Fishbone resonance structure in the attosecond transient absorption spectrum of graphene

Abstract

We investigate the attosecond transient absorption spectrum (ATAS) of graphene by numerically solving four-band density-matrix equations, which demonstrates apparent ``fishbone'' resonance structures. To gain insight into these interesting structures, we exploit a simplified model that considers only the electrons of $\mathrm{\ensuremath{\Gamma}}$ and $\mathrm{M}$ points in the Brillouin zone. With the help of this model, we can analytically express the ATAS 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 $\mathrm{\ensuremath{\Gamma}}$ and $\mathrm{M}$ points. Lorentzian and Fano line shapes in the absorption spectrum are addressed. The fishbone structure consists of a 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 fishbone 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 fishbone structures of the ATAS.