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The Journal of Chemical Physics

First-, third-, and fifth-order resonant spectroscopy of an anharmonic displaced oscillators system in the condensed phase

Yoshitaka Tanimura, Ko Okumura

Published Feb 8, 1997• 60 citations
Publisher

Abstract

We have obtained Nth-order response functions for a two-level system described by displaced anharmonic potential surfaces coupled to a heat bath. The anharmonicity of the potentials has been taken into account as a perturbation of harmonic potentials. The heat-bath was assumed to be an ensemble of harmonic oscillators. Coupling between the two-level system and the bath was assumed to be bilinear. The calculations were done analytically using the Liouville-space generating functional, which had been obtained by way of the path-integral approach. The response functions have been defined in terms of line-shape functions with these line-shape functions being expressed in terms of the bath spectral density and the temperature. We have carried out model calculations of the first-, third-, and fifth-order optical processes for various parameters of anharmonicity to show that anharmonicity plays a minor role in linear absorption, impulsive pump–probe, and photon echo experiments, but plays a major role, in some cases, in fifth-order two-dimensional resonant spectroscopy which is proposed in this paper.

DOI
10.1063/1.473099

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We have obtained Nth-order response functions for a two-level system described by displaced anharmonic potential surfaces coupled to a heat bath. The anharmonicity of the potentials has been taken into account as a perturbation of harmonic potentials. The heat-bath was assumed to be an ensemble of harmonic oscillators. Coupling between the two-level system and the bath was assumed to be bilinear. The calculations were done analytically using the Liouville-space generating functional, which had been obtained by way of the path-integral approach. The response functions have been defined in terms of line-shape functions with these line-shape functions being expressed in terms of the bath spectral density and the temperature. We have carried out model calculations of the first-, third-, and fifth-order optical processes for various parameters of anharmonicity to show that anharmonicity plays a minor role in linear absorption, impulsive pump–probe, and photon echo experiments, but plays a major role, in some cases, in fifth-order two-dimensional resonant spectroscopy which is proposed in this paper.
The quantum Fokker-Planck equation is derived for a system nonlinearly coupled to a harmonic oscillator bath. The system-bath interaction is assumed to be linear in the bath coordinates but quadratic in the system coordinate. The relaxation induced dynamics of a harmonic system are investigated by simulating the higher-order correlation functions of the Raman polarizability and the dipole moment, which represent the nonlinear optical responses of Raman or infrared spectroscopy. The 5th-order Raman response shows that, in addition to the frequency fluctuations induced by the bath, higher-order energy transfer between the system and bath plays a role. The nonlinearity of the system-bath interaction yields also an interesting feature in the 7th-order Raman echo or the 3rd-order infrared photon echo response: The calculations predict a finite signal for the case of a harmonic potential and a linear coordinate dependence of the polarizability or dipole while for linear system-bath coupling this response vanishes completely due to destructive interference of different Liouville space pathways.
The relaxation processes in a quantum system nonlinearly coupled to a harmonic Gaussian-Markovian heat bath are investigated by the quantum Fokker-Planck equation in the hierarchy form. This model describes frequency fluctuations in the quantum system with an arbitrary correlation time and thus bridges the gap between the Brownian oscillator model and the stochastic model by Anderson and Kubo. The effects of the finite correlation time and the system-bath coupling strength are studied for a harmonic model system by numerically integrating the equation of motion. The one-time correlation function of the system coordinate, which is measured in conventional Raman and infrared absorption experiments, already reflects the inhomogeneous character of the relaxation process. The finite correlation time of the frequency fluctuations, however, is directly evident only in the two- and three-time correlation function as probed by multidimensional spectroscopic techniques such as the Raman echo and the fifth-order 2D Raman experiment.
