on the phase space behavior of the compass-like state as a whole (see the contour plots in Fig. \[planck\](a) and Fig. \[planck\](c) for the symmetric and slight asymmetric case respectively). However, this is no more true if we look at the effect of the slight asymmetry on the interference pattern and we zoom in the center of the phase space plot (see Fig. \[planck\](b) and Fig. \[planck\](d)). The effect of the potential asymmetry on the interference pattern is equivalent to a phase-space shift, because one has a displacement of a single tile such that, approximately, the maximum of a tile comes to coincide with a zero of the unperturbed state. This fact just confirms that one has an extreme sensitivity with respect to small changes of the potential because the two states becomes quasi-orthogonal to a very good approximation ($\sim 0.1$). In fact, one has that $$|\langle\chi_{\beta}^{\mathrm{sym}}|\chi_{\beta}^{\mathrm{asym}}\rangle|^2\!\!=\!\!\int\!\!\!\!\int \!W_{\mathrm{sym}}(x,p)
W_{\mathrm{asym}}(x,p)\; dx\; dp;\label{wigoverlap}$$ when the two interference patterns are displaced by a tile, one has a destructive interference effect, the above integration tends to zero and the two states become quasi-orthogonal. Thus, the size of sub-Planck structures sets a sensitivity limit on probing small potential modifications.

![Oscillations of the central sub-Planck region 