Multiscale Molecular Dynamics Simulations of Polaritonic Chemistry

Strong Coupling beyond the Light-Line

Strong coupling of molecules placed in an optical microcavity may lead to the formation of hybrid states called polaritons; states that inherit characteristics of both the optical cavity modes and the molecular resonance. Developing a better understanding of the matter characteristics of these hybrid states has been the focus of much recent attention. Here, as we will show, a better understanding of the role of the optical modes supported by typical cavity structures is also required. Typical microcavities used in molecular strong coupling experiments support more than one mode at the frequency of the material resonance. While the effect of strong coupling to multiple photonic modes has been considered before, here we extend this topic by looking at strong coupling between one vibrational mode and multiple photonic modes. Many experiments involving strong coupling make use of metal-clad microcavities, ones with metallic mirrors. Metal-clad microcavities are well-known to support coupled plasmon modes in addition to the standard microcavity mode. However, the coupled plasmon modes associated with a metal-clad optical microcavity lie beyond the light-line and are thus not probed in typical experiments on strong coupling. Here we investigate, through experiment and numerical modeling, the interaction between molecules within a cavity and the modes both inside and outside the light-line. Making use of grating coupling and a metal-clad microcavity, we provide an experimental demonstration that such modes undergo strong coupling. We further show that a common variant of the metal-clad microcavity, one in which the metal mirrors are replaced by distributed Bragg reflector also show strong coupling to modes that exist in these structures beyond the light-line. Our results highlight the need to consider the effect of beyond the light-line modes on the strong coupling of molecular resonances in microcavities and may be of relevance in designing strong coupling resonators for chemistry and materials science investigations.

Effect of many modes on self-polarization and photochemical suppression in cavities

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.

Light-Matter Hybrid-Orbital-Based First-Principles Methods: The Influence of Polariton Statistics

A detailed understanding of strong matter–photon interactions requires first-principle methods that can solve the fundamental Pauli–Fierz Hamiltonian of nonrelativistic quantum electrodynamics efficiently. A possible way to extend well-established electronic-structure methods to this situation is to embed the Pauli–Fierz Hamiltonian in a higher-dimensional light–matter hybrid auxiliary configuration space. In this work we show the importance of the resulting hybrid Fermi–Bose statistics of the polaritons, which are the new fundamental particles of the “photon-dressed” Pauli–Fierz Hamiltonian for systems in cavities. We show that violations of these statistics can lead to unphysical results. We present an efficient way to ensure the correct statistics by enforcing representability conditions on the dressed one-body reduced density matrix. We further present a general prescription how to extend a given first-principles approach to polaritons and as an example introduce polaritonic Hartree–Fock theory. While being a single-reference method in polariton space, polaritonic Hartree–Fock is a multireference method in the electronic space, i.e., it describes electronic correlations. We also discuss possible applications to polaritonic QEDFT. We apply this theory to a lattice model and find that, the more delocalized the bound-state wave function of the particles is, the stronger it reacts to photons. The main reason is that within a small energy range, many states with different electronic configurations are available as opposed to a strongly bound (and hence energetically separated) ground-state wave function. This indicates that under certain conditions coupling to the quantum vacuum of a cavity can indeed modify ground state properties.

Ab initio polaritonic potential-energy surfaces for excited-state nanophotonics and polaritonic chemistry

Advances in nanophotonics, quantum optics, and low-dimensional materials have enabled precise control of light-matter interactions down to the nanoscale. Combining concepts from each of these fields, there is now an opportunity to create and manipulate photonic matter via strong coupling of molecules to the electromagnetic field. Toward this goal, here we demonstrate a first principles framework to calculate polaritonic excited-state potential-energy surfaces, transition dipole moments, and transition densities for strongly coupled light-matter systems. In particular, we demonstrate the applicability of our methodology by calculating the polaritonic excited-state manifold of a formaldehyde molecule strongly coupled to an optical cavity. This proof-of-concept calculation shows how strong coupling can be exploited to alter photochemical reaction pathways by influencing avoided crossings with tuning of the cavity frequency and coupling strength. Therefore, by introducing an ab initio method to calculate excited-state potential-energy surfaces, our work opens a new avenue for the field of polaritonic chemistry.

