3.1 Characteristic of traditional OODR spectroscopy
Fig. 3 shows traditional OODR spectra when the frequency of 852 nm pump laser are locked one by one to the three resonance peaks and two crossover (CO) peaks of the SAS between 6S1/2 (F=4) → 6P3/2 transition, and the 917 nm probe laser is scanned over the whole 6P3/2 - 6D5/2 transition. Horizontal x axis is calibrated by the known frequency interval values of 6D5/2 state provided by Ref [11], and its frequency detuning is relative to the 6P3/2 (F’=5) → 6D5/2 (F”=6) resonance in Fig. 3. Under the condition of two-photon resonance considering the Doppler effect ( Δpump + v/c*υpump + Δprobe - v/c*υprobe = 0, Δpump and Δprobe are frequency detuning of pump laser and probe laser, υpump and υprobe are frequency of pump laser and probe laser, v is the speed of atoms in the direction of the laser beams, c is speed of light ), these spectral signals from left to right correspond in turn to the hyperfine transitions between the energy levels F’-F” = 5-6, 5-5, 4-5, 5-4, 4-4, 3-4, 4-3, 3-3, 3-2 as shown in Fig. 3. It is clear that the frequency intervals of OODR spectra are insensitive to frequency variations of the 852 nm pump laser: when pump laser resonates at different frequency positions between the 6S1/2F=4-6P3/2 transition, it only results in a shift of the OODR spectrum as a whole and a change in the relative amplitude of these signals, while maintaining the same frequency intervals. For example, when the pump laser is resonant on the F-F’ = 4-5 transition, more atoms with zero velocity in direction of pump laser beam are populated in the F’ = 5 hyperfine state, so the signals for the F’-F” = 5-6, 5-5, 5-4 transitions are relatively bigger. Similarly, while the pump laser is tuned to the side of the F-F’ = 4-3 transition, the signals for the F’-F” = 3-4, 3-3, 3-2 will gradually grow larger. The above experimental results corroborate that the frequency intervals in the OODR spectrum are immune to frequency drift of the 852 nm pump laser, it is advantageous for precise measurement of the hyperfine structure of higher excited states 6D5/2. However, using the traditional OODR spectrum to measure the hyperfine structure of the excited state, some extra frequency calibration tools are required, such as EOM, AOM, frequency comb, etc. [11-20].
3.2 Dual-excited state spectroscopy and measurement of hyperfine structure of higher excited state
Different from the traditional OODR spectrum [14-16], here for the first time we adopt the completely inverse working mode of two lasers: the 852 nm pump laser is scanned over the 6S1/2 F=4 - 6P3/2 transition by a piezoelectric ceramic transducer (PZT) driving the grating of ECDL with the ~20 Hz scanning frequency, while the 917 nm probe laser is resonant on the different frequency position between the 6P3/2 - 6D5/2 transition, the OODR spectra with a completely flat background for characterizing hyperfine structure of the excited states 6D5/2 are also obtained in detector PD3 under the condition of two-photon resonance as shown in Fig. 4. Moreover, due to the frequency scanning of the 852 nm pump laser, the SAS for the 6S1/2 (F=4) → 6P3/2 transition without Doppler background is obtained as the difference signal between PD1 and PD2 detectors simultaneously, which can be as a frequency calibration tool with the known hyperfine splitting frequency intervals of the 6P3/2 excited state as shown in Fig. 1 [31], and its detuning is set to the 6S1/2(F=4) → 6P3/2( F’=5) resonance line in Fig. 4. When the frequency of 917 nm probe laser is tuned between the 6P3/2 - 6D5/2 transition, it only causes an overall shift of the OODR spectrum indicated by lines 1-4 from top to bottom, which is similar to the Fig. 3, and also shows that the frequency intervals of OODR spectra are insensitive to frequency drift of the 917 nm probe laser.
The higher excited state energy levels of an atom are, the smaller their hyperfine splittings are, and so it is key to fully distinguish hyperfine energy levels experimentally. In a ladder-type atomic system, the linewidth of excited state spectroscopy is usually more narrow when the pump and probe laser beams are counter-propagating through the atomic medium in a room-temperature vapor cell due to atomic coherence effect [23, 24]. Theoretically, the natural linewidth for two-color excited state spectrum under the condition of weak laser fields is given by [32]:
(2)
Where k1 and k2 are the wave vectors of pump laser and probe laser, respectively. Using the Γ1 = 5.2 MHz natural linewidth of the 6P3/2 state and the Γ2 = 3.1 MHz natural linewidth of the 6D5/2 state, Eq. (2) predicts the natural linewidth of excited state spectrum of ~3.67 MHz. In our experiment, all spectral lines obeyed dipole selection rule between the excited states 6P3/2 and 6D5/2 in OODR spectrum are clearly distinguishable as shown in Fig. 5, and the narrowest spectral linewidth is about ~4.0 MHz, which is much smaller than that for co-propagating pump and probe laser beams in a Cs vapor cell [14]. Thus, it is convenient that four frequency intervals of the hyperfine splitting for the 6D5/2 state can be simultaneously determined in a single measurement.
