long time.

![(color online). Particle number versus central position of the Gaussian wave packet: $k_0/a = 20, a g = 10$. For larger bandwidth (narrower wave packet in time domain), the particle number distribution is narrower, showing that particles are localized around the past event horizon. []{data-label="Fig:pn"}](f4.png){width="8.0cm"}

![(color online). Energy of the wave packets versus the central frequency: $\sigma/a = 1.0, a V_0 =0, a g = 10$. The energy is almost constant in the high central frequency limit. []{data-label="Fig:energy"}](f5.png){width="8.0cm"}

If we assume that the reflectivity $R_{\omega}$ of the mirror is a power law of $\omega$ as $\omega \rightarrow 0$ ($R_{\omega} \sim \omega^{\gamma}$) then in order to obtain finite particle number we must have $\gamma > 1$. As a concrete example, we choose $$R_{\omega} = \sin^2 \theta_{\omega} = \frac{g^2\omega^2}{1+g^2\omega^2},$$ where $g$ is a parameter characterizing the low-frequency cutoff. Fig. \[Fig:pn\] shows the particle number $N(f)$ versus the central position of the Gaussian wave packet. We can see that the particle-number distribution is symmetric with respect to $V_0 = 0$. In addition, for larger bandwidth (narrower wave packet in time domain), the distribution is more localized around $V_0 = 0$. These two facts indicate that the particle flux radiated by the 