Oscillating potential well in the complex plane and the adiabatic theorem

A quantum particle in a slowly changing potential well V(x,t)=V(x-x0(ϵt)), periodically shaken in time at a slow frequency ϵ, provides an important quantum mechanical system where the adiabatic theorem fails to predict the asymptotic dynamics over time scales longer than ∼1/ϵ. Specifically, we consider a double-well potential V(x) sustaining two bound states spaced in frequency by ω0 and periodically shaken in a complex plane. Two different spatial displacements x0(t) are assumed: the real spatial displacement x0(ϵt)=Asin(ϵt), corresponding to ordinary Hermitian shaking, and the complex one x0(ϵt)=A-Aexp(-iϵt), 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 ϵ is close to an odd resonance of ω0. However, a different physical mechanism underlying nonadiabatic transitions is found in the two cases. For the Hermitian shaking, an avoided crossing of quasienergies 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 quasienergy crossing.