Long-lived qubit entanglement by surface plasmon polaritons in a Weyl semimetal

We investigate spontaneous entanglement of two qubits mediated by nonreciprocal surface plasmon polaritons (SPPs) in a Weyl semimetal. In the absence of external magnetic fields, the topology of the Weyl semimetal even gives rise to nonreciprocal SPPs that are topologically protected and reside inside the photonic gap. We utilize this nonreciprocal SPP as a mediator of entanglement of two spatially separated qubits. Our two main findings are (1) the nonreciprocal SPP gives better quantum entanglement than the reciprocal one, and (2) the entanglement achieved in the Weyl semimetal is sufficiently long lived compared to the entanglement using SPPs in conventional metals.

Figure 1

(a) The electronic energy dispersion of the Weyl semimetal. The Weyl nodes are separated by the wave vector $\mathbf{b}$. (b) Illustration of SPP propagating on the interface between the Weyl semimetal (WSM) and a normal metal (NM). The electromagnetic field of an SPP decays in the $z$ direction and propagates in the $x$ direction. We also illustrate the qubits separated by a distance $d$. The qubits are placed above the WSM with a distance $\mathcal{Z}$.

Figure 2

The frequency of bulk photon propagating inside the Weyl semimetal with $xz$ plane of propagation. Here we use ${\mathrm{\Omega }}_b=0.5$.

Figure 3

(a) The frequency of TM bulk photon propagating inside the Weyl semimetal and the frequency of SPP at the interface between the Weyl semimetal and normal metal. Here the bulk plasmon frequency of the normal metal (silver) is $293{\omega }_p(9.6$ eV). (b) Frequency plot similar to (a), but for the bulk plasmon frequency of the normal metal of $5{\omega }_p$. (c), (d) Frequency plot similar to (a) and (b), respectively, but we use the Dirac semimetal $[{\mathrm{\Omega }}_b=0$, instead of the Weyl semimetal (${\mathrm{\Omega }}_b\ne 0$)]. The inset in (c) is the frequency difference between the frequency of the SPP and the bulk TM photon. The order of $\mathrm{\Delta }\mathrm{\Omega }$ is ${10}^{-4}$, and $\mathrm{\Delta }\mathrm{\Omega }$ increases with increasing $Q$. Here we use ${\mathrm{\Omega }}_b=0.5$.

Figure 4

The concurrence for the Weyl semimetal (dashed line) and the Dirac semimetal (solid line) as a function of normalized time $T=2\mathrm{\Gamma }t/{\hslash }^2$, where $\mathrm{\Gamma }$ is defined for each material. The bulk plasmon frequency of the normal metal is (a) $5{\omega }_p$ and (b) $293{\omega }_p$. The frequency for both cases is $\mathrm{\Omega }=1.1$.

Figure 5

The matrix elements of the density matrix of the qubits as a function of $T$ for the case of the Weyl semimetal (dashed lines) and the Dirac semimetal (solid lines). The bulk plasmon frequency of the normal metal is $5{\omega }_p$ and $\mathrm{\Omega }=1.1$.

Figure 6

The concurrence for the Weyl semimetal for several values of $\mathrm{\Omega }$ as a function of $t$. The bulk plasmon frequency of the normal metal is (a) $5{\omega }_p$ and (b) $293{\omega }_p$.

Figure 7

(a) $\mathrm{\Gamma }/\hslash$ given in eV as a function $\mathrm{\Omega }$ for several ${\mathrm{\Omega }}_b$ values. In this plot we use a normal metal with the bulk plasmon frequency of $10{\omega }_p$. The frequency of TM bulk photon propagating inside the Weyl semimetal and the frequency of the SPP for (b) ${\mathrm{\Omega }}_b=0.5$ and (c) ${\mathrm{\Omega }}_b=1.5$.