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Present status of the fine-structurefrequencies of the 23P helium level

G. Giusfredi, P. Cancio Pastor, P. De Natale, D. Mazzotti,C. de Mauro, L. Fallani, G. Hagel, V. Krachmalnicoff, andM. Inguscio

Abstract: A new measurement of the fine-structure frequencies of the 23P level in 4Heis presented. The result for the largest interval 23P0–23P1 is 29 616 952.7(1.0) kHz, and2 291 167.7(11.0) kHz for the smallest one, 23P1–23P2. Taking into account this new result,an agreement among different experiments at the 1 kHz level is found. Implications of thissituation for the determination of the fine-structure constant α are discussed.

PACS Nos.: 31.15.Pf, 31.30.Jv, and 32.10.Hq

Résumé : Nous présentons une nouvelle mesure des fréquences de structure fine du niveau23P dans 4He. Le résultat est 29 616 952,7(1,0) kHz pour le plus grand intervalle, 23P0–23P1,alors qu’il est 2 291 167,7(11,0) kHz pour le plus petit, 23P1–23P2. Tenant compte de cenouveau résultat, nous trouvons un accord au niveau de 1 kHz pour différentes expériences.Nous discutons des conséquences de cette situation sur la valeur de la constante de structurefine, α.

[Traduit par la Rédaction]

1. Introduction

Helium is the simplest three-body calculable system. The level energy (i.e., the energy differences)can be calculated in terms of a power series of the fine-structure (FS) constant α, by using quantumelectrodynamics (QED) theory. Precise spectroscopic measurements of these energies can be used asa test of those calculations. Among these, comparison between theory and measurements for the FSsplittings in the 23PJ manifold of states in helium has the potential to provide an accurate determinationof the FS constant [1,2]. In fact, theoretical [3,4] and experimental [5] effort during the 1970s and 1980sgave an α value from He FS with an accuracy of 900 parts per billion (ppb) (1 billion = 109) although laserspectroscopy and modern computational techniques had not yet fully developed. Today, the accuracy

Received 11 October 2004. Accepted 15 November 2004. Published on the NRC Research Press Web site athttp://cjp.nrc.ca/ on 9 May 2005.

G. Giusfredi, P. Cancio Pastor,1 P. De Natale, and D. Mazzotti. Istituto Nazionale di Ottica Applicata, L.goE. Fermi 6, Firenze, I-50125, Italy and European Laboratory for non-Linear Spectroscopy (LENS), V.N. Carrara1, Sesto Fiorentino, I-50019, Italy.C. de Mauro,2 L. Fallani, G. Hagel, V. Krachmalnicoff, and M. Inguscio. European Laboratory for non-Linear Spectroscopy and Dipartimento di Fisica, Universitá di Firenze,V.N. Carrara 1, Sesto Fiorentino, I-50019,Italy.

1 Corresponding author (e-mail: pcp@ino.it).2Also at: Dipartimento di Fisica, Universitá di Siena, V. Roma 56, Siena, Italy.

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Fig. 1. Experimental set-up.

goal is given by the uncertainty with which α is known. In the last CODATA adjustment of the physicalconstants [6], α is given with an accuracy of 3.3 ppb, mainly determined by the anomalous magneticmoment (g − 2) of the electron (see Sect. 3 for details). A part of the α value from the ratio h/MCs, α

determinations from other physical systems are accurate at the level of tens of ppb, and discrepanciesamong them are of a similar magnitude. Therefore, an α value from He with 8 ppb uncertainty can helpto get a more consistent determination of this constant. Moreover, it would contribute to the significanceof determining α by a variety of different methods: it is a sensitive test of the consistency of the physicsacross a range of energy scales and physical phenomena. Due to the fact that FS energies are mainlydetermined by the leading α2 term, theory and experiment of FS splittings must reach an accuracy ofthe order of 16 ppb (or 1.6 × 10−8) to be competitive in terms of α.

