Resonant states reveal strong light-matter coupling in nanophotonic cavities

TL;DR

This work introduces resonant states (RSs) as the natural framework for describing light-mather coupling in open photonic environments by leveraging complex-frequency eigenmodes of the system. It derives an effective Hamiltonian that couples a single RS of the bare cavity to multiple material resonances, showing that coupling both hybridizes modes and shifts the cavity’s eigenfrequency, a feature absent in many conventional models. The authors demonstrate the method on planar Fabry-Perot and core-shell nanoparticle systems, extract individual coupling rates via an inverse-eigenproblem, and extend the formalism to multiple material resonances using a resonant-state expansion. By tracking RS trajectories in the complex frequency plane, the approach unambiguously separates weak, hidden strong, and observable strong coupling regimes, enabling robust design and analysis of light-matter interactions in open photonic structures.

Abstract

Photonic resonances are a powerful tool for controlling light-matter interactions. However, unlocking many of the most scientifically intriguing and technologically promising phenomena requires entering the strong coupling regime, where light and matter fully mix, unlocking emergent properties of the coupled states. Nowadays, distinguishing between weak and strong coupling primarily relies on studying the optical response of the hybrid system at real frequencies, which only provides indirect estimates of the underlying resonant dynamics. In contrast, the actual resonances live at complex frequencies. Resolving this contradiction, we show that photonic resonant states provide the framework to unambiguously quantify the strength of light-matter interaction, enabling a rigorous distinction between weak and strong coupling regimes. Assuming a single dominant resonant state of the bare photonic resonator, we derive an effective Hamiltonian that captures the interaction between the photonic resonator and an arbitrary number of material resonances. Our analysis reveals that, unlike most coupled-oscillator models commonly employed in the literature, hybridization not only introduces off-diagonal coupling but also shifts the bare eigenfrequency of the photonic mode. We demonstrate the accuracy of this approach by studying planar and spherical silver resonators filled with a molecular material whose properties were extracted from quantum-chemical simulations. Our work paves the way towards a unified description of light-matter coupling in open photonic environments.

Figures (7)

