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The Journal of Chemical Physics

Towards microscopic assignment of oscillative signatures in two-dimensional electronic photon-echo signals of vibronic oligomers: A vibronic dimer model

Leah Z. Sharp, Dassia Egorova

Published Oct 8, 2013• 14 citations
Publisher

Abstract

We employ multi-level Redfield theory and the equation-of-motion phase-matching approach to model two-dimensional electronic photon-echo signals of a vibronic dimer weakly coupled to an uncorrelated harmonic bath. The vibronic manifold of the dimer is formed by Franck-Condon active vibrational modes (one mode per monomer). We study the effect of the coupled exciton-vibrational dynamics on the two-dimensional photon-echo profiles. Our main focus is the relationship of the system coherences to the oscillatory behavior of the cross-peak intensities with population time. The work represents the first step towards detailed understanding of oscillative signatures in two-dimensional electronic photon-echo signals of vibronic oligomers.

DOI
10.1063/1.4822425

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quant-pharXiv:2602.04061•Feb 5, 2026
We employ multi-level Redfield theory and the equation-of-motion phase-matching approach to model two-dimensional electronic photon-echo signals of a vibronic dimer weakly coupled to an uncorrelated harmonic bath. The vibronic manifold of the dimer is formed by Franck-Condon active vibrational modes (one mode per monomer). We study the effect of the coupled exciton-vibrational dynamics on the two-dimensional photon-echo profiles. Our main focus is the relationship of the system coherences to the oscillatory behavior of the cross-peak intensities with population time. The work represents the first step towards detailed understanding of oscillative signatures in two-dimensional electronic photon-echo signals of vibronic oligomers.
Stick-spectrum expressions for electronic two-dimensional (2D) photon-echo (PE) signal of a generic multi-level system are presented and employed to interrelate oscillations in individual peaks of 2D PE signal and the underlying properties (eigenstates and coherent dynamics) of excitonic or vibronic systems. When focusing on the identification of the origin of oscillations in the rephasing part of 2D PE it is found, in particular, that multiple frequencies in the evolution of the individual peaks do not necessarily directly reflect the underlying system dynamics. They may originate from the excited-state absorption contribution to the signal, or arise due to multi-level vibrational structure of the electronic ground state, and represent a superposition of system frequencies, while the latter may evolve independently. The analytical stick-spectrum predictions are verified and illustrated by numerical calculations of 2D PE signals of an excitonic trimer and of a displaced harmonic oscillator with unequal vibrational frequencies in the two electronic states. The excitonic trimer is the smallest excitonic oligomer where excited-state absorption may represent a superposition of excited-state coherences and significantly influence the phase of the observed oscillations. The displaced oscillator is used to distinguish between the frequencies of the ground-state and of the excited-state manifolds, and to demonstrate how the location of a cross peak in 2D pattern of the PE signal "predetermines" its oscillatory behavior. Although the considered models are kept as simple as possible for clarity, the stick-spectrum analysis provides a solid general basis for interpretation of oscillatory signatures in electronic 2D PE signals of much more complex systems with multi-level character of the electronic states.
In this article the third-order response of an excitonically coupled dimer is studied. The three-pulse photon echo signals were calculated by extracting polarization components from the total polarization in the corresponding phase-matched directions. The total nonlinear response was obtained by numeric propagation of the density matrix, with the exciton-vibrational coupling modeled via Redfield relaxation theory. The full two-dimensional three-pulse photon echo signals and the peak shift were analyzed in terms of the density-matrix dynamics of coherence dephasing and population relaxation. The location of the two-exciton state was found to be essential for proper modeling of the three-pulse photon echo. In particular, an oscillation in the three-pulse photon echo peak shift is found if the two-exciton state is displaced. The oscillations can be related to the dynamics of the one-exciton coherences.
The two-dimensional three-pulse photon echo signals from a dimer system are investigated in detail. A perturbative approach is used to calculate the response of the system via a numerical propagation of the density matrix in exciton state representation. Exciton vibrational coupling is modeled by Redfield relaxation theory. The main goal of this paper is to dissect the two-dimensional spectrum of the dimer to give better understanding of how the dynamics influence the spectral features. We will show how different Liouville pathways lead to the appearance/disappearance of diagonal and cross peaks. The inclusion of a Gaussian shaped electric field will be contrasted to the use of delta-pulses in the impulsive limit. The impulsive limit is found to be a satisfactory approximation at long population times, while at shorter times, in the pulse-overlap region, more realistic electric fields are called for.
Signatures of chemical exchange and spectral diffusion in 2D photon-echo line shapes of molecular aggregates are studied using model calculations for a dimer whose Hamiltonian parameters are stochastically modulated. Cross peaks induced by chemical exchange and by exciton transport have different dynamics and distinguish two models which have the same absorption spectrum (a two-state jump bath modulation model of a dimer and a four-state jump bath model of a single chromophore). Slow Gaussian-Markovian spectral diffusion of a symmetric dimer induces new peaks which are damped as the dipole moment is equilibrated. These effects require an explicit treatment of the bath and may not be described by lower-level theories such as the Redfield equations, which eliminate the bath.
Two-dimensional (2D) photon-echo spectra of a single subunit of the Fenna-Matthews-Olson (FMO) bacteriochlorophyll trimer of Chlorobium tepidum are simulated, employing the equation-of-motion phase-matching approach (EOM-PMA). We consider a slightly extended version of the previously proposed Frenkel exciton model, which explicitly accounts for exciton coherences in the secular approximation. The study is motivated by a recent experiment reporting long-lived coherent oscillations in 2D transients [Engel et al., Nature 446, 782 (2007)] and aims primarily at accurate simulations of the spectroscopic signals, with the focus on oscillations of 2D peak intensities with population time. The EOM-PMA accurately accounts for finite pulse durations as well as pulse-overlap effects and does not invoke approximations apart from the weak-field limit for a given material system. The population relaxation parameters of the exciton model are taken from the literature. The effects of various dephasing mechanisms on coherence lifetimes are thoroughly studied. It is found that the experimentally detected multiple frequencies in peak oscillations cannot be reproduced by the employed FMO model, which calls for the development of a more sophisticated exciton model of the FMO complex.
Long-lived beatings in two-dimensional electronic spectroscopy (2DES) remain difficult to interpret within standard excitonic open-system models, which typically assume factorized initialization and predict rapid coherence decay. We show that persistent beatings can arise from a correlation-driven mechanism that requires both slow bath memory and ultrafast pulse sequences that propagate system-bath correlations across optical interactions. In this regime, the pulse sequence unitarily dresses the bath-memory contribution and activates nonsecular population-coherence transfer during field-free evolution, sustaining coherence signatures far beyond factorized or weak-memory descriptions. Rather than addressing what is oscillating (excitonic versus vibronic) or quantum-versus-classical semantics, this work reframes long-lived beatings as a protocol-level dynamical effect: correlation-mediated retrieval under ultrafast control.