We benchmark a selection of semiclassical and perturbative dynamics techniques by investigating the correlated evolution of a cavity-bound atomic system to assess their applicability to study problems involving strong light-matter interactions in quantum cavities. The model system of interest features spontaneous emission, interference, and strong coupling behaviour, and necessitates the consideration of vacuum fluctuations and correlated light-matter dynamics. We compare a selection of approximate dynamics approaches including fewest switches surface hopping, multi-trajectory Ehrenfest dynamics, linearized semiclasical dynamics, and partially linearized semiclassical dynamics. Furthermore, investigating self-consistent perturbative methods, we apply the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the second Born approximation. With the exception of fewest switches surface hopping, all methods provide a reasonable level of accuracy for the correlated light-matter dynamics, with most methods lacking the capacity to fully capture interference effects.Here we broaden our scope by investigating the performance of a comprehensive class of approximate quantum dynamics methods for simulating spontaneous emission in an optical cavity, including Ehrenfest meanfield theory 33,34 , Tully's surface hopping algorithm 35 , fully linearized 36 and partially linearized 37,38 semilclassical dynamics techniques, and a selection of approximate closures for the quantum mechanical Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Through benchmark comparisons with exact numerical results, we assess the accuracy and efficiency of each method and highlight the possibilities and theoretical challenges involved with extending these approaches towards realistic systems.The remainder of this work is divided into four sections: Sec. II gives a short overview of general quantum mechanical arXiv:1909.07177v2 [quant-ph]

We describe vacuum fluctuations and photon-field correlations in interacting quantum mechanical light-matter systems, by generalizing the application of mixed quantum-classical dynamics techniques. We employ the multi-trajectory implementation of Ehrenfest mean field theory, traditionally developed for electron-nuclear problems, to simulate the spontaneous emission of radiation in a model quantum electrodynamical cavitybound atomic system. We investigate the performance of this approach in capturing the dynamics of spontaneous emission from the perspective of both the atomic system and the cavity photon field, through a detailed comparison with exact benchmark quantum mechanical observables and correlation functions. By properly accounting for the quantum statistics of the vacuum field, while using mixed quantum-classical (mean field) trajectories to describe the evolution, we identify a surprisingly accurate and promising route towards describing quantum effects in realistic correlated light-matter systems.

The exact factorization approach, originally developed for electron-nuclear dynamics, is extended to light-matter interactions within the dipole approximation. This allows for a Schrödinger equation for the photonic wavefunction, in which the potential contains exactly the effects on the photon field of its coupling to matter. We illustrate the formalism and potential for a two-level system representing the matter, coupled to an infinite number of photon modes in the Wigner-Weisskopf approximation, as well as a single mode with various coupling strengths. Significant differences are found with the potential used in conventional approaches, especially for strong-couplings. We discuss how our exact factorization approach for light-matter interactions can be used as a guideline to develop semiclassical trajectory methods for efficient simulations of light-matter dynamics.

We find and analyze the exact time-dependent potential energy surface driving the proton motion for a model of cavity-induced suppression of proton-coupled electron-transfer. We show how, in contrast to the polaritonic surfaces, its features directly correlate to the proton dynamics and discuss cavity-modifications of its structure responsible for the suppression. The results highlight the interplay between non-adiabatic effects from coupling to photons and coupling to electrons, and suggest caution is needed when applying traditional dynamics methods based on polaritonic surfaces. Impressive experimental advances [1][2][3][4][5] have led to a rekindling of interest in cavity quantum electrodynamics. Rapidly expanding applications to molecules and nanostructures require us to go beyond the simplest few-level-single-mode models explored in the early days of quantum mechanics, with the interplay of coupled electronic, nuclear, and photonic excitations revealing a plethora of new phenomena from enhanced conductivity in semiconductors to photochemical suppression of chemical reactions to cavity-enhanced superconductivity to superradiance, see e.g. Refs. [6][7][8][9][10][11][12].There is now the possibility to manipulate real matter with cavity parameters providing tunable dials for photo-chemical control of reactions, replacing shaped laser pulses as photonic reagents [1,13,14]. The hope is attain strong light-matter coupling and control without large power sources, possibly reducing unintended byproducts such as multiphoton absorption and ionization channels.The cavity clearly modifies the potential that the matter evolves in, and various constructs have been put forward to serve in lieu of the Born-Oppenheimer (BO) surfaces that have proved so instrumental for our understanding of cavity-free dynamics. In particular, "polaritonic surfaces" that arise from diagonalizing the electron-photon Hamiltonian parametrized by nuclear coordinates have been instructive in interpreting some of the novel phenomena mentioned above [15][16][17][18][19]. Another construct are the "cavity-BO surfaces" where the photonic displacement-field coordinates are treated on the same footing as the nuclear coordinates [7,20]. A complete dynamical picture of how the electronic and photonic degrees of freedom influence the nuclear dynamics can only be obtained when several of such surfaces in the chosen manifold together with their couplings are considered: quite typically at a given time the nuclear wavepacket locally straddles several surfaces, or distinct parts of the nuclear wavepacket are associated with different surfaces. To go beyond using the surfaces only for qualitative interpretation, and to implement them in dynamics schemes, couplings be-tween the surfaces must be included [8,21], and there is interplay between non-adiabatic effects arising from photon-matter coupling and electron-nuclear coupling. Practical necessity calls for approximations which usually work best when this choice of surfaces in some sense represents a "zeroth o...

