Anomalously small blackbody radiation shift in Tl+ frequency standard

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by  Z. Zuhrianda

Anomalously small blackbody radiation shift in Tl+ frequency standard
Z. Zuhrianda1, M. S. Safronova1, and M. G. Kozlov2
1 Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA and
2 Petersburg Nuclear Physics Institute, Gatchina, Leningrad District, 188300, Russia
(Dated: February 1, 2012)
Theoperationofatomicclocksisgenerallycarriedoutatroomtemperature,whereasthedefinition
2 ofthesecond refers totheclock transition in an atom at absolute zero. Thisimplies thattheclock
1 transition frequency should be corrected in practice for the effect of finite temperature of which
0 the leading contributor is the blackbody radiation (BBR) shift. In the present work, we used
2 configuration interaction + coupled-cluster method to evaluate polarizabilities of the 6s2 1S0 and
an s6hsi6fpt 3oPf0thseta6tses6pof3TP0l+−io6ns;2w1eSfi0nTdl+α0t(r1aSn0s)it=ion19a.6t a3.0u0.Kandisα∆0(ν3BPB0R)==2−10.4.0a1.5u7.(.1T6)heHrze.suTlthinisgrBesBuRlt
J demonstrates that near cancelation of the 1S0 and 3P0 state polarizabilities in divalent B+, Al+,
In+ ions of group IIIB [Safronova et al., PRL 107, 143006 (2011)] continues for much heavier Tl+,
1
leading to anomalously small BBR shift for this system. This calculation demonstrates that the
3 BBR contribution to the fractional frequency uncertainty of the Tl+ frequency standard at 300 K
is1×10−18. WefindthatTl+ hasthesmallest fractional BBR shift amongall presentorproposed
]
h frequency standardswith the exception of Al+.
p
- PACSnumbers: 06.30.Ft,31.15.ac, 31.15.ap,31.15.am
m
o
t I. INTRODUCTION ns2 1S0 andmetastablensnp3P0 states arenearlyequal
a toeachotherinB+,Al+,andIn+,allofwhicharegroup
.
s IIIB ions. Asaresult,these threeionshaveanomalously
ic ledRteocenutnpardevcaednecnesteidniamtpomroivcemanedntsopintictahlepahcycsuicrsachyavoef small BBR shifts of the ns2 1S0−nsnp 3P0 clock tran-
s sitions. The fractional BBR shifts for these ions are at
y optical frequency standards that are essential for many
least 10 times smaller than those of any other present or
h applications including measurements of the fundamental
proposed optical frequency standards at the same tem-
p constantsandsearchoftheirvariationwithtime,testing
[ perature, and are less than 0.3% of the Sr clock shift.
ofphysicspostulates,inertialnavigation,magnetometry, Optical frequency standard based on 204Tl+
v1 tcrloacckkinwgitohfadereepc-osrpdacloewprfroabcetsi,oannadl forethqueresn[c1y].uAncnerotpatiinctayl 6s2 1S0 mF = 0 – 6s6p 3P0 mF′ = 0 transition
1 of 8.6×10−18, based on quantum logic spectroscopy of was proposed in Ref. [7]. The radioactive isotope of
3 an Al+ ion was demonstrated in 2010 [2]. 204Tl has a half-life of 3.78 years,a spin of 2, and a very
6 small magnetic moment of 0.0908 nuclear magnetons
6 Anydefinitionofthesecondshouldbebasedonaclock making it ideal object for very high-resolution laser
. decoupledfromitsparticularenvironment. Thermalfluc- spectroscopy [7]. Because of its small nuclear magnetic
1
0 tuations of the electromagnetic field, i.e. blackbody ra- moment the natural linewidth of the clock transition in
2 diation(BBR),arepervasiveandcanonlybesuppressed 204Tl+ is expected to be orders of magnitude smaller
1 bycoolingtheclock. TheBBRatanynon-zerotempera- than estimated for stable Tl isotopes [7]. The BBR
: ture induces small shifts in atomic energy levels through in this frequency standard have not been previously
v
i the AC Stark effect. The operation of atomic clocks is estimated. Since three group IIIB ions exhibit very
X generally carriedout at roomtemperature and the clock smallBBR shifts, it is veryinteresting to evaluate if this
r transition frequency should be corrected in practice for trend holds for much heavier Tl+.
