Non-perturbative calculation of 2D spectra in heterogeneous systems: Exciton relaxation in the FMO complex

Higher-Order Multidimensional and Pump-Probe Spectroscopies

Transient absorption and coherent two-dimensional spectroscopy are widely established methods for the investigation of ultrafast dynamics in quantum systems. Conventionally, they are interpreted in the framework of perturbation theory at the third order of interaction. Here, we discuss the potential of higher-(than-third-)order pump-probe and multidimensional spectroscopy to provide insight into excited multiparticle states and their dynamics. We focus on recent developments from our group. In particular, we demonstrate how phase cycling can be used in fluorescence-detected two-dimensional spectroscopy to isolate higher-order spectra that provide information about highly excited states such as the correlation of multiexciton states. We discuss coherently detected fifth-order 2D spectroscopy and its power to track exciton diffusion. Finally, we show how to extract higher-order signals even from ordinary pump-probe experiments, providing annihilation-free signals at high excitation densities and insight into multiexciton interactions.

Pulse overlap artifacts and double quantum coherence spectroscopy

The double quantum coherence (DQC) signal in nonlinear spectroscopy gives information about the many-body correlation effects not easily available by other methods. The signal is short-lived, consequently, a significant part of it is generated during the pulse overlap. Since the signal is at two times the laser frequency, one may intuitively expect that the pulse overlap-related artifacts are filtered out by the Fourier transform. Here, we show that this is not the case. We perform explicit calculations of phase-modulated two-pulse experiments of a two-level system where the DQC is impossible. Still, we obtain a significant signal at the modulation frequency, which corresponds to the DQC, while the Fourier transform over the pulse delay shows a double frequency. We repeat the calculations with a three-level system where the true DQC signal occurs. We conclude that with realistic dephasing times, the pulse-overlap artifact can be significantly stronger than the DQC signal. Our results call for great care when analyzing such experiments. As a rule of thumb, we recommend that only delays larger than 1.5 times the pulse length should be used.

Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals

Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.

Non-Markovianity between Site Pairs in FMO Complex Using Discrete-Time Quantum Jump Model

The Fenna-Mathews-Olson (FMO) complex present in green sulfur bacteria is known to mediate the transfer of excitation energy between light-harvesting chlorosomes and membrane-embedded bacterial reaction centers. Due to the high efficiency of this transport process, it is an extensively studied pigment-protein complex system with the eventual aim of modeling and engineering similar dynamics in other systems and using it for real-time application. Some studies have attributed the enhancement of transport efficiency to wavelike behavior and non-Markovian quantum jumps resulting in long-lived and revival of quantum coherence, respectively. Since dynamics in these systems reside in the quantum-classical regime, quantum simulation of such dynamics will help in exploring the subtle role of quantum features in enhancing the transport efficiency, which has remained unsettled. Discrete simulation of the dynamics in the FMO complex can help in efficient engineering of the heat bath and controlling the environment with the system. In this work, using the discrete quantum jump model we show and quantify the presence of higher non-Markovian memory effects in specific site pairs when internal structures and environmental effects are in favor of faster transport. As a consequence, our study leans toward the connection between non-Markovianity in quantum jumps with the enhancement of transport efficiency.

Multidimensional electronic spectroscopy in high-definition-Combining spectral, temporal, and spatial resolutions

Over the past two decades, coherent multidimensional spectroscopies have been implemented across the terahertz, infrared, visible, and ultraviolet regions of the electromagnetic spectrum. A combination of coherent excitation of several resonances with few-cycle pulses, and spectral decongestion along multiple spectral dimensions, has enabled new insights into wide ranging molecular scale phenomena, such as energy and charge delocalization in natural and artificial light-harvesting systems, hydrogen bonding dynamics in monolayers, and strong light-matter couplings in Fabry-Pérot cavities. However, measurements on ensembles have implied signal averaging over relevant details, such as morphological and energetic inhomogeneity, which are not rephased by the Fourier transform. Recent extension of these spectroscopies to provide diffraction-limited spatial resolution, while maintaining temporal and spectral information, has been exciting and has paved a way to address several challenging questions by going beyond ensemble averaging. The aim of this Perspective is to discuss the technological developments that have eventually enabled spatially resolved multidimensional electronic spectroscopies and highlight some of the very recent findings already made possible by introducing spatial resolution in a powerful spectroscopic tool.

