  Oscillator-strength sum rule in light-induced transitions is one general form
of quantum-mechanical identities. Although this sum rule is well established in
equilibrium photo-physics, an experimental corroboration for the validation of
the sum rule in a nonequilibrium regime has been a long-standing unexplored
question. The simple band structure of graphene is an ideal system for
investigating this question due to the linear Dirac-like energy dispersion.
Here, we employed both ultrafast terahertz and optical spectroscopy to directly
monitor the transient oscillator-strength balancing between quasi-free
low-energy oscillators and high-energy Fermi-edge ones. Upon photo-excitation
of hot Dirac fermions, we observed that the ultrafast depletion of high-energy
oscillators precisely complements the increased terahertz absorption
oscillators. Our results may provide an experimental priori to understand, for
example, the intrinsic free-carrier dynamics to the high-energy
photo-excitation, responsible for optoelectronic operation such as
graphene-based phototransistor or solar-energy harvesting devices.
