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The Journal of Chemical Physics

Parallels between multiple population-period transient spectroscopy and multidimensional coherence spectroscopies

Champak Khurmi, Mark A. Berg

Published Aug 12, 2008• 29 citations
Publisher

Abstract

We have recently shown that homogeneous and heterogeneous kinetics can be distinguished by experiments that compare the evolution of the population of a state over two time intervals [E. van Veldhoven et al., ChemPhysChem 8, 1761 (2007)]. This paper elaborates on the analogy between these multiple population-period transient spectroscopy (MUPPETS) experiments and more familiar spectroscopies based on the evolution of coherences. Using a modified inverse-Laplace transform, a standard kinetics decay is re-expressed as a "rate spectrum." A nonexponential decay creates a linewidth in this spectrum. Mechanisms for line broadening in rate spectra are compared to those for line broadening in frequency-domain spectra. Homogeneous and heterogeneous kinetics are defined precisely and are shown to be the counterparts of homogeneous and inhomogeneous line broadenings in frequency-domain spectra. Homogeneous line broadening mechanisms are further divided into equilibrium and nonequilibrium mechanisms, with equilibrium mechanisms more prevalent in frequency spectra and nonequilibrium mechanisms more prevalent in rate spectra. Spectral representations of two-dimensional MUPPETS experiments are developed that are equivalent to two-dimensional coherence spectroscopies. In particular, spectra equivalent to hole-burning and to correlation spectra are defined. Frequency-domain spectra are often modeled as an inhomogeneous distribution of identical homogeneous line shapes. A parallel homogeneous-heterogeneous model for kinetics is defined. Within this model, MUPPETS has sufficient information to completely separate the homogeneous and heterogeneous contributions to a nonexponential decay, even when the homogeneous contribution is nonexponential.

DOI
10.1063/1.2960589

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We have recently shown that homogeneous and heterogeneous kinetics can be distinguished by experiments that compare the evolution of the population of a state over two time intervals [E. van Veldhoven et al., ChemPhysChem 8, 1761 (2007)]. This paper elaborates on the analogy between these multiple population-period transient spectroscopy (MUPPETS) experiments and more familiar spectroscopies based on the evolution of coherences. Using a modified inverse-Laplace transform, a standard kinetics decay is re-expressed as a "rate spectrum." A nonexponential decay creates a linewidth in this spectrum. Mechanisms for line broadening in rate spectra are compared to those for line broadening in frequency-domain spectra. Homogeneous and heterogeneous kinetics are defined precisely and are shown to be the counterparts of homogeneous and inhomogeneous line broadenings in frequency-domain spectra. Homogeneous line broadening mechanisms are further divided into equilibrium and nonequilibrium mechanisms, with equilibrium mechanisms more prevalent in frequency spectra and nonequilibrium mechanisms more prevalent in rate spectra. Spectral representations of two-dimensional MUPPETS experiments are developed that are equivalent to two-dimensional coherence spectroscopies. In particular, spectra equivalent to hole-burning and to correlation spectra are defined. Frequency-domain spectra are often modeled as an inhomogeneous distribution of identical homogeneous line shapes. A parallel homogeneous-heterogeneous model for kinetics is defined. Within this model, MUPPETS has sufficient information to completely separate the homogeneous and heterogeneous contributions to a nonexponential decay, even when the homogeneous contribution is nonexponential.
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.
High inhomogeneous broadening of impurity spectra in amorphous media reflects one type of dispersion, characteristic for disordered systems: dispersion of electronic transition frequency. In principle, there can exist a pronounced dispersion of other parameters: homogeneous line width (Γ), Debay-Waller factor (α), hole burning efficiency (ϕ), etc. The problem of the burning efficiency dispersion (BED) was extensively discussed in the recent years ([1-6] and references therein). The nonexponentiality of the burning kinetics was treated in some publications [1-3,6] as an evidence of such dispersion. But burning kinetics is always nonexponential in strongly inhomogeneously broadened systems, if a burning laser line width Δνl is much less then Γ. That was found experimentally and treated theoretically in one of the first publications on a hole burning [7] (see, also [8]). All other sources of the nonexponentiality are, in fact, additional to this one. So, it’s not easy to attain experimentally the BED parameters on such background. The procedure of a numerical fit ting of the burning kinetic with BED, developed in [4,5], is correct, but it looks unreliable due to many fitting parameters.
We present an ultrafast coherent spectroscopy data acquisition scheme that samples slices of the time domain used in multidimensional coherent spectroscopy to achieve faster data collection than full spectra. We derive analytical expressions for resonance lineshapes using this technique that completely separate homogeneous and inhomogeneous broadening contributions into separate projected lineshapes for arbitrary inhomogeneous broadening. These lineshape expressions are also valid for slices taken from full multidimensional spectra and allow direct measurement of the parameters contributing to the lineshapes in those spectra as well as our own.
Because of inevitable size distribution, shape variations, different concentration of impurities and defects, and fluctuations of local environment and charge distribution, every ensemble of nanocrystals dispersed in some solid or liquid medium possesses inhomogeneously broadened absorption and emission spectra. Therefore, a number of properties inherent in molecular and atomic inhomogeneously broadened spectra can be a priori foreseen for nanocrystals. These include spectral hole-burning, fluorescence line narrowing under selective excitation, and decay time distribution. At the same time, spectrally selective techniques developed for inhomogeneously broadened molecular and atomic structures have been successfully applied to nanocrystals providing evaluation of individual parameters smeared as a result of inhomogeneous broadening. This chapter gives a brief overview of specific phenomena inherent in all spectrally inhomogeneous media and a survey of the relevant experimental techniques including nonlinear pump-and-probe spectroscopy, fluorescence excitation spectroscopy, and single molecule spectroscopy. Population-induced optical nonlinearity and spectral hole-burning Every real system consisting of particles with discrete energy levels exhibits absorption saturation under intense optical excitation. The only example of a nonsaturable system is an ensemble of ideal harmonic oscillators that possess an infinite number of equally spaced energy levels, optical transitions allowed only for a couple of neighboring levels, and the probability of optical transitions being proportional to the level number (Stepanov and Gribkovskii 1963).
Theory based on a configuration coordinate model is presented to clarify solute–solvent relaxation effect on the line shape of a time-resolved hole-burning spectrum. The time-resolved hole-burning spectrum is affected by both a hole created in a ground-state population and an excess excited-state population. The solute–solvent relaxation causes thermalization in both populations and then the shift and broadening of the spectrum is expected observable. The line shapes of three typical cases are mainly considered: (1) The excited state is assumed to have an infinitely long lifetime; (2) the excited state has a finite lifetime, the population being assumed to relax toward an electronic state other than the ground state, and (3) the population created in the excited state relaxes to the ground state, where a hole-filling effect is taken into account. Results obtained by numerical calculations show that a time evolution of the ground-state hole is clearly observed when the low-energy tail of the absorption band is excited under the condition that the population decay to the other electronic state is very fast. Further, it is predicted that the hole-filling effect causes a considerable deformation of the spectral shape when the solvent relaxation rate is comparable to the excited-state population decay rate.
Polarization and Rayleigh type optical mixing spectra of a homogeneously and inhomogeneously broadened two-level system are calculated. We show that the nonlinear signal has contributions from the population pulsation and saturation effect. In the homogeneous system these effects can induce an asymmetric spectrum when the probe frequency is different from the resonant value. In the inhomogeneous system the spectral response is always symmetric independent of the probe frequency value. Line saturation and dynamic stark splitting are substantially reduced in the inhomogeneous case. The mentioned differences can be used to identify the type of broadening of a given spectral line.