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We report on the first experimental observation of a roton-type mode softening in a long-range interacting Bose-Einstein condensate reminiscent of the roton mode in 4He. The long-range interactions are realized by illuminating the Bose-Einstein condensate placed inside an optical high-finesse cavity by a laser beam which induces AC electric dipoles. These dipoles interact on a long-range scale via the exchange of virtually excited intracavity photons. We experimentally observe the mode softening of a single excitation mode in the dispersion relation using a variant of Bragg spectroscopy. The roton-type spectrum shows the tendency of the dilute quantum gas to spatially order. By increasing the strength of the cavity-mediated longrange interactions, we can provoke an instability in the dispersion relation of the ultracold atomic gas, which leads to the observation of a supersolid phase. The latter is characterized by its superfluid character and a non-trivial chequerboard-type density structure breaking a discrete spatial Z2-symmetry of the system. We provide detailed theoretical descriptions of the optical potentials, the mean-field solutions in the superfluid and the supersolid phases and our variant of Bragg spectroscopy. Due to cavity decay, we realize an open-system version of this phase transition. On the one hand, it enables us to monitor the atomic density fluctuations driving the superfluidsupersolid phase transition in real-time. On the other hand, however, the constant loss of photons represents a continuous measurement on the system, which is expected to lead to increased density fluctuations due to measurement backaction. We measure the density fluctuations in our system to scale qualitatively and quantitatively differently near the critical point compared to the closed-system realization. We describe the atomic and photonic fluctuations in the system by a quantum Langevin approach and obtain a quantitative agreement of our model with the experimental data.

Roton-type mode softening in a dissipative quantum many-body system with cavity-mediated long-range interactions