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

We observe the microwave transitions of calcium from the $4snf$ states to the $4s(n+1)d,4sng,4snh,4sni$, and $4snk$ states for $18\le n\le 23$ using delayed field ionization as the state selective detection technique. The observed intervals between the $\ell \ge 5$ states can be analyzed to extract the ${\mathrm{Ca}}^{+}$ ionic dipole $\left({\alpha }_d\right)$ and quadrupole $\left({\alpha }_q\right)$ polarizabilities using two nonadiabatic core polarization models. Using these two models we determine the ionic dipole and quadrupole polarizabilities to be $75.3a_0^3<{\alpha }_d<76.9a_0^3$ and $206a_0^5<{\alpha }_q<1590a_0^5$, respectively.

Figure 1

Laser excitation scheme of the experiment.

Figure 2

Timing sequence of the experiment.

Figure 3

One-photon $4s22f\to 4s22g$ resonance. The linewidth of the resonance is $\sim 1$ MHz which is a transform limited linewidth of a 1 $\mu \mathrm{s}$ microwave pulse.

Figure 4

Two-photon $4s18f\to 4s18h$ resonances. The two resonances are separated by the $K$ splitting of the $4s18h$ state.

Figure 5

Three-photon $4s19f\to 4s19i$ resonance at relative microwave power 0.63 and at relative rf power 1.0. The $K$ splitting of the $4s19i$ states cannot be resolved due to the Earth's magnetic-field broadening.

Figure 6

Extrapolation of the three-photon $4s19f\to 4s19i$ transition to zero rf and microwave powers. (a) At relative microwave power 0.63, resonances were observed at different rf powers to obtain the resonance frequency at zero rf power. (b) Several zero rf power resonances were obtained at different microwave powers and extrapolated to zero rf and microwave powers.

Figure 7

Four-photon $4s19f\to 4s19k$ resonance at 0.178 relative microwave power and 0.794 relative rf power. The $K$ splitting of the $4s19k$ states is on the order of 1 MHz and cannot be resolved.

Figure 8

One-photon $4s19f\to 4s20d$ resonance.

Figure 9

Adiabatic plot of the measured $nh-ni$ $\left(▴\right)$ and $ni-nk$ $\left(\blacksquare \right)$ intervals using Eq. (7). There are three data points for the $ni-nk$ $\left(\blacksquare \right)$ intervals, $18\le n\le 20$, and six data points for the $nh-ni$ $\left(▴\right)$ intervals, $18\le n\le 23$. A fit to the straight line yields the $y$ intercept and slope, which are ${\alpha }_d$ and ${\alpha }_q$, respectively. The resulting fit values are ${\alpha }_d=75.32\left(4\right)a_0^3$ and ${\alpha }_q=-257\left(8\right)a_0^5$.

Figure 10

Energy levels of the Rydberg states converging to the ${\mathrm{Ca}}^{+}4s$ and $Np$ states and the continua associated with the latter. A $4sn\ell$ Rydberg state is dipole coupled to $Np{n}^{\prime }(\ell \pm 1)$ states which span an energy range $\Delta$. If $\Delta \ll W_{4s-Np}$ the adiabatic approximation is valid.

Figure 11

Nonadiabatic plot of the measured $nh-ni$ $\left(▴\right)$ and $ni-nk$ $\left(\blacksquare \right)$ intervals using Eq. (14). There are three data points for the $ni-nk$ $\left(\blacksquare \right)$ intervals, $18\le n\le 20$, and six data points for the $nh-ni$ $\left(▴\right)$ intervals, $18\le n\le 23$. A linear fit (solid line) gives values for the $y$ intercept and slope of 76.99(7)$a_0^3$ and 228(12)$a_0^5$, respectively. When we take into account the overcorrection of $k_d$, we obtain the data points $(•)$ and the lower fit line (broken line), which leads to our final values of ${\alpha }_d=76.91\left(5\right)a_0^3$ and ${\alpha }_q=206\left(9\right)a_0^5$.