Classical Divergence of Nonlinear Response Functions

Abstract

The time divergence of classical nonlinear response functions reveals the fundamental difficulty of dynamic perturbation based on classical mechanics. The nature of the divergence is established for systems in regular motions using asymptotic decomposition of Fourier integrals. The asymptotic analysis shows that the divergence cannot be removed by phase-space averaging such as the Boltzmann distribution function. The implications of this study are discussed in the context of the conceptual development of quantum-classical correspondence in dynamic response.

Vibrational coherence and energy transfer in two-dimensional spectra with the optimized mean-trajectory approximation

“…[7][8][9][10][11][12][13][14][15] Numerically exact quantum dynamical calculations of response functions are generally impractical while purely classical calculations may be qualitatively incorrect except at the shortest time scales. [16][17][18][19][20][21][22][23] These challenges motivate the development of approximate methods for computing response functions. One successful strategy is to treat a subsystem of interest with an implementation of quantum dynamics and to treat the rest of the system with classical or semiclassical mechanics.…”

“…[24][25][26][27][28][29][30][31][32][33] An alternative approach is to treat all degrees of freedom consistently with a semiclassical approximation to quantum dynamics. 19,20,[34][35][36][37][38][39][40] Kryvohuz and Cao 41 have developed a semiclassical approximation of response functions in terms of classical dynamics in action-angle variables 42 with action variables subject to quantization conditions. This approach results from expressing response functions exactly as phase space integrals of Wigner-transformed operators and then evaluating the Wigner transforms within a semiclassical approximation.…”

Quantum interferences and their classical limit in laser driven coherent control scenarios

“…The phenomenon requires a nonlinear response of the system to the laser field, and hence it applies to both quantum or classical systems with anharmonic potentials. It is worth reemphazing that the matter interference effects and other entirely quantum contributions can have an important quantitative effect on the response [29,30]. However, the qualitative nonlinear response to the laser field that gives rise to laser controllable interference contributions does not necessarily rely upon them.…”

“…For this it is convenient to represent b O H ðtÞ and Eq. (29) in the phase space of c-number position x and momentum p variables, and then take the classical (h ¼ 0) limit. A phase space picture of b O H ðtÞ can be obtained using the Wigner transform [21,22] At the initial time common observables such as position, momentum, angular momentum and the Hamiltonian are semiclassically admissible.…”

Thermal weights for semiclassical vibrational response functions

“…[1][2][3][4][5][6] Numerically exact quantum dynamical calculations of response functions are generally impractical for large systems, while classical calculations of nonlinear response functions can be qualitatively incorrect at long times. [7][8][9][10][11] Semiclassical approximations to quantum dynamics [12][13][14][15][16][17][18][19][20][21][22][23][24][25] have the capacity to incorporate quantum effects with a computational effort comparable to that of a purely classical calculation. One semiclassical strategy applied to the evaluation of spectroscopic response functions [26][27][28] draws on the old quantum theory 29 by propagating classical trajectories at quantized action values.…”