Measurement of the Hyperfine Quenching Rate of the Clock Transition in 171 Yb
C.-Y. Xu, J. Singh, J. C. Zappala, K. G. Bailey, M. R. Dietrich, J. P. Greene, W. Jiang, N. D. Lemke, Z.-T. Lu, P. Mueller, T. P. O'Connor
ddraft version: October 6, 2018
Measurement of the Hyperfine Quenching Rate of the Clock Transition in Yb C.-Y. Xu ( 徐 晨昱 ),
1, 2
J. Singh, ∗ J. C. Zappala,
1, 2
K. G. Bailey, M. R. Dietrich, J. P. Greene, W. Jiang ( 蒋 蔚 ), N. D. Lemke, Z.-T. Lu ( 卢 征 天 ),
1, 2
P. Mueller, and T. P. O’Connor Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Department of Physics and Enrico Fermi Institute,University of Chicago, Chicago, Illinois 60637, USA (Dated: October 6, 2018)We report the first experimental determination of the hyperfine quenching rate of the 6 s S ( F =1 / − s p P ( F = 1 /
2) transition in
Yb with nuclear spin I = 1 /
2. This rate determines thenatural linewidth and the Rabi frequency of the clock transition of a Yb optical frequency standard.Our technique involves spectrally resolved fluorescence decay measurements of the lowest lying P , levels of neutral Yb atoms embedded in a solid Ne matrix. The solid Ne provides a simple way totrap a large number of atoms as well as an efficient mechanism for populating P . The decay ratesin solid Ne are modified by medium effects including the index-of-refraction dependence. We findthe P hyperfine quenching rate to be (4 . ± . × − s − for free Yb, which agrees withrecent ab initio calculations.
PACS numbers: 31.30.Gs, 32.50.+d, 32.70.Cs
The conservation of angular momentum strictly for-bids single-photon transitions between two atomic statesif both electronic angular momenta are equal to zero, i.e. J = 0 (cid:61) J (cid:48) = 0. This restriction can be circumventedby state mixing due to the hyperfine interaction [1]. Theconsequent increase in the transition rate is referred toas hyperfine quenching (HFQ), a feeble mechanism thattypically plays a significant role in the radiative decay ofonly the lowest lying P , levels of divalent atoms.The earliest studies of the HFQ effect focused on thespectra originating from nebulae [2]. More recently, theisotopic dependence of these astronomical spectra havebeen used to infer HFQ rates [3] and, conversely, isotoperatios that result from stellar nucleosynthesis [4]. In thelaboratory, the 1 s p P , levels in He-like ions were thefirst to be measured and are the most thoroughly stud-ied [5]. The HFQ rates of a handful of many-electronions have also been measured [5–8]. However, the ratehas never been measured in any neutral atoms due todifficulties involved in populating the relevant levels andsubsequently observing their slow decay.In neutral atoms, efforts have been made in modern ab initio calculations of the HFQ rate [9, 10], motivatedby the promising application of neutral divalent atoms tooptical clocks [11], quantum computing [12], and quan-tum simulation of many-body systems [13]. In the caseof optical clocks, the HFQ rate determines the naturallinewidth and the Rabi frequency of the “clock tran-sition” ns S − nsnp P in fermionic isotopes. TheHFQ rate calculations require accurate knowledge ofthe atomic structure of the many-electron atoms. For6 s S ( F = 1 / − s p P ( F = 1 /
2) in
Yb with nu-clear spin I = 1 /
2, the HFQ rate ( A HFQ ) involves the ma-trix element of the electric-dipole operator ( D ) betweenintermediate levels ( γ ) and the ground level 6 s S , aswell as the hyperfine interaction ( H HFI ) matrix element between these levels ( γ ) and 6 s p P , A HFQ ( S − P ) ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) γ (cid:10) S || D || γ (cid:11)(cid:10) γ || H HFI || P (cid:11) E ( γ ) − E ( P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1)where E ( γ ) are the energies of atomic levels relative tothe ground level [9]. Among the intermediate levels, theHFQ of 6 s p P is predominantly caused by the admix-ture of the lowest lying 6 s p P and 6 s p P [9], fromwhich the transitions to the ground level are both E1 al-lowed. Our measurement, therefore, serves as a sensitivebenchmark for these calculations.We employ a novel technique of probing atoms embed-ded in solid Ne to extract the HFQ rate A HFQ ( S − P )in free Yb. Interrogating atoms trapped in a solidoffers both high atomic density and long observationtime. In addition, while matrix isolated Yb atoms qual-itatively resemble free atoms, they exhibit an enhancedinter-system crossing 5 d s D ← s p P , enabling ef-ficient population of 6 s p P by pumping the strong FIG. 1: Low-lying atomic levels and transitions of Yb in solidNe. P can be efficiently populated by virtue of an enhancedinter-system crossing D ← P . The radiative decay of P is observed in both Yb and
