Oscillating potential well in complex plane and the adiabatic theorem

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.