In order to minimize the influence of nonlinear frequency scanning of grating external cavity on measurement accuracy, the frequency scanning range of the 852.3 nm pump laser is reduced as much as possible, and only a complete set of OODR spectra between the excited states 6P3/2 - 6D5/2 transition can be obtained as shown in Fig. 5. When the frequency of 852.3 nm pump laser is scanned over the 6S1/2F=4 - 6P3/2 transition, the known 125.564 MHz frequency interval between the F=4 → F’=3,4 and F=4 → F’=3,5 crossover resonance peaks of the SAS is used to calibrate the horizontal axis of OODR spectrum in Fig. 5(a). Then we use multi-peak Lorentz curves to fit the OODR spectra to locate the exact central position of each signal, and calculate the frequency intervals between them. Finally, considering the influence of wavelength mismatch between the pump and probe laser in a ladder-type atomic system, these values of frequency intervals are multiplied by a k2/k1 factor to get the hyperfine intervals between the F”=6-2 of the 6D5/2 state. Due to the limitation of the transition selection rule ∆MF = ±1, in order to get the frequency interval between the F”=2 and F”=1 of the 6D5/2 state, the 852.3 nm laser is again scanned over the 6S1/2F=3 - 6P3/2 transition, we use the known 176.256 MHz frequency interval between the F=3 → F’=2,3 and F=3 → F’=3,4 crossover resonance peaks of the SAS to calibrate the horizontal axis of OODR spectrum in Fig. 5(b). Each hyperfine splitting interval of the 6D5/2 state is averaged over 30 - 40 individual spectra, and their histograms of the hyperfine splitting intervals with corresponding normal distribution curves are shown in Fig. 6. Table 1 shows good agreement between our measured hyperfine splitting intervals of the 6D5/2 state and that of previous literatures [11, 13, 14, 30].
Table 1 Measured values of hyperfine splitting of the excited state 6D5/2 for the 133Cs atoms.
Hyperfine intervals of the 6D5/2 state
|
Measured
(this work)
(MHz)
|
Ref. [11]
(MHz)
|
Ref.[13]a
(MHz)
|
Ref.[14]
(MHz)
|
Ref. [30]
(MHz)
|
F”=1 → F”= 2
|
9.15 (37)
|
8.97 (39)
|
9.23
|
9.4 (2)
|
-
|
F”=2 → F”=3
|
14.05 (56)
|
14.07 (36)
|
13.85
|
14.8 (2)
|
-
|
F”=3 → F”= 4
|
18.39 (38)
|
18.57 (21)
|
18.49
|
18.5 (2)
|
-
|
F”=4 → F”= 5
|
23.01 (59)
|
22.40 ( 8)
|
23.15
|
23.1 (2)
|
22.1 (7)
|
F”=5 → F”=6
|
27.35 (67)
|
27.93 (35)
|
27.82
|
27.5 (1)
|
29.1 (5)
|
a The values of hyperfine splitting have been calculated from the reported Ahfs and Bhfs values.
At present, the experimental error in measurement of frequency intervals mainly results from the nonlinear frequency scanning of our homemade 852.3 nm pump laser, a high quality PZT driving the grating of ECDL will further improve the measuring accuracy of hyperfine structure. Another error comes from light shift of atomic energy level being proportional to the laser intensity, which is confirmed by the fact that Autler-Townes splitting of hyperfine components is obviously observed in OODR spectrum when the intensity of 852.3 nm pump laser increases in experiment. By processing of the same experimental data repeatedly, we know that the error in determining the exact position of OODR spectral lines by multi-peak Lorentz fitting is relatively small (< ~0.1 MHz). In addition, the Zeeman shift due to the residual magnetic field is estimated to less than ~0.04 MHz, which is negligible, compared to the other uncertainties in our measurement.
3.3 Determination of magnetic dipole and electric quadrupole coupling constants
Using Eq. (1) and the above measured hyperfine splitting intervals of the 6D5/2 state (I = 7/2 and J = 5/2) as shown in Table 1, a set of coupled linear equations are obtained:
6Ahfs + 18/35 Bhfs = -27.35 (67) MHz
5Ahfs + 1/28 Bhfs = -23.01 (59) MHz
4Ahfs - 8/35 Bhfs = -18.39 (38) MHz
3Ahfs- 9/28 Bhfs = -14.05 (56) MHz
2Ahfs - 2/7 Bhfs = -9.15 (37) MHz (3)
Applying the method of least squares, we determine the magnetic dipole coupling constant Ahfs and electric quadrupole coupling constant Bhfs, and propagate the uncertainties through these formulas. The Ahfs agrees well with previous reports, while the consistency of Bhfs is poor in different literature due to its small influence on the hyperfine splitting of D state by indicated equations (3), as shown in Table 2.
Table 2 The magnetic dipole (Ahfs) and electric quadrupole (Bhfs) hyperfine coupling constants of the 6D5/2 state for the 133Cs atoms.
State
|
Reference
|
Method
|
Ahfs (MHz)
|
Bhfs (MHz)
|
133Cs
6D5/2
|
2022, This work
|
Dual-excited state spectroscopy (self-calibration)
|
-4.60 (5)
|
0.23 (47)
|
2021, Herd et al.[13]
|
Two-photon spectroscopy
+ frequency comb
|
-4.629 (14)
|
-0.10 (15)
|
2006, Kortyna et al.[14]
|
OODR + EOM
|
-4.66 (4)
|
0.9 (8)
|
2005, Ohtsuka et al.[11]
|
Two-photon spectroscopy
+ EOM
|
-4.56 (9)
|
-0.35 (18)
|
1994, Georgiades
et al.[29]
|
Two-photon spectroscopy
in cold atoms + AOM
|
-4.69 (4)
|
0.18 (73)
|
1975, Tai et al.[1]
|
Cascade fluorescence
spectroscopy + decoupling
|
-3.6 (10)
|
-
|