A closed quantum system that beats like a clock forever

Using the Bateman dual oscillator — a framework that pairs a damped oscillator with an energy-gaining counterpart to keep the total system conservative — the authors show that tracing out the hidden half produces a memory-laden reduced dynamics that sustains indefinite oscillations in the observable subsystem. Because the effect arises from non-equilibrium subsystem dynamics rather than ground-state symmetry breaking, it sidesteps the no-go theorems that forbid equilibrium time crystals. The same equations also carve out logarithmic-spiral, self-similar fractal trajectories, linking dissipation, memory, and temporal order in one closed quantum system.