Absolute frequency measurement of rubidium 5S-6P transitions

TL;DR

This paper presents highly precise absolute frequency measurements of rubidium 5S-6P transitions using saturation absorption spectroscopy, improving accuracy and verifying literature values for isotope shifts and hyperfine constants.

Contribution

It provides the first high-precision absolute frequency measurements of rubidium 5S-6P transitions with uncertainties below 500 kHz.

Findings

Achieved measurement uncertainties of 450 kHz and 20 kHz for the two transitions.

Verified literature values for isotope shifts and hyperfine constants.

Derived hyperfine splitting with high accuracy.

Abstract

We report on measurements on the 5S-6P rubidium transition frequencies for rubidium isotopes with an absolute uncertainty of better than \SI{450}{kHz} for the 5S $\rightarrow$ 6P$_{1/2}$ transition and \SI{20}{kHz} for the 5S $\rightarrow$ 6P$_{3/2}$ transition, achieved by saturation absorption spectroscopy. From the results we derive the hyperfine splitting with an accuracy of \SI{460}{kHz} and \SI{30}{kHz}, respectively. We also verify the literature values for the isotope shifts as well as magnetic dipole constant and the electric quadrupole constant.

Figures

Tables

Table 1. Table 1: Measured absolute transition frequencies between the 5S 1/2 and 6P 3/2 states of 85 Rb and 87 Rb.

	Transition	Frequency [THz]
87Rb	F=2 $\to$ F’=1	713.281601641(16)
	F=2 $\to$ F’=1/2	713.281627545(16)
	F=2 $\to$ F’=2	713.281653455(16)
	F=2 $\to$ F’=1/3	713.281671206(16)
	F=2 $\to$ F’=2/3	713.281696845(16)
	F=2 $\to$ F’=3	713.281740464(16)
85Rb	F=3 $\to$ F’=2	713.282822529(16)
	F=3 $\to$ F’=2/3	713.282832859(16)
	F=3 $\to$ F’=3	713.282843436(16)
	F=3 $\to$ F’=2/4	713.282852661(16)
	F=3 $\to$ F’=3/4	713.282863062(16)
	F=3 $\to$ F’=4	713.282882547(16)
85Rb	F=2 $\to$ F’=1	713.285854135(18)
	F=2 $\to$ F’=1/2	713.285859657(18)
	F=2 $\to$ F’=2	713.285864366(18)
	F=2 $\to$ F’=1/3	713.285869902(18)
	F=2 $\to$ F’=2/3	713.285874803(18)
	F=2 $\to$ F’=3	713.285885399(18)
87Rb	F=1 $\to$ F’=0	713.288419955(20)
	F=1 $\to$ F’=0/1	713.288430802(20)
	F=1 $\to$ F’=1	713.288442281(20)
	F=1 $\to$ F’=0/2	713.288456485(20)
	F=1 $\to$ F’=1/2	713.288468381(20)
	F=1 $\to$ F’=2	713.288493970(20)

Table 2. Table 2: Measured absolute transition frequencies between the 5S 1/2 and 6P 1/2 states of 85 Rb and 87 Rb.

	Transition	Frequency [THz]
87Rb	F=2 $\to$ F’=1	710.95797053(45)
	F=2 $\to$ F’=1/2	710.95810465(45)
	F=2 $\to$ F’=2	710.95823892(45)
85Rb	F=3 $\to$ F’=2	710.95924220(45)
	F=3 $\to$ F’=2/3	710.95930147(45)
	F=3 $\to$ F’=3	710.95936053(45)
85Rb	F=2 $\to$ F’=2	710.96227673(45)
	F=2 $\to$ F’=2/3	710.96233612(45)
	F=2 $\to$ F’=3	710.96239520(45)
87Rb	F=1 $\to$ F’=1	710.96480730(45)
	F=1 $\to$ F’=1/2	710.96494083(45)
	F=1 $\to$ F’=2	710.96507276(45)

Table 3. Table 3: Hyperfine splitting. The literature values were given without errorbars.

