Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions

Long-range interactions in quantum gases are predicted to give rise to an excitation spectrum of roton character, similar to that observed in superfluid helium. We investigate the excitation spectrum of a Bose-Einstein condensate with cavity-mediated long-range interactions, which couple all particles to each other. Increasing the strength of the interaction leads to a softening of an excitation mode at a finite momentum, preceding a superfluid to supersolid phase transition. We study the mode softening spectroscopically across the phase transition using a variant of Bragg spectroscopy. The measured spectrum is in very good agreement with ab initio calculations and, at the phase transition, a diverging susceptibility is observed. The work paves the way towards quantum simulation of long-range interacting many-body systems.