Quantum open system description of a hybrid plasmonic cavity

TL;DR

This work develops a self-consistent quantum open-system description for lossy plasmonic cavities that support hybrid EC–SPP polaritons. By combining Hopfield diagonalization with a Dyson-self-energy dressed photon propagator, it obtains complex eigenfrequencies that encode both renormalization and damping through a Drude–Lorentz susceptibility. Tracing out the environments yields a GKSL master equation with leakage, UP–LP relaxation, and dephasing, and the authors derive exact population/coherence dynamics alongside a linear mean-field treatment that includes coherent and Raman-like drives. The framework provides analytic expressions for steady-state polariton populations, the UP–LP quench rate, and linewidths, enabling straightforward response-spectrum and time-domain analyses and offering a route to dissipation engineering in nanophotonic systems.

Abstract

We present a unified quantum open system framework for lossy plasmonic cavities in which coherent dynamics, relaxation, dephasing, and irreversible absorption are treated on equal footing. The Dyson equation for the cavity photon propagator in the random-phase approximation yields a complex self-energy S that accounts for both the renormalization and the damping of hybrid plasmon-photon modes (polaritons, in a quasi-particle description). Tracing out the electronic and photonic environments leads to a Liouvillian for the upper (UP) and lower (LP) polaritonic branches, incorporating leakage through the imaginary part of the self-energy, internal UP-LP scattering rates, and dephasing. Time evolution equations for polariton populations, interbranch coherence, and driven amplitudes in closed form also provide analytic expressions for their steady-state values, the quench rate of UP-LP oscillations and polaritonic lineshapes, valid in the limit of low polaritonic density, but covering light-matter ultrastrong coupling. The theory establishes a self-consistent description of dissipative polariton dynamics in plasmonic and nanophotonic cavities, directly applicable to response spectra, time-domain measurements, and dissipation engineering.

Figures (4)

• Figure 1: (a) Sketch of a resonant plasmonic cavity illuminated at normal incidence. A lattice of gold nanoparticles on the illuminated face excites an SPP mode with frequency $\omega_\mathrm{pl}$, set by the lattice period, that propagates along the dielectric/reflector (assumed gold) interface, while the cavity supports an EC mode at $\omega_\mathrm{c}$. (b) Schematic representation of EC--SPP hybridization and (c) dispersion of eigenfrequencies $\omega_\pm$ for $\omega_\mathrm{c} = 1.5$ eV, plasma frequency $\Omega_\mathrm{pl} \approx 8.5$ eV, adimensional EC-SPP coupling constant $\eta = 0.02$, and zero losses, \ref{['eq:H_QED_total', 'eq:polaritonic_Hamiltonian', 'eq:QED_eigenvalues']}.
• Figure 2: (a) Diagrammatic view of the Dyson equation for $D$, with amplitude damping represented as the imaginary part of the self-energy. The free photon propagator $D_0$ is dressed by the polarization bubble $\Pi(\omega)$. The vertical dashed cut through the bubble represents on-pole intermediate states and yields $2\,\mathrm{Im}\,\mathcal{S}(\omega)$ (optical theorem 1995Peskin), i.e., physical loss channels (absorption in the metal). (b) EC–SPP hybridization in the strong-coupling regime with losses, which reduce (and possibly set to zero) the Rabi frequency splitting.
• Figure 3: (a) The eigenvalues $\omega_\pm$ as function of the normalized coupling strength $g/\omega_\mathrm{c}$. The inset shows that, in presence of losses, the polaritonic splitting does not exist for very low values of $g/\omega_\mathrm{c}$. (b) The normalized square polaritonic energy separation as function of the loss rate $\gamma_\mathrm{D}$ normalized to the value for gold, for three values of $g$. (c) The loss rate $\gamma_\mathrm{D,\,EP}$ that defines the exceptional point, above which the polaritonic splitting cannot exist, plotted against normalized coupling strength.
• Figure 4: (a) Time evolution of the polaritonic population densities $n_\pm(t) / V_\mathrm{eff}$ for the plasmonic cavity with a gold reflector (overdamped regime, $\Gamma = \gamma_\mathrm{D,\, Au} \approx 25$ meV, lifetime $\approx 0.026$ ps). Because $\Gamma \gg \Delta$, turning on the UP--LP coherent coupling at $t=500$ ps with zero detuning ($\delta=0$) does not generate oscillatory exchange between the branches: each mode behaves as an independently driven, strongly damped oscillator. (b) Same as in (a), but for a low-loss cavity ($\Gamma = 0.25V$ meV, lifetime $\approx 2.6$ ps), for which $\Delta \gg \Gamma$ and long-lived relaxation oscillations appear with period $P \approx \pi/\Delta \simeq 0.4$ ps. Panels (c) and (d) show two-dimensional color maps of $n_{+}(t) / V_\mathrm{eff}$ and $n_{-}(t) / V_\mathrm{eff}$ as functions of time and detuning $\delta$, displaying the characteristic Lorentzian profile predicted by \ref{['eq:n_ss']}. For all simulations, $\Delta = 5$ meV ($\approx 7.6$ ps$^{-1}$) is turned on at $t = 500$ ps, UP$\rightarrow$LP relaxation rate $\gamma_{\downarrow} = 1$ ps$^{-1}$, $T = 300$ K, and $\omega_{+} - \omega_{-} = 0.1$ eV (yielding $\gamma_{\uparrow} = 0.019$ ps$^{-1}$). The population density has been evaluated for a $30 \times 30\,\mu$m$^2$ and $1\,\mu$m-thick cavity (effective volume $V_\mathrm{eff} = 9 \times 10^{-10}$ cm$^{-3}$, just as an example).