Branching fractions for P 3/2 decays in Ba +
Zhiqiang Zhang, K. J. Arnold, S. R. Chanu, R. Kaewuam, M. S. Safronova, M. D. Barrett
BBranching fractions for P / decays in Ba + Zhiqiang Zhang, K. J. Arnold,
1, 2
S. R. Chanu, R. Kaewuam, M. S. Safronova,
3, 4 and M. D. Barrett
1, 5, ∗ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore Temasek Laboratories, National University of Singapore, 5A Engineering Drive 1, 117411 Singapore Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA Joint Quantum Institute, National Institute of Standards and Technologyand the University of Maryland, College Park, Maryland, 20742 Department of Physics, National University of Singapore, 2 Science Drive 3, 117551 Singapore
Branching fractions for decays from the P / level in Ba + have been measured with a singlelaser-cooled ion. Decay probabilities to S / , D / and D / are determined to be 0 . . . D / lifetime, for which we obtain30.14(40) s. PACS numbers: 06.30.Ft, 06.20.fb
I. INTRODUCTION
Singly-ionized Barium has been well studied over theyears with a wide range of precision measurements [1–8]providing valuable benchmark comparisons with theory[9–14]. Such comparisons have been motivated in partby the proposed parity nonconservation (PNC) measure-ment using the S / − D / transition in Ba + [15].Recently, we have proposed that accurately measuredproperties of Ba + can be combined to provide an ac-curate model of the dynamic differential scalar polariz-ability ∆ α ( ω ) for the S / − D / clock transition [16].With such a model, ac-Stark shifts of the clock transitioncould provide an in situ calibration of laser intensitiesand comparison of polarizabilities with another species.A limiting factor for the proposal was the accuracy of ex-isting values for P / branching ratios. Moreover, recentmeasurements of P / branching fractions [17] indicatedpossible problems with existing P / values [8]. Conse-quently, we have remeasured these values with an orderof magnitude improvement in their uncertainties.Our measurements only involve optical pumping andstate detection. Consequently, results are insensitive tolaser intensity, frequency, and polarization changes pro-vided they do not significantly affect optical pumpingtimes and detection. Our implementation also allows thebranching ratios to be determined in two different ways,which we do by way of a consistency check on our results. II. EXPERIMENT OVERVIEW
The relevant level structure for Ba + is given inFig. 1(a). Doppler cooling is provided by driving thetransitions at 493 and 650 nm, with light scattered at ∗ [email protected] S / P / P / D / D / Ba +
585 nm 455 nm 493 nm650 nm614 nm ˆyˆzˆx ˆB = cos( ✓ ) ˆx + sin( ✓ ) p ( ˆy + ˆz ) (a) (b) FIG. 1. (a) Level structure showing the Doppler cooling anddetection transitions at 493 and 650 nm, and optical pumpingtransitions at 455, 585, and 614 nm, (b) Beam configurationsrelative to the trap. The magnetic field is specified with re-spect to the axes shown and has θ ≈ ◦ .
650 nm collected onto a single photon counting module(SPCM) for detection. The remaining three transitionsat 455, 585, and 614 nm facilitate optical pumping as re-quired. Throughout the report, we refer to S / and D / states collectively as the bright state, and D / states asthe dark state.The experiments are carried out in a linear Paul trapwith axial end-caps as described in [18, 19]. The laserconfigurations relative to the trap are shown in Fig. 1(b).An applied magnetic field of ∼ .
