Precision measurement of branching fractions of 138 Ba + : Testing many body theories below one percent level
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Precision measurement of branching fractions of Ba + : Testing many body theoriesbelow one percent level D. De Munshi , T. Dutta , R. Rebhi and M. Mukherjee , ∗ Centre for Quantum Technologies, National University Singapore, Singapore 117543. Department of Physics, National University Singapore, Singapore 117551. (Dated: June 23, 2018)The branching fractions from the excited state 6 P / of singly charged barium ion has beenmeasured with a precision 0 .
05% in an ion trap experiment. This measurement along with theknown value of the upper state life-time allowed the determination of the dipole matrix elementsfor the transitions P − S and P − D to below one percent level. Therefore, for the first time itis now possible to compare the many body calculations of these matrix elements at level which isof significance to any parity non-conservation experiment on barium ion. Moreover, these dipolematrix elements are the most significant contributors to the parity violating matrix element betweenthe S − D transition, contributing upto 90% to the total. Our results on the dipole matrix elementsare 3 . ± .
014 and 3 . ± .
016 for the S − P and P − D transitions respectively. PACS numbers: 32.70.Cs, 37.10.Ty, 06.30.Ft
Trapping and laser cooling of ions provide perturbationfree environment to measure atomic state life-time [1],light-shift [2], branching ratio [3] and other fundamentalproperties of atoms with high precision [4]. This leadsto the use of trapped and laser cooled ions for quantumstate manipulation [5, 6], atomic clocks [7] and to studyfundamental interactions [8]. The study of fundamentalinteractions via atomic properties include measurementsof the Lamb shift [9], the parity non-conservation (PNC)in atomic system [10], the conserved vector current hy-pothesis [11], the electron-electric dipole moment (e-EDM) [12] etc .. As most of the original experimentshave been carried out with atomic beams, they sufferedfrom large systematic uncertainties due to limited con-trol over the environment. These systematics are largelyabsent for stored atomic systems, in addition, they pro-vide long observation time. Therefore, in recent years,trapped and laser cooled ions have emerged as potentialcandidate to perform high precision experiments of fun-damental importance like atomic parity violation [13, 14]and e-EDM [8]. Barium ion is particularly suitable forthe investigation of PNC as was pointed out by Fort-son [13] because of its large nuclear charge and ease oflaser cooling and trapping.The best atomic PNC measurement performed so far isthat of cesium with a precision of 0.3% [10]. However, thenuclear anapole moment obtained from this measurementshows a discrepancy with other nuclear data strongly sug-gesting the need to measure atomic PNC in other speciesin order to verify or to go beyond the Standard Modelof particle physics. In this context, a number of exper-imental groups are pursuing an ion trap based atomicPNC experiment which has been proposed to be capableof limiting systematic uncertainty to below one percentlevel. In addition to the experiment, one also needs thetheoretical value of the parity violating dipole matrix ele-ment with a similar precision. In principle different vari- ants of the coupled cluster theory [15–18] are capable ofproviding such precision, provided the many-body wave-functions are accurately known. Precision measurementof atomic properties of the low lying energy levels allowthese theory to be tuned to provide high accuracy wave-functions. Therefore measuring branching fractions ortransition probabilities and life-time of Ba + with preci-sion below one percent are of utmost need. Moreover,for PNC measurement in barium ion between the states | S i and | D i , the contribution to the parity non-conserving dipole matrix element ǫ P NC comes from thesum over all high p − states given by, ǫ P NC = X n,j h D / | ˆ d | nP j ih nP j | H P NC | S / i W S / − W nP j + h.c., (1)where, ˆ d , H P NC and W are the dipole operator, PNCoperator and electronic binding energy respectively.The principle quantum number and the total angularmomentum quantum numbers are denoted by n and j . The state | P i contributes about 90% [16, 18] tothe PNC matrix element ǫ P NC via the matrix elements h P | D | D i and h P | D | S i as shown in eq. (1).In this letter, we present measurements of the branch-ing fractions for the dipole transitions from the 6 P stateof barium ion with a precision well below 0 .