Photochemistry in the strong coupling regime: A trajectory surface hopping scheme

The strong coupling regime between confined light and organic molecules turned out to be promising in modifying both the ground state and the excited states properties. Under this peculiar condition, the electronic states of the molecule are mixed with the quantum states of light. The dynamical processes occurring on such hybrid states undergo several modifications accordingly. Hence, the dynamical description of chemical reactivity in polaritonic systems needs to explicitly take into account the photon degrees of freedom and nonadiabatic events. With the aim of describing photochemical polaritonic processes, in the present work, we extend the direct trajectory surface hopping scheme to investigate photochemistry under strong coupling between light and matter.

Conductivity and Photoconductivity of a p-Type Organic Semiconductor under Ultrastrong Coupling

During the past decade, it has been shown that light-matter strong coupling of materials can lead to modified and often improved properties which has stimulated considerable interest. While charge transport can be enhanced in n-type organic semiconductors by coupling the electronic transition and thereby splitting the conduction band into polaritonic states, it is not clear whether the same process can also influence carrier transport in the valence band of p-type semiconductors. Here we demonstrate that it is indeed possible to enhance both the conductivity and photoconductivity of a p-type semiconductor rr-P3HT that is ultrastrongly coupled to plasmonic modes. It is due to the hybrid light-matter character of the virtual polaritonic excitations affecting the linear response of the material. Furthermore, in addition to being enhanced, the photoconductivity of rr-P3HT shows a modified spectral response due to the formation of the hybrid polaritonic states. This illustrates the potential of engineering the vacuum electromagnetic environment to improve the optoelectronic properties of organic materials.

Cavity molecular dynamics simulations of liquid water under vibrational ultrastrong coupling

We simulate vibrational strong coupling (VSC) and vibrational ultrastrong coupling (V-USC) for liquid water with classical molecular dynamics simulations. When the cavity modes are resonantly coupled to the O-H stretch mode of liquid water, the infrared spectrum shows asymmetric Rabi splitting. The lower polariton (LP) may be suppressed or enhanced relative to the upper polariton (UP) depending on the frequency of the cavity mode. Moreover, although the static properties and the translational diffusion of water are not changed under VSC or V-USC, we do find the modification of the orientational autocorrelation function of H2O molecules especially under V-USC, which could play a role in ground-state chemistry.

Dynamics and spectroscopy of molecular ensembles in a lossy microcavity

The radiative and nonradiative relaxation dynamics of an ensemble of molecules in a microcavity are investigated with emphasis on the impact of the cavity lifetime on reactive and spectroscopic properties. Extending a previous study [I. S. Ulusoy et al., J. Phys. Chem. A 123, 8832-8844 (2019)], it is shown that the dynamics of the ensemble and of single molecules are influenced by the presence of a cavity resonance as long as the polariton splitting can be resolved spectroscopically, which critically depends on the lifetime of the system. Our simulations illustrate how the branching between nonradiative intersystem crossing and radiative decay through the cavity can be tuned by selecting specific cavity photon energies resonant at specific molecular geometries. In the case of cavity-photon energies that are not resonant at the Franck-Condon geometry of the molecules, it is demonstrated numerically and analytically that collective effects are limited to a handful of molecules in the ensemble.

