  Access to collective excitations lies at the heart of our understanding of
quantum many-body systems. We study the Higgs and Goldstone modes in a
supersolid quantum gas that is created by coupling a Bose-Einstein condensate
symmetrically to two optical cavities. The cavity fields form a U(1)-symmetric
order parameter that can be modulated and monitored along both quadratures in
real time. This enables us to measure the excitation energies across the
superfluid-supersolid phase transition, establish their amplitude and phase
nature, as well as characterize their dynamics from an impulse response.
Furthermore, we can give a tunable mass to the Goldstone mode at the crossover
between continuous and discrete symmetry by changing the coupling of the
quantum gas with either cavity.
