A path towards single molecule vibrational strong coupling in a Fabry–Pérot microcavity

Interaction between light and molecular vibrations leads to hybrid light-matter states called vibrational polaritons. Even though many intriguing phenomena have been predicted for single-molecule vibrational strong coupling (VSC), several studies suggest that these effects tend to be diminished in the many-molecule regime due to the presence of dark states. Achieving single or few-molecule vibrational polaritons has been constrained by the need for fabricating extremely small mode volume infrared cavities. In this theoretical work, we propose an alternative strategy to achieve single-molecule VSC in a cavity-enhanced Raman spectroscopy (CERS) setup, based on the physics of cavity optomechanics. We then present a scheme harnessing few-molecule VSC to thermodynamically couple two reactions, such that a spontaneous electron transfer can now fuel a thermodynamically uphill reaction that was non-spontaneous outside the cavity.


Introduction
Strong coupling (SC) ensues when the rate of coherent energy exchange between the matter degrees of freedom (DOF) and a conned electromagnetic eld exceeds the losses from either of them. [1][2][3][4] This interplay leads to the emergence of hybrid lightmatter states called polaritons, 5-7 which inherit properties from both the photonic and the matter constituents. For molecular systems, due to the small magnitude of the transition dipole moment of most individual molecules, SC is typically achieved by having an ensemble of N [ 1 molecules interact with a cavity mode. 5,8 In this collective case, in addition to two polariton states, SC leads to (N − 1) dark states which are predominantly molecular in character. 9 In both the electronic and vibrational regimes, harnessing these hybrid light-matter states has led to the emergence of a plethora of polaritonbased devices 10 such as ampliers, 11,12 tunneling diodes, 13 routers, 14 and ultrafast switches; 15,16 and novel phenomena like enhanced energy and charge transport, [17][18][19] modication and control of a chemical reaction without external pumping, 20,21 and remote catalysis. 22 Theoretical models of polaritons oen use a single molecule with a collective super radiant coupling to the cavity to explain the experimentally observed effects of collective SC on physical and chemical phenomena. [23][24][25][26] However, several theoretical studies, that account for the large number of molecules coupled to the cavity, suggest that SC could be rendered less effective in the collective regime owing to the entropic penalty from the dark states. 6,27,28 For enhanced polaritonic effects, the state-ofthe-art is either to use polariton condensates 29,30 or to achieve single-molecule SC. 31 In the electronic regime, both polariton condensation [32][33][34][35] and single-molecule SC 31, [36][37][38] have been achieved. There have been theoretical proposals of ways to achieve a vibrational polariton condensate. 39 However, to the best of our knowledge, in the vibrational regime, neither condensation nor single-molecule SC has yet been experimentally demonstrated. The bottleneck for single molecule SC in the vibrational case is the fabrication of low-mode volume cavities in the infrared (IR) regime. 40 This calls for alternate strategies to attain vibrational SC with a single or few molecules.
In this work, we propose using optomechanics as a way to achieve SC for molecular vibrations. Over the last decade, optomechanics has emerged as a powerful tool in quantum technologies with applications 41 such as backaction cooling of a mechanical oscillator, 42,43 parametric amplication, 44-46 optomechanically induced transparency, 47,48 and generation of nonclassical quantum states. 49,50 Aspelmeyer and co-workers have demonstrated SC in an optomechanical architecture, for a micromechanical resonator coupled to an optical cavity setup. 51 It has been shown recently that surface-enhanced and cavity-enhanced Raman spectroscopy (SERS/CERS) can be understood through the theoretical framework of cavity optomechanics. [52][53][54][55][56][57] Here we exploit this observation to demonstrate that a single molecule in a CERS setup, under strong illumination of a red-detuned laser can be a viable platform to achieve the long-standing goal of single and fewmolecule vibrational polaritons. Few-molecule polaritons do not suffer from the deleterious effects of a macroscopic number of dark states, and hence are better candidates for harnessing the properties of polaritons. 5 As a proof-of-concept application of few-molecule vibrational polaritons, we will introduce the intriguing concept of coupling chemical reactions via the latter. Biological systems use coupled chemical reactions and thermodynamics to their advantage by driving energetically uphill processes, such as active transport, using spontaneous reactions, like the dissociation of ATP. 58 Humans have looked towards nature for inspiration and translated biological knowledge into innovative products and processes. 59,60 We shall show how the delocalization of the polariton modes inside the cavity can be exploited to design a biomimetic of ATP-driven molecular machines.