Yb samples. a r X i v : . [ phy s i c s . a t o m - ph ] J un s S − s p P transition and subsequent spontaneousdecay (Fig. 1) [14]. We choose solid Ne as the matrix be-cause it is less polarizable than heavier noble gas solidsand more technically accessible than solid He. While Heonly solidifies under at least 25 bar pressure, Ne readilyforms a solid with face-centered-cubic crystal structureat 24.5 K and 1 bar [15].The main challenge of performing this measurement isto properly account for various medium effects. First, themedium may open additional radiative or non-radiativedecay channels on an excited atom. Second, the mediummay alter the HFQ rate of a free atom by modifying theatomic wavefunctions and shifting the energies in Eq. (1).Third, Fermi’s golden rule dictates that the spontaneousemission rate of a transition depends cubically on thetransition frequency that may be shifted in medium. Fi-nally, the spontaneous emission rate also depends on theenvironment of the emitter. Such a phenomenon, knownas the Purcell effect, is one of the hallmarks of quantumelectrodynamics (QED). In cavity QED, the decay rate ismodified by the geometry of the surrounding vacuum en-vironment [16–19]. Within a medium, however, the decayrate depends on the index of refraction because it mod-ifies both the photon dispersion relation and the energyfluctuation of the QED vacuum. Although the index-of-refraction effect has been known for some time, there isstill considerable tension in its understanding [20, 21].We address these effects as follows. First, we measurethe 6 s p P decay rate for isotopically pure Yb and
Yb ( I = 0) in solid Ne (Fig. 2). The difference betweenthese two rates separates the HFQ contribution from anymedium quenching mechanisms that are independent ofisotopes. Second, the Yb transition frequencies in solidNe are used to calculate the energy and frequency depen-dent corrections. Third, we measure the decay rate of Yb FIG. 2: The time-dependent fluorescence intensity of Yb P (red solid circles) and Yb P (blue open circles) insolid Ne near the center of the emission peak. The influenceof the HFQ effect is evident. s p P in solid Ne and compare it with the experimen-tal value in vacuum [22] to provide a direct calibrationof the index-of-refraction effect. After making these cor-rections, we then obtain the HFQ rate of a free atom.The samples are prepared with a similar setup we usedpreviously [14]. Before the deposition on the liquid-He cooled sapphire substrate, Ne gas (99.999 %) flowsthrough a noble gas purifier (LDetek LDP1000) and a 77K charcoal trap in order to minimize the growth defectsand increase the sample transparency. We co-deposit Ybusing an atomic beam generated by an effusive oven. Toavoid the formation of Yb clusters, we keep the Yb-to-Ne ratio below 5 ppm and the temperature below 5 Kto suppress the mobility of the atoms. Samples with iso-topically pure Yb (95 %, Oak Ridge batch 196043)and
Yb (97 %, Oak Ridge batch 124501) are sepa-rately made. While the enriched Yb is available for sev-eral even isotopes,
Yb contains the least concentrationof odd isotopes.We use a 385 nm light-emitting diode (LED) to excitethe S − P transition [23] and subsequently populate P . The fluorescence is detected by a 1.5 nm resolutionoptical spectrometer (Ocean Optics USB4000-UV-VIS).Fig. 3(a) shows the emission spectrum of Yb P (solidcircles) and Yb P (open circles) in solid Ne after theLED is switched off. We record the fluorescence decay atselect wavelengths for 100 s for Yb and 300 s for
Yb.The decay near the center of the emission peak (565 nm)
FIG. 3: (a) The S ← P emission spectrum of Yb(red solid circles) and