a
the BBR shift. Experimental measurements of the BBR The BBR frequency shift of the clock transition can
shiftsaresufficientlydifficultthatnodirectmeasurement be related to the difference of the static electric-dipole
hasyetbeenreportedforopticalfrequencystandards. At polarizabilities between the clock states, ∆α0, by [5]
room temperature, the BBR shift of a clock transition
turnsouttomakeoneofthelargestirreduciblecontribu- 4
1 T(K)
tions to the uncertainty budget of optical atomic clocks ∆νBBR =− (831.9 V/m)2 ∆α0(1+η), (1)
2 (cid:18) 300 (cid:19)
[3]. The present status of the theoretical and experi-
mentaldeterminationsofthe BBRshifts inallfrequency where η is a small dynamic correction due to the fre-
standards was recently reviewed in [3, 4].
quency distribution and only the electric-dipole transi-
The BBR frequency shift of a clock transition can be tion part of the contribution is considered. The M1 and
related to the difference of the static electric-dipole po- E2contributions havebeen estimated for Al+ andfound
larizabilities between the two clock states [5]. Recent tobenegligible[6]. Therefore,thecalculationoftheBBR
work[6]demonstratedthatthe polarizabilitiesofground shift reduces to accurate calculation of the static polar-
2
izabilities of the clock states and dynamic correction η. MBPT and all-order terms were summed over the entire
Inthiswork,weevaluatepolarizabilitiesofthe6s2 1S0 N =35, l≤5 basis set.
and6s6p3P0 statesinTl+,correspondingBBRshiftand Themultiparticlerelativisticequationforthreevalence
its uncertainty. Dynamic correction to the BBR shift is electrons is solved within the CI framework [12] to find
evaluatedandfoundnegligible. Wealsocalculateanum- the wave functions and the low-lying energy levels:
ber of electric-dipole matrix elements in Tl+ for transi-
tions between low-lying levels. We note that our cal- Heff(En)Φn =EnΦn.
culation of all of these properties is independent on the The effective Hamiltonian is defined as
particularisotopenumberwellwithinthe quotedlevelof
precision. Therefore, all these results apply to any Tl+ Heff(E)=HFC+Σ(E),
isotope.
whereHFC isthe Hamiltonianinthefrozen-coreapprox-
imation and the energy-dependent operator Σ(E) takes
into account virtual core excitations. The Σ(E) part of
II. METHOD
theeffectiveHamiltonianisconstructedusingthesecond-
order perturbation theory in the CI+MBPT approach
Correlationcorrectionsbetweenafewvalenceelectrons
[8] and linearized coupled-cluster single-double method
can be accurately treated by the configuration interac-
in the CI+all-order approach [9]. The Σ(E) = 0 in the
tion (CI) method. Since the valence-valence correla-
pureCIcalculation. Constructionofthe effectiveHamil-
tions are very large, the CI method provides better de-
tonianinCI+MBPTandCI+all-orderapproximationsis
scription of these correlations than the perturbative ap-
described in detail in Refs. [8, 9].
proaches. However, excitations of the core [1s2,...,5d10]
electronscannotbedirectlyincludedintheCIapproach
due to enormous size of such problem. An elegant ap- III. RESULTS
proach to the inclusion of the core-valence correlations
withintheCIframeworkwasdevelopedin[8],wherecore- Comparison of the energy levels (in cm−1) obtained
valencecorrelationswereincorporatedintotheCIbycon-
in the CI, CI+MBPT, and CI+all-orderapproximations
structinganeffectiveHamiltonianusingthesecond-order
withexperimentalvalues[13,14]isgiveninTableI.Cor-
many-body perturbation theory (CI+MBPT). Recently,
respondingrelativedifferencesofthesethreecalculations
we have developed the relativistic CI+all-order method
with experiment are given in the last three columns in
[9] combining CI with coupled-cluster (CC) approach.
%. Two-electron binding energies are given in the first
This method, first suggestedin [10], was successfully ap-
row of Table I, energies in other rows are counted from
plied to the calculation of divalent atom properties in
the ground state. We also observedsignificant, by a fac-
Refs. [6, 9]. The coupled-cluster method used here is
tor of 4 or better, improvement in the precision of the
known to describe the core-core and core-valence corre-
energy levels with CI+all-order method in comparison
lations very well as demonstrated by its great success
withtheCI+MBPTone. Forexample,CI+MBPTvalue
in predicting alkali-metal atom properties [11]. There-
for the two-electron binding energy differs from experi-
fore, combinationof the CI and all-ordercoupled-cluster
ment by 1.8%, while our all-order value differs from the
methods allows to account for all dominant correlations
experiment by only 0.4% (see line one of Table I). The
to all orders. To evaluate uncertainty of our results, we
experimental value of the two-electron binding energy
useallthreeoftheapproachesandcomparetheresultsof
is obtained as the sum of the Tl+ and Tl2+ ionization
the CI, CI+MBPT, and CI+all-order calculations. We limits given in [13], 164765(5) cm−1 and 240600 cm−1.