31 mT with the orien-tation given in Fig. 1(b) is sufficient to avoid any darkstates, independent of the polarization state of any of thelasers.Probabilities for decay from P / to S / , D / , and D / are here denoted p , p , and p , respectively. Theyare inferred from population measurements following theoptical pumping sequences shown in Fig. 2 where it isgiven that each sequence is preceded by preparation intothe appropriate bright or dark state. The remainingbright state population for the sequences shown in (a), a r X i v : . [ phy s i c s . a t o m - ph ] M a r (b), and (c), are p , q and 1 − r , respectively, where p = p p + p , q = p p + p , (1)and r = p p (1 − p )(1 − p ) . (2)Since (cid:80) p k = 1, only two measurements are needed touniquely determine p k . In addition, each of the sequencesin Fig. 2 can be repeated m times within a single exper-iment to provide measurements of p m , q m , and 1 − r m .This can provide better statistics with a similar approachhaving been demonstrated with Ca + [20]. Here we pro-vide measurements of p and q , and use measurements of p m and 1 − r m over a range of m to check for consistency. III. MEASUREMENTSA. Measurement of q
Measurements of q involved optical pumping withlasers at 650, 455, and 614 nm with measured pumpingtime constants of 0.3, 1.0, and 1 . µ s, respectively. How-ever, as the 455-nm laser was only locked by a waveme-ter, occasional frequency excursions of ∼
10 MHz couldoccur, significantly changing the pumping rate. Conse-quently, the pump rate was also measured within eachcycle of the experiment. In addition, measurements of adark and bright state were also made to assess the de-tection efficiency in each case. Thus, a single experimentcycle consisted of four interleaved experiments.Explicit pulse sequences for each of the four experi-ments are given in table I. The cooling pulse in all fourcases had a duration of 400 µ s and included repump lightat 614 nm to ensure the atom is pumped to the brightstate. The first two experiments use lasers at 455 and650 nm to optically pump to D / ; partial pumping witha 1 µ s pulse monitors pumping rate of the 455-nm transi-tion, and complete pumping with a 30 µ s pulse preparesthe ion in the dark state. The third experiment usescomplete optical pumping of the 455-nm transition aloneto determine q , and the final experiment provides a mea-surement of the bright state. The full cycle was repeatedin blocks of 1000. Between each block, 10 measurementsof the counts in 1 ms were made for both the bright stateand background, where the latter was taken with the493-nm laser turned off.Over the course of two days, ∼ . × − for detecting dark. These events are not, how-ever, Poisson distributed. In all cases where two or moredark events occured, they were clustered in consecutiveexperiments suggesting a collision event had inhibited TABLE I. Interleaved experiments made when measuring q and the pulse sequences used for each. Cooling pulses hada duration of 400 µ s and included repump light at 614 nm inaddition to the detection light at 493 and 650 nm.Expt. 1 Expt. 2 Expt. 3 Expt. 4Cooling Cooling Cooling Cooling650 (20 µ s) 650 (20 µ s) 650 (20 µ s) -650/455 (1 µ s) 650/455 (30 µ s) 455 (30 µ s) -Detect. Detect. Detect. Detect. bright detection through multiple 400 µ s cooling cycles.Consequently any blocks of 1000 cycles that have morethan one dark state detection event during expt. 4 wereomitted. This eliminates only 15 blocks, or ∼ .
1% ofthe data.From the remaining data we obtain an estimate q =0 . q estimated from blocks of size 10 are given inFig. 3(a) where the uncertainties for each point are theexpected projection noise, and the reduced χ is 0.84.The fractional standard deviation of mean values esti-mated from blocks of variable size N is given in Fig. 4(b)to further demonstrate that the measurement continuesto average down in accordance with the projection noiselimit given by the solid line. B. Measurement of p
As illustrated in Fig. 2 (a), measurement of p entailsoptical pumping with lasers at 493 and 585 nm, whichhad measured pumping time constants of approximately500 and 300 ns, respectively. The 585-nm light was gen-erated by an 1170-nm diode laser that was locked to afrequency comb [21] and frequency doubled by a fiber-coupled, waveguide doubler. The result was a very stablescattering rate compared to that with the 455-nm laserused in the measurement of q . Thus, the typical experi-ment cycle consisted of only two interleaved experiments:a bright state detection, to monitor collision events, andthe measurement of p . The latter consisted of 400 µ s ofDoppler cooling, 40 µ s of optical pumping of the 493-nmtransition to D / , 40 µ s of optical pumping with 585-nm light, and then detection. The pumping rate of the585-nm transition was additionally monitored in-betweeneach block to confirm its stability.As for the measurement of q , data is conditioned onthe bright state detection experiment removing 16 out of43100 blocks. From the remaining ∼
