The existence of a paradoxical supersolid phase of matter, possessing the apparently incompatible properties of crystalline order and superfluidity, was predicted 50 years ago 1-3 . Solid helium was the natural candidate, but there supersolidity has not been observed yet, despite numerous attempts 4-7 . Ultracold quantum gases have recently shown the appearance of the periodic order typical of a crystal, due to various types of controllable interactions 8-12 . A crucial feature of a Ddimensional supersolid is the occurrence of up to D+1 gapless excitations reflecting the Goldstone modes associated with the spontaneous breaking of two continuous symmetries: the breaking of phase invariance, corresponding to the locking of the phase of the atomic wave functions at the origin of superfluid phenomena, and the breaking of translational invariance due to the lattice structure of the system. The occurrence of such modes has been the object of intense theoretical investigations 1,13-17 , but their experimental observation is still missing. Here we demonstrate the supersolid symmetry breaking through the appearance of two distinct compressional oscillation modes in a harmonically trapped dipolar Bose-Einstein condensate, reflecting the gapless Goldstone excitations of the homogeneous system. We observe that the two modes have different natures, with the higher frequency mode associated with an oscillation of the periodicity of the emergent lattice and the lower one characterizing the superfluid oscillations. Our work paves the way to explore the two quantum phase transitions between the superfluid, supersolid and solid-like configurations that can be accessed by tuning a single interaction parameter.

Motivated by a recent experiment [L.Chomaz et al., Nature Physics 14, 442 (2018)], we perform numerical simulations of a dipolar Bose-Einstein Condensate (BEC) in a tubular, periodic confinement at T=0 within Density Functional Theory, where the beyond-mean-field correction to the ground state energy is included in the Local Density Approximation. We study the excitation spectrum of the system by solving the corresponding Bogoliubov-de Gennes equations. The calculated spectrum shows a roton minimum, and the roton gap decreases by reducing the effective scattering length. As the roton gap disappears, the system spontaneously develops a periodic, linear structure formed by denser clusters of atomic dipoles immersed in a dilute superfluid background. This structure shows the hallmarks of a supersolid system, i.e. (i) a finite non-classical translational inertia along the tube axis and (ii) the appearance of two gapless modes, i.e. a phonon mode associated to density fluctuations and resulting from the translational discrete symmetry of the system, and the Nambu-Goldstone gapless mode corresponding to phase fluctuations, resulting from the spontaneous breaking of the gauge symmetry. A further decrease in the scattering length eventually leads to the formation of a periodic linear array of self-bound droplets.Dipolar Bose Einstein condensates (BECs) attracted great attentions in recent years, since the first experimental realizations of BECs with strongly magnetic atomic gases [1][2][3]. This interest is motivated by the particular properties of such systems which are characterized by anisotropic and long-range dipole-dipole interactions in addition to the short-range contact interactions, resulting in a geometry dependent stability diagram [4] where the system (which is intrinsically unstable in 3D) becomes stable against collapse if the confinement along the polarization axis is much tighter that the in-plane confinement. The properties of dipolar BECs have been the subject of numerous experimental and theoretical studies, extensively reviewed in Ref. [5,6].Recent experiments [7,8] on the stability of a dipolar BEC of 164 Dy trapped in a flat "pancake" trap showed the formation of droplets arranged in an ordered structure, their collapse being prevented by the tight confinement along the short axis. This effect is the equivalent of the Rosensweig instability of classical ferrofluids [9].Remarkably, recent experiments[10] showed that selfbound droplets can be realized in a dipolar Bose gas depending upon the ratio between the strenghts of the longrange dipolar attraction and the short range contact repulsion. These droplets, whose densities are higher by about one order of magnitude than the density of the weakly interacting condensate, are stable even in free space, after the external trapping potential is removed.The possibility of self-bound dipolar droplets has been explained theoretically in Ref. [11][12][13], where it has been shown that the binding arises from the interplay between the two-body dipolar interac...

Distintictive features of supersolids show up in their rotational properties. We calculate the moment of inertia of a harmonically trapped dipolar Bose-Einstein condensed gas as a function of the tunable scattering length parameter, providing the transition from the (fully) superfluid to the supersolid phase and eventually to an incoherent crystal of self-bound droplets. The transition from the superfluid to the supersolid phase is characterized by a jump in the moment on inertia, revealing its first order nature. In the case of elongated trapping in the plane of rotation we show that the the moment of inertia determines the value of the frequency of the scissors mode, which is significantly affected by the reduction of superfluidity in the supersolid phase. The case of isotropic trapping is instead well suited to study the formation of quantized vortices, which are shown to be characterized, in the supersolid phase, by a sizeable deformed core, caused by the presence of the sorrounding density peaks.

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