Existence and nonexistence of an intrinsic tunneling time

Using a time operator, we define a tunneling time for a particle going through a barrier. This tunneling time is the average of the phase time introduced by other authors. In addition to the delay time caused by the resonances over the barrier, the present tunneling time is also affected by the branch point at the edge of the energy continuum. We find that when the particle energy is near the branch point, the tunneling time becomes strongly dependent on the width of the incoming wave packet, which implies that there is no intrinsic tunneling time. This effect is related to the quantum uncertainty in the particle’s momentum.

Figure 1

Relation between the age difference, $t_{\text{tunnel}}$, $t_{\text{outside}}$, and the delay times $\Delta {\tau }_A$ and $\Delta {\tau }_B$.

Figure 2

Integration contours and poles of the age difference.

Figure 3

The tunneling time $t_{\text{tunnel}}$ vs the momentum of the particle $k_0$, for $a=15$, $m=1$, $2mV=1$, and $L_0=150$ (solid line) and $L_0=300$ (dotted line).

Figure 4

Main plot: the age difference $t_{{\psi }_2,{\psi }_1}$ (solid line) and the age difference with no barrier $t_{{\psi }_2,{\psi }_1}^{\left(0\right)}$ (dashed line) vs $k_0$, for $L_0=150$, $a=15$, $m=1$, and $2mV=1$. Inset: a zoom-in view around $k_0=0.5$.