  We study a quantum many-body variant of the parametric oscillator, by
investigating the driven sine-Gordon model with a modulated tunnel coupling via
a semi-classical Truncated Wigner Approximation (TWA). We first analyze the
parametric resonant regime for driving protocols that retain our model gapped,
and compare the TWA to a Time-Dependent Gaussian Variational Ansatz (TGVA). We
then turn to a drive which closes the gap, resulting in an enhanced energy
absorption. While the TGVA approach breaks down in this regime, we can apply
TWA to explore the dynamics of the mode-resolved energy density, and the
higher-order correlations between modes in the prethermal heating regime. For
weak driving amplitude, we find an exponentially fast energy absorption in the
main resonant mode, while the heating of all remaining modes is almost
perfectly suppressed on short time scales. At later times, the highly excited
main resonance provides effective resonant driving terms for its higher
harmonics through the non-linearities in the Hamiltonian, and gives rise to an
exponentially fast heating in these particular modes. We capture the strong
correlations induced by these resonant processes by evaluating higher order
connected correlation functions. Our results can be experimentally probed in
ultracold atomic settings, with parallel one-dimensional quasi-condensates in
the presence of a modulated tunnel coupling.
