Exciton-phonon coupling strength in single-layer MoSe2 at room temperature

Shaping exciton polarization dynamics in 2D semiconductors by tailored ultrafast pulses

The ultrafast formation of strongly bound excitons in two-dimensional semiconductors provides a rich platform for studying fundamental physics as well as developing novel optoelectronic technologies. While extensive research has explored the excitonic coherence, many-body interactions, and nonlinear optical properties, the potential to study these phenomena by directly controlling their coherent polarization dynamics has not been fully realized. In this work, we use a sub-10 fs pulse shaper to study how temporal control of coherent exciton polarization affects the generation of four-wave mixing in monolayer WS e 2 under ambient conditions. By tailoring multiphoton pathway interference, we tune the nonlinear response from destructive to constructive interference, resulting in a 2.6-fold enhancement over the four-wave mixing generated by a transform-limited pulse. This demonstrates a general method for nonlinear enhancement by shaping the pulse to counteract the temporal dispersion experienced during resonant light-matter interactions. Our method allows us to excite both 1s and 2s states, showcasing a selective control over the resonant state that produces nonlinearity. By comparing our results with theory, we find that exciton-exciton interactions dominate the nonlinear response, rather than Pauli blocking. This capability to manipulate exciton polarization dynamics in atomically thin crystals lays the groundwork for exploring a wide range of resonant phenomena in condensed matter systems and opens up new possibilities for precise optical control in advanced optoelectronic devices.

Phonon-assisted Auger decay of excitons in doped transition metal dichalcogenide monolayers

The competition between the radiative and nonradiative lifetimes determines the optical quantum yield and plays a crucial role in the potential optoelectronic applications of transition metal dichalcogenides (TMDCs). Here, we show that, in the presence of free carriers, an additional nonradiative decay channel opens for excitons in TMDC monolayers. Although the usual Auger decay channel is suppressed at low doping levels by the simultaneous momentum and energy conservation laws, exciton-phonon coupling relaxes this suppression. By solving a Bethe-Salpeter equation, we calculate the phonon-assisted Auger decay rates in four typical TMDCs as a function of doping, temperature, and dielectric environment. We find that even for a relatively low doping of 1012 cm-2, the nonradiative lifetime ranges from 16 to 165 ps in different TMDCs, offering competition to the radiative decay channel.

Mixed quantum-classical modeling of exciton-phonon scattering in solids: Application to optical linewidths of monolayer MoS2

We present a mixed quantum-classical framework for the microscopic and non-Markovian modeling of exciton-phonon scattering in solid-state materials and apply it to calculate the optical linewidths of monolayer MoS2. Within this framework, we combine reciprocal-space mixed quantum-classical dynamics with models for the quasiparticle band structure as well as the electron-hole and carrier-phonon interactions, parametrized against ab initio calculations, although noting that a direct interfacing with ab initio calculations is straightforward in principle. We introduce various parameters for truncating the Brillouin zone to select regions of interest. Variations of these parameters allow us to determine linewidths in the limit of asymptotic material sizes. The obtained asymptotic linewidths are found to agree favorably with experimental measurements across a range of temperatures. As such, our framework establishes itself as a promising route toward unraveling the non-Markovian and microscopic principles governing the nonadiabatic dynamics of solids.

Perspectives: Light Control of Magnetism and Device Development

Accurately controlling magnetic and spin states presents a significant challenge in spintronics, especially as demands for higher data storage density and increased processing speeds grow. Approaches such as light control are gradually supplanting traditional magnetic field methods. Traditionally, the modulation of magnetism was predominantly achieved through polarized light with the help of ultrafast light technologies. With the growing demand for energy efficiency and multifunctionality in spintronic devices, integrating photovoltaic materials into magnetoelectric systems has introduced more physical effects. This development suggests that sunlight will play an increasingly pivotal role in manipulating spin orientation in the future. This review introduces and concludes the influence of various light types on magnetism, exploring mechanisms such as magneto-optical (MO) effects, light-induced magnetic phase transitions, and spin photovoltaic effects. This review briefly summarizes recent advancements in the light control of magnetism, especially sunlight, and their potential applications, providing an optimistic perspective on future research directions in this area.

