  In this letter we study how fast the energy density of a quantum gas can
increase in time, when the inter-atomic interaction characterized by the
$s$-wave scattering length $a_\text{s}$ is increased from zero with arbitrary
time dependence. We show that, at short time, the energy density can at most
increase as $\sqrt{t}$, which can be achieved when the time dependence of
$a_\text{s}$ is also proportional to $\sqrt{t}$, and especially, a universal
maximum energy growth rate can be reached when $a_\text{s}$ varies as
$2\sqrt{\hbar t/(\pi m)}$. If $a_\text{s}$ varies faster or slower than
$\sqrt{t}$, it is respectively proximate to the quench process and the
adiabatic process, and both result in a slower energy growth rate. These
results are obtained by analyzing the short time dynamics of the short-range
behavior of the many-body wave function characterized by the contact, and are
also confirmed by numerical solving an example of interacting bosons with
time-dependent Bogoliubov theory. These results can also be verified
experimentally in ultracold atomic gases.
