Coherence Quantum Beats in Two-Dimensional Electronic Spectroscopy Yuan-Chung Cheng and Graham R. Fleming* Department of Chemistry and QB3 Institute, UniVersity of California, Berkeley and Physical Bioscience DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720 ReceiVed: NoVember 12, 2007; In Final Form: February 13, 2008 We study the coherence quantum beats in two-dimensional (2D) electronic spectroscopy of a coupled dimer system using a theoretical method based on a time-nonlocal quantum master equation and a recently proposed scheme for the evaluation of the third-order photon echo polarization [Gelin, M. F.; Egorova, D.; Domcek, W. J. Chem. Phys. 2005, 123, 164112]. The simulations show that the amplitude and peak shape beating in the 2D spectra is a result of the interplay between the rephasing and non-rephasing contributions to the 2D signals and can be used to elucidate the coherence dynamics in a multichromophoric system. In addition, the results suggest that the rephasing and non-rephasing 2D spectra contain complementary information, and a study of both of them could provide more dynamical information from 2D electronic spectroscopy. 1. Introduction The advent of two-dimensional (2D) electronic spectroscopy 1-3 has provided a direct probe of electronic couplings and excitation energy-transfer pathways in complex multilevel quantum sys- tems. The 2D technique correlates the absorption and emission frequencies of the material system and thereby provides a map showing electronic couplings between electronic excitations and spectral diffusion between two transition energies. In particular, 2D cross peaks can elucidate electronic couplings and energy- transfer dynamics in complex multichromophoric systems, 4-6 while analysis of 2D line shape can reveal solvation dynamics and solute-solvent interactions in the condensed phase. 7-9 Applications to molecular J aggregates, 8 photosynthetic light- harvesting complexes, 10-12 and semiconductor quantum wells 13 have demonstrated that 2D electronic spectroscopy is a versatile and powerful probe for dynamical information. A powerful aspect of 2D electronic spectroscopy is its capability to exploit the phase and coherence information in the time evolution of the optical polarization induced by the optical pulses. Pisliakov et al. first showed that these excitonic coherence effects will be manifested in quantum beats in 2D electronic spectra and can be related quantitatively to the coherence dynamics in the system. 5 Recently, quantum beats in 2D electronic spectra in the Fenna-Matthews-Olson bac- teriochlorophyll (FMO) complex of green sulfur bacteria were observed, and the analysis of the beating patterns of diagonal peaks provided direct evidence of excitonic coherence in the system, that is, energy transfer in the FMO complex is described by wave-like motion rather than incoherent hopping. 12 However, while the beats in the cross peaks in 2D spectra are well characterized theoretically, 5,6,14,15 the origins of the amplitude and especially the peak shape beats in the diagonal peaks remain unclear. Since the cross peaks in 2D spectra are usually lower in amplitude and overlap with stronger diagonal peaks, an analysis based on the diagonal peaks is critical. Therefore, it is important that we understand the nature of the coherence beats in the diagonal peaks and formulate a prescription that can be applied to quantitatively extract coherence dynamical informa- tion from experimental 2D spectral data. Recently, a method for the efficient calculation of third-order photon echo polarization was developed and applied to simulate a three-pulse photon echo peak shift and 2D electronic spectroscopy of a two-level electronic system coupled to explicit vibrational degrees of freedom. 16,17 This approach was later extended to treat two-exciton states for describing general third- order experiments on multichromophoric systems. 18 In an earlier work, we applied this density-matrix-based method to show that quantitative analysis of the time evolution of the cross peaks in 2D electronic spectroscopy can provide a complete understand- ing of the population and coherence dynamics for the system under study. 6 Because this approach is based on a time-nonlocal quantum master equation formalism that explicitly includes a field-matter interaction and non-Markovian effects, 19 all pos- sible pulse-overlap effects and interferences between contribu- tions from different Liouville pathways to the signal are included in the calculation. Therefore, this density-matrix-based method is ideal for investigating the electronic coherence effects and origin of quantum beats in the diagonal peaks in 2D electronic spectroscopy. In this work, we apply the density matrix based method to study the 2D electronic spectroscopy of a model dimer system. Focusing on the time evolution of the amplitude and peak shape in 2D spectra, we aim to understand the origin of the amplitude and peak shape beating in 2D spectra and provide insights that can aid experimental studies. We also demonstrate that the rephasing and non-rephasing spectra of the model dimer system, when treated separately, can provide complementary dynamical information hidden in the total 2D spectra. 2. Theoretical Methods 2.1. 2D Electronic Spectroscopy. 2D electronic spectroscopy is a four-wave mixing experiment in which three laser fields interact with the sample to create a polarization and the signal in the phase-matching direction k s )-k 1 + k 2 + k 3 is heterodyne-detected and Fourier-transformed with respect to the coherence time τ (the time delay between the first and second pulses) and the rephasing time t (the time delay between the * To whom correspondence should be addressed. E-mail: GRFleming@ lbl.gov. 4254 J. Phys. Chem. A 2008, 112, 4254-4260 10.1021/jp7107889 CCC: $40.75 © 2008 American Chemical Society Published on Web 04/01/2008