Abstract
Despite the potential paradigm breaking capability of microcavities to control chemical processes, the extent to which photonic devices change properties of molecular materials is still unclear, in part due to challenges in modeling hybrid lightmatter excitations delocalized over many length scales. We overcome these challenges for a photonic wire under strong coupling with a molecular ensemble. Our simulations provide a detailed picture of the effect of photonic wires on spectral and transport properties of a disordered molecular material. We find stronger changes to the probed molecular observables when the cavity is redshifted relative to the molecules and energetic disorder is weak. These trends are expected to hold also in higherdimensional cavities, but are not captured with theories that only include a single cavitymode. Therefore, our results raise important issues for future experiments and model building focused on unraveling new ways to manipulate chemistry with optical cavities.
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Introduction
Strong light–matter interactions hosted by nanostructures and optical microcavities can induce significant and qualitative changes to chemical processes^{1,2,3} including photoconductivity^{4,5}, energy transport^{6,7,8}, and optical nonlinearities^{9,10,11}. Much of the observed phenomenology stems from the hybridization of the collective material polarization and the resonances of an optical cavity, which leads to the formation of delocalized polariton modes when the energy exchange between the collective molecular excitations and the photonic structure is faster than the dissipative processes acting on each subsystem^{9,12,13,14,15}. Polaritonic states are always accompanied by molecular states weakly coupled to light^{16}. The latter are also sometimes called “dark states”, as the optical response of strongly coupled molecular ensembles is dominated by polaritonic excitations. The weakly coupled states form a reservoir containing the vast majority of the states of the system with significant molecular character^{16,17,18}, and therefore, they play an essential role in equilibrium and nonequilibrium molecular phenomena in optical cavities^{13,19,20,21,22,23}.
Polaritons dominate the optical response of a strongly coupled device, but the reservoir states are much more numerous^{16,18}. Therefore, it is puzzling that significant changes in thermal reaction rates and branching ratios of some organic systems^{24,25,26,27,28,29} were observed under conditions of infrared collective strong coupling. The dominance of molecular reservoir modes over the polaritonic^{18,30,31} suggests there is no simple explanation of the cavity effect on thermal reactions based on transitionstate theory^{32,33,34}. This notion has motivated the hypothesis that cavityinduced changes to chemical reactions originate from dynamical effects of the electromagnetic environment on intramolecular dynamics^{35,36,37} (see also^{38,39,40,41}, for theoretical treatments of strong light–matter interaction effects on chemical reaction dynamics when a single molecule is strongly coupled to a cavityphoton mode and dark states are absent).
Given the complexity of polaritonic systems with states delocalized over length scales of the order of the optical wavelength, much of what is known about collective strong coupling relies on quantum mechanical simulations of effective models^{42,43} where the cavity is modeled as a single bosonic mode and the molecular system has permutational symmetry, i.e., it consists of an ensemble of identical N_{M} ≫ 1 twolevel system with equal transition energy and dipole moment^{13,14,18}. In these models, hybrid light–matter excitations (lower and upper polaritons, LP and UP, respectively) extending over the entire system emerge from the interaction of the homogeneous cavity field with the molecular bright mode corresponding to the totally symmetric combination of molecular states where a single molecule is excited. The other N_{M}−1 molecular modes are degenerate and correspond to the dark reservoir discussed above. Permutationally invariant models deviating from this TavisCummings (TC) picture via the introduction of excitonphonon interactions^{18,44,45,46} have also been thoroughly investigated.
The effects of material imperfections on polaritonic and dark states were probed in early work by Houdre et al.^{17} who showed that the presence of energetic and structural disorder lead to weak photonic intensity borrowing by the reservoir states, but only minor changes to the TC picture when the collective light–matter interaction Ω_{R}/2 (Rabi splitting) is much greater than the mean fluctuation σ of the material transition energies. Recently, Scholes revisited the polariton coherence protection in singlemode (0D) cavities^{47}, whereas Botzung et al.^{48} provided quantitative localization properties of the reservoir states of an energetically disordered emitter ensemble under strong coupling to a single spatially homogeneous cavity mode. Both studies showed in agreement with ref. ^{17} that polariton coherence is largely unaffected by energetic disorder weaker than the collective light–matter coupling, but also noted that dark modes inherit spatial delocalization (see also refs. ^{19,49} for identification of signatures of darkstate delocalization in energy transport simulations, and ref. ^{50} where properties of polaritonic and dark excitations in a leaky singlemode cavity are given).