Two-dimensional spectroscopy beyond the perturbative limit: The influence of finite pulses and detection modes

Ultra-fast and multi-dimensional spectroscopy gives a powerful looking glass into the dynamics of molecular systems. In particular, two-dimensional electronic spectroscopy (2DES) provides a probe of coherence and the flow of energy within quantum systems, which is not possible with more conventional techniques. While heterodyne-detected (HD) 2DES is increasingly common, more recently fluorescence-detected (FD) 2DES offers new opportunities, including single-molecule experiments. However, in both techniques, it can be difficult to unambiguously identify the pathways that dominate the signal. Therefore, the use of numerically modeling of 2DES is vitally important, which, in turn, requires approximating the pulsing scheme to some degree. Here, we employ non-perturbative time evolution to investigate the effects of finite pulse width and amplitude on 2DES signals. In doing so, we identify key differences in the response of HD and FD detection schemes, as well as the regions of parameter space where the signal is obscured by unwanted artifacts in either technique. Mapping out parameter space in this way provides a guide to choosing experimental conditions and also shows in which limits the usual theoretical approximations work well and in which limits more sophisticated approaches are required.

Fifth-order two-quantum absorptive two-dimensional electronic spectroscopy of CdSe quantum dots

Two-quantum variants of two-dimensional electronic spectroscopy (2DES) have previously been used to characterize multi-exciton interactions in molecules and semiconductor nanostructures though many implementations are limited by phasing procedures or non-resonant signals. We implement 2DES using phase-cycling to simultaneously measure one-quantum and two-quantum spectra in colloidal CdSe quantum dots. In the pump-probe geometry, fully absorptive spectra are automatically acquired by measuring the sum of the rephasing and nonrephasing signals. Fifth-order two-quantum spectroscopy allows for direct access to multi-exciton states that may be obscured in excited state absorption signals due to population relaxation or third-order two-quantum spectra due to the non-resonant response.

Effects of high pulse intensity and chirp in two-dimensional electronic spectroscopy of an atomic vapor

The effects of high pulse intensity and chirp on two-dimensional electronic spectroscopy signals are experimentally investigated in the highly non-perturbative regime using atomic rubidium vapor as clean model system. Data analysis is performed based on higher-order Feynman diagrams and non-perturbative numerical simulations of the system response. It is shown that higher-order contributions may lead to a fundamental change of the static appearance and beating-maps of the 2D spectra and that chirped pulses enhance or suppress distinct higher-order pathways. We further give an estimate of the threshold intensity beyond which the high-intensity effects become visible for the system under consideration.

Simple Quantum Dynamics with Thermalization

In this paper, we introduce two simple quantum dynamics methods. One is based on the popular surface-hopping method, and the other is based on rescaling of the propagation on the bath ground-state potential surface. The first method is special, as it avoids specific feedback from the simulated quantum system to the bath and can be applied for precalculated classical trajectories. It is based on the equipartition theorem to determine if hops between different potential energy surfaces are allowed. By comparing with the formally exact Hierarchical Equations Of Motion approach for four model systems we find that the method generally approximates the quantum dynamics toward thermal equilibrium very well. The second method is based on rescaling of the nonadiabatic coupling and also neglect the effect of the state of the quantum system on the bath. By the nature of the approximations, they cannot reproduce the effect of bath relaxation following excitation. However, the methods are both computationally more tractable than the conventional fewest switches surface hopping, and we foresee that the methods will be powerful for simulations of quantum dynamics in systems with complex bath dynamics, where the system-bath coupling is not too strong compared to the thermal energy.