The FS structure of the 23P level presents two intervals: namely, ν01 between the 23P0 and 23P1levels of about 29.6 GHz, and ν12 between the 23P1 and 23P2 levels of about 2.3 GHz. An experimentaland theoretical uncertainty of about 16 ppb implies accuracies of about 0.5 and 0.04 kHz for ν01and ν12, respectively. While, today, this accuracy can be reached for ν01, it is a challenge for thesmallest interval. Therefore, measurements and QED calculations of ν01 are used for α determination,whereas ν12 results are used to test the He FS theory. Since 1990, new methods and computationaltechniques have allowed theoretical determinations of these separations with an uncertainty of a fewhundreds hertz [7, 8]. In the same period, laser spectroscopy at 1083 nm between the metastable 23S1and 23PJ (J = 0, 1, 2) levels [9–12], or 1083 nm laser pumping and microwave spectroscopy [13,14]have brought the experimental uncertainty to the 1 kHz level for both intervals. In this paper, we reporta new measurement of these FS splittings. Basically, we follow the same approach as in our previousmeasurements [10, 11], but some experimental improvements have been implemented. With this newHe spectrometer, we have recently measured, with an unprecedented accuracy, the absolute frequencyof 1083 nm He transitions by using an optical frequency comb synthesizer (OFS) [15]. The new FSmeasurements when compared with other experiments and with theory, will be discussed, in view ofgetting an α value from helium.

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Fig. 2. Allan deviation of the master laser frequency locked to the I2 reference, measured with respect to anOFS.

2. Our new measurement

2.1. Experiment

In Florence, we determine the 23P FS frequencies by frequency difference of the 23S1 → 23PJ

(J = 0, 1, 2) transitions at 1083 nm [10,11,16].This approach combines sub-Doppler laser spectroscopywith direct frequency measurements in the microwave domain.

Although, at present, we are able to measure the 1083 nm optical transition frequencies with highaccuracy [15], we take advantage from the cancellation or the minimization of common systematicuncertainties by measuring directly their differences to obtain the FS separations. Thus, if we have twolaser frequencies whose difference can be accurately controlled, we may use one as a fixed reference(master laser) and tune the second (slave laser) across the atomic resonances. In fact, our approachis basically an heterodyne technique, where all the transitions are measured with respect to the samefrequency reference, that can take any arbitrary but stable value.

In the experimental realization, we obtain the two frequencies by phase-locking two DBR diodelasers at 1083 nm (master and slave), i.e., phase-locking their beat note to a microwave oscillator [17],as shown in Fig. 1. A detailed description of this phase-lock, lying outside the scope of this paper,can be found in ref. 17: we just report the measured root Allan variance of the frequency difference,0.1τ−1/2

√Hz. We lock the master frequency to the hyperfine transitions of the iodine molecule at the

doubled frequency (i.e., around 541 nm). This stable reference was chosen not only for iodine is awell-established metrological molecule, but also due to the lack of stable references around 1083 nm.To achieve the iodine lock, we developed a new laser source at 541 nm [18], by amplifying up to 1 Wthe master laser with an Yb fiber amplifier, and by a single-pass frequency doubling in a periodicallypoled KTP non-linear crystal. More than 6 mW of green power were generated and used to perform FMsaturation spectroscopy of the I2 hyperfine transitions. We stabilize the master frequency to be locked tothe center of one isolated component by controlling the laser cavity length. In Fig. 2, the Allan deviationof the beat-note between the I2-locked master and the nearest tooth of an OFS [15] is plotted. The masterstability is 3.5 × 10−11 in 1 s and up to 5 × 10−12 for long time scales (1000 s). This new referenceimproves the master stability by about one order of magnitude in short time scales and cancels the

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Fig. 3. The third derivative of the saturated fluorescence dip for the 23S1 → 23P1 transition. The continuousline is the fit to the expected function. Residuals of this fit are also shown.

long-term frequency drifts [19] present in the reference used in the previous FS measurements [10,11].

The second important issue of our experimental approach is the use of fluorescence saturationspectroscopy in an atomic beam of metastable He to detect narrow resonances around 1083 nm. Themetastabilization to the 23S1 level is achieved by means of a DC discharge in the atomic beam. Aflux of about 1015 atoms str−1 cm−3 of metastable atoms is produced in this way, with a residualtransversal velocity divergence of about 90 MHz (FWHM), in terms of the Doppler broadened width.A saturated absorption excitation of these atoms to the 23P levels is allowed by the 1083 nm linearlypolarized slave-laser light that crosses them at right angles in both the forward and backward directions.Diode-laser linewidth narrowing by means of an extended cavity configuration [17] was used to resolvethe natural linewidth (1.6 MHz) of these saturated dips. Fluorescence from the 23P is detected byusing an Ag–O–Cs photocathode. We increase the signal-to-noise ratio (S/N) and eliminate the residualDoppler fluorescence background by means of a standard laser-frequency-modulation technique: a thirdharmonic demodulated (3D) lineshape as shown in Fig. 3 is recorded.

