Two-dimensional spectroscopy and harmonically coupled anharmonic oscillators

Phase-locking of time-delayed attosecond XUV pulse pairs

We present a setup for the generation of phase-locked attosecond extreme ultraviolet (XUV) pulse pairs. The attosecond pulse pairs are generated by high harmonic generation (HHG) driven by two phase-locked near-infrared (NIR) pulses that are produced using an actively stabilized Mach-Zehnder interferometer compatible with near-single cycle pulses. The attosecond XUV pulses can be delayed over a range of 400 fs with a sub-10-as delay jitter. We validate the precision and the accuracy of the setup by XUV optical interferometry and by retrieving the energies of Rydberg states of helium in an XUV pump-NIR probe photoelectron spectroscopy experiment.

Stochastic simulation of anharmonic dissipation. II. Harmonic bath potentials with quadratic couplings

The workhorse simulating the dissipative dynamics is mainly based on the harmonic bath potentials together with linear system-bath couplings, but a realistic bath always assumes anharmonicity. In this work, we extend the linear dissipation model to include quadratic couplings and suggest a stochastic simulation scheme for the anharmonic dissipation. We show that the non-Gaussian noises induced by the anharmonic bath can be rigorously constructed, and the resulting stochastic Liouville equation has the same form as that for the linear dissipation model. As a preliminary application, we use this stochastic method to investigate the vibration-induced symmetry breaking in two-level electronic systems and find that the characteristic function of the non-Gaussian noises determines the absorption and fluorescence spectra.

Phase-space algorithm for simulating quantum nonlinear response functions of bosons using stochastic classical trajectories

Using the positive P-representation of the density matrix, we develop an algorithm for calculating the quantum many-body nonlinear response functions of a system of bosons driven impulsively by external fields. The formalism maps the quantum time evolution of N boson degrees of freedom into a stochastic dynamics of 4N classical degrees of freedom. The first- and the third-order response functions are calculated by propagating the parameters of the P-representation using a set of coupled Langevin equations with multiplicative noise. These parameters serve as classical variables. Two classical ways for computing the response functions are presented. In the nonequilibrium method, an observable is calculated for weak impulsive pulses, and the response functions are obtained by taking its derivatives with respect to the pulse amplitudes. In the alternative, equilibrium simulation, the response functions are expressed in terms of time-correlation functions involving the P-representation parameters and stability matrices representing the perturbation of the trajectories. The stability matrices can be propagated simultaneously with the Langevin equations for the parameters. The formalism is generalized for a many-body boson system coupled to a harmonic bath.

Modeling, calculating, and analyzing multidimensional vibrational spectroscopies

Spectral line shapes in a condensed phase contain information from various dynamic processes that modulate the transition energy, such as microscopic dynamics, inter- and intramolecular couplings, and solvent dynamics. Because nonlinear response functions are sensitive to the complex dynamics of chemical processes, multidimensional vibrational spectroscopies can separate these processes. In multidimensional vibrational spectroscopy, the nonlinear response functions of a molecular dipole or polarizability are measured using ultrashort pulses to monitor inter- and intramolecular vibrational motions. Because a complex profile of such signals depends on the many dynamic and structural aspects of a molecular system, researchers would like to have a theoretical understanding of these phenomena. In this Account, we explore and describe the roles of different physical phenomena that arise from the peculiarities of the system-bath coupling in multidimensional spectra. We also present simple analytical expressions for a weakly coupled multimode Brownian system, which we use to analyze the results obtained by the experiments and simulations. To calculate the nonlinear optical response, researchers commonly use a particular form of a system Hamiltonian fit to the experimental results. The optical responses of molecular vibrational motions have been studied in either an oscillator model or a vibration energy state model. In principle, both models should give the same results as long as the energy states are chosen to be the eigenstates of the oscillator model. The energy state model can provide a simple description of nonlinear optical processes because the diagrammatic Liouville space theory that developed in the electronically resonant spectroscopies can easily handle three or four energy states involved in high-frequency vibrations. However, the energy state model breaks down if we include the thermal excitation and relaxation processes in the dynamics to put the system in a thermal equilibrium state. The roles of these excitation and relaxation processes are different and complicated compared with those in the resonant spectroscopy. Observing the effects of such thermal processes is more intuitive with the oscillator model because the bath modes, which cause the fluctuation and dissipation processes, are also described in the coordinate space. This coordinate space system-bath approach complements a realistic full molecular dynamics simulation approach. By comparing the calculated 2D spectra from the coordinate space model and the energy state model, we can examine the role of thermal processes and anharmonic mode-mode couplings in the energy state model. For this purpose, we employed the Brownian oscillator model with the nonlinear system-bath interaction. Using the hierarchy formalism, we could precisely calculate multidimensional spectra for a single and multimode anharmonic system for inter- and intramolecular vibrational modes.

