Fig. 4: Theoretical calculation of HHG from massless Dirac fermions in graphene.

a Photo-excited charge carrier population in the conduction band \({\rho }_{{{{{{\rm{CB}}}}}}}\) under linearly-polarized laser field along the x-direction (\({E}_{{{{{{\rm{x}}}}}}}\)). b Detailed profile of carrier population around K point (region marked with a white square in a). Dashed lines are constant energy contours on a Dirac cone, which describes conduction band states vertically separated by energy \({{mE}}_{{{{{{\rm{ph}}}}}}}\). c Transition dipole moment profile. Color scale and arrow show the magnitude and direction of transition dipole moment. d, e Microscopic channels for carrier dynamics at the states on the \({k}_{{{{{{\rm{y}}}}}}}\)-axis. Under linearly-polarized excitation along the x-direction (d), the interband transition (black arrows labeled with \({M}_{{{{{{\rm{x}}}}}}}^{{{{{{\rm{inter}}}}}}}\)) creates and recombine photo-excited electrons and holes (red filled and empty circles, respectively). Simultaneously, the intraband transition drives carriers along the \({k}_{{{{{{\rm{x}}}}}}}\)-direction (orange arrows labeled with \({M}_{{{{{{\rm{x}}}}}}}^{{{{{{\rm{intra}}}}}}}\)). Under elliptically polarized excitation (e), the intraband transition also drives carriers along the \({k}_{{{{{{\rm{y}}}}}}}\)-direction (blue arrows labeled with \({M}_{{{{{{\rm{y}}}}}}}^{{{{{{\rm{intra}}}}}}}\)). f Polarization profile of the fifth harmonics under laser ellipticity \({\varepsilon }_{{{{{{\rm{exc}}}}}}}\) = 0.2 from selected pathways. Black, orange, and blue solid lines show the polarization profile from quantum pathways via \({M}_{{{{{{\rm{x}}}}}}}^{{{{{{\rm{inter}}}}}}}\) + \({M}_{{{{{{\rm{y}}}}}}}^{{{{{{\rm{inter}}}}}}}\), \({M}_{{{{{{\rm{x}}}}}}}^{{{{{{\rm{inter}}}}}}}\) + \({M}_{{{{{{\rm{x}}}}}}}^{{{{{{\rm{intra}}}}}}}\), and \({M}_{{{{{{\rm{x}}}}}}}^{{{{{{\rm{inter}}}}}}}\) + \({M}_{{{{{{\rm{y}}}}}}}^{{{{{{\rm{intra}}}}}}}\), respectively. Black empty circles show experimental results for the same laser excitation condition.