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The Journal of Chemical Physics

Classical mechanical photon echo of a solvated anharmonic vibration

Ryan B. Williams, Roger F. Loring

Published Aug 1, 2000• 28 citations
Publisher

Abstract

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.

DOI
10.1063/1.481997

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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.
We calculate the signal in a time-resolved, coherent light scattering experiment on a dilute solution of a nonpolar solute dissolved in a nonpolar solvent. We model the solute and solvent molecules as electronic two level systems, and assume that only the solute interacts with radiation. The electronic degree of freedom is treated perturbatively within the Born–Oppenheimer approximation. The nuclear degrees of freedom are treated classically, and their effects are modeled by a molecular dynamics simulation. We assume the electronic absorption spectrum is dominated by inhomogeneous broadening. The validity of this assumption for our model is verified through the calculation of the absorption line shape with and without the inhomogeneous broadening assumption. A quantitative measure of the homogeneous dephasing time is obtained from a calculation of the photon echo signal for our model. The time dependence of the light scattering signal is shown to reflect both dynamics of the fluid in its electronic ground state and in its electronic excited state. We analyze the relative importance of contributions to the signal from these two types of dynamics as a function of experimental time scale and laser frequency.
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.
Two- and three-pulse photon echo signals are calculated for various model systems. The use of an experimental solvation correlation function as the solvent fluctuation correlation function leads to two conclusions. First, inertial solvent motion plays a major role in the electronic dephasing process. Second, simple models such as Markovian or exponential models for the solvent fluctuation correlation function may not provide an adequate description of the echo signal. The real and imaginary parts of the echo response, which may be measured via heterodyne detected stimulated photon echoes, are compared with conventional photon echo signals.
We show how in the framework of the multimode Brownian oscillator model the system−bath correlation function can be derived from conventional and time-gated stimulated photon echo experiments and consideration of the linear optical spectra. Experiments are performed on the infrared dye DTTCI in room temperature solutions of ethylene glycol, methanol, and acetonitrile. The obtained correlation function is the sum of several Brownian oscillators, of which four are attributed to intrachromophore vibrational dynamics and the other three to solute−solvent dynamics. The ultrafast part of the correlation function on the time scale of the excitation pulses is interpreted as a free induction decay-like effect due to impulsive excitation of spectrally broad underlying vibrational structure in the dye's electronic transition. The slower parts are assigned to multiple time scale solute−solvent dynamics. The effect of vibrational coherences on the echo measurements is also analyzed; this analysis permits the dissection of the correlation function into a part due to intrachromophore dynamics and a part due to solvation dynamics. The spectral densities associated with these latter oscillators are located in the far infrared, in the same spectral region as probed by the optical Kerr effect. The measurements, however, provide no definite answer to the question of whether these spectral densities are the same.
This review emphasizes the physical principles underlying the dephasing of nonstationary vibrational states of molecules with multiple vibrational degrees of freedom. The motion of atoms within molecules can be described to a very good approximation by a many-level system of coupled anharmonic quantized oscillators. The separation of electronic and nuclear timescales, combined with weak symmetry breaking inherent in molecular structures, guarantees that the Hamiltonian can always be cast in a form that is local in the quantum state space of the molecular vibrations. The resulting nonexponential dephasing dynamics can be controlled with external fields to stabilize nonstationary quantum states, and both quantum–classical and fully quantum control formalisms are described. Coupled molecular vibrations interacting with external fields also offer prospects for quantum computing because vibrational level spacings can be made very large compared to thermal noise, and relaxation mechanisms such as infrared fluorescence are many orders of magnitude slower than the timescales required for coherence transfer. Finally, a toy model that provides insights into the interaction of vibrational degrees of freedom with 'solvent' modes is also discussed, and exhibits nonexponential dynamics in certain regimes.
The time-integrated three-pulse photon echo is discussed in terms of the line-shape functions which are shown to be directly related to the solvation dynamics. By using the short-time approximation to the line-shape function, an algebraic form of the three-pulse photon echo signal S PE (τ, T ) is obtained as a function of two delay times, τ and T, between pulses. On the basis of the approximate form of the photon echo signal, the echo peak shift with respect to the second delay time (population evolution period) is derived and found to be linearly proportional to the time-dependent fluorescence Stokes shift function for times greater than the bath correlation time, τ c . In the case of intermediate inhomogeneous broadening, the asymptotic echo peak shift magnitude is found to be useful in estimating the static inhomogeneous width. To illustrate the theoretical results, an analysis of the echo peak shift measurement of IR144 in a room temperature glass, PMMA, is presented.