Vibropolaritonic Reaction Rates in the Collective Strong Coupling Regime: Pollak–Grabert–Hänggi Theory

Abstract

Following experimental evidence that vibrational polaritons, formed from collective vibrational strong coupling (VSC) in optical microcavities, can modify ground-state reaction rates, a spate of theoretical explanations relying on cavity-induced frictions have been proposed through the Pollak−Grabert−Hanggi (PGH) theory, which goes beyond transition state theory (TST). However, by considering only a single reacting molecule coupled to light, these works do not capture the ensemble effects present in experiments. Moreover, the relevant light−matter coupling should have been √N times smaller than those used by preceding works, where N ≈ 10 6 to 10 12 is the ensemble size. In this work, we explain why this distinction is significant and can nullify effects from these cavity-induced frictions. By analytically extending the cavity PGH model to realistic values of N, we show how this model succumbs to the polariton "large N problem", that is, the situation whereby the single reacting molecule feels only a tiny 1/N part of the collective light−matter interaction intensity, where N is large.

Understanding the Energy Gap Law under Vibrational Strong Coupling

“…Physically, this implies that the cavity’s coupling to the single decaying molecule dominates the relaxation dynamics, and further interactions between the N – 1 nondecaying molecules and the single decaying one, through the cavity, appear only as higher-order processes. This effect, which is further explained by Du and co-workers, is essentially the message from the polariton “large N problem” (more to follow).…”

“…49 Therefore, for some constant and small g√N (relative to ω vib ), we expect VSC to enhance the decay rate at small N, since this implies having a system with larger g. Physically, this implies that the cavity's coupling to the single decaying molecule dominates the relaxation dynamics, and further interactions between the N − 1 nondecaying molecules and the single decaying one, through the cavity, appear only as higher-order processes. This effect, which is further explained by Du and co-workers, 50 (see Supplementary Note 4); effectively, this approach assumes a continuum of final dark states with discrete final polariton states or, equivalently, that most of the energy is being distributed through the dark mode. With this alternative, more intuitive approach, the polariton modes' contribution to the decay rate may be separately identified.…”

On the Mechanism of Polaritonic Rate Suppression from Quantum Transition Paths

“…The resulting system Hamiltonian, H s , is H s = P 2 / ( 2 M ) + U false( R false) + p c 2 / 2 + ω c 2 2 ( q c + 2 / ( ℏ ω c 3 ) χ μ false( R false) ) 2 where P and R are the proton momentum and position, p c and q c are the corresponding photon coordinates, ω c is the photon frequency, ℏ is Planck’s constant, μ( R ) is the proton dipole operator, U ( R ) is the potential energy of the proton coordinate, and χ is a parameter which controls the coupling strength between light and matter. The coupling of the cavity to the system dipole should be interpreted as being dependent on the number of reactive molecules in the cavity, which, under the assumption that the dipolar molecules’ motion is independent and isotropic, can be decoupled . The resultant Born–Oppenheimer surface is given by E ( R , q c ) = H s − P 2 / ( 2 M ) − p c 2 / 2 Note that we do not consider the effects of cavity leakage here, which has been shown to be important in certain regimes. , …”

“…The bath is envisioned to include all nonreactive modes of the system, including molecular and solvent modes. Additionally, the bath captures interactions between the reactive mode of the molecule and other reactive molecules whose dipoles are aligned with the cavity . The bath is approximated by an infinite set of harmonic oscillators that relax quickly in comparison to the system dynamics, which allows the bath effects to be addressed perturbatively via the Born–Markov approximation. − The coupling operator involves R , the position operator of the proton, via the tensor product with B = ∑ k c k R k , a sum over position coordinates, R k , of the bath harmonic oscillators, with coupling strength parameters, c k , determined by the spectral density, J false( ω false) = π 2 ∑ k c k 2 ω k δ false( ω − ω k false) = η ω e − false| ω false| / ω b in which ω k is the frequency of bath oscillator k , ω b is the bath cutoff frequency, and η is the system–bath coupling strength.…”

“…It is much smaller than the coupling strengths often employed in classical theoretical treatments; however, it is still relatively very strong coupling compared to that in experiments. Note that we are considering a single molecule coupled to the cavity, and thus the relevant coupling strength is related to the polaritonic splitting by a factor dependent on the number of solutes …”