Collective light-matter interaction in plasmonic waveguide quantum electrodynamics

Abstract

Rabi oscillations characterize light-matter hybridization in the waveguide quantum electrodynamics~(WQED) framework, with their associated decay rates reflecting excitation damping, yet their behavior remains unresolved when collective emitters are coupled to a collective waveguide mode. This scenario reveals a conceptually novel collective-light-collective-matter interaction, realizable when a timed-Dicke state~(TDS) of subwavelength emitters couples to a slow, delocalized surface-plasmon mode, forming a hybridized plasmon-polariton~(HPP). The HPP acquires its directionality from the TDS via momentum matching. It also exhibits plasmonic characteristics, with excitation frequencies following the surface-plasmon dispersion relation. We obtain a Rabi oscillation and a long-time decay that describe the HPP and use them to reveal weak- and strong-coupling regimes through the emergence of normal-mode splitting. By performing a finite-time Lyapunov-exponent analysis, we show that the HPP also exhibits instantaneous decay and identify three distinct decay regimes: early-time rapid, transient-time oscillatory, and long-time classical. Finally, by analyzing the emission spectrum, we observe an anticrossing of the peak doublets~(a feature also seen in cavity QED setups) which originates from quantum vacuum effects and the resulting non-Markovian HPP evolution in our WQED.

Figures (4)

• Figure 1: Interaction between TDS and surface-plasmon field: An ensemble of $N$ QEs, equidistantly spaced by $d$ and prepared in a TDS, have ground ($\ket{g}$) and excited ($\ket{\Psi_\text{D}}:=\ket{\Psi_\text{D}(N|k)}$) states, with frequency separation $\omega_\text{eg}$. The TDS is prepared by an external laser $\bm{k}$ and is situated on top of a metallic layer with optical properties $\varepsilon_\text{m}(\bm{r}',\tilde{\omega})$ and $\mu_\text{m}(\bm{r}',\tilde{\omega})=1$, where $\bm{r}'$ includes the lower half-plane $z<0$. Interaction between the TDS and the surface plasmon evolves as the HPP in the direction $\bm{k}$ at frequency $\omega_\text{eg}$. The surface plasmon has a group velocity $v_\text{g}$, excitation frequency $\omega_\text{spp}:= \Re[\tilde{\omega}_\text{spp}]$, and is delocalized enough to cover the QE ensemble. The top-left inset shows the top view of this interaction, while the top-right inset shows the energy diagrams of the TDS and surface plasmon.
• Figure 2: Panel (a) shows the temporal evolution of $\alpha_\text{D}(t)$ governed by Eq. \ref{['Eq:Dynamic']}, corresponding to different values of effective frequency $\Omega_\text{s}/\gamma_\text{spp}$. Panel (b) is the HPP evolution in the phase space $(\alpha_\text{D}(t),\partial_t\alpha_\text{D}(t))$; the circulation signifies the non-Markovian SPP evolution for different $\Omega_\text{s}/\gamma_\text{spp}$. We have used $v_\text{g}=0.1\mathcal{L}\gamma_\text{spp}$, $\mathcal{L}\approx100~\mu$m and defined $\tau_0=\gamma_\text{spp}^{-1}$. See the text for other parameter values.
• Figure 3: HPP's spectral evolution. (a) $|\alpha_\text{D}(\omega_\text{p})|$ ($\omega_\text{p}$ denoting the Fourier component) obtained via Fourier transformation of Eq. \ref{['Eq:Dynamic']} for different values of $\Omega_\text{s}$. The transition from single- to double-peaks due to normal-mode splitting is observed. (b) Variation of $\alpha_{\text{max},\text{D}}$ in the $\Omega_\text{s}-\omega_\text{p}$ plane shows a superlinear dependence of $\Omega_\text{s}$ on $\omega_\text{p}$ for $\Omega_\text{s}>\gamma_\text{spp}$, i.e., in the strong-coupling regime. (c) Evolution of $|\alpha_{\text{max},\text{D}}|$ for different $\Omega_\text{s}$; $\alpha_{\text{max},\text{D}}$ decays exponentially at lower $\Omega_\text{s}$ but exhibits oscillations for higher values. (d) HPP's collective oscillation $\omega_\text{s}$ and long-time decay $\Gamma_\text{s}$ for various $\Omega_\text{s}$. Blue circles and squares denote $\tilde{\omega}\text{s}$ for $\Omega\text{s}=0.5\gamma_\text{spp}$, red for $\Omega_\text{s}=1\gamma_\text{spp}$, black for $\Omega_\text{s}=2\gamma_\text{spp}$, and violet for $\Omega_\text{s}=4\gamma_\text{spp}$. Parameters are the same as in Fig. \ref{['fig:Temporal_Dynamics']}.
• Figure 4: (a) Dynamics of Lyapunov exponent $\lambda$ for various ranges of effective Rabi frequency, showing stable three decay regimes: early-time fast, transient-time oscillatory, and long-time classic. (b) Displays the emission spectrum for different detunings $\Delta:=\omega_\text{eg}-\omega_\text{spp}$ and Fourier frequency components $\omega_\text{p}$ for $\Omega_\text{s}=4\gamma_\text{spp}$. We characterize the evolution of the positive branch ($\tilde{\omega}_+$, dashed white curve) and negative branch ($\tilde{\omega}_-$, dashed red curve), and obtain with peak splitting up to $\Omega=200$ meV. Parameters are the same as Fig. \ref{['fig:Temporal_Dynamics']}.