Few-Mode Field Quantization of Arbitrary Electromagnetic Spectral Densities

We develop a framework that provides a few-mode master equation description of the interaction between a single quantum emitter and an arbitrary electromagnetic environment. The field quantization requires only the fitting of the spectral density, obtained through classical electromagnetic simulations, to a model system involving a small number of lossy and interacting modes. We illustrate the power and validity of our approach by describing the population and electric field spatial dynamics in the spontaneous decay of an emitter placed in a complex hybrid plasmonic-photonic structure.

Figure 1

Sketch of the model and quantization approach. Fitting an original spectral density, $J(\omega )$ (usually obtained by numerical simulations) to a few-mode model $J_{\text{mod}}(\omega )$, Eq. (5), provides quantized EM modes described by the master equation Eq. (4).

Figure 2

(a) Sketch of the model system consisting of a silver dimer nanoantenna embedded in a dielectric microsphere. (b) Purcell factor $J(\omega )/J_0(\omega )$ for the system (thick black line), for the fitted model described by Eq. (5) with 20 modes (orange line), and for a model without interactions with the same number of modes (dashed light blue line). Thin gray lines indicate the energy positions of the eigenstates of $\tilde{\mathbf{H}}$ for the fitted interacting system. The red dotted line indicates the frequency of the emitter used in Fig. 3 and Fig. 4. The insets show the electric field distributions for the two modes indicated by the arrows (see main text for details).

Figure 3

Emitter and mode dynamics for the spontaneous emission problem. Solid lines show the emitter excited-state population $⟨{\sigma }^{+}{\sigma }^{-}⟩(t)$ for the exact calculation (black) and fitted models with (orange) and without (light blue) interactions. Dashed lines indicate the populations $⟨{\tilde{a}}_{\alpha }^{†}{\tilde{a}}_{\alpha }⟩(t)$ of modes $\alpha =6$ (blue) and $\alpha =7$ (red) (see text for details), and the sum over all other mode populations (magenta).

Figure 4

Electric field intensity $⟨{\overrightarrow{E}}^{(-)}(\overrightarrow{r})·{\overrightarrow{E}}^{(+)}(\overrightarrow{r})⟩$ for the initially excited emitter as in Fig. 3. (a) Field intensity at four points, shown by numbered white circles in panel (c), normalized to their maximum values $I_{\max }$, given by $2.0\times {10}^8\mathrm{W}/{\mathrm{m}}^2$, $1.8\times {10}^6\mathrm{W}/{\mathrm{m}}^2$, $1.1\times {10}^6\mathrm{W}/{\mathrm{m}}^2$, and $4.0\times {10}^3\mathrm{W}/{\mathrm{m}}^2$, respectively. (b) Field intensity distribution in the $x-z$ plane at time $t=10\mathrm{fs}$, and (c) at $t=100\mathrm{fs}$.