Nonequilibrium effects of cavity leakage and vibrational dissipation in thermally-activated polariton chemistry

Strong light-matter interaction, or simply strong coupling (SC), occurs when an optical cavity mode and a material excitation coherently exchange energy faster than either species decays.(Kavokin et al., 2017; Törmä and Barnes, 2015; Bitton et al., 2019; Mueller et al., 2020) This interaction results in light-matter eigenstates called polaritons. In addition to inheriting both photonic and material properties, the hybrid states have different energies than their constituents.

These features make polaritons a promising platform for modifying the physicochemical properties of molecular systems.(Ribeiro et al., 2018a; Flick et al., 2018) Typically, SC in such systems is achieved by the interaction between a cavity mode and $N\gg 1$ quasidegenerate excitations in a molecular ensemble. This collective SC produces two polariton states and $N-1$ dark states. At light-matter resonance, the so-called lower and upper polaritons are respectively redshifted and blueshifted from the bare molecular (photonic) excitations by the collective light-matter coupling strength $g\sqrt{N}$, where $g$ is the single-molecule light-matter coupling strength and $\Omega=2g\sqrt{N}$ is the so-called Rabi splitting. In contrast, the dark states remain unchanged in energy. Nevertheless, the polaritons can dissipate into the dark states,(Agranovich et al., 2003; Virgili et al., 2011; del Pino et al., 2015; Neuman and Aizpurua, 2018; Xiang et al., 2018; Ribeiro et al., 2018b; Scholes et al., 2020) enriching the relaxation dynamics of the SC regime.

In light of the considerable impact that polariton formation has on molecular excited states,(Galego et al., 2015; Kowalewski et al., 2016; Szidarovszky et al., 2018; Haugland et al., 2020) and of the photonic character of polaritons, it is not surprising that SC is emerging as a versatile tool for manipulating photochemistry. It has been shown that, in organic materials, polaritons can lower the activation barrier of spin-conversion processes,(Martínez-Martínez et al., 2018; Stranius et al., 2018; Martínez-Martínez et al., 2019; Yu et al., 2020; Ye et al., 2020) enhance conductivity,(Orgiu et al., 2015; Schachenmayer et al., 2015; Feist and Garcia-Vidal, 2015; Liu et al., 2019; Krainova et al., 2020) change the dynamics of photoinduced charge transfer,(Herrera and Spano, 2016; Groenhof and Toppari, 2018; Wang et al., 2020; Mandal et al., 2020; Hoffmann et al., 2020; Mauro et al., 2020) and alter photoisomerization yield.(Hutchison et al., 2012; Galego et al., 2016; Fregoni et al., 2018; Mandal and Huo, 2019; Fregoni et al., 2020; Antoniou et al., 2020) In some situations, dissipation associated with SC plays a key role in the modification of photochemical processes. Notably, cavity decay (photon leakage from the cavity) can enable the suppression of photodegradation by diverting population from polaritons to the electronic ground state instead of to the product.(Munkhbat et al., 2018; Felicetti et al., 2020; Nefedkin et al., 2020) The effect of cavity decay on photochemical reactivity has also been demonstrated theoretically for photodissociation(Ulusoy and Vendrell, 2020; Davidsson and Kowalewski, 2020) and photoisomerization(Fregoni et al., 2020; Antoniou et al., 2020). Another contributor to the suppression of photodegradation is the relaxation between polaritons and dark states.(Munkhbat et al., 2018; Nefedkin et al., 2020) This dissipative coupling can even mediate polariton-assisted remote energy transfer.(Coles et al., 2014; Zhong et al., 2017; Du et al., 2018; Sáez-Blázquez et al., 2018; Xiang et al., 2020) Moreover, relaxation from polaritons to dark states can be harnessed to realize remote control of photoisomerization.(Du2019)