However, the majority of photonic materials employed in polariton chemistry research are multimode FabryPerot (FP) or plasmonic cavities with a continuous spectrum^{51,52}. For example, in an FP cavity, each electromagnetic mode is characterized by its polarization (TE or TM), cavity order m = 1, 2, 3... (equivalent to the longitudinal wavevector k_{z}), and the essentially continuous inplane wavevector q = (q_{x}, q_{y}). In systems such as FP cavities, the disorder is known to severely restrict polariton spacetime coherence^{16,53} and may lead to weak or strong Anderson localization as well as diffusive and ballistic transport depending on the initial wavepacket, disorder strength, cavity geometric parameters, and magnitude of light–matter interactions^{54}. Numerical simulations have shown that polariton wave functions can be localized over different length scales depending on their mean wavevector^{55,56,57} (see ref., ^{58} for a recent study of multimodecavity effects on polariton relaxation).
Despite their prominence, much less attention has been paid to multimode strong coupling effects on the reservoir states of structurally and energetically disordered molecular ensembles. These states form the majority of the excitations with significant molecular character, and therefore, they largely determine the equilibrium and transport properties of a molecular subsystem under collective strong coupling with an optical cavity.
Here, we employ numerical simulations of the microscopic states of a multimode photonic wire under strong coupling with a disordered molecular ensemble to investigate the influence of the cavity on the local molecular density of states and the exciton return probability. These quantities essentially reflect the properties of the reservoir modes (since they are much more numerous than the polaritonic) and allow us to quantitatively probe the effects of multimode optical cavities on the molecular ensemble.
Our results have significant implications for future model building in polariton chemistry since they provide detailed illustrations of qualitative shortcomings of singlemode representations of multimode devices. We also suggest practical principles to enhance cavity effects on molecular properties likely holding for generic systems, e.g., we find that for systems with equal collective light–matter interaction, polariton effects on the molecular ensemble are largest when the cavity is redshifted, the distance between molecules is largest and energetic disorder is weak. Wave function localization theory^{53,54,59} and simple spectral overlap arguments explain these facts which provide another piece to the puzzle^{60} of the optical cavity effect on chemical reactions since some experimental observations suggest that the influence of photonic materials on chemical reactions is greater when the cavitymatter detuning vanishes^{61}.
Results and discussion
Microscopic model
We model an optical microcavity with O(μm) longitudinal and lateral confinement lengths L_{z} and L_{y} along the z and y axes, respectively^{62}. The length of the long axis is L ≫ L_{z}, L_{y}. The molecules are distributed homogeneously along x and have equal y, z coordinates (see Fig. 1). Ideal reflective surfaces confine the EM field along z and y, whereas periodic boundary conditions are assumed along x. We include a single polarization of the EM field parallel to the direction of each molecule’s transition dipole moment. The bare cavity modes have frequency \(\omega =(c/\sqrt{\epsilon })k\), where c is the speed of light, ϵ is the static dielectric constant of the intracavity medium and k is the magnitude of the threedimensional wavevector k = (2πm_{x}/L,n_{y}π/L_{y}, n_{z}π/L_{z}), where \({m}_{x}\in {\mathbb{Z}}\), and n_{y} and n_{z} are positive integers. The parameters L, L_{y}, L_{z}, and ϵ are chosen such that the vast majority of the molecules are only resonant with cavity modes in their lowestenergy band (n_{z} = n_{y} = 1). Therefore, we ignore all other bands, and only include photons with \({{{{{{{\bf{k}}}}}}}}=(2\pi m/L,\pi /{L}_{{{{{{{{\rm{y}}}}}}}}},\pi /{L}_{{{{{{{{\rm{z}}}}}}}}}),\,m\in {\mathbb{Z}}\). From now on, we label the cavity modes by q ≡ k_{x}, omit any reference to k_{y} and k_{z} and identify q as the light wavevector degree of freedom. The empty cavity Hamiltonian is given by:
where \({q}_{0}=\sqrt{{(\pi /{L}_{{{{{{{{\rm{z}}}}}}}}})}^{2}+{(\pi /{L}_{{{{{{{{\rm{y}}}}}}}}})}^{2}}\). The q = 0 cavity mode has lowest energy \({E}_{{{{{{{{\rm{C}}}}}}}}}(0)=\hslash c{q}_{0}/\sqrt{\epsilon }\).
