  The Higgs mechanism, i.e., spontaneous symmetry breaking of the quantum
vacuum, is a cross-disciplinary principle, universal for understanding dark
energy, antimatter and quantum materials, from superconductivity to magnetism.
Yet, Higgs modes in one-band superconductors (SCs) are currently under debate
due to their competition with charge-density fluctuations. A distinct Higgs
mode, controllable by terahertz (THz) laser pulses, can arise in multi-band,
unconventional SCs via strong {\em interband} Coulomb interaction, but is yet
to be accessed. Here we both discover and demonstrate quantum control of such
collective mode in iron-based high-temperature superconductors. Using
two-pulse, phase coherent THz spectroscopy, we observe a tunable and coherent
2$\Delta_{\mathrm{SC}}$ amplitude oscillation of the complex order parameter in
such SC with coupled lower and upper bands. The nonlinear dependence of the
amplitude mode oscillations on the THz driving fields is distinct from any
one-band and conventional SC results: we observe a large nonlinear change of
resonance strength, yet with a persisting mode frequency. We argue that this
result provides compelling evidence for a transient coupling between the
electron and hole amplitude modes via strong interband coherent interaction. To
support this scenario, we perform quantum kinetic modeling of a hybrid Higgs
mechanism without invoking extra disorder or phonons. In addition to
distinguishing between collective modes and charge fluctuations, the light
quantum control of multiband SCs can be extended to probe and manipulate
many-body entanglement and hidden symmetries in different quantum materials.