Much less is understood about the ability to modify thermally activated reactions using SC(Shalabney et al., 2015; Casey and Sparks, 2016) of molecular vibrations, commonly known as vibrational SC (VSC). Recent experiments(Hirai et al., 2020a) demonstrate that reactions can be enhanced or suppressed by VSC without external pumping (e.g., laser excitation). Reactions affected by VSC include organic substitution,(Thomas et al., 2016, 2019; Lather et al., 2019) organic rearrangement,(Hirai et al., 2020b) and hydrolysis.(Hiura2019ChemRxiv) However, the intriguing VSC reaction kinetics still lacks a theoretical underpinning. Transition-state theory, the most commonly used reaction-rate theory, has been unsuccessful at explaining the VSC reactions.(Galego et al., 2019; Campos-Gonzalez-Angulo and Yuen-Zhou, 2020; Zhdanov, 2020; Li et al., 2020a) While a model of nonadiabatic electron transfer under VSC captures some of the observed trends,(Campos-Gonzalez-Angulo et al., 2019) additional connections to experiments have yet to be made. Common to the attempted theoretical approaches is the following assumption: internal thermalization (i.e., within the states of each chemical species) is much faster than reactive transitions (i.e., between the states of different chemical species). Said differently, states of each chemical species are assumed to remain at internal thermal equilibrium throughout the reaction.

Nonequilibrium effects may therefore be relevant in thermally activated reactions modified by VSC. In the context of adiabatic reactions, recent findings(Li et al., 2020b) reveal that non-Markovian dynamics along the cavity-mode coordinate can induce trapping of population in a high-energy region of the potential energy surface, preventing internal thermalization and promoting backward reactive transitions. Alternatively, it would be interesting to explore how VSC affects reactions with significant nonequilibrium effects outside the cavity. For instance, some organic reactions involve the formation of vibrationally hot intermediate species that react before they fully thermalize.(Oyola and Singleton, 2009; Glowacki et al., 2010; Carpenter, 2013) This may happen in desilylation reactions, which feature reactive intermediates(Climent and Feist, 2020) and appear in studies reporting suppression of product formation(Thomas et al., 2016, 2019, 2020) and rise in product selectivity(Thomas et al., 2019) using VSC. If nonequilibrium effects are indeed relevant to the VSC reactions, then another intriguing topic is the role of dissipation, which is what enables thermal equilibrium to be reached.

Here, we carry out a kinetic study of how the dissipative processes that VSC introduces to the chemical system—specifically, cavity decay plus incoherent energy exchange among polaritons and dark states—affects thermally activated reactions with significant nonequilibrium effects. The class of reactions we consider is electron transfer, which is modeled using Marcus-Levich-Jortner (MLJ) theory.(Marcus, 1964; Levich, 1966; Jortner, 1976) We examine reactions where vibrational relaxation and reactive transitions occur on similar timescales. For the case of VSC, we use a generalization of the MLJ model to calculate rates of reactive transitions. Rates associated with internal thermalization are calculated using a standard treatment(del Pino et al., 2015) of polariton relaxation. By comparing reaction kinetics under VSC with those in the bare case and under weak light-matter coupling, we show that dissipation associated with VSC can alter reaction kinetics by suppressing nonequilibrium effects. It follows that the presence of nonequilibrium effects in the bare reaction can be important in determining the influence of VSC on reactivity. Before concluding this paper, we discuss our results in the context of the aforementioned experiments.