The molecular ensemble is represented by a set of N_{M} = L/a twolevel systems with transition frequencies sampled from a Gaussian distribution (representing lowfrequency fluctuations of the solvent environment around each molecule) with mean E_{M} and variance σ^{2}, so the transition energy for the ith molecule is E_{i} = E_{M} + σ_{i}, where \({\left\langle {\sigma }_{i}\right\rangle }_{d}=0\) and \({\langle {\sigma }_{i}{\sigma }_{j}\rangle }_{d}={\sigma }^{2}{\delta }_{ij}\). We also include structural disorder in the form of random deviations of the molecular center of mass positions relative to a perfect crystal arrangement with lattice spacing a, and by allowing the singlemolecule transition dipole moment to deviate weakly from the mean value μ_{0} > 0. The ith molecule position is x_{i} = (i − 1)a + Δx_{i} (mod N_{M}a), where Δx_{i} = f_{i}a, and f_{i} is sampled from a uniform distribution over \([f,f]\subset {\mathbb{R}}\). The parameter f controls the maximal (1 + 2f)a and minimal (1 − 2f)a distances between neighboring molecules, respectively. The transition dipole moment of the ith molecule is given by the random variable μ_{i} sampled from a normal distribution with mean μ_{0} > 0 and variance \({\sigma }_{\mu }^{2}\). The structural disorder is typically neglected in numerical treatments of disorder effects on polaritons, but is included here since when Ω_{R} ≫ σ, polariton localization at low energies may be primarily driven by fluctuations in the molecular position and dipole moments (relative to a perfectly ordered system)^{53,54}. Nevertheless, for the studied model, except for infinitesimal values of σ and simultaneous σ_{μ}/μ_{0} close to or greater than 1, energetic disorder plays a more important role, and therefore, we take σ_{μ} and f to be constant throughout this work. A detailed discussion of energetic and structural disorder effects on molecular observables is given in Supplementary Note 2 including a quantitative comparison of structural and energetic disorder on the exciton escape probability χ_{M} and molecular local density of states entropy variation ΔS[ρ_{M}] (see Supplementary Figure 1).
Assuming a is sufficiently large, the Hamiltonian for the bare molecules is:
where E_{M} = ℏω_{M}, and \({b}_{i}^{+}\,({b}_{i}^{})\) creates (annihilates) an excitation at the ith molecule. These operators can be written as \({b}_{i}^{+}=\left{1}_{i}\right\rangle \left\langle 0\right\) and \({b}_{i}^{}=\left0\right\rangle \left\langle {1}_{i}\right\), where \(\left0\right\rangle\) is the state where all molecules and cavity modes are in their groundstate, and \(\left{1}_{i}\right\rangle\) is the state where only the ith molecule is excited. The total Hamiltonian is given by:
where H_{LM} contains the light–matter interaction. We employ the Coulomb gauge^{63} in the rotatingwaveapproximation since we take Ω_{R}/2 < 0.1E_{M}^{64}. It follows that,
where \({{{\Omega }}}_{{{{{{{{\rm{R}}}}}}}}}={\mu }_{0}\sqrt{\hslash {\omega }_{0}\rho /2\epsilon }\), ρ = N_{M}/LS, and S = L_{y}L_{z}. We ignored the diamagnetic contribution (A^{2} term) to H_{LM} since its effects are negligible under the studied conditions^{65}.
For a given Ω_{R} and a = 1/ρS, N_{M}, and N_{C} are free parameters. Any particular choice is equivalent to imposing low and highenergy cutoffs to the EM field. Specifically, L = N_{M}a defines the resolution of the cavity in reciprocal space Δq = 2π/L, and N_{C} defines the maximal cavitymode energy \({E}_{\max }\). Simulation results are independent of these cutoffs as long as L is larger than the longest coherence length of the system and \({E}_{\max }\) is greater than any relevant energy scale. Alternatively, thermodynamic limit (N_{M}, L→∞ with fixed ρ) independence of molecular observables to the number of included degrees of freedom can be imposed to obtain a minimal number of molecular and cavity modes. Such computationally optimal number of modes can be strongly dependent on the molecular observable of interest and may also vary significantly with Ω_{R} and σ.
We employ N_{C} = N_{M} in our simulations below, since in this case, the thermodynamic limit is reached with a small number of disorder realizations for the observables and range of parameters probed in our studies. A significant fraction of cavity modes is highly offresonant with the molecular system in every case studied (N_{M} ≥ 1001), and we checked that a smaller number of cavity modes \({N}_{{{{{{{{\rm{C}}}}}}}}}^{0}\) would suffice to obtain thermodynamic limit predictions. However, as mentioned above, \({N}_{{{{{{{{\rm{C}}}}}}}}}^{0}\) depends on various parameters and we leave for future work a detailed analysis of optimal multimodecavity representations^{66} for the study of the effects of photonic devices on molecules.
