Third-order Response Functions Research Articles

Transient Chiral Dynamics in the Fenna-Matthews-Olson Complex Revealed by Two-Dimensional Circular Dichroism Spectroscopy.

Chirality plays a pivotal role across scientific disciplines with profound implications spanning light-matter interactions, molecular recognition, and natural evolutionary processes. This study delves into the active influence of molecular chirality on exciton energy transfer within photosynthetic protein complexes, focusing on the Fenna-Matthews-Olson (FMO) complex. Employing two-dimensional circular dichroism (2DCD) spectroscopy, we investigate the transient chiral dynamics of excitons during energy transfer processes within the FMO complex. Our approach, incorporating pulse information into population dynamics based on the third-order response function, facilitates the calculation of 2DCD spectra and dynamics. This enables the extraction of chiral contributions to excitonic energy transfer and the examination of electronic wave functions. We demonstrate that 2DCD spectra offer excitation energies that are better resolved than those from conventional two-dimensional electronic spectroscopy. These findings deepen our understanding of exciton energy transfer mechanisms in natural photosynthesis, emphasizing the potential of 2DCD spectroscopy as a powerful tool for unraveling the chiral contribution to exciton dynamics.

Theory of 2D electronic spectroscopy of water soluble chlorophyll-binding protein (WSCP): Signatures of Chl b derivate.

We investigate how electronic excitations and subsequent dissipative dynamics in the water soluble chlorophyll-binding protein (WSCP) are connected to features in two-dimensional (2D) electronic spectra, thereby comparing results from our theoretical approach with experimental data from the literature. Our calculations rely on third-order response functions, which we derived from a second-order cumulant expansion of the dissipative dynamics involving the partial ordering prescription, assuming a fast vibrational relaxation in the potential energy surfaces of excitons. Depending on whether the WSCP complex containing a tetrameric arrangement of pigments composed of two dimers with weak excitonic coupling between them binds the chlorophyll variant Chl a or Chl b, the resulting linear absorption and circular dichroism spectra and particularly the 2D spectra exhibit substantial differences in line shapes. These differences between Chl a WSCP and Chl b WSCP cannot be explained by the slightly modified excitonic couplings within the two variants. In the case of Chl a WSCP, the assumption of equivalent dimer subunits facilitates a reproduction of substantial features from the experiment by the calculations. In contrast, for Chl b WSCP, we have to assume that the sample, in addition to Chl b dimers, contains a small but distinct fraction of chemically modified Chl b pigments. The existence of such Chl b derivates has been proposed by Pieper et al. [J. Phys. Chem. B 115, 4042 (2011)] based on low-temperature absorption and hole-burning spectroscopy. Here, we provide independent evidence.

Time-Resolved X-ray Absorption Spectroscopy: An MCTDH Quantum Dynamics Protocol.

Expressions for linear and nonlinear spectroscopy simulation in the X-ray window in which the time evolution of a photoexcited molecular system is treated via quantum dynamics are derived. By leveraging on the peculiar properties of core-excited/ionized states, first- and third-order response functions are recast in the limit of time-scale separation between the extremely short core-state lifetime and the (comparably longer) electronic-state transfer and nuclear vibrational motion. This work is a natural extension of Segatta et al. (J. Chem. Theory Comput. 2023, 19, 2075-2091), in which some of the present authors coupled MCTDH quantum dynamics to spectroscopy simulation at different levels of sophistication. Full quantum dynamics and approximate expressions are compared by simulating X-ray transient absorption spectroscopy at the carbon K-edge in the pyrene molecule.

Measuring the Ultrafast Correlation Dynamics of a Multilevel System Using the Center Line Slope Analysis in Two-Dimensional Electronic Spectroscopy.

