  A time-dependent potential barrier has been used to probe the arrival-time
distribution of the wave packet of a hot electron by raising the barrier to
block the packet upon arrival of the packet at the barrier. To see whether the
barrier precisely detects the distribution, it is necessary to study an error
caused by a finite rising speed of the barrier. For this purpose, we study
transmission of an electron wave packet through the dynamical barrier, and
identify two regimes, the semiclassical regime and the quasistatic regime. In
each regime, we calculate the arrival-time distribution reconstructed by using
the barrier and quantify the error in the detection, the difference of the
temporal uncertainty between the wave-packet distribution and the reconstructed
distribution. Our finding suggests that for precise detection, the time scale,
in which the barrier height rises over the energy distribution of the wave
packet and the tunneling energy window of the barrier, has to be much shorter
than the temporal uncertainty of the wave packet. The analytical results are
confirmed with numerical calculations.