Cross-phase modulation in two-dimensional spectroscopy

Developing from transient absorption (TA) spectroscopy, two-dimensional (2D) spectroscopy with pump-probe geometry has emerged as a versatile approach for alleviating the difficulty in implementing 2D spectroscopy with other geometries. However, the presence of cross-phase modulation (XPM) in TA spectroscopy introduces significant spectral distortions, particularly when the pump and probe pulses overlap. We demonstrate that this phenomenon is extended to the 2D spectroscopy with pump-probe geometry and the XPM is induced by the interference of the two pump pulses. We present the oscillatory behavior of XPM in the 2D spectrum and its displacement with respect to the waiting time delay through both experimental measurements and numerical simulations. Additionally, we explore the influence of probe pulse chirp on XPM and discover that by compressing the chirp, the impact of XPM on the desired signal can be reduced.

Ultrafast Charge Transfer and Recombination Dynamics in Monolayer-Multilayer WSe2 Junctions Revealed by Time-Resolved Photoemission Electron Microscopy

The ultrafast carrier dynamics of junctions between two chemically identical, but electronically distinct, transition metal dichalcogenides (TMDs) remains largely unknown. Here, we employ time-resolved photoemission electron microscopy (TR-PEEM) to probe the ultrafast carrier dynamics of a monolayer-to-multilayer (1L-ML) WSe2 junction. The TR-PEEM signals recorded for the individual components of the junction reveal the sub-ps carrier cooling dynamics of 1L- and 7L-WSe2, as well as few-ps exciton-exciton annihilation occurring on 1L-WSe2. We observe ultrafast interfacial hole (h) transfer from 1L- to 7L-WSe2 on an ∼0.2 ps time scale. The resultant excess h density in 7L-WSe2 decays by carrier recombination across the junction interface on an ∼100 ps time scale. Reminiscent of the behavior at a depletion region, the TR-PEEM image reveals the h density accumulation on the 7L-WSe2 interface, with a decay length ∼0.60 ± 0.17 μm. These charge transfer and recombination dynamics are in agreement with ab initio quantum dynamics. The computed orbital densities reveal that charge transfer occurs from the basal plane, which extends over both 1L and ML regions, to the upper plane localized on the ML region. This mode of charge transfer is distinctive to chemically homogeneous junctions of layered materials and constitutes an additional carrier deactivation pathway that should be considered in studies of 1L-TMDs found alongside their ML, a common occurrence in exfoliated samples.

Optically Addressing Exciton Spin and Pseudospin in Nanomaterials for Spintronics Applications

Oriented exciton spins that can be generated and manipulated optically are of interest for a range of applications, including spintronics, quantum information science, and neuromorphic computing architectures. Although materials that host such excitons often lack practical coherence times for use on their own, strategic transduction of the magnetic information across interfaces can combine fast modulation with longer-term storage and readout. Several nanostructure systems have been put forward due to their interesting magneto-optical properties and their possible manipulation using circularly polarized light. These material systems are presented here, namely two-dimensional (2D) systems due to the unique spin-valley coupling properties and quantum dots for their exciton fine structure. 2D magnets are also discussed for their anisotropic spin behavior and extensive 2D magnetic states that are not yet fully understood but could pave the way for emergent techniques of magnetic control. This review also details the experimental and theoretical tools to measure and understand these systems along with a discussion on the progress of optical manipulation of spins and magnetic order transitions.

Subtractive Nanopore Engineered MXene Photonic Nanomedicine with Enhanced Capability of Photothermia and Drug Delivery for Synergistic Treatment of Osteosarcoma

Two-dimensional (2D) nanomaterials as drug carriers and photosensitizers have emerged as a promising antitumor strategy. However, our understanding of 2D antitumor nanomaterials is limited to intrinsic properties or additive modification of different materials. Subtractive structural engineering of 2D nanomaterials for better antitumor efficacy is largely overlooked. Here, subtractively engineered 2D MXenes with uniformly distributed nanopores are synthesized. The nanoporous defects endowed MXene with enhanced surface plasmon resonance effect for better optical absorbance performance and strong exciton-phonon coupling for higher photothermal conversion efficiency. In addition, porous structure improves the binding ability between drug and unsaturated bonds, thus promoting drug-loading capacity and reducing uncontrolled drug release. Furthermore, the porous structure provides adhesion sites for filopodia, thereby promoting the cellular internalization of the drug. Clinically, osteosarcoma is the most common bone malignancy routinely treated with doxorubicin-based chemotherapy. There have been no significant treatment advances in the past decade. As a proof-of-concept, nanoporous MXene loaded with doxorubicin is developed for treating human osteosarcoma cells. The porous MXene platform results in a higher amount of doxorubicin-loading, faster near-infrared (NIR)-controlled doxorubicin release, higher photothermal efficacy under NIR irradiation, and increased cell adhesion and internalization. This facile method pioneers a new paradigm for enhancing 2D material functions and is attractive for tumor treatment.

