Unified Light-Matter Coupling

• Unified light-matter coupling is a framework that integrates diverse regimes—from weak to deep-strong—using quantum models like the Rabi and Hopfield Hamiltonians.
• It employs methods such as Green’s functions, nonperturbative transformations, and ab initio QEDFT to predict polariton formation and emergent optical nonlinearities.
• The framework guides the design of quantum devices by analyzing photon-matter hybridization in various platforms, including nanocavities, bulk solids, and free-space systems.

Unified description of light-matter coupling encapsulates the diverse physical regimes and platforms—ranging from molecules in microcavities, nanophotonic resonators, crystalline solids, to bulk materials—where the quantum states of light (photons) and matter (electrons, phonons, excitons, etc.) become hybridized. This hybridization manifests as polariton formation, emergent optical nonlinearities, and profound modifications of material properties, and it is described by theoretical frameworks that unify weak, strong, ultrastrong, and deep-strong coupling in a single formalism. Core approaches include quantized Hamiltonians (Rabi, Hopfield), gauge-invariant minimal coupling, Green’s function methods, nonperturbative transformations, and generalized density-functional theories. The unified description provides the foundation for both analyzing fundamental phenomena and designing quantum devices exploiting light-matter hybridization.

1. Universal Model Hamiltonians and Their Regimes

The quantum Rabi model forms the foundation of light-matter coupling, encapsulating the essential interaction between a cavity photon mode (annihilation operator $a$) and a two-level system (Pauli operators $\sigma_x,\sigma_z$) with full consideration of counter-rotating terms: $$H_{\mathrm{Rabi}} = \frac{1}{2}\omega_0\,\sigma_z + \omega_c\,a^\dagger a + g\,\sigma_x(a + a^\dagger) + \frac{g^2}{\omega_0}$$ where $g$ is the coupling strength and $g^2/\omega_0$ the $A^2$ diamagnetic term (Yu et al., 2022). This model transitions through the well-defined regimes:

• Weak coupling: $g \ll \gamma$ (decay), simple Lorentzian response.
• Strong coupling: $g > \gamma$, manifests as vacuum Rabi splitting; polaritons form.
• Ultrastrong coupling: $g/\omega_0 \gtrsim 0.1$; counter-rotating terms reshape the spectrum, yielding nonperturbative ground states, virtual excitations, Bloch–Siegert shifts, and squeezing (Forn-Díaz et al., 2018, Yu et al., 2022, Mueller et al., 9 May 2025).
• Deep-strong coupling: $g > \omega_0$; the system ground state and excitation spectrum are dramatically altered.

For multi-emitter or extended systems, generalizations include:

• Tavis–Cummings and Dicke models: $N$ two-level systems, collective bright/dark states, superradiant transitions.
• Hopfield bosonic models: bosonic matter modes $b_k$, photon-matter, matter-matter, and photon-photon interactions (Mueller et al., 9 May 2025, Kolaric et al., 2018).

The continuum limit and crystalline environments further require Peierls-substitution approaches and inclusion of band structure, Berry connection, and quantum metric contributions.

2. Linear Response, Polarizability, and Quantum Geometry

Unified frameworks rely on the frequency-dependent polarizability $\alpha(\omega)$, accessible via Kubo–Green formalism: $$\alpha(\omega)=\frac{d^2}{\hbar}\sum_{n>0}|⟨n|σ_x|0⟩|^2\left[ -\frac{1}{\omega - \omega_{n0} - iΓ_n} + \frac{1}{\omega + \omega_{n0} - iΓ_n} \right]$$ where the sum is over Rabi eigenstates, and $Γ_n$ denotes transition linewidths (Yu et al., 2022).

• Weak coupling ($g \rightarrow 0$): Lorentz oscillator, single resonance.
• Deep strong coupling: $\omega_{10} \rightarrow 0$, all transitions decouple at $\omega > 0$, rendering the system effectively transparent.

