Fishbone resonance structure in the attosecond transient absorption spectroscopy of graphene

We investigate the attosecond transient absorption spectroscopy (ATAS) of graphene by numerically solving four-band density-matrix equations, which demonstrates apparent fish bone resonance structures. To gain insight into these interesting structures, we exploit a simplified model that only considers the electrons of $\Gamma$ and M points in the Brillouin zone. With the help of this model, we can analytically express the ATAS spectrum as the sum of zeroth- and first-order Bessel functions in the variables of the strength and frequency of the infrared pump field as well as the effective mass of electrons at the $\Gamma$ and M points. Lorentzian and Fano line shapes in the absorption spectrum are addressed. The fish bone structure consists of periodic V-shaped structure that can be explained by first-order Bessel functions and its tilt angle is solely determined by the frequency of the pump laser. The periodicity of the V-shaped structure in the fish bone originates from the periodic dependence of the Lorentzian and Fano line shapes of the absorption spectrum on the time delay between the pump and probe lasers. Compared with the numerical results, our analytical theory can qualitatively or even quantitatively predict the zeroth- and first-order fringes in the fish bone structures of the ATAS spectrum. The gauge issues in the numerical simulations are also discussed.