A metal nanoparticle's localized surface plasmon resonance and a molecular (or J-aggregate) exciton are both damped oscillators. When the energy exchanged between them per cycle beats the rate at which either loses energy, they no longer exist as separate excitations — they hybridize into two new eigenmodes, polaritons, with a characteristic anticrossing in energy.
The system is modeled as two coupled damped harmonic oscillators (a non-Hermitian 2×2 Hamiltonian in the basis {plasmon, exciton}):
H = [ Epl − i·γpl/2 g ]
[ g Eex − i·γex/2 ]
E± = (Epl+Eex)/2 − i(γpl+γex)/4
± sqrt[ g² + ( (Epl−Eex)/2 − i(γpl−γex)/4 )² ]
Re(E±) gives the upper/lower polariton energies plotted as the anticrossing curve; −2·Im(E±) gives each branch's linewidth. At zero detuning the observable mode splitting is the vacuum/classical Rabi splitting, ħΩ_R = 2g (when g exceeds the dissipative mismatch |γpl−γex|/4 — the strong-coupling criterion shown live above).
Each eigenmode is a superposition of the bare plasmon and exciton states, |LP⟩ = Cpl|plasmon⟩ + Cex|exciton⟩, with |Cpl|² + |Cex|² = 1 — the Hopfield coefficients shown for the lower branch. Near zero detuning both fractions approach 50%, meaning the polariton genuinely is "half light, half matter": it inherits the plasmon's sub-wavelength field confinement and the exciton's material nonlinearity, which is what makes plasmon–exciton strong coupling useful for room-temperature polariton lasing, single-photon nonlinearities and chemistry under vibrational strong coupling.
- Coupling g — the plasmon–exciton interaction energy; raising it widens the anticrossing gap.
- Detuning Δ — the bare energy mismatch; sweeping it traces the characteristic avoided-crossing "X" shape.
- γpl, γex — dissipation (radiative + non-radiative decay) of each bare mode; larger values wash out the splitting and can push the system into weak coupling even at fixed g.
- Spectrum panel — the extinction lineshape, modeled as two Lorentzians centered at Re(E±) with widths from Im(E±) and weights from the Hopfield fractions, exactly as measured in a dark-field scattering or extinction experiment.