Four-time Correlation Functions Research Articles

Simulation of Time- and Frequency-Resolved Four-Wave-Mixing Signals at Finite Temperatures: A Thermo-Field Dynamics Approach

We propose a new approach to simulate four-wave-mixing signals of molecular systems at finite temperatures by combining the multiconfigurational Ehrenfest method with the thermo-field dynamics theory. In our approach, the four-time correlation functions at finite temperatures are mapped onto those at zero temperature in an enlarged Hilbert space with twice the vibrational degrees of freedom. As an illustration, we have simulated three multidimensional spectroscopic signals, time- and frequency-resolved fluorescence spectra, transient-absorption pump-probe spectra, and electronic two-dimensional (2D) spectra at finite temperatures, for a conical intersection-mediated singlet fission model of a rubrene crystal. It is shown that a detailed dynamical picture of the singlet fission process can be extracted from the three spectroscopic signals. An increasing temperature leads to lower intensities of the signals and broadened vibrational peaks, which can be attributed to faster singlet-triplet population transfer and stronger bath-induced electronic dephasing at higher temperatures.

Nonlinear response functions in an exponential trap model

The nonlinear response to an oscillating field is calculated for a kinetic trap model with an exponential density of states above the glass transition temperature T0. For temperatures not too close to T0, the results are similar to those obtained for the model with a Gaussian density of states. The choice of the dynamical variable that couples to the field and in particular its dependence on the trap energies generally has a strong impact on the shape of the dynamic response. The modulus of the frequency dependent third-order response either shows a peak or exhibits a monotonous decay from a finite low-frequency limit to a vanishing response at high frequencies depending on the dynamical variable. If a peak is observed, its height can show different temperature dependencies with the common feature of a scaling behavior near T0. Additionally, in some but not all cases the static nonlinear susceptibility diverges at T0. A recently proposed approximation that relates the cubic response to a four-time correlation function does not give reliable results due to a wrong estimate of the low-frequency limit of the response.

Photon statistics of intense entangled photon pulses

Time- and frequency-gated two-photon counting is given by a four-time correlation function of the electric field. This reduces to two times with purely time gating. We calculate this function for entangled photon pulses generated by parametric down-conversion. At low intensity, the pulses consist of well-separated photon pairs, and crossover to squeezed light as the intensity is increased. This is illustrated by the two-photon absorption signal of a three-level model, which scales linearly for a weak pump intensity where both photons come from the same pair, and gradually becomes nonlinear as the intensity is increased. We find that the strong frequency correlations of entangled photon pairs persist even for higher photon numbers. This could help facilitate the application of these pulses to nonlinear spectroscopy, where these correlations can be used to manipulate congested signals.

Higher-order correlation functions and nonlinear response functions in a Gaussian trap model

The four-time correlation function of a general dynamical variable obeying gaussian statistics is calculated for the trap model with a gaussian density of states. It is argued that for energy-independent variables this function is reminiscent of the four-time functions that have been discussed earlier in the interpretation of the results of four-dimensional NMR experiments on supercooled liquids. Using an approximative relation between the four-time correlation function and the cubic response function the nonlinear susceptibility is calculated and the results are compared with the corresponding ones resulting from an exact calculation. It is found that the results of the approximation change the qualitative behavior of the modulus of the susceptibility. Whereas in the exact calculation a peak is found in the modulus in most cases, depending on temperature and the additional model parameters no such peak occurs in the approximation. This difference has its origin mainly in an incorrect estimate of the static response. The results are discussed in relation to recent experimental findings.