In a two-dimensional (2D) optical spectrum of a multilevel system, there are diagonal peaks and off-diagonal cross-peaks that correlate the different levels. The time-dependent properties of these diagonal peaks and cross-peaks contain much information about the dynamics of the multilevel system. The time-dependent diagonal peakshape that depends on the spectral diffusion dynamics of the associated transition and characterized by the frequency-fluctuation correlation function (FFCF) is well studied. However, the time-dependent peakshape of a cross-peak that provides the correlation dynamics between different transitions is much less studied or understood. We derived the third-order nonlinear response functions that describe the cross-peaks in a 2D electronic spectrum of a multilevel system that arise from processes sharing a common ground state and/or from internal conversion and population transfer. We can use the center line slope (CLS) analysis to characterize the cross-peaks in conjunction with the diagonal peaks. This allows us to recover the frequency-fluctuation cross-correlation functions (FXCFs) between two transitions. The FXCF and its subsidiary quantities such as the initial correlation and the initial covariance between different transitions are important for studying the correlation effects between states in complex systems, such as energy-transfer processes. Furthermore, knowledge of how various molecular processes over different timescales affect simultaneously different transitions can also be obtained from the measured FXCF. We validated and tested our derived equations and analysis process by studying, as an example, the 2D electronic spectra of metal-free phthalocyanine in solution. We measured and analyzed the diagonal peaks of the Qx and Qy transitions and the cross-peaks between these two transitions of this multilevel electronic system and obtained the associated FFCFs and FXCFs. In this model system, we measured negative components of FXCF over the tens of picosecond timescale. This suggests that in phthalocyanine, the Qx and Qy transitions coupling with the solvent molecule motion are anticorrelated to each other.

Vibrational response functions for multidimensional electronic spectroscopy in nonadiabatic models.

The interplay of nuclear and electronic dynamics characterizes the multidimensional electronic spectra of various molecular and solid-state systems. Theoretically, the observable effect of such interplay can be accounted for by response functions. Here, we report analytical expressions for the response functions corresponding to a class of model systems. These are characterized by coupling between the diabatic electronic states and the vibrational degrees of freedom, resulting in linear displacements of the corresponding harmonic oscillators, and by nonadiabatic couplings between pairs of diabatic states. In order to derive the linear response functions, we first perform the Dyson expansion of the relevant propagators with respect to the nonadiabatic component of the Hamiltonian, then derive and expand with respect to the displacements the propagators at given interaction times, and finally provide analytical expressions for the time integrals that lead to the different contributions to the linear response function. The approach is then applied to the derivation of third-order response functions describing different physical processes: ground state bleaching, stimulated emission, excited state absorption, and double quantum coherence. Comparisons between the results obtained up to sixth order in the Dyson expansion and independent numerical calculation of the response functions provide evidence of the series convergence in a few representative cases.

Vibrational response functions for multidimensional electronic spectroscopy in the adiabatic regime: A coherent-state approach.

Multi-dimensional spectroscopy represents a particularly insightful tool for investigating the interplay of nuclear and electronic dynamics, which plays an important role in a number of photophysical processes and photochemical reactions. Here, we present a coherent state representation of the vibronic dynamics and of the resulting response functions for the widely used linearly displaced harmonic oscillator model. Analytical expressions are initially derived for the case of third-order response functions in an N-level system, with ground state initialization of the oscillator (zero-temperature limit). The results are then generalized to the case of Mth order response functions, with arbitrary M. The formal derivation is translated into a simple recipe, whereby the explicit analytical expressions of the response functions can be derived directly from the Feynman diagrams. We further generalize to the whole set of initial coherent states, which form an overcomplete basis. This allows one, in principle, to derive the dependence of the response functions on arbitrary initial states of the vibrational modes and is here applied to the case of thermal states. Finally, a non-Hermitian Hamiltonian approach is used to include in the above expressions the effect of vibrational relaxation.

Theory of two-dimensional spectroscopy with intense laser fields.