1% therebymaking it possible to compare with the existing theoryto the precision that is required for any PNC measure-ment with barium ion. The branching fraction and tran-sition probabilities of a fast decaying excited state canbe measured by different techniques like ultra-fast excita-tions [3], optical nutation [19] or by simple photon count-ing at different wavelengths. The first approach requirescomplicated laser pulse sequence and suffers from system-atics due to synchronization, the second one is prone tosystematics due to the measurement of the actual inten-sity at the ion position and the last technique is limitedby the availability of well calibrated detectors. We per-formed the branching fraction measurement on bariumion using a similar method as recently proposed and per-formed for calcium by Ramm et al. [20]. This methodhas been shown to be largely free of common systematicslike magnetic field fluctuation, intensity fluctuations etc. .In the following, a brief description of the experimentalsetup, followed by procedure, results and discussion willbe made.A schematic of our setup is shown in Figure(1). Theion trap is a linear blade trap with radial parameter r = 0 . z = 2 . Top-tica SHGpro providing light at 493 nm (green) whichaddresses the main cooling transition between S and P . In order to minimize laser frequency drifts duringthe whole experiment, the laser frequency is locked to areference cavity which is then locked to one of the clos-est molecular transitions of Te [21] by modulationtransfer spectroscopy. The frequency difference betweenresonance of Te and Ba + is bridged by an acusto-optic modulator (AOM) named AO1 in double-pass con-figuration and another AOM (AO2) in single-pass. Asthe cooling transition is not entirely closed due to thepresence of a metastable D state, the ion populationis re-pumped into the cooling cycle by a 650 nm (red) Toptica DLpro laser. The 650 nm laser is locked to areference cavity which has a common zerodur c (cid:13) spacerwith the cavity of 493 nm, thereby cancelling their rel-ative drifts. In order to avoid population trapping intothe Zeeman dark states, a magnetic field of about 1 . .
4) as-pheric lens from
Asphericon . The spontaneously emittedphotons are counted by a Hamamatsu photo multipliertube (PMT) after being filtered by an interference filterat 493 nm with a bandwidth of 20 nm from
Semrock .If we consider the probability of spontaneous emissionof photons from the state | P / i to | S / i as r , theprobability of emission into the state | D / i is 1 − r . LaserPolarization AO1 650 nmDL ProReferenceCavityIon Trap WavemeterAO2AO3 493 nmSHG ProDDSBoardPMTMCSCard Ni-USBDAQ PCMagneticFieldVacuumChamberCollectionLens Te Cell Feedback493nm986nm650nmElectrical Communication PathsviaLabviewOpticalMixing S D P t =7.92 ns FIG. 1. (Color online) Schematic diagram of our experiment:During the experiment, AO1 and AO3 are switched using aDDS controlled by NIDAQ. Relevant atomic levels of Ba + ions are also shown. The branching fractions reported hereare r and 1 − r for the two decay channels while the upperstate life-time is the best literature value available [22]. Now if only the green beam is on, there will be on average h n i number of photons emitted before the ion settles tothe state | D / i . Therefore the number of average greenphotons emitted is [20] h N g i = h n i − r − r . (2)Once we have transferred the ion with only the greenlaser beam, we can perform a similar back transfer fromthe | D / i to the | P / i state by applying the red laserbeam alone. During this transfer and back transfer, wemeasure the average number of green photons emitted.Thus we get r = N g N g + N r , (3)where r is the branching fraction for the 6 P / − S / transition and N r is the total counts of green photonswhile the red laser is kept on. As is evident from eq.(2)and eq.(3), there is no dependence of r on the intensityand detuning of the excitation laser or the efficiency ofthe detector. The time sequence used to implement thescheme is shown in figure (2). The average count N g and N r are measured for 10 µ s and 20 µ s in steps (a)and (e) respectively. An equivalent time is also spent tocollect background counts for subtraction to obtain theactual counts of N g and N r originating from the ions.The time sequence is controlled by National InstrumentDAQ (NIDAQ/USB 6363), while the photon counts areregistered into a multi channel scalar (MCS) from Ortec with a resolution of 100 ns. To work in the linear re-gion of the detector’s response, the intensity of the greenlaser has been kept low throughout these measurements. (f)(e)(c) (d) (e) (b)(a) (a) (c) C oun t s p e r n s b i n w i d t h Time/ sec
85 90 95 100 105
FIG. 2. (Color online)Experimental sequence and green pho-ton count measurement. The top panel shows the exper-imental time sequence of the green and red pulses whilethe corresponding photon counts are shown in the middlepanel. The sequence consists of (a) green photon measure-ment (green laser on), (b) green background counts (greenlaser on), (c) cooling (both lasers on), (d) optical pumpingto D -state (green laser on), (e) red-repumping while greenphoton counting (red laser on), (f) dark count measurement(both lasers off) and (c) another cooling pulse. The lowerpanel shows zoomed part of both the decay curves due totransfer and back-transfer of population. The pulsing of the lasers is done by intensity modulat-ing the AO1 for the green and AO3 for the red using adirect digital synthesizer (DDS) which drives the AOMsvia amplifiers.We repeated the experimental sequence as mentionedin figure (2) for 27 different experimental sets, each ofthese measurements have 2 × cycles. The total timerequired is mainly limited by the required uncertaintywhich we targeted to be below 0 .
