Variational Lang-Firsov Approach Plus Møller-Plesset Perturbation Theory with Applications to Ab Initio Polariton Chemistry.

We apply the Lang-Firsov (LF) transformation to electron-boson coupled Hamiltonians and variationally optimize the transformation parameters and molecular orbital coefficients to determine the ground state. Møller-Plesset (MP-n, with n = 2 and 4) perturbation theory is then applied on top of the optimized LF mean-field state to improve the description of electron-electron and electron-boson correlations. The method (LF-MP) is applied to several electron-boson coupled systems, including the Hubbard-Holstein model, diatomic molecule dissociation (H2, HF), and the modification of proton transfer reactions (malonaldehyde and aminopropenal) via the formation of polaritons in an optical cavity. We show that with a correction for the electron-electron correlation, the method gives quantitatively accurate energies comparable to that by exact diagonalization or coupled-cluster theory. The effects of multiple photon modes, spin polarization, and the comparison to the coherent state MP theory are also discussed.

Theoretical Advances in Polariton Chemistry and Molecular Cavity Quantum Electrodynamics.

When molecules are coupled to an optical cavity, new light-matter hybrid states, so-called polaritons, are formed due to quantum light-matter interactions. With the experimental demonstrations of modifying chemical reactivities by forming polaritons under strong light-matter interactions, theorists have been encouraged to develop new methods to simulate these systems and discover new strategies to tune and control reactions. This review summarizes some of these exciting theoretical advances in polariton chemistry, in methods ranging from the fundamental framework to computational techniques and applications spanning from photochemistry to vibrational strong coupling. Even though the theory of quantum light-matter interactions goes back to the midtwentieth century, the gaps in the knowledge of molecular quantum electrodynamics (QED) have only recently been filled. We review recent advances made in resolving gauge ambiguities, the correct form of different QED Hamiltonians under different gauges, and their connections to various quantum optics models. Then, we review recently developed ab initio QED approaches which can accurately describe polariton states in a realistic molecule-cavity hybrid system. We then discuss applications using these method advancements. We review advancements in polariton photochemistry where the cavity is made resonant to electronic transitions to control molecular nonadiabatic excited state dynamics and enable new photochemical reactivities. When the cavity resonance is tuned to the molecular vibrations instead, ground-state chemical reaction modifications have been demonstrated experimentally, though its mechanistic principle remains unclear. We present some recent theoretical progress in resolving this mystery. Finally, we review the recent advances in understanding the collective coupling regime between light and matter, where many molecules can collectively couple to a single cavity mode or many cavity modes. We also lay out the current challenges in theory to explain the observed experimental results. We hope that this review will serve as a useful document for anyone who wants to become familiar with the context of polariton chemistry and molecular cavity QED and thus significantly benefit the entire community.

Theory for Cavity-Modified Ground-State Reactivities via Electron-Photon Interactions.

We provide a simple and intuitive theory to explain how coupling a molecule to an optical cavity can modify ground-state chemical reactivity by exploiting intrinsic quantum behaviors of light-matter interactions. Using the recently developed polarized Fock states representation, we demonstrate that the change of the ground-state potential is achieved due to the scaling of diabatic electronic couplings with the overlap of the polarized Fock states. Our theory predicts that for a proton-transfer model system, the ground-state barrier height can be modified through light-matter interactions when the cavity frequency is in the electronic excitation range. Our simple theory explains several recent computational investigations that discovered the same effect. We further demonstrate that under the deep strong coupling limit of the light and matter, the polaritonic ground and first excited eigenstates become the Mulliken-Hush diabatic states, which are the eigenstates of the dipole operator. This work provides a simple but powerful theoretical framework to understand how strong coupling between the molecule and the cavity can modify ground-state reactivities.

Ultrafast imaging of polariton propagation and interactions.

Semiconductor excitations can hybridize with cavity photons to form exciton-polaritons (EPs) with remarkable properties, including light-like energy flow combined with matter-like interactions. To fully harness these properties, EPs must retain ballistic, coherent transport despite matter-mediated interactions with lattice phonons. Here we develop a nonlinear momentum-resolved optical approach that directly images EPs in real space on femtosecond scales in a range of polaritonic architectures. We focus our analysis on EP propagation in layered halide perovskite microcavities. We reveal that EP-phonon interactions lead to a large renormalization of EP velocities at high excitonic fractions at room temperature. Despite these strong EP-phonon interactions, ballistic transport is maintained for up to half-exciton EPs, in agreement with quantum simulations of dynamic disorder shielding through light-matter hybridization. Above 50% excitonic character, rapid decoherence leads to diffusive transport. Our work provides a general framework to precisely balance EP coherence, velocity, and nonlinear interactions.

