Vibrational Echoes:  Dephasing, Rephasing, and the Stability of Classical Trajectories

Abstract

The experimental observables in coherent, multiple pulse infrared spectroscopic measurements can be calculated from the nonlinear response functions describing the nuclear dynamics of molecular and condensed phase systems. Within classical mechanics, these nonlinear response functions can be expressed in terms of the monodromy matrices that quantify the stability of classical trajectories. We use an ensemble of noninteracting, anharmonic oscillators to examine the effects of the divergence in time of the classical stability matrix on the analytic properties of the third-order response function, relevant to vibrational echo spectroscopy. The twopulse echo measurement is designed to rephase a macroscopic variable, that is, to reverse the effects of destructive interference among the dynamics of microscopic systems characterized by a static distribution of energies. Within classical mechanics, this rephasing is shown to preserve the growth with time of the nonlinear response function that is the signature of the divergence of nearby trajectories. For systems with nearly classical nuclear motions, the vibrational echo measurement may then be interpreted as a probe of the stability of atomic trajectories.

Classical nonlinear response of a chaotic system. I. Collective resonances

“…The nonlinear response function for classical integrable dynamics have been shown to have power-like divergence at long times which can be eliminated by invoking a quantum description [7,8,9,10,11]. Divergence of nonlinear response in integrable systems does not imply unphysical behavior by itself, since integrable dynamics is an idealization.…”

“…It has been pointed out [8,9] that the divergence or the classical response functions in integrable systems originates from quasiperiodic nature of the underlying motion. Combined with the picture of almost integrable dynamics established by KAM theory, this demonstrates the unphysical divergence of the response functions is stable with respect to at least weak deviations from integrability.…”

“…Classical nonlinear response functions have been shown to diverge with time in the general case [11], although in certain cases the divergence can be removed by thermal averaging using the canonical distribution [7,8]. The latter effect can be viewed as a result of destructive interference of different paths (dephasing), while the divergence is associated with the rephasing.…”

“…A number of studies have been devoted to the nonlinear response in stable (integrable) dynamical systems [7,8,9,10,11]. Classical nonlinear response functions have been shown to diverge with time in the general case [11], although in certain cases the divergence can be removed by thermal averaging using the canonical distribution [7,8].…”

Quantum-Classical Correspondence in Response Theory

“…The exact evaluation of this expression is a challenge even for small number of degrees of freedom N, and the complexity of calculations growths exponentially with N. The latter motivates the investigation of semiclassical approach to the calculation of response functions [3,4,5,6]. The classical limit of the quantum response function is usually obtained by replacing commutation relations with Poisson brackets and neglecting terms of higher order in the Plank constant [7].…”