Tunable phononic coupling in excitonic quantum emitters

Engineering the coupling between fundamental quantum excitations is at the heart of quantum science and technologies. An outstanding case is the creation of quantum light sources in which coupling between single photons and phonons can be controlled and harnessed to enable quantum information transduction. Here we report the deterministic creation of quantum emitters featuring highly tunable coupling between excitons and phonons. The quantum emitters are formed in strain-induced quantum dots created in homobilayer WSe2. The colocalization of quantum-confined interlayer excitons and terahertz interlayer breathing-mode phonons, which directly modulates the exciton energy, leads to a uniquely strong phonon coupling to single-photon emission, with a Huang-Rhys factor reaching up to 6.3. The single-photon spectrum of interlayer exciton emission features a single-photon purity >83% and multiple phonon replicas, each heralding the creation of a phonon Fock state in the quantum emitter. Due to the vertical dipole moment of the interlayer exciton, the phonon-photon interaction is electrically tunable to be higher than the exciton and phonon decoherence rate, and hence promises to reach the strong-coupling regime. Our result demonstrates a solid-state quantum excitonic-optomechanical system at the atomic interface of the WSe2 bilayer that emits flying photonic qubits coupled with stationary phonons, which could be exploited for quantum transduction and interconnection.

High-order pump-probe and high-order two-dimensional electronic spectroscopy on the example of squaraine oligomers

Time-resolved spectroscopy is commonly used to study diverse phenomena in chemistry, biology, and physics. Pump-probe experiments and coherent two-dimensional (2D) spectroscopy have resolved site-to-site energy transfer, visualized electronic couplings, and much more. In both techniques, the lowest-order signal, in a perturbative expansion of the polarization, is of third order in the electric field, which we call a one-quantum (1Q) signal because in 2D spectroscopy it oscillates in the coherence time with the excitation frequency. There is also a two-quantum (2Q) signal that oscillates in the coherence time at twice the fundamental frequency and is fifth order in the electric field. We demonstrate that the appearance of the 2Q signal guarantees that the 1Q signal is contaminated by non-negligible fifth-order interactions. We derive an analytical connection between an nQ signal and (2n + 1)th-order contaminations of an rQ (with r < n) signal by studying Feynman diagrams of all contributions. We demonstrate that by performing partial integrations along the excitation axis in 2D spectra, we can obtain clean rQ signals free of higher-order artifacts. We exemplify the technique using optical 2D spectroscopy on squaraine oligomers, showing clean extraction of the third-order signal. We further demonstrate the analytical connection with higher-order pump-probe spectroscopy and compare both techniques experimentally. Our approach demonstrates the full power of higher-order pump-probe and 2D spectroscopy to investigate multi-particle interactions in coupled systems.

Exciton Lifetime and Optical Line Width Profile via Exciton-Phonon Interactions: Theory and First-Principles Calculations for Monolayer MoS2

Exciton dynamics dictates the evolution of photoexcited carriers in photovoltaic and optoelectronic devices. However, interpreting their experimental signatures is a challenging theoretical problem due to the presence of both electron-phonon and many-electron interactions. We develop and apply here a first-principles approach to exciton dynamics resulting from exciton-phonon coupling in monolayer MoS₂ and reveal the highly selective nature of exciton-phonon coupling due to the internal spin structure of excitons, which leads to a surprisingly long lifetime of the lowest-energy bright A exciton. Moreover, we show that optical absorption processes rigorously require a second-order perturbation theory approach, with photon and phonon treated on an equal footing, as proposed by Toyozawa and Hopfield. Such a treatment, thus far neglected in first-principles studies, gives rise to off-diagonal exciton-phonon self-energy, which is critical for the description of dephasing mechanisms and yields exciton line widths in excellent agreement with experiment.

Negative-mass exciton polaritons induced by dissipative light-matter coupling in an atomically thin semiconductor

Dispersion engineering is a powerful and versatile tool that can vary the speed of light signals and induce negative-mass effects in the dynamics of particles and quasiparticles. Here, we show that dissipative coupling between bound electron-hole pairs (excitons) and photons in an optical microcavity can lead to the formation of exciton polaritons with an inverted dispersion of the lower polariton branch and hence, a negative mass. We perform direct measurements of the anomalous dispersion in atomically thin (monolayer) WS₂ crystals embedded in planar microcavities and demonstrate that the propagation direction of the negative-mass polaritons is opposite to their momentum. Our study introduces the concept of non-Hermitian dispersion engineering for exciton polaritons and opens a pathway for realising new phases of quantum matter in a solid state.