Quantum dynamical effects of vibrational strong coupling in chemical reactivity.

Recent experiments suggest that ground state chemical reactivity can be modified when placing molecular systems inside infrared cavities where molecular vibrations are strongly coupled to electromagnetic radiation. This phenomenon lacks a firm theoretical explanation. Here, we employ an exact quantum dynamics approach to investigate a model of cavity-modified chemical reactions in the condensed phase. The model contains the coupling of the reaction coordinate to a generic solvent, cavity coupling to either the reaction coordinate or a non-reactive mode, and the coupling of the cavity to lossy modes. Thus, many of the most important features needed for realistic modeling of the cavity modification of chemical reactions are included. We find that when a molecule is coupled to an optical cavity it is essential to treat the problem quantum mechanically to obtain a quantitative account of alterations to reactivity. We find sizable and sharp changes in the rate constant that are associated with quantum mechanical state splittings and resonances. The features that emerge from our simulations are closer to those observed in experiments than are previous calculations, even for realistically small values of coupling and cavity loss. This work highlights the importance of a fully quantum treatment of vibrational polariton chemistry.

Microscopic Theory of Multimode Polariton Dispersion in Multilayered Materials.

We develop a microscopic theory for the multimode polariton dispersion in materials coupled to cavity radiation modes. Starting from a microscopic light-matter Hamiltonian, we devise a general strategy for obtaining simple matrix models of polariton dispersion curves based on the structure and spatial location of multilayered 2D materials inside the optical cavity. Our theory exposes the connections between seemingly distinct models that have been employed in the literature and resolves an ambiguity that has arisen concerning the experimental description of the polaritonic band structure. We demonstrate the applicability of our theoretical formalism by fabricating various geometries of multilayered perovskite materials coupled to cavities and demonstrating that our theoretical predictions agree with the experimental results presented here.

Quasi-diabatic propagation scheme for simulating polariton chemistry.

We generalize the quasi-diabatic (QD) propagation scheme to simulate the non-adiabatic polariton dynamics in molecule-cavity hybrid systems. The adiabatic-Fock states, which are the tensor product states of the adiabatic electronic states of the molecule and photon Fock states, are used as the locally well-defined diabatic states for the dynamics propagation. These locally well-defined diabatic states allow using any diabatic quantum dynamics methods for dynamics propagation, and the definition of these states will be updated at every nuclear time step. We use several recently developed non-adiabatic mapping approaches as the diabatic dynamics methods to simulate polariton quantum dynamics in a Shin-Metiu model coupled to an optical cavity. The results obtained from the mapping approaches provide very accurate population dynamics compared to the numerically exact method and outperform the widely used mixed quantum-classical approaches, such as the Ehrenfest dynamics and the fewest switches surface hopping approach. We envision that the generalized QD scheme developed in this work will provide a powerful tool to perform the non-adiabatic polariton simulations by allowing a direct interface between the diabatic dynamics methods and ab initio polariton information.

Nuclear gradient expressions for molecular cavity quantum electrodynamics simulations using mixed quantum-classical methods.

We derive a rigorous nuclear gradient for a molecule-cavity hybrid system using the quantum electrodynamics Hamiltonian. We treat the electronic-photonic degrees of freedom (DOFs) as the quantum subsystem and the nuclei as the classical subsystem. Using the adiabatic basis for the electronic DOF and the Fock basis for the photonic DOF and requiring the total energy conservation of this mixed quantum-classical (MQC) system, we derived the rigorous nuclear gradient for the molecule-cavity hybrid system, which is naturally connected to the approximate gradient under the Jaynes-Cummings approximation. The nuclear gradient expression can be readily used in any MQC simulations and will allow one to perform the non-adiabatic on-the-fly simulation of polariton quantum dynamics. The theoretical developments in this work could significantly benefit the polariton quantum dynamics community with a rigorous nuclear gradient of the molecule-cavity hybrid system and have a broad impact on the future non-adiabatic simulations of polariton quantum dynamics.

Incorporating Lindblad decay dynamics into mixed quantum-classical simulations.

We derive the L-mean-field Ehrenfest (MFE) method to incorporate Lindblad jump operator dynamics into the MFE approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the L-MFE method that satisfies this exact mapping. This establishes a method that uses independent trajectories that exactly reproduce Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantum-classical simulations.

Resonant Cavity Modification of Ground-State Chemical Kinetics.