43 M experiments,we obtain the estimate p = 0 . p estimated from blocks of size 10 are given inFig. 4 (a) where the uncertainties are the expected pro-jection noise, and the associated reduced χ is 0.76. Thefractional standard deviation of mean values estimatedfrom blocks of variable size N is given in Fig. 4 (b) to S / P / P / D / D / S / P / P / D / D / S / P / P / D / D / S / P / P / D / D / S / P / P / D / D / S / P / P / D / D / (a) (b) (c) FIG. 2. Optical pumping schemes used in this work. (a), (b), and (c) provide measurements of p , q , and 1 − r , respectively,with p, q, r as defined in the text. Population is given to be distributed in S / and D / at the start for (a) and (b), andoptically pumped to D / for (c). In each case the two steps given can be repeated with m repetitions giving populations p m , q m , and 1 − r m in the bright states S / and D / . experiment block iteration ( q ) experiment block size s t d d e v ( q i ) / q (a)(b) FIG. 3. Results for the measurements of q . (a) Mean values of q estimated from blocks of size N = 10 , with ∆ q denoting thevalue relative to the mean of all data. Uncertainties for eachpoint are the expected projection noise for the given N , andthe shaded region is the uncertainty in the mean of all data.(b) Fractional standard deviation of mean values estimatedfrom blocks of size N . further demonstrate the measurement continues to aver-age down in accordance with the projection noise limitgiven by the solid line. C. Measurements of p m and 1-r m In a similar manner to the previous two subsections,consistency checks were carried out by cycling the pump-ing schemes in Fig. 2 (a) and (c) to measure p m and1 − r m for a range of m . Values for the estimated p and r for each value of m are plotted in Fig. 5 whereeach measurement consists of approximately 10 experi-ments. Taking the weighted average over m in each casegives p = 0 . r = 0 . m = 1 result in the estimate for p .The value for p is in good agreement with m = 1 re-sult p = 0 . experiment block iteration ( p ) experiment block size s t d d e v ( p i ) / p (a)(b) FIG. 4. Results for the measurements of p . (a) Mean valuesof p estimated from blocks of size N = 10 , with ∆ p denotingthe value relative to the mean of all data. Uncertainties foreach point are the expected projection noise for the given N , and the shaded region is the uncertainty in the mean ofall data. (b) Fractional standard deviation of mean valuesestimated from blocks of size N . value of r is also in good agreement with r = 0 . r = p (1 − q ) p + q − pq , (3)and the measured values of p and q from the previoussections. D. The D lifetime
A recent measurement [22] gave a D / lifetime of25.6(0.5) s, which was significantly less than previousexperimental reports [23–27] and numerous calculations[28]. As it factors into consideration of systematic ef-fects, we provide our own measurement for reference pur-poses. Our method continuously monitored a string of ( p ) m ( r ) (a)(b) FIG. 5. Estimated values of p and r from measurements of p m and 1 − r m where m is the number of pumping cycles withina single experiment as discussed in the text. Values for p arerelative to the m = 1 result found in Sect. III B. Values for r are given relative to the value found using Eq. 3 and values of p and q from Sect. III A, and III B. In both cases, the dashedline is the weighted mean over m and the shaded region thecorresponding uncertainty. The weighted mean for p excludesthe m = 1 result. four ions initialized with typically one or two ions in thebright state and registered when each ion became bright.Detection light was imaged onto an electron multiplyingcharged-coupled device (EMCCD) camera with an imag-ing system having sufficient resolution to resolve the stateof individual ions from a 10 ms exposure with typicallybetter than 99% efficiency. Images were captured every20 ms until all ions were detected bright, which was ver-ified by a further ten images. In these experiments, the614 light was blocked at all times.The initial state of the ion-string was set by opticalpumping to | S / , m = 1 / (cid:105) and driving | S / , m =1 / (cid:105) ↔ | D / , m = 1 / (cid:105) with a clock laser at 1762 nm.Ideally, the initial string would have one bright ion to fa-cilitate sympathetic cooling for other ions in D / . How-ever, the configuration could only be done probabilisti-cally, which depends on the spatial profile of the beam,and pulse duration of the clock laser. Instances in whichno ions are transferred to D / simply trigger a restart.Records starting with all ions dark are still used, so longas ions appear bright one at a time and the crystal isstable.Within the detection records, there are four identifi-able events in addition to the expected transition to thebright state: detection errors, swaps, melts, and darktransitions. Detection errors are identified by a