With the rapid development of laser technology, the generation of isolated pulses with a timescale down to 43 as is achieved. This breakthrough enables the exploration of electron dynamics on an ultrashort timescale. One promising method for investigating the sub-femtosecond dynamics of electronic systems is the attosecond transient absorption spectrum (ATAS). In the previous study, the electron dynamics in atoms are investigated using ATAS, and the offset time in the Autler-Townes split is observed. Recently, the research on ATAS is extended to graphene, where we also observe the offset time in the fishbone structure of ATAS in graphene. In this work, we will reveal the underlying mechanism of the offset time in graphene.

Graphene is a two-dimensional single layer of carbon atoms arranged in a honeycomb lattice structure. In this study, we consider four-energy bands of graphene to calculate the time-dependent density matrix equations. To uncover the underlying mechanism of the offset time, we simplify the two-dimensional four-band model into a single-electron model located at the Van Hove singularity on the one-dimensional two-band structure. Using this simplified model, an analytical model of offset time is established.

Our numerical simulation results reveal the fishbone resonance structure around the Van Hove singularity points, and the offset time is observed. Using the simplified model, our numerical results for ATAS are qualitatively consistent with those calculated by the four bands model. This consistency suggests that the simplified model is a viable tool for investigating the offset time. Based on the simplified model, we derive an analytical model of the offset time, and the offset time predicted by the analytical model is qualitatively consistent with the simulation results of four-band density matrix equations. We also extend our calculation to more pump laser wavelengths and laser pulse periods, and the simulation results indicate that the analytical model can predict the outcome of numerical simulations.

We numerically simulate the ATAS of graphene using the four-band density matrix equations of graphene. Our simulation results reveal the existence of the offset time in zero-order fringes of the fishbone structure similar to the atomic offset time. To reveal the underlying mechanism of the offset time, we simplify the four-band model to a single-electron model located at the van Hove singularity on the one-dimensional two-energy band, and establish an analytical model of the offset time based on this model. The offset time predicted by our analytical model can qualitatively predict numerical simulation results under different laser wavelengths and laser pulse periods. Our analytical theory can be verified experimentally, facilitating the study of ultrafast electron dynamics in two-dimensional crystals.