Recent experiments have suggested that ground-state chemical kinetics can be suppressed or enhanced by coupling molecular vibrations with a cavity radiation mode. Here, we develop an analytical rate theory for cavity-modified chemical kinetics based on the Pollak-Grabert-Hänggi theory. Unlike previous work, our theory covers the complete range of solvent friction values, from the energy-diffusion-limited to the spatial-diffusion-limited regimes. We show that chemical kinetics is enhanced when bath friction is weak and suppressed when bath friction is strong. For weak bath friction, the resonant photon frequency (at which the maximum modification of the chemical rate is achieved) is close to the reactant well. In the strong friction limit, the resonant photon frequency is instead close to the barrier frequency. Finally, we observe that rate changes as a function of the photon frequency are much sharper and more sizable in the weak friction limit than in the strong friction limit.

Resolving ambiguities of the mode truncation in cavity quantum electrodynamics.

This work provides the fundamental theoretical framework for few-mode cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the photonic mode truncation. We first propose a general framework to resolve ambiguities for an arbitrary truncation in a given gauge. Then, we specifically consider the case of mode truncation, deriving gauge invariant expressions for both the Coulomb and dipole gauge Hamiltonians that naturally reduce to the commonly used single-mode Hamiltonians when considering a single-mode truncation. We finally provide the analytical and numerical results of both atomic and molecular model systems coupled to the cavity to demonstrate the validity of our theory.

Theory of vibrational polariton chemistry in the collective coupling regime.

We theoretically demonstrate that the chemical reaction rate constant can be significantly suppressed by coupling molecular vibrations with an optical cavity, exhibiting both the collective coupling effect and the cavity frequency modification of the rate constant. When a reaction coordinate is strongly coupled to the solvent molecules, the reaction rate constant is reduced due to the dynamical caging effect. We demonstrate that collectively coupling the solvent to the cavity can further enhance this dynamical caging effect, leading to additional suppression of the chemical kinetics. This effect is further amplified when cavity loss is considered.

Ab initio symmetric quasi-classical approach to investigate molecular Tully models.

We perform on-the-fly non-adiabatic molecular dynamics simulations using the symmetrical quasi-classical (SQC) approach with the recently suggested molecular Tully models: ethylene and fulvene. We attempt to provide benchmarks of the SQC methods using both the square and triangle windowing schemes as well as the recently proposed electronic zero-point-energy correction scheme (the so-called γ correction). We use the quasi-diabatic propagation scheme to directly interface the diabatic SQC methods with adiabatic electronic structure calculations. Our results showcase the drastic improvement of the accuracy by using the trajectory-adjusted γ-corrections, which outperform the widely used trajectory surface hopping method with decoherence corrections. These calculations provide useful and non-trivial tests to systematically investigate the numerical performance of various diabatic quantum dynamics approaches, going beyond simple diabatic model systems that have been used as the major workhorse in the quantum dynamics field. At the same time, these available benchmark studies will also likely foster the development of new quantum dynamics approaches based on these techniques.

Polariton induced conical intersection and berry phase.

We investigate the Polariton induced conical intersection (PICI) created from coupling a diatomic molecule with the quantized photon mode inside an optical cavity, and the corresponding Berry Phase effects. We use the rigorous Pauli-Fierz Hamiltonian to describe the quantum light-matter interactions between a LiF molecule and the cavity, and use the exact quantum propagation to investigate the polariton quantum dynamics. The molecular rotations relative to the cavity polarization direction play a role as the tuning mode of the PICI, resulting in an effective CI even within a diatomic molecule. To clearly demonstrate the dynamical effects of the Berry phase, we construct two additional models that have the same Born-Oppenheimer surface, but the effects of the geometric phase are removed. We find that when the initial wavefunction is placed in the lower polaritonic surface, the Berry phase causes a π phase-shift in the wavefunction after the encirclement around the CI, indicated from the nuclear probability distribution. On the other hand, when the initial wavefunction is placed in the upper polaritonic surface, the geometric phase significantly influences the couplings between polaritonic states and therefore, the population dynamics between them. These BP effects are further demonstrated through the photo-fragment angular distribution. PICI created from the quantized radiation field has the promise to open up new possibilities to modulate photochemical reactivities.

Theory of Mode-Selective Chemistry through Polaritonic Vibrational Strong Coupling.

Recent experiments have demonstrated remarkable mode-selective reactivities by coupling molecular vibrations with a quantized radiation field inside an optical cavity. The fundamental mechanism behind such effects, on the other hand, remains elusive. In this work, we provide a theoretical explanation of the basic principle of how cavity frequency can be tuned to achieve mode-selective reactivities. We find that the dynamics of the radiation mode leads to a cavity frequency-dependent dynamical caging effect of a reaction coordinate, resulting in suppression of the rate constant. In the presence of competitive reactions, it is possible to preferentially cage a reaction coordinate when the barrier frequencies of competing reactions are different, resulting in a selective slow down of a given reaction. Our theoretical results illustrate the cavity-induced mode-selective chemistry through polaritonic vibrational strong couplings, revealing the fundamental mechanism for changing chemical selectivities through cavity quantum electrodynamics.