Model
Our theoretical model considers a single molecule in condensed phase placed inside a UV-vis cavity, such that the cavity frequency is off-resonant and lower than any optically allowed transitions of the molecule. In this regime, the coupling between the cavity and the molecule is purely parametric through the molecule's polarizability, with the vibration of the molecule causing a dispersive shi in the cavity resonance. 52 Due to better spatial overlap between the mode prole of the cavity and the molecule, we consider using a dielectric Fabry-Perót cavity. However, the formalism presented here is valid for other cavity types. 61,62 We show that the cavity-molecule system, when pumped with a laser, off-resonant to electronic transition of the molecule, and red-detuned from the cavity, with the detuning in the order of the molecule's vibrational frequency (Fig. 1b), yields an effective Hamiltonian resembling the vibrational polaritonic Hamiltonian. Importantly, the light-matter coupling can be tuned by varying the laser power. By deploying efficient off-resonant light-matter interactions, this scheme represents an attractive opportunity to explore strong lightmatter coupling phenomena without worrying about resonant interactions that could induce deleterious photodamage of molecules.
In this setup, an anti-Stokes Raman process creates a composite photon mode by absorption of a red-detuned laser photon and subsequent emission of a cavity photon. When the laser-cavity detuning is parked at the vibrational frequency of the molecule, the composite photon is in resonance with the molecular vibration and couples to it (Fig. 1b), with an effective light-matter coupling strength that can be tuned by changing the laser intensity. When the coupling strength exceeds the decay rates of the system, we see polariton peaks in the spectra. Importantly, and contrary to the perturbative description of anti-Stokes Raman scattering, the presently studied effect occurs even at T = 0, in the absence of initial populations in the vibrational excited states of the molecules.
We model the photon mode and the vibration of the molecule as harmonic oscillators with frequencies u cav and U v , and annihilation operators a and b, respectively. The cavity has a decay rate of k and the vibrational mode has a decay rate of g. In this work, the losses will be modeled using Lindblad master equations. 63 Since the polarizability, a, of the molecule, to the leading order, depends on its vibrational displacement, 56 , the Hamiltonian for the cavity-molecule system is 52 is the vacuum cavity-molecule coupling, with V c and 3 0 as the mode volume of the cavity and vacuum permittivity, respectively. This Hamiltonian formally resembles an optomechanical setup, 52 where the displacement of a mechanical oscillator modulates the frequency of the cavity. The cavity then acts back on the oscillator through radiation Fig. 1 Experimental setup for achieving few-molecule vibrational SC in a cavity-enhanced Raman spectroscopy (CERS) setup. (a) A single molecule with a vibrational mode at frequency U v is placed inside a UV-vis cavity (u cav ) and the cavity is illuminated with a laser red-detuned from the cavity (u cav − U v ). The cavity mode and the laser detuned to any electronic transition of the molecule. The laser and the cavity together constitute a composite photon mode that strongly couples to the vibrational mode of the molecule. (b) Schematic depicting the coupling of the composite photon mode denoted by annihilation operatorl †ã with the molecular vibration denoted with annihilation operator b. Herel denotes the 'laser-like' andã denotes the 'cavity-like' normal mode of the laser-cavity subsystem. The composite photon mode and the molecular vibration strongly couple with effective coupling g eff , to give the polaritons. pressure force, which is a function of the cavity's photon occupation.
