Spectroscopy of ${}^{87}\mathrm{Sr}$ triplet Rydberg states

A combined experimental and theoretical spectroscopic study of high-$n, 30\lesssim n\lesssim 100$, triplet $S$ and $D$ Rydberg states in ${}^{87}\mathrm{Sr}$ is presented. ${}^{87}\mathrm{Sr}$ has a large nuclear spin $I=9/2$, and at high-$n$ the hyperfine interaction becomes comparable to, or even larger than, the fine structure and singlet-triplet splittings, which poses a considerable challenge both for precision spectroscopy and for theory. For high-$n S$ states, the hyperfine shifts are evaluated nonperturbatively, taking advantage of earlier spectroscopic data for the $I=0$ isotope ${}^{88}\mathrm{Sr}$, which results in good agreement with the present measurements. For the $D$ states, this procedure is reversed by first extracting from the present ${}^{87}\mathrm{Sr}$ measurements the energies of the ${}^3{D}_{1,2,3}$ states to be expected for isotopes without hyperfine structure (${}^{88}\mathrm{Sr}$), which allows the determination of corrected quantum defects in the high-$n$ limit.

Figure 1

The $n$ scaling of the fine-structure splitting $\mathrm{\Delta }{E}_J^{\left(0\right)}$ (), the spin singlet-triplet splitting $\mathrm{\Delta }{E}_S^{\left(0\right)}$ (), and the level separation $\mathrm{\Delta }{E}_n^{\left(0\right)}$ () between like states in ${}^{88}\mathrm{Sr}$ that differ in $n$ by one. Also shown is the strength of the hyperfine interaction in ${}^{87}\mathrm{Sr}$ (). The splittings $\mathrm{\Delta }{E}_J^{\left(0\right)}$ and $V_{\mathrm{hf}}$ refer to ${}^3{D}_{}$ states with $J=1$ and $J=2$ and are evaluated using the measured data and their extrapolation.

Figure 2

Solid lines show hyperfine energy shifts of the $5sns{}^{1,3}{S}_{}$ states in ${}^{87}\mathrm{Sr}$ relative to the eigenvalues of $H_0(88,m_{87})$ [see Eq. (1)]. The state labels for the mixed $F=I$ submanifold [Eqs. (10, 11, 12)] indicate the state with the largest overlap. Dashed lines show hyperfine energy shifts for $F=I$ states when mixing of adjacent $n$ levels due to the hyperfine interaction is neglected, i.e., setting $O_{n,n^{\prime }}={\delta }_{n,n^{\prime }}$ in Eq. (12).

Figure 3

Solid lines show the fractional part of quantum defect $\mu \left({\nu }_{F=4}\right)$ evaluated relative to the $F=4$ ionization threshold [see Eq. (14)] as a function of the effective quantum number ${\nu }_{F=4}$ for different $F$ manifolds of ${}^{87}\mathrm{Sr}$ in the $D$ sector. Each state is labeled by its dominant ${}^{2S+1}{D}_J$ state component.

Figure 4

(a) Diagram of the experimental arrangement showing the 461-nm cooling beams and the counterpropagating 689- and 319-nm Rydberg excitation lasers. (b) Two-photon excitation scheme utilizing either the (i) $5s5p{}^3{P}_1,F=11/2$ or (ii) $5s5p{}^3{P}_1,F=9/2$ intermediate states. The detunings ${\mathrm{\Delta }}_{11/2}\sim 12\mathrm{MHz}$ and ${\mathrm{\Delta }}_{9/2}\sim 36\mathrm{MHz}$ remain fixed. (c) Arrangement of the electrodes used for ionizing Rydberg atoms and guiding the electrons towards the MCP detector.

Figure 5

(a) Wavelength dependence of the offset $\delta$ between the measured and published transition frequencies used to calibrate the wavemeter. The black line shows the linear fit used to obtain the offset at $638\mathrm{nm}$ and the shaded region the uncertainty in the wavemeter calibration obtained from Monte Carlo simulations (see the text). The inset shows the offset of the 689-nm transition in ${}^{88}\mathrm{Sr}$ measured at different times.

Figure 6

Quantum defects ${\mu }_{n,S,L,J}^{\left(0\right)}$ for the $5sns{}^3{S}_1$ levels: measurements from earlier work [20, 38] ($•$), present measurements of the $5sns{}^3{S}_1,F=11/2$ states derived from the earlier ionization limit [39] ($▴$), and present measurements with modified ionization limit (see the text) ($▼$). Also shown are the predictions using the Rydberg-Ritz formula from [18] () and the modified Rydberg-Ritz formula (). The inset shows the higher-$n$ region on an expanded scale.

Figure 7

Shown in blue are the measured spectra for $5snd{}^{3}D$ states of ${}^{87}\mathrm{Sr}$ in the vicinity of (a) $n=50$, (b) $n=60$, and (c) $n=98$. Energies are given relative to the $5s50s, 5s60s$, and $5s98s{}^3{S}_1,F=11/2$ states, respectively. Rydberg excitation was performed following scheme (ii) in (a) and (b) and scheme (i) in (c). The vertical bars above the data show the calculated positions for the various hyperfine states (see the text). The measured levels and splittings are given in Table 2.

Figure 8

Quantum defects ${\mu }_{n,S,L,J}^{\left(0\right)}$ for the $5snd{}^3{D}_{1,2,3}$ levels: measurements from earlier work [20, 38] ($•$), present measurements ($▴$), predictions using the Rydberg-Ritz formulas developed previously [18] (), and predictions based on the present updated Rydberg-Ritz formulas (see the text) (). The insets show the high-$n$ region on an expanded scale.