Surface acoustic wave induced phenomena in two-dimensional materials

Surface acoustic wave (SAW)-matter interaction provides a fascinating key for inducing and manipulating novel phenomena and functionalities in two-dimensional (2D) materials. The dynamic strain field and piezo-electric field associated with propagating SAWs determine the coherent manipulation and transduction between 2D excitons and phonons. Over the past decade, many intriguing acoustic-induced effects, including the acousto-electric effect, acousto-galvanic effect, acoustic Stark effect, acoustic Hall effect and acoustic exciton transport, have been reported experimentally. However, many more phenomena, such as the valley acousto-electric effect, valley acousto-electric Hall effect and acoustic spin Hall effect, were only theoretically proposed, the experimental verification of which are yet to be achieved. In this minireview, we attempt to overview the recent breakthrough of SAW-induced phenomena covering acoustic charge transport, acoustic exciton transport and modulation, and coherent acoustic phonons. Perspectives on the opportunities of the proposed SAW-induced phenomena, as well as open experimental challenges, are also discussed, attempting to offer some guidelines for experimentalists and theorists to explore the desired exotic properties and boost practical applications of 2D materials.

High-sensitivity fluorescence-detected multidimensional electronic spectroscopy through continuous pump-probe delay scan

Fluorescence-detected multidimensional electronic spectroscopy (fMES) promises high sensitivity compared to conventional approaches and is an emerging spectroscopic approach toward combining the advantages of MES with the spatial resolution of a microscope. Here, we present a visible white light continuum-based fMES spectrometer and systematically explore the sensitivity enhancement expected from fluorescence detection. As a demonstration of sensitivity, we report room temperature two-dimensional coherence maps of vibrational quantum coherences in a laser dye at optical densities of ∼2-3 orders of magnitude lower than conventional approaches. This high sensitivity is enabled by a combination of biased sampling along the optical coherence time axes and a rapid scan of the pump-probe waiting time T at each sample. A combination of this approach with acousto-optic phase modulation and phase-sensitive lock-in detection enables measurements of room temperature vibrational wavepackets even at the lowest ODs. Alternative faster data collection schemes, which are enabled by the flexibility of choosing a non-uniform undersampled grid in the continuous T scanning approach, are also demonstrated.

Observation of Phonon Cascades in Cu-Doped Colloidal Quantum Wells

Electronic doping has endowed colloidal quantum wells (CQWs) with unique optical and electronic properties, holding great potential for future optoelectronic device concepts. Unfortunately, how photogenerated hot carriers interact with phonons in these doped CQWs still remains an open question. Here, through investigating the emission properties, we have observed an efficient phonon cascade process (i.e., up to 27 longitudinal optical phonon replicas are revealed in the broad Cu emission band at room temperature) and identified a giant Huang-Rhys factor (S ≈ 12.4, more than 1 order of magnitude larger than reported values of other inorganic semiconductor nanomaterials) in Cu-doped CQWs. We argue that such an ultrastrong electron-phonon coupling in Cu-doped CQWs is due to the dopant-induced lattice distortion and the dopant-enhanced density of states. These findings break the widely accepted consensus that electron-phonon coupling is typically weak in quantum-confined systems, which are crucial for optoelectronic applications of doped electronic nanomaterials.

Simultaneous capturing phonon and electron dynamics in MXenes

Plasmonic MXenes are of particular interest, because of their unique electron and phonon structures and multiple surface plasmon effects, which are different from traditional plasmonic materials. However, to date, how electronic energy damp to lattice vibrations (phonons) in MXenes has not been unraveled. Here, we employed ultrafast broadband impulsive vibrational spectroscopy to identify the energy damping channels in MXenes (Ti3C2Tx and Mo2CTx). Distinctive from the well-known damping pathways, our results demonstrate a different energy damping channel, in which the Ti3C2Tx plasmonic electron energy transfers to coherent phonons by nonthermal electron mediation after Landau damping, without involving electron-electron scattering. Moreover, electrons are observed to strongly couple with A1g mode (~60 fs, 85-100%) and weakly couple with Eg mode (1-2 ps, 0-15%). Our results provide new insight into the electron-phonon interaction in MXenes, which allows the design of materials enabling efficient manipulation of electron transport and energy conversion.