Yb (blue open circles) in solid Neafter the 385 nm LED is switched off. The peak is shiftedfrom the vacuum position at 578.4 nm. The spectra are sep-arately normalized so that the peaks appear to have similarheight. (b) The decay rate of Yb P (red solid circles)and Yb P (blue open circles), and the difference of thedecay rates between the two isotopes (purple solid triangles)at select wavelengths. The error bars are about the size ofthe markers. is shown in Fig. 2. At each wavelength, the decay rate ofeach isotope is obtained by fitting the data to an expo-nential function and is plotted in Fig. 3(b). The uncer-tainty about the size of the markers is determined fromthe fitting error and the sample variance based on three Yb and two
Yb samples with different Yb densi-ties and optical transparencies. The strong wavelengthdependence of the decay rates is in part due to the fre-quency cube dependence. The remainder is likely causedby the interaction with phonons of different energies. Thesum of multiple exponential functions, which describesmultiple types of trapping sites in solid Ne, gives a bet-ter fit at some wavelengths, but the weighted average ofthe multiple rates is not significantly different from therate of the single exponential fit.The decay rate of Yb P near 565 nm is approx-imately 2 × − s − . Since its single-photon decay invacuum is strictly forbidden, this rate reflects the over-all medium quenching of an excited atom. One possi-ble quenching mechanism may be that the atomic wave-functions are perturbed by the crystal field in solid Ne.To model this perturbation, we assume that this field israndomly oriented and has a constant strength. Elevenlow-lying levels are included, between which the reducedmatrix elements of the electric-dipole operator have beencalculated [24]. We sum over M J states and the orienta-tion of the field, leading to a Stark-like coupling betweenlevels. In order to account for the observed decay rate, itrequires a 20 MV/m crystal field so that the perturbed P wavefunction has an admixture of P with a mixingcoefficient of 1 . × − and of P with 1 . × − . Sucha crystal field strength is not unexpected in solid Ne [15].For the Yb P decay, since the nuclear spins areunpolarized and the crystal field is randomly oriented,the effects of the medium quenching and the HFQ addincoherently. We plot the difference of the decay ratesbetween the two isotopes in Fig. 3(b) (solid triangles).As expected, this differential rate is mostly independentof the wavelength and represents the HFQ contribution.We take the average of the rates weighted by the emissionintensity and find the HFQ rate of Yb P in solidNe to be (6 . ± . × − s − . The uncertainty isconservatively chosen to be half of the full range.We first examine the medium’s influence on the HFQmechanism described in Eq. (1). From the crystal fieldstrength estimation, we are assured that the atomic wave-functions are essentially intact. However, the mediumalters the energy differences in the denominators. TheHFQ of P is predominantly caused by the admixtureof the lowest lying P and P [9]. In solid Ne, wetake E ( P ) = (565 nm) − , E ( P ) = (546 nm) − , and E ( P ) = (396 nm) − in the emission mode [14, 23]. E ( P ) − E ( P ) in solid Ne is equal to 616 cm − andis changed from its vacuum value (704 cm − ) by a fac-tor of 0.875. Therefore, the HFQ rate is enhanced by afactor of 1.306 if the P term dominates the sum. Simi- larly, E ( P ) − E ( P ) is changed by 0.971, and the rateenhanced by 1.061. Assuming a uniform probability dis-tribution of the relative contribution from P and P ,we take the mid-point as the mean and 1 / √
12 of the fullrange as the uncertainty [25] and obtain an enhancementfactor of 1 . ± . A ) of a transition, A m A v = (cid:18) ω m ω v (cid:19) G ( n ) , (2)where ω is the transition frequency, the subscript m ( v )refers to medium (vacuum), and the scale factor G isa function of the index of refraction ( n ). To extract A HFQ ,v ( S − P ), we use ω Ne ( S − P ) = (565 ± − and ω v ( S − P ) = (578 . − to calculate the fre-quency dependent correction. The uncertainty is due tothe spectrometer calibration and the sample variance.We determine G Ne by measuring the P decay for thefollowing reasons. The HFQ transition S ( F = 1 / − P ( F = 1 /
