Fig. 1: Superfluid fraction in superfluids and supersolids.

Sketches of the superfluid fraction from the application of a phase twist in a bosonic system at zero temperature. a, In a homogeneous superfluid, a phase twist with amplitude Δφ results in a constant gradient of the phase, that is, a constant velocity, whereas in a supersolid (b,c), the kinetic energy can be minimized by accumulating most of the phase variation in the low-density regions. The grey and green areas represent the number density and the kinetic energy density, respectively, whereas the phase profile is plotted in red. The superfluid fraction is the ratio of the area under the green curve to that of the homogeneous case. b, Leggett’s approach, which—for an annular system—would correspond to a stationary rotation, leads to a monotonous increase of the phase. c, Our method, based on an alternating oscillation of the phase, leads to Josephson oscillations. Both kinetic energy and superfluid fraction are the same for b and c.