  We present solid evidence for the existence of a well-defined Higgs amplitude
mode in two-dimensional relativistic field theories based on analytically
continued results from quantum Monte Carlo simulations of the Bose-Hubbard
model in the vicinity of the superfluid-Mott insulator quantum critical point,
featuring emergent particle-hole symmetry and Lorentz-invariance. The Higgs
boson, seen as a well-defined low-frequency resonance in the spectral density,
is quickly pushed to high energies in the superfluid phase and disappears by
merging with the broad secondary peak at the characteristic interaction scale.
Simulations of a trapped system of ultra-cold $^{87}$Rb atoms demonstrate that
the low-frequency resonance feature is lost for typical experimental
parameters, while the characteristic frequency for the onset of strong response
is preserved.