Molecular Polaritons Generated from Strong Coupling between CdSe Nanoplatelets and a Dielectric Optical Cavity.

We demonstrate the formation of CdSe nanoplatelet (NPL) exciton-polaritons in a distributed Bragg reflector (DBR) cavity. The molecule-cavity hybrid system is in the strong coupling regime with an 83 meV Rabi splitting, characterized from angle-resolved reflectance and photoluminescence measurements. Mixed quantum-classical dynamics simulations are used to investigate the polariton photophysics of the hybrid system by treating the electronic and photonic degrees of freedom (DOF) quantum mechanically and the nuclear phononic DOF classically. Our numerical simulations of the angle-resolved photoluminescence (PL) agree extremely well with the experimental data, providing a fundamental explanation of the asymmetric intensity distribution of the upper and lower polariton branches. Our results also provide mechanistic insights into the importance of phonon-assisted nonadiabatic transitions among polariton states, which are reflected in the various features of the PL spectra. This work proves the feasibility of coupling nanoplatelet electronic states with the photon states of a dielectric cavity to form a hybrid system and provides a new platform for investigating cavity-mediated physical and chemical processes.

Cavity frequency-dependent theory for vibrational polariton chemistry.

Recent experiments demonstrate the control of chemical reactivities by coupling molecules inside an optical microcavity. In contrast, transition state theory predicts no change of the reaction barrier height during this process. Here, we present a theoretical explanation of the cavity modification of the ground state reactivity in the vibrational strong coupling (VSC) regime in polariton chemistry. Our theoretical results suggest that the VSC kinetics modification is originated from the non-Markovian dynamics of the cavity radiation mode that couples to the molecule, leading to the dynamical caging effect of the reaction coordinate and the suppression of reaction rate constant for a specific range of photon frequency close to the barrier frequency. We use a simple analytical non-Markovian rate theory to describe a single molecular system coupled to a cavity mode. We demonstrate the accuracy of the rate theory by performing direct numerical calculations of the transmission coefficients with the same model of the molecule-cavity hybrid system. Our simulations and analytical theory provide a plausible explanation of the photon frequency dependent modification of the chemical reactivities in the VSC polariton chemistry.

Ring polymer quantization of the photon field in polariton chemistry.

We use the ring polymer (RP) representation to quantize the radiation field inside an optical cavity to investigate polariton quantum dynamics. Using a charge transfer model coupled to an optical cavity, we demonstrate that the RP quantization of the photon field provides accurate rate constants of the polariton mediated electron transfer reaction compared to Fermi's golden rule. Because RP quantization uses extended phase space to describe the photon field, it significantly reduces the computational costs compared to the commonly used Fock state description of the radiation field. Compared to the other quasi-classical descriptions of the photon field, such as the classical Wigner based mean-field Ehrenfest model, the RP representation provides a much more accurate description of the polaritonic quantum dynamics because it alleviates the potential quantum distribution leakage problem associated with the photonic degrees of freedom (DOF). This work demonstrates the possibility of using the ring polymer description to treat the quantized radiation field in polariton chemistry, offering an accurate and efficient approach for future investigations in cavity quantum electrodynamics.

Resolution of Gauge Ambiguities in Molecular Cavity Quantum Electrodynamics.

This work provides the fundamental theoretical framework for molecular cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the electronic state truncation. We conjecture that such ambiguity arises because not all operators are consistently constrained in the same truncated electronic subspace for both gauges. We resolve this ambiguity by constructing a unitary transformation operator that properly constrains all light-matter interaction terms in the same subspace. We further derive an equivalent and yet convenient expression for the Coulomb gauge Hamiltonian under the truncated subspace. We finally provide the analytical and numerical results of a model molecular system coupled to the cavity to demonstrate the validity of our theory.

Polarized Fock States and the Dynamical Casimir Effect in Molecular Cavity Quantum Electrodynamics.

We present a new theoretical framework, polarized Fock states (PFSs), to describe the coupled molecule-cavity hybrid system in quantum electrodynamics. Through the quantum light-matter interactions under the dipole Gauge, the molecular permanent dipoles polarize the photon field by displacing the photonic coordinate. Hence, it is convenient to use these shifted Fock states (termed the PFSs) to describe light-matter interactions under the strong coupling regimes. These PFSs are nonorthogonal to each other and are light-matter entangled states. They allow an intuitive understanding of several phenomena that go beyond the prediction of the quantum Rabi model, while also offering numerical convenience to converge the results with much fewer states. With this powerful new theoretical framework, we explain how molecular permanent dipoles lead to the generation of multiple photons from a single electronic excitation (down-conversion), effectively achieving the dynamical Casimir effect through the nuclear vibration instead of cavity mirror oscillations.