  A quantum particle in a slowly-changing potential well
$V(x,t)=V(x-x_0(\epsilon t))$, periodically shaken in time at a slow frequency
$\epsilon$, provides an important quantum mechanical system where the adiabatic
theorem fails to predict the asymptotic dynamics over time scales longer than $
\sim 1 / \epsilon$. Specifically, we consider a double-well potential $V(x)$
sustaining two bound states spaced in frequency by $\omega_0$ and
periodically-shaken in complex plane. Two different spatial displacements
$x_0(t)$ are assumed: the real spatial displacement $x_0(\epsilon t)=A \sin
(\epsilon t)$, corresponding to ordinary Hermitian shaking, and the complex one
$x_0(\epsilon t)=A-A \exp( -i \epsilon t)$, corresponding to non-Hermitian
shaking. When the particle is initially prepared in the ground state of the
potential well, breakdown of adiabatic evolution is found for both Hermitian
and non-Hermitian shaking whenever the oscillation frequency $\epsilon$ is
close to an odd-resonance of $\omega_0$. However, a different physical
mechanism underlying nonadiabatic transitions is found in the two cases. For
the Hermitian shaking, an avoided crossing of quasi-energies is observed at odd
resonances and nonadiabatic transitions between the two bound states, resulting
in Rabi flopping, can be explained as a multiphoton resonance process. For the
complex oscillating potential well, breakdown of adiabaticity arises from the
appearance of Floquet exceptional points at exact quasi energy crossing.