Analysis of time resolved femtosecond and femtosecond/picosecond coherent anti-Stokes Raman spectroscopy: Application to toluene and Rhodamine 6G

The third-order polarization for coherent anti-Stokes Raman scattering (CARS) from a pure state is described by 48 terms in perturbation theory, but only 4 terms satisfy the rotating wave approximation. They are represented by Feynman dual time-line diagrams and four-wave mixing energy level diagrams. In time-resolved (tr) fs and fs/ps CARS from the ground vibrational state, one resonant diagram, which is the typical CARS term, with three field interactions-pump, Stokes, followed by probe-on the ket is dominant. Using the separable, displaced harmonic oscillators approximation, an analytic result is obtained for the four-time correlation function in the CARS third-order polarization. Dlott's phenomenological expression for off-resonance CARS from the ground vibrational state is derived using a three-state model. We calculated the tr fs and fs/ps CARS for toluene and Rhodamine 6G (R6G), initially in the ground vibrational state, to compare with experimental results. The observed vibrational features and major peaks for both tr fs and fs/ps CARS, from off-resonance (for toluene) to resonance (for R6G) pump wavelengths, can be well reproduced by the calculations. The connections between fs/ps CARS, fs stimulated Raman spectroscopy, and impulsive stimulated scattering for toluene and R6G are discussed.

Simple aspects of femtosecond stimulated Raman spectroscopy

Femtosecond stimulated Raman spectroscopy (FSRS), using an overlapping pair of narrow band Raman pump and broadband probe pulses with heterodyne detection along the probe pulse direction, is a new nonlinear spectroscopic technique to record vibrational spectra of even highly fluorescent molecules and to study vibrational dynamics on excited electronic states of molecules, as in photoisomerization. FSRS is described by diagrammatic third-order perturbation theory with wave packet analysis. The phase matching condition gives rise to forty-eight terms for FSRS, but the resonant condition reduces it to just eight terms, which can be depicted by Feynman dual time-line diagrams, or closed time path loop diagrams, or the complementary four-wave mixing energy level diagrams. The eight terms fall into four sets-SRS(I), SRS(II), IRS(I), IRS(II)-where SRS stands for stimulated Raman scattering and IRS stands for inverse Raman scattering. The SRS(I) set can also account for spontaneous Raman scattering, but the remaining SRS(II), IRS(I) and IRS(II) terms are only present in stimulated scattering with the presence of a probe field. The SRS(I) set accounts for the Stokes Raman lines while the IRS(I) term accounts for the anti-Stokes lines, relative to the Raman pump frequency, in the FSRS spectrum. The remaining SRS(II) and IRS(II) terms give rise to broad baselines. Using a harmonic oscillator model, analytic results are obtained for the four-time correlation functions in the third-order polarizations. The issue of high time and high frequency resolution in time-resolved FSRS spectra is discussed. Calculations are made with the theory to compare with experimental results for: (a) resonance FSRS of fluorescent Rhodamine 6G and (b) 2D-FSRS from a coherent vibrational state that has been prepared by an impulsive, off-resonant pump pulse on CDCl3. The calculated results compared well with experimental results, and in the case of 2D-FSRS on CDCl3 there is a dominant cascade effect contributing to the FSRS spectra.

Inverse Raman bands in ultrafast Raman loss spectroscopy

Ultrafast Raman loss spectroscopy (URLS) is equivalent to anti-Stokes femtosecond stimulated Raman spectroscopy (FSRS), using a broadband probe pulse that extends to the blue of the narrow bandwidth Raman pump, and can be described as inverse Raman scattering (IRS). Using the Feynman dual time-line diagram, the third-order polarization for IRS with finite pulses can be written down in terms of a four-time correlation function. An analytic expression is obtained for the latter in the harmonic approximation which facilitates computation. We simulated the URLS of crystal violet (CV) for various resonance Raman pump excitation wavelengths using the IRS polarization expression with finite pulses. The calculated results agreed well with the experimental results of S. Umapathy et al., J. Chem. Phys. 133, 024505 (2010). In the limit of monochromatic Raman pump and probe pulses, we obtain the third-order susceptibility for multi-modes, and for a single mode we recover the well-known expression for the third-order susceptibility, χ(IRS) ((3)), for IRS. The latter is used to understand the mode dependent phase changes as a function of Raman pump excitation in the URLS of CV.

