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Theoretical and Mathematical Physics

Relaxation in the system of vibrational levels of a harmonic oscillator

Elmar G. Petrov, V. I. Teslenko

Published Jan 1, 1979• 4 citations
Publisher
DOI
10.1007/bf01030263

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The Journal of Chemical Physics•Sep 15, 1995•68 citations
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.
A recently formulated continuum limit semiclassical initial value series representation (SCIVR) of the quantum dynamics of dissipative systems is applied to the study of vibrational relaxation of model harmonic and anharmonic oscillator systems. As is well known, the classical dynamics of dissipative systems may be described in terms of a generalized Langevin equation. The continuum limit SCIVR uses the Langevin trajectories as input, albeit with a quantum noise rather than a classical noise. Combining this development with the forward-backward form of the prefactor-free propagator leads to a tractable scheme for computing quantum thermal correlation functions. Here we present the first implementation of this continuum limit SCIVR series method to study two model problems of vibrational relaxation. Simulations of the dissipative harmonic oscillator system over a wide range of parameters demonstrate that at most only the first two terms in the SCIVR series are needed for convergence of the correlation function. The methodology is then applied to the vibrational relaxation of a dissipative Morse oscillator. Here, too, the SCIVR series converges rapidly as the first two terms are sufficient to provide the quantum mechanical relaxation with an estimated accuracy on the order of a few percent. The results in this case are compared with computations obtained using the classical Wigner approximation for the relaxation dynamics.
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.