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.

Overcoming positivity violations for density matrices in surface hopping.

Fewest-switches surface hopping (FSSH) has emerged as one of the leading methods for modeling the quantum dynamics of molecular systems. While its original formulation was limited to adiabatic populations, the growing interest in the application of FSSH to coherent phenomena prompts the question of how one should construct a complete density matrix based on FSSH trajectories. A straightforward solution is to define adiabatic coherences based on wavefunction coefficients. In this paper, we demonstrate that inconsistencies introduced in the density matrix through such treatment may lead to a violation of positivity. We furthermore show that a recently proposed coherent generalization of FSSH results in density matrices that satisfy positivity while yielding improved accuracy throughout much (but not all) of parameter space.

Ehrenfest modeling of cavity vacuum fluctuations and how to achieve emission from a three-level atom.

A much-needed solution for the efficient modeling of strong coupling between matter and optical cavity modes is offered by mean-field mixed quantum-classical dynamics, where a classical cavity field interacts self-consistently with quantum states of matter through Ehrenfest's theorem. We previously introduced a modified mean-field approach, referred to as decoupled mean-field (DC-MF) dynamics, wherein vacuum fluctuations of the cavity field are decoupled from the quantum-mechanical ground state as a means to resolve an unphysical drawing of energy from the vacuum fluctuations by a two-level atom. Here, we generalize DC-MF dynamics for an arbitrary number of (nondegenerate) atomic levels and show that it resolves an unphysical lack of emission from a three-level atom predicted by conventional mean-field dynamics. We furthermore show DC-MF to provide an improved description of reabsorption and (resonant) two-photon emission processes.

Modulating Singlet Fission by Scanning through Vibronic Resonances in Pentacene-Based Blends.

Vibronic coupling has been proposed to play a decisive role in promoting ultrafast singlet fission (SF), the conversion of a singlet exciton into two triplet excitons. Its inherent complexity is challenging to explore, both from a theoretical and an experimental point of view, due to the variety of potentially relevant vibrational modes. Here, we report a study on blends of the prototypical SF chromophore pentacene in which we engineer the polarizability of the molecular environment to scan the energy of the excited singlet state (S1) continuously over a narrow energy range, covering vibrational sublevels of the triplet-pair state (1(TT)). Using femtosecond transient absorption spectroscopy, we probe the dependence of the SF rate on energetic resonance between vibronic states and, by comparison with simulation, identify vibrational modes near 1150 cm-1 as key in facilitating ultrafast SF in pentacene.

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.

Theory of Apparent Circular Dichroism Reveals the Origin of Inverted and Noninverted Chiroptical Response under Sample Flipping.

Circular dichroism (CD) finds widespread application as an optical probe for the structure of molecules and supramolecular assemblies. Its underlying chiral light-matter interactions effectively couple between photonic spin states and select quantum-mechanical degrees of freedom in a sample, implying an intricate connection with photon-to-matter quantum transduction. However, effective transduction implementations likely require interactions that are antisymmetric with respect to the direction of light propagation through the sample, yielding an inversion of the chiroptical response upon sample flipping, which is uncommon for CD. Recent experiments on organic thin films have demonstrated such chiroptical behavior, which was attributed to "apparent CD" resulting from an interference between the sample's linear birefringence and linear dichroism. However, a theory connecting the underlying optical selection rules to the microscopic electronic structure of the constituent molecules remains to be formulated. Here, we present such a theory based on a combination of Mueller calculus and a Lorentz oscillator model. The theory reaches good agreement with experimental CD spectra and allows for establishing the (supra)molecular design rules for maximizing or minimizing this chiroptical effect. It furthermore highlights that, in addition to antisymmetrically, it can manifest symmetrically such that no chiroptical response inversion occurs, which is a consequence of a helical stacking of molecules in the light propagation direction.

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.

Analytic and numerical vibronic spectra from quasi-classical trajectory ensembles.

The truncated Wigner approximation to quantum dynamics in phase space is explored in the context of computing vibronic line shapes for monomer linear optical spectra. We consider multiple model potential forms including a shifted harmonic oscillator with both equal and unequal frequencies on the ground and excited state potentials as well as a shifted Morse potential model. For the equal-frequency shifted harmonic oscillator model, we derive an analytic expression for the exact vibronic line shape that emphasizes the importance of using a quantum mechanical distribution of phase space initial conditions. For the unequal-frequency shifted harmonic oscillator model, we are no longer able to obtain an exact expression for the vibronic line shape in terms of independent deterministic classical trajectories. We show how one can rigorously account for corrections to the truncated Wigner approximation through nonlinear responses of the line shape function to momentum fluctuations along a classical trajectory and demonstrate the qualitative improvement in the resulting spectrum when the leading-order quantum correction is included. Finally, we numerically simulate absorption spectra of a highly anharmonic shifted Morse potential model. We find that, while finite quantization and the dissociation limit are captured with reasonable accuracy, there is a qualitative breakdown of the quasi-classical trajectory ensemble's ability to describe the vibronic line shape when the relative shift in Morse potentials becomes large. The work presented here provides clarity on the origin of unphysical negative features known to contaminate absorption spectra computed with quasi-classical trajectory ensembles.

