Quantum Pointwise Dimension (QUPID): An algorithm for identifying classical chaos in quantum dynamics

Abstract

The Pointwise Dimension is a diagnostic that reveals the effective dimensionality of classical trajectories in a $2N$-dimensional phase space. The essence of this diagnostic is to find a scaling according to which density changes with radius $r$ around a point of interest in phase space. Particularly, regular trajectories are confined to an $N$-dimensional invariant torus, while chaotic trajectories explore a $(2N-1)$-dimensional energy surface. In classical mechanics, this trajectory density can be found by counting recurrences to within the sphere of radius $r$. In quantum mechanics, probability density is intrinsic, making the Pointwise Dimension compelling, but surprisingly this has not been studied in the quantum regime. In this work, we implement the Pointwise Dimension in the quantum regime using the Husimi representation of the quantum wavepacket. This implementation entails a long-time average of the time-dependent Husimi representation to obtain a stationary phase space density, and a radial spatial integration of this density over a sphere of radius $r$. The classical-quantum correspondence is excellent for both autonomous and non-autonomous systems, adding a new diagnostic to the study of quantum chaos. Finally, we explore the possibility, both classically and quantum mechanically, of obtaining the Pointwise Dimension from a time correlation function.