In this section, we perform kinetic simulations of thermally activated nonadiabatic electron transfer under VSC. We simulate a set of representative reactions whose reactive transitions are comparable in timescale to internal thermalization. As discussed in Sec. I, we are interested in understanding how such reactions are affected by VSC-induced dissipative processes. To this end, we choose reaction parameters that highlight the impact of cavity decay, as well as relaxation among polaritons and the dark state. In addition, parameters are chosen such that the dynamics of states with more than one cavity-vibrational excitation are insignificant compared to the dynamics of states with one or zero of such excitation (Sec. II.2). Parameters specific to each reaction are listed in Table 1. Throughout the simulations, we use $\hbar\omega_{\text{v}}=2000$ cm^-1 and assume a resonant cavity mode (i.e., $\omega_{\text{c}}=\omega_{\text{v}}$). We also fix the temperature at $T=298$ K. Unless otherwise stated, we use the following parameters to characterize internal thermalization: $\gamma=0.01$ ps^-1 , $\kappa=1$ ps^-1, $\eta=0.001$, and $\omega_{\text{cut}}=0.1\omega_{\text{v}}$. We note that the chosen cavity decay rate ($\kappa$) is much faster than the chosen vibrational decay rate ($\gamma$) and that both rates are similar to those found in VSC experiments.(Xiang et al., 2018; Ribeiro et al., 2018b; Xiang et al., 2020) In calculations of reactions under VSC, we choose $g=(0.03\omega_{\text{v}})/\sqrt{2}$, which yields a Rabi splitting of $\Omega=0.06\omega_{\text{v}}$.

For comparison, we also simulate the reactions in the bare case (see Appendix A for kinetic model) and in the case of weak light-matter coupling (see Appendix B for kinetic model and its derivation). In the latter, we set the light-matter coupling to 1% of the value used for VSC. For all simulations in the weak-coupling regime, cavity decay is fast compared to the overall reaction dynamics (which can have a different timescale than the reactive transitions; see Sec. IV for further discussion). Therefore, the weak light-matter coupling effectively induces relaxation between vibrational and cavity states (Appendix B).(Metzger et al., 2019; Pelton, 2015)

In all simulations, the initial population is a thermal distribution of reactant eigenstates. The distribution is determined by the Boltzmann probability function [i.e., $p_{(\bm{\phi};\mathbf{m})}\propto\exp(-E_{\bm{\phi};\mathbf{m}}/k_{B}T)$] and normalized to 1. In accordance with our truncation of the cavity-vibrational subspace (Sec. II.2), the initial distribution consists only of states with $\leq 1$ excitation in this subspace.

After calculating the population evolution for each state in the system, the total population $N_{\varphi}(t)$ of each reactive species $\varphi$ at time $t$ is computed according to

The projection operator

counts the number of molecules belonging to reactive species $\varphi$.

We first simulate a reaction where the vibrationally hot product undergoes the reverse reaction as fast as it decays (Fig. 1a). The main reactive transition involves a $0\rightarrow 1$ vibrational excitation upon going from reactant to product. The vibrationally hot product either returns to the reactant via the reverse transition or relaxes to the ground state, with both processes occurring at similar rates. This scenario represents a nonequilibrium effect where the product does not fully thermalize before it undergoes the reverse reaction. Under VSC, the reaction is enhanced compared to the bare case (Fig. 1c, blue solid line vs black dashed line). The observed modification is consistent with the following mechanism: dissipative processes associated with VSC speed up internal thermalization and thereby force the product to decay instead of transforming back into reactant. To test this mechanism, we carry out simulations for various values of cavity decay rate $\kappa$ (Fig. 1d) and anharmonic system-bath coupling parameter $\eta$ (Fig. 1e). The latter determines the rates of relaxation among polaritons and the dark state (Fig. 1b, red dotted arrows). As cavity decay rate $\kappa$ is lowered to zero, most of the reaction enhancement goes away (Figs. 1e). In contrast, the modification largely survives as $\eta$ is decreased. These trends reveal not only that the reaction is enhanced by accelerated internal thermalization, but also that cavity decay plays the main role in speeding up the thermalization. Specifically, cavity decay of the polaritons (Fig. 1b, yellow dashed arrows) enables the vibrationally hot product to cool much faster than it can undergo a reverse reactive transition. Relaxation from the dark state to polaritons (Fig. 1b, red dotted arrows) plays a supporting role: it transfers population to the (polariton) states that decay via cavity leakage.