In later sections, we characterize the dependence of local molecular observables on the energetic disorder, cavity detuning, and mean intermolecular distance (or photonic lattice constant) for fixed N_{M} = N_{C} and Ω_{R}. The study of lattice constant effects with fixed N_{M} = N_{C} is a distinctive feature of our work in relationship to refs. ^{53,56} (who analyzed polariton coherence in the thermodynamic limit with fixed a) and is motivated by the following question: are there significant differences in the polariton effects on molecules showing equal Ω_{R} but different molecular densities (1/a in our photonic wire)? Note that a change in the lattice constant a to f × a (f > 0) leaves Ω_{R} fixed under the mentioned conditions if an only if \({\mu }_{0}\mapsto \sqrt{f}{\mu }_{0}\) (as Ω_{R} ∝ μ_{0}a^{−1/2}), as expected since a reduced molecular density requires greater transition dipole moment per molecule to preserve a given Rabi splitting. Nonetheless, a potential unintended consequence of changing a with fixed N_{C} in cavity lattice models is that the photon density of states ρ_{C}(ω) = ∑_{q}δ(ω − ω_{q}) is also modified. The thermodynamic limit, in fact, \({\rho }_{{{{{{{{\rm{C}}}}}}}}}(\omega )\to [L/(2\pi )]{\int}_{{\mathbb{R}}}{{{{{{{\rm{d}}}}}}}}q\,\delta (\omega {\omega }_{q})\) is proportional to L, and only ρ_{C}(ω)/L is independent of L. In Fig. 2, we show that the effects of optical microcavities in the molecular ensemble measured in our study are essentially independent of L as long as a sufficiently large number of modes is introduced in the theory. This suggests that the change in photon density of states that result from varying a (with fixed N_{C} = N_{M} and Ω_{R}) is immaterial to our conclusions. In Supplementary Note 1, we provide further discussion and numerical evidence that further supports these points (see Supplementary Tables I and II).
Observables
Let the eigenstates and eigenvalues of H be denoted by ψ and E_{ψ}, respectively. The ensembleaveraged molecular local density of states (LDOS) gives the conditional probability that an excited molecule will be detected with energy E:
\({P}_{n\psi }= \left\langle {1}_{n} \psi \right\rangle { }^{2}\) and \(\langle {P}_{n\psi }\rangle =\mathop{\sum }\nolimits_{n = 1}^{{N}_{{{{{{{{\rm{M}}}}}}}}}} \left\langle {1}_{n} \psi \right\rangle { }^{2}/{N}_{{{{{{{{\rm{M}}}}}}}}}\) is the average probability to find a specific molecule excited when the system is in the eigenstate \(\left\psi \right\rangle\). In the absence of light–matter interactions, the molecular LDOS \({\rho }_{{{{{{{{\rm{M}}}}}}}}}^{(0)}(E)\) is a Gaussian distribution centered at E_{M} and width σ. We quantify the photonic effect on ρ_{M}(E) by evaluating the cavityinduced change in its Shannon entropy
ΔS[ρ_{M}] allows us to quantify the cavity effect on the molecular ensemble energy fluctuations. Roughly speaking, ΔS[ρ_{M}] provides a measure of molecular excitedstate delocalization in energy space. Therefore, we expect ΔS[ρ_{M}] to be a nondecreasing function of Ω_{R} that is greater than or equal to zero, since polaritons have energies separated from the bare molecule excitedstates by approximately ± Ω_{R}/2 (Fig. 1). However, the reduced bare photonic DOS relative to the molecular at energies where the molecular DOS is maximal suggests weakly coupled reservoir states dominate the molecular LDOS and the cavitydriven change in S[ρ_{M}] is expected to be small.
The mean probability that an initially excited molecule at time t = 0 will be detected in the same state at t > 0 is the timedependent exciton survival (return) probability \({P}_{{{{{{{{\rm{M}}}}}}}}}(t)=\mathop{\sum }\nolimits_{j = 1}^{{N}_{{{{{{{{\rm{M}}}}}}}}}}{P}_{j}(t)/{N}_{{{{{{{{\rm{M}}}}}}}}}\), where
The exciton return probability Π_{M} is the t→∞ limit of P_{M}(t):
The exciton escape probability χ_{M} is simply related to Π_{M} via:
This quantity provides the ensembleaveraged probability that energy initially stored as a localized molecular exciton migrates to a distinct molecule or is converted into a cavity photon after an infinite amount of time.
