Phase-space crystals (PSCs) represent a novel class of nonequilibrium ordered many-body states in which atoms form periodic structures in phase space under periodic driving. This review systematically introduces the theoretical framework of PSCs, the mechanisms of many-body interactions, and the topological dynamics of PSCs, with particular emphasis on their realization in ultracold atomic systems.
Any target Hamiltonian with an arbitrary lattice structure in phase space can, in principle, be synthesized by appropriately designing a time-periodic driving field. The eigenstates of the target Hamiltonian can be bosonic code states, such as Gottesman-Kitaev-Preskill (GKP) states. Thus, PSCs realized with cold atoms in optical lattices may provide a platform for fault-tolerant bosonic quantum computation.
From the perspective of stroboscopic dynamics, the effective interaction between two atoms can be obtained by averaging their interaction over one driving period. This mapping shows that the short-range contact interaction between cold atoms in real space can be transformed into a long-range, Coulomb-like interaction in phase space. Combined with the underlying lattice structure, this long-range interaction makes PSCs a promising platform for simulating strongly correlated many-body models relevant to condensed-matter physics, including the Bose–Hubbard and Heisenberg models, as well as many-body localization.
Unlike conventional crystals in real space, PSCs possess an intrinsically noncommutative geometry, analogous to that experienced by the magnetron motion of charged particles in a magnetic field. Consequently, the collective vibrational modes of PSCs, analogous to phonons in conventional crystals, can exhibit novel topological transport phenomena. In particular, these modes can support chiral edge states without explicitly breaking time-reversal symmetry, giving rise to an “anomalous Chern insulator” that differs from both conventional Chern insulators and topological insulators.
To assess the experimental feasibility of realizing PSCs with cold atoms, we analyze their stability in open environments subject to thermal noise. By introducing a crystal-order parameter and performing a linear stability analysis, we estimate critical thresholds for the dissipation rate, interaction strength, and temperature. The resulting stability conditions indicate that, although experimentally challenging, the realization of PSCs remains feasible with state-of-the-art ultracold-atom platforms.
Time crystals (TCs) and PSCs are closely related but distinct concepts. Both concern subharmonic modes that emerge in periodically driven systems. Unlike TCs, which require the spontaneous breaking of discrete time-translation symmetry (DTTS), PSCs do not necessarily require such DTTS breaking; instead, they focus on the many-body physics arising from interacting subharmonic modes. Extending this many-body framework into the fully quantum regime opens promising opportunities for exploring strongly correlated topological phases of matter. Together, TCs and PSCs provide complementary perspectives on nonequilibrium many-body physics in periodically driven systems.