Tunneling times of a macroscopic quantum wave packet in different energy regimes

Abstract Tunneling time as an important quantity to characterize tunneling process has not been well defined yet in quantum mechanics. In this paper, we adopt the definition of tunneling time given in [S.-C. Li, New J. Phys. 24 , 083033 (2022)] to investigate the time spent by a macroscopic quantum wave packet passing through a Gaussian barrier. We show a general scenario of tunneling times in different energy regimes and find that it has a good agreement with the scenario for a rectangular barrier, which verifies the validity of the above definition of tunneling time for the Gaussian barrier. The influence of both incident energy of wave packet and barrier parameters on tunneling time is discussed thoroughly. The connection between tunneling time and tunneling probability for the Gaussian barrier is established as well. In particular, we use the product of the variances of position and momentum to quantify the deformation of wave packet caused by tunneling process and demonstrate that different behaviors of the variances strongly support our oft-emphasized division of energy regimes.