Multiset Matrix Product State Calculations Reveal Mobile Franck-Condon Excitations Under Strong Holstein-Type Coupling.

We show that the dynamics of (vertical) Franck-Condon excitations in the regime where Holstein-coupled vibrational modes mix strongly with electronic degrees of freedom sharply contrasts with the known self-localized behavior of vibrationally relaxed excitations. Instead, the strongly coupled modes are found to periodically induce resonances between interacting electronic sites, during which effective excitation transfer occurs, allowing Franck-Condon excitations to attain substantial mean square displacements under conditions where relaxed excitations are essentially trapped to a single site. In demonstrating this behavior, we employ a multiset matrix product state formalism. We find this tensor network state method to be a remarkably efficient and accurate approach for the notoriously difficult problem posed by the Holstein model in the regime where the electronic coupling, the vibrational quantum, and the vibrational reorganization energy are comparable in magnitude.

Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches.

The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. We also discuss issues associated with the non-normality of HEOM.

Vibronic exciton theory of singlet fission. III. How vibronic coupling and thermodynamics promote rapid triplet generation in pentacene crystals.

We extend the vibronic exciton theory introduced in our previous work to study singlet fission dynamics, in particular addressing recent indications of the importance of vibronic coupling in this process. A microscopic and non-perturbative treatment of electronic and selected vibrational degrees of freedom in combination with Redfield theory allows us to dynamically consider clusters of molecules under conditions close to those in molecular crystals that exhibit fission. Using bulk pentacene as a concrete example, our results identify a number of factors that render fission rapid and effective. Strong coupling to high-frequency Holstein modes generates resonances between the photo-prepared singlet and product triplet states. We furthermore find the large number of triplet combinations associated with bulk periodic systems to be critical to the fission process under such vibronically resonant conditions. In addition, we present results including, in an approximate manner, the effects of Peierls coupling, indicating that this factor can both enhance and suppress fission depending on its interplay with vibronic resonance and thermodynamics.

Generalization of fewest-switches surface hopping for coherences.

Fewest-switches surface hopping (FSSH) is perhaps the most widely used mixed quantum-classical approach for the modeling of non-adiabatic processes, but its original formulation is restricted to (adiabatic) population terms of the quantum density matrix, leaving its implementations with an inconsistency in the treatment of populations and coherences. In this article, we propose a generalization of FSSH that treats both coherence and population terms on equal footing and which formally reduces to the conventional FSSH algorithm for the case of populations. This approach, coherent fewest-switches surface hopping (C-FSSH), employs a decoupling of population relaxation and pure dephasing and involves two replicas of the classical trajectories interacting with two active surfaces. Through extensive benchmark calculations of a spin-boson model involving a Debye spectral density, we demonstrate the potential of C-FSSH to deliver highly accurate results for a large region of parameter space. Its uniform description of populations and coherences is found to resolve incorrect behavior observed for conventional FSSH in various cases, in particular at low temperature, while the parameter space regions where it breaks down are shown to be quite limited. Its computational expenses are virtually identical to conventional FSSH.

Triplet Separation Drives Singlet Fission after Femtosecond Correlated Triplet Pair Production in Rubrene.

Singlet fission, a multistep molecular process in which one photon generates two triplet excitons, holds great technological promise. Here, by applying a combination of transient transmittance and two-dimensional electronic spectroscopy with 5 fs laser pulses, we resolve the full set of fission steps before the onset of spin dephasing. In addition to its role as a viable singlet fission material, single-crystalline rubrene is selected because its energetics and transition dipole alignment uniquely allow for the unambiguous identification of the various fission steps through their contributions to distinct spectroscopic features. The measurements reveal that the neighboring correlated triplet pair achieves its maximum population within 20 fs. Subsequent growth of the triplet signal on picosecond time scales is attributable to spatial separation of the triplets, proceeding nonadiabatically through weakly coupled but near-resonant states. As such, we provide evidence in crystalline rubrene for a singlet fission step that, until now, has not been convincingly observed.

Vibronic exciton theory of singlet fission. II. Two-dimensional spectroscopic detection of the correlated triplet pair state.

Singlet fission, the molecular process through which photons are effectively converted into pairs of lower energy triplet excitons, holds promise as a means of boosting photovoltaic device efficiencies. In the preceding article of this series, we formulated a vibronic theory of singlet fission, inspired by previous experimental and theoretical studies suggesting that vibronic coupling plays an important role in fission dynamics. Here, we extend our model in order to simulate two-dimensional electronic spectra, through which the theory is further validated based on a comparison to recent measurements on pentacene crystals. Moreover, by means of such spectral simulations, we provide new insights into the nature of the correlated triplet pair state, the first product intermediate in the fission process. In particular, we address a controversy in the literature regarding the identification, energies, and transition dipole moments of its optical transitions towards higher-lying triplet states.

