Quantum-Classical Correspondence in Response Theory

Abstract

In this thesis, theoretical analysis of correspondence between classical and quantum dynamics is studied in the context of response theory. Thesis discusses the mathematical origin of time-divergence of classical response functions and explains the failure of classical dynamic perturbation theory. The method of phase space quantization and the method of semiclassical corrections are introduced to converge semiclassical expansion of quantum response function. The analysis of classical limit of quantum response functions in the Weyl-Wigner representation reveals the source of time-divergence of classical response functions and shows the non-commutativity of the limits of long time and small Planck constant. The classical response function is obtained as the leading term of the h-expansion of the Weyl-Wigner phase space representation and increases without bound at long times as a result of ignoring divergent higher order contributions. Systematical inclusion of higher order contributions improves the accuracy of the h expansion at finite times. The time interval for the quantum-classical correspondence is estimated for quasiperiodic dynamics and is shown to be inversely proportional to anharmonicity. The effects of dissipation on the correspondence between classical and quantum response functions are studied. The quantum-classical correspondence is shown to improve if coupling to the environment is introduced. In the last part of thesis the effect of quantum chaos on photon echo-signal of two-electronic state molecular systems is studied. The temporal photon echo signal is shown to reveal key information about the nuclear dynamics in the excited electronic state surface. The suppression of echo signals is demonstrated as a signature of level statistics that corresponds to the classically chaotic nuclear motion in the excited electronic state.

Thermal weights for semiclassical vibrational response functions

“…These are the OMT approximation 28,[45][46][47] based on the consideration of double-sided Feynman diagrams [48][49][50][51] and the semiclassical Wigner transform (SWT) approximation 26 based on the application of Wigner transforms with action-angle variables.…”

“…This formalism was used to establish connections between the structures of classical and quantum response functions. 26 In Sec. IV, we apply this semiclassical approximation for Wigner transforms 26 in action-angle variables to the canonical linear response function.…”

“…26 In Sec. IV, we apply this semiclassical approximation for Wigner transforms 26 in action-angle variables to the canonical linear response function. Invoking additional harmonic approximations to the Wigner transforms of the coordinate and density operators recovers the OMT expression for the linear response function.…”

“…Kryvohuz and Cao 26 have constructed a semiclassical approximation to microcanonical vibrational response functions that employs action-quantized classical trajectories. This approach is based on formulating quantum response functions as phase-space averages of Wigner transforms expressed in action-angle variables.…”

“…[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. 30,31 The calculation of a canonical spectroscopic response function may be divided into two related parts: determining the thermal weights for initial conditions and performing the time propagation.…”