Polariton-Mediated Electron Transfer via Cavity Quantum Electrodynamics

We investigate the polariton-mediated electron transfer reaction in a model system with analytic rate constant theory and direct quantum dynamical simulations. We demonstrate that the photoinduced charge transfer reaction between a bright donor state and dark acceptor state can be significantly enhanced or suppressed by coupling the molecular system to the quantized radiation field inside an optical cavity. This is because the quantum light-matter interaction can influence the effective driving force and electronic couplings between the donor state, which is the hybrid light-matter excitation, and the molecular acceptor state. Under the resonance condition between the photonic and electronic excitations, the effective driving force can be tuned by changing the light-matter coupling strength; for an off-resonant condition, the same effect can be accomplished by changing the molecule-cavity detuning. We further demonstrate that using both the electronic coupling and light-matter coupling helps to extend the effective couplings across the entire system, even for the dark state that carries a zero transition dipole. Theoretically, we find that both the counter-rotating terms and the dipole self-energy in the quantum electrodynamics Hamiltonian are important for obtaining an accurate polariton eigenspectrum as well as the polariton-mediated charge transfer rate constant, especially in the ultrastrong coupling regime.

Exciton-phonon polaritons in organic microcavities: Testing a simple ansatz for treating a large number of chromophores

Polaritons in an ensemble of permutationally symmetric chromophores confined to an optical microcavity are investigated numerically. The analysis is based on the Holstein-Tavis-Cummings Hamiltonian which accounts for the coupling between an electronic excitation on each chromophore and a single cavity mode, as well as the coupling between the electronic and nuclear degrees of freedom on each chromophore. A straightforward ensemble partitioning scheme is introduced, which, along with an intuitive ansatz, allows one to obtain accurate evaluations of the lowest-energy polaritons using a subset of collective states. The polaritons include all three degrees of freedom-electronic, vibronic, and photonic-and can therefore be described as exciton-phonon polaritons. Applications focus on the limiting regimes where the Rabi frequency is small or large compared to the nuclear relaxation energy subsequent to optical excitation, with relaxation occurring mainly along the vinyl stretching coordinate in conjugated organic chromophores. Comparisons are also made to the more conventional vibronic polariton approach, which does not take into account two-particle excitations and vibration-photon states.

Relevance of the Quadratic Diamagnetic and Self-Polarization Terms in Cavity Quantum Electrodynamics

Experiments at the interface of quantum optics and chemistry have revealed that strong coupling between light and matter can substantially modify the chemical and physical properties of molecules and solids. While the theoretical description of such situations is usually based on nonrelativistic quantum electrodynamics, which contains quadratic light-matter coupling terms, it is commonplace to disregard these terms and restrict the treatment to purely bilinear couplings. In this work, we clarify the physical origin and the substantial impact of the most common quadratic terms, the diamagnetic and self-polarization terms, and highlight why neglecting them can lead to rather unphysical results. Specifically, we demonstrate their relevance by showing that neglecting these terms leads to the loss of gauge invariance, basis set dependence, disintegration (loss of bound states) of any system in the basis set limit, unphysical radiation of the ground state, and an artificial dependence on the static dipole. Besides providing important guidance for modeling of strongly coupled light-matter systems, the presented results also indicate conditions under which those effects might become accessible.

Benchmarking semiclassical and perturbative methods for real-time simulations of cavity-bound emission and interference

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 behavior 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 (FSSH), multitrajectory Ehrenfest dynamics, linearized semiclassical 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.

Ultrafast Dynamics of Nonequilibrium Organic Exciton-Polariton Condensates

Exciton-polariton condensation in organic materials, arising from the coupling of Frenkel excitons to the electromagnetic field in cavities, is a phenomenon resulting in low-threshold coherent light emission among other fascinating properties. The exact mechanisms leading to the thermalization of organic exciton-polaritons toward condensation are not yet understood, partly due to the complexity of organic molecules and partly to the canonical microcavities used in condensation studies, which limit broadband studies. Here, we exploit an entirely different cavity design, i.e., an array of plasmonic nanoparticles strongly coupled to organic molecules, to successfully measure the broadband ultrafast dynamics of the strongly coupled system. Sharp features emerge in the transient spectrum originating from the formation of a condensate with a well-defined molecular vibrational composition. These measurements represent the first direct experimental evidence that molecular vibrations drive condensation in organic systems and provide a benchmark for modeling the dynamics of organic-based exciton-polariton condensates.