Π_{M} provides a measure of excitedstate delocalization in realspace and coherent energy diffusion efficiency: in systems where all Hamiltonian eigenstates are delocalized P_{nψ} ∝ 1/N_{M} for all ψ and n, and therefore, Π_{M} vanishes in the thermodynamic limit, whereas in noninteracting systems with maximally localized excitedstates, each molecule corresponds to a Hamiltonian eigenstate, and therefore P_{nψ}(t) = δ_{nψ} and Π_{M} = 1. In our model, the case where Π_{M} = 1 corresponds to the molecules outside of the cavity (since we assume direct intermolecular interactions are insignificant), while we find Π_{M}→0 when energetic disorder vanishes. Dark states are expected to have a much large contribution to Π_{M} than polaritons since the latter inherit greater delocalization from their strong mixing with cavity modes.
Note that for a disordered multimode system there is no unambiguous definition of polariton and dark states since there is no energy gap between the LP band and the reservoir modes which are weakly coupled to the cavity. However, given a definition of polariton and weakly coupled modes, it is possible to decompose Π_{M} and χ_{M} into a sum of contributions from polariton and reservoir modes and gauge the sensitivity of polariton and darkstate delocalization to increasing disorder. To analyze our numerical simulations, we choose polaritons to consist of all states where the total photonic content is >15%, but <85%, whereas dark states are all eigenstates with total molecular content >85%. We briefly discuss this choice in the section “Density and energetic disorder dependence”.
Thermodynamic limit convergence
Before presenting a detailed quantitative study of the photonic effects on the mentioned molecular observables, we first show that with a relatively small number of molecules and modes we obtain robust predictions for cavityinduced changes in molecular properties. We set L_{y} = 400 nm, L_{z} = 200 nm, and dielectric constant ϵ = 3. This gives the lowestenergy cavity mode with E_{C}(0) ≡ ℏω_{0} = 2.0eV. The highest energy wavevector is \({q}_{\max }\approx \pi /a\) with a = 10, 25, and 50 nm. To probe the thermodynamic limit of ρ_{M}(E) and Π_{M}, we take Ω_{R} = 0.3 eV, E_{M} = 2.0 eV, σ = 0.2 Ω_{R}, f = 0.1, and σ_{μ}/μ_{0} = 0.05. To compute the molecular LDOS we employed 400 bins of width 5 meV spanning the interval [1.5 eV, 3.5 eV].
In Fig. 2, we show ρ_{M}(E) obtained with five realizations of a system with N_{M} = N_{C} = 1001 and 5001 and a = 10 nm. Despite the small number of realizations, Fig. 2 shows that the model with N_{M} = 1001 leads to ρ_{M}(E) nearly indistinguishable from that with N_{M} = 5001.
These observations are quantitatively confirmed in Fig. 3, where we find that when N_{M} = N_{C} > 1001, both S[ρ_{M}] and Π_{M} depend only on the molecular density 1/a. This feature ensures that we have reached the thermodynamic limit for these observables, and justifies our utilization of N_{M} = N_{C} = 1001 in subsequent sections of this article.
Microscopic states
Our computations reveal quasiextended and localized lowenergy polariton states in qualitative agreement with earlier work focused on disorder effects on 1D polaritonic states^{55,57} (although, we note that the Hamiltonians used in these studies break timereversal symmetry and therefore lead to quantitatively distinct properties relative to our work since coherent localization is generally weakened when timereversal symmetry is absent). These qualitative features of polariton localization in photonic wires have been thoroughly investigated so we provide no detailed discussion, except to mention that there is a clear analogy between the polariton states obtained in photonic wires and the local exciton ground states and the quasiextended exciton states of polymers^{67,68}. This analogy is a consequence of Anderson localization universality according to which, in general, there are no extended states over a 1D system in the thermodynamic limit. Instead, any small amount of disorder leads to a breakdown of longrange order.
In photonic wires strongly coupled to a resonant material, wave function localization, and more generally, the expected distancedependent decay of intermolecular correlations (mediated by the optical cavity) emerge only when both disorder and multiple electromagnetic field modes are included in the light–matter Hamiltonian.
