  The ratio of the transverse and longitudinal component of polarization
transfer to protons in quasi-elastic $(\vec{e}, e^{\prime} \vec{p}\,)$
reaction, $P^{\prime}_x/P^{\prime}_z$, is sensitive to the proton's
electromagnetic form factor ratio, $G_E/G_M$. To explore density-dependent
in-medium modifications, a comparison of polarization transfer ratios involving
protons from distinct nuclear shells, each with different local nuclear
densities, has been proposed. In this study, we present such comparisons
between four shells, $1s_{1/2}$, $1p_{3/2}$ in $^{12}\mathrm{C}$ and
$1d_{3/2}$, $2s_{1/2}$ in $^{40}\mathrm{Ca}$. In an effort to account for other
many-body effects that may differ between shells, we use state-of-the-art
relativistic distorted-wave impulse-approximation (RDWIA) calculation and
present the double ratios, $(P^{\prime}_x/P^{\prime}_z)_{\rm
Data}/(P^{\prime}_x/P^{\prime}_z)_{\rm RDWIA}$ as well as the super ratios,
$\left[(P^{\prime}_x/P^{\prime}_z)_{\rm A}/(P^{\prime}_x/P^{\prime}_z)_{\rm
B}\right]_{\rm Data}/\left[(P^{\prime}_x/P^{\prime}_z)_{\rm
A}/(P^{\prime}_x/P^{\prime}_z)_{\rm B}\right]_{\rm RDWIA}$, for chosen shells A
and B, as a function of effective local nuclear densities. We find that double
ratios for individual shells show a dependence on the probed effective nuclear
densities. Studying the ratios, we observed a systematic variation between
pairs of higher- and lower-density shells.