Exploring a free energy landscape by means of multidimensional infrared and terahertz spectroscopies

A model for the dipolar crystal system is employed to explore a role of free energy landscape (FEL), in which dipolar molecules are posted on two-dimensional lattice sites with two-state libratinal dynamics. All dipole-dipole interactions are included to have frustrated interactions among the dipoles. For the regular and distorted lattice cases, the FEL is calculated from the interaction energies and the total polarizations for all possible dipolar states at various temperatures. At high temperatures, the shape of the calculated FEL is smooth and parabolic, while it becomes bumpy at low temperatures exhibiting multiple local minima. To study dynamical aspects of the system, the single flip dynamics and the single-double mixed flips dynamics of dipoles are examined from a master equation approach. As the observables of linear absorption and two-dimensional (2D) infrared, far infrared, and terahertz spectroscopies, the first- and third-order response functions of polarization are calculated for different physical conditions characterized by the FEL. While the linear absorption signals decay in time in a similar manner regardless of the FEL profiles, the 2D signals exhibit prominent differences for those profiles. This indicates that we may differentiate the FEL profiles by changing two-time valuables in 2D spectroscopy. As illustrated in the single-double flips case, the FEL study by means of 2D spectroscopy, however, relies on the dynamics which is set independently from the FEL. The Smoluchowski equation is applied to examine the description of the collective dynamics on the microscopically calculated FEL. We found that the one-dimensional and 2D signals calculated from the Smoluchowski equation agree with those from master equation only at temperatures where the FEL becomes parabolic shape.

Dynamics of a Multimode System Coupled to Multiple Heat Baths Probed by Two-Dimensional Infrared Spectroscopy

Reduced equation of motion for a multimode system coupled to multiple heat baths is constructed by extending the quantum Fokker-Planck equation with low-temperature correction terms (J. Phys. Soc. Jpn. 2005, 74, 3131). Unlike such common approaches used to describe intramolecular multimode vibration as a Bloch-Redfield theory and a stochastic theory, the present formalism is defined by the molecular coordinates. To explore the correlation among different modes through baths, we consider two cases of system-bath couplings. One is a correlated case in which two modes are coupled to a single bath, and the other is an uncorrelated case in which each mode is coupled to a different bath. We further classify the correlated case into two cases, the plus- and minus-correlated cases, according to distinct correlation manners. For these, one-dimensional and two-dimensional infrared (2D-IR) spectra are calculated numerically by solving the equation of motion. It is demonstrated that 2D-IR spectroscopy has the ability to analyze the correlation of fluctuation-dissipation processes among different modes.

Calculating fifth-order Raman signals for various molecular liquids by equilibrium and nonequilibrium hybrid molecular dynamics simulation algorithms

The fifth-order two-dimensional (2D) Raman signals have been calculated from the equilibrium and nonequilibrium (finite field) molecular dynamics simulations. The equilibrium method evaluates response functions with equilibrium trajectories, while the nonequilibrium method calculates a molecular polarizability from nonequilibrium trajectories for different pulse configurations and sequences. In this paper, we introduce an efficient algorithm which hybridizes the existing two methods to avoid the time-consuming calculations of the stability matrices which are inherent in the equilibrium method. Using nonequilibrium trajectories for a single laser excitation, we are able to dramatically simplify the sampling process. With this approach, the 2D Raman signals for liquid xenon, carbon disulfide, water, acetonitrile, and formamide are calculated and discussed. Intensities of 2D Raman signals are also estimated and the peak strength of formamide is found to be only five times smaller than that of carbon disulfide.

