Cavity-Induced Modifications of Molecular Structure in the Strong-Coupling Regime

In most theoretical descriptions of collective strong coupling of organic molecules to a cavity mode, the molecules are modeled as simple two-level systems. This picture fails to describe the rich structure provided by their internal rovibrational (nuclear) degrees of freedom. We investigate a first-principles model that fully takes into account both electronic and nuclear degrees of freedom, allowing an exploration of the phenomenon of strong coupling from an entirely new perspective. First, we demonstrate the limitations of applicability of the Born-Oppenheimer approximation in strongly coupled molecule-cavity structures. For the case of two molecules, we also show how dark states, which within the two-level picture are effectively decoupled from the cavity, are indeed affected by the formation of collective strong coupling. Finally, we discuss ground-state modifications in the ultrastrong-coupling regime and show that some molecular observables are affected by the collective coupling strength, while others depend only on the single-molecule coupling constant.

Popular Summary

When organic molecules interact with light modes confined in nanostructures, hybrid light-matter states (polaritons) can be created via so-called “strong coupling.” The participating molecules are affected by the interaction, and this effect can be used for controlling chemical reactions or modifying a material’s properties. Up until now, there has been no consensus on how these modifications occur, and accordingly, there has been no way to accurately predict these modifications or design novel structures to exploit them. Here, we demonstrate that it is possible to understand and predict these modifications by adapting well-known techniques from chemical physics. The extent of the modifications turns out to depend sensitively on which observable is interrogated.

We consider two model molecules, and we simplify our analyses by limiting the number of electrons to just one moving in one dimension. Progress up until now has been hindered by the fact that organic molecules possess a large number of internal (rovibrational) degrees of freedom and do not behave like simple two-level emitters. Here, we employ a first-principles approach that takes into account electronic, nuclear, and electromagnetic degrees of freedom on an equal footing. We demonstrate that the Born-Oppenheimer approximation, which underlies most of our understanding of chemical structure and dynamics and assumes that nuclear and electronic motion can be separated, can break down under strong coupling. However, we can predict exactly how this approximation breaks down; in addition, it also holds true in many important cases. By exploiting this approach, we show that molecular structure is modified even for so-called dark states, which are typically thought to be uncoupled from electromagnetic modes. By calculating absorption cross sections for different coupling strengths, we recover surprising effects such as correlated nuclear motion in separated molecules.

Our findings provide the basis for understanding how to manipulate chemical structure and reactions through strong coupling. We expect that our results will have wide-ranging implications for the design of novel nanostructures.

Figure 1

Illustration of energy level structure of a bare complex molecule and the hybrid states that result in the strong-coupling regime with a photonic mode of energy $\hslash {\omega }_c$, resonant with the molecular excitation.

Figure 2

Bare-molecule potential energy surfaces of the two first electronic states in the BOA for (a) the rhodamine 6G-like model molecule and (c) the anthracenelike model molecule. The vibrational levels and associated nuclear probability densities are represented on top of the PES. (b), (d) Absorption spectrum for the (b) R6G-like and (d) anthracenelike molecule in arbitrary units.

Figure 3

Strongly coupled electronic PES (solid lines) in the singly excited subspace, for the anthracenelike molecule for (a) $g=0.001\mathrm{a}.\mathrm{u}.$ and (b) $g=0.008\mathrm{a}.\mathrm{u}.$. The dashed lines show the corresponding uncoupled states: A molecule in the first excited state, $E_e(R)$, and a molecule in the ground state with one photon present, $E_g(R)+{\omega }_c$.

Figure 4

Absorption cross sections of a single molecule, calculated using the full Hamiltonian without approximation (solid green lines) and within the BOA (dashed black lines). Results are shown for the (a) R6G-like and (b) anthracene-like model molecules, for several values of the coupling strength $g$.

Figure 5

(a) Uncoupled potential energy surfaces of two anthracenelike molecules in the singly excited subspace: $E_{eg0}(R_1,R_2)$ (orange), $E_{eg0}(R_1,R_2)$ (blue), and $E_{gg1}(R_1,R_2)$ (green). (b) Coupled PES for $g=0.002\mathrm{a}.\mathrm{u}.$ and (c) $g=0.013\mathrm{a}.\mathrm{u}.$, corresponding to the lower polariton (orange), dark state (blue), and upper polariton (green). For clarity, only parts where $R_1<R_2$ are shown (note that the system is symmetric under the exchange $R_1\leftrightarrow R_2$).

Figure 6

Absorption cross section of two molecules driven coherently, calculated using the full Hamiltonian without approximation (solid green lines) and within the BOA (dashed black lines). Results are shown for the (a) R6G-like and (b) anthracenelike model molecules, for several values of the coupling strength $g$. The values of $g$ are scaled by $1/\sqrt{2}$ with respect to the single-molecule case (Fig. 4) in order to obtain the same total Rabi frequency ${\mathrm{\Omega }}_R$.

Figure 7

Coupling between nuclear motion in different molecules for the lower (LP) and upper polariton (UP) and dark-state (DS) PES. Results are shown as the ratio $\beta /\alpha$ between the prefactors of the off-diagonal $R_1{R}_2$ and diagonal $R_i^2$ terms in Eq. (11), for the R6G-like model molecule.

Figure 8

Non-Born-Oppenheimer correction terms coupling the “lower polariton” and “upper polariton” PES for a single anthracenelike model molecule for a coupling strength of $g=0.002\mathrm{a}.\mathrm{u}.$. Solid colored lines: results from a full numerical calculation. Dashed black lines: results from the model Eq. (a4). Note that while all results are given in atomic units, the units of the $\widehat{P}$ and ${\widehat{P}}^2$ terms are not identical, and thus not directly comparable.

Figure 9

Solid lines: mutual information in the nuclear propability distribution of the vibrational ground states of the coupled PES as predicted by Eq. (b2) using the values of $\beta /\alpha$ shown in Fig. 7. Dashed lines: mutual information in the nuclear probability distribution of the steady state under driving of a single molecule, Eq. (b1).