Quantum Trails and Memory Effects in the Phase Space of Chaotic Quantum Systems

The eigenstates of a chaotic system can be enhanced along underlying unstable periodic orbits in so-called quantum scars, making it more likely for a particle launched along one such orbits to be found still there at long times. Unstable periodic orbits are, however, a negligible part of the phase space, and a question arises regarding the structure of the wave function elsewhere. Here, we address this question and show that a weakly dispersing dynamics of a localized wave packet in phase space leaves a "quantum trail" on the eigenstates, that is, makes them vary slowly when moving along trajectories in phase space, even if not periodic. The quantum trails underpin a remarkable dynamical effect: for a system initialized in a localized wave packet, the long-time phase-space distribution is enhanced along the short-time trajectory, which can result in ergodicity breaking. We provide the general intuition for these effects and prove them in the stadium billiard, for which an unwarping procedure allows us to visualize the phase space on the two-dimensional space of the page.