Influence of weak vibrational-electronic couplings on 2D electronic spectra and inter-site coherence in weakly coupled photosynthetic complexes

Coherence oscillations measured in two-dimensional (2D) electronic spectra of pigment-protein complexes may have electronic, vibrational, or mixed-character vibronic origins, which depend on the degree of electronic-vibrational mixing. Oscillations from intrapigment vibrations can obscure the inter-site coherence lifetime of interest in elucidating the mechanisms of energy transfer in photosynthetic light-harvesting. Huang-Rhys factors (S) for low-frequency vibrations in Chlorophyll and Bacteriochlorophyll are quite small (S ≤ 0.05), so it is often assumed that these vibrations influence neither 2D spectra nor inter-site coherence dynamics. In this work, we explore the influence of S within this range on the oscillatory signatures in simulated 2D spectra of a pigment heterodimer. To visualize the inter-site coherence dynamics underlying the 2D spectra, we introduce a formalism which we call the "site-probe response." By comparing the calculated 2D spectra with the site-probe response, we show that an on-resonance vibration with Huang-Rhys factor as small as S = 0.005 and the most strongly coupled off-resonance vibrations (S = 0.05) give rise to long-lived, purely vibrational coherences at 77 K. We moreover calculate the correlation between optical pump interactions and subsequent entanglement between sites, as measured by the concurrence. At 77 K, greater long-lived inter-site coherence and entanglement appear with increasing S. This dependence all but vanishes at physiological temperature, as environmentally induced fluctuations destroy the vibronic mixing.

Simulating signatures of two-dimensional electronic spectra of the Fenna-Matthews-Olson complex: By using a numerical path integral

A framework for simulating electronic spectra from photon-echo experiments is constructed by using a numerical path integral technique. This method is non-Markovian and nonperturbative and, more importantly, is not limited by a fixed form of the spectral density functions of the environment. Next, a two-dimensional (2D) third-order electronic spectrum of a dimer system is simulated. The spectrum is in agreement with the experimental and theoretical results previously reported [for example, M. Khalil, N. Demirdöven, and A. Tokmakoff, Phys. Rev. Lett. 90, 047401 (2003)]. Finally, a 2D third-order electronic spectrum of the Fenna-Matthews-Olson (FMO) complex is simulated by using the Debye, Ohmic, and Adolphs and Renger spectral density functions. It is shown that this method can clearly produce the spectral signatures of the FMO complex by using only the Adolphs and Renger spectral density function. Plots of the evolution of the diagonal and cross-peaks show that they are oscillating with the population time.

Strong-pump strong-probe spectroscopy: effects of higher excited electronic states

The present paper is devoted to the simulation of (integral and dispersed) pump-probe signals in the nonperturbative regime for a series of material systems with multiple electronic states and excited-state absorption. We show that strong-pump strong-probe spectroscopy permits the probing of vibrational wavepackets in high-lying and/or short-lived excited electronic states with a time resolution which is not limited by the pulse durations. The field strength can be regarded as an additional experimentally controllable parameter, which can be tuned to maximize the spectroscopic information for a given material system.

An Efficient N(3)-Scaling Propagation Scheme for Simulating Two-Dimensional Infrared and Visible Spectra

In this paper, we develop and test a new approximate propagation scheme for calculating two-dimensional infrared and visible spectra. The new scheme scales one order more efficiently with the system size than the existing schemes. A Trotter type of approximation is used for the matrix exponent that describes the time evolution of the quantum system. This is needed for calculating the third-order response functions governing the two-dimensional infrared and visible spectra. The method is tested on a model dimer system, the amide I spectrum of the Gramicidin A antimicrobial peptide, the spectrum of the OH stretching vibration of bulk water, and a homogeneous J-aggregate. Errors due to the approximation are hardly observable in the calculated spectra. Scaling simulations with different system sizes are used to demonstrate that the new scheme is indeed scaling with the system size to the third power, one order faster than the existing methods.

Exciton dynamics in chromophore aggregates with correlated environment fluctuations

We study the effects of correlated molecular transition energy fluctuations in molecular aggregates on the density matrix dynamics, and their signatures in the optical response. Correlated fluctuations do not affect single-exciton dynamics and can be described as a nonlocal contribution to the spectral broadening, which appears as a multiplicative factor in the time-domain response function. Intraband coherences are damped only by uncorrelated transition energy fluctuations. The signal can then be expressed as a spectral convolution of a local contribution of the uncorrelated fluctuations and the nonlocal contribution of the correlated fluctuations.

