  We analyze intrinsic nonlinearities in two-dimensional polaritonic materials
interacting with an optical wave. Focusing on the case of graphene, we show
that the second-order nonlinear optical conductivity due to carrier density
fluctuations associated with the excitation of a plasmon polariton is closely
related to the ponderomotive force due to the oscillating optical field. This
relation is first established through an elegant thermodynamic approach for a
Drude-like plasma, in the frequency range where intraband scattering is the
dominant contribution to conductivity. Subsequently, we extend our analysis to
the interband regime, and show that for energies approximately half the Fermi
energy, the intraband contribution to the ponderomotive force diverges. In
practice, thermal broadening regularizes this divergence as one would expect,
but even at room temperature typically leaves a strong ponderomotive
enhancement. Finally, we study the impact of nonlocal corrections and find that
nonlocality does not lead to further broadening (as one would expect in the
case of Landau damping), but rather to a splitting of the ponderomotive
interband resonance. Our analysis should prove useful to the open quest for
exploiting nonlinearities in graphene and other two-dimensional polaritonic
materials, through effects such as second harmonic generation and photon drag.