singleimage that is different from previous and subsequent im-ages. Swaps, as the name would imply, are a changein the ion configuration without a change in number ofbright ions. Melts are characterized by all ions appear-ing dark for several images. They are presumably theresult of a high impact collision, which destabilizes theion crystal and may take several seconds to recrystallize.Finally, dark transitions are identified by a decrease in time (s) o cc u r a n c e s FIG. 6. Histogram of observed decay times from D / .Black dots are the expected values from the estimated τ D =30 . χ for the observed histogram is 0.99. the number of bright ions and are the result of very rare,off-resonant excitation of the 493-nm cooling light.Approximately 32 hours of data were collected fromthe same string of four ions with a total 2429 detectionrecords starting with at least one dark ion. An additional48 records were discarded as quantum jumps could notbe unambiguously identified due to: (a) multiple ionsappearing bright when starting from an all dark state(9); (b) the all bright verification step failing indicatinga detection error triggered the last event (25); (c) thenumber of bright ions changing after a melting event (11);or (d) multiple ions transitioning at the same time (3).Within the 2429 detection records there were 5620bright transitions, 179 detection errors, 253 swaps, 50melts, and 20 dark transitions. As all quantum jumpsare recorded, the average of all bright transition times, τ D = 30 . τ D and num-ber of events recorded. The reduced χ for the observedhistogram is 0.97.Note that it is important that all quantum jumps areobserved in a given record when using this approach.When monitoring the decay of N ions, the time for thefirst decay is governed by an exponential distributionwith parameter N λ , where λ is the decay rate for oneion. Hence, an observed set of times { T k } has a distribu-tion P ( { T k } ) = e − NλT e − ( N − λ ( T − T ) · · · e − λ ( T N − T N − ) = (cid:89) k e − λT k , (4)which is the expected N independent, identical distribu-tions for a single decay. This would not be the case ifsome events were missed.With the exception of the most recent report [22], thevalue of τ D is consistent with previous reports [23–27] and o cc u r a n c e s FIG. 7. Histogram of the estimated time constants associ-ated with the 455-nm pumping rate determined throughoutthe measurement of p . These can be compared to the totalpumping time of 40 µ s used in the experiment. a number of theoretical estimates [28]. Collision-induceddecays are not likely significant in this experiment; ofthe 5620 observed decays, only 12 included an observablechange in the ion configuration. In any case, such effectswould decrease the observed lifetime. IV. SYSTEMATICS
The experiments only involve optical pumping andstate detection, which results in relatively few consid-erations for systematic effects. These are finite opticalpumping times, unwanted scattering from leakage light,finite lifetime of D / and D / , collisions, and detectionerrors. The effect of finite lifetimes for D / and D / have the same effect as leakage light at 650 and 614 nm,respectively, and are considered in this context. Simi-larly collisions and detection errors are also consideredin a similar context. A. Finite pumping times
With the exception of the lasers at 455 and 614 nm,which were locked to a wavemeter, all others lasers werelocked to stable references giving correspondingly stablepumping rates. In the latter cases pumping times weremore than 20 time-constants of the associated pumpingrates, corresponding to a probability of < × − forpumping errors. Hence only the 455- and 614-nm pump-ing times might be of concern.As noted in Sect. III B, the 455-nm pumping rate wasmonitored in an interleaved experiment during the mea-surement of p by measuring the population after partialpumping for a fixed time. From these measurements thepumping rate was estimated and a histogram of the as-sociated time constant is given in Fig. 7. Almost all ex-periment cycles have pumping times more than 20 time-constants of the measured pumping rates rendering thiseffect is negligible at the (cid:46) − level. Repumping on the614-nm transition occurs during the 400 µ s cooling whichis far longer than the pumping time constant and thus insensitive to variations. The pumping rates for transi-tions at 455 and 614 nm were not monitored during themeasurements of p m and 1 − r m in Sect. III C. Howeverthe consistency of those results with the independentlymeasured p and q demonstrate that the finite pumpingtimes are not important to the accuracy reported. B. Leakage pumping rates
Here we describe the effects of stray light from eachof the lasers and the experiments used to bound theireffects. If a laser was not needed for any measurementsin Sect. III, it is given that it was physically blocked andnot merely switched off via an acousto-optic modulator(AOM).