Two-dimensional Raman spectra of atomic solids and liquids

We calculate third- and fifth-order Raman spectra of simple atoms interacting through a soft-core potential by means of molecular-dynamics (MD) simulations. The total polarizability of molecules is treated by the dipole-induced dipole model. Two- and three-body correlation functions of the polarizability at various temperatures are evaluated from equilibrium MD simulations based on a stability matrix formulation. To analyze the processes involved in the spectroscopic measurements, we divide the fifth-order response functions into symmetric and antisymmetric integrated response functions; the symmetric one is written as a simple three-body correlation function, while the antisymmetric one depends on a stability matrix. This analysis leads to a better understanding of the time scales and molecular motions that govern the two-dimensional (2D) signal. The 2D Raman spectra show novel differences between the solid and liquid phases, which are associated with the decay rates of coherent motions. On the other hand, these differences are not observed in the linear Raman spectra.

Multidimensional infrared spectroscopy for molecular vibrational modes with dipolar interactions, anharmonicity, and nonlinearity of dipole moments and polarizability

We present an analytical expression for the linear and nonlinear infrared spectra of interacting molecular vibrational motions. Each of the molecular modes is explicitly represented by a classical damped oscillator on an anharmonic multidimensional potential-energy surface. The two essential interactions, the dipole-dipole (DD) and the dipole-induced-dipole (DID) interactions, are taken into account, and each dipole moment and polarizability are expanded to nonlinear order with respect to the nuclear vibrational coordinate. Our analytical treatment leads to expressions for the contributions of anharmonicity, DD and DID interactions, and the nonlinearity of dipole moments and polarizability elements to the one-, two-, and three-dimensional spectra as separated terms, which allows us to discuss the relative importance of these respective contributions. We can calculate multidimensional signals for various configurations of molecules interacting through DD and DID interactions for different material parameters over the whole range of frequencies. We demonstrate that contributions from the DD and DID interactions and anharmonicity are separately detectable through the third-order three-dimensional IR spectroscopy, whereas they cannot be distinguished from each other in either the linear or the second-order IR spectroscopies. The possibility of obtaining the intra- or intermolecular structural information from multidimensional spectra is also discussed.

Multidimensional vibrational spectroscopy for tunneling processes in a dissipative environment

Simulating tunneling processes as well as their observation are challenging problems for many areas. In this study, we consider a double-well potential system coupled to a heat bath with a linear-linear (LL) and square-linear (SL) system-bath interactions. The LL interaction leads to longitudinal (T1) and transversal (T2) homogeneous relaxations, whereas the SL interaction leads to the inhomogeneous dephasing (T2*) relaxation in the white noise limit with a rotating wave approximation. We discuss the dynamics of the double-well system under infrared (IR) laser excitations from a Gaussian-Markovian quantum Fokker-Planck equation approach, which was developed by generalizing Kubo's stochastic Liouville equation. Analytical expression of the Green function is obtained for a case of two-state-jump modulation by performing the Fourier-Laplace transformation. We then calculate a two-dimensional infrared signal, which is defined by the four-body correlation function of optical dipole, for various noise correlation time, system-bath coupling parameters, and temperatures. It is shown that the bath-induced vibrational excitation and relaxation dynamics between the tunneling splitting levels can be detected as the isolated off-diagonal peaks in the third-order two-dimensional infrared (2D-IR) spectroscopy for a specific phase matching condition. Furthermore, this spectroscopy also allows us to directly evaluate the rate constants for tunneling reactions, which relates to the coherence between the splitting levels; it can be regarded as a novel technique for measuring chemical reaction rates. We depict the change of reaction rates as a function of system-bath coupling strength and a temperature through the 2D-IR signal.

