of maxima and minima. The phase space area of these ’tiles’ are given by $\hbar^2$ divided by the effective phase-space area $A$ globally occupied by the state, which can be significantly larger than $\hbar$. These structures are very sensitive to environmental decoherence [@zurek; @ph; @wisn; @pathak1; @pathak2; @ghosh1] and may have important application in Heisenberg-limited measurements and quantum parameter estimation [@toscano; @dalvit]. A classical analogue of sub-Planck scale structures, regarded as sub-Fourier sensitivity, has been studied in Ref. [@praxmeyer], and these structures have also been found in diatomic molecular systems [@ghosh], entangled cat states [@manan], optical fibers [@fibre], and Kirkwood-Rihaczek distribution [@jay]. Recently, a connection between these structures and teleportation fidelity has been also established [@scott].

Sub-Planck structures have been extensively studied in the case of superpositions of harmonic oscillator coherent states and also for superpositions of generalized coherent states which occur in the time evolution of systems with nonlinear potentials at fractional revival times. It is therefore important to extend the study of this phenomenon to other nonlinear potentials in order to investigate which are the relevant parameters affecting this subtle quantum behavior, and if Heisenberg-limited sensitivity can be approached also in other nonlinear systems.

Here we focus our 