Simulating exciton migration and annihilation dynamics in higher order spectroscopies of a molecular trimer

We present simulations of 3rd and 5th order multidimensional time-dependent signals of a molecular trimer. The developed model allows for the simulation of both signal types within one theoretical framework. We therefore extended standard Redfield theory of exciton migration by a new term to include the process of exciton-exciton annihilation.


Introduction
2D electronic spectroscopy (2D-ES) has been successfully used to probe relaxation dynamics of single excitons by resolving 3 rd order signals in both emission and excitation frequencies as well as in population times [1]. The most common experiments measure so-called rephasing and non-rephasing signals along kI= k1-k2+k3 and kII= -k1+k2+k3, respectively. In this notation, k1,2,3 refers to the directions of the three involved excitation pulses.
Dynamics involving multiple excitons can be studied by another 3 rd order signal, called 2Q-2D electronic spectroscopy, and measured along kIII= k1+k2-k3. This technique, however, is experimentally challenging, suffers from pulse related artefacts and measures biexcitonic states only as part of an optical coherence, but not directly in their population decay behavior [2]. Dostal et al. [3] recently suggested that certain 5 th order signals, measured along kIV= 2k1-2k2+k3 and kV= -2k1+2k2+k3, probe decays of biexcitonic states more directly by explicitly populating biexcitonic energy levels.
Here, we present a master equation approach to treat 5 th and 3 rd order dynamics on an equal footing. The standard Redfield theory of excitonic transport is extended by a new term describing exciton-exciton annihilation (EEA, see section 2). We thus provide a comprehensive theoretical framework for a better understanding of multiple exciton effects, essential for reconstructing annihilation-free spectroscopic signals.

Results
Both 3 rd and 5 th order signals are modeled in a resonantly coupled molecular complex consisting of three subunits, each possessing two energy levels and an associated transition dipole moment µi. A trimer is the simplest aggregate allowing transfer between biexciton states, while retaining a tractable number of multiexcitonic states, as necessary for a detailed theoretical description of the nonlinear optical response. The dynamics inside excitonic manifolds, described by the Frenkel exciton Hamiltonian [4], are modulated by local fluctuations of the excited state energy on each monomeric subunit, accounted for by three harmonic coordinates ̂. Those coordinates are uncorrelated by assumption and characterized by a single correlation function <̂( )̂(0) >. Relaxation rates between eigenstates of the Frenkel exciton Hamiltonian are calculated by using standard Redfield theory. This Frenkel exciton model describes migration within excitonic manifolds. However, it neglects biexcitonic effects, such as EEA. The latter is regarded as an incoherent process, where excitation energy is resonantly transferred between two adjacent sites. This transfer stimulates one site back to the ground state whereas the other is promoted to a higher lying state. From there it decays via a phonon-assisted non-radiative process back to the first excited state [4]. EEA is therefore a process which reduces the number excitations inside an aggregate. In other words, it induces a transition between the n th manifold and the (n-1) th manifold. We therefore model EEA as a unidirectional transfer to lower excitonic manifolds by developing an addition to Redfield theory based on a quantum master equation in Lindblad form [4]. Fig. 1(a) illustrates this idea for a trimeric molecular structure. All transitions between exciton manifolds due to EEA are indicated by arrows. The quantum master equation developed in this work is capable, in combination with the response function formalism [4], to simulate dynamics occurring in both 3 rd and 5 th order signals of a molecular trimer. Examples of such transfer processes, occurring in both signal types during the waiting time t2, are illustrated in Fig. 1(b) with the aid of double sided Feynman diagrams. All of those dynamical processes are captured by the developed quantum master equation. Simulations of spectroscopic signals employing the above described quantum master equation are shown in  Fig 2(a,b) at t2= 0 fs, assuming a linear chain of three transition dipole moments fixed to a head-to-tail arrangement along the longest molecular axis (J-aggregate).

Fig. 2.
Simulated absorption, pump probe and 2D-ES spectra of both 3 rd and 5 th order signals at t2= 0fs are depicted in (a) and (b). (c) Plot of the different decay times τD obtained by fitting transients of peak F1 and F2 along t2 by a mono-exponential decay for different annihilation rates τA.
The absorption spectrum, depicted in the top panel of Fig. 2(a), shows one main absorption band at the lowest excitonic eigenenergy Ea1, typical for J-aggregates. Two main features are observed in the pump probe and 3 rd order 2D-ES spectrum shown in Fig 2(a): a strong positive peak P1 and a weak negative peak P2. The positive peak corresponds to ground state bleach (GSB) and stimulated emission (SE) contributions associated with the main absorption band. The negative peak stems from excited state absorption (ESA) pathways. The 5 th order spectrum, shown in Fig 2(b), shows a similar two peak structure. Here, positive features are attributed to ESA and negative features to GSB and SE pathways, as the sing is inverted compared to 3 rd 2D-ES, due to two additional interactions of the dipole operator with the density matrix. The similarity between the two maps arises from the fact that the symmetry of the wave functions in both 1 st and 2 nd manifold are identical for a J-type configuration. The lowest, symmetric, state therefore carries the most oscillator strength in both cases, giving rise to a single positive and negative feature in both maps. To study the effect of annihilation onto 5 th order spectra, all annihilation rates ri were fixed to the same value ra = 1/ τA and τA was changed from 10 fs to 100 fs in 10 fs steps. From the obtained 5 th order 2D-ES maps, transients were taken at peak positions F1 and F2 and were fitted with a monoexponential decay. The obtained decay times τD, normalized to the input annihilation time τA, are plotted in Fig.2 (c) versus τA. For small values of τA a strong deviation is observed, as the dominant rate is given by the relaxation rate inside the 1 st manifold. By increasing τA it becomes the dominant timescale and τD therefore gradually approaches τA. This corroborates the view that 5 th order spectroscopy is a promising tool to understand the general structure of biexciton levels and biexcitonic effects such as EEA, as the 5 th order spectrum is distinctly modified by changing the annihilation rate.