Density and energetic disorder dependence
In Fig. 4, we report the changes induced by strong light–matter interactions in the molecular LDOS entropy (Eq. (6)) and the molecular excitedstate escape probability χ_{M} (Eq. (10)) as a function of energetic disorder for various mean intermolecular distances at zero detuning and Ω_{R} = 0.3 eV. Both ΔS[ρ_{M}] and χ_{M} show a generic decay with increasing disorder (except when a = 10 nm and σ/Ω_{R} < 0.4 where χ_{M} has a shallow local minimum). This behavior is not surprising: when the molecular system is perfectly ordered, all excitations are maximally delocalized across the entire system (including the weakly coupled molecular states with negligible photonic content) and both ΔS[ρ_{M}] and χ_{M} take their largest possible value. For even small values of σ/Ω_{R} < 0.1, scattering induced by energetic disorder induces weak and strong localization of polaritonic and reservoir modes, respectively. This notion is corroborated by Fig. 5a, b, which show the polariton and darkstate contributions to the exciton survival probability, respectively (according to the criterion that polaritons are hybrid states where the photonic content is greater than 15%, but less than 85%, and reservoir modes have >85% molecular composition—while quantitative results depend on these choices, qualitative features are robust). According to our results, reservoir modes are highly sensitive to disorder, undergoing strong localization even when σ/Ω_{R} ≤ 0.1. Conversely, the polaritonic contribution to the survival probability is minimal at small values of σ/Ω_{R}, but becomes significant when the energetic disorder is sufficiently large, further confirming that both weakly and strongly coupled excitations become localized over distances smaller than the size of the system. These trends agree with general ideas of wave function localization theory, namely that exciton localization becomes more prominent, in general, as σ increases.
The consistent increase of ΔS[ρ_{M}] and χ_{M} with the intermolecular distance is more intriguing, since in the thermodynamic limit polariton scattering induced by energetic fluctuations is expected to be independent of a when q is nearly conserved^{53}. Moreover, while elastic scattering induced by structural fluctuations is expected to be stronger as a approaches 1/q, we show numerically in Supplementart Note 2 that unless σ vanishes, energetic disorder provides the dominant contribution to polariton and reservoir localization.
The origin of the enhancement of photonic wire effects on the molecular ensemble with a for fixed Ω_{R} may be understood by noting that if the cavity modes are integrated out in a path integral representation of the partition function of the system^{69,70}, the effective action for the molecular system acquires retarded twobody intermolecular interactions mediated by the cavity with coupling constant proportional to the product of the magnitudes of the singlemolecule transition dipole moments^{33,63,71}. For systems with fixed N_{M} and Ω_{R}, the mean transition dipole moment is proportional to \(1/\sqrt{\rho }\propto \sqrt{a}\), (since \({{{\Omega }}}_{{{{{{{{\rm{R}}}}}}}}}\propto \sqrt{\rho {\mu }^{2}}\) and ρ ∝ 1/a). Therefore, when Ω_{R} and N_{M} are fixed, the cavity influence on each molecule becomes stronger with decreasing value of density. Evidence for this feature can be obtained from the darkstate contribution to Π_{M} in Fig. 5 (right). This shows the reservoir modes become significantly more delocalized when a is increased from 10 to 50 nm. Note that, while the polaritonic contribution to Π_{M} in Fig. 5 also increases with a, the gain in escape probability is dominated by the reservoir modes since their contribution to Π_{M} is nearly one order of magnitude larger than the polaritonic.
Cavity detuning effects
In several chemical reactions where strong influence by optical cavities has been reported, the obtained data suggest that the photonic material maximizes its effect on the reactive system when a specific transition of the reactant is onresonance with the q = 0 mode of the optical cavity^{24,25,26,29}. This feature motivates our study of S[ρ_{M}] and χ_{M} as a function of cavity detuning. In Fig. 6, we present the results given by our multimode theory. The same figure also includes the predictions of theories including a single cavity mode, but a discussion of these results is left to the next section.
As shown by Fig. 6, under strong coupling with a photonic wire, the local molecular DOS entropy and the exciton escape probability are maximally affected by the optical cavity when the lowestenergy photon mode is redshifted relative to the molecular system. In our disordered multimode model, the photonic material is least effective at modifying the lowenergy properties of the molecular ensemble when the photons are all blueshifted relative to the excitons. This trend is robust with respect to changes in the molecular density, and the observed dependence of χ_{M} and S[ρ_{M}] on the density follows the pattern discussed in the previous subsection.
The maximization of the effects of multimode photonic devices on lowenergy observable properties (i.e., depending only on the ground and first excitedstates of the excitonic subsystem) of the molecular system when E_{C}(0) < E_{M} can be understood as follows. First, when the mean molecular excitedstate energy is smaller than or equal to all allowed cavityphoton energies (E_{C}(0) − E_{M} ≥ 0), a significant fraction of the molecular excitons will be offresonant with all cavity modes and the spectral overlap of photonic and molecular modes will be weaker (See bottom of Fig. 1b for a comparison of the case where E_{C}(0) = E_{M} and E_{C}(0) < E_{M} showing that a large number of molecules will be offresonant with all cavity modes when E_{C}(0) = E_{M}). This notion is confirmed in Fig. 7 (bottom left and right), where we show that the number of polaritons N_{P} becomes largest at negative detuning, whereas the number of reservoir modes N_{DS} is minimized in redshifted cavities. Similarly, Fig. 7 (top right) demonstrates that in redshifted cavities, dark states become more delocalized in comparison to cavities with vanishing or positive detuning (\({{{\Pi }}}_{{{{{{{{\rm{M}}}}}}}}}^{DS}\) is smallest for E_{C}(0) − E_{M} < 0). This feature can also be ascribed to the greater mixing of the reservoir modes with the cavity when the latter is redshifted.
