Survey of hyperfine structure measurements in alkali atoms

Atomic beam (AB). In an atomic beam atoms propagate along a given direction with a small spread in the orthogonal plane, see Ramsey (1956). Usually, an exciting laser propagates perpendicularly to the beam propagation leading to a very small Doppler broadening.
Vapor cell (VC). In a glass/quartz cell the atoms are in the vapour phase and their vapour pressure and the atomic density are controlled by the cell temperature.
Magneto-optical trap (MOT). The combined action of laser cooling and magnetic field confinement produces dense atomic samples having a greatly reduced Doppler linewidth. Such samples allow to detect weak absorption features, as for highly excited states, and to perform experiments with long interaction times leading to increased precision.
Fountain (FOUNT). In a fountain, atoms from a MOT are launched vertically by radiation pressure or a moving optical lattice, see Metcalf and van der Straten (1999). Excitation and detection take place at the same vertical position, the first one at the launching time and the second after the parabolic motion. Very high parabolic evolutions are used in order to increase the interrogation time. This approach is applied to the atomic clocks.
Optical Dipole Trap (ODT) In the experiment by  Neuzner et al. (2015) a single ⁸⁷Rb atom is trapped by an optical dipole trap created by a 2-D optical lattice. Cavity-enhanced state detection of the optical absorption produces a good signal-to-noise ratio even for a single atom. Light-shift correction is carefully applied.
Thermoionic diode (TD). In presence of a weak electrical discharge in a VC, the light excited atoms are ionized by electron collisions. These ions diffuse into the space charge region of the diode, compensate partially the space charge, and increase the thermionic diode current, as described in Herrmann et al. (1985)

The experimental techniques used to measure the reported hyperfine measurements are classified in the following. Several research groups have introduced a specific name for their technique. While our classification scheme is concise, detailed presentations of the techniques can be found in textbooks, such as Ramsey (1956); Kopfermann (1958); Foot (2005); Budker, Kimball, and DeMille (2008); Inguscio and Fallani (2013).
Coherent-control spectroscopy (CCS). This saturated absorption spectroscopy is based on copropagating pump and probe acting on a three-level V system. Similar to saturated absorption spectroscopy in a two-level system and to the electromagnetic-induced transparency in a $\Lambda$ level scheme, a pump laser originates an atomic coherence in a branch of the V scheme. The absorption profile of a probe laser is modified. The frequency difference between pump and probe lasers furnishes the excited state energy splitting, as in Das and Natarajan (2005)
Delayed detection (DD). The natural linewidth of a spectroscopic resonance is reduced by monitoring the atom evolution for times longer than the spontaneous emission lifetime. This refinement was combined with other techniques, such as as laser-induced fluorescence by Shimizu et al. (1987) or hyperfine quantum-beats by Deech et al. (1977); Krist et al. (1977); Yei, Sieradzan, and Havey (1993).
Double-resonance optical pumping (DROP). A double-resonance optical excitation on a ladder three-level scheme is applied. An increased signal-to-noise ratio is obtained by detecting the population of the ground state rather than the excited state one. In the copropagating geometry, one laser excites zero velocity atoms on the lower transition and a second laser is scanned in frequency, as in Moon, Lee, and Suh (2009). In the counterpropaganting geometry, it is often combined with electomagnetic-induced transparency.
Electromagnetic induced transparency (EIT). This coupling/probe spectroscopy is based on the very narrow coherent feature produced in the absorption spectrum of three-level $\Lambda$ or ladder systems. For the $\Lambda$ scheme a very narrow linewidth is determined by the long relaxation rate of the ground state coherence. Counterpropagating lasers and one laser locked to an atomic transition are used to produce sub-Doppler resolution, as presented in Krishna et al. (2005); Wang et al. (2013, 2014a) .
Frequency comb spectroscopy (FC), The atomic absorption peaks are determined with a reference to a frequency comb. The absolute precision of the frequency readings is largely increased, see for instance Udem et al. (1999); Das et al. (2006a).
Frequency modulated laser (FML). When the exciting laser is modulated at the hyperfine splitting frequency, cross-over resonances are induced in three-level $\Lambda$ or V systems, as in Noble et al. (2006). The advantage of this technique is that only a single laser is required.
