Controlling photon bunching and antibunching of two quantum emitters near a core-shell sphere

The collective spontaneous emission of two point-dipole emitters near a plasmonic core-shell nanosphere is theoretically investigated. Based on the expansion of mode functions in vector spherical harmonics, we derive closed analytical expressions for both the cooperative decay rate and the dipole-dipole interaction strength associated with two point dipoles close to a sphere. Considering a plasmonic nanoshell containing a linearly amplifying medium inside the core, the second-order correlation function for the two emitters shows that it is possible to tune the photon emission, selecting either photon bunching or antibunching as a function of the polarization and position of the sphere. This result opens vistas to applications involving tunable single-photon sources in engineered artificial media.

Figure 1

Two point-dipole emitters located in the vicinity of a coated sphere. The core-shell sphere has inner radius $a$ and outer radius $b$, with electric permittivities ${\varepsilon }_{\mathrm{c}}$ for the core and ${\varepsilon }_{\mathrm{s}}$ for the shell. The dipole emitters are characterized by the atomic dipole moment ${\mathbf{d}}_q$ and are located at ${\mathbf{r}}_q$, with $q=\{1,2\}$. The interatomic distance is $r_{12}=|{\mathbf{r}}_{12}|=|{\mathbf{r}}_1-{\mathbf{r}}_2|$, with $\alpha$ being the angle between ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$.

Figure 2

Optical cross sections associated with light scattering by a core-shell nanosphere in free space. A dielectric core of refractive index $n_{\mathrm{c}}=3.5-ı\kappa$ and radius $a=180\mathrm{nm}$ is coated with a silver shell [63, 64] of thickness $b-a=20\mathrm{nm}$. The plots in (a) and (b) show the total scattering cross section ${\sigma }_{\mathrm{sca}}$ for $\kappa =0$ (no gain) and $\kappa =0.015$ (a gain-assisted core), respectively. We show multipole contributions to the light scattering: electric dipole (ED), magnetic dipole (MD), electric quadrupole (EQ), and magnetic quadrupole (MQ). The plot in (c) shows the differential scattering cross section associated with the MQ resonance ($\lambda =780\mathrm{nm}$) for ${\kappa }_{\mathrm{c}}=0$ (left panel) and $\kappa =0.015$ (right panel).

Figure 3

Two identical dipole emitters with $|{\mathbf{r}}_1|=|{\mathbf{r}}_2|$ in the vicinity of a core-shell sphere with a gain material inside. The incoming electromagnetic wave is such that $\mathbf{k}·{\mathbf{r}}_{12}=0$. We consider two basic configurations: the interatomic distance vector ${\mathbf{r}}_{12}$ is orthogonal or parallel to the incident electric field ${\mathbf{E}}_{\mathrm{in}}$. The detector is fixed in the $z$ direction.

Figure 4

Two dipole emitters with transition wavelength ${\lambda }_0=780\mathrm{nm}$ in the vicinity of a dielectric nanosphere ($n_{\mathrm{c}}=3.5$) of radius $a=180\mathrm{nm}$ coated with an Ag nanoshell of radius $b=200\mathrm{nm}$ for a fixed interatomic distance $r_{12}=800\mathrm{nm}$. The emitters are equally distant from the center of the sphere, where $(0,0,z)$ is the position of the midpoint of the line connecting the emitters. ${\gamma }_1^{\left(0\right)}={\gamma }_2^{\left(0\right)}$ is the single-emitter decay rate in free space. The polarization of the incident electric field ${\mathbf{E}}_{\mathrm{in}}$ is along $x$ direction. (a) Comparison between the single-emitter decay rate ${\gamma }_1={\gamma }_2$ (with and without the cross-damping decay rate ${\gamma }_{12}$ contribution) in the vicinity of a coated sphere for two polarizations: ${\mathbf{r}}_{12}$ is parallel $\left(\right|\left|\right)$ or orthogonal ($\perp$) to ${\mathbf{E}}_{\mathrm{in}}$. (b) The cross-damping decay rate ${\gamma }_{12}$. (c) The dipole-dipole interaction ${\delta }_{12}$. (d) The normalized intensity-intensity correlation function $g^{\left(2\right)}(\tau =0)$ for detectors in $z$ direction. The dash-dotted line corresponds to $g^{\left(2\right)}\left(0\right)$ for independent emitters (${\gamma }_{12}=0={\delta }_{12}$) in vacuum.

