can be re-expressed as a delta function over the frequencies of the associated modes, $\beta _{kl}=\mathcal T_{k}\delta \left( \omega
_{l}-\omega _{k}\right) $. Then the rule $\left\{ \delta \left(
\omega _{l}-\omega _{k}\right) \right\} ^{2}\rightarrow \left(
1/2\pi \right) \delta \left( \omega _{l}-\omega _{k}\right) \int
dt$ may be used to calculate a continuous rate of creation of particle-antiparticle pairs [@So82; @Kim02], $$w=\frac{d}{dt}\left\langle N\right\rangle =\frac{1}{2\pi }\int
\left| \mathcal T_{k}\right| ^{2}d\omega .$$

The physical meaning of the particle creation process is the following: a wave packet of negative frequencies incident on an electric field oriented along the $z-$direction will be partly reflected by the electric field and partly transmitted to $z\rightarrow \infty $ as a wave with positive frequencies. This process is nothing else than tunnelling [@So82; @Gr85; @Wa88]. However, in the following we restrict the tunnelling probability to the transmission probability through the potential barrier but exclude any non-zero transmission probability above the potential barrier.

In the WKB approximation the transmission probability $\mathcal T$ can be approximately given as $\left| \mathcal T\right| ^{2}=\exp \left( -2\int_{barrier}\left|
k_{z}\right| dz\right) =\exp\left(-2\sigma \right)$, where $\sigma
= \int_{barrier}\left| k_{z}\right| dz$ and $k_z$ is the longitudinal momentum of the electron [@So82; @Wa88; @Kim02].

For a fixed frequency $\omega $, the 