• Figure 1: Mode hybridization: (a) When a system parameter $\rho$ of a photonic resonator is modified, the system dynamics change accordingly, expressed here in terms of a changing (complex) eigenfrequency as a function of $\rho$. Examples for possible choices of $\rho$ are the spacing between the plates for a Fabry-Perot cavity or the radius of a core or a shell for a spherical resonator. (b) When a material characterized by a single Lorentz-oscillator is introduced to the photonic resonator, the photonic mode, which can be considered as an independent oscillator, and material resonance couple to each other, leading to hybridization. The new hybrid modes distribute their energy across the coupled system, leading to shifts in the eigenfrequencies from the bare system without coupling. (c) The simple relation between coupling coefficients and frequency splitting gets spoiled when multiple material resonances are introduced. Column (1) illustrates the considered system, column (2) expresses the material properties, and column (3) shows the (un-)coupled eigenfrequencies.
• Figure 2: Optical response of a planar cavity filled with a resonant medium in terms of observables commonly used as hallmarks of strong coupling. From left to right (1-4), the coupling strength is changed by scaling the oscillator strength of an artificial single-pole Lorentz medium by a factor $\eta$ (material and geometry parameters are given in Table \ref{['tab:params']}). (a) The solid black line shows the spectrally resolved transmission through the cavity upon illumination from below. For illustrative purposes, the contributions of the two dominant RSs are indicated proportional to $\left| \omega - \tilde{\omega}_\mathrm{m}\right|^{-2}$ (orange and blue lines), which corresponds to the Lorentzian line shapes the RSs would have in isolation. (b) The cavity mode is tuned by changing the cavity thickness $d$. Note how (b1) resembles an avoided crossing, and (a1) correspondingly shows two clearly separated peaks, despite the system being in the weak coupling regime. The perceived splitting results from the interference between two modes that are at the same real frequency but have different linewidths. This phenomenon is known in the framework of Fano resonances limonov_fano_2017. The broad cavity mode (increasing transmission) can be understood as the background, on which a narrow absorption dip arises due to the material resonance. (c) Absorption upon the same illumination as in (a).
• Figure 3: Parametrized pole trajectories: The thickness of a planar cavity containing a single-pole Lorentz medium is changed, while the RS eigenfrequencies are traced. The specific material parameters and geometry are given in Table \ref{['tab:params']}. As in Figure \ref{['fig:observable']}, the panels 1-4 correspond to increasing oscillator strength (scaled by $\eta$) of the material resonance, and thus increasing coupling strength. (a) Dispersion of the resonance frequencies (i.e., the real part of the eigenfrequencies) with changes in $d$. This provides a more precise view of the hybrid modes lifting the fog of damping-induced linewidth broadening [compare to Figures \ref{['fig:observable']}(b1-b4)]. Lines of different colors correspond to cavity modes of different orders (blue being the fundamental mode). Each panel contains two branches, corresponding to distinct RSs resulting from the hybridization. In (a1), these branches cross, while they avoid the crossing in (a2-a4). The correspondence to Figures \ref{['fig:observable']}(b1-b4) is further emphasized by the gray shaded region, covering $\Re\{\hbar \tilde{\omega}\} \pm \Im \{\hbar \tilde{\omega}\}$ of the fundamental mode, which can be understood as the linewidth of the corresponding Lorentzian. (b) Trajectories of the eigenfrequencies in the complex frequency plane as a function of the cavity thickness $d$ (indicated by the color of the lines). The dispersion of the RSs associated with the lowest-order cavity mode is most prominent, as the eigenfrequency varies strongly within the chosen thickness interval. At the same time, the RSs associated with higher-order cavity modes barely move, resulting in short line segments. These segments only have an appreciable length in subfigure (a4) due to the larger interval in which the cavity thickness $d$ is varied. The inset in (b1) shows additional details making the higher-order RSs more clearly visible (same color code as in the top row). This perspective of the complex plane clearly visualizes the attraction and repulsion of the hybrid modes in the weak and strong coupling regime, respectively. The triangle markers in (b1) are used in the main text to refer to sections of the trajectories when explaining these phenomena. FP and M denote hybrid modes that are dominated by Fabry-Perot and material contributions, respectively. The dashed line indicates the dispersion of the bare cavity mode. A black cross (x) indicates the position of the single resonance of the Lorentz permittivity. A circle (o) marks its complex zero. RSs associated with higher-order cavity modes appear close to the material resonance, where an infinite number of modes are supported due to the diverging permittivity, forming an accumulation point. The peculiar feature in the upper polariton branch in subfigure (b2; magnified in inset) results from the vicinity to the transition from weak to strong coupling, as it causes the eigenfrequency to locally vary almost exclusively in its imaginary part with changing $d$.
• Figure 4: Parametrized pole trajectories for a 3D finite system. In analogy to Figure \ref{['fig:osc_reduction']}, the evolution of the transverse magnetic dipolar resonances of a nanoparticle (NP) are traced. The NP consists of a SURMOF core with variable radius $r_\mathrm{core}$ and a fixed 20 nm silver shell. The plasmon resonance predominantly localized on the outer surface of the metallic shell [(bright) green line at the bottom of each panel] has vastly higher damping than the material resonance. The high damping results from radiation loss as the plasmon couples strongly to free space. At the same time it has low overlap with the SURMOF core. As a consequence, their interaction is negligible. In contrast, the other mode is predominantly localized at the inner surface of the metallic shell, characterized by a lower radiation loss and a better overlap with the SURMOF core. It is that mode that can be strongly coupled to the material excitation. As the light matter coupling is varied across the columns, the plasmon mode predominantly localized at the inner surface of the shell [blue/teal lines in (a)/(b) respectively] traverses from the weak to the observable strong coupling regime.
• Figure 5: Pole trajectories: A Fabry-Perot cavity made from two silver films separated by a varying thickness $d$ is filled with a SURMOF material whose dielectric function can be described effectively by material resonances. (a) Real and imaginary part of the permittivity of the SURMOF material inside the cavity as extracted from quantum-chemical simulations. (b) Analogous to Figure \ref{['fig:osc_reduction']}(a), the trajectories of the complex eigenfrequencies of the RSs are parametrized by changing the cavity thickness $d$ as indicated by their color. The background color corresponds to the real part of the SURMOF permittivity. The three material resonances (marked by x) generate one accumulation point each. With increasing cavity thickness, the modes dominated by the higher frequency material resonances leave the accumulation points, moving left towards the complex zeros of $\varepsilon_r$ (open circles) created by the superposition of neighboring material resonances. (c) Projection of the pole trajectories onto the real frequency axis: The RSs are colored distinctly according to the Fabry-Perot mode they belong to. The dashed area marks the real frequency interval in which $\Re\{\varepsilon_r\}<0$.