Particle localization in a double-well potential and pseudo-supersymmetry

We show that a non-periodical smooth driving force can produce coherent destruction of tunnelling for a particle in a double-well potential having a couple of (lowest) energy levels very close to each other and well enough separated from the other levels (two-level approximation). An analysis of the time behavior of the occupation probability shows that the probability oscillates between a minimal value exceeding 1/2 and a maximal value close to 1. In particular, for specific parameter values of the driving force, it becomes a function monotonously increasing up to a value more then 1/2. These non-periodical external fields were obtained with the help of an extension of the transformation operator method to the case of a (non-Hermitian) Dirac-like Hamiltonian describing a two-level system. Two pseudosuperpartners and pseudosupersymmetry generators closing a quadratic pseudosuperalgebra were constructed. This allows us to conclude that a quadratic pseudosupersymmetry may be associated with the two-level system.