Reduced Density-Matrix Approach to Strong Matter-Photon Interaction

We present a first-principles approach to electronic many-body systems strongly coupled to cavity modes in terms of matter-photon one-body reduced density matrices. The theory is fundamentally nonperturbative and thus captures not only the effects of correlated electronic systems but accounts also for strong interactions between matter and photon degrees of freedom. We do so by introducing a higher-dimensional auxiliary system that maps the coupled fermion-boson system to a dressed fermionic problem. This reformulation allows us to overcome many fundamental challenges of density-matrix theory in the context of coupled fermion-boson systems and we can employ conventional reduced density-matrix functional theory developed for purely fermionic systems. We provide results for one-dimensional model systems in real space and show that simple density-matrix approximations are accurate from the weak to the deep-strong coupling regime. This justifies the application of our method to systems that are too complex for exact calculations and we present first results, which show that the influence of the photon field depends sensitively on the details of the electronic structure.

Modification of Enzyme Activity by Vibrational Strong Coupling of Water

Vibrational strong coupling (VSC) has recently emerged as a completely new tool for influencing chemical reactivity. It harnesses electromagnetic vacuum fluctuations through the creation of hybrid states of light and matter, called polaritonic states, in an optical cavity resonant to a molecular absorption band. Here, we investigate the effect of vibrational strong coupling of water on the enzymatic activity of pepsin, where a water molecule is directly involved in the enzyme's chemical mechanism. We observe an approximately 4.5-fold decrease of the apparent second-order rate constant kcat /Km when coupling the water stretching vibration, whereas no effect was detected for the strong coupling of the bending vibration. The possibility of modifying enzymatic activity by coupling water demonstrates the potential of VSC as a new tool to study biochemical reactivity.

Modifying the Nonradiative Decay Dynamics through Conical Intersections via Collective Coupling to a Cavity Mode

The coupling of a molecular ensemble to the confined electromagnetic modes of a microcavity can strongly modify the photophysics and photochemistry of the molecules upon photoexcitation. We investigate here how collective coupling effects lead to modifications of the mechanisms and rates of photochemical processes, in particular, photodissociation and nonradiative decay in NaI and pyrazine, respectively. We show that, after direct excitation into the lower polaritonic states, the lower-energy light-matter hybrid states, the dynamics of the molecular ensemble coupled to light is very similar to the dynamics of the corresponding isolated molecules. Conversely, excitation into the upper polaritonic states results in more complex dynamics that involve as a first step the population transfer toward the manifold of intermediate dark states. These dynamics differ substantially from those of the isolated molecules and may result in measurable time delays for nonradiative decay or excited-state reaction mechanisms. Similarly, we describe how addition of a buffer of nonreactive molecules coupled to the cavity mode can be used to delay the onset of the photochemical processes of the reactive part of the ensemble, where the buffer medium is more effective in inhibiting the reactive process than only reactive molecules in the cavity.

Investigating New Reactivities Enabled by Polariton Photochemistry

We perform quantum dynamics simulations to investigate new chemical reactivities enabled by cavity quantum electrodynamics. The quantum light-matter interactions between the molecule and the quantized radiation mode inside an optical cavity create a set of hybridized electronic-photonic states, so-called polaritons. The polaritonic states adapt the curvatures from both the ground and the excited electronic states, opening up new possibilities to control photochemical reactions by exploiting intrinsic quantum behaviors of light-matter interactions. With quantum dynamics simulations, we demonstrate that the selectivity of a model photoisomerization reaction can be controlled by tuning the photon frequency of the cavity mode or the light-matter coupling strength, providing new ways to manipulate chemical reactions via the light-matter interaction. We further investigate collective quantum effects enabled by coupling the quantized radiation mode to multiple molecules. Our results suggest that in the resonance case, a photon is recycled among molecules to enable multiple excited state reactions, thus effectively functioning as a catalyst. In the nonresonance case, molecules emit and absorb virtual photons to initiate excited state reactions through fundamental quantum electrodynamics processes. These results from quantum dynamics simulations reveal basic principles of polariton photochemistry as well as promising reactivities that take advantage of intrinsic quantum behaviors of photons.

