Majorana oscillations and parity crossings in semiconductor nanowire-based transmon qubits

We show that the microwave (MW) spectra in semiconductor-nanowire-based transmon qubits provide a strong signature of the presence of Majorana bound states in the junction. This occurs as an external magnetic field tunes the wire into the topological regime and the energy splitting of the emergent Majorana modes oscillates around zero energy owing to their wave function spatial overlap in finite-length wires. In particular, we discuss how these Majorana oscillations, and the concomitant fermion parity switches in the ground state of the junction, result in distinct spectroscopic features—in the form of an intermittent visibility of absorption lines—that strongly deviate from standard transmon behavior. In contrast, nonoscillating zero modes, such as topologically trivial Andreev bound states resulting from sufficiently smooth potentials, exhibit an overall standard transmon response. These differences in the MW response could help determine whether the junction contains topological Majoranas or not.

Figure 1

Sketch and spectrum of a NW-based superconducting island. (a) Simplified transmon/CPB circuit. $E_J$ is implicit in the Josephson potential $V_J\left(\phi \right)$, while the combination of a shunting capacitor $C_J$ and the gate capacitance $C_g$ define the charging energy $E_C=e^2/2(C_J+C_g)$. The presence of MBSs ${\gamma }_i$ gives rise to two new energy scales: the Majorana splitting $\delta /2$ and the Majorana coupling $E_M$. (b) Spectrum of the island versus $n_g=V_g/\left(2e{C}_g\right)$ in the CPB regime ($E_J/E_C=0.5$, $E_M/E_C\sim 0.12$) and for $B\sim B_c$. Blue/orange colors denote fermionic even/odd parities, while intermediate gradient signals parity mixing. The dotted lines show a slightly different $B$ field with $\delta =0$. (c) Same as (b) but in a transmon regime with $E_J/E_C=25$, $E_M/E_C\sim 6.5$ (dashed lines show the original transmon lines at $B=0$). Panel (d) shows the same transmon case but for $\delta \ne 0$ ($B=1.4B_c$). (e), (f) Magnetic field dependence for the CPB in (b), at $n_g=0.25$ and $n_g=0.5$, respectively. In this latter case, the topological transition (where $E_J\to 0$) can be seen as a closing of the qubit frequency (asterisk). (g), (h) Same as (e), (f) but for the transmon (arrows mark parity crossings). (i), (j) Parity polarization $\langle {\widehat{\tau }}_z\rangle$ of the two lowest states in (g), (h). At parity crossings, the polarization jumps. Rest of parameters: $L_S=2.2\mu \mathrm{m}$, $\tau =0.8$, $\mu =0.5\mathrm{meV}$, $\mathrm{\Delta }=0.25\mathrm{meV}$.

Figure 2

MW spectroscopy of a NW-based transmon. (a) Contour plot of MW absorption spectrum $S_N\left(\omega \right)$ versus $\omega$ and $B/B_c$ at $n_g=0.5$. Bright lines signal allowed transitions in the MW response. Spectral holes in the $n=1$ transition line and abrupt jumps in the $n>1$ ones, where lines suddenly disappear/appear, coincide with parity crossings in the GS owing to Majorana oscillations. (b) Transition frequencies and spectral weights (shadowed widths). At minima of the oscillations (arrows), the spectral weight of different transitions gets exchanged. (c)–(h) $n_g$ dependence. Same parameters as the transmon in Fig. 1.

Figure 3

MW spectroscopy of a junction containing nontopological robust zero modes. (a) The BdG spectrum of a NW with a smooth $\mu \left(x\right)$ (which interpolates between $\mu =0.5$ meV and $\mu =2$ meV at each NW segment) shows robust zero modes for $B<B_c$. Inset: blowup near $B\lesssim B_c$ where the zero mode starts to develop oscillations. (b), (c) The $B$-field dependence of the MW response (shown here at $n_g=0.5$) of a transmon containing such wires is distinct depending on whether it is governed by the trivial mode pinned to zero at $B\lesssim 0.8B_c$ or the oscillating Majoranas at $B>B_c$. While the former leads to an overall standard transmon response (d), (g), the latter shows parity switches, (e), (h) and (f), (i), as discussed in Fig. 2. Same parameters as Fig. 2.