We drive the cavity with a laser red-detuned (u L = u cav − D, D > 0) from the cavity resonance. The full Hamiltonian for a laser mode with annihilation operator l coupled to the cavitymolecule subsystem is given as with the cavity-laser coupling J ¼ Here, s rt L is related to the laser power P ¼ n L ħu L s rtL with n L = hl † li being the mean photon number in the laser mode. 64 We make the rotating wave approximation (RWA) in the laser-cavity coupling and diagonalize the laser-cavity subsystem. The operatorsl andã represent the new 'laser-like' and 'cavitylike' normal modes of the laser-cavity subsystem with frequenciesũ L andũ cav , respectively. The Hamiltonian aer making the approximation and change of basis is is the laser-cavity mixing angle. 65 Considering the red-detuned case for the cavity-laser detuning, D, in the order of O(U v ), we drop the off-resonant contributions in the cavity-molecule interaction term, simplifying H full to We will later set the laser-cavity detuning D = U v . We dene a composite laser-cavity photon mode with annihilation operator A ph ¼l †ã ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðñ L Àñ a Þ p (Fig. 1b), where ñ L = hl †l i and ñ a = hã †ã i are the mean photon occupations in the 'laserlike' and 'cavity-like' normal modes, respectively. The Heisenberg equations of motion for the operators A ph and b in the mean-eld approximation, 66 yield the effective Hamiltonian when n L [ ñ a and J ( D (see ESI note-1 †). Here we have suggestively dened u ph h D. For an input laser drive with power P, the interaction strength between the composite photon and the molecular vibration transforms to Pk ħu L r , consistent with the results obtained from the classical treatments of the laser mode. 41,55,57,64 When u ph = U v , H eff resembles a vibrational polaritonic Hamiltonian, where the composite photon mode is resonant with the vibrational DOF (Fig. 1b). 5 Here the coupling strength is tunable by changing the pumping power of the laser. This can, in principle, foster the SC regime when the coupling strength supersedes the decay processes in the system. To look at parameter sets yielding this regime and to compute spectra, we simulate the dynamics of the density matrix (r) of the system using Lindblad master equations 56,63 given as The last four terms on the right-hand side are the Lindblad-Kossakowski terms dened as Here, L A models the incoherent decay from the composite photon mode. The incoherent decay, thermal pumping, and pure dephasing of the vibrational mode by the environment at temperature T are modeled by the L b ; L b † and L b † b terms, respectively, where n th v =(e ħUv/kBT −1) −1 is the Bose-Einstein distribution function at transition energy ħU v . Additionally, in the limit of large photon number in the laser (n L [ 1), assuming the photon occupation to be constant, and thus the laser mode to be non-lossy, the decay rate of the composite photon equals the cavity decay rate k (see ESI note-2 †).
The simulations have been performed using the QuTip package 67,68 and the results are presented in Fig. 2 for the molecule Rhodamine 6G, 52 Here, g r = 10 −4 U v (1.7 × 10 −5 eV) and g pd are the rates for vibrational relaxation and pure dephasing, respectively. 6 The temperature chosen for the simulations is T = 298 K. The uence of the lasers is chosen to be below ∼10 MW cm −2 (ref. 52 and 69) with a beam area of A = 5 mm 2 . Fig. 2a shows the effective light-matter coupling, g eff , as a function of laser uence, P/A, and single photon coupling strength, g 0 . In Fig. 2b, the vibrational spectrum of the molecule S b ðuÞ ¼ Re½ Ð N 0 e Àiut hx v ðtÞx v ð0Þi ss dt, splits, demonstrating SC. We see the Rabi-splitting increases with laser power, thus giving us additional control over the light-matter coupling strength. Fig. 2c shows the spectra with one, two, and four molecules for constant laser power. Fig. 2d is the vibrational spectrum of the molecule as a function of the cavity-laser detuning. The avoided crossing at the detuning (D) equal to the vibrational frequency (U v ) demonstrates maximal hybridization between the photonic and matter DOF. Finally, Fig. 2e

Polariton-assisted thermodynamic driving
The matter component of the polariton modes is delocalized over many molecules under collective SC. 5,72 This delocalization can be exploited more effectively with few-molecule polaritons, owing to reduced involvement of dark modes, 9,31 which remain parked essentially at the same energy as the original molecular transitions. In this work, we consider the molecular species undergoing electron-transfer reactions, 73,74 modeled using Marcus-Levich-Jortner (MLJ) theory. [75][76][77] Our system consists of two reactive molecules A and B of different species placed inside an optomechanical cavity. Here molecule A features a spontaneous reaction (with negative free energy change, DG A < 0), while molecule B features an endergonic reaction (DG B > 0, with DG B > k B T). We demonstrate thermodynamic coupling between the two molecular species via the composite photon mode, such that the spontaneous electron transfer in A can drive B to react. Schematically, electron transfer in A creates a vibrationally hot product (Fig. 3a), which, outside the cavity, just decays to the product ground state. However, inside the cavity, in the timescale of the Rabi frequency, this excitation can be captured by the photon mode, which then can excite the reactant in B to its vibrational excited state. The electron transfer in B can then proceed spontaneously from the reactant's excited state (Fig. 3b). Notably, this scheme can also be generalized to other types of reactions. We emphasize that throughout this process there is no direct photoexcitation (preparation of vibrationally excited state population) of the molecules by light. This feature arises from the off-resonant nature of the parametrically induced strong light-matter coupling, and represents an opportunity to avoid undesired photochemical processes in molecules due to resonant multiphoton excitations.
