An analysis of the adiabatic switching method: Foundations and applications

Adiabatic Switching Extended To Prepare Semiclassically Quantized Rotational-Vibrational Initial States for Quasiclassical Trajectory Calculations

An approximation-free adiabatic switching method to generate semiclassically quantized ensembles of rovibrational states of polyatomic molecules for use as initial conditions in quasiclassical trajectory calculations is presented. Vibrational states are prepared, starting from an ensemble of classical states corresponding to the desired quantum state of the normal-mode Hamiltonian by slowly switching on the anharmonicity in internal coordinates, thereby avoiding rotational contamination. To generate rovibrational states, an extension is proposed: The vibrationally quantized molecules are slowly spun up to the desired quantized angular momentum. The ensembles obtained with adiabatic switching for CH₄ are insensitive to the choice of internal coordinates and stationary; furthermore, their mean energies agree remarkably well with the quantum mechanical values: The zero-point energy and 15 vibrational levels of the first three polyads are within 20 cm^-1, the rotational levels are between J = 1 and 50 within 1%, and the standard deviation is always <1%. Adiabatic switching produces classical state ensembles with significantly better properties than normal-mode sampling, making them more appropriate in quasiclassical trajectory calculations.

Multiresonant control of two-dimensional dynamical systems

It is shown that many two degree of freedom (2D) nonlinear dynamical systems can be controlled by continuous phase-locking (double autoresonance) between the two canonical angle variables of the system and two independent external oscillating perturbations having slowly varying frequencies. Conditions for stability of the 2D autoresonance and classification of systems with doubly autoresonant solutions in the vicinity of a stable equilibrium are outlined in terms of the Hessian matrix elements of the unperturbed system. The doubly autoresonant states in a generic, driven 2D system can be accessed by starting in equilibrium and simultaneous passage through two linear resonances in the system, provided that the driving amplitudes exceed a threshold scaling as alpha(3/4) , alpha being the characteristic chirp rate of the driving frequencies. The formation of nearly periodic trajectories in linearly nondegenerate, 2D driven systems with a single stable equilibrium is suggested as an application. Examples of autoresonant excitation and formation of nearly periodic states in other types of driven systems are presented, including a three-particle Toda chain, a particle in a 2D double-well potential, and a 3D oscillator.

Quantum-"classical" correspondence in a nonadiabatic transition system

A nonadiabatic transition system which exhibits "quantum chaotic" behavior [H. Fujisaki and K. Takatsuka, Phys. Rev. E 63, 066221 (2001)] is investigated from quasiclassical aspects. Since such a system does not have a naive classical limit, we take the mapping approach [Stock and Thoss, Phys. Rev. Lett. 78, 578 (1997)] to represent the quasiclassical dynamics of the system. We numerically show that there is a sound correspondence between the quantum chaos and classical chaos for the system.

Calculations of periodic trajectories for the Henon-Heiles Hamiltonian using the monodromy method

The monodromy method, for calculating classical periodic trajectories, is applied to the famous Henon-Heiles potential, which is invariant under the group D(3). The monodromy method is computationally very efficient and is used to find many families of periodic trajectories, including a number of simple bifurcations from the main families of the Henon-Heiles potential.