The second reaction we investigate has a reactive transition that starts from a vibrational excited state and occurs on the same timescale as vibrational decay (Fig. 2a). Although thermodynamically unfavorable, this reaction can serve as an instructional example for thermodynamically favorable situations where the most reactive channel involves a vibrational excited state in the reactant. Due to a large energy difference between reactant and product electronic states, there is only one reactive transition, which is accompanied by a $1\rightarrow 0$ vibrational deexcitation. This transition happens at a rate comparable to the decay of the initial vibrational state. Since vibrational decay is extremely fast compared to its reverse process, the timescale of reactant internal thermalization is that of vibrational decay, i.e., that of the reactive transition. Intuitively speaking, the population of the reactive vibrational excited state does not reach its fully replenished (thermalized) value prior to subsequent reactive transitions. In the presence of VSC, the reaction experiences an enhancement (Fig. 2c, blue solid line vs black dashed line). By varying $\kappa$ (Fig. 2d) and $\eta$ (Fig. 2e), we again find that the reaction modification is attributed mostly to cavity decay and less so to relaxation among polaritons and the dark state. Cavity decay facilitates rapid thermalization between the ground state and polaritons (Fig. 2b, yellow dashed arrows), while the dark state quickly reaches thermal equilibrium through incoherent energy transfer with the polaritons (Fig. 2b, blue dotted arrows). As a result of these relaxation dynamics, the excited cavity-vibrational states are refilled with population before the next reactive transition takes place.

While the first two reactions are enhanced by VSC, suppression is found for the third reaction, which involves a vibrationally hot intermediate that reacts before it can fully thermalize (Fig. 3a).(Oyola and Singleton, 2009; Glowacki et al., 2010; Carpenter, 2013) The first reactive transition is from reactant to intermediate and involves a $0\rightarrow 1$ vibrational excitation. Next, the vibrationally hot intermediate can either make a reactive transition to product or relax to the ground state, with similar rates for both processes. The former process is followed by vibrational decay, while the latter is followed by an intermediate-to-product reactive transition that is much slower than vibrational decay. Overall, a significant amount of intermediate population first makes a fast reactive transition and then cools, and the remaining intermediate population first cools and then makes a slow reactive transition. With VSC, the population kinetics of the reactant is unchanged (Fig. 3c, blue solid line vs black dashed line) because the vibrationally hot intermediate, with or without VSC, is transformed into a different state much quicker than it can transition back to reactant. However, there is greater accumulation of the intermediate (Fig. 3d, blue solid line vs black dashed line) and suppressed formation of the product under VSC (Fig. 3e, blue solid line vs black dashed line). Carrying out the same analysis as done with the previous two reactions (see Figs. 3f-3i), we conclude that the rapid decay of polaritons via cavity leakage (Fig. 3b, yellow dashed arrows), assisted by the relaxation from the dark state to polaritons (Fig. 3b, green dotted arrows), causes the intermediate to reach thermal equilibrium before it can go through the faster reactive transition.

To obtain additional insight regarding nonequilibrium effects and how VSC alters reactivity, we study the reactions under weak light-matter coupling. Figs. 1c, 2c, 3d-3e show population kinetics for VSC (blue solid line), weak light-matter coupling (aquamarine dotted line), and the bare case (black dashed line). For all reactions, weak light-matter coupling leads to significantly modified kinetics. The direction of change is the same as that of VSC. This finding is consistent with the fact that, similar to VSC, weak light-matter coupling opens up relaxation between vibrations and the cavity, and enables internal thermalization of the reactive species to be sped up by cavity decay. It also makes sense that the magnitude of change is less than that of VSC, since the vibration-cavity relaxation resulting from weak light-matter interaction is not as fast as cavity decay.(Metzger et al., 2019) Nevertheless, it is interesting that one only needs to enter the weak-coupling regime to manipulate reactivity. The same conclusion has been reached in a study of how cavity decay allows light-matter coupling to protect molecules from photodegradation.(Felicetti et al., 2020)