In crystalline materials, minimal coupling via Peierls’ substitution generates both paramagnetic (linear, dipole) and diamagnetic (quadratic, $A^2$) terms. The geometric nature of light-matter coupling appears most clearly in flat-band and moiré crystals, where standard band velocities vanish but the quantum metric $g^n_{\mu\nu}(k)$ and Berry connection $A^\mu_{nm}(k)$ mediate intra- and inter-band couplings, respectively (Topp et al., 2021). This leads to nonzero diamagnetic response even in perfectly flat bands and enables Floquet-topological phase engineering in moiré systems.

3. Unification Across Structures: Cavity, Free-Space, and Bulk

A central result is the seamless unification of cavity QED, nanophotonic, and bulk solid-state regimes:

• Cavity/Nanophotonic Resonators: The Hopfield or Jaynes–Cummings model (inc. Lindblad dissipation) quantifies Rabi splitting, Purcell enhancement, quantum nonlinear optics, and collective effects in molecular ensembles, microcavities, and metasurfaces (Kolaric et al., 2018, Yu et al., 2022, Jeannin et al., 2020).
• Free-Space Coupling: Input–output theory generalizes to continuous photon baths, yielding scattering matrices, nonperturbative emission, and enabling extension of the strong/ultrastrong regime outside confined geometries. The same unified Hamiltonian governs both discrete and continuous photonic environments, with key distinctions arising only in the spectral distribution and boundary conditions (Huppert et al., 2016).
• Bulk Materials: The generalized Hopfield lattice formalism treats each unit cell as a quantum dipole coupled to light, including full matter-matter (dipole–dipole), light–matter, and photon–photon ($A^2$) terms (Mueller et al., 9 May 2025). Diagonalization yields polariton bands throughout the Brillouin zone. The crossover to ultrastrong coupling is natural in this picture, and criticality (e.g., soft-mode–driven phase transitions to ferroelectric, insulator-metal, or excitonic insulator states) is described as a bulk polaritonic instability.

The table below summarizes typical GW Hamiltonians, regimes, and their applicability:

Hamiltonian	Regime	Typical Systems
Jaynes–Cummings/Rabi	Cavity QED, molecules	Microcavities, quantum dots
Hopfield (bosonic)	Bulk solids, polaritons	Phonon, exciton, plasmon
Peierls-minimal-coupling	Crystallines, moiré	TBG, flat-band 1D chains
Input–output (cont.)	Free-space, open systems	Plasmonic emitters, 2DEGs

4. Multimode and Open-Resonator Frameworks

Modern platforms often exhibit multi-mode photonic environments and significant radiative and non-radiative loss. Unified multimode Hopfield diagonalization and resonant-state/pseudomode techniques treat realistic, leaky, dispersive systems:

• Multi-Mode Hopfield Approach: Coupling between $M$ photonic modes $a_n$ and matter $b$ yields rich polaritonic spectra. Overlap matrices $\eta_{nm}$ quantify field spatial structure and mixing, leading to S-shaped anticrossings and tunable near-field distributions in nanostructured THz, plasmonic, or dielectric resonators (Cortese et al., 2022).
• Resonant-State/Eigenmode Formulation: Open, lossy cavities and nanoparticles are rigorously handled via resonant state expansion (RS, quasinormal modes, complex frequencies). The effective non-Hermitian Hamiltonian captures off-diagonal couplings, frequency shifts, and exact criteria for crossover from weak to observable strong coupling, based on spectral trajectory anticrossing in the complex plane and explicit relationships between coupling, linewidths, and eigenfrequencies (Fischbach et al., 5 Nov 2025).

5. Nonperturbative, Ab Initio, and “Photon-Free” Frameworks

For ab initio quantum-electrodynamical density-functional theory (QEDFT) and many-body treatments:

• Photon-Free Effective Hamiltonians: By nonperturbative elimination (polaron-like or adiabatic transformation) of the photon field from the Pauli–Fierz (or minimal-coupling) Hamiltonian, one arrives at an equivalent H_eff acting only on the matter Hilbert space, exact in weak, strong, homogeneous, and adiabatic limits (Ashida et al., 2020, Schäfer et al., 2021):

$$H_{\rm eff} = -\frac{1}{2} \sum_i \nabla_i^2 + V + W + \sum_{\alpha} \frac{\tilde{\omega}_\alpha}{2} - \sum_{\alpha} \frac{\lambda_\alpha^2}{2 N_e \tilde{\omega}_\alpha^2} [\epsilon_\alpha \cdot \sum_i (-i \nabla_i)]^2$$

This allows QEDFT calculations at computational cost comparable to standard electronic DFT and generalization to periodic solids and time-dependent phenomena (Schäfer et al., 2021).

