Probing anomalous relaxation by coherent multidimensional optical spectroscopy

Water inside the Selectivity Filter of a K+ Ion Channel: Structural Heterogeneity, Picosecond Dynamics, and Hydrogen Bonding

Water inside biological ion channels regulates the key properties of these proteins, such as selectivity, ion conductance, and gating. In this article, we measure the picosecond spectral diffusion of amide I vibrations of an isotope-labeled KcsA potassium channel using two-dimensional infrared (2D IR) spectroscopy. By combining waiting time (100-2000 fs) 2D IR measurements of the KcsA channel including 13C18O isotope-labeled Val76 and Gly77 residues with molecular dynamics simulations, we elucidated the site-specific dynamics of water and K+ ions inside the selectivity filter of KcsA. We observe inhomogeneous 2D line shapes with extremely slow spectral diffusion. Our simulations quantitatively reproduce the experiments and show that water is the only component with any appreciable dynamics, whereas K+ ions and the protein are essentially static on a picosecond timescale. By analyzing simulated and experimental vibrational frequencies, we find that water in the selectivity filter can be oriented to form hydrogen bonds with adjacent or nonadjacent carbonyl groups with the reorientation timescales being three times slower and comparable to that of water molecules in liquid, respectively. Water molecules can reside in the cavity sufficiently far from carbonyls and behave essentially like "free" gas-phase-like water with fast reorientation times. Remarkably, no interconversion between these configurations was observed on a picosecond timescale. These dynamics are in stark contrast with liquid water, which remains highly dynamic even in the presence of ions at high concentrations.

Water inside the selectivity filter of a K+ ion channel: structural heterogeneity, picosecond dynamics, and hydrogen-bonding

Water inside biological ion channels regulates the key properties of these proteins such as selectivity, ion conductance, and gating. In this Article we measure the picosecond spectral diffusion of amide I vibrations of an isotope labeled KcsA potassium channel using two-dimensional infrared (2D IR) spectroscopy. By combining waiting time (100 - 2000 fs) 2D IR measurements of the KcsA channel including 13C18O isotope labeled Val76 and Gly77 residues with molecular dynamics simulations, we elucidated the site-specific dynamics of water and K+ ions inside the selectivity filter of KcsA. We observe inhomogeneous 2D lineshapes with extremely slow spectral diffusion. Our simulations quantitatively reproduce the experiments and show that water is the only component with any appreciable dynamics, whereas K+ ions and the protein are essentially static on a picosecond timescale. By analyzing simulated and experimental vibrational frequencies, we find that water in the selectivity filter can be oriented to form hydrogen bonds with adjacent, or non-adjacent carbonyl groups with the reorientation timescales being three times slower and comparable to that of water molecules in liquid, respectively. Water molecules can reside in the cavity sufficiently far from carbonyls and behave essentially like "free" gas-phase-like water with fast reorientation times. Remarkably, no interconversion between these configurations were observed on a picosecond timescale. These dynamics are in stark contrast with liquid water that remains highly dynamic even in the presence of ions at high concentrations.

Measuring the Ultrafast Correlation Dynamics of a Multilevel System Using the Center Line Slope Analysis in Two-Dimensional Electronic Spectroscopy

In a two-dimensional (2D) optical spectrum of a multilevel system, there are diagonal peaks and off-diagonal cross-peaks that correlate the different levels. The time-dependent properties of these diagonal peaks and cross-peaks contain much information about the dynamics of the multilevel system. The time-dependent diagonal peakshape that depends on the spectral diffusion dynamics of the associated transition and characterized by the frequency-fluctuation correlation function (FFCF) is well studied. However, the time-dependent peakshape of a cross-peak that provides the correlation dynamics between different transitions is much less studied or understood. We derived the third-order nonlinear response functions that describe the cross-peaks in a 2D electronic spectrum of a multilevel system that arise from processes sharing a common ground state and/or from internal conversion and population transfer. We can use the center line slope (CLS) analysis to characterize the cross-peaks in conjunction with the diagonal peaks. This allows us to recover the frequency-fluctuation cross-correlation functions (FXCFs) between two transitions. The FXCF and its subsidiary quantities such as the initial correlation and the initial covariance between different transitions are important for studying the correlation effects between states in complex systems, such as energy-transfer processes. Furthermore, knowledge of how various molecular processes over different timescales affect simultaneously different transitions can also be obtained from the measured FXCF. We validated and tested our derived equations and analysis process by studying, as an example, the 2D electronic spectra of metal-free phthalocyanine in solution. We measured and analyzed the diagonal peaks of the Qx and Qy transitions and the cross-peaks between these two transitions of this multilevel electronic system and obtained the associated FFCFs and FXCFs. In this model system, we measured negative components of FXCF over the tens of picosecond timescale. This suggests that in phthalocyanine, the Qx and Qy transitions coupling with the solvent molecule motion are anticorrelated to each other.