Figure 5

Same quantities as in Fig. 4 but now with a gain-assisted core ($n_{\mathrm{c}}=3.5-ı0.015$). (a) Total decay rates for emitters when ${\mathbf{r}}_{12}$ is orthogonal ($\perp$) or parallel ($\left|\right|$) to the incident electric field ${\mathbf{E}}_{\mathrm{in}}\left|\right|{\mathbf{e}}_p$ as a function of $z$ (the position of the midpoint of the line connecting the two emitters; see inset). (b) The cross-damping decay rate ${\gamma }_{12}$. (c) The dipole-dipole interaction ${\delta }_{12}$. (d) The normalized intensity-intensity correlation function $g^{\left(2\right)}(\tau =0)$ for detectors in the $z$ axis. The inset shows the possibility of tuning $g^{\left(2\right)}\left(0\right)$ by changing the polarization of the laser beam.

Figure 6

Two-time second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ for two point-dipole emitters (${\lambda }_0=780\mathrm{nm}$) near a core-shell nanosphere with the same parameters as in Fig. 5: a doped AlGaAs nanosphere ($a=180\mathrm{nm}$) coated with an Ag layer ($b=200\mathrm{nm}$). The emitters are in equivalent position in relation to the sphere and the midpoint of the interatomic distance $r_{12}=800\mathrm{nm}$ is fixed at $z=-3.04b$. We consider the detectors in $z$ direction. The plots show the cases of a sphere with and without gain: two emitters in vacuum, two independent emitters in vacuum (${\gamma }_{12}={\delta }_{12}=0$), single emitter near a sphere with gain $(\kappa =0.015)$, and single emitter in vacuum. $\phi$ is the angle between ${\mathbf{r}}_{12}$ and ${\mathbf{E}}_{\mathrm{in}}\left|\right|{\mathbf{e}}_p$. (a) $g^{\left(2\right)}\left(\tau \right)$ for ${\mathbf{r}}_{12}\left|\right|{\mathbf{e}}_p$ ($\phi =0^{\mathrm{o}}$). (b) $g^{\left(2\right)}\left(\tau \right)$ for ${\mathbf{r}}_{12}\perp {\mathbf{e}}_p$ ($\phi ={90}^{\mathrm{o}}$). (c) $g^{\left(2\right)}\left(\tau \right)$ for various angles $\phi$ when the gain coefficient is $\kappa =0.015$. (d) $g^{\left(2\right)}\left(0\right)$ as a function of $\phi$ for different values of the gain coefficient $\kappa$.

Figure 7

One-time second-order correlation function $g^{\left(2\right)}\left(0\right)$ associated with two emitters in the vicinity of a gain-assisted sphere ($n_{\mathrm{c}}=3.5-ı\kappa$) of radius $a=180\mathrm{nm}$ coated with a silver shell of radius $b=200\mathrm{nm}$ as a function of the interatomic distance $r_{12}$. The midpoint of the distance $r_{12}$ is fixed at $z=-3.04b$. The plots (a) and (b) show two different laser polarizations: ${\mathbf{r}}_{12}\perp {\mathbf{e}}_p$ and ${\mathbf{r}}_{12}\left|\right|{\mathbf{e}}_p$, respectively, where ${\mathbf{e}}_p$ is the unity vector along the direction of the incident electric field. The photon-bunching effect occurs for $r_{12}\approx 4b$ and $\kappa =0.015$ in (a) and can be switched to a strong antibunching by changing the polarization to (b).