1. Stray light at 455 nm
The 455-nm laser was derived from a 910-nm diodelaser, which was switched by a double-pass acousto-opticmodulator (AOM) before a single-pass frequency dou-bling crystal. Hence the extinction ratio was very high.This was tested by optical pumping into S / , and wait-ing 10 ms before detection and cooling (with light at614 nm blocked). In ∼
672 k cycles we observed oneevent pumping to D / with three others attributed tocollisions. The latter are identified by a sudden drop incounts that slowly recovers to the normal level.The main effect of stray light at 455 nm would be to in-fluence bright state detection, by pumping into the darkstate, and measurements made of r m , by repumping dur-ing pumping of the 614-nm transition. In any case, givena pumping rate of (cid:46) / hour and the typical experimenttime of < µ s including pumping and detection, theprobability of unwanted pumping is (cid:46) × − .
2. Stray light at 614 nm
Stray light from the 614-nm laser effectively decreasesthe lifetime of the D / level, which was measured tobe 30.14(40) s on a four ion string, as described inSect. III D. To test the leakage rate, we repeated this life-time measurement with the 614-nm light unblocked butthe switching AOM off. A neutral density filter attenu-ating the 614-nm beam was removed to give maximumpumping rate with the AOM on, which was measuredto be ∼
420 ns at the time of this test. From 239 de-cay events, we get an estimated lifetime of 29.7(1.9) s.This implies a scattering rate of 0 . ± . × − s − con-sistent with zero. Taking 3 × − s − as an estimatedupper bound, the probability of scattering during the ex-periment is (cid:46) × − . During the actual data runsthe measured pumping time constant was ∼ µ s so theleakage scattering rate likewise would be smaller.
3. Stray light at 585 nm
Similar to the 455-nm laser, the switch AOM for the585-nm beam was placed before the doubling stage re-sulting in high extinction. As an initial test, the ionwas left continuously scattering with the 585-nm light‘off’, but unblocked, for about half an hour with no darkevents. A further test was done by pumping to D / andwaiting for 10 ms before detection and cooling (with lightat 614 nm blocked). In 480 k experiments, we observedonly one pumping event and two collisions. As with straylight at 455 nm, given a pumping rate of (cid:46) − s − andtypical experiment time of < µ s including pumpingand detection, the probability of unwanted pumping isthen (cid:46) × − .
4. Stray light at 650 nm
Stray light from the 650-nm laser effectively decreasesthe lifetime of of D / . Unlike light at 455, 614, and585 nm, stray light at 650 nm does not affect results atthe detection step and thus one is only concerned withscattering during the pumping steps. A test for any leak-age light at 650 nm was done by pumping to D / andwaiting for τ w = 20 ms with light at 455 nm on before de-tection and cooling. If a scattering event on the 650-nmtransition results in a decay into S / , it is then pumpedto D / by light at 455 nm with probability (1 − q ) ∼ . p d = 2 . × − . From the 79.8(4.6) s lifetime of D / , we would expect a 2.5(3) × − probability of de-cay to S / in the 20 ms, of which 1 − q will be pumpeddark by light at 455 nm. The observed rate is thus dom-inated by scattering on the 650-nm transition, and therate of scattering out of D / is γ s = p d / ( τ w (1 − q )) =0 .
15 s − . This effectively reduces the lifetime of D / toabout 6 . p , there is little opportunity forleakage light at 650 nm to cause a problem because lightat 585 nm will depopulate D / on a timescale of 1 µ s.So the error caused by leakage light at 650 nm can onlybe ∼ × − . During measurements of q , decay of thepopulation from D / during the 30 µ s of optical pumpingon the 455-nm transition will be more significant. Sincethe 455-nm optical pumping rate is much shorter than thepumping time, we make the approximation that any D / decay is immediately pumped to D / with probability1 − q . The error is then γ s × µ s × (1 − q ) ≈ × − .
