  The neutron density distributions and neutron skin thicknesses in
$^{40,48}$Ca are determined from the angular distributions of the cross
sections and analyzing powers of polarized proton elastic scattering at $E_p =
295$ MeV. Based on the framework of the relativistic impulse approximation with
the density-dependent effective $NN$ interaction, the experimental data is
successfully analyzed, providing precise information of neutron and proton
density profiles of $^{40,48}$Ca with small uncertainties. The extracted
neutron and proton density distributions give neutron skin thicknesses in
$^{40,48}$Ca for $-0.010^{+0.022}_{-0.024}$ fm and $0.168^{+0.025}_{-0.028}$
fm, respectively. The results of the density profiles and the neutron skin
thickness in $^{48}$Ca are directly compared with the {\it ab initio}
coupled-cluster calculations with interactions derived from chiral effective
field theory, as well as relativistic and non-relativistic energy density
functional theories.