Correlated spectral fluctuations quantified by line shape analysis of fifth-order two-dimensional electronic spectra

Correlated spectral fluctuations were suggested to coordinate excitation transport inside natural light harvesting complexes. We demonstrate the capacities of 2D line shapes from fifth-order coherent electronic signals (R5-2D) to report on such fluctuations in molecular aggregates and present a stochastic approach to fluctuations in correlated site and bi-exciton binding energies in the optical dynamics of Frenkel excitons. The model is applied to R5-2D line shapes of a homodimer, and we show that the peak tilt dynamics are a measure for site energy disorder, inter-site correlation, and the strength of bi-exciton binding energy fluctuations.

Asymptotically Exact Theory for Nonlinear Spectroscopy of Random Quantum Magnets

We study nonlinear response in quantum spin systems near infinite-randomness critical points. Nonlinear dynamical probes, such as two-dimensional (2D) coherent spectroscopy, can diagnose the nearly localized character of excitations in such systems. We present exact results for nonlinear response in the 1D random transverse-field Ising model, from which we extract information about critical behavior that is absent in linear response. Our analysis yields exact scaling forms for the distribution functions of relaxation times that result from realistic channels for dissipation in random magnets. We argue that our results capture the scaling of relaxation times and nonlinear response in generic random quantum magnets in any spatial dimension.

Center Line Slope Analysis in Two-Dimensional Electronic Spectroscopy

Center line slope (CLS) analysis in 2D infrared spectroscopy has been extensively used to extract frequency–frequency correlation functions of vibrational transitions. We apply this concept to 2D electronic spectroscopy, where CLS is a measure of electronic gap fluctuations. The two domains, infrared and electronic, possess differences: In the infrared, the frequency fluctuations are classical, often slow and Gaussian. In contrast, electronic spectra are subject to fast spectral diffusion and affected by underdamped vibrational wavepackets in addition to Stokes shift. All these effects result in non-Gaussian peak profiles. Here, we extend CLS-analysis beyond Gaussian line shapes and test the developed methodology on a solvated molecule, zinc phthalocyanine. We find that CLS facilitates the interpretation of 2D electronic spectra by reducing their complexity to one dimension. In this way, CLS provides a highly sensitive measure of model parameters describing electronic–vibrational and electronic–solvent interaction.

Exciton coherence length fluctuations in chromophore aggregates probed by multidimensional optical spectroscopy

The coherent third order optical response of molecular aggregates with fluctuating frequencies, couplings, and transition dipole moments is studied. We derived stochastic nonlinear exciton equations (SNEEs) by combining the quasiparticle picture of excitons with the path integral over stochastic bath paths described by the stochastic Liouville equations. Coherent two-dimensional (2D) spectra are calculated for a tetramer model system whose transition dipole orientations undergo two-state stochastic jumps on an arbitrary timescale. Correspondence between domains of ordered dipoles, which determine the exciton coherence length and the absorption peaks, is established. Signatures of domain coherence length fluctuations are observed in the cross peak dynamics of the 2D spectra in specific pulse polarization configurations.

Strong field line shapes and photon statistics from a single molecule under anomalous noise

We revisit the line-shape theory of a single molecule with anomalous stochastic spectral diffusion. Waiting time profiles for bath induced spectral jumps in the ground and excited states become different when a molecule, probed by continuous-wave laser field, reaches the steady state. This effect is studied for the stationary dichotomic continuous-time-random-walk spectral diffusion of a single two-level chromophore with power-law distributions of waiting times. Correlated waiting time distributions, line shapes, two-point fluorescence correlation function, and Mandel Q parameter are calculated for arbitrary magnitude of laser field. We extended previous weak field results and examined the breakdown of the central limit theorem in photon statistics, indicated by asymptotic power-law growth of Mandel Q parameter. Frequency profile of the Mandel Q parameter identifies the peaks of spectrum, which are related to anomalous spectral diffusion dynamics.

Multiplicative noise induces zero critical frequency

Stochastic Bloch equations which model the fluorescence of two-level molecules and atoms, NMR experiments, and Josephson junctions are investigated to illustrate the profound effect of multiplicative noise on the critical frequency of a dynamical system. Using exact solutions and the cumulant expansion we find two main effects: (i) even very weak noise may double or triple the number of critical frequencies, which is related to an instability of the system, and (ii) strong multiplicative noise may induce a nontrivial zero critical frequency thus wiping out the overdamped phase.

Stochastic Liouville equations for coherent multidimensional spectroscopy of excitons

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.

Anomalous lineshapes and aging effects in two-dimensional correlation spectroscopy

Multitime correlation functions provide useful probes for the ensembles of trajectories underlying the stochastic dynamics of complex systems. These can be obtained by measuring their optical response to sequences of ultrashort optical pulse. Using the continuous time random walk model for spectral diffusion, we analyze the signatures of anomalous relaxation in two-dimensional four wave mixing signals. Different models which share the same two point joint probability distribution show markedly different lineshapes and may be distinguished. Aging random walks corresponding to waiting time distributions with diverging first moment show dependence of 2D lineshapes on initial observation time, which persist for long times.