  Starting from the quasiparticle random phase approximation based on the
Skyrme interaction SLy5, we study the effects of phonon-phonon coupling~(PPC)
on the low-energy electric dipole response in $^{40-58}$Ca. Using the same set
of parameters we describe available experimental data for $^{40,44,48}$Ca and
give prediction for $^{50-58}$Ca. The inclusion of the PPC results in the
formation of low-energy $1^-$ states. There is an impact of the PPC effect on
low-energy $E1$~strength of $^{40,44,48}$Ca. The PPC effect on the electric
dipole polarizability is discussed. We predict a strong increase of the summed
$E1$~strength below 10MeV, with increasing neutron number from $^{48}$Ca till
$^{58}$Ca.
