Microwave spectroscopy of the calcium $4snf\rightarrow4s(n+1)d$, $4sng$, $4snh$, $4sni$, and $4snk$ transitions

We use a delayed field ionization technique to observe the microwave transitions of calcium Rydberg states, from the $4snf$ states to the $4s(n+1)d$, $4sng$, $4snh$, $4sni$, and $4snk$ states for $18\leq n\leq23$. We analyze the observed intervals between the $\ell$ and $(\ell+1)$, $\ell\ge5$, states of the same $n$ to determine the Ca$^+$ $4s$ dipole and quadrupole polarizabilities. We show that the adiabatic core polarization model is not adequate to extract the Ca$^+$ $4s$ dipole and quadrupole polarizabilities and a non adiabatic treatment is required. We use the non adiabatic core polarization model to determine the ionic dipole and quadrupole polarizabilities to be $\alpha_d=76.9(3)$ a$_0^3$ and $\alpha_q=206(9)$ a$_0^5$, respectively.