Entangling two distant nanocavities via a waveguide

In this paper, we present a scheme for generating continuous-variable entanglement between two spatially separated nanocavities in photonic crystals, which are mediated by a coupled-resonator optical waveguide. The entanglement degree and purity of the generated states are investigated as varying the cavity frequencies, the cavity-waveguide coupling strength, and the location of the second cavity. It is shown that a steady and pure entanglement between separated nanocavities can be generated only with a weak cavity-waveguide coupling when the cavities are resonant with the band center of the waveguide. Various cases with different cavity frequencies and coupling strengths, which affect the degree of entanglement, are also investigated, and interestingly sudden death and sudden birth of entanglement occur for strong couplings.

Figure 1

A schematic plot of two spatially separated nanocavities coupled to a CROW structure at the sites $n_i$ $(i=1,2)$ in photonic crystals.

Figure 2

(a) The spectral density $J_{\mathit{ii}}(\omega )$ for the cavities at different locations $n_i$. (b) The spectral density $J_{12}(\omega )$ for the different sites $n_2$ of the second cavity and the first cavity location $n_1=1$. Here the two cavity frequencies are taken to be equal and also equal to the resonator frequency in the waveguides: ${\omega }_1={\omega }_2={\omega }_0$. The cavity-waveguide coupling ${\xi }_1={\xi }_2=0.2{\xi }_0$, and the hopping rate in the waveguide ${\xi }_0=0.05{\omega }_0$.

Figure 3

The temporal behavior of the logarithmic negativity $E_N$ and the corresponding purity $P$ of the two-mode cavity states for different values of the site $n_2$ of the second cavity and the cavity-waveguide coupling $\eta$. The cavity frequency is resonant with the band center of the waveguide, i.e., ${\omega }_c={\omega }_0$, the hopping rate ${\xi }_0=0.05{\omega }_0$, the squeezing parameter $r=1.0$, and the site of the first cavity $n_1=1$.

Figure 4

The temporal behavior of the logarithmic negativity $E_N$ (red, lower lines) and the corresponding purity $P$ (green, upper lines) of the two-mode cavity states for different values of the site $n_2$ of the second cavity and the coupling $\eta$. The cavity frequency is resonant with the band center of the waveguide (${\omega }_c={\omega }_0$), and other parameters are the same as in Fig. 3.

Figure 5

The temporal behavior of the logarithmic negativity $E_N$ for the different values of the cavity-waveguide coupling $\eta$. The cavity frequency is resonant with the band center of the waveguide (${\omega }_c={\omega }_0$), and the other parameters are the same as in Fig. 3.

Figure 6

The temporal behavior of the logarithmic negativity $E_N$ and the corresponding purity $P$ of the two-mode cavity states. In (a) and (b), the site of the second cavity $n_2=10$, the cavity-waveguide coupling $\eta =0.2$ (red dashed line), $0.4$ (green dotted line), and $0.6$ (blue solid line). In (c) and (d), the coupling $\eta =0.4$, the site $n_2=6$ (red dashed line), $10$ (green dotted line), and $16$ (blue solid line). The cavity frequency ${\omega }_c=1.2{\omega }_0$ outside of the waveguide band, and the other parameters are the same as in Fig. 3.

Figure 7

The temporal behavior of the logarithmic negativity $E_N$ and the corresponding purity $\mathrm{P}$ of the two-mode cavity states for the cavity-waveguide coupling $\eta =0.1$ (red dashed line), 0.2 (green dotted line), and 0.4 (blue solid line). The cavity frequency ${\omega }_c=1.03{\omega }_0$ within the waveguide band and the site of the second cavity $n_2=5$. The inset in (a) depicts the average photon numbers $\langle a_i^{†}{a}_i\rangle$ of the two cavities for the coupling $\eta =0.4$. The other parameters are the same as in Fig. 3.

Figure 8

The temporal behavior of the logarithmic negativity $E_N$ for the cavity frequency ${\omega }_c=1.06{\omega }_0$ outside the band of the waveguide and the second cavity locating at the site $n_2=5$. The other parameters are the same as in Fig. 3.