Hyperfine optical pumping and focus (HOPF). Hyperfine transitions induced by microwaves or by optical pumping change the relative populations of the hyperfine levels of the ground state. In HOPF this modification is detected by measuring the atomic beam intensity at the exit of a magnet that focuses or defocuses atoms with different magnetic quantum numbers. The focused atoms are analyzed by a mass spectrometer, as in Liberman et al. (1980).
Hyperfine quantum beats (HQB). Quantum beats are based on coherent pulsed excitation of excited hyperfine levels producing a time decay of the excited state populations modulated by the hyperfine frequency splitting. Polarized excitation and detection are required, as in Deech et al. (1977). Bellini, Bartoli, and Hänsch (1997) applied delayed pulses of a frequency comb in order to probe the coherent hyperfine superposition of excited states.
Ion detection (ION). This detection technique is very sensitive because a single ion can be detected. It is applied within different schemes, such as the resonant laser ionization (RIS), the selective electric field ionization of a Rydberg state or the thermoionic diode operation.
Laser-induced fluorescence spectroscopy (LIF). The emitted fluorescence is monitored as a function of the laser frequency. In Doppler spectroscopy, a high resolution is achieved by a careful analysis of the absorption lineshapes, as in Truong et al. (2015). In an AB with the laser propagation perpendicular to the atomic motion, the resolution is limited by the natural linewidth. By applying a sudden change to the laser phase and monitoring the atomic evolution at a later time $T$, subnatural width resolution reaching $\approx 1/(2T)$ is achieved in Shimizu et al. (1987). For short-lived atoms with low density using a fast beam and a collinear laser propagation, LIF is combined with nuclear decay to increase the signal to noise ratio as in Duong et al. (1987); Lynch et al. (2016).
Level crossing by magnetic or electric fields. An energy crossings of excited state levels vs an external parameter, either magnetic field (MLC) or electric field (ELC), is monitored, see for instance Nagourney, Happer, and Lurio (1978); Auzinsh et al. (2007). A precise determination of the applied field is required.
Maser (MA). The emission frequency of a maser operating on a hyperfine ground state transition is measured in Tetu, Fortin, and Savard (1976).
Magnetic field decoupling (MFD). Starting from an initial anisotropic Zeeman sublevel population distribution, the hyperfine constants are derived from the polarization of the fluorescent emission monitored vs an applied magnetic field decoupling the nuclear and electronic angular momenta, as presented in van Wijngaarden and Sagle (1991b).
Microwave spectroscopy (MWS). The population modifications induced by transitions between hyperfine levels, mainly in the microwaves, are detected. In order to increase the signal-to-noise ratio, MWS is combined to other techniques, such as HOPF, LIF, or selective electric field ionisation for Rydberg states as in Goy et al. (1982). The FOUNT+MWS combination applied to optical clocks leads to an extremely high precision, as in Guéna et al. (2014); Ovchinnikov, Szymaniec, and Edris (2015). A Ramsey optical interferometer is used for the potassium ground state measurement of  Arias et al. (2019); Peper et al. (2019).
Optical radiofrequency or microwave double resonance (ORFDR). The radio-frequency induced transitions between excited states are detected through the modification of the LIF, either in its spectrum or in its polarization, as in Farley, Tsekeris, and Gupta (1977); Lam, Gupta, and Happer (1980). Optical pumping is applied to modify the population distribution and increase the detected signal.
Optical-optical double-resonance (OODR). A two-colour excitation via an intermediate step produces the population of excited states. The sub-Doppler resolution is obtained by operating in a MOT in Fort et al. (1995a), by applying the lasers in a counterpropagating geometry in Stalnaker et al. (2010), by a counterpropagating laser geometry selecting a single class of velocities different from zero in Lee and Moon (2015), or by using saturated absorption to lock on the first transition and excite only the atoms at zero velocity in Yang et al. (2016). Detection is based mainly on the spontaneous emission from the intermediate or final state. In presence of an optical pumping process, the population distribution perturbed by the second step excitation is monitored, as in Wang et al. (2014b).
Optical spectroscopy (OS). Doppler limited high optical resolution spectroscopy from an alkali cell as in Truong et al. (2015), or Doppler-free in a MOT as in Antoni-Micollier et al. (2017); Arias et al. (2019).
