  Quantum phase transitions in spin systems are supposed to be accompanied by a
soft collective mode, which has not been seen in experiments. Here, we directly
measure the low energy excitation modes of a well-known realization of the
Ising model in transverse field, LiHoF$_4$, using microwave spectroscopy
techniques to probe energies well below what is accessible via neutron
scattering experiments. Instead of the single excitation expected for a simple
quantum Ising system, we find and characterize a remarkable array of
`electronuclear' modes, arising from coupling of the spin-1/2 Ising electronic
spins to a bath of spin-7/2 Ho nuclear spins. The lowest-lying electronuclear
mode softens at the approach to the quantum critical point from below and
above, a softening that can be quenched with the application of a longitudinal
magnetic field. The electronuclear mode structure has direct implications for
the Ising systems that serve as the building blocks of adiabatic quantum
computers and quantum annealers.