5. Stray light at 493 nm
The 493-nm beam is double-passed by two AOMs andthe measured extinction was in excess of 130 dB. Wenonetheless attempted to measure this by pumping to S / , waiting 10 ms before optical pumping on the 585-nm transition before detection and cooling, all with light at 614 nm blocked. From 40 k cycles we saw no shelvingevents. Thus the scattering rate out of S / would be (cid:46) − s − . As with stray light at 650 nm, this does notaffect detection, and hence the error is (cid:46) × − . C. Collisions and detection errors
For the branching ratio measurements reported herewe used a Bayesian detection scheme [29, 30] in which thebright (dark) state probability is estimated in real timeand terminates when the bright (or dark) state probabil-ity falls below a preset threshold, here set to 10 − . Countrates during the experiments were typically in the range ∼ −
40 counts/ms for a bright state, and a more stable0.4 counts/ms for a dark state.From the measurements sequences used for q , the ob-served probability of detecting dark during expt. 4 was7 . × − , which is taken as an estimate of a brightstate detection error. Similarly, the probability of detect-ing a bright state during expt. 2 was 1 . × − , whichis taken as an estimate of a dark state detection error. Inboth cases the observed error rates are significantly abovethe 10 − thresholds set in the Bayesian detection algo-rithm. Possible causes for excess detection errors wouldbe collisions and the D / lifetime for bright and darkstates, respectively.The bright state control experiments taken during themeasurements of p , and q and records for the D / life-time measurement gave clear evidence of collisions. Fromthe collision events identified in the measurements of p and q , the estimated collision rate would be roughly 1every 48 minutes. Similarly data from the D / lifetimesuggests a collision rate of 1 every 6 minutes, which isroughly consistent given the larger number of ions. Col-lision events occurring on this timescale, however, wouldnot significantly contribute to the observed error rates,more so that these events were detected and removed.However, those events were collisions with sufficient im-pact to interrupt cooling over many experimental cycles.Less impactful collisions that interrupt only a single ex-periment would likely have a higher rate. We also notethat varying fluorescent rates can influence the validityof the threshold. Simulations and independent experi-ments indicate this can be as much as a factor of fourfor a 10% drop in fluorescence. In any case, the observedbright state error rate is not significant at the level of thestatistical uncertainties in the reported results.For the dark state control experiment, decay duringdetection would likely trigger a bright state result. Theprobability of this occurring depends on the distributionof detection times. For our detection parameters, the av-erage time to detect a dark state is ∼ µ s and the ex-pected probability to decay within this time is 1.3 × − .Although clearly a contributing factor, it is still smallerthan the observed error rate. This could be due to statepreparation errors associated with pumping into the darkstate but, in any case, the observed dark state error rate TABLE II. Probabilities for decay from P / to S / , D / ,and D / , which are denoted p , p , and p , respectively. Val-ues are determined from the measured values of p and q givenin Eq. 5. Theory values were estimated as discussed in [17]. p p p Expt. 0.741717(71) 0.028031(23) 0.230253(62)Theory 0.7423(18) 0.0280(2) 0.2297(15) of < × − is not significant at the level of the statis-tical uncertainties in the reported results. V. DISCUSSION
The dominant systematic in the branching ratios re-ported here are from detection errors and are not signif-icant at the level of the statistical uncertainties. Thusour final values are given by p = 0 . , and q = 0 . . (5)From these we can calculate the probabilities for decayfrom P / to S / , D / , and D / , denoted p , p , and p , respectively. Values are given in table II along withthose calculated from theory for comparison. As dis-cussed in [17], theory values were calculated using a lin-earized coupled-cluster approach including single-doubleexcitations with uncertainties estimated from additionalcalculation methods.The experimental values of p k differ by 2-3 standarddeviations of the results