In addition to the reduced light–matter spectral overlap, hybrid modes formed from strongly interacting cavity photons with q ≈ 0 tend to localize over smaller distances than polaritons dominated by wavevectors with larger magnitude, since in states with longdistance coherence the wavevector uncertainty δq is much smaller than q^{16,53,54}. Therefore, in devices where the molecules are only resonant with q ≈ 0 transitions (zero detuning), any small amount of disorder will lead to polariton modes with finite δq > q. The latter are localized over an interval with a length smaller than the wavelength, and any delocalization effects on the molecular system induced by light–matter hybridization will be weaker relative to the case where the molecules are resonant with cavity modes with larger q^{16,54,57}. This effect is easier to notice for the systems with a = 25 and 50 nm in Fig. 7. For these intermolecular distances, the number of polaritons decreases significantly when E_{C}(0) − E_{M} approaches zero from negative detuning, but the polaritonic contribution \({{{\Pi }}}_{{{{{{{{\rm{M}}}}}}}}}^{P}\) to the survival probability increases slightly. This confirms that each polariton becomes on average significantly more localized when the cavity is onresonance with the molecular system at q = 0 in comparison to the redshifted case.
Note that, although our numerical results are strictly valid for onedimensional systems, both arguments employed to explain the observed trends are independent of the dimensionality of the photonic and molecular system. Therefore, the detuning behavior reported in Fig. 6 is expected to hold generically for more realistic treatments of the molecules and the photonic device.
Our observations do not contradict experimental results on polariton effects on chemical reactions, since we only investigated the influence of strong light–matter interaction effects on lowenergy properties of the molecular system involving only the ground and first excitedstates of the material, while the energy scales and relevant microscopic states governing thermal chemical reaction rates can be much higher (especially for some of the reactive systems studied experimentally where barrier crossing timescales range from minutes to hours).
Additionally, we assumed that only the lowestenergy cavityphoton band interacts with the molecular system whereas in infrared optical cavities it is typical for the molecular transitions to have N > 1 resonances^{72}. The zerodetuning condition is then satisfied only by one of the resonant cavity bands, whereas the remaining N − 1 have resonances at q ≠ 0. Further work is required to establish whether the inclusion of multiple polariton bands allows recovery by lowenergy effective models of the approximate experimental trend that zerodetuning cavities exert a stronger influence on the local properties of a molecular system.
Comparison to singlemode theories
Before concluding, we contrast the main results of this article with the behavior predicted by optical cavity models that include only a single mode.
In systems with equal Ω_{R}, σ, and detuning, Fig. 6 shows that singlemode (0D) cavities act on strongly coupled molecular systems in a qualitatively distinct fashion relative to a multimode photonic device. This is unsurprising, since the photonic modes of 0D microcavities are isolated. Therefore, the spectral overlap between the cavity and the molecular subsystem is maximized when they are resonant. This leads to a maximal cavitymediated exciton escape probability at zero detuning (Fig. 6), since if the retainedmode were offresonant with the molecular system, the spectral overlap of the material and the cavity would be weaker. This behavior is exclusive to 0D systems with isolated modes and does not apply to cavities with a continuous spectrum, such as those employed in almost all polariton chemistry experiments. In FP cavities or photonic wires, negative cavity detuning gives enhanced light–matter spectral overlap, and a larger number of propagating (delocalized) polariton modes relative to a cavity with zero detuning. Hence, the lowenergy properties of molecular ensembles are expected to be more affected by a photonic material when it is redshifted relative to the molecules. A related conclusion was reached in Ref. ^{31} upon analysis of the viability of a hypothetical mechanism for cavity effects on charge transfer rates^{73}.
Additional examples of qualitative disagreements between single and multimodecavity theories are displayed in Fig. 8, where we compare the cavitymodified ρ_{M}(E) (blue) and bare \({\rho }_{{{{{{{{\rm{M}}}}}}}}}^{0}(E)\) (red) at zero (left panel) and negative (right panel) detuning with equal Ω_{R} and N_{M}. The top panel (Fig. 8a, b) contains the results obtained assuming a single electromagnetic mode interacts with the molecular system, whereas the middle (Fig. 8c, d) and bottom (Fig. 8e, f) panels show the predicted ρ_{M}(E) for multimode cavities where the intermolecular distances are in average equal to 10 nm and 25 nm, respectively.