Transition 85Rb	Frequency [MHz] Grundevik et al. (1977)	Frequency [MHz]
6P1/2 F=2 - F=3	117	118.40(46)
6P1/2	68.25	69.07(46)
6P3/2 F=1 - F=2	9	10.048(25)
6P3/2 F=2 - F=3	21	20.967(25)
6P3/2 F=3 - F=4	39	39.127(25)
6P3/2	32.5	32.703(25)
Transition 87Rb	Frequency [MHz] Grundevik et al. (1977)	Frequency [MHz]
6P1/2 F=1 - F=2	265	265.15(46)
6P1/2	165.625	165.72(46)
6P3/2 F=0 - F=1	24	23.739(26)
6P3/2 F=1 - F=2	52	51.654(26)
6P3/2 F=2 - F=3	87	87.009(26)
6P3/2	63.331	63.340(26)

Table 4. Table 4: Calculated values for the magnetic dipole constant A and the electric quadrupole constant B.

Transition	Constant	literature [MHz]	this work [MHz]
85Rb 5S $\to$ 6P3/2	A	8.179(12) Arimondo et al. (1977)	8.174(119)
85Rb 5S $\to$ 6P3/2	B	8.190(49) Arimondo et al. (1977)	8.113(119)
87Rb 5S $\to$ 6P3/2	A	27.700(17) Safronova and Safronova (2011)	27.716(108)
87Rb 5S $\to$ 6P3/2	B	3.953(24) Safronova and Safronova (2011)	3.983(108)
85Rb 5S $\to$ 6P1/2	A	39.11(3) Arimondo et al. (1977)	39.47(50)
87Rb 5S $\to$ 6P1/2	A	132.56(3) Safronova and Safronova (2011)	132.83(50)

Full text

Introduction

The advent of optical frequency combs has revolutionized the world of high precision spectroscopy and has enabled measurements of atomic transition frequencies with exceptional accuracy Udem et al. (2002, 2009); Cundiff and Ye (2003). Precise knowledge of these frequencies facilitates better calculations of atomic models and the derivation of more accurate values of physical quantities such as the Lamb shift Huang et al. (2018), hyperfine structure constants Lee and Moon (2015), magnetic dipole or electric quadrupole constant. Furthermore, precise knowledge of these values is desirable to experimentally investigate novel ways of manipulating atomic states, such as the coherent excitation of Rydberg atoms for quantum information processing Saffman et al. (2010); Levine et al. (2018), generation of atomic memories Patton and Fischer (2013) or implementing novel quantum gates Sárkány et al. (2015).

For quantum information purposes rubidium Rydberg atoms are a widely used species. A common way to excite Rubidium atoms from the 5S ground state to Rydberg states with high principal quantum numbers n is via a three level ladder scheme 5S$\rightarrow$5P$\rightarrow$nS or nD using a pair of lasers with wavelengths 780nm and 480nm.This scheme, however, commonly relies on the generation of $480\text{\,}\mathrm{nm}$ light by frequency-doubling a $960\text{\,}\mathrm{nm}$ laser, which limits the available laser power Mack et al. (2011).

A promising alternative is the excitation using the 6P state as intermediate state Simonelli et al. (2017); Gutiérrez et al. (2017). Due to the five times larger lifetime $\tau$ = $121\text{\,}\mathrm{ns}$ compared to the 5P state Gomez et al. (2004), dephasing during the excitation is reduced and the coherence time of this transition is expected to be larger Sárkány et al. (2015, 2018). The commercial availability of $420\text{\,}\mathrm{nm}$ ECDL lasers renders the Rydberg excitation (ladder) scheme via the 6P intermediate state a viable alternative to the commonly used excitation state via the 5P state. Additionally, the light driving the 6P $\rightarrow$ nS transition at $1016\text{\,}\mathrm{nm}$ can easily be generated with high power using external cavity diode lasers (ECDL) and allows high Rabi frequencies in the excitation to Rydberg states.

