Quantum theory of surface lattice resonances

Abstract

The collective interactions of nanoparticles arranged in periodic structures give rise to high-$Q$ in-plane diffractive modes known as surface lattice resonances. While these resonances and their broader implications have been extensively studied within the framework of classical electrodynamics and linear response theory, a quantum optical theory capable of describing the dynamics of these structures, especially in the presence of material nonlinearities beyond \textit{ad hoc} few-mode approximations, is largely missing. To this end, we consider a lattice of metallic nanoparticles coupled to the electromagnetic field and derive the quantum input--output relations within the electric dipole approximation. As applications, we analyze coupling between the nanoparticle array and external quantum emitters, and show how the formalism extends to molecular optomechanics, where the high $Q$-factors of SLRs enable coupling to collective vibrational modes. We further consider arrays composed of saturable excitonic emitters, demonstrating how emitter nonlinearities can be used to switch the SLR condition between electronic transitions. Using a perturbative approach that accounts for population dynamics, we show how these effects can be probed in pump--probe experiments and give rise to nonlinear phase-matching phenomena. Our work provides a microscopic framework for modeling SLRs interacting with quantum emitters without phenomenological descriptions of the electromagnetic environment.

Figures (9)

• Figure 1: (a) 1D array of MNPs with lattice spacing $a$, modeled as quantum harmonic oscillators with annihilation operator $\hat{A}_j$, which are excited by an electric input field $\hat{\mathbf E}_\mathbf{k}^\mathrm{in}(\mathbf{r} , t)$ incident at a wavevector $\mathbf{k}$. In-plane diffractive modes at wavevectors $\mathbf{k}_\mathrm{RA}$ lead to the formation of SLRs (red shaded areas indicate regions of field enhancement). (b) Extinction spectrum for an array of $M=8\cdot 10^3$ MNPs with a resonance wavelength of $\lambda_0=500\:\mathrm{nm}$, a lattice spacing of $a=550\:\mathrm{nm}$, and a single-particle radiative linewidth of $\Gamma_0^\mathrm{rad}=0.5\:\mathrm{eV}$. The dashed white lines show the dispersion of the Rayleigh anomalies (RAs) corresponding to the first diffractive order ($m=1$). The lower panel shows a cross section through the extinction profile at $|\mathbf{k}_\parallel|=0$ as indicated by the vertical green dashed line in the upper panel. The peak corresponds to the SLR while the dip arises from the RA. Here, the polarization of the incoming light field and the orientation of the dipoles are chosen orthogonal to the direction of the chain, i.e., $\mathbf{E}_0\parallel\boldsymbol\mu _0$, $\boldsymbol\mu_0 \perp \mathbf{r}_\Lambda$.
• Figure 2: Intensity profile $|\mathbf E (\mathbf r)|^2$ radiated by a 1D chain of MNPs (indicated as silver dots) with a resonance wavelength of $\lambda_0=500\:\mathrm{nm}$ and a radiative linewidth of $\Gamma_0^\mathrm{rad}=0.5\:\mathrm{eV}$, for a lattice spacing of (a) $a=300\:\mathrm{nm}$, and (b) $a=550\:\mathrm{nm}$. The field distribution is plotted in the $xz$-plane and we have assumed normally-incident illumination of the array ($\theta_\mathrm{inc}=0$) with a polarization vector along the $y$ direction, i.e., $\mathbf{E}_0 \parallel \boldsymbol\mu_0\parallel \mathbf{e}_y$. In (a), the electric field is evaluated at the frequency corresponding to the original LSP resonance ($\omega_0=2\pi c/\lambda_0$), while in (b) it is evaluated at the red-shifted SLR frequency.
• Figure 3: Molecular optomechanics with SLRs. (a) Sketch of array of molecular dipoles, illustrated as diatomic oscillators, characterized by their induced Raman dipoles $\hat{\mathbf{p}}_R^j$, interacting with the electric field produced by a set of MNPs, characterized by dipoles $\hat{\boldsymbol{\mu}}_j$. (b) Sketch of relevant interacting modes as well as processes leading to the enhancement of Stokes- and anti-Stokes scattering by the SLR mode. (c) and (d) show extinction spectra of the nanoparticle array in the red- and blue-detuned regime (regarding the laser frequency as compared to the SLR frequency at $k_\parallel = 0$). We assumed a nanoparticle linewidth of $\Gamma_0^\mathrm{rad}=0.5\,\mathrm{eV}$, a (collective) induced Raman dipole moment of $p=0.3\mu_0$, and have set $\Gamma_\mathrm{vib}=0$.
• Figure 4: Comparison of cavity (left column) and Raman dipole extinction spectrum (right column) for (a), (b) $\Gamma_\mathrm{vib}>\gamma_\mathrm{p}^\mathrm{rad}$ and (c), (d) $\Gamma_\mathrm{vib}<\gamma_\mathrm{p}^\mathrm{rad}$ in the red- and blue-detuned regimes. The curves are normalized to the bare (grey) extinction spectra, obained in the absence of optomechanical interaction. We have set $\omega_0=\omega_\ell\pm\omega_\mathrm{vib}=1\,\mathrm{eV}$ and $\Gamma_0^\mathrm{rad}=0.1\,\mathrm{eV}$. In (a) and (b), we have set $\Gamma_\mathrm{vib}=10\gamma_\mathrm{p}^\mathrm{rad}=0.1\Gamma^0_\mathrm{rad}$, while in (c) and (d) these values are reversed.
• Figure 5: (a) Nonlinear SLR with array of saturable emitters: Pumping excitation into the $\ket{2}$ state shuts off the polarizability of the $\ket{1}\to\ket{2}$ transition while opening up the dipole transition from $\ket{2}\to\ket{3}$ whose frequency matches the first diffractive mode of the lattice. (b) Extinction spectrum of an array of $M=10^3$ three-level systems if all of the population is in the ground state (top) and if half of the population is pumped to the $\ket{2}$ state (bottom). The lattice spacing is $a=415\:\mathrm{nm}$. We chose the transition frequencies $\hbar\omega_{12}\approx 1.5\:\mathrm{eV},\, \hbar\omega_{13}\approx 3.0\:\mathrm{eV}$, as well as the radiative linewidths $\Gamma_{12}^\mathrm{rad}=\Gamma_{23}^{\mathrm{rad}}=0.25\:\mathrm{eV}$. The $\omega_{12}$ and $\omega_{23}$ transitions corresponds to a resonance wavelengths of $\lambda_{12}=800\:\mathrm{nm}$ and $\lambda_{23}=400\:\mathrm{nm}$, respectively. The dotted white lines show the frequencies corresponding to the Rayleigh anomalies. The vertical dashed green lines mark cross sections through the extinction spectrum at $k_\parallel =0$ which are shown in (c).