Figure 8a, b shows that the molecular LDOS is essentially unaffected by a singlemode optical cavity, and ρ_{M}(E) is indistinguishable from \({\rho }_{{{{{{{{\rm{M}}}}}}}}}^{0}(E)\). On the other hand, Figs. 8c−f show the effect of a multimode photonic wire on the molecular LDOS is finite and particularly prominent at the tails of ρ_{M}(E). This feature leads to finite ΔS[ρ_{M}] for a multimode cavity (in contrast to a singlemode theory, for which ΔS[ρ_{M}] = 0). In addition, the singlemode cavity is completely insensitive to the intermolecular distance, whereas our multimode photonic wire captures the stronger effects of the electromagnetic field on molecular ensembles with greater dipole moment per molecule required to preserve the Rabi splitting when the molecular density is smaller (see discussion under “Density and energetic disorder dependence”).
Contrary to the predictions of singlemode theories^{48}, we also find that, except for a narrow interval of values of σ/Ω_{R} where the escape probability does not change too much (Fig. 4), the energetic disorder tends to localize both dark and polaritonic excitations, thus resulting in a reduced escape probability for molecular excitedstates.
Conclusions
We investigated spectral and transport properties of disordered molecular excitons under collective strong coupling with a photonic wire. Our results suggest that lowenergy effects of polaritons on bulk properties of the molecular ensemble are largest when the singlemolecule transition dipole moment is large, energetic disorder is small, and when the cavity is redshifted (negatively detuned) relative to the molecular system. In negatively detuned devices, there exists enhanced spectral overlap between the cavity and the molecular system. In addition, polariton modes with significant molecular and photonic content have a mean wavevector far from zero, and delocalization is protected from the small fluctuations induced by structural and energetic fluctuations.
The detuning dependence of cavity effects on molecular properties is particularly relevant when viewed in the context of recent research on polariton effects on thermal chemical reactions^{24,25,26,27}. Several of the reported experiments imply that photonic materials are more effective at changing reaction rates when the molecular system is resonant with the lowestenergy mode of a cavity band (zero detuning). Our results suggest otherwise but do not contradict experiments, since we only assess lowenergy properties of the molecular system, whereas the studied slow chemical reactions are complex events often involving higher excitedstates than those included in our model.
Increasing the number of molecules, cavityphoton modes, dimensionality, and introducing dynamical disorder and cavity leakage is unlikely to qualitatively change any of the trends reported in this work. We expect the behavior of molecular observables with cavity detuning, energetic disorder, and density (for a fixed Ω_{R}), to be generic for lowenergy polariton models. For example, while wave function delocalization is significantly more favored in 2D and 3D^{74} relative to the case studied in this paper, it remains true that the spectral overlap between excitons and the optical cavity will be weaker at zero or positive detuning when compared to negative. Dynamical disorder (timedependent fluctuations of the exciton frequency, position, and dipole magnitude/orientation) induces exciton localization and ultrafast coherence decay^{75} further reducing the efficiency of intermolecular exciton transport reported here, but is unlikely to affect any of the qualitative trends reported.
To conclude, we reiterate that although 0D singlemode cavity theories are insightful and predictive of many features of polaritonic optical response, these models give incorrect qualitative predictions for the cavity effects on molecular ensembles analyzed in this work. Specifically, singlemode models fail to capture the detuning, density, and disorder strength dependence of the exciton escape probability and molecular LDOS entropy gain upon collective strong coupling with a photonic device. These shortcomings must be recognized in future model building aimed at predicting novel ways to control chemistry with optical microcavities.
Methods
Exact diagonalization was performed for multiple realizations of the random light–matter Hamiltonian in Eq. (3), and the eigenstates of each realization were employed to compute averages of the molecular ensemble quantities ρ_{M}(E), S[ρ_{M}], and Π_{M} as described in the main text. The criteria used to identify states as polaritonic or weakly coupled (dark) states is discussed under Results and Discussion and specified in the captions of Figs. 5 and 7.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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R.F.R. acknowledges generous startup funds from the Emory University Department of Chemistry.
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Ribeiro, R.F. Multimode polariton effects on molecular energy transport and spectral fluctuations. Commun Chem 5, 48 (2022). https://doi.org/10.1038/s42004022006600
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DOI: https://doi.org/10.1038/s42004022006600
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