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The Journal of Chemical Physics

Theory of nonlinear optical experiments with harmonic oscillators

John T. Fourkas, Hitoshi Kawashima, Keith Adam Nelson

Published Sep 15, 1995• 68 citations
Publisher

Abstract

We develop a theory of photon-echo phenomena in harmonic vibrational modes. Although classical harmonic oscillators cannot produce any nonlinear optical signal in the linear response limit, we demonstrate that quantum harmonic oscillators that are coupled to any physically reasonable bath can give rise to novel nonlinear optical behavior, even in the perturbative limit. We show that photon echoes in high-frequency vibrational modes are strongly affected by both population relaxation and pure dephasing. The time dependence of the echo signal is shown to be highly sensitive to the amount of inhomogeneous broadening in the vibrational line. As an example, we develop the simple model of population relaxation resulting from linear coupling to the bath and pure dephasing resulting from quadratic coupling to the bath. Counter to the classical picture, echo signal is present when the only coupling to the bath is linear, but absent when the only coupling is quadratic.

DOI
10.1063/1.470680

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We develop a theory of photon-echo phenomena in harmonic vibrational modes. Although classical harmonic oscillators cannot produce any nonlinear optical signal in the linear response limit, we demonstrate that quantum harmonic oscillators that are coupled to any physically reasonable bath can give rise to novel nonlinear optical behavior, even in the perturbative limit. We show that photon echoes in high-frequency vibrational modes are strongly affected by both population relaxation and pure dephasing. The time dependence of the echo signal is shown to be highly sensitive to the amount of inhomogeneous broadening in the vibrational line. As an example, we develop the simple model of population relaxation resulting from linear coupling to the bath and pure dephasing resulting from quadratic coupling to the bath. Counter to the classical picture, echo signal is present when the only coupling to the bath is linear, but absent when the only coupling is quadratic.
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.
A theory of optical echo spectroscopies of large polyatomic molecules in condensed phases is developed. Using phase space correlation functions, we examine the interrelationships among the following optical measurements: ordinary photon echo, stimulated photon echo, accumulated photon echo, incoherent accumulated photon echo, and pump–probe absorption. Conditions for the elimination of inhomogeneous broadening in these experiments are specified. A multimode Brownian oscillator model is used to account for high frequency molecular vibrations, as well as solvent modes, and electronic dephasing processes. The effects of quantum beats, spectral diffusion, and homogeneous dephasing on the echo signals are studied and compared in detail with pump–probe and hole burning spectroscopy.
Vibrational relaxation of a harmonic oscillator nonlinearly coupled to a heat bath is investigated by the Gaussian–Markovian quantum Fokker–Planck equation approach. The system–bath interaction is assumed to be linear in the bath coordinate, but linear plus square in the system coordinate modeling the elastic and inelastic relaxation mechanisms. Interplay of the two relaxation processes induced by the linear–linear and square–linear interactions in Raman or infrared spectra is discussed for various system–bath couplings, temperatures, and correlation times for the bath fluctuations. The one-quantum coherence state created through the interaction with the pump laser pulse relaxes through different pathways in accordance with the mechanisms of the system–bath interactions. Relations between the present theory, Redfield theory, and stochastic theory are also discussed.
The vibrational photon echo measurement employs short pulses of coherent infrared radiation to discriminate among line broadening processes of different time scales. The measurement has been analyzed with the optical Bloch equations, in which the coupling of a quantum three-level system to a bath is represented by dephasing times. A detailed treatment of vibrational dynamics in a molecular liquid requires a Hamiltonian model. As such models are more readily treated in classical than in quantum mechanics, we consider the calculation of the vibrational echo in classical mechanics. Vibrational echoes are calculated for a solvated anharmonic oscillator. The classical mechanical echo is shown to share attributes of a quantum mechanical calculation.
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.
Oscillatory dephasing is a characteristic signature of non-Markovian relaxation phenomena. From its analysis one can obtain valuable information about the dynamics of the surroundings of a given system. In this paper we calculate the transient four-wave mixing signal in the short pulse limit from two-level systems coupled to a bath of harmonic oscillators for arbitrary coupling strengths and in the presence of inhomogeneous broadening. The strong and weak coupling cases having already been analyzed, we focus on the study of the intermediate case where oscillatory dephasing is observed. We specifically show that even in the presence of inhomogeneous broadening, information about the surroundings can still be extracted in the intermediate coupling regime. The formation and the evolution of the photon echo in such experiments are also analyzed. \textcopyright{} 1996 The American Physical Society.