Turbulence in a matter-wave supersolid

Quantum turbulence associated with wave and vortex dynamics is numerically investigated for a two-dimensional trapped atomic Rydberg-dressed Bose-Einstein condensate (BEC). When the coupling constant of the soft-core interaction is over a critical value, the superfluid (SF) system can transition into a hexagonal supersolid (SS) state. Based on the Gross-Pitaevskii equation approach, we have discovered a new characteristic k^-13/3 scaling law for wave turbulence in the SS state, that coexists with the waveaction k^-1/3 and energy k^-1 cascades commonly existing in a SF BEC. The new k^-13/3 scaling law implies that the SS system exhibits a negative, minus-one power energy dispersion (E ~ k^-1) at the wavevector consistent with the radius of the SS droplet. For vortex turbulence, in addition to the presence of the Kolmogorov energy k^-5/3 and Saffman enstrophy k^-4 cascades, it is found that large amount of independent vortices and antivortices pinned to the interior of the oscillating SS results in a strong k^-1 scaling at the wavevector consistent with the SS lattice constant.