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arXiv:0709.1221 · The Journal of Chemical Physics

Anomalous lineshapes and aging effects in two-dimensional correlation spectroscopy

František Šanda, Shaul Mukamel

Published Oct 18, 2007• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.2793786

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Journal of Nonlinear Optical Physics & Materials•Dec 1, 2018•4 citations
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.
Recursive relations are derived for nonlinear optical response functions of a two-level chromophore with stochastic frequency fluctuations described by a continuous-time random walk. Stationary ensembles are constructed and signatures of anomalous relaxation in the photon echo signal are discussed for a two state jump modulation with a power law waiting time density function psi(t) approximately t(-alpha-1). Stretched exponential decay of the photon echo signal is predicted for 0<alpha<1 and power law asymptotics for 1<alpha<2.
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.
Two-dimensional (2D) line shapes derived from four types of coherent third-order nonlinear spectroscopies as a function of two independent time or frequency variables are compared. The signals scattered into two wave-vector-matching directions in the limits of high-time or high-frequency resolution are calculated for an inhomogeneously broadened ensemble of two-level systems with phenomenological dephasing and population relaxation times. While one-dimensional (1D) line shape analysis for this model is ambiguous, the contours of these 2D line shapes are shown to be characteristic of the relative time scales of the three relevant line-broadening parameters. These broadening mechanisms are also apparent by comparing 2D spectral profiles along the diagonal and antidiagonal frequency axes. For time domain experiments, in analogy with NMR correlation spectroscopy, the radiated signal is observed as a function of the two coherence periods and is 2D Fourier transformed to obtain a line shape. For experiments based on the photon echo, the ellipticity of the absolute value 2D line shape is related to the ratio of the inhomogeneous width to the dephasing rate, whereas experiments based on the transient grating method are shown to be functionally 1D. For frequency domain experiments, the contours of the 2D line shape can yield information on static and both dynamic broadening mechanisms. While the addition of spectral diffusion to such calculations will modify these results for frequency domain experiments, characterizing these dynamics in the time domain naturally leads to a 3D experiment analogous to the spin diffusion experiment in NMR.
Two-dimensional (2D) spectroscopy is a powerful pump-pump-probe technique for revealing couplings between quantum states and disentangling the different contributions to the optical response of a system. We present an efficient method for 2D spectroscopy simulations in the Markovian limit for the environment, capable of handling arbitrary pulse shapes and reproducing time-ordering and overlapping pulse effects, while maintaining a computational cost that scales linearly with the number of sampling points. We leverage this framework to investigate how 2D spectra are affected by spectral phase distortions and highly non-Gaussian pulse shapes, such as those produced experimentally by hollow-core fibers or non-collinear optical amplifiers. We show that realistic pulses can induce the appearance of additional spectral features, lineshape distortions and oscillating contributions in the system's dynamics. Notably, even weak temporal pulse tails arising from uncorrected high-order spectral phase terms cause visible changes in the 2D spectra. We also find that homodyne detection schemes employed in experiments can mitigate the presence of such pulse effects. These results emphasize the importance of including realistic pulses in 2D spectroscopy simulations to identify pulse-induced effects and minimize ambiguities in the interpretation of experimental data.
A two level model of a single molecule undergoing spectral diffusion dynamics and interacting with a sequence of two short laser pulses is investigated. Analytical solution for the probability of n=0,1,2 photon emission events for the telegraph and Gaussian processes is obtained. We examine under what circumstances the photon statistics emerging from such pump-probe setup provides new information on the stochastic process parameters and what are the measurement limitations of this technique. The impulsive and selective limits, the semiclassical approximation, and the fast modulation limit exhibit general behaviors of this new type of spectroscopy. We show that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful results, the information on the photon statistics is contained in the molecule's dipole correlation function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained within the selective limit depends on the both spectral shifts and rates and exhibits oscillations, which are not found in the corresponding line shape.
Effects of spectral diffusion on the third-order nonlinear susceptibility for two- and three-level quantum systems, immersed in a thermal reservoir, are evaluated with a four-wave mixing (FWM) signal through the density matrix formalism. For this, inhomogeneously broadened two- and three-level quantum systems are used. In these models, the distribution of natural frequencies follows a Lorentzian function. The absorptive and dispersive nonlinear optical responses are determined for the considered quantum models. The results presented in this work show potential applications in the development of optical switches and provide a potential method to measure spectral diffusion relaxation times in three-level quantum systems.