Spontaneous emission interference in negative-refractive-index waveguides

The spontaneous decay of a $V$-type three-level atom placed in a negative-refractive-index waveguide is analyzed. We find that in thin waveguides, highly efficient surface-guided modes are supported, which do not occur in positive-index waveguides. In addition, at low absorption, the mode density and thus spontaneous emission into particular regular-guided modes is enhanced by several orders of magnitude as compared to regular dielectric waveguides. The asymmetries between emission into the different modes and the enhancement of particular guided modes allow us to induce strong spontaneous emission interference between transitions with orthogonal transition dipole moments.

Figure 1

(Color online) Schematic setup for spontaneous decay in a waveguide with negative index of refraction material. (a) The waveguide consists of an inner layer with material parameters ${ϵ}_3$, ${\mu }_3$, and thickness $d_3$. It is surrounded by infinite outer layers with material parameters ${ϵ}_i$ and ${\mu }_i\left(i∊\left\{1,2\right\}\right)$. In the middle layer, an atom is embedded at position ${\mathbf{r}}_0$. (b) Electronic structure of the embedded atom, which is a three-level atom in $V$-type configuration.

Figure 2

(Color online) (a) Radiation modes with total rate ${\Gamma }_r$ (red solid line) and contribution ${\Gamma }_{rz}^p$ of the $p$ mode for $z$ polarization (blue dashed). (b) Waveguide modes with total rate ${\Gamma }_g$ (red solid line) and contribution ${\Gamma }_{gz}^p$ of the $p$ mode for $z$ polarization (blue dashed). The atom is located at $z_0^{\prime }=0.25d_3^{\prime }$, and the material parameters are ${\epsilon }_1={\epsilon }_2={\mu }_1={\mu }_2=1.0$, ${\omega }_{pe}={\omega }_{pm}=1.32{\omega }_0$, ${\omega }_{Te}={\omega }_{Tm}={\omega }_0$, ${\omega }_a=1.09{\omega }_0$, and ${\gamma }_e={\gamma }_m={10}^{-10}{\omega }_0$. Here, ${\omega }_0$ is a scaling parameter. Then, ${\mu }_3\left({\omega }_a\right)={\epsilon }_3\left({\omega }_a\right)\approx -1.99+1.73\times {10}^{-9}i$.

Figure 3

(Color online) Decay rates into surface-guided modes with total decay-rate contribution ${\Gamma }_s$ (solid red line) and contribution ${\Gamma }_{sz}^p$ of the $p$ mode (dashed blue line) for $z$ polarization. The parameters are as in Fig. 2.

Figure 4

(Color online) The influence of dispersion and absorption on the results. The red solid line shows the frequency dependence of the total decay rate for parameters as in Fig. 2 but with $d_3$ set to the first peak in ${\Gamma }_z^p$. The blue dashed line is plotted for same parameters but includes small absorption $\gamma ={\gamma }_e={\gamma }_m={10}^{-8}{\omega }_0$. The black horizontal dotted lines indicate the peak amplitudes of about $7100{\gamma }_0$ and $710{\gamma }_0$, respectively, of the two curves, verifying the scaling of the peak amplitudes with $1/\sqrt{\gamma }$.

Figure 5

(Color online) Spontaneous emission into strongly absorbing NRI waveguides. (a) Radiation mode contributions ${\Gamma }_{rz}^p$ (solid red line), ${\Gamma }_{rx}^p$ (dashed blue line) and ${\Gamma }_{rx}^s$ (dotted black line). (b) Waveguide mode contributions ${\Gamma }_{gz}^p$ (solid red line), ${\Gamma }_{gx}^p$ (dashed blue line) and ${\Gamma }_{gx}^s$ (dotted black line). The atom is located at $z_0^{\prime }=0.5d_3^{\prime }$ and the material parameters are ${\epsilon }_1={\epsilon }_2={\mu }_1={\mu }_2=1.0$, ${\omega }_{pe}=1.25{\omega }_0$, ${\omega }_{pm}=1.189{\omega }_0$, ${\omega }_{Te}={\omega }_{Tm}={\omega }_0$, ${\omega }_a=1.08{\omega }_0$, and ${\gamma }_e={\gamma }_m={10}^{-3}{\omega }_0$. Here, ${\omega }_0$ is a scaling parameter. Then, ${\epsilon }_3\left({\omega }_a\right)\approx -2.38+2.19\times {10}^{-2}i$ and ${\mu }_3\left({\omega }_a\right)\approx -1.48+1.61\times {10}^{-2}i$.

Figure 6

(Color online) Decay rates into surface-guided modes. The total decay rate ${\Gamma }_s$ has contribution only from ${\Gamma }_{sx}^p$ of the $p$ mode for $x$ polarization. The parameters are as in Fig. 5.

Figure 7

(Color online) Quantum interference induced by a negative-refraction-index waveguide structure characterized by the interference strength $\kappa$. (a) Parameters as in Fig. 2 except for $z_0^{\prime }=d_3^{\prime }/2$, ${\omega }_{pm}=1.189{\omega }_0\left[{\mu }_3\left({\omega }_a\right)\approx -1.20+1.27\times {10}^{-9}i\right]$. (b) Strong absorption case. Parameters are as in Fig. 5 with ${\epsilon }_3\left({\omega }_a\right)\approx -2.38+2.19\times {10}^{-2}i$ and ${\mu }_3\left({\omega }_a\right)\approx -1.48+1.61\times {10}^{-2}i$.