The first two terms in Eq. (\[Woe\]) are the two localized CS Wigner functions, situated at the left and right sides of the well. The interference fringes are associated to the third terms of the two expressions above, where the following quantities appear $$\begin{aligned}
\nonumber
W_{oe}(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}\chi_{\beta}^{o}(x-z,0)\chi_{\beta}^{e}(x+z,0)
e^{-2ipz/\hbar}dz,\\
W_{mn}(x,p)=\frac{1}{\pi\hbar}\int_{-\infty}^{\infty}\psi_{m}(x-z)\psi_{n}(x+z,0)
e^{-2ipz/\hbar}dz.\nonumber\end{aligned}$$ The oscillatory interference ripples are parallel to the line joining the two split sub-CSs, i.e., the interference pattern is oscillatory only in $p$ space and also squeezed. The number of ripples decreases as the two sub-CSs come closer.

At $T_{rev}/8$ (Fig. \[wigSymAll\](b)), the initial wave-packet splits into four distinct sub-wave-packets (Eq. (\[oneeightcl\])). The Wigner function contains four distinct parts $W_{1},W_{2},W_{3}$ and $W_{4}$ (see Fig. \[wigSymAll\](b)) and six interference terms, among which two diagonal partners ($W_{13}$ and $W_{24}$) overlap at the center of the phase space and generate a smaller chess-board interference pattern. This structure is the signature of sub-Planck scale structures [@zurek; @ghosh] for a compass-like state. The central interference pattern can also be seen as the superposition of interferences of two orthogonally situated cat states, and it is formed by small “tiles” much smaller than the individual CS peaks.

![image](fig3.eps){width="7.in"}

Finally, at $t=T_{rev}/12$, the initial wave-packet splits into six CSs 