Dipole-dipole shift of quantum emitters coupled to surface plasmons of a nanowire

Placing quantum emitters close to a metallic nanowire, an effective interaction can be achieved over distances large compared to the resonance wavelength due to the strong coupling between emitters and the surface plasmon modes of the wire. This leads to modified collective decay rates, as well as to Lamb and dipole-dipole shifts. In this paper, we present a general method for calculating these level shifts, which we subsequently apply to the case of a pair of atoms coupled to the guided modes of a nanowire.

Figure 1

Collective atomic decays and dipole-dipole shifts of a pair of interacting two-level atoms, described in the Dicke basis. States $|ee\rangle$ and $|gg\rangle$ mean both atoms are in the excited and in the ground states, respectively. The symmetrized and antisymmetrized single-excitation states are defined as $|S\rangle =1/\sqrt{2}\left(\right|eg\rangle +|ge\rangle )$ and $|AS\rangle =1/\sqrt{2}\left(\right|eg\rangle -|ge\rangle )$. If there is a strong effective coupling (high ${\Gamma }_{12}$) between the atoms, the difference between the collective decay rates becomes large, leading to a superradiance phenomenon.

Figure 2

Dipole-dipole shift $\delta {\omega }_{12}$ due to the presence of the wire and the ${\Gamma }_{12}$ coupling between the atoms, scaled by the total single atom decay rate ${\Gamma }_{11}$, as a function of interatomic distance $\Delta z$ in units of the vacuum radiation wavelength ${\lambda }_0$. $\delta {\omega }_{12}^{\text{appr}}$ is a good analytic approximation of ${\delta }_{12}$, for distances comparable to ${\lambda }_0$ and higher. Here, $\delta {\omega }_{12}/{\Gamma }_{11}$ shows an oscillating behavior shifted by $\pi /2$ with respect to ${\Gamma }_{12}/{\Gamma }_{11}$. For interemitter distances comparable to the wire radius, there is a substantial increase in $\delta {\omega }_{12}$ and it begins to strongly deviate from $\delta {\omega }_{12}^{\text{appr}}$.

Figure 3

First term and the second, integral term of the dipole-dipole shift, as seen in Eq. (18) and the actual wire-induced dipole-dipole shift $\delta {\omega }_{12}$, scaled by the total single-emitter decay rate ${\Gamma }_{11}$. Because of the increasing static ($\omega =0$) contributions in the Green's tensor, for small interemitter distances, the integral term becomes comparable to the first term, becoming no longer negligible.