The simulations discussed so far suggest that the VSC-induced modifications to the above reactions arise mainly from the suppression of nonequilibrium effects. To strengthen this claim, we simulate the same reactions except the vibrational decay rate is made 100 times larger (i.e., $\gamma=1\text{ ps}^{-1}=\kappa$). With this condition, we find that the VSC and bare kinetics (Fig. 4, blue solid line vs black dashed line) are either virtually identical or much more similar than in the case of slow vibrational decay. Thus, the changes in reactivity due to VSC are considerably reduced when, outside the cavity, internal thermalization is already rapid compared to reactive transitions.

While our focus has been on nonequilibrium effects and their suppression by VSC-related dissipation, we should note that the calculated reaction kinetics do exhibit some changes as a consequence of VSC modifying activation energies and redistributing vibronic coupling among the cavity-vibrational eigenmodes (Sec. II.2). Because cavity decay has been identified as the major contributor to the altered kinetics in the above reactions, the impact of the minor contributors could be revealed by comparing simulations for VSC and no cavity decay (i.e., $\kappa=0$ ps^-1) with those for the bare case. For the reaction of Fig. 1, the kinetics is noticeably different for the two conditions (Fig. 1d, yellow solid line vs black dashed line). It seems that the difference cannot be fully accounted for by the presence of relaxation among polaritons and dark state (Fig. 1e). Hence, modified energy differences and Franck-Condon factors likely impart a small but appreciable influence on the reaction. This conclusion is corroborated by the slight enhancement observed for the VSC reaction when vibrational decay is made faster than all reactive transitions (Fig. 4a, blue solid line vs black dashed line); in this regime, dissipative processes introduced by VSC should have no effect. For further discussion of how changes in energetics and Franck-Condon factors due to VSC manifest as changes in reactivity, we refer the reader to Refs. (Campos-Gonzalez-Angulo et al., 2019) and (Phuc et al., 2020) (see also Ref. (Semenov and Nitzan, 2019), which discusses these topics but in the context of SC to electronic states).

A noteworthy difference between our simulations and the experiments of VSC reactions is the number $N$ of molecules involved in VSC. In our simulations $N=2$. In the experiments, $N$ is estimated to be $10^{6}$ to $10^{12}$.(Campos-Gonzalez-Angulo et al., 2019; del Pino et al., 2015; Vurgaftman et al., 2020)

