Coherent manipulation of a Majorana qubit by a mechanical resonator

We propose a hybrid system composed of a Majorana qubit and a nanomechanical resonator, implemented by a spin-orbit-coupled superconducting nanowire, using a set of static and oscillating ferromagnetic gates. The ferromagnetic gates induce Majorana bound states in the nanowire, which hybridize and constitute a Majorana qubit. Due to the oscillation of one of these gates, the Majorana qubit can be coherently rotated. By tuning the gate voltage to modulate the local spin-orbit coupling, it is possible to reach the resonance of the qubit-oscillator system for relatively strong couplings.

Figure 1

(a) Schematic diagram of the proposed Majorana qubit-nanomechanical resonator hybrid system. A semiconductor nanowire is placed on the surface of an $s$-wave superconductor. Three ferromagnetic gates are on top of the nanowire, of which FM1 and FM3 are sufficiently long (of the order of 1–10 $\mu \mathrm{m}$) and static, while FM2 is relatively short (of the order of 100 nm) and free to oscillate as a harmonic oscillator. The ferromagnetic gates induce a local Zeeman splitting $B_0$ in the underlying nanowire and can also be used to modulate the local Rashba SOC strength by applying an electric voltage $V$. (b) Wave amplitude ${\left|\mathrm{\Psi }\right|}^2$ of the four coupled MBS at a static state in an InSb nanowire with the setup shown in (a). The red dashed and red solid curves respectively correspond to the wave amplitudes of the lowest two eigenstates (close to zero energy). These two states constitute the Majorana qubit. The dotted curve with the scale on the right-hand side of the frame indicates the profile of the inhomogeneous Zeeman splitting along the nanowire. In the calculation $l_{12}=l_{34}=150$ nm, $l_{23}=400$ nm, $B_0=1$ meV, and $\mathrm{\Delta }=0.5$ meV. The gate voltage $V$ is zero and the Rashba SOC is homogeneous along the nanowire, with a strength ${\alpha }_0^{\mathrm{R}}=20$ meV nm.

Figure 2

Majorana coupling strength $g_n$ ($g_t$) versus $l_n$ ($l_t$), the length of the inner $N$ ($T$) domain between the two outer $T$ ($N$) domains. The derivative $g_n^{\prime }$ versus $l_n$ is also shown, with the scale on the right-hand side of the frame. The necessary parameters for the calculation are specified in the main text.

Figure 3

Qubit splitting $\varepsilon$ and the qubit-phonon coupling $g$ (the scale is on the right-hand side of the frame) versus ${\alpha }_V^{\mathrm{R}}$, the Rashba SOC strength in the topological $T$ domains modulated by the gate voltage $V$.

Figure 4

Time evolution of the occupations of the Majorana qubit (solid curves) and phonons (dashed curves), which are in Rabi resonance. The qubit relaxation time is set as (a) 50 $\mu \mathrm{s}$ and (b) 150 $\mu \mathrm{s}$. The calculations for both (a) and (b) are performed with two different resonator quality factors: $Q={10}^6$ and $Q=3\times {10}^5$.