Within the framework of MLJ theory, the molecules can exist in either of the two diabatic electronic states: jR i i corresponding to the reactant and jP i i corresponding to the product for molecule i˛{A, B} (Fig. 3a). For molecule B, in this case the switching between jR B i and jP B i through electron transfer contributes to useful mechanical work, manifested in changes of nuclear conguration. [78][79][80] The electronic states for each molecule are dressed with a local high-frequency intramolecular vibrational coordinate represented by annihilation operator a x,i , for x˛{R,P}, and coupled to a low-frequency effective solvent mode treated classically with rescaled momentum and position as p S,i and q S,i . respectively. We assume that the high-frequency modes of both species, being resonant, are the only ones that couple to the composite photon. 81 Upon reaction, the high-frequency mode undergoes a change in its equilibrium conguration according to, a R,i = D † i a P,i D i , where D i ¼ exp½ða † P;i À a P;i Þ ffiffiffiffi S i p is the displacement operator, and S i is the Huang-Rhys factor. 39 The Hamiltonian describing the system is given as H = H 0 + V react , where K, and A = 5 mm 2 unless otherwise mentioned. Here is the bare Hamiltonian corresponding to the composite photon mode consisting of the laser and the cavity, H x,i represents the high-frequency mode and the solvent mode associated with molecule x i , with d S,i and U S,i being the displacement and frequency along the solvent coordinate, respectively, and DG i , the free energy difference for the molecular species i. Additionally, x;i A ph Þ is the effective coupling between the photonic and molecular DOF. For simplicity, we assume that the reaction involves a vibrational mode with nearly identical frequency and light-matter coupling strength for species A and B in both the reactant and product electronic states (U x,i = U y,j hU v and g x,i = g y,j h g). Finally, the diabatic couplings between the electronic states jR i i and jP i i are given by V react , where J i is the coupling strength.
We can solve H 0 parametrically as a function of the solvent coordinates to construct the potential energy surfaces(PES) (Fig. 4a and b). Considering these diabatic couplings V react to be perturbative, H 0 can be diagonalized to obtain the two polariton modes, a x A ,y B (±) , with frequencies such that, c x,A + c y,B = 0 and jc x,A j 2 + jc y,B j 2 = 1. Here, the detuning between the composite photon mode and the molecule. Here we have chosen the composite photon mode to be resonant with the intramolecular vibration, i.e., d = 0.
We now dene multi-particle states jf;n f i that span the Hilbert space of the system, where jfi = jx A , y B i, x, y˛{R,P} corresponds to the electronic DOF, and jn f i = jn + xA,yB ,n − xA,yB ,n D xA,yB i to the cavity-vibrational mode of each electronic state jfi. 6 To describe the reaction, we look at the population dynamics in the electronic states. The kinetic master equations governing the time evolution of the system are given as 6 dp ðf;nfÞ ðtÞ dt ¼ À  Step-1: we start from the reactant electronic states and vibrational ground states in both molecules (jR A ;0i, jR B ;0i).
Step-2: as molecule A reacts spontaneously (jR A ;0i/jP A ;1i), the polariton modes, jP A ,R B ;1 ± i, being delocalized also promote vibrational excitation in B from jR B ;0i to jR B ;1i.
Step-3: this allows B to react from its excited state (jR B ;1i/jP B ;0i).
Step-4: finally, both A and B relax to jP A ;0i and jR B ;0i, respectively, after which the molecule A needs to be replaced for the next cycle (PES have been drawn not to scale, to emphasize the mechanism.).