Also in Fig. 3, the quality of the fit to the expected function used to measure the line-center frequencywith respect to the master frequency is shown. As in the previous FS measurements [10, 11], theline-center uncertainty is limited by the master-laser stability and by the S/N ratio. Apart from thelonger integration time due to the cancellation of master frequency drifts with the new reference, otherexperimental improvements were implemented to increase the S/N ratio: a “noise-eater” system tocontrol the slave-laser interaction power at the µW level, and a new optical system to get better controlof the alignment and the wavefront of the slave laser that interacts with the He atoms. Basically, thissystem is a telescope (f :2f :f configuration, f = 1 m), with a pinhole in the focal telescopic plane thatallows us to obtain a plane-wavefront Gaussian TEM00 mode for the laser beams that interact with theatomic beam, placed in the image f plane. The stringent parallelism required for the counter propagatingbeams was obtained by using a cat’s-eye optical system located after the atomic beam. All these changesin the experiment have improved the line-center precision at least by a factor of two with respect to theprevious measurements. Moreover, as we explain below, they help to minimize some systematic effectsthat affect our measurements.

We followed this line-center acquisition procedure alternatively for the three 1083 nm He transitions,

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Table 1. Budget of measurements and corrections. Uncertainties are given in parentheses in the unitsof the last quoted digit.

ν12 (kHz) ν01 (kHz)

2002 2004 2001 2002

Statistical value 2 291 008.2(30) 2 291 000.8(10) 29 616 940.4(10) 29 616 938.6(25)

RS + LS + 2nd DS + 165.5(150) + 161.1(150) + 12.5(5) + 13.5(5)

Zeeman shift (0.1) (0.1) (0.1) (0.1)Final value 2 291 173.7(153) 2 291 161.9(150) 29 616 952.9(11) 29 616 952.1(25)

to randomize time-dependent systematic errors. Then, the FS frequencies were determined by takingthe difference of the measured line-centers, and the final statistical values were calculated by averagingseveral tens of measurements, as indicated in the first line of Table 1. In Table 1, two runs of measure-ments for each interval have been considered, depending on different experimental conditions. Theseconditions strongly affect the systematic corrections that must be applied for each run, and therefore,the measurements cannot be considered all together. The runs are indicated in Table 1 labeled with theyear when the measurements were done. In total, we collected 245 measurements for ν01 and 259 forν12, recorded in different days.

2.2. Systematic errors

Although, in principle, the first-order Zeeman shift is eliminated by selecting transitions connectingm = 0 states, at the kHz level of accuracy even the second-order effect is relevant. As a consequence,we chose to work with zero magnetic field: a cylinder of high magnetic permittivity metal (mu-metal)embeds the interaction region to shield it against DC and AC magnetic fields, while a pair of Helmholtzcoils inserted therein allows for the fine cancellation. We minimize it in situ by measuring the Helmholtzcoils current that produces the minimum fluorescence signal, according to the non-linear Hanle effect.We then measured the magnetic field to be below 0.03 µT in a cube of 1 cm3 around the interactionregion. The related systematic error, with the conservative assumption that it linearly scales with theresidual field, is less than 0.1 kHz.

Saturation spectroscopy was chosen to cancel the first-order Doppler shift (1st DS) due to thelongitudinal atomic velocity anisotropy. However, a mutual misalignment of the counter propagatingslave-laser beams can shift the He frequencies by a quantity proportional to the longitudinal atomicvelocity. We measured a mean longitudinal atomic velocity of about 3200 m/s by recording the longitu-dinal Doppler profile of the 23S1 → 23P2 transition and fitting it with a “generalized Maxwellian” [10].As a result, a 1st DS of about 1.6 MHz/mrad can be present due to slave-laser beams overlap, i.e., due tothe cat’s-eye focal distance. It was accurately calibrated by measuring the relative 1st DS between thewell-aligned and the misaligned retroreflected beam with respect to the lens–mirror distance. We assigna conservative error of 1 µrad to this calibration, which means a maximum 1st DS of 1.6 kHz in themaster–slave relative frequency for all three 1083 nm He transitions. Thus, this shift cancels in the FSfrequencies if the alignment between different transitions is maintained, as in our case. Moreover, sincethe shift was randomized by periodic realignment of the retroreflected beam, no systematic correctionis applied for the FS frequencies.