2) and the intercombination transition S − P are both of E1 type. Their transition wavelengths aresufficiently close that the wavelength dependence of theindex of refraction is insignificant. The P decay ratein vacuum is precisely known A v ( S − P ) = (1 . ± . × s − [22]. Compared to this rate, the mediumquenching rate ( ∼ × − s − ) is negligible, which allowsus to use the measured total decay rate for A Ne ( S − P ).For the P lifetime measurement in solid Ne, samplesof natural Yb are used. We excite the S − P transi- FIG. 4: (a) The S ← P emission spectrum of Yb in solidNe induced by the 543 nm laser. The peak is shifted from thevacuum position at 555.8 nm. (b) The decay rate of P atselect wavelengths in solid Ne (green circles). The error barsare about the size of the markers. The decay rate in vacuum,1 . × s − , is off the scale. The black square with anerror bar indicates the predicted P decay rate in solid Neusing the RC model and the frequency cube dependence. tion by a 543 nm diode pumped solid state laser (OptoEngine MGL-III-543). The fluorescence light is coupledinto a monochromator (McPherson 225) and detectedby a photomultiplier tube counting module (Sens-TechP10PC-2) mounted at the exit of the monochromator.A dead time correction is applied for the counting rate.Fig. 4(a) shows the steady-state emission spectrum of P in solid Ne with 1 nm resolution.We chop the laser at 50 kHz with 50 % duty cycleusing an acousto-optic modulator and record the decayat select wavelengths with 50 ns resolution. The decayrate at each wavelength is plotted in Fig. 4(b). The av-erage of the rates weighted by the emission intensity is A Ne ( S − P ) = (1 . ± . × s − , where theuncertainty is half of the full range. From Eq. (2) forthe P decay with ω Ne ( S − P ) = (546 ± − and ω v ( S − P ) = (555 . − , we obtain the transition-independent G Ne = 1 . ± . P decay, we arrive at A HFQ ,v ( S − P ) =(4 . ± . × − s − for free Yb. All the correctionswe have made are summarized in Table I.We compare this result to two available calculations:6 . × − s − (no uncertainty provided) [26] and 4 . × − s − (a few percent uncertainty) [9]. Authors of ref-erence [26] have used experimentally measured hyperfineparameters in their calculation and have included onlytwo intermediate levels in Eq. (1). Authors of reference[9] have computed the sum with multiple intermediatelevels and have independently calculated the hyperfineconstants with better than 1 % accuracy as a verificationof the quality of their technique. Our measurement is ingood agreement with reference [9].We are also able to compare our experimentally deter-mined G Ne to theoretical predictions. One theory sup-ported by recent experiments for E1 transitions [27, 28]is the real cavity (RC) model [29]. It treats the emit-ter as residing in an empty spherical cavity carved outof a lossless, homogeneous, and isotropic medium withpermittivity (cid:15) = n (cid:15) . The macroscopic field in the di-electric is canonically quantized. The model predicts thefollowing scaling with n , G RC ( n ) = n (cid:20) n (cid:18) E loc E mac (cid:19) RC (cid:21) . (3)The factor n comes from the in-medium photon disper- TABLE I: A summary of corrections due to medium effectsfor the extraction of the P HFQ rate of free
Yb based onthe measurements in solid Ne.Correction Scale factor UncertaintyEnergy difference, Eq.(1) 0.845 6.0 %Medium quenching 0.771 4.2 %Index-of-refraction effect, Eq.(2) 0.838 3.0 %Frequency cube, Eq.(2) 0.932 0.5 %Total 0.508 7.9 % sion relation. The macroscopic field operator ˆ E mac isrenormalized by 1 /n due to the in-medium energy den-sity (cid:15) ˆ E /
2. The ratio of the local field inside the cavity E loc to the macroscopic field far outside the cavity E mac is found to be ( E loc /E mac ) RC = 3 (cid:15)/ (2 (cid:15) + (cid:15) ) using theboundary conditions on the sphere.Given the growth conditions of our solid Ne samplesand both the wavelength- and the temperature- depen-dence of the index of refraction, we take n Ne = 1 . ± . G RCNe = 1 . ± . G Ne = 1 . ± . P rate in solid Ne using the RC model and thefrequency cube dependence is also indicated in Fig. 4(b)(solid square).In heavier noble gas solids, we find that the Yb transi-tions suffer from exacerbated medium effects. In solid Ar,they manifest in a stronger wavelength dependence of the P decay rate. Our measurements show ω Ar ( S − P ) =(562 ± − , A Ar ( S − P ) = (1 . ± . × s − ,and thus G Ar = 1 . ± .
17. The larger uncertaintymakes solid Ar a less attractive medium for transition-rate measurements. Nevertheless, this result still agreeswith the RC model prediction G RCAr = 1 . ± .
05 with n Ar = 1 . ± .
