Title: Magic wavelengths of Ca$^{+}$ ion for linearly and circularly polarized light,
Abstract: The dynamic dipole polarizabilities of the low-lying states of Ca$^{+}$ for
linearly and circularly polarized light are calculated by using relativistic
configuration interaction plus core polarization (RCICP) approach. The magic
wavelengths, at which the two levels of the transitions have the same ac Stark
shifts, for $4s$-$4p_{j,m}$ and $4s$-$3d_{j,m}$ magnetic sublevels transitions
are determined. The present magic wavelengths for linearly polarized light
agree with the available results excellently. The polarizability for the
circularly polarized light has the scalar, vector and tensor components. The
dynamic polarizability is different for each of magnetic sublevels of the
atomic state. Additional magic wavelengths have been found for the circularly
polarized light. We recommend that the measurement of the magic wavelength near
850 nm for $4s-4p_{\frac32,m=\pm\frac32,\pm\frac12}$ could be able to determine
the oscillator strength ratio of $f_{4p_{\frac32} \to 3d_{\frac32}}$ and
$f_{4p_{\frac32} \to 3d_{\frac52}}$.