Energy Ordering from Nonlinear Quantum Dissipation

Energy-ordered occupation is deeply embedded in quantum physics, from the Aufbau principle governing the filling of electronic states in atoms and molecules to the emergence of low-energy configurations in quantum many-body systems. However, the dynamical mechanism by which a generic quantum state develops such an energy hierarchy remains a fundamental question. Here we show that such an energy hierarchy can emerge dynamically from nonlinear quantum dissipation. Rather than being imposed as a principle or generated through coupling to a thermal reservoir, an Aufbau-like ordering of energy levels emerges intrinsically under quantum Landau-Lifshitz-Gilbert dynamics from a generic initial mixed state. This convergence is a nontrivial consequence of Lyapunov monotonicity and instability of disordered population configurations. The resulting dynamics establish an intrinsic nonlinear mechanism for organizing density-matrix populations and provides a route toward selective preparation of low-energy subspaces. Numerical simulations confirm analytical predictions and illustrate convergence toward low-energy sectors.