1% for the branchingfraction measurement. In order to check for systemat-ics we have performed about 50 similar experiments un-der different experimental conditions like varying mag-netic field, laser intensity, added micro-motion, differentCoulomb crystal structure etc. . None of the above var-ied condition showed variation in the value of r abovethe 1 σ statistical variation. The laser intensity, howeverneeds to be within certain range as too high value wouldparalyze the PMT in short time scales and too low valuewould make the decay exponent too long for countingin reasonable time. In order to check for any birefrin-gence in our detection setup leading to disproportionatered to green counts due to any possible polarization de-pendence, we purposely changed the polarization angleof the linearly polarized green and the red beams show-ing no significant deviation beyond the statistical uncer- TABLE I. Error budget for the S − P branching fraction mea-surement. parameter shift uncertainty Detector dead time 70 ns a +5 × − × − Photon counting (statistical) 1 . × − Photon counting(finite measurement time) 6 . × − . × − Hamamatsu H7421-40 tainty. Even though the PMT is in the non-paralyzingregime, a significant contribution to our systematic un-certainty comes from the detector dead time. The deadtime of the PMT leads to a calculable shift as well asuncertainty. The systematic shifts and uncertainties aretabulated in table(I). As is evident, the main limitationto the measured uncertainty is statistical and hence, lim-ited by the finite measurement time. A relatively largeshift of r is due to the detector dead-time of 70 ns com-bined with lower resolution in our MCS binning (about100 ns). Nevertheless, the uncertainties are similar tothe statistical uncertainties thereby not limiting the finalvalues.The measured values of the branching fractions, alongwith the best literature value of the upper state life-time,7 . ± .
08 ns [22], provides the transition probability aswell as the matrix elements of the relevant transitionsby following the procedure in [15]. In table(II), we showour results along with the values measured or calculatedtill date. The best measured experimental data on thesetransitions are limited to about 5% uncertainties on thematrix elements. On the contrary, since the experimentalproposal by Fortson [13] for the possibility of measuringthe atomic PNC in barium ion, the accuracies on the the-oretical values of the matrix elements has improved sig-nificantly aiming towards below one percent level wheremany-electron correlation effects become significant. Ascan be seen from figure (3), the theory values scatterwithin the experimental uncertainties while the claimedtheoretical uncertainties are significantly lower than ex-periments [18]. Our measurements, for the first time,provides the values below one percent limit, thereby al-lowing the theories to be compared at a similar uncer-tainty level. The S − P transition probability is ratherclose to the theory values of [15–17] but it is quite offfrom the value of [18]. On the other hand, the P − D transition probability is close to the values calculatedby [15, 17, 18] while deviating significantly from [16].These theories mostly consider all orders in perturbationbut limited to certain number of collective excitations,therefore it is now possible to make a comparative studyof these different approaches in view of the experimen-tal data. The branching fractions themselves are impor-tant for estimating the abundance of barium in solar and TABLE II. A comparison of the values of the branching fractions, transition probabilities and matrix elements for Ba + betweendifferent experiments and theories arranged chronologically. Transition involved Branching fraction Transition probabil-ity × s − Transition matrix(a.u.) reference | P i − | S i . ± .
021 0 . ± .
07 3 . ± .
11 [25]0 . ± .
095 3 . ± .
15 [26]0 . .
300 [17]0 . ± .
012 0 . ± .
024 3 . ± .
038 [23]0 . ± .
09 3 . ± .
14 [19]0 . .
309 [16]0 . .
333 [15]0 . ± .
095 3 . ± .
15 [24]0 .
978 3 .
405 [18]0 . ± . . ± .
009 3 . ± .
014 this work | P i − | D i . ± .
021 0 . ± .
04 2 . ± .
18 [25]0 . ± .
083 3 . ± .
37 [26]0 .
334 3 .
007 [17]0 . ± .
012 0 . ± .
018 2 . ± .
084 [23]0 . ± .
019 3 . ± .
085 [19]0 .
37 3 .
165 [16]0 .
326 2 .
971 [15]0 . ± .
031 2 . ± .
15 [24]0 .
331 2 .
993 [18]0 . ± . . ± . . ± .