While the transition frequencies for the excitation scheme 5S $\rightarrow$ 5P $\rightarrow$ nS, nD are known to $6\text{\,}\mathrm{kHz}$ accuracy Steck (2001) , there has been so far no absolute data available for the transition 5S $\rightarrow$ 6P $\rightarrow$ nS, nD, which is however necessary for the implementation of quantum information protocols using this excitation path. Knowing the 6P transition frequencies and the fine and hyperfine structure sub-levels with high accuracy, the transition frequencies from the 6P intermediate state to Rydberg states can then be calculated using the quantum defect theory Mack et al. (2011); Li et al. (2003); Han et al. (2006); Afrousheh et al. (2006). Measurements of the hyperfine splitting of the 85Rb and 87Rb 6P levels have been performed before Arimondo et al. (1977); Bize et al. (1999), but the most accurate value for the absolute frequencies in literature have uncertainties of $850\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ ($0.0005\text{\,}\mathrm{nm}$) Kramida et al. (2018). Here, we report on the measurements of the absolute frequencies for this transitions and on measurement schemes to determine the relative frequencies. We have measured the absolute frequencies of the 5S $\rightarrow$ 6P3/2 resonance with an uncertainty of $\leq 20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and better than $450\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ for the 5S $\rightarrow$ 6P1/2 transition, improving the literature values by five and four orders of magnitude, respectively. In addition to verifying the literature values of the isotope shifts and the magnetic dipole and electric quadrupol constants, we have also determined the hyperfine splittings, depicted in Fig. 1, from the measured transition frequencies, which brings their accuracies to the same respective orders of magnitude, improving the literature values by three and two orders of magnitudes.

Experimental Setup

Our experimental setup is a saturation spectroscopy as shown in Fig. 2. The laser source is an ECDL with a linewidth of $<400\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ as measured by the beating signal between the ECDL laser and the frequency comb. The frequency comb is phase-locked to a solid state $1550\text{\,}\mathrm{nm}$ laser (I15, NKT) with a linewidth of $<$100\text{,}\mathrm{H}\mathrm{z}. The linewidth of the comb is $<2\text{,}\mathrm{k}\mathrm{H}\mathrm{z}$, which was measured by beating it to a second$1550\text{,}\mathrm{nm}$ laser.

The pump and probe beams counter-propagate through the heated rubidium vapor cell ($321\text{\,}\mathrm{K}$). Polarizing beam splitters ensure crossed linear polarizations to avoid interference effects within the cell. A double-layer magnetic $\mu$-metal shielding leads to a reduction of magnetic fields to less than $0.03\text{\,}\mathrm{G}$, which has been verified with a Gaussmeter (GM07 Gaussmeter, Hirst Magnetic Instruments LTD). The power ratio between probe and pump beams was adjusted in order to optimize the signal-to-noise ratio of the Lamb dips.The optimal ratio was found to be near (10:1), using a probe power of $0.488\text{\,}\mathrm{m}\mathrm{W}$ ($14.9\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$) and a pump beam power of $56\text{\,}\mathrm{\SIUnitSymbolMicro W}$ ($1.2\text{\,}\mathrm{m}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$). The intensity and the frequency of the pump beam were modulated with a frequency of $50\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ using an $80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ acousto-optical modulator (AOM). The probe beam signal from the photodiode was subsequently demodulated at the same frequency using a lock-in Amplifier (HF2LI, Zurich Instruments), which results in a fully Doppler free spectroscopy signal Ye et al. (1996), as depicted exemplarily in Fig. 3 (blue line) for the 87Rb 5S${}_{1/2}(F=1)\rightarrow$ 6P${}_{1/2}(F^{\prime}=1)$ transitions. Since the AOM shifts the pump beam by $80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ the measured Lamb dips and cross-overs are red-shifted by $-40\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. This offset has been corrected in the final data analysis.

The relative frequencies between the 5S $\rightarrow$ 6P1/2 transitions were measured by scanning the laser with a wavemeter and simultaneously recording the wavelength of the laser.

Additionally, we can impose $3\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ sidebands by frequency modulation (FM) of the laser diode current. After appropriate demodulation, this leads to a Pound-Drever-Hall-like (PDH) error signal, which can be used to stabilize the laser onto the transition resonance frequency Drever et al. (1983). Fig. 3 shows an exemplary spectrum acquired by the Lock-in amplifier (blue) and the FM error signals (red) for the 87Rb 5S${}_{1/2}(F=1)\rightarrow$ 6P1/2 transitions.