Resonant ionization spectroscopy (RIS). Atoms in specific states are ionized by multistep laser absorption, and the ions are detected. It was introduced by Andreev, Letokhov, and Mishin (1987) for measuring of the ground state hyperfine structure in francium. The resonant tuning of the intermediate step provides the spectroscopic resolution of its hyperfine structure. High resolution and sensitivity due to ion detection, are associated with this technique as in the MOT experiment by Gabbanini et al. (1999) or for exotic isotopes in an atomic beam of exotic isotopes using the collinear laser spectroscopy as in Lynch et al. (2014). These last authors directed the ion to an alpha-decay detection station for clear identification in order to reduce isobaric and ground state contamination in their francium isotopes studies. In order to increase the frequency resolution, in recent francium accelerated atomic beam experiments Neugart et al. (2017) the excitation laser is split into two beams, co-propagating and counter-propagating with the atoms in order to increase the frequency resolution.
Saturated absorption spectroscopy (SAS). This technique is based on a pump and probe laser applied to the same transition. It produces spectra with a natural linewidth resolution. The counter-propagating geometry compensates for the Doppler broadening. Main limitations are imposed by the laser stability, as analyzed in Das and Natarajan (2008); Glaser et al. (2020) .
Stark spectroscopy (SS). Stark spectroscopy is based on the electric field shift of atomic level energies. It is used mainly for Rydberg states as in Stevens et al. (1995). Information on lower energy states may be derived by the difference in level Stark shifts.
Two-photon sub-Doppler spectroscopy (TPSDS). A single-colour two photon not-resonant excitation explores highly excited states. The sub-Doppler resolution is obtained by operating with counterpropagating beams in a VC as described by Herrmann et al. (1985); Hagel et al. (1999) or in a MOT as in Georgiades, Polzik, and Kimble (1994).

Data for this atom are reported in Table LABEL:Table:Li. As for all alkalis, several spectroscopic investigations are stimulated by the interest in laser cooling, but only a few experiments are performed in a MOT. The recent ⁶Li MOT experiments by Wu et al. (2018); Li et al. (2020); Rui et al. (2021) at Shanghai may open a new trend. For the ground state of both lithium isotopes, the "old” atomic beam measurements based on MWS report the highest precision, not reached by the several ones based on laser spectroscopy. Otto et al. (2002) performed measurements for a few high-$n$ states of ⁷Li, as well as for other alkalis.
For the ^6,7Li 2${}^{2}P_{1/2,3/2}$ state measurements by Umfer, Windholz, and Musso (1992), the error bar is derived on the basis of Eqs. (9) from the off-diagonal constants presented in the following. For the $A$ value of the 3${}^{2}S_{1/2}$ state of both isotopes measured by Lien et al. (2011), we estimated the error bar not reported by the authors.
A wide experimental effort concentrates on the $2^{2}P_{1/2}$ state of both isotopes. Instead, the $2^{2}P_{3/2}$ state of ⁶Li remains with the data obtained before laser spectroscopy. For ⁷Li, three experimental investigations of this doublet, by Orth, Ackermann, and Otten (1975),  Nagourney, Happer, and Lurio (1978) and  Umfer, Windholz, and Musso (1992), consider the contribution of the off-diagonal hyperfine couplings linked to the small fine-structure splitting. Orth, Ackermann, and Otten (1975) include in their analysis the previous level crossing data by Brog, Eck, and Wieder (1967). The analysis by  Nagourney, Happer, and Lurio (1978) combines their own data and the previous ones.  Umfer, Windholz, and Musso (1992) acquired enough data for their own analysis.  Orth, Ackermann, and Otten (1975) using Eqs. (9) and their measured $A_{3/2,1/2}=11.85(35)$ MHz found $a_{c}=-9.838(48)$, $a_{d}=-1.876(12)$, $a_{o}=8.659(37)$.  Nagourney, Happer, and Lurio (1978) reported $a_{c}=-9.838(48)$, $a_{d}=-1.975(22)$, $a_{o}=8.659(37)$, and  Umfer, Windholz, and Musso (1992) obtained $a_{c}=-9.93(37)$, $a_{d}=-1.72(20)$, $a_{o}=8.69(31)$ (all in MHz). For the same doublet, Beloy and Derevianko (2008) performed an accurate evaluation of the off-diagonal hyperfine couplings. They produced corrections to the measured dipole constant in the $A(^{2}P_{1/2})$ accurately measured by  Orth, Ackermann, and Otten (1975),  Walls et al. (2003), and  Das and Natarajan (2008). For this last one, the 27.0 kHz correction should be compared to the author’s original 3 kHz experimental uncertainty.