Self-trapped excitons in soft semiconductors

Self-trapped excitons (STEs) have attracted tremendous attention due to their intriguing properties and potential optoelectronic applications. STEs are formed from the lattice distortion induced by the strong electron (exciton)-phonon coupling in soft semiconductors upon photoexcitation, which features in broadband photoluminescence (PL) emission spectra with a large Stokes shift. Recently, significant progress has been achieved in this field but many remain challenges that need to be solved, including the understanding of the underlying physical mechanism, tuning of the performance, and device applications. Along these lines, for the first time, systematic experimental characterizations and advanced theoretical calculations are presented in this review to shed light on the physical mechanism. The possibility of tuning the STEs through multiple degrees of freedom is also presented, along with an overview of the STE-based emerged applications and future research perspectives.

Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods

We benchmark a set of quantum-chemistry methods, including multitrajectory Ehrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfest dynamics, against exact quantum-many-body techniques by studying real-time dynamics in the Holstein model. This is a paradigmatic model in condensed matter theory incorporating a local coupling of electrons to Einstein phonons. For the two-site and three-site Holstein model, we discuss the exact and quantum-chemistry methods in terms of the Born-Huang formalism, covering different initial states, which either start on a single Born-Oppenheimer surface, or with the electron localized to a single site. For extended systems with up to 51 sites, we address both the physics of single Holstein polarons and the dynamics of charge-density waves at finite electron densities. For these extended systems, we compare the quantum-chemistry methods to exact dynamics obtained from time-dependent density matrix renormalization group calculations with local basis optimization (DMRG-LBO). We observe that the multitrajectory Ehrenfest method, in general, only captures the ultrashort time dynamics accurately. In contrast, the surface-hopping method with suitable corrections provides a much better description of the long-time behavior but struggles with the short-time description of coherences between different Born-Oppenheimer states. We show that the multiconfigurational Ehrenfest method yields a significant improvement over the multitrajectory Ehrenfest method and can be converged to the exact results in small systems with moderate computational efforts. We further observe that for extended systems, this convergence is slower with respect to the number of configurations. Our benchmark study demonstrates that DMRG-LBO is a useful tool for assessing the quality of the quantum-chemistry methods.

Hybridized Exciton-Photon-Phonon States in a Transition Metal Dichalcogenide van der Waals Heterostructure Microcavity

Excitons in atomically thin transition-metal dichalcogenides (TMDs) have been established as an attractive platform to explore polaritonic physics, owing to their enormous binding energies and giant oscillator strength. Basic spectral features of exciton polaritons in TMD microcavities, thus far, were conventionally explained via two-coupled-oscillator models. This ignores, however, the impact of phonons on the polariton energy structure. Here we establish and quantify the threefold coupling between excitons, cavity photons, and phonons. For this purpose, we employ energy-momentum-resolved photoluminescence and spatially resolved coherent two-dimensional spectroscopy to investigate the spectral properties of a high-quality-factor microcavity with an embedded WSe_{2} van der Waals heterostructure at room temperature. Our approach reveals a rich multibranch structure which thus far has not been captured in previous experiments. Simulation of the data reveals hybridized exciton-photon-phonon states, providing new physical insight into the exciton polariton system based on layered TMDs.

A reciprocal-space formulation of surface hopping

Surface hopping has seen great success in describing molecular phenomena where electronic excitations tend to be localized, but its application to materials with band-like electronic properties has remained limited. Here, we derive a formulation of fewest-switches surface hopping where both the quantum and classical equations of motion are solved entirely in terms of reciprocal-space coordinates. The resulting method is directly compatible with band structure calculations and allows for the efficient description of band-like phenomena by means of a truncation of the Brillouin zone. Using the Holstein and Peierls models as examples, we demonstrate the formal equivalence between real-space and reciprocal-space surface hopping and assess their accuracy against mean-field mixed quantum-classical dynamics and numerically exact results. Having very similar equations of motion, reciprocal-space surface hopping can be straightforwardly incorporated in existing (real-space) surface hopping implementations.