Vibronic exciton theory of singlet fission. I. Linear absorption and the anatomy of the correlated triplet pair state.

Recent time-resolved spectroscopic experiments have indicated that vibronic coupling plays a vital role in facilitating the process of singlet fission. In this work, which forms the first article of a series, we set out to unravel the mechanisms underlying singlet fission through a vibronic exciton theory. We formulate a model in which both electronic and vibrational degrees of freedom are treated microscopically and non-perturbatively. Using pentacene as a prototypical material for singlet fission, we subject our theory to comparison with measurements on polarization-resolved absorption of single crystals, and employ our model to characterize the excited states underlying the absorption band. Special attention is given to the convergence of photophysical observables with respect to the basis size employed, through which we determine the optimal basis for more expensive calculations to be presented in subsequent work. We furthermore evaluate the energetic separation between the optically prepared singlet excited state and the correlated triplet pair state, as well as provide a real-space characterization of the latter, both of which are of key importance in the discussion of fission dynamics. We discuss our results in the context of recent experimental studies.

Laser-Limited Signatures of Quantum Coherence.

Quantum coherence is proclaimed to promote efficient energy collection by light-harvesting complexes and prototype organic photovoltaics. However, supporting spectroscopic studies are hindered by the problem of distinguishing between the excited state and ground state origin of coherent spectral transients. Coherence amplitude maps, which systematically represent quantum beats observable in two-dimensional (2D) spectroscopy, are currently the prevalent tool for making this distinction. In this article, we present coherence amplitude maps of a molecular dimer, which have become significantly distorted as a result of the finite laser bandwidth used to record the 2D spectra. We argue that under standard spectroscopic conditions similar distortions are to be expected for compounds absorbing over a spectral range similar to, or exceeding, that of the dimer. These include virtually all photovoltaic polymers and certain photosynthetic complexes. With the distortion of coherence amplitude maps, alternative ways to identify quantum coherence are called for. Here, we use numerical simulations that reproduce the essential photophysics of the dimer to unambiguously determine the excited state origin of prominent quantum beats observed in the 2D spectral measurements. This approach is proposed as a dependable method for coherence identification.

Mixed Quantum-Classical Dynamics under Arbitrary Unitary Basis Transformations.

A common approach to minimizing the cost of quantum computations is by unitarily transforming a quantum system into a basis that can be optimally truncated. Here, we derive classical equations of motion subjected to similar unitary transformations and propose their integration into mixed quantum-classical dynamics, allowing this class of methods to be applied within arbitrary bases for both the quantum and classical coordinates. To this end, canonical positions and momenta of the classical degrees of freedom are combined into a set of complex-valued coordinates amenable to unitary transformations. We demonstrate the potential of the resulting approach by means of surface hopping calculations of an electronic carrier scattering onto a single impurity in the presence of phonons. Appropriate basis transformations, capturing both the localization of the impurity and the delocalization of higher-energy excitations, are shown to faithfully capture the dynamics within a fraction of the classical and quantum basis sets.

Theory predicts 2D chiral polaritons based on achiral Fabry-Pérot cavities using apparent circular dichroism.

Realizing polariton states with high levels of chirality offers exciting prospects for quantum information, sensing, and lasing applications. Such chirality must emanate from either the involved optical resonators or the quantum emitters. Here, we theoretically demonstrate a rare opportunity for realizing polaritons with so-called 2D chirality by strong coupling of the optical modes of (high finesse) achiral Fabry-Pérot cavities with samples exhibiting "apparent circular dichroism" (ACD). ACD is a phenomenon resulting from an interference between linear birefringence and dichroic interactions. By introducing a quantum electrodynamical theory of ACD, we identify the design rules based on which 2D chiral polaritons can be produced, and their chirality can be optimized.

Dark State Concentration Dependent Emission and Dynamics of CdSe Nanoplatelet Exciton-Polaritons.

The recent surge of interest in polaritons has prompted fundamental questions about the role of dark states in strong light-matter coupling phenomena. Here, we systematically vary the relative number of dark states by controlling the number of stacked CdSe nanoplatelets confined in a Fabry-Pérot cavity. We find the emission spectrum to change significantly with an increasing number of nanoplatelets, with a gradual shift of the dominant emission intensity from the lower polariton branch to a manifold of dark states. Through accompanying calculations based on a kinetic model, this shift is rationalized by an entropic trapping of excitations by the dark state manifold, while a weak dark state dispersion due to local disorder explains their nonzero emission. Our results point toward the relevance of the dark state concentration to the optical and dynamical properties of cavity-embedded quantum emitters with ramifications for Bose-Einstein condensate formation, polariton lasing, polariton-based quantum transduction schemes, and polariton chemistry.