reported in [8], but are in reason-able agreement with the values predicted in our previouswork [17]. The latter used measured branching fractionsfor decays from P / , experimental matrix elements from[4], and theoretical estimates of matrix element ratios.With the above branching fractions and the reduced ma-trix element (cid:104) P / (cid:107) r (cid:107) S / (cid:105) = 4 . . u . reported in[4], matrix elements associated with P / decays and thelifetime can now be obtained using only experimentallydetermined values. The remaining matrix elements arefound to be (cid:104) P / (cid:107) r (cid:107) D / (cid:105) = 4 . . u ., (6a) (cid:104) P / (cid:107) r (cid:107) D / (cid:105) = 1 . . u ., (6b)which yields a lifetime of τ ( D / ) = 6 . π × . (cid:104) P / (cid:107) r (cid:107) S / (cid:105) .The matrix elements given in Eq. 6 are in excellentagreement with theoretical calculations [16], but differfrom estimates obtained using experimental results com-bined with ratios of matrix elements calculated from the- ory [17]. The ratios in question are R = (cid:104) P / (cid:107) r (cid:107) S / (cid:105)(cid:104) P / (cid:107) r (cid:107) S / (cid:105) , (7a) R = (cid:104) P / (cid:107) r (cid:107) D / (cid:105)(cid:104) P / (cid:107) r (cid:107) D / (cid:105) , (7b)and R = (cid:104) P / (cid:107) r (cid:107) D / (cid:105)(cid:104) P / (cid:107) r (cid:107) D / (cid:105) , (7c)which can now be obtained from the experimental resultsin Ref. [4], the P / branching ratio reported in Ref. [17],and values given in Eq. 5. Specifically we have R = (cid:115)(cid:18) − pp (cid:19) (cid:18) − ¯ p ¯ p (cid:19) (cid:18) ω ω ω ω (cid:19) / R , (8)and R = (cid:114) q − q (cid:18) ω ω (cid:19) / R , (9)where ¯ p is the branching fraction for the P / level re-ported in [17].Values for R , R /R , R /R , and R /R obtainedfrom experimental results are given in table III. Uncer-tainties are derived in the usual manner by taking thequadrature sum of independent contributions with theexception of R for which relative uncertainties from eachmatrix element are added to allow for possible correlationin their determination. Theory values are also given forcomparison with uncertainties determined from the max-imum discrepancy between different computation meth-ods [17]. As noted in [17], calculated ratios are expectedto be accurate as they depend only weakly on correla-tion corrections. Consequently, the 1 . σ and 1 . σ in thedifference for R and R /R between experiment andtheory maybe statistically significant. We would there-fore recommend a 2 σ estimate of uncertainties for derivedquantities such as the matrix elements in Eq. 6 and thecorresponding estimates of τ and Γ.In general, theoretical calculations for Ba + are in ex-cellent agreement with experimental results derived fromthe matrix elements reported in [4] and branching ratiosreported here and in [17]. In all cases the matrix elementsare better than 1% of the experimentally determined val-ues. As branching fractions provide ratios of matrix ele-ments, all values are tied to the values of (cid:104) P / (cid:107) r (cid:107) S / (cid:105) and (cid:104) P / (cid:107) r (cid:107) S / (cid:105) reported in [4]. Thus, it would be ofinterest to confirm those results by a different method-ology such as that demonstrated in [31, 32]. However,it would likely be difficult to reach a comparable level ofaccuracy via that approach.In summary we have provided new branching ratiomeasurements for decays from P / . The new values are TABLE III. Matrix elements ratios R , R /R , R /R , and R /R , where R k are as given in Eq. 7. Experimental valuesare derived from the measured values of p and q given inEq. 5, matrix elements reported in [4], and the P / branchingfraction reported in [17]. Theory values were estimated asdiscussed in [17]. Expt Theory R R /R R /R R /R more than an order of magnitude more accurate thanthose given in previous reports and measurements havebeen done in two different ways to check for consistency. Together with results in [4] and [17], they provide acomplete set of experimentally determined matrix ele-ments for all the dominant contributions to the differ-ential scalar polarizability, ∆ α ( ω ) of the S / ↔ D / clock transition. In addition, we have provided a newmeasurement of the D / lifetime, which also improvesupon previous values. ACKNOWLEDGMENTS
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