Impact of Dark Excitons on the Population and Relaxation Kinetics of Two-Dimensional Biexcitons in [CH3(CH2)3NH3]2Pb1-xMnxBr4 (x = 0-0.09)

Two-dimensional (2D) semiconductors have emerged as an excellent platform for studying various excitonic matter under strong quantum and dielectric confinements. However, such effects can be seriously overestimated for Coulomb binding of two excitons to form a biexciton by a naive interpretation of the corresponding photoluminescence (PL) spectrum. By using 2D halide perovskite single crystals of [CH₃(CH₂)₃NH₃]₂Pb_1-xMn_xBr₄ (x = 0-0.09) as a model system, we investigated both population and relaxation kinetics of biexcitons as a function of excitation density, temperature, polarization, and Mn doping. We show that the biexciton is formed by binding of two dark excitons, which are partially bright, but they radiatively recombine to yield a bright exciton in the final state. This renders the spectral distance between the exciton peak and the biexciton peak as very different from the actual biexciton binding energy (ϕ) because of large bright-dark splitting. We show that Mn doping introduces paramagnetism to our 2D system and improves the biexciton stability as evidenced by increase in ϕ from 18.8 ± 0.7 to 20.0 ± 0.7 meV and the increase of the exciton-exciton capture coefficient C from 2.4 × 10^-11 to 4.3 × 10^-11cm²/ns within our doping range. The precisely determined ϕ values are significantly smaller than the previously reported ones, but they are consistent with the instability of the biexciton against thermal dissociation at room temperature. Our results demonstrate that electron-hole exchange interaction must be considered for precisely locating the biexciton level; therefore, the ϕ values should be reassessed for other 2D halide perovskites that even do not exhibit any dark exciton PL.

2D Electronic Spectroscopic Techniques for Quantum Technology Applications

2D electronic spectroscopy (2DES) techniques have gained particular interest given their capability of following ultrafast coherent and noncoherent processes in real-time. Although the fame of 2DES is still majorly linked to the investigation of energy and charge transport in biological light-harvesting complexes, 2DES is now starting to be recognized as a particularly valuable tool for studying transport processes in artificial nanomaterials and nanodevices. Particularly meaningful is the possibility of assessing coherent mechanisms active in the transport of excitation energy in these materials toward possible quantum technology applications. The diverse nature of these new target samples poses significant challenges and calls for a critical rethinking of the technique and its different realizations. With the confluence of promising new applications and rapidly developing technical capabilities, the enormous potential of 2DES techniques to impact the field of nanosystems, quantum technologies, and quantum devices is here delineated.

Multidimensional electronic spectroscopy in high-definition-Combining spectral, temporal, and spatial resolutions

Over the past two decades, coherent multidimensional spectroscopies have been implemented across the terahertz, infrared, visible, and ultraviolet regions of the electromagnetic spectrum. A combination of coherent excitation of several resonances with few-cycle pulses, and spectral decongestion along multiple spectral dimensions, has enabled new insights into wide ranging molecular scale phenomena, such as energy and charge delocalization in natural and artificial light-harvesting systems, hydrogen bonding dynamics in monolayers, and strong light-matter couplings in Fabry-Pérot cavities. However, measurements on ensembles have implied signal averaging over relevant details, such as morphological and energetic inhomogeneity, which are not rephased by the Fourier transform. Recent extension of these spectroscopies to provide diffraction-limited spatial resolution, while maintaining temporal and spectral information, has been exciting and has paved a way to address several challenging questions by going beyond ensemble averaging. The aim of this Perspective is to discuss the technological developments that have eventually enabled spatially resolved multidimensional electronic spectroscopies and highlight some of the very recent findings already made possible by introducing spatial resolution in a powerful spectroscopic tool.

A reciprocal-space formulation of mixed quantum-classical dynamics

We derive a formulation of mixed quantum-classical dynamics for modeling electronic carriers interacting with phonons in reciprocal space. For dispersionless phonons, we start by expressing the real-space classical coordinates in terms of complex variables. Taking these variables as a Fourier series then yields the reciprocal-space coordinates. Evaluating the electron-phonon interaction term through Ehrenfest's theorem, we arrive at a reciprocal-space formalism that is equivalent to mean-field mixed quantum-classical dynamics in real space. This equivalence is numerically verified for the Holstein and Peierls models, for which we find the reciprocal-space Hellmann-Feynman forces to involve momentum-derivative contributions in addition to the position-derivative terms commonly seen in real space. To illustrate the advantage of the reciprocal-space formulation, we present a proof of concept for the inexpensive modeling of low-momentum carriers interacting with phonons using a truncated reciprocal-space basis, which is not possible within a real-space formulation.