\section{Conclusion}
In summary, we have presented a general
and detailed study of the tunneling time by investigating the collisions between
a macroscopic quantum wave packet and a square barrier. We explore the dependence of the tunneling time on the incident energy of the wave packet and find three different time scales in different energy regimes. In particular, the counterintuitive behavior that the tunneling time increases rapidly with the incidence velocity is shown in the negative incident energy regime, where the tunneling probability is very small. We also investigate the effect of the barrier width on the maximum tunneling time and find a critical barrier width, which together with the incident energy requirement determines the boundary between the quantum and classical regimes. We demonstrate the important role of both wave--particle duality and the uncertainty principle in the study of tunneling time. 
Our results are not only qualitatively consistent with the Larmor clock experiments but also support the claim of instantaneous tunneling in the study of strong-field ionization. Our study shows a comprehensive physical picture for understanding and studying the tunneling time and provides a new perspective to resolve the controversy over the tunneling time in strong-field ionization.


This work is supported by the National Natural Science Foundation of China (Grant No. 11974273).