Two-dimensional vibrational and electronic spectroscopic observables of isotropically oriented molecular samples in solution are sensitive to laser field intensities and polarization. The third-order response function formalism predicts a signal that grows linearly with the field strength of each laser pulse, thus lacking a way of accounting for non-trivial intensity-dependent effects, such as saturation and finite bleaching. An analytical expression to describe the orientational part of the molecular response, which, in the weak-field limit, becomes equivalent to a four-point correlation function, is presented. Such an expression is evaluated for Liouville-space pathways accounting for diagonal and cross peaks for all-parallel and cross-polarized pulse sequences, in both the weak- and strong-field conditions, via truncation of a Taylor series expansion at different orders. The results obtained in the strong-field conditions suggest how a careful analysis of two-dimensional spectroscopic experimental data should include laser pulse intensity considerations when determining molecular internal coordinates.

On-the-fly ab initio semiclassical evaluation of third-order response functions for two-dimensional electronic spectroscopy.

Ab initio computation of two-dimensional electronic spectra is an expanding field, whose goal is improving upon simple, few-dimensional models often employed to explain experiments. Here, we propose an accurate and computationally affordable approach, based on the single-trajectory semiclassical thawed Gaussian approximation, to evaluate two-dimensional electronic spectra. Importantly, the method is exact for arbitrary harmonic potentials with mode displacement, changes in the mode frequencies, and inter-mode coupling (Duschinsky effect), but can also account partially for the anharmonicity of the involved potential energy surfaces. We test its accuracy on a set of model Morse potentials and use it to study anharmonicity and Duschinsky effects on the linear and two-dimensional electronic spectra of phenol. We find that in this molecule, the anharmonicity effects are weak, whereas the Duschinsky rotation and the changes in the mode frequencies must be included in accurate simulations. In contrast, the widely used displaced harmonic oscillator model captures only the basic physics of the problem but fails to reproduce the correct vibronic lineshape.

Multi-faceted spectroscopic mapping of ultrafast nonadiabatic dynamics near conical intersections: A computational study.

We studied spectroscopic signatures of the nonadiabatic dynamics at conical intersections formed by the lowest excited singlet states in pyrazine. We considered two ab initio models of conical intersections in the excited states of pyrazine developed by Sala et al. [Phys. Chem. Chem. Phys. 16, 15957 (2014)]: a two-state (B2u and B3u), five-mode model and a three-state (B2u, B3u, and Au), nine-mode model. We simulated the signals of three widely used techniques: time- and frequency-resolved fluorescence spectroscopy, transient absorption pump-probe spectroscopy, and electronic two-dimensional spectroscopy. The signals were calculated through third-order response functions, which, in turn, were evaluated with the numerically accurate multiple Davydov ansatz. We establish spectroscopic signatures of the optically dark Au state and demonstrate that the key features of the photoinduced dynamics, such as electronic/nuclear populations, electronic/nuclear coherences, and electronic/nuclear energy transfer processes, are imprinted in the spectroscopic signals. We show that a fairly complete picture of the nonadiabatic dynamics at conical intersections can be obtained when several spectroscopic techniques are combined. Provided that the time resolution is sufficient, time- and frequency-resolved fluorescence may provide the best visualization of the nonadiabatic dynamics near conical intersections.

Theory of coherent two-dimensional vibrational spectroscopy.

Two-dimensional (2D) vibrational spectroscopy has emerged as one of the most important experimental techniques useful to study the molecular structure and dynamics in condensed phases. Theory and computation have also played essential and integral roles in its development through the nonlinear optical response theory and computational methods such as molecular dynamics (MD) simulations and electronic structure calculations. In this article, we present the fundamental theory of coherent 2D vibrational spectroscopy and describe computational approaches to simulate the 2D vibrational spectra. The classical approximation to the quantum mechanical nonlinear response function is invoked from the outset. It is shown that the third-order response function can be evaluated in that classical limit by using equilibrium or non-equilibrium MD simulation trajectories. Another simulation method is based on the assumptions that the molecular vibrations can still be described quantum mechanically and that the relevant molecular response functions are evaluated by the numerical integration of the Schrödinger equation. A few application examples are presented to help the researchers in this and related areas to understand the fundamental principles and to use these methods for their studies with 2D vibrational spectroscopic techniques. In summary, this exposition provides an overview of current theoretical efforts to understand the 2D vibrational spectra and an outlook for future developments.