Tracking Polariton Relaxation with Multiscale Molecular Dynamics Simulations

When photoactive molecules interact strongly with confined light modes in optical cavities, new hybrid light-matter states form. They are known as polaritons and correspond to coherent superpositions of excitations of the molecules and of the cavity photon. The polariton energies and thus potential energy surfaces are changed with respect to the bare molecules, such that polariton formation is considered a promising paradigm for controlling photochemical reactions. To effectively manipulate photochemistry with confined light, the molecules need to remain in the polaritonic state long enough for the reaction on the modified potential energy surface to take place. To understand what determines this lifetime, we have performed atomistic molecular dynamics simulations of room-temperature ensembles of rhodamine chromophores strongly coupled to a single confined light mode with a 15 fs lifetime. We investigated three popular experimental scenarios and followed the relaxation after optically pumping (i) the lower polariton, (ii) the upper polariton, or (iii) uncoupled molecular states. The results of the simulations suggest that the lifetimes of the optically accessible lower and upper polaritons are limited by (i) ultrafast photoemission due to the low cavity lifetime and (ii) reversible population transfer into the "dark" state manifold. Dark states are superpositions of molecular excitations but with much smaller contributions from the cavity photon, decreasing their emission rates and hence increasing their lifetimes. We find that population transfer between polaritonic modes and dark states is determined by the overlap between the polaritonic and molecular absorption spectra. Importantly, excitation can also be transferred "upward" from the lower polariton into the dark-state reservoir due to the broad absorption spectra of the chromophores, contrary to the common conception of these processes as a "one-way" relaxation from the dark states down to the lower polariton. Our results thus suggest that polaritonic chemistry relying on modified dynamics taking place within the lower polariton manifold requires cavities with sufficiently long lifetimes and, at the same time, strong light-matter coupling strengths to prevent the back-transfer of excitation into the dark states.

Exact Potential Energy Surface for Molecules in Cavities

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 we discuss cavity modifications of its structure responsible for the suppression. The results highlight the interplay between nonadiabatic effects from coupling to photons and coupling to electrons and suggest caution is needed when applying traditional dynamics methods based on polaritonic surfaces.

Strong plasmon-molecule coupling at the nanoscale revealed by first-principles modeling

Strong light-matter interactions in both the single-emitter and collective strong coupling regimes attract significant attention due to emerging applications in quantum and nonlinear optics as well as opportunities for modifying material-related properties. Exploration of these phenomena is theoretically demanding, as polaritons exist at the intersection between quantum optics, solid state physics, and quantum chemistry. Fortunately, nanoscale polaritons can be realized in small plasmon-molecule systems, enabling treatment with ab initio methods. Here, we show that time-dependent density-functional theory calculations access the physics of nanoscale plasmon-molecule hybrids and predict vacuum Rabi splitting. By considering a system comprising a few-hundred-atom aluminum nanoparticle interacting with benzene molecules, we show that cavity quantum electrodynamics holds down to resonators of a few cubic nanometers in size, yielding a single-molecule coupling strength exceeding 200 meV due to a massive vacuum field of 4.5 V · nm⁻¹. In a broader perspective, ab initio methods enable parameter-free in-depth studies of polaritonic systems for emerging applications.

Light-matter interaction is described by many different approaches and approximations depending on the coupling strength. Here the authors model a plasmonic system of aluminum nanoparticle interacting with benzene molecules using TDDFT and show its validity to a strong plasmon-molecule coupling regime.

Collective Jahn-Teller Interactions through Light-Matter Coupling in a Cavity

The ultrafast nonradiative relaxation of a molecular ensemble coupled to a cavity mode is considered theoretically and by real-time quantum dynamics. For equal coupling strength of single molecules to the cavity mode, the nonradiative relaxation rate from the upper to the lower polariton states is found to strongly depend on the number of coupled molecules. The coupling of both bright and dark polaritonic states among each other constitutes a special case of (pseudo-)Jahn-Teller interactions involving collective displacements the internal coordinates of the molecules in the ensemble, and the strength of the first order vibronic coupling depends exclusively on the gradient of the energy gaps between molecular electronic states. For N>2 molecules, the N-1 dark light-matter states between the two optically active polaritons feature true collective conical intersection crossings, whose location depends on the internal atomic coordinates of each molecule in the ensemble, and which contribute to the ultrafast nonradiative decay from the upper polariton.