For such large $N$, the particular reactions studied here are unlikely to be modified by VSC. To understand this statement and to identify reactions that would be significantly affected by many-molecule VSC, we present the following argument. Without loss of generality, consider a reaction featuring a single forward reactive transition, which involves a $0\rightarrow 1$ vibrational excitation. This hypothetical reaction resembles that depicted in Fig. 1a. Assuming the steady-state approximation (SSA) for the vibrationally excited product and ignoring vibrational gain (i.e., the reverse process of vibrational decay), the reaction rate can be written as $k_{R\rightarrow P}=k_{f}[k_{d}/(k_{r}+k_{d})]$, where $k_{f}$ is the rate of the forward reactive transition, $k_{r}$ is the rate of the reverse reactive transition, and $k_{d}$ is the decay rate of the vibrationally excited product. The quantity in square brackets, $k_{d}/(k_{r}+k_{d})$, is the efficiency with which the vibrationally excited product decays rather than returns to reactant. In the reaction with VSC, there are $N+1$ forward reactive transitions, each involving the $0\rightarrow 1$ excitation of a polariton or dark state; there are two and $N-1$ such transitions, respectively. We first focus on the two polaritonic forward reactive transitions. Recall that the Franck-Condon factor for a 0-1 vibronic transition is proportional to the displacement between initial and final vibrational states [Eq. (20) with $m_{q}^{\prime}=1$ and $m_{q}=0$]. Since the overlap between a polariton mode and each bare vibrational mode is $O(N^{-1/2})$ [Eq. (10)], the displacements—and therefore the 0-1 Franck-Condon factors—associated with the polaritons are $O(N^{-1/2})$ times the corresponding bare values [Eqs. (14)-(15)]. Then the polaritonic reactive transitions have rates that scale as $O(N^{-1})$ because the rate of a reactive transition depends on the square of a Franck-Condon factor [Eq. (18)]. Suppose that cavity decay is much faster than all other transitions from the polaritons, including reverse reactive transitions and relaxation to dark states. This scenario, which has been observed for cavity-coupled W(CO)₆ in nonpolar solvents,(Xiang et al., 2019) results in a decay efficiency of 1 for product states with an excitation in a polariton mode. By applying the SSA to such states and neglecting cavity gain (i.e., the reverse process of cavity decay), we arrive at the following effective rate for the polaritonic reactive transitions: $k_{R\rightarrow P}^{(p)}=\epsilon k_{f}/N$, where $\epsilon$ is a dimensionless constant that accounts for changes in the reaction rate caused by the polaritons having modified energies. In contrast, we assume that the reactive transitions involving the dark states do not afford changes in the reaction rate, given that the dark states have similar decay dynamics to the bare vibrations.(Agranovich et al., 2003; del Pino et al., 2015; Dunkelberger et al., 2016, 2018; Xiang et al., 2018; Ribeiro et al., 2018b; Xiang et al., 2019) Furthermore, relaxation from dark states to polaritons should be negligible: this dissipative process is mediated by local system-bath interactions (see Sec. II.2) and has rates scaling as $O(N^{-1})$,(del Pino et al., 2015; Du et al., 2018; Martínez-Martínez et al., 2018) corresponding to the probability of each molecule in either polariton [Eq. (10)]. So, in order for the reaction to be modified by VSC, the reaction rate ($k_{R\rightarrow P}^{(p)}$) due to the polaritonic reactive transitions should be, at least, comparable to the bare reaction rate ($k_{R\rightarrow P}$), i.e.,

With arbitrary reactions in mind, criterion (25) says that for systems with experimentally relevant values of $N$, VSC can change the kinetics of thermally activated reactions if

1. 1.

   the polaritons give rise to $\epsilon\gg 1$, which means that activation energies are significantly reduced compared to the bare case (this scenario is discussed in Ref. (Campos-Gonzalez-Angulo et al., 2019)), and/or

2. 2.

   in the bare case, reverse reactive transitions are extremely fast compared to internal thermalization, i.e., populations are efficiently trapped in the reactants.

When at least one of these two conditions is satisfied, a large-$N$ generalization of the kinetic model presented in Sec. II.2 should predict altered reactivity under VSC. Indeed, Ref. (Campos-Gonzalez-Angulo et al., 2019) theoretically demonstrates the ability for condition 1 to enable VSC catalysis of ground-state reactions when $N=10^{7}$. While such a demonstration has not been carried out for condition 2, Ref. (Polak et al., 2020) (see Fig. 6a of that work) has shown that condition 2 enables electronic SC to boost the efficiency of harvesting triplet excitons into singlet polaritons, ultimately leading to enhanced photoluminescence compared to the bare organic material (in this case, the reverse reactive transition is the very rapid fission of singlets into triplets).