A disagreement exceeding the error bars exists for the 3${}^{2}S_{1/2}$ ⁷Li hyperfine constant $A$ measured in Stevens et al. (1995), compared to those of Bushaw et al. (2003); Lien et al. (2011). We do not include that measurement in the w.a.
The discrepancy for the $3^{2}P_{3/2}$ ⁷Li constants between the previous values byBudick et al. (1966); Isler, Marcus, and Novick (1969) and the  Nagourney, Happer, and Lurio (1978) ones remains unexplained as pointed out by these authors. Therefore, instead of the recommended value from Arimondo, Inguscio, and Violino (1977), we consider all the previous measurements. The $\chi^{2}_{red}$ correction is applied to the statistical error. An analysis of the off-diagonal elements was performed by Nagourney, Happer, and Lurio (1978) for the $3^{2}P$ ⁷Li doublet, leading to $a_{c}=-3.10(67)$, $a_{d}=-0.54(27)$, $a_{o}=2.61(40)$.

The sodium results of Table LABEL:Table:Na are emblematic of the progress achieved in hyperfine constant measurements. In chronological order, the two-photon sub-Doppler spectroscopy was applied in 1978 to probe the $4^{2}D$ excited states, by Biraben and Beroff (1978) and  Burghardt et al. (1978), extended by this last research group to the $3^{2}D$ states in Burghardt, Hoffmann, and Meisel (1988). In 1989,  Kasevich et al. (1989) published the first hyperfine ground state determination in an atomic fountain with 1 mHz precision, close to the previous best atomic beam value of Table LABEL:Table:Na, opening the road to further amazing improvements in fountain atomic clocks.  Zhu, Oates, and Hall (1993) performed the first hyperfine constant measurement in a MOT exploring the $5^{2}P$ states with a relative precision $\approx 1\times 10^{-4}$ among the best ones for excited states. In the same year,  Yei, Sieradzan, and Havey (1993) performed subnatural linewidth measurements of hyperfine coupling constants using the delayed detection in polarization quantum beat spectroscopy. In 2003, the excitation of ultracold atoms in a MOT on the $3^{2}P-4^{2}P$ electric quadrupole transition allowed  Bhattacharya, Haimberger, and Bigelow (2003) to perform high resolution spectroscopy of the 4${}^{2}P_{1/2}$ level.  Das and Natarajan (2006b) introduced the CCS approach for the first excited state, 3${}^{2}P_{3/2}$. However, no competitive new data for the ground state are available.
The 3${}^{2}P_{1/2,3/2}$ states have been examined by several authors, with an increasing precision and a good agreement among the different results. For the 4${}^{2}D$ states of Biraben and Beroff (1978) and for the 7, 8 and 9${}^{2}P_{3/2}$ ones of Jiang, Jönsson, and Lundberg (1982), the sign has been determined here from the scaling law. In sodium, there is not enough data for a global analysis. For the large majority of excited states above the $5^{2}P$ ones, no new data have been published after 1982.

Our previous Li and Na remarks on the ground state values do not apply to the ³⁹K ground state, as shown in Table  LABEL:Table:K. For this atom, several recent measurements exist, with the frequency comb spectral resolution applied to determine the absolute transition frequencies, and from them, the fine and hyperfine splittings. A high precision is achieved, usually with a good agreement among data from different research groups, e. g. for several $n=5-8$ ${}^{2}S_{1/2}$ states with measurements performed over a long time span.  Otto et al. (2002) explored the ${}^{2}S$ states of the 39 and 41 isotopes up to $n=14$ using two-photon sub-Doppler spectroscopy.
The spatial dependence of the HQB polarized fluorescence intensity in a given magnetic field was used by Głódź and Kraińska-Miszczak (1985a) to derive for the first time the signs of the $A$ constants in the $6^{2}D$ states of ³⁹K. For other ${}^{2}D$ states, the investigations by Belin et al. (1975b),  Sieradzan et al. (1997)and  Głódź and Kraińska-Miszczak (1985b) produced only the absolute sign of the $A$ and $B$ constants. Their signs are determined here on the basis of the scaling laws.