  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.
