Time-Quasicrystals Dynamically Protected Topological Phases in Driven Dissipative Systems

The classification of phases of matter has been revolutionised by topological order and non equilibrium phenomena such as time crystals. However, a unified framework connecting temporal order, topology, and dissipation remains elusive. Here we introduce the concept of "Time-Quasicrystals"—non-equilibrium topological phases that exhibit quasi-periodic temporal order protected by dissipative dynamics. These phases emerge in driven dissipative quantum systems through the interplay of Floquet engineering and non-local dissipation, leading to an effective non-Hermitian Floquet Hamiltonian with topological edge modes in the frequency domain. We demonstrate three hallmarks of Time-Quasicrystals: (1) fractal-like temporal correlations with multiple incommensurate frequencies, (2) a quantised topological invariant robust against disorder, and (3) algebraically decaying memory of phase perturbations. We provide explicit experimental realisations in Rydberg atom arrays and circuit quantum electrodynamics platforms. Our work establishes Time-Quasicrystals as a bridge between topological phases, time crystals, and quantum simulation, opening pathways to dissipative protected quantum memory and novel non-equilibrium quantum phenomena.