Quantum-Chaotic Evolution Reproduced from Effective Integrable Trajectories

Abstract

Classically integrable approximants are here constructed for a family of predominantly chaotic periodic systems by means of the Baker-Hausdorff-Campbell formula. We compare the evolving wave density for the corresponding exact quantum systems using semiclassical approximations based alternatively on the chaotic and on the integrable trajectories. It is found that the latter reproduce the quantum oscillations and provide superior approximations even when the initial coherent state is placed in a broad chaotic region. Time regimes are then accessed in which the propagation based on the system's exact chaotic trajectories breaks down.

Complexified phase spaces, initial value representations, and the accuracy of semiclassical propagation

“…The reason for this might be that the main contributions come from the regular trajectories, since, as states earlier, for chaotic trajectories the pre-factors diverge very fast. Indeed, an artificial erasure of chaos in a strongly chaotic system was shown to provide better semiclassical results than the ones obtained from the system's original chaotic dynamics [16].…”

“…In addition to providing high-accuracy results for the complicated processes inherent to the chemical sciences, the H-K propagator was also scrutinized by the physical community, who attested for its remarkable accuracy and implementation ease (e.g. [12,13,14,15,16]). The reason behind such ease is mostly due to the fact that, as all IVRs, it sums over all initial positions and momenta instead of root-searching for specific trajectories, a procedure required by cruder propagators such as vV-G's.…”

Harmonic oscillator kicked by spin measurements: a Floquet-like system without classical analogous

“…Often, these external influences are time-periodic and impulsive, as is the case of the archetypal delta-kicked mechanical systems. Their quantum versions, such as tops [2][3][4][5] and oscillators [6][7][8], among others [9], are, arguably, even more intriguing, not only under the perspective of quantum chaos [10]. These systems have been shown to display quantum phase transitions, fractal bands in the quasi-energy spectrum, and crystalline patterns in both phase space and in the time domain.…”