where p (f;n f ) (t) represents the population in jf;n f i state, and kðf 0 ; n 0 f 0 f; n f Þ is the rate constant for population transfer from jf;n f i to f 0 ; n 00 f i due to processes like reactive transitions between the electronic states accompanied by solvent reorganization and decay through the cavity and vibrational DOF. The rate constant for the reactive transition at a temperature T within the framework of MLJ theory is given as 81 Here, For the simulations in this work, the Franck-Condon factors have been computed numerically from eigenstates obtained using the standard discrete-variable representation (DVR) of Colbert and Miller. 82 The reactive transitions transfer populations across different electronic states, while the cavity and vibrational decays lead to dynamics within the same electronic state. In these simulations, since k B T ( ħU v , we restrict ourselves to the rst excitation manifold in the photon-vibration DOF. With the bare vibrational decay rate for the intramolecular vibrations for molecule A and B as g A and g B , respectively, and bare cavity decay rate as k, we have k(f,1 q,f jf,0) = jc q0 j 2 k + jc qA j 2 g A + jc qB j 2 g B , where 1 q,f represents a single excitation in the polaritons (q = ±) or the dark (q = D) mode, and the c qj 's correspond to the expansion coefficients of the excited eigenmode in terms of the cavity and vibrational modes. 83 Finally, the anharmonic couplings between the vibrational mode of interest and an other bath of low frequency modes leads to transitions between the polaritons and the dark mode, 6 1ħU v , T = 298 K, and h = 0.0001 unless otherwise specified.
where Q(U) is the heavyside step function, n th (U) is the Bose-Einstein distribution function at the transition energy ħU = ħ(U q ′ − U q ), and J ðUÞ is the spectral density of the low frequency modes. Assuming the spectral density to be ohmic, 84 we have J ðUÞ ¼ hU exp½ÀðU=U cut Þ 2 , where h is a dimensionless parameter modeling the anharmonic system-bath interactions and U cut is the cut-off frequency for the low-frequency modes.
The results of the simulations are presented in Fig. 4. Here, ħU R,i = ħU P,i = 0.22 eV (call ħU v ), 7ħU v (0.15 eV), ħU cut = 0.1ħU v (2.2 × 10 −2 eV), and h = 0.0001. 6 The decay rates have been chosen to be similar to those typically found in VSC experiments. 66,85,86 The diabatic couplings are chosen to be ħJ A = ħJ B = 0.005ħU v =(1.1 × 10 −3 eV) and T = 298 K. We start from the initial electronic state jR A ,R B ;0i. Independently, the reaction of molecule A is spontaneous due to its negative free energy change, DG A < 0, while molecule B remains in its thermodynamically stable conformer jR B ;0i ( Fig. 4a and b). This reects the dynamics of the species outside of the cavity. Placing both the molecules inside the cavity couples the two reactions via the photonic mode enabling the spontaneity of the reaction of molecule A to thermodynamically 'li' B to its unstable conguration jP B ;0i producing mechanical work. However, aer molecule A has fully reacted (change in nuclear conguration), inevitably B has to relax again to its stable conguration jR B ;0i, completing one cycle of the mechanical motion of B (Fig. 3b). The maximum population obtained in jP B i before molecule B relaxes back to jR B i increases with the light-matter coupling strength (g eff ), tunable with the uence of the driving laser (Fig. 4d). For the cycle to be repeated, molecule A needs to be 'recharged' or 'replaced'. To achieve this, we envision a ow setup, as schematically depicted in Fig. 4c, that can circulate molecule A inside and out of the cavity. Continued circulation of the A molecules is essential for the molecular machine of B to be oscillating between reactant and product and producing mechanical work ( Fig. 4a and b). This phenomenon realizes a heat engine producing mechanical work in molecule B, using the (chemical) energy ow from a 'source' to a 'sink' bath in the form of molecule A. 87

Conclusion
We have shown that the physics of cavity optomechanics can be harnessed in CERS to achieve single to few-molecule vibrational SC using laser-driven UV-vis cavities. We show that the coupling strength and hence the Rabi splitting is tunable with the laser intensity, and it is achievable with realistic pump powers and cavity-molecule couplings. SC in the few molecules regime can avail enhanced polaritonic effects owing to the reduced entropic penalty from the dark states. By using the MLJ theory for electron transfer, we show that the photon-mediated coupling between two reactions, one spontaneous and one non-spontaneous, can be exploited to thermodynamically drive the non-spontaneous process using the spontaneous one. This effect is analogous to harnessing ATP to drive uphill biological processes like the active transport of ions across a membrane against their concentration gradient and can be used to design bio-inspired molecular machines. 88 Moving forward, an experimental realization of the scheme for vibrational SC presented here would be a signicant step towards utilizing polaritons for chemistry.

Conflicts of interest
The authors declare no competing interests.