02 [32, 33]. In solid Xe, the Yb P lifetimeis shorter than 50 µ s due to a much stronger crystal field.Therefore, the HFQ measurement becomes impossible.In conclusion, we have measured the HFQ rate of the S ( F = 1 / − P ( F = 1 /
2) transition in
Yb basedon the matrix isolation technique using solid Ne andspectrally resolved fluorescence decay measurements. Wehave accounted for medium effects using measurementsof both the Yb P decay and the Yb P decay insolid Ne. The average P decay rate across the emissionpeak in solid Ne agrees with the RC model prediction. Inorder to carry out a more precise study on the index-of-refraction effect, one needs to consider the phonon inter-action to better understand the wavelength dependence.Finally, the most suitable naturally abundant candi-dates for the study of the HFQ effect using this techniqueare Mg, Ca, Zn, Sr,
Cd,
Cd,
Yb,
Yb,
Hg, and
Hg. For each of these candidates, a nat-urally abundant nuclear spin-0 isotope is available, andthe transition from the ground level to the lowest lying P is optically accessible. Lighter atoms are more tightlybound which likely means that the medium induced cor-rections are smaller but the efficiency of populating P is worse. On the other hand, lighter atoms also havehigher P levels which may provide an alternative andmore efficient path for the P population.We would like to thank T. Oka, S. T. Pratt and R.W. Dunford for helpful discussions and the use of theirequipment. This work is supported by Department ofEnergy, Office of Nuclear Physics, under Contract No.DEAC02-06CH11357. J. S. and N. D. L. are supportedby Argonne Director’s postdoctoral fellowship. ∗ Present address: Technische Universit¨at M¨unchen,Exzellenzcluster Universe, 85748 Garching, Germany.[1] I. S. Bowen, Rev. Mod. Phys. , 55 (1936).[2] R. H. Garstang, J. Opt. Soc. Am. , 845 (1962).[3] T. Brage, P. G. Judge, and C. R. Proffitt, Phys. Rev.Lett. , 281101 (2002).[4] R. H. Rubin et al. , Astrophys. J. , 784 (2004).[5] W. R. Johnson, Can. J. Phys. , 429 (2011).[6] E. Tr¨abert et al. , New J. Phys. , 023017 (2011).[7] S. Schippers et al. , Phys. Rev. A , 012513 (2012).[8] T. Becker et al. , Phys. Rev. A , 051802 (2001).[9] S. G. Porsev and A. Derevianko, Phys. Rev. A , 042506(2004).[10] R. Santra, K. V. Christ, and C. H. Greene, Phys. Rev. A , 042510 (2004).[11] A. Derevianko and H. Katori, Rev. Mod. Phys. , 331(2011).[12] A. J. Daley, Quantum Inf. Process. , 865 (2011).[13] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[14] C.-Y. Xu et al. , Phys. Rev. Lett. , 093001 (2011).[15] N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, 1976).[16] H. Walther et al. , Rep. Prog. Phys. , 1325 (2006).[17] K. Drexhage, J. Lumin. , 693 (1970).[18] D. J. Heinzen et al. , Phys. Rev. Lett. , 1320 (1987).[19] F. A. Inam et al. , New J. Phys. , 073012 (2011). [20] D. Toptygin, J. Fluoresc. , 201 (2003).[21] K. Dolgaleva and R. W. Boyd, Adv. Opt. Photon. , 1(2012).[22] K. Beloy et al. , Phys. Rev. A , 051404 (2012).[23] R. Lambo et al. , J. Chem. Phys. , 204315 (2012).[24] S. G. Porsev, Y. G. Rakhlina, and M. G. Kozlov, Phys.Rev. A , 2781 (1999).[25] Evaluation of measurement data – Guide to the expres-sion of uncertainty in measurement (Bureau Interna-tional des Poids et Mesures, 2008). For a uniform proba-bility distribution defined in the interval [ − a/ , a/ a/ √ , 769 (2001).[27] G. L. J. A. Rikken and Y. A. R. R. Kessener, Phys. Rev.Lett. , 880 (1995).[28] F. J. P. Schuurmans et al. , Phys. Rev. Lett. , 5077(1998).[29] R. J. Glauber and M. Lewenstein, Phys. Rev. A , 467(1991).[30] D. N. Batchelder, D. L. Losee, and R. O. Simmons, Phys.Rev. , 767 (1967).[31] A. Dewaele et al. , Phys. Rev. B , 094112 (2003).[32] W. Schulze and D. M. Kolb, J. Chem. Soc. Farad. T. 2 , 1098 (1974).[33] A. C. Sinnok and B. L. Smith, Phys. Rev.181