NMR Multi-Time Correlation Functions of Ion Dynamics in Solids

Abstract We show that NMR multi-time correlation functions provide interesting new insights into the nature of lithium and silver ion dynamics in solids. For solid ion conductors, they usually probe the elementary jumps of the long-range charge transport. NMR two-time correlation functions yield rates and activation energies of the ionic hopping motion. Moreover, they reveal that the ionic relaxation exhibits a strong nonexponentiality in the studied crystals and glasses. NMR three-time correlation functions enable quantitative determination of the origin of the nonexponentiality. For various solid-state electrolytes, the analysis shows that pronounced dynamical heterogeneities govern the ionic hopping motion. If dynamical heterogeneities exist, NMR four-time correlation functions allow one to measure the time scale of exchange processes within the rate distributions. For silver ion dynamics in a crystalline and a glassy material, the results indicate that initially slow ions exhibit random new rates from broad rate distributions after very few jump events.

Multitime response functions and nonlinear spectra for model quantum dissipative systems

Using iterative evaluation of the real-time path integral expression, we calculate four-time correlation functions for one-dimensional systems coupled to model dissipative environments. We use these correlation functions to calculate response functions relevant to third order infrared or seventh order Raman experiments for harmonic, Morse, and quadratic-quartic potentials interacting with harmonic and two-level-system dissipative baths. Our calculations reveal the role of potential features (anharmonicity and eigenvalue spectrum), both on short and long time scales, on the response function. Further, thermal excitation causes dramatic changes in the appearance of the response function, introducing symmetry with respect to the main diagonal. Finally, coupling to harmonic dissipative baths leads to decay of the response function (primarily along the tau(3) direction) and a broadening of the peaks in its Fourier transform. At high temperatures two-level-system baths are less efficient in destroying coherence than harmonic baths of similar parameters.

Femtosecond stimulated Raman scattering for polyatomics with harmonic potentials: Application to rhodamine 6G

The perturbation theory of stimulated Raman scattering (SRS), with Raman pump on minus pump off and heterodyne detection along the probe direction, is reviewed. It has four third-order polarization terms, labeled as SRS or inverse Raman scattering (IRS): SRS(I), SRS(II), IRS(I), and IRS(II). These four polarizations have a wave packet interpretation. The polarizations, with homogenous and inhomogeneous broadening included, can be written as integrals over four-time correlation functions, and analytic formulas are derived for the latter for multidimensional harmonic potential surfaces with Franck-Condon displacements in the modes which facilitates the calculation of the SRS cross sections. The theory is applied to understand recent experimental results on the femtosecond SRS (FSRS) of a fluorescent dye, rhodamine 6G (R6G), where the Raman pump pulse is about 1 ps long, and the probe pulse is about 10 fs. The calculations compared very well with the R6G experimental results for off-resonance and resonance FSRS spectra spanning both Stokes and anti-Stokes bands, and for negative and positive pump-probe delay times on resonance.

Nonexponential polymer segmental motion in the presence and absence of ions: H2 NMR multitime correlation functions for polymer electrolytes poly(propylene glycol)-LiClO4

The authors measure 2H NMR multitime correlation functions to investigate the segmental motion of poly(propylene glycol) containing various amounts of the salt LiClO4. 2H NMR two-time correlation functions indicate that addition of salt affects not only the time scale of the segmental motion, but also the degree of the nonexponential relaxation behavior. To quantify the origin of the nonexponential segmental motion, the authors analyze 2H NMR three-time correlation functions. In general, nonexponential relaxation can result from homogeneous dynamics, i.e., intrinsic nonexponentiality, and from heterogeneous dynamics, i.e., existence of a distribution of correlation times G(ln tau). For the studied high and low salt concentrations, including neat poly(propylene glycol), the analysis shows that both homogeneous and heterogeneous contributions are important. 2H NMR four-time correlation functions allow the authors to measure the lifetime of the dynamical heterogeneities. For the studied salt concentrations, the rate exchange occurs on the same time scale as the segmental motion, indicating short-lived dynamical heterogeneities. To arrive at these results, the authors reconsider the interpretation of (2)H NMR three-time correlation functions. Results of analytical calculations and computer simulations show that it is necessary to extend the previous way of analysis so as to include effects due to correlated back-and-forth jumps.

