Surface-plasmon-enhanced quantum field entanglement through anisotropic Purcell factors

We theoretically propose a mechanism for the enhancement of quantum field entanglement generated by four-wave mixing through anisotropic Purcell factors in a three-level atomic system. With anisotropic Purcell factors, the dependence of the entanglement on the relative polarization direction between two dipole moments is investigated. It is found that, for the two-mode quantum field entanglement, if the bisector of the two orthogonal dipole moments lies in the large (small) Purcell factor axis, the entanglement would be enhanced (decreased) with changing the crossing damping accordingly. Moreover, larger anisotropism of Purcell factors leads to further enhancement of the entanglement. We also reveal a mechanism for increasing the duration of four-mode quantum field entanglement through the anisotropic Purcell factors. With the help of the custom-designed plasmon nanostructure which creates the anisotropic Purcell factors, we demonstrate that the quantum field entanglement can be effectively modulated at subwavelength scale. Our paper provides a promising way towards the entangled source in nanophotonic structures and may have potential applications in on-chip devices for quantum information.

Figure 1

(a) The schematic of a three-level $\mathrm{\Lambda }$-type atomic system with the crossing damping $\kappa$ for the FWM process. (b) The sketch of the propagation direction and the phase-matching condition for the FWM process (forward propagation classical light ${\mathrm{\Omega }}_1$, backward propagation classical light ${\mathrm{\Omega }}_2$, and four quantized optical modes $a,b,c,\text{and}d$). (c) The spatial orientation of dipole moments for the FWM process related channels. ${\mathrm{\Gamma }}_{xx}$ (${\mathrm{\Gamma }}_{zz}$) describes the decay rate along the $x$ ($z$) direction. ${\theta }_{32}$ and ${\theta }_{31}$ are the angles between the dipole moments ${\mu }_{32}$ and ${\mu }_{31}$ and the $x$ axis, respectively.

Figure 2

Two-mode field entanglement modulated by the anisotropic Purcell factors with the two orthogonal dipole moments. (a) Enhancing (blue dotted curve) or suppressing (red dashed curve) the two-mode field entanglement as compared with the isotropic Purcell factor (black solid curve). The orientations of the two orthogonal dipole moments for the fixed anisotropic Purcell factors are shown in the inset, for ${\theta }_{31}={45}^{\circ }$ and ${\theta }_{32}={135}^{\circ }$ (blue dotted curve), ${\theta }_{31}={45}^{\circ }$ and ${\theta }_{32}={135}^{\circ }$ (black solid curve), and ${\theta }_{31}=-{45}^{\circ }$ and ${\theta }_{32}={45}^{\circ }$ (red dashed curve), respectively. (b) Two-mode field entanglement with respect to the different extent of the anisotropism of Purcell factors. Here, ${\mathrm{\Delta }}_{31}=-131{\gamma }_0$, $\delta =5{\gamma }_0$, and ${\mathrm{\Omega }}_1=100{\gamma }_0$.

Figure 3

The two-mode field entanglement with varying (a) coupling field detuning ${\mathrm{\Delta }}_{31}$ and (b) coupling field Rabi frequency ${\mathrm{\Omega }}_1$. Entanglement increases with larger coupling field detuning and smaller Rabi frequency. ${\theta }_{31}={45}^{\circ },\text{and}{\theta }_{32}={135}^{\circ }$. Other parameters remain the same as those in Fig. 2.

Figure 4

Four-mode field entanglement modulation induced by the anisotropic Purcell factors (a), (c) and isotropic Purcell factors (b) with the two orthogonal dipole moments. Longer duration exists with greater anisotropism of Purcell factors. The results with isotropic Purcell factors are in line with those in Ref. [35]. Here, (a), (b) ${\theta }_{31}={45}^{\circ }\text{and}{\theta }_{32}={135}^{\circ }$ and (c) ${\theta }_{31}=-{45}^{\circ }\text{and}{\theta }_{32}={45}^{\circ }$. ${\mathrm{\Delta }}_{31}=-131{\gamma }_0$, $\delta =5{\gamma }_0$, ${\mathrm{\Omega }}_1=131{\gamma }_0$, and ${\mathrm{\Omega }}_2=1.2{\mathrm{\Omega }}_1$.

Figure 5

Four-mode field entanglement with increasing extent of the anisotropism of Purcell factors, for (a) $\kappa =-0.2{\gamma }_0$, (b) $\kappa =-0.6{\gamma }_0$, and (c) $\kappa =-0.9{\gamma }_0$. Duration time of entanglement becomes longer when $\kappa$ increases. ${\theta }_{31}=-{45}^{\circ },\text{and}{\theta }_{32}={45}^{\circ }$. Other parameters remain the same as those in Fig. 4.

Figure 6

Schematic (a) and absorption (b) of a resonant gold nanocavity composed of three pairs of nanoantennas. Also shown are the distributions of anisotropic Purcell factors (c) ${\mathrm{\Gamma }}_{xx}/{\gamma }_0$ and (d) ${\mathrm{\Gamma }}_{zz}/{\gamma }_0$ on the $xy$ plane with $z=105$ nm at the wavelength of $\lambda =895$ nm (the origin of the coordinate is in the center point of the nanocavity).

Figure 7

Two-mode field entanglement (a) for various locations on the $xy$ plane at $z=75$ nm and (b) for $z=75,105,\text{and}135$ nm on the $xy$ plane with $x=0$ nm and $y=112.5$ nm. Here, ${\theta }_{31}={45}^{\circ },\text{and}{\theta }_{32}={135}^{\circ }$.