n ϵ$_{n}$ cm − 1 d$_{n,x}$ | d$_{n}$ | d$_{n,y}$ | d$_{n}$ | d$_{n,z}$ | d$_{n}$ | | d$_{n}$ | d$_{0}$ h$_{n}$ α$_{n}$ grad 1 9850 1 0 0 0 . 65 0 0 2 10000 − 0 . 94 0 . 34 0 2 . 15 10 60 3 10150 − 0 . 94 0 . 34 0 0 . 9 10 120 Table I: Parameters of the model trimer. The parameter ϵ$_{n}$ represents the transition energy of n − th monomer, transition dipole moments d$_{n}$ are taken relative to some value d 0 . Pa- rameters h$_{n}$ and α$_{n}$ are explained on Fig. 2. ometry of the models, together with the meaning of the parameters is presented in Fig. 2. In Tab. I we summa- rize the main parameters of the model. Parameters J , h and d$_{0}$ from Tab. I are not independent. For given h$_{n}$ and d$_{n}$ we could in principle calculate the value of reso- nance coupling J . Because we are not interested in the absolute amplitude of the absorption or 2D spectra we assume d$_{0}$ to be fixed by the values of h$_{n}$ and d$_{n}$ to yield the expected value of J . All three resonance couplings J between the molecules are set to J = 200 cm − 1 for the calculations presented here. The values of the tran- sition dipole moments determine the initial condition for the population dynamics. We assume that the excita- tion light intensity and the value of the transition dipole moment are such that the system is only weakly excited. The total population of the excited state band is normal- ized to 0 . 01 . The relative values of the transition dipole moments are chosen so that the linear absorption spec- trum (see Fig. 3) shows peaks of roughly the same height. Two peaks originating from the energetically lowest and the energetically highest states dominate the spectrum, the third level contributes as a shoulder to lowest energy peak. Two parameters that influence the coupling for the model system to the bath are reorganization energy λ and correlation time τ$_{c}$. We vary these parameters in the range that can conceivably represent chlorophylls in pho- tosynthetic complexes (see e. g. Refs. [38, 39]). Two-dimensional coherent spectrum, Ξ( ω$_{t}$, T, ω$_{τ}$ ) , is obtain from the non-linear signal by Fourier transforming the time and pulse delay dependent signal electric field E$_{S}$ ( t, T, τ ) along the t and τ variables [8, 11] as The Fourier transform in τ yields an ω$_{τ}$ dependence that is formally similar to linear absorption spectrum, while the transform in t yields generalized absorption and stim- ulated emission from a non-equilibrium state created by the first two laser pulses. 2D spectrum thus represents a 2D absorption/emission and absorption/absorption cor- relation plot. During the pulse delay time T the system evolves both in the electronically excited state and in the ground state, but no optical signal is generated. Relax- ation of populations in the electronically excited band leads to evolution of non-diagonal 2D spectral features, so-called cross-peaks. Cross-peaks appearing at T = 0 are a signature of excitonic origin of the observed ex- cited states. The 2D cross-peaks oscillate in T as long as the corresponding electronic coherence elements of the reduced density matrix are oscillating. The life time of the electronic coherences can thus be estimated directly from the T dependent sequence of 2D spectra [12, 14]. In this section we study dynamics of model aggregate viewed via population and coherence dynamics and via 2D coherent spectrum. We define a simple model ag- gregate for which we calculate excited state dynamics in- cluding evolution of coherences between electronic states, linear absorption and 2D spectra at chosen population times. Calculations of linear absorption, which require only knowledge of the time evolution of optical coher- ences, are performed using the secular time local equa- tion, since it is known to yield exact result at least for some models [35]. Population dynamics is calculated us- ing all four methods we discussed in Section III, and the results are compared. The simplest model of an aggregate that can exhibit all effects observed in Ref. [14] is a trimer. The ge- Let us now concentrate on short time behavior of the populations and coherences in more retail. In the short First, we compare relaxation dynamics of populations of excited state of our aggregate after excitation by an ultrashort laser pulse. TL equations of motion where solved by standard numerical methods for ordinary dif- ferential equations provided by the Mathematica R ○ soft- ware. For the TNL equations we used fast Fourier trans- form method. Figure 4 presents the first 1 ps of the population dynamics after a δ − pulse excitation of the trimer from Tab. I at the temperature T = 300 K. Re- organization energy λ = 120 cm − 1 and correlation time τ$_{c}$ = 50 fs are the same at all three monomers. The dynamics with the same parameters for a selected coher- ences element ρ$_{13}$ ( t ) is presented in Fig. 5. The overall conclusion is that all four methods yield a similar general behavior for the populations, with some difference at the short time evolution and also slightly different long time equilibrium. Examination of the Figure 5 leads us to the conclusion that the methods yield two different results - a short coherence life time for the time local methods, and a relatively longer life time in case of the time non-local methods. The behavior of the coherence ρ$_{13}$ ( t ) repre- sents a general tendency that we have observed for all electronic coherences over a wide range of parameters. V. NUMERICAL RESULTS AND DISCUSSION A. Population relaxation and evolution of coherences B. Two-dimensional Coherent Spectroscopy Ξ( ω$_{t}$, T, ω$_{τ}$ ) = ∞ ˆ −∞ dt ∞ ˆ −∞ dτE$_{s}$ ( t, T, τ ) e iω$_{t}$t − iω$_{τ}$ $^{τ}$. (45) 8