On the lifetime of dynamical heterogeneities associated with the ionic jump motion in glasses: Results from molecular dynamics simulations and NMR experiments

Recording four-time correlation functions in experiments and computer simulations, we measure the lifetime of dynamical heterogeneities associated with the ionic jump motion in phosphate glasses. 109Ag NMR four-time correlation functions for a silver iodide doped silver phosphate glass indicate that the silver ions exhibit a random new rate from a broad distribution after very few jumps, implying that silver sites featuring fast and slow ionic jumps are intimately mixed rather than grouped into extended domains. Likewise, four-time correlation functions from molecular dynamics simulations of LiPO3 glass show that the dynamical heterogeneities observed for the lithium ionic jumps are short lived. Hence, rapid rate exchange is an important feature of the ion dynamics in an experimental and a model phosphate glass, i.e., the ionic diffusion is comprised of a random series of fast and slow jumps. In addition, we perform simulations for various lattice models to elucidate the interpretation of NMR four-time correlation functions for solid ion conductors, which can differ from that in previous applications of this method.

Rotational motion in the molecular crystals meta- and ortho-carborane studied by deuteron nuclear magnetic resonance

Spin-lattice and spin-spin-relaxation times, one- and two-dimensional spectra as well as two- and four-time correlation functions were measured for the molecular crystals ortho- and meta-carborane using deuteron nuclear magnetic resonance. It is found that in their noncubic phases these crystals exhibit highly anisotropic motions. In order to allow for a quantitative description of the motional geometry of the carboranes several stochastic models are formulated. By comparison of the model calculations with the experimental results it is found that the dynamics of these quasi-icosahedrally shaped molecules is governed by a composite reorientation process. Here the molecules perform threefold jumps around a molecule-fixed axis which itself can be tilted in four different directions with respect to a crystal-fixed axis. The tilt angle increases significantly with increasing temperature. On the basis of measurements of four-time stimulated-echo functions, implications for dynamic heterogeneity also in comparison with that of supercooled liquids are discussed.

Lifetime of dynamic heterogeneities in a binary Lennard-Jones mixture

A four-time correlation function was calculated using a computer simulation of a binary Lennard-Jones mixture. The information content of the four-time correlation function is similar to that of four-time correlation functions measured in NMR experiments. The correlation function selects a subensemble and analyzes its dynamics after some waiting time. The lifetime of the subensemble selected by the four-time correlation function is calculated, and compared to the lifetimes of slow subensembles selected using two different definitions of mobility, and to the alpha relaxation time.

Complex lithium ion dynamics in simulatedLiPO3glass studied by means of multitime correlation functions

Molecular dynamics simulations are performed to study the lithium jumps in ${\mathrm{LiPO}}_{3}$ glass. In particular, we calculate higher-order correlation functions that probe the positions of single lithium ions at several times. Three-time correlation functions show that the nonexponential relaxation of the lithium ions results from both correlated back-and-forth jumps and the existence of dynamical heterogeneities, i.e., the presence of a broad distribution of jump rates. A quantitative analysis yields that the contribution of the dynamical heterogeneities to the nonexponential depopulation of the lithium sites increases upon cooling. Further, correlated back-and-forth jumps between neighboring sites are observed for the fast ions of the distribution, but not for the slow ions and, hence, the back-jump probability depends on the dynamical state. Four-time correlation functions indicate that an exchange between fast and slow ions takes place on the time scale of the jumps themselves; i.e., the dynamical heterogeneities are short lived. Hence, sites featuring fast and slow lithium dynamics, respectively, are intimately mixed. In addition, a backward correlation beyond the first neighbor shell for highly mobile ions and the presence of long-range dynamical heterogeneities suggest that fast ion migration occurs along preferential pathways in the glassy matrix. In the melt, we find no evidence for correlated back-and-forth motions and dynamical heterogeneities on the length scale of the next-neighbor distance.