We can control the frequency of the laser in three ways: The first is to stabilize the laser frequency with the FM error signal to one of the transitions using a digital laser locking module (DigiLock, TOPTICA). Alternatively, the laser frequency can be stabilized and scanned by the wavelength meter, which is calibrated with a laser that is frequency stabilized to the 5S1/2 (F=2) $\rightarrow$ 5P3/2 (F’=3) transition of 87Rb ($780.246\,291\text{\,}\mathrm{nm}$) via an FM spectroscopy. Those methods were used to measure the absolute and the relative frequencies for both transitions. For the third method the beam of the ECDL and the modes of a narrow linewidth frequency comb (TOPTICA) were superimposed and frequency-filtered by a grating in a beat detection unit (DFC BC and DFC MD, TOPTICA) and monitored on a photodetector with a bandwidth of $50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. The resulting beating signal was acquired with a digital oscilloscope (Picoscope 5442A, Pico Technology). Using the beating signal the $420\text{\,}\mathrm{nm}$ laser was phase-locked to the frequency comb, using a phase frequency detector (PFD, Toptica) with a filter, that is tunable between 2 and $38\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$.

A typical phase-locked beating signal with a width of $\approx 1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ can be seen in Fig. 4. The frequency comb is offset-free and its repetition rate $f_{\text{rep}}$=$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ is locked to a GPS-based $10\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ frequency reference Puppe et al. (2016) with an accuracy of $10^{-10}$. The comb is phase-locked to the $1500\text{\,}\mathrm{nm}$ laser using a phase frequency detector, which results in comb modes with a linewidth of $<$5\text{,}\mathrm{kHz}$$, measured by locking the comb to another frequency comb (FC 1500, Menlo). This method to measure the frequencies was only applicable to the $5S\rightarrow$ 6P3/2 transition, since the comb doesn’t support the corresponding wavelength to the $5S\rightarrow$ 6P1/2 transition.

Fig. 5 shows a typical saturation spectrum for the 5S $\rightarrow$ 6P3/2 transition.

Measurement of the 5S $\rightarrow$ 6P3/2 transitions

For the measurement of the absolute transition frequencies the ECDL laser was beated with the frequency comb. The laser was locked to the FM error signal of each transition. Simultaneously, the wavemeter recorded the frequency and a digital oscilloscope saved the beat frequency. Knowing these values and the repetition rate of the comb, we were not only able to calculate the absolute frequencies, but also to determine the related comb mode. In order to reduce the statistical error, each measurement was repeated 60 times and subsequently averaged. To characterize the locking accuracy, we locked the $420\text{\,}\mathrm{nm}$ laser to an arbitrary frequency for $1\text{\,}\mathrm{h}$ and recorded the beating signal between laser and frequency comb every $15\text{\,}\mathrm{s}$. We found that the lock frequency of the laser deviates less than $11.6\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{h}$ from its mean value.

The relative frequencies were determined by locking the laser to the frequency comb at a given locking point using the PFD. Starting at the previously measured frequencies and scanning this locking point between 2 and $38\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ for the corresponding comb mode, the relative frequencies of the 5S${}_{1/2}\rightarrow$ 6P3/2 transitions can be determined with an accuracy of $<1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ resulting from the linewidth of the ECDL laser locked to the frequency comb. Figure 6 shows the resulting spectra from these scans, which have been calibrated using the absolute frequencies from the wavemeter measurement. The gaps in the measured spectra are caused by the locking scheme with the PFD, which can only be tuned between 2 and $38\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ for each comb mode, since there has to be a frequency difference to the next comb mode. This results in $4\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ gaps whenever the comb mode has to be changed. All spectra are fitted using a superposition of Pseudo-Voigt profiles, yielding a linewidth of $\approx 2.7\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ (FWHM). The errors arising from the fit routine was calculated separately for each spectrum and were found to be smaller than $20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

The uncertainty in the data for the absolute frequency comprises several known error sources of physical and technical nature. The laser linewidth of the phase-locked laser ($1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$), technical noise and physical deviations cause the fitting error to contribute to the uncertainty of the measured frequency. The overall uncertainty resulting from these effects was determined independently for each measured transition and was found to be $\leq 20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. The absolute transition frequencies are summarized in Tab. 1.

We have also performed measurements of the absolute transition spectra using the wavemeter to lock the laser. The experimental sequence was as follows: First, the wavelength meter was calibrated as described above. Second, the laser frequency was swept linearly at a rate of $50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{s}$ over a range of $250\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ while the signal of the lock-in amplifier was recorded. In order to reduce the statistical error, each trace has been measured 100 times and then averaged. The deviations between this measurement and the one including the frequency comb was found to be smaller than $500\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$.