Dual frequency comb photon echo spectroscopy

Dual frequency comb (DFC) nonlinear spectroscopy is an emerging technique that can be used to study a variety of molecular nonlinear responses by exploiting the automatic pulse-to-pulse time delay scanning and fast data acquisition characteristics of the DFC techniques. Here, we propose a DFC-based photon echo spectroscopy (DFC-PES), where two optical frequency comb lasers have slightly different repetition rates and are also allowed to have different carrier frequencies. We first demonstrate our theoretical approach for DFC linear spectroscopy. Then, the signals expected from the proposed DFC-PES are theoretically calculated. The slight offset in the pulse repetition rates enables asynchronous optical sampling and automatic scanning of the time intervals between different field–matter interaction events, making frequency tuning unnecessary. Analytic expression of the third-order photon echo signal is demonstrated in two frequency dimensions for a simple two-level model system. The signal has a well-defined and simple connection to the underlying third-order response function and exhibits the expected down-conversion features from the optical frequency into the radio frequency region with a down-conversion factor that is experimentally controllable.

Monitoring Charge Separation Processes in Quasi-One-Dimensional Organic Crystalline Structures.

We perform the transient absorption spectroscopy experiments to investigate the dynamics of the low-energy collective electron-hole excitations in α-copper phthalocyanine thin films. The results are interpreted in terms of the third-order nonlinear polarization response function. It is found that, initially excited in the molecular plane, the intramolecular Frenkel exciton polarization reorients with time to align along the molecular chain direction to form coupled Frenkel-charge-transfer exciton states, the eigenstates of the one-dimensional periodic molecular lattice. The process pinpoints the direction of the charge separation in α-copper phthalocyanine and similar organic molecular structures. Being able to observe and monitor such processes is important both for understanding the physical principles of organic thin film solar energy conversion device operation and for the development of organic optoelectronics in general.

Resonantly excited exciton dynamics in two-dimensional MoSe2 monolayers

We report on the exciton and trion density dynamics in a single layer of MoSe$_2$, resonantly excited and probed using three-pulse four-wave mixing (FWM), at temperatures from 300K to 77K . A multi-exponential third-order response function for amplitude and phase of the heterodyne-detected FWM signal including four decay processes is used to model the data. We provide a consistent interpretation within the intrinsic band structure, not requiring the inclusion of extrinsic effects. We find an exciton radiative lifetime in the sub-picosecond range consistent to what has been recently reported. After the dominating radiative decay, the remaining exciton density, which has been scattered from the initially excited bright radiative state into dark states of different nature by exciton-phonon scattering or disorder scattering, shows a slower dynamics, covering 10ps to 10ns timescales. This includes direct bright transitions with larger in-plane momentum, as well as indirect dark transitions to indirect dark states. We find that exciton-exciton annihilation is not relevant in the observed dynamics, in variance from previous finding under non-resonant excitation. The trion density at 77K reveals a decay of the order of 1ps, similar to what is observed for the exciton. After few tens of picoseconds, the trion dynamics resembles the one of the exciton, indicating that trion ionization occurs on this timescale.

Nonlinear Correction to Absorption Spectrum under Irradiation of Microwave Field in Conventional BCS Superconductors

We investigate the absorption spectrum of s-wave superconductors under microwave pump field irradiation. The third-order response function is calculated in the dirty limit with the electron–phonon interaction included at finite temperatures. We find that the nonlinear correction to the linear absorption shows peculiar behavior when the pump field frequency is smaller than the superconducting gap. At finite temperatures, a negative nonlinear correction exists, which is caused by thermally excited quasiparticles. The vertex correction by impurity scattering is found to contain a dissipation mechanism by inelastic scattering (interaction between electrons and acoustic phonons) or nonlocality. We need this mechanism to obtain finite absorption in a nonequilibrium stationary state under a monochromatic external field. Although this term originates from the deformation of a one-particle state, there is also a final-state interaction (the amplitude mode). The latter term represents two-photon excitation and is a...

