Topological quantum hodographs

Abstract

In quantum mechanics, the wave function encodes all information about a particle through its probability density and phase. While stationary states are characterized by conserved quantum numbers such as energy and, in central potentials, angular momentum, non-stationary superpositions - particularly in anisotropic or time-dependent fields - generally lack equally universal descriptors of their spatiotemporal dynamics. Here we introduce quantum hodographs: the trajectories traced in time by the expectation value of observable vector quantities. For a free electron in a superposition of three plane waves, all hodographs of the probability current lie on a universal cubic surface with conical singularities. Rational frequency (energy) difference ratios produce non-contractible loops with well-defined winding numbers. In anisotropic harmonic oscillators the Ehrenfest trajectories - hodographs of the expectation value of particle radius-vector - form three-dimensional Lissajous knots, echoing the classical Thomson vortex-atom model. Externally driven quantum systems allow controllable initiation of knotted hodographs. We propose an optical modulation spectroscopy scheme for reconstructing these topological features in trapped ions and single-electron systems. The topological indices of the loops and knots are robust to parameter variations, offering a new tool for characterizing complex quantum dynamics.