  The particles produced from the vacuum in the dynamical Casimir effect are
highly entangled. In order to quantify the correlations generated by the
process of vacuum decay induced by moving mirrors, we study the entanglement
evolution in the dynamical Casimir effect by computing the time-dependent
R\'enyi and von Neumann entanglement entropy analytically in arbitrary
dimensions. We consider the system at parametric resonance, where the effect is
enhanced. We find that, in (1+1) dimensions, the entropies grow logarithmically
for large times, $S_A(\tau)\sim\frac{1}{2}\log(\tau)$, while in higher
dimensions (n+1) the growth is linear, $S_A(t)\sim \lambda\,\tau$ where
$\lambda$ can be identified with the Lyapunov exponent of a classical
instability in the system. In $(1+1)$ dimensions, strong interactions among
field modes prevent the parametric resonance from manifesting as a Lyapunov
instability, leading to a sublinear entropy growth associated with a constant
rate of particle production in the resonant mode. Interestingly, the
logarithmic growth comes with a pre-factor with $1/2$ which cannot occur in
time-periodic systems with finitely many degrees of freedom and is thus a
special property of bosonic field theories.