As in the previous measurements, we find that the major limitation to our experimental accuracyis due to the recoil imparted by photons to atoms (RS). A qualitative and quantitative analysis ofthis effect, lying outside the scope of this paper, can be found in refs. 10, 11, 20: we note only thatthe line-center blue-shift is a consequence of line-shape asymmetry, due to the fact that the velocityand position distribution of atoms are modified by the radiation forces during the interaction time.The magnitude of the RS depends on the number of photons scattered by any atom. This number is

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limited by the pumping-time in the case of the 23S1 → 23P0,1 open transitions (i.e., transitionsfor which the selection rules make at least one of the Zeeman 23S1 sublevels uncoupled to the laserlight). However, a big RS is expected for the 23S1 → 23P2 closed transition, where photons can bescattered during the whole interaction time. As a consequence, the RS, different for all three transitions,does not cancel in the differences and a correction for the FS intervals must be considered. Due tothe difficulty of measuring the RS with high precision, we decided to calculate it by considering thephysical model of the effect that takes into account the real experimental conditions. The populationof the excited 23PJ (J = 0, 1, 2) level was calculated by solving numerically the 1D-optical Blochequations integrated with respect to the atom position in the interacting standing wave, the transversalvelocity (i.e., detuning), and the interaction time (i.e., laser beam waist × atomic longitudinal velocity).The calculation was made in the atomic frame: the second-order Doppler correction (2nd DS) wasdirectly included in the detuning frequency. In Bloch equations, not only the mechanical effects of thelight are included: AC Stark shift (light shift, LS), due to close non-resonant 23P levels, has also beenconsidered. This effect was small for the 100 µW power used in the measurements, but not negligible atthe kHz accuracy level. Therefore, a frequency correction for each transitions, due to RS, LS, and 2nd DSall together, was calculated by fitting the third derivative of the integrated fluorescence signal from thecalculated population taking into account the longitudinal velocity distribution and the spatial detectorsensitivity, as in the experiment. We verified that the shifts predicted by the model were in agreementwith the observed values for different interaction times [20]. The systematic correction applied to eachFS interval (see Table 1) was calculated as a difference of the shift for each transition involved in theFS splitting. Concerning this correction, the improvement in these new FS measurements is obtainedby better control of the experimental parameters that not only give the exact magnitude of the effect butalso limit its final uncertainty. In fact, power and beam waist of the slave laser and atomic longitudinalvelocity were continuously measured and monitored during each run of measurements. The differentvalues given for the RS + LS + 2nd DS correction in Table 1 are due to the changes of these parametersin the experiment during different runs. The uncertainties given for these corrections were calculatedfrom the variations of the simulation results, by changing the parameters in their experimental precisionrange. They are mainly limited by the slave beam-waist and atomic longitudinal velocity variations.The larger uncertainty assigned to the RS + LS + 2nd DS correction for the ν12 interval reflects thelarger sensitivity to these variations, because it contains a closed transition.

2.3. Results

The final results were calculated by a weighted average of the values listed in the last line of Table 1.3

We obtained

ν12 = 2 291 167.7(110) kHz

ν01 = 29 616 952.7(10) kHz

These results improve by a factor of two the accuracy of our previous FS measurements [10, 11].We have plotted these results in Fig. 4 together with other sub-MHz measurements [9–14]. For thelargest interval, ν01, our new value is in agreement with our previous measurement [10] and with themost precise one from Hessels and co-workers [13]. The achieved 1 kHz final accuracy is limited bythe present experimental approach used to measure the interval: it is more than 1000 times smaller thanthe natural linewidth of the 1083 nm transitions. Instead, for the smallest interval, ν12, although it is inagreement with previous 1 kHz accurate measurements [12,14], it is 11 times less accurate than those,due to systematic uncertainties. Anyway, other experiments, as for example, precision spectroscopy

3 Uncertainties are given in parentheses in the units of the last quoted digit.

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Fig. 4. Comparison between experiment and theory for both 23P FS intervals. The hatched bar gives theweighted-mean value and uncertainty of the experiments. Labels in the figures are taken from referencesas follows: Minardi99 from ref. 10, Cancio98 from ref. 11, Shiner00 from ref. 12, Hessels00 from ref. 14,Hessels01 from ref. 13, and Drake02 from ref. 7.

with cooled atoms, could minimize the systematics that affect this interval, and therefore, the accuracycould be improved.