Measurement of the 5S $\rightarrow$ 6P1/2 transitions

Since the output of the frequency comb near $420\text{\,}\mathrm{nm}$ is limited to a range between $418.8\text{\,}\mathrm{nm}$ and $420.3\text{\,}\mathrm{nm}$ it was not possible to obtain beating signals for the 5S${}_{1/2}\,\rightarrow$ 6P1/2 transition at $421\text{\,}\mathrm{nm}$. Therefore, the frequency measurements for this manifold are based on the DigiLock module and wavemeter scans. As determined for the 5S${}_{1/2}\,\rightarrow$ 6P3/2 transition, these methods show deviations of about $500\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ from each other, so we deem this to be the maximum error. Similarly, as described above, the frequencies were determined by locking the laser to the error signal using the DigiLock. Then the laser was locked to a calibrated wavemeter and swept at a rate of $50\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}\mathrm{/}\mathrm{s}$ over a range of $500\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ to measure the spectra. Again, each trace was measured 100 times and subsequently averaged, to reduce statistical errors. The resulting spectra are depicted in Fig. 7. We typically measured linewidths (FWHM) of $\approx 2.8\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, roughly twice as large as expected from the natural linewidth of $1.135\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ Safronova and Safronova (2011). The positions of the peaks were evaluated by fitting superpositions of Pseudo-Voigt profiles to the spectra. The error due to the fitting routine is on the order of a few kHz and can be neglected compared to the deviations due to the linewidth of the laser. To estimate the error caused by the wavemter we have characterized its locking accuracy. Therefore we have locked the $420\text{\,}\mathrm{nm}$ laser to an arbitrary frequency within the output range of the frequency comb for $1\text{\,}\mathrm{h}$ and recorded the beat signal between laser and a frequency comb mode every $15\text{\,}\mathrm{s}$. We found the frequency of the calibrated wavemeter to be normally distributed with a standard deviation of $160\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. The total error of the frequencies was calculated to be $<$450 kHz, combining the laser linewidth of $<400\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and the relative error of the callibrated wavemeter of $160$ kHz. The results for the transition frequencies are summarized in Tab. 2.

Discussion

Since pressure broadening is negligible ($<1\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$) and the Zeeman shift caused by the magnetic field is below $10\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, the broadening of the natural linewidth discrepancy is assumed to be mostly the consequence of the transversal Doppler effect (1 MHz/0.12∘) and power broadening.

With the measured data we calculated the hyperfine splitting for both transitions. The errors are calculated via propagation of uncertainty based on the transition frequency errors. In Tab. 3 a comparison between the literature values and the newly determined values is shown.

The total isotope shifts were found to be $41.935(60)\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ for the 6P3/2 isotopes and $41.237(62)\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ for the 6P1/2 isotopes, which agree well with the literature values Grundevik et al. (1977); Aldridge et al. (2011).

With the determined hyperfine splitting values, the magnetic dipole constant A and the electric quadrupole constant B can be calculated for the 5S $\rightarrow$ 6P transitions. The values for each 85Rb und 87Rb are listed in Tab. 4 and are also in good agreement with the literature data.

Conclusion

In summary, we have performed high precision saturated absorption spectroscopy of 85Rb and 87Rb using a diode laser. The laser was stabilized and scanned by a wavelength meter for the 6P1/2 transition and locked to a narrow linewidth frequency comb for the 6P3/2 transition. This allows for absolute frequency measurements with an uncertainty of $<20\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ for the 6P3/2 transition and $<450\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ for the 6P1/2 transition. The lower uncertainty in the measurement of the 5S $\rightarrow$ 6P3/2 transition results from the laser locking to a narrow line frequency comb, while for the measurement of the 5S$\rightarrow$ 6P1/2 transition it is limited by the stability of the wavelength meter. From the measured data we derive hyperfine splitting values with unprecedented accuracy and verified the literature values for the isotope shifts, the magnetic dipole constant and the electric quadrupole constant.

Acknowledgements

This work has been supported by the DFG (SSP 1929 GiRyd and CIT) and BMBF (FKZ: 13N14903). C.G. would like to thank the Evangelische Studienstiftung Villigst e.V.