$ of the cavity transmission for a forward and backward sweep of ${\omega }_{\mathrm{RF}}$, for $P_{\mathrm{in}}=P_{\mathrm{in}}^{\mathrm{crit}}/4$ (blue dotted lines), $P_{\mathrm{in}}=P_{\mathrm{in}}^{\mathrm{crit}}$ (full red lines), and $P_{\mathrm{in}}=4P_{\mathrm{in}}^{\mathrm{crit}}$ (green dashed lines). (b) Cavity Husimi function $Q\left(\alpha \right)$ at points A, B, and C of Fig. 4a, at a time $\Delta t=175\mathrm{ns}$ after switching off an input power imposing a coherent cavity state $\left|i\right.〉$. We have used the same parameters as in Fig. 2, $\rho =\gamma /2\sqrt{{\gamma }_{\mathrm{in}}{\gamma }_{\mathrm{out}}}$ and $Q_{\mathrm{cav}}=\hslash {\omega }_{\mathrm{cav}}/2\pi \gamma =10000$.