Two-dimensional spectroscopy for a two-dimensional rotator coupled to a Gaussian–Markovian noise bath

The dynamics of a system in the condensed phase are more clearly characterized by multitime correlation functions of physical observables than by two-time ones. We investigate a two-dimensional motion of a rigid rotator coupled to a Gaussian–Markovian harmonic oscillator bath to probe this issue. The analytical expression of a four-time correlation function of a dipole that is the observable of two-dimensional microwave or far-infrared spectroscopy is obtained from a generating functional approach. The spectra in the absence of damping are discrete and reveal transitions between eigenstates of the angular momentum quantized due to the cyclic boundary condition. For a weakly damped case, the two-dimensional spectrum predicts three echolike peaks corresponding to transition processes between the rotational energy levels, which cannot be observed in one-dimensional (linear-absorption) spectroscopy related to the two-time correlation function of the dipole [J. Phys. Soc. Jpn. 71, 2414 (2002)]. The two-dimensional spectra are more sensitive to the noise effects than the one-dimensional spectra. It is because the effects of the initial thermal distribution determine the profile of the continuous line shape in one-dimensional spectroscopy, while such thermal effects are canceled through the higher-order optical transition process in two-dimensional spectroscopy. If the rotator system is strongly coupled to the colored noise bath, the system exhibits one overdamped and other oscillatory motions. We observe peaks arising from interaction between these two modes in the two-dimensional spectra, which are difficult to distinguish in one-dimensional spectra.

Interference fringes with maximal contrast at finite coherence time.

Interference fringes can result from the measurement of four-time fourth-order correlation functions of a wave field. These fringes have a statistical origin and, as a consequence, they show the greatest contrast when the coherence time of the field is finite. A simple acoustic experiment is presented in which these fringes are observed, and it is demonstrated that the contrast is maximal for partial coherence. Random telegraph phase noise is used to vary the field coherence in order to highlight the problem of interpreting this interference; for this noise, the Gaussian moment theorem may not be invoked to reduce the description of the interference to one in terms of first-order interference.

Nonequilibrium initial conditions of a Brownian oscillator system observed by two-dimensional spectroscopy

We study effects of a nonequilibrium initial condition of a Brownian oscillator system upon two-, three-, and four-time correlation functions of an oscillator coordinate as a subject of multidimensional spectroscopy. A nonequilibrium initial condition is set by a displacement of a Gaussian wave packet in an oscillator potential. Such situation may be found in a vibrational motion of molecules after a sudden bond breaking between a fragmental molecule and a targeting vibrational system or a movement of wave packet in an electronic excited state potential surface created by a laser pump pulse. Multitime correlation functions of oscillator coordinates for a nonequilibrium initial condition are calculated analytically with the use of generating functional from a path integral approach. Two-, three-, and four-time correlation functions of oscillator coordinates correspond to the third-, fifth-, and seventh-order Raman signals or the first-, second-, and third-order infrared signals. We plotted these correlation functions as a signal in multidimensional spectroscopy. The profile of the signal depends on the initial position and momentum of the wave packet in the fifth- and seventh-order Raman or the second and third order infrared measurement, which makes it possible to measure the dynamics of the wave packet directly in the phase space by optical means.

Multiple-time correlation functions in spin-3/2 solid-state NMR spectroscopy

Stimulated echo spectroscopy of nonselectively excitable I = 3/2 nuclei offers new perspectives for the investigation of ultraslow motions predominantly in inorganic solids and solid-like materials. Conditions for the generation of pure, quadrupole modulated multipolar spin orders and for the detection of two- and four-time correlation functions are discussed. The case of spins I > 3/2 is also briefly considered. Copyright 2000 Academic Press.

The rate memory of a polymer close toT gas elucidated by reduced 4-D NMR echo experiments

We analyze the reorientational dynamics of a polymer close to the glass transition. Via comparison of two-time and four-time correlation functions, obtained from multidimensional nuclear magnetic resonance13C echo experiments, we are particularly sensitive to the fluctuations within the heterogeneous distribution of mobilities. From variation of the experimental parameters we obtain a detailed picture of the interplay between jump and diffusion dynamics, including the different types of motional heterogeneities and the geometry of the system.