Optical N-wave-mixing spectroscopy with strong and temporally well-separated pulses: the doorway-window representation

We have extended the doorway-window representation of optical pump-probe spectroscopy with weak pulses toward N-wave-mixing spectroscopy with temporally well-separated pulses of arbitrary strength. The expressions for the signals in the strong-pulse doorway-window representation are derived in the framework of the nonperturbative theory of N-wave-mixing spectroscopy. The strong-pulse doorway-window representation is complementary to the equation-of-motion phase-matching approach. The latter fully accounts for pulse-overlap effects in signals induced by weak pulses but is computationally more expensive. The performance of the doorway-window approximation for temporally well-separated strong pulses is illustrated for an electronic two-level system with an underdamped Condon-active vibrational mode.

Strong and Long Makes Short: Strong-Pump Strong-Probe Spectroscopy

We propose a new time-domain spectroscopic technique that is based on strong pump and probe pulses. The strong-pump strong-probe (SPSP) technique provides temporal resolution that is not limited by the durations of the pump and probe pulses. By numerically exact simulations of SPSP signals for a multilevel vibronic model, we show that the SPSP signals exhibit electronic and vibrational beatings on time scales which are significantly shorter than the pulse durations. This suggests the possible application of SPSP spectroscopy for the real-time investigation of molecular processes that cannot be temporally resolved by pump-probe spectroscopy with weak pump and probe pulses.

Chiroptical nature of two-exciton states of light-harvesting complex: Doubly resonant three-wave-mixing spectroscopy

Photosynthetic light-harvesting complex is a coupled multichromophore system. Due to electronic couplings between neighboring chlorophylls in the complex, the one- and two-exciton states are delocalized and they can be written as linear combinations of singly and doubly excited configurations, respectively. Despite that the chiroptical properties of one-exciton states in such a multichromophore system have been investigated by using linear optical activity measurement techniques; those of two-exciton states have not been studied before due to a lack of appropriate measurement methods. Here, we present a theoretical description on chiroptical chi((2)) spectroscopy and show that it can be used to investigate such properties of a photosynthetic light-harvesting system, which is the Fenna-Matthews-Olson complex, consisting of seven bacteriochlorophylls in its protein subunit. To simulate the doubly resonant sum- and difference-frequency-generation spectra of the complex, one- and two-exciton transition dipoles were calculated. Carrying out quantum chemistry calculations of electronically excited states of a model bacteriochlorophyll system and taking into account the dipole-induced dipole electronic transition processes between the ground state and two-exciton states, we could calculate the two-dimensional sum-frequency-generation spectra revealing dominant second-order chiroptical transition pathways and involved one- and two-exciton states. It is believed that the present computational scheme and the theoretically proposed doubly resonant two-dimensional three-wave-mixing spectroscopy would be of use to shed light on the chiroptical natures of two-exciton states of arbitrary coupled multichromophore systems.

Photon echo studies of photosynthetic light harvesting

The broad linewidths in absorption spectra of photosynthetic complexes obscure information related to their structure and function. Photon echo techniques represent a powerful class of time-resolved electronic spectroscopy that allow researchers to probe the interactions normally hidden under broad linewidths with sufficient time resolution to follow the fastest energy transfer events in light harvesting. Here, we outline the technical approach and applications of two types of photon echo experiments: the photon echo peak shift and two-dimensional (2D) Fourier transform photon echo spectroscopy. We review several extensions of these techniques to photosynthetic complexes. Photon echo peak shift spectroscopy can be used to determine the strength of coupling between a pigment and its surrounding environment including neighboring pigments and to quantify timescales of energy transfer. Two-dimensional spectroscopy yields a frequency-resolved map of absorption and emission processes, allowing coupling interactions and energy transfer pathways to be viewed directly. Furthermore, 2D spectroscopy reveals structural information such as the relative orientations of coupled transitions. Both classes of experiments can be used to probe the quantum mechanical nature of photosynthetic light-harvesting: peak shift experiments allow quantification of correlated energetic fluctuations between pigments, while 2D techniques measure quantum beating directly, both of which indicate the extent of quantum coherence over multiple pigment sites in the protein complex. The mechanistic and structural information obtained by these techniques reveals valuable insights into the design principles of photosynthetic light-harvesting complexes, and a multitude of variations on the methods outlined here.