Rydberg spectrum of a single trapped ${\mathrm{Ca}}^{+}$ ion: A Floquet analysis

We compute the Rydberg spectrum of a single ${\mathrm{Ca}}^{+}$ ion in a Paul trap by incorporating various internal and external coupling terms of the ion to the trap in the Hamiltonian. The coupling terms include spin-orbit coupling in ${\mathrm{Ca}}^{+}$, charge (electron and ionic core) coupling to the radio frequency and static fields, ion-electron coupling in the Paul trap, and ion center-of-mass coupling. The electronic Rydberg states are precisely described by a one-electron model potential for $e^{-}+{\mathrm{Ca}}^{2+}$, and accurate eigenenergies, quantum defect parameters, and static and tensor polarizabilities for a number of excited Rydberg states are obtained. The time-periodic rf Hamiltonian is expanded in the Floquet basis, and the trapping-field-broadened Rydberg lines are compared with recent observations of ${\mathrm{Ca}}^{+}\left(23P\right)$ and ${\mathrm{Ca}}^{+}\left(52F\right)$ Rydberg lines.

Figure 1

Stark map of ${\mathrm{Ca}}^{+}$ near the $52F$ level showing the mixing of Rydberg states with different angular momenta in an electric field. As the inset indicates, at small fields, relevant to experimental values, the $52F$-state energy shift is quadratic in the field and there is no field mixing.

Figure 2

The Mollow triplet effect around the zero detuning of $3D_{3/2}$ to $52F_{5/2}$ transition line, when ${\mathrm{\Omega }}_{\mathrm{rf}}/2\pi =3.5$ (solid curve) and 5.2 (dashed curve) MHz. The rf and static field gradients are $\alpha =8.52\times {10}^6 \mathrm{V}/{\mathrm{m}}^2$ and $\beta =3.32\times {10}^4 \mathrm{V}/{\mathrm{m}}^2$, respectively. The residual electric field is not considered in these calculations, i.e., $E_{\mathrm{geom}}=0$. (a) The calculations limited to 1-photon processes, by including 3 Floquet channels, i.e., $(-1,0,+1)$. (b) The calculations limited to 2-photon processes (5 Floquet channels). (c) Up to 12-photon processes allowed (25 Floquet channels). The Gaussian convolution is performed on the calculated results by considering a 5-MHz laser linewidth.

Figure 3

The normalized oscillator strength for the ${\mathrm{Ca}}^{+}(3D_{3/2}\to 52F_{5/2})$ resonant transition at various $E_{\mathrm{geom}}$. Correspondingly, $\alpha =8.52\times {10}^6 \mathrm{V}/{\mathrm{m}}^2, \beta =3.32\times {10}^4 \mathrm{V}/{\mathrm{m}}^2$, and ${\mathrm{\Omega }}_{\mathrm{rf}}/2\pi =3.5\mathrm{MHz}$. Up to $\pm 12$ photons in absorption and emission are included in the Floquet calculations for convergence. The Gaussian convolution is performed on the calculated results by considering a 5-MHz laser linewidth.

Figure 4

The oscillator strength for the ${\mathrm{Ca}}^{+}(3D_{3/2}\to 52F_{5/2}$) resonant transition with $E_{\mathrm{geom}}=0.24$ V/cm (a) and $E_{\mathrm{geom}}=0.84$ V/cm (b). The transverse and longitudinal trap frequencies are ${\omega }_{\mathrm{radial}}/2\pi =200\mathrm{kHz}$ and ${\omega }_{\mathrm{axial}}/2\pi =90\mathrm{kHz}$, respectively. The parameters used are the same as those in Fig. 3. Experimental data are courtesy of the Mainz group [36] and the error bars depict the quantum projection noise. The Gaussian convolution is performed on the calculated results by considering a 5-MHz laser linewidth.

Figure 5

The normalized oscillator strength for the ${\mathrm{Ca}}^{+}(3D_{3/2}\to 23P_{1/2}$) resonance transition with $E_{\mathrm{geom}}=1.6$ V/cm, ${\mathrm{\Omega }}_{\mathrm{rf}}/2\pi =14.56\mathrm{MHz}, \alpha =3.161\times {10}^8 \mathrm{V}/{\mathrm{m}}^2$, and $\beta =1.286\times {10}^6 \mathrm{V}/{\mathrm{m}}^2$ (solid black line) and $E_{\mathrm{geom}}=0.1$ V/cm, ${\mathrm{\Omega }}_{\mathrm{rf}}/2\pi =5.98\mathrm{MHz}, \alpha =1.298\times {10}^8 \mathrm{V}/{\mathrm{m}}^2$, and $\beta =1.286\times {10}^6 \mathrm{V}/{\mathrm{m}}^2$ (dashed-dotted red line). These results in black and red should be compared with the experimental spectra in Figs. 2(b) and 2(c) in Ref. [6], respectively. In particular, the formation of Rydberg side bands at about $\pm 45$ and $\pm 55\mathrm{MHz}$ are visible in the inset plots and detected in the experiment. A 1-MHz laser linewidth is used for the Gaussian convolution of the calculated spectra.