Teleportation-induced entanglement of two nanomechanical oscillators coupled to a topological superconductor

A one-dimensional topological superconductor features a single fermionic zero mode that is delocalized over two Majorana bound states located at the ends of the system. We study a pair of spatially separated nanomechanical oscillators tunnel coupled to these Majorana modes. Most interestingly, we demonstrate that the combination of electron-phonon coupling and a finite charging energy on the mesoscopic topological superconductor can lead to an effective superexchange between the oscillators via the nonlocal fermionic zero mode. We further show that this electron teleportation mechanism leads to entanglement of the two oscillators over distances that can significantly exceed the coherence length of the superconductor.

Figure 1

Schematics of the proposed setup. Two nanoelectromechanical oscillators (blue) each tunnel coupled to one end of a one-dimensional topological superconductor. The one-dimensional topological superconductor is sketched as a nanowire (yellow) placed on top of a mesoscopic superconductor (gray). At each end of the wire a single Majorana quasiparticle (orange) is located. A gate voltage $V_g$ is applied (across a gate capacitance $C_g$) to the mesoscopic superconductor resulting in a finite charging energy $E_c=e^2/2C_g$ of the superconductor. The nanoelectromechanical oscillators are modeled as normal metal leads at the chemical potentials ${\mu }_{L/R}$.

Figure 2

Anomalous cotunneling process decomposed into two (virtual) steps. Looking only at input and output states, one electron tunnels from the left to the right lead (blue). The state of the superconductor is unchanged. This is the only anomalous second-order process contributing to the current that does not contain states with $E>E_c$. The oval denotes a Cooper pair in the condensate. The level in the middle is the subgap-fermion $c$. The tunnel processes depicted here appear in the effective tunnel Hamiltonian Eq. (2).

Figure 3

(a) Logarithmic negativity as a function of $E_c$ and time for $V/\Omega =0.5$. For small $E_c$ the generated entanglement is higher but decays faster than for large $E_c$. (b) Logarithmic negativity as a function of $V$ and time for $E_c/\Omega =4$. For lower bias voltages the entanglement is higher. In both cases, the other parameters are $T_{el}=0$, $L_c/\Omega =1$, and $t_0{t}_x/\sqrt{m\Omega }=0.1$.