potential stretches or squeezes the wave-packet near the left side of the well. Moreover it spreads in $x$ space and becomes more squeezed in $p$ space. The asymmetry affects also the interference pattern of the states: the interference ripples are not anymore placed at the center, the diagonal interference patterns overlap partially and destroy the chess-board and benzene-like structures. Moreover, the interference pattern and the localized sub-CSs partially overlap and become much less distinguishable.

Quantitative analysis of the sub-Planck scale structures
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Let us now make a quantitative analysis and check if the smallest structures in the interference patterns, e.g., the alternate positive and negative tiles have really sub-Planck scale dimensions. Therefore we have to verify if the phase-space area occupied by the tiles ‘$a$’ scales, as predicted, as $\sim\hbar^2/A$ ($\sim 1/A$ in atomic units), where $A$ is the classical action of the state. $A$ is approximately given by the product of the effective support of its state in position and momentum: $A\sim \Delta x\times \Delta p$, where $\Delta
x=\sqrt{\langle x^2 \rangle - \langle x \rangle^2}$ and $\Delta
p=\sqrt{\langle p^2 \rangle - \langle p \rangle^2}$.

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