Theory of plasmonic effects in nonlinear optics: The case of graphene

We develop a microscopic large-$N$ theory of electron-electron interaction corrections to multi-legged Feynman diagrams describing second- and third-order nonlinear response functions. Our theory, which reduces to the well-known random phase approximation in the linear-response limit, is completely general and is useful to understand all second- and third-order nonlinear effects, including harmonic generation, wave mixing, and photon drag. We apply our theoretical framework to the case of graphene, by carrying out microscopic calculations of the second- and third-order nonlinear response functions of an interacting two-dimensional (2D) gas of massless Dirac fermions. We compare our results with recent measurements, where all-optical launching of graphene plasmons has been achieved by virtue of the finiteness of the quasi-homogeneous second-order nonlinear response of this inversion-symmetric 2D material.

Effects of finite laser pulse width on two-dimensional electronic spectroscopy

We combine the hierarchical equations of motion method and the equation-of-motion phase-matching approach to calculate two-dimensional electronic spectra of model systems. When the laser pulse is short enough, the current method reproduces the results based on third-order response function calculations in the impulsive limit. Finite laser pulse width is found to affect both the peak positions and shapes, as well as the time evolution of diagonal and cross peaks. Simulations of the two-color two-dimensional electronic spectra also show that, to observe quantum beats in the diagonal and cross peaks, it is necessary to excite the related excitonic states simultaneously.

Modeling of ultrafast time-resolved fluorescence applied to a weakly coupled chromophore pair.

We present theory for calculating the third-order non-linear response function of a molecular aggregate in the weak inter-chromophore coupling regime. This approach is based on the perturbative expansion of the system evolution with respect to the resonance coupling, while the system-bath interaction is treated non-perturbatively by means of cumulant expansion. An explicit expression for the time-resolved fluorescence signal is then obtained. This allows us to investigate the ultrafast time-dependent Stokes shift, signatures of coherent dynamics, and the excitonic polaron formation in the excited state of the aggregate. Numerical simulations of the time-resolved fluorescence spectra of a pair of coupled molecules demonstrate these effects.

Davydov Ansatz as an efficient tool for the simulation of nonlinear optical response of molecular aggregates.

We have developed a variational approach to the description of four-wave-mixing signals of molecular aggregates, in which the third-order response functions are evaluated in terms of the Davydov Ansätze. Our theory treats both singly and doubly excited excitonic states, handling the contributions due to stimulated emission, ground state bleach, and excited state absorption. As an illustration, we simulate a series of optical two-dimensional spectra of model J-aggregates. Our approach may become suitable for the computation of femtosecond optical four-wave-mixing signals of molecular aggregates with intermediate-to-strong exciton-phonon and exciton-exciton coupling strengths.

Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide

Carbon disulfide is the most popular material for applications of nonlinear optical (NLO) liquids, and is frequently used as a reference standard for NLO measurements. Although it has been the subject of many investigations, determination of the third-order optical nonlinearity of CS2 has been incomplete. This is in part because of several strong mechanisms for nonlinear refraction (NLR), leading to a complex pulse width dependence. We expand upon the recently developed beam deflection technique, which we apply, along with degenerate four-wave mixing and Z-scan, to quantitatively characterize (in detail) the NLO response of CS2, over a broad temporal range, spanning 6 orders of magnitude (∼32 fs to 17 ns). The third-order response function, consisting of both nearly instantaneous bound-electronic and noninstantaneous nuclear contributions, along with the polarization and wavelength dependence from 390 to 1550 nm, is extracted from these measurements. This paper provides a self-consistent, quantitative picture of the third-order NLO response of liquid CS2, establishing it as an accurate reference material over this broad temporal and spectral range. These results allow prediction of the outcome of any NLR experiment on CS2.

Terahertz Kerr effect in gallium phosphide crystal

We report on the observation of the Kerr effect induced by an intense terahertz pulse in a 〈100〉 gallium phosphide crystal. The temporal and angular behaviors of the phase retardation have been measured and agree well with theoretical predictions. From these measurements, we extracted the two nonzero tensor elements of the third-order response function of the crystal in the terahertz range.