order to see the effects of the asymmetry on sub-Planck scale structures. In both cases we verify quantitatively that the dimension of the sub-Planck structures scales as $\hbar^2/A$, as expected, also for the PT potential.

We then study the sensitivity of sub-Planck scale structures to perturbations, by considering two different situations: i) “internal” perturbations, i.e., a small change of the potential which is turned from symmetric into a slightly asymmetric one; ii) “external” perturbations, as for example a weak force, yielding a small phase space displacement of the state. Both cases will confirm that also in the PT potential, sub-Planck structures are responsible for a very high sensitivity to displacements.

The paper is organized as follows. In Sec II, the symmetric and asymmetric properties of PT potential are reviewed and the revival dynamics of a coherent state (CS) wave-packet are discussed. Existence of sub-Planck scale structures in phase space is demonstrated for both symmetric and asymmetric cases in Sec III. In Sec IV we discuss the sensitivity of sub-Planck scale structures due to a small asymmetry of the potential and then the one associated with a small phase-space displacement of the state. We end with some conclusions in Sec V.

