in $x$-space at $p=0$. Solid line refers to the symmetric case ($\rho=\kappa=50$), while the dotted line refers to the slightly asymmetric case ($\rho=50$, $\kappa=46$) as shown in Fig. \[planck\]. Here, $x$ is in atomic unit.[]{data-label="2dsmall"}](fig6.eps){width="3.2in"}

This sensitivity is also visualized in Fig. \[2dsmall\], showing the section at $p=0$ of the Wigner function of the states at $T_{rev}/8$ for the two potentials. The Wigner distribution is negative, due to the non-classicality of the state, and a shift of the maximum of one curve to a minimum of the other is clearly visible. This means a sub-Planck scale shift along $x$ ($\sim
1/\Delta p$) of about $0.01247\;a.u.$, consistent with that in Fig. \[planck\]. A small difference in amplitudes due to the asymmetric effect is also captured in this enlarged $2$D view.

**II.** The perturbation studied above is difficult to implement in practice because one has to change to potential by keeping its PT form. In typical experimental situations a small perturbation can be applied through a weak constant force, which will physically shift the state in phase space. Therefore we consider what happens if we shift in phase space the compass-like state of a symmetric PT potential. We apply the displacement operator 