Probing electronic coupling in excitonically coupled heterodimer complexes by two-color three-pulse photon echoes

Following the earlier work of Yang et al. [J. Chem. Phys. 110 (1999) 2983] analytical expressions for the downhill and uphill resonant two-color three-pulse photon echo peak shift (2C-3PEPS) of a heterodimer system are derived in the impulsive limit. It is shown how to obtain information about coupling between the components of the dimer from the combined one- and two-color peak shift measurements. Further analytical relations are derived which enable site specific information about the environment of the components, including the relative difference of the inhomogeneity and the difference between the energy-gap correlation functions on the heterodimer sites to be obtained. The simulations show only a very small influence of the laser pulse length on the measured values of coupling coefficient and other relevant quantities suggesting that current 2C-3PEPS measurements can find practical application in directly measuring couplings in excitonically coupled heterodimer complexes.

Two-dimensional Raman and infrared vibrational spectroscopy for a harmonic oscillator system nonlinearly coupled with a colored noise bath

Multidimensional vibrational response functions of a harmonic oscillator are reconsidered by assuming nonlinear system-bath couplings. In addition to a standard linear-linear (LL) system-bath interaction, we consider a square-linear (SL) interaction. The LL interaction causes the vibrational energy relaxation, while the SL interaction is mainly responsible for the vibrational phase relaxation. The dynamics of the relevant system are investigated by the numerical integration of the Gaussian-Markovian Fokker-Planck equation under the condition of strong couplings with a colored noise bath, where the conventional perturbative approach cannot be applied. The response functions for the fifth-order nonresonant Raman and the third-order infrared (or equivalently the second-order infrared and the seventh-order nonresonant Raman) spectra are calculated under the various combinations of the LL and the SL coupling strengths. Calculated two-dimensional response functions demonstrate that those spectroscopic techniques are very sensitive to the mechanism of the system-bath couplings and the correlation time of the bath fluctuation. We discuss the primary optical transition pathways involved to elucidate the corresponding spectroscopic features and to relate them to the microscopic sources of the vibrational nonlinearity induced by the system-bath interactions. Optical pathways for the fifth-order Raman spectroscopies from an "anisotropic" medium were newly found in this study, which were not predicted by the weak system-bath coupling theory or the standard Brownian harmonic oscillator model.

Two-dimensional femtosecond spectroscopy

The simplest two-dimensional (2D) spectra show how excitation with one (variable) frequency affects the spectrum at all other frequencies, thus revealing the molecular connections between transitions. Femtosecond 2D Fourier transform (2D FT) spectra are more flexible and share some of the remarkable properties of their conceptual parent, 2D FT nuclear magnetic resonance. When 2D FT spectra are experimentally separated into real absorptive and imaginary refractive parts, the time resolution and frequency resolution can both reach the uncertainty limit set for each resonance by the sample itself. Coherent four-level contributions to the signal provide new molecular phase information, such as relative signs of transition dipoles. The nonlinear response can be picked apart by selecting a single coherence pathway (e.g., specifying the relative signs of energy level difference frequencies during different time intervals as in the photon echo). Because molecules are frozen on the femtosecond timescale, femtosecond 2D FT experiments can separate a distribution of instantaneous molecular environments and intramolecular geometries as inhomogeneous broadening. This review provides an introduction to two-dimensional Fourier transform experiments exploiting second- and third-order vibrational and electronic nonlinearities.

Higher-order optical correlation spectroscopy in liquids

Linear optical spectroscopies have long been used to study the behavior of liquids. Laser technology has progressed to the point that it has become possible to perform nonlinear optical experiments that probe higher-order correlation functions in liquids, opening a new window into our understanding of the microscopic details of solution-phase processes. Here we review advances that have been made in recent years in employing higher-order electronic and vibrational spectroscopies to study liquid-state dynamics and structure.