Continuous observation of a quantum system yields a measurement record that faithfully reproduces the classically predicted trajectory provided that the measurement is sufficiently strong to localize the state in phase space but weak enough that quantum backaction noise is negligible. We investigate the conditions under which classical dynamics emerges, via a continuous position measurement, for a particle moving in a harmonic well with its position coupled to internal spin. As a consequence of this coupling, we find that classical dynamics emerges only when the position and spin actions are both large compared to $\ensuremath{\hbar}$. These conditions are quantified by placing bounds on the size of the covariance matrix which describes the delocalized quantum coherence over extended regions of phase space. From this result, it follows that a mixed quantum-classical regime (where one subsystem can be treated classically and the other not) does not exist for a continuously observed spin-
21 particle. When the conditions for classicality are satisfied (in the large-spin limit), the quantum trajectories reproduce both the classical periodic orbits as well as the classically chaotic phase space regions. As a quantitative test of this convergence, we compute the largest Lyapunov exponent directly from the measured quantum trajectories and show that it agrees with the classical value.
Understanding how complex systems transition between order and chaos is a central challenge of nonequilibrium physics. While weak perturbations of classical integrable systems give rise to a mixed phase space of coexisting regular and chaotic trajectories, analogous behavior in interacting quantum many-body systems has remained elusive. Here we develop and experimentally implement a hybrid quantum-classical feedback protocol that autonomously discovers and stabilizes long-lived regular trajectories in a superconducting quantum processor. Each iteration combines short-time quantum evolution with classical optimization that projects the dynamics back onto a low-entanglement variational manifold, effectively distilling coherence from chaotic evolution. The stabilized trajectories reveal a quantum many-body mixed phase space emerging from nonlinear variational dynamics, without a direct analogue in classical or few-body quantum systems. Our results establish a versatile framework for algorithmic discovery and control of coherent dynamics previously inaccessible to experiment.
The hydrodynamic analogy for quantum mechanics is developed in a quantum phase-space representation. The key technical step in this development is the formulation of a phase-space flux (or current) density expression. Lagrangian fluid trajectories in phase space are explicitly found as solutions to a set of first-order ordinary differential equations which are formulated without making semiclassical approximation to the quantum dynamics. These fluid trajectories may be used to construct sharp structures in quantum phase space in the same way that one uses classical trajectories in classical mechanics. While these sharp structures do not themselves represent states, their properties ``control'' the dynamics of states in a manner familiar from the theory of first-order flows. The formalism reduces to Liouville dynamics in the classical limit, \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. In addition to several analytic applications to the pure- and mixed-state dynamics, the approach is numerically implemented for tunneling dynamics in a double-well problem. A new picture for quantum tunneling emerges where the tunneling transport occurs when density spirals outward from one well along real-valued fluid trajectories and then into the other well. The tunneling is found to occur through localized portals, or ``flux gates'' in the phase space. We advocate the phase-space hydrodynamic model as a general tool to understand classical quantum correspondence.