We report extensive numerical studies of the dynamics of a classical particle in an anharmonic one-dimensional potential while it is harmonically coupled to three different many-particle systems. The studies address the comparatively simpler dynamical problem when the energy of the anharmonic oscillator is sufficiently low. The first model is one in which the many-particle system is a chain of harmonic oscillators that have nearest-neighbor harmonic interactions. The anharmonic oscillator is connected to every harmonic oscillator via harmonic springs. The second model is identical to the first except that each oscillator is subjected to an additional one-body harmonic potential. The third model is identical to the second except that the nearest-neighbor couplings between the individual harmonic oscillators are absent, i.e., the oscillators are decoupled with respect to one another. In the first model we find that the lowest frequency of the anharmonic oscillator, while possessing a lower bound, increases as $\ensuremath{\omega}\ensuremath{\propto}\sqrt{\mathrm{NK}},$ where NNN is the number of oscillators in the bath and KKK is the force constant describing the strength of interaction between the anharmonic oscillator and the interacting harmonic many-body system. In the second model we find aspects of dynamical correlations as seen in the canonical ensemble and a dominant frequency that increases as NK.\sqrt{\mathrm{NK}}.NK​. We recover the dynamical correlations of the anharmonic oscillator as obtained within the framework of a canonical ensemble [S. Sen, R. S. Sinkovits, and S. Chakravarti, Phys. Rev. Lett. 77, 4855 (1996)] in the third model for sufficiently many harmonic oscillators. We comment briefly on an alternate way to correctly model heat baths for numerical studies on the dynamics of physical systems using canonical ensembles.
An approximate excited state vibrational Hamiltonian, He, which accounts for both linear and diagonal quadratic electron-phonon coupling is derived and shown to be adequate for mode frequency changes up to ~30%. The associated linear response function which is valid for finite temperature in the absence of damping is obtained. 2-point correlation function or, equivalently, the line broadening function, g(t;T), that provides a good physical description of pure electronic dephasing and vibrational damping for linearly coupled modes is postulated. A linear response function that accounts for pure electronic dephasing and damping for systems with either linearly or quadratically coupled modes is proposed. Applications of the new linear response function include the single site absorption and hole-burned spectra of the special pair band of the bacterial reaction center and the temperature dependence of the single site absorption spectrum of Al-phthalocyanine tetrasulphonate in glassy ethanol. A 4-point correlation function which is valid for finite temperature in the absence of damping using the approximate excited state vibrational Hamiltonian, He, is derived. An impulsive stimulated photon echo signal for a single-mode system with very large inhomogeneous broadening is calculated using the 4-point correlation function. An expression for ordinary photon echo with two finite pulses is derived. 4-point correlation function for linearly coupled modes is expressed in terms of g(t;T) and can be used to calculate all non-linear spectroscopic signals. Applications of the 4-point correlation function are limited to impulsive 2-pulse photon echo (IPE) signals. IPE signals are generated for different values of the Huang-Rhys factor (S) at a fixed temperature and at different temperatures for a fixed value of S in order to see how the echo profile changes as one varies S or temperature. An initial fast non-exponential decay, which reflects the contribution of the phonon sideband, followed by vibrational oscillations (quantum beats) due to linear electron-phonon coupling and a slow exponential decay, on picosecond time scales, due to the zero-phonon line are observed in the IPE signals. Application of IPE to the special pair absorption band of the bacterial reaction center is also included.
Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.
In this work, we develop an extended dissipaton theory that generalizes the environmental couplings beyond the conventional linear and quadratic forms, enabling the treatment of arbitrary order of bath couplings. Applying this theoretical framework to the condensed-phase non-Condon spectroscopy, we demonstrate the interplay of anharmonicity, non-Condon and solvent effects on optical spectra. Precise simulations are carried out with high efficiency on linear absorption spectra involving the above mentioned correlated effects. We exhibit how an anharmonic potential modulates the vibronic feature, offering insights into the role of nonlinear environmental couplings in spectroscopic signatures and exemplifying the success of the extended dissipaton formalism as an exact and efficient method for higher-order bath couplings.