• Green’s Function, Density of States, and Input–Output Theory: The continuum local density of states (LDOS) approach yields exact, parameter-free descriptions of polariton spectra in dispersive, absorbing environments via resolvent (Green’s function) methods (Gunasekaran et al., 2023). This enables mapping between quantum electrodynamics in dispersive media (mQED) and Lindblad master equations via the pseudomode approach, with loss rates and coupling strengths extracted directly from the measured Purcell spectrum (Wang et al., 2021).

6. Nonlinear and Collective Phenomena

In the ultrastrong coupling regime, nonlinear response and collective effects naturally emerge and are unified within these frameworks:

• Nonlinear Optical Response: The full Power–Zienau–Woolley (PZW) Hamiltonian plus fermionic quadratures yields ab initio predictions of harmonic generation, saturation, and optical bistability, including ENZ effects and local-field enhancements not captured by single-particle or perturbative treatments (Krieguer et al., 11 Dec 2024, Jeannin et al., 2020).
• Saturation and Bistability: Saturation intensity exhibits qualitatively distinct doping dependence in weak vs. strong coupling, with optical bistability emerging above a well-defined cooperativity threshold, unified from atomic to solid-state systems (Jeannin et al., 2020).
• Ground-State Engineering and Phase Transitions: As density or oscillator strength increases, zero-point energy shifts modify the bulk modulus, and soft-mode instabilities induce ferroelectricity, insulator–metal, or exciton condensation transitions (Mueller et al., 9 May 2025).

7. Practical Design Principles and Applications

Unified frameworks provide recipe-like guidance for engineering and diagnosing strong and ultrastrong coupling in real systems:

• Cavity/nanophotonic systems: Engineering Rabi splitting via adjustment of oscillator strength, mode volume, and loss; leveraging Purcell and ENZ enhancements for nonlinear optics (Krieguer et al., 11 Dec 2024, Kolaric et al., 2018).
• Bulk and quantum materials: Identifying conditions for collective polariton soft modes, leveraging geometric quantum metric contributions for robust light-matter coupling in vanishing band-velocity systems (e.g., TBG, flat bands) (Topp et al., 2021).
• Mode hybridization control: Utilizing multimode mixing, field overlap matrix engineering, and local field reconfiguration for bespoke polariton fields at subwavelength resolution (Cortese et al., 2022).
• Open, leaky architectures: Applying resonant-state formalism and continuum DOS models for robust description and optimization in real (lossy, open) photonic environments (Fischbach et al., 5 Nov 2025, Gunasekaran et al., 2023).

Scalable design rules emerge, for instance, maximizing field overlap in structured devices, dynamically tuning material excitations for desired polariton admixtures, and exploiting critical phenomena for novel material functionalities.

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The unified description of light-matter coupling now spans the full parameter landscape from weak perturbative regimes to ultrastrong and deep-strong coupling, in closed cavities, open systems, and bulk solids, rigorously connecting the spectrum, losses, nonlinearities, and collective behaviors to underlying microscopic and geometric structure. This multidimensional theoretical framework enables detailed predictive modeling and control of quantum phenomena at the interface of photonics, condensed matter, chemistry, and quantum information science (Forn-Díaz et al., 2018, Yu et al., 2022, Mueller et al., 9 May 2025, Fischbach et al., 5 Nov 2025, Cortese et al., 2022, Krieguer et al., 11 Dec 2024, Gunasekaran et al., 2023, Topp et al., 2021, Jeannin et al., 2020, Ashida et al., 2020, Wang et al., 2021, Schäfer et al., 2021).