  The Sauter--Schwinger effect predicts the creation of electron--positron
pairs from the vacuum due to a quasiconstant electric field
$E_{\mathrm{strong}}$. The pair-creation yield can be exponentially enhanced
without destroying the tunneling-like nature of this mechanism by adding a
weaker temporal Sauter pulse $E_{\mathrm{weak}}/\cosh^{2}(\omega t)$ with
$\omega$ above a certain threshold $\omega_{\mathrm{crit}}$. In this original
form of the so-called dynamically assisted Sauter--Schwinger effect,
$\omega_{\mathrm{crit}}$ is independent of $E_{\mathrm{weak}}\ll
E_{\mathrm{strong}}$. Via the semiclassical solution (contour integral) of the
Riccati equation in 1+1 spacetime dimensions, we find that a Gaussian-shaped
pulse $E_{\mathrm{weak}}\exp[-(\omega t)^{2}]$ assists tunneling in a similar
way but with $\omega_{\mathrm{crit}}$ depending on $E_{\mathrm{weak}}$. This
remarkable sensitivity to the pulse shape arises due to the different pole
structures of the vector potentials for complex times. We also study dynamical
assistance by an oscillation $E_{\mathrm{weak}}\cos(\omega t)$ as a model for
counterpropagating laser beams and find another dependence
$\omega_{\mathrm{crit}}(E_{\mathrm{weak}})$.
  The largeness of the Schwinger limit $E_{\mathrm{crit}}^{\mathrm{QED}}\approx
10^{18}\,\mathrm{V/m}$ has rendered the observation of this nonperturbative
pair-creation mechanism impossible so far. In order to facilitate a better
understanding of this effect and its dynamical assistance via experiments, we
propose an analog of the many-body Dirac Hamiltonian in direct-bandgap
semiconductors. The nonrelativistic Bloch-electron Hamiltonian is restricted to
the valence and conduction bands in reciprocal space, which correspond to the
two relativistic energy continua. Similar models have been considered
before---but mainly for constant external fields. [...]
