Larmor time of a wave packet tunneling through a barrier with an infinitesimal magnetic field

Recently, experimental measurements of the Larmor time of incident wave packets have been conducted, once again drawing attention to such a time. These experiments measured the tunneling time of Bose-condensed $^{87}\mathrm{Rb}$ atoms, with theoretical analyses employing free particles. In this paper, we investigate the tunneling process of wave packets, which, unlike free particles with a definite incident energy, have a continuous distribution of energies. We show how the Larmor time ${\ensuremath{\tau}}_{z(y)}$ for a wave packet tunneling through a barrier depends on the barrier width $d$, the barrier's characteristic wave vector ${k}_{0}$, the incident wave packet's central wave vector ${k}_{c}$, and its width $\ensuremath{\sigma}$ in wave-number space. When the barrier width $d$ is small, we find that ${\ensuremath{\tau}}_{z(y)}$ varies monotonically. For wide barriers ($d>2.0\ensuremath{\pi}$), we find an anomaly where wave packets with lower incident energy or higher barriers have shorter tunneling times, which is consistent with recent experimental results. Additionally, for narrow wave packets in wave-number space ($\ensuremath{\sigma}<\ensuremath{\gamma}$), where $\ensuremath{\gamma}$ is the width of the fitted Fano peak, the results are generally consistent with those for plane waves. However, as the wave packet widens in wave-number space, the maximum points of ${\ensuremath{\tau}}_{z(y)}$ will undergo a nonlinear decrease and $\ensuremath{\sigma}$ significantly affects the maximum Larmor time.