  We report analytic results on quantum capacitance (C$_{q}$) measurements and
their optical tuning in dual-gated device with potassium-doped multi-layered
black phosphorous (BP) as the channel material. The two-dimensional (2D)
layered BP is highly anisotropic with a semi-Dirac dispersion marked by linear
and quadratic contributions. The C$_{q}$ calculations mirror this asymmetric
arrangement. A further increase to the asymmetry and consequently C$_{q}$ is
predicted by photon-dressing the BP dispersion. To achieve this and tune
C$_{q}$ in a field-effect transistor (FET), we suggest a configuration wherein
a pair of electrostatic (top) and optical (back) gates clamp a BP channel. The
back gate shines an optical pulse to rearrange the dispersion of the 2D BP.
Analytic calculations are done with Floquet Hamiltonians in the off-resonant
regime. The value of such C$_{q}$ calculations, in addition, to its role in
adjusting the current drive of an FET is discussed in context of
metal-insulator and topological phase transitions and enhancements to the
thermoelectric figure of merit.