Manipulating azobenzene photoisomerization through strong light-molecule coupling

The formation of hybrid light–molecule states (polaritons) offers a new strategy to manipulate the photochemistry of molecules. To fully exploit its potential, one needs to build a toolbox of polaritonic phenomenologies that supplement those of standard photochemistry. By means of a state-of-the-art computational photochemistry approach extended to the strong-coupling regime, here we disclose various mechanisms peculiar of polaritonic chemistry: coherent population oscillations between polaritons, quenching by trapping in dead-end polaritonic states and the alteration of the photochemical reaction pathway and quantum yields. We focus on azobenzene photoisomerization, that encompasses the essential features of complex photochemical reactions such as the presence of conical intersections and reaction coordinates involving multiple internal modes. In the strong coupling regime, a polaritonic conical intersection arises and we characterize its role in the photochemical process. Our chemically detailed simulations provide a framework to rationalize how the strong coupling impacts the photochemistry of realistic molecules.

Manipulation of the photochemistry of molecules is traditionally achieved through synthetic chemical modifications. Here the authors use computational photochemistry to show how to control azobenzene photoisomerization through hybrid light–molecule states (polaritons).

Conical Intersections Induced by Quantum Light: Field-Dressed Spectra from the Weak to the Ultrastrong Coupling Regimes

In classical laser fields with frequencies resonant with the electronic excitation in molecules, it is by now known that conical intersections are induced by the field and are called light-induced conical intersections (LICIs). As optical cavities have become accessible, the question arises whether their quantized modes could also lead to the appearance of LICIs. A theoretical framework is formulated for the investigation of LICIs of diatomics in such quantum light. The eigenvalue spectrum of the dressed states in the cavity is studied, putting particular emphasis on the investigation of absorption spectra of the Na₂ molecule, that is, on the transitions between dressed states, measured by employing a weak probe pulse. The dependence of the spectra on the light-matter coupling strength in the cavity and on the frequency of the cavity mode is studied in detail. The computations demonstrate strong nonadiabatic effects caused by the appearing LICI.

Coherent Light Harvesting through Strong Coupling to Confined Light

When photoactive molecules interact strongly with confined light modes, new hybrid light-matter states may form: the polaritons. These polaritons are coherent superpositions of excitations of the molecules and of the cavity photon. Recently, polaritons were shown to mediate energy transfer between chromophores at distances beyond the Förster limit. Here we explore the potential of strong coupling for light-harvesting applications by means of atomistic molecular dynamics simulations of mixtures of photoreactive and non-photo-reactive molecules strongly coupled to a single confined light mode. These molecules are spatially separated and present at different concentrations. Our simulations suggest that while the excitation is initially fully delocalized over all molecules and the confined light mode, it very rapidly localizes onto one of the photoreactive molecules, which then undergoes the reaction.

Polariton chemistry: controlling molecular dynamics with optical cavities

Molecular polaritons are the optical excitations which emerge when molecular transitions interact strongly with confined electromagnetic fields. Increasing interest in the hybrid molecular-photonic materials that host these excitations stems from recent observations of their novel and tunable chemistry. Some of the remarkable functionalities exhibited by polaritons include the ability to induce long-range excitation energy transfer, enhance charge conductivity, and inhibit or accelerate chemical reactions. In this review, we explain the effective theories of molecular polaritons which form a basis for the interpretation and guidance of experiments at the strong coupling limit. The theoretical discussion is illustrated with the analysis of innovative applications of strongly coupled molecular-photonic systems to chemical phenomena of fundamental importance to future technologies.