The standard description of cavity-modified molecular reactions typically involves a single (resonant) mode, while in reality, the quantum cavity supports a range of photon modes. Here, we demonstrate that as more photon modes are accounted for, physicochemical phenomena can dramatically change, as illustrated by the cavity-induced suppression of the important and ubiquitous process of proton-coupled electron-transfer. Using a multi-trajectory Ehrenfest treatment for the photon-modes, we find that self-polarization effects become essential, and we introduce the concept of self-polarization-modified Born–Oppenheimer surfaces as a new construct to analyze dynamics. As the number of cavity photon modes increases, the increasing deviation of these surfaces from the cavity-free Born–Oppenheimer surfaces, together with the interplay between photon emission and absorption inside the widening bands of these surfaces, leads to enhanced suppression. The present findings are general and will have implications for the description and control of cavity-driven physical processes of molecules, nanostructures, and solids embedded in cavities.

The exact time-dependent potential energy surface driving the nuclear dynamics was recently shown to be a useful tool to understand and interpret the coupling of nuclei, electrons, and photons in cavity settings. Here, we provide a detailed analysis of its structure for exactly solvable systems that model two phenomena: cavity-induced suppression of proton-coupled electron-transfer and its dependence on the initial state, and cavity-induced electronic excitation. We demonstrate the inadequacy of simply using a weighted average of polaritonic surfaces to determine the dynamics. Such a weighted average misses a crucial term that redistributes energy between the nuclear and the polaritonic systems, and this term can in fact become a predominant term in determining the nuclear dynamics when several polaritonic surfaces are involved. Evolving an ensemble of classical trajectories on the exact potential energy surface reproduces the nuclear wavepacket quite accurately, while evolving on the weighted polaritonic surface fails after a short period of time. The implications and prospects for application of mixed quantum-classical methods based on this surface are discussed.

Simulating photon dynamics in strong light–matter coupling situations via classical trajectories is proving to be powerful and practical. Here, we analyze the performance of the approach through the lens of the exact factorization approach. Since the exact factorization enables a rigorous definition of the potentials driving the photonic motion, it allows us to identify that the underestimation of photon number and intensities observed in earlier work is primarily due to an inadequate accounting of light–matter correlation in the classical Ehrenfest force rather than errors from treating the photons quasiclassically per se. The latter becomes problematic when the number of photons per mode begins to exceed a half.

The use of electric fields to modify chemical reactions is a promising, emerging technique in catalysis. However, there exist few guiding principles, and rational design requires assumptions about the transition state or explicit atomistic calculations. Here, we present a linear free energy relationship, familiar in other areas of physical organic chemistry and catalysis, that microscopically relates field-induced changes in the activation energy to those in the reaction energy, connecting kinetic and thermodynamic behaviors. We verify our theory using first-principles electronic structure calculations of a symmetric SN2 reaction and the dehalogenation of an aryl halide on gold surfaces and observe hallmarks of linear free energy relationships, such as the shifting to early and late transition states. We also report and explain a counterintuitive case, where the constant of proportionality relating the activation and reaction energies is negative, such that stabilizing the product increases the activation energy, that is, opposite of the Bell–Evans–Polanyi principle.