The strong coupling between plasmons and excitons represents a fundamental quantum electrodynamics (QED) phenomenon where light-matter interactions lead to hybridized states with unique properties. This regime occurs when the coupling strength exceeds the dissipation rates of the individual systems, resulting in coherent energy exchange and the formation of new quasiparticles called polaritons. Theoretical frameworks for describing this coupling draw from quantum optics, condensed matter physics, and nanophotonics, with key concepts including Rabi splitting, the Jaynes-Cummings model, and polariton dispersion relations.

Plasmons are collective oscillations of free electrons in metallic nanostructures, while excitons are bound electron-hole pairs in semiconductors or molecules. When these systems interact strongly, their individual excitations hybridize, leading to an anti-crossing behavior in the energy spectrum. The energy separation at resonance is termed the vacuum Rabi splitting, a direct measure of the coupling strength. For a two-level system coupled to a single plasmon mode, the splitting energy ℏΩ is given by ℏΩ = 2g, where g is the coupling rate. In the case of N emitters, the collective enhancement leads to ℏΩ = 2g√N, demonstrating the superradiant nature of the coupling.

The Jaynes-Cummings model provides a foundational QED framework for describing the plasmon-exciton system. The Hamiltonian is written as H = ℏω_a σ⁺σ⁻ + ℏω_c a⁺a + ℏg(a⁺σ⁻ + aσ⁺), where ω_a and ω_c are the exciton and plasmon frequencies, a⁺ and a are the plasmon creation and annihilation operators, and σ⁺ and σ⁻ are the excitonic raising and lowering operators. The rotating wave approximation is typically applied, neglecting counter-rotating terms. Diagonalization of this Hamiltonian yields the polariton eigenstates with energies E_± = ℏ(ω_a + ω_c)/2 ± ℏ√(Δ² + 4g²)/2, where Δ = ω_a - ω_c is the detuning. At zero detuning, the eigenstates are maximally hybridized with an energy splitting of 2ℏg.

For extended systems with continuous dispersion, such as plasmonic films or waveguides coupled to excitonic materials, the Hopfield transformation is employed to describe the polariton branches. The lower and upper polariton energies follow the dispersion relations ω_LP,UP(k) = [ω_pl(k) + ω_ex] / 2 ± √[ (ω_pl(k) - ω_ex)² / 4 + g(k)² ], where ω_pl(k) and ω_ex are the plasmon and exciton dispersions, and g(k) is the wavevector-dependent coupling strength. The mixing angle θ_k quantifies the excitonic/plasmonic character of the polaritons, with tan(2θ_k) = 2g(k) / (ω_pl(k) - ω_ex).

In the strong coupling regime, the system dynamics are governed by coherent Rabi oscillations rather than irreversible energy transfer. The temporal evolution shows periodic exchange between plasmon and exciton populations with frequency Ω_R = 2g/ℏ. For quantum dots or molecules coupled to plasmonic nanocavities, the Purcell effect enhances the spontaneous emission rate by a factor F = 3Q(λ³)/4π²V, where Q is the quality factor and V is the mode volume. However, in strong coupling, this picture modifies due to the formation of new eigenstates.

Theoretical treatments must account for loss mechanisms through open quantum system approaches. The dissipative dynamics are often modeled using the Lindblad master equation dρ/dt = -i/ℏ[H,ρ] + ∑_i γ_i (L_i ρ L_i⁺ - {L_i⁺L_i,ρ}/2), where ρ is the density matrix, γ_i are decay rates, and L_i are jump operators for plasmon decay, exciton recombination, and dephasing. The strong coupling criterion requires g > (γ_pl + γ_ex)/4, where γ_pl and γ_ex are the plasmon and exciton linewidths.

For systems beyond the single-mode approximation, such as plasmonic nanoparticles supporting multiple resonant modes, the Tavis-Cummings model generalizes the Hamiltonian to include multiple excitons coupled to multiple plasmon modes. The interaction terms become ℏ∑_{i,j} g_{ij}(a_j⁺σ_i⁻ + a_jσ_i⁺), where g_{ij} describes the coupling between the i-th exciton and j-th plasmon mode. This leads to more complex polaritonic spectra with multiple anticrossings.

First-principles calculations based on density functional theory (DFT) and finite-difference time-domain (FDTD) methods can provide microscopic insights into the coupling mechanisms. The coupling strength g is determined by the overlap integral g ∝ ∫ d³r E_pl(r) · μ_ex ψ_ex(r), where E_pl is the plasmon field, μ_ex is the excitonic transition dipole, and ψ_ex is the exciton wavefunction. For spherical quantum dots near a metal nanoparticle, analytical expressions for g can be derived using quasistatic approximations, showing dependence on nanoparticle size, quantum dot position, and dielectric environment.

Nonlinear effects become significant at high excitation densities, where polariton-polariton interactions lead to phenomena such as bistability and parametric amplification. The interaction strength is quantified by the nonlinear coefficient U, which for exciton-polaritons scales with the exciton binding energy and for plasmon-polaritons depends on the Kerr nonlinearity of the metal. Theoretical descriptions incorporate these effects through additional terms in the Hamiltonian like ℏU a⁺a⁺aa or ℏU σ⁺σ⁺σ⁻σ⁻.

Recent theoretical advances have explored strong coupling in complex nanostructures, including hyperbolic metamaterials, where the plasmon density of states is greatly enhanced, and in topological photonic systems, where polaritons inherit robust edge states. Numerical methods such as boundary element simulations and finite-element Maxwell-Bloch equations enable precise modeling of these systems while maintaining quantum coherence effects.

Theoretical understanding of plasmon-exciton strong coupling continues to evolve with developments in quantum optics and nanophotonics. Challenges remain in accurately describing systems with strong losses, many-body interactions, and spatial inhomogeneities. Advanced techniques combining QED with macroscopic Maxwell equations or non-Markovian open quantum systems are being developed to address these complexities while maintaining predictive power for emerging nanoscale light-matter systems.