Phase space crystal vibrations: Chiral edge states with preserved time-reversal symmetry

It was recently discovered that atoms subject to a time-periodic drive can give rise crystal structure in phase space. In this work, we point out the atom-atom interactions collective phonon excitations of phase-space via pairing interaction with intrinsically complex phases lead Chern insulator, accompanied by topologically robust chiral transport along edge crystal. This topological is realized even scenarios where time-reversal transformation symmetry, which surprising because breaking symmetry strict precondition for standard setting real-space crystals. Our work has also important implications dynamics 2D charged particles strong magnetic field.