3. Discussion and perspectives

From the comparison of the FS measurements, as shown in Fig. 4, we note that there is agreementbetween values, at the level of a few kHz accuracy, for both intervals. Due to the different experimentalapproaches and to the different systematics that affect these measurements, we can conclude that theexperimental values of the 23P FS frequencies are now well determined at this level of accuracy.Therefore, we have calculated a weighted-mean value for these separations, taking into account onlymeasurements with an accuracy better than 3 kHz 3

ν12 = 2 291 175.3(8) kHz (350 ppb)

ν01 = 29 616 951.8(6) kHz (20 ppb)

They must be considered as the most accurate measurements of the helium FS splittings so far.As we indicated in the Introduction, the FS constant, α can be determined by bringing the ν01

measurement into agreement with its theoretical value. Pachucki [8] gives a simple formula for gettingthis determination, by considering all QED and relativistic corrections of the order α3 up to α5 as anα-independent correction δν01 of the measured value ν01:

ν01 − δν01 = α2 × 556 200 289.5(1) MHz

At present, two theoretical groups are working at the complicated QED theory for helium [7, 8].Although each one uses a different strategy to solve the problem, both included, in their calculations,terms and operators already considered by the other, sometimes controlled and corrected with the group’s

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Fig. 5. Comparison of α determinations from different physical systems with respect to the presentCODATA accepted value [6].

own theory. Therefore, we have considered the FS theory reported in [7] as the most updated theoreticaldetermination for the 23P FS energies.4 From this paper, we have calculated a frequency correctionδ ν01 = −1472.13 (50) kHz. If we use the simple formula above, we obtain

α−1 = 137. 035 9872(18) (13 ppb)

In Fig. 5, we have plotted this value with others coming from other physical systems. The α valuefrom He is the third more accurate determination, but there is a discrepancy of more than 7 standarddeviations between this value and the present CODATA recommended value. Such a discrepancy impliesthat this determination is not statistically compatible with the others, following the CODATA criteria [6]to consider some new value in a new adjustment of this constant, although, as we can see in Fig. 5,there is an agreement with some other determinations (as, for example, the one from the Josephsoneffect). The origin of this difference can be attributed to the poor agreement between measurementsand theoretical determinations of the FS intervals, as shown in Fig. 4. This disagreement for the ν01interval is about 9 times larger than their combined standard deviations. A larger discrepancy, about24 combined standard deviations, is found for the smallest interval, ν12, which is used as a test of theQED theory for these FS intervals. Because measured values are well established at the kHz level, asindicated above, and since the contribution from uncalculated α6 and higher QED corrections has beenestimated to be of the order of about 1 kHz, we conclude that the discrepancies can be explained withsome QED contribution that has not been well-evaluated or has not yet been taken into account. With

4 Personnel communication. 2004. G.W.F Drake confirmed that his calculations in ref. 7, already contain the termscalculated by Pachucki et al. in ref. 8. The magnitude of these corrections are now being checked using Drake’stheory.

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respect to the former cause, Drake and co-workers4 are controlling some high-order recoil correctionsto 23P FS calculated by Pachucki et al. [8]. But the discrepancy could also be due to an additionalspin-dependent contribution that has not yet been taken into account, as pointed out by Drake [7].

Looking for these physical effects might help to give a better understanding of the finite nuclear sizeand nuclear structure of helium,5 which can be given by spectroscopic measurements. As an example,precise measurements of isotope shifts (IS), when compared with theoretical determinations, can be usedto determine the difference in the nuclear charge radii of both isotopes [21–23]. To this end, in Florence,we are doing hyperfine structure (HFS) measurements in 3He and IS measurements between 3He–4Heat 1083 nm. A new improvement has been implemented in the experimental apparatus to perform theserelative frequency measurements: now we phase-lock the master-laser frequency to the nearest tooth ofthe OFS. In this way, not only better frequency stability than with the iodine-lock is achieved, but wealso get absolute frequency control of the master, i.e., absolute frequency measurements of the 1083 nmtransitions. Moreover, the possibility of simultaneously locking two 1083 nm lasers, that are resonantwith two different He transitions, to two different OFS-teeth [24], will allow us to measure directly FS,HFS, and IS in helium, cancelling common systematic errors, and improving the final accuracy.

In summary, measured values for the fine structure frequencies of the 23P He manifold are inagreement at the kHz accuracy level. Improvement of this accuracy requires the use of spectroscopictechniques, which go beyond the limit given by the life-time of these levels. The comparison of thesevalues with theoretical determinations is already providing a test of QED at unprecedented levels ofaccuracy for the three-body system. However, the remaining discrepancies must be resolved before ameaningful value for α can be derived from this atomic system.

Acknowledgements

G. Hagel ackmowledges financial support from the European Union under contract No. HPRICT1999-00111 and from Istituto Nazionale di Ottica Applicata. This work was partially funded by Min-istero Italiano della Istruzione e della Ricerca under Fondi Italiani per la Ricerca di Base contractRBNE01KZ94.

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