Despite the above and other differences between this theoretical work and the experiments, the suppression of nonequilibrium effects is a general phenomena and could be relevant to the mechanism by which VSC modifies reaction kinetics in the latter. According to our calculations, the suppression of nonequilibrium effects can lead to significant increase or decrease in reactivity, depending on the specific properties of the reaction. Both types of modifications have been found experimentally.(Hirai et al., 2020a) The resemblance does not stop there. Recall the experimental studies showing that the desilylation of PTA using TBAF is suppressed by VSC.(Thomas et al., 2016, 2020) This reaction involves an intermediate species.(Climent and Feist, 2020) Based on kinetic measurements, the authors propose that VSC forces the reaction to go through a new pathway, which involves a different intermediate. Compared to the intermediate of the bare reaction, the intermediate in the VSC reaction forms more easily but reacts less readily. We would like to propose an alternative mechanism based on our results. In the bare case, there could be nonequilibrium effects that cause slow conversion of reactant to intermediate (similar to the reaction of Fig. 2) and fast conversion of intermediate to product (like in the reaction of Fig. 3; see Refs. (Oyola and Singleton, 2009), (Glowacki et al., 2010), and (Carpenter, 2013)). Under VSC, the intermediate would be the same as in the bare case, but nonequilibrium effects would be suppressed, thereby accelerating the reactant-to-intermediate transformation and decelerating the intermediate-to-product transformation. This mechanism is consistent with the experimentally proposed one. Future works should test the possibility of VSC-induced suppression of nonequilibrium effects in the desilylation and other reactions.

When evaluating this possibility, it is important to keep in mind that nonequilibrium effects can affect reactivity even if internal thermalization occurs on a much shorter timescale than the reaction. Whether or not nonequilibrium effects are present is determined by how fast internal thermalization is relative to reactive transitions, which may be faster than the (overall) reaction. These guiding principles are supported by our simulations. For the bare reactions shown in Figs. 1 and 3, internal thermalization occurs on a 100 ps timescale while the reaction occurs on a 10 ns timescale (Figs. 1a, 1c, 3a, 3c). Nevertheless, as already discussed in Sec. III, the ability of VSC—in particular, the associated relaxation processes—to change the reaction kinetics relies on having (in the bare case) reactive transitions with similar rates as internal thermalization. The above principles, when made specific to adiabatic reactions, can be stated as follows:(Leitner and Wolynes, 1997) regardless of the actual reaction rate or that predicted by transition-state theory, the influence that nonequilibrium effects have on reactivity depends on how the rate of internal thermalization compares to the reactive flux (i.e., the rate of product formation per unit population) at the transition-state barrier.

In this work, we study how reactions with significant nonequilibrium effects are influenced by the dissipative channels that VSC introduces to the chemical system. By using the MLJ formalism of electron transfer as our reaction model, we present a kinetic framework that captures reactive transitions, vibrational decay, cavity decay, and relaxation among polaritons and dark states. We then simulate reactions where internal thermalization and reactive transitions occur on the same timescale. The considered reactions respectively exhibit three representative nonequilibrium effects: vibrationally hot product transforming back to reactant while thermalizing (Fig. 1), slow replenishing of reactive vibrational excited state after a reactive transition (Fig. 2), and vibrationally hot intermediate reacting before it can fully thermalize (Fig. 3). Under VSC, the first two reactions are enhanced whereas the last reaction is suppressed. These modifications largely occur because nonequilibrium effects are suppressed by VSC-induced dissipative processes. The suppression of nonequilibrium effects is driven by cavity decay, which causes polaritons to thermalize much faster than reactive transitions occur. Relaxation between polaritons and dark states allows the latter to indirectly experience accelerated thermalization due to cavity decay. When we substantially increase the vibrational decay rate, i.e., make internal thermalization much faster than reactive transitions (for the cavity-free case), the bare and VSC reactivities become much closer (Fig. 4). Finally, we discuss our work in the context of recent experiments,(Hirai et al., 2020a) which demonstrate that thermally activated reactions can be enhanced or suppressed by the collective VSC of a large number of molecules. We identify types of reactions for which our theory would predict modified kinetics induced by many-molecule VSC. We also highlight resemblances between our results and experimental observations. These resemblances suggest that VSC-triggered suppression of nonequilibrium effects could play a role in the experiments.

The data that support the findings of this study are available from the corresponding author upon reasonable request.