referthereadertoRefs.[6,8,9]forthedescriptionofthe
Ref.[13] notesthat the ionizationlimitfor Tl2+, derived
methodsandoutlineonlymainpointsofthecalculations
from the first 3 members of the 2S series was shifted by
below. 300 cm −1 to give effective quantum number for 5g 2G
We start with solving Dirac-Fock (DF) equations
thatisnearlyhydrogenic. Therefore,thereissomeuncer-
Hˆ0ψc =εcψc, teanienrtgyy(i<∼n0T.l1+%.)associatedwiththetwo-electronbinding
whereH0 istherelativisticDFHamiltonian[8,9]andψc We also compared the transition energies between the
and εc are single-electron wave functions and energies. 6s6p 3P0 level and 4 levels relevant to the calculation of
The self-consistent calculations were performed for the the 6s6p 3P0 polarizability. These values, calculated in
[1s2,...,5d10] closed core and the 6s, 7s, 6p, 7p, and 6d the CI+all-order approximation are compared with ex-
orbitals. Wehaveconstructedthe B-splinebasissetcon- periment in Table II. We find that these transition en-
sistingofN =35orbitalsforeachofthes, p , p , ... ergies are substantially more accurate than the energy
1/2 3/2
partial waves up to l ≤ 5; core, 6s, 7s, 6p, 7p, and 6d levels counted from the ground state listed in Table I.
orbitals were replaced by the exact DF functions for in- In the present calculation, the Tl+ scalar polarizabil-
creased accuracy. The basis set is formed in a spher- ity α0 is separated into a valence polarizability αv0, ionic
ical cavity with radius 60 a.u. The CI space is effec- core polarizability α , and a small term α ( that mod-
c vc
tivelycompleteandincludes20spand21dfgorbitals. All ifies ionic core polarizability due to the presence of two
3
abilityisdeterminedbysolvingtheinhomogeneousequa-
TABLEI:Comparisonbetweenexperimental[13,14]andthe-
tion of perturbation theory in the valence space, which
oreticalenergylevelsincm−1. Two-electron bindingenergies
is approximated as
are given in the first row, energies in other rows are counted
from the ground state. Results of the CI, CI+MBPT, and ′
CI+all-order calculations are given in columns labeled CI, (Ev−Heff)|Ψ(v,M )i=Deff,q|Ψ0(v,J,M)i (2)
MBPT, and All. Corresponding relative differences of these
for a state v with the total angular momentum J and
threecalculations withexperimentaregiveninthelastthree ′
projection M [15]. The wave function Ψ(v,M ), where
columns in %.
′
M = M + q, is composed of parts that have angular
Differences (%) momentaofJ′ =J,J±1fromwhichthescalarandtensor
State Expt. CI MBPT All
CI MBPT All
polarizability of the state |v,J,Mi can be determined
66ss27s1S3S01 410055326259 37962190425 411028607361 410076102258 −−172 12..87 00..48 c[1o5r]r.ectTiohnes.effective dipole operator Deff includes RPA
6s7s1S0 108000 96304 110845 108904 −11 2.6 0.8 Unlessstatedotherwise,weuseatomicunits (a.u.) for
6s6d1D2 115166 101238 118678 116194 −12 3.1 0.9 all matrix elements and polarizabilities throughout this
6s6d3D1 116152 103334 119000 116857 −11 2.5 0.6
6s6d3D2 116436 103555 119339 117284 −11 2.5 0.7 paper: the numerical values of the elementary charge,
6s6d3D3 116831 103911 119688 117758 −11 2.5 0.8 e, the reduced Planck constant, ¯h = h/2π, and the
6p23P0 117408 108495 120875 118450 −8 3.0 0.9 electron mass, me, are set equal to 1. The atomic
66pp2233PP12 112258383187 111147976211 112392470514 112269484309 −−89 33..21 00..98 uαn/iht [fHorz/p(oVla/rmiz)a2b]=ili2t.y48c8a3n2×be10c−o8nαve(rate.ud.)t.o TSIheunciotnsvveira-
6s8s3S1 133568 120147 136369 134187 −10 2.1 0.5 sion coefficient is 4πǫ0a30/h in SI units and the Planck
6s8s1S0 134292 121089 137132 134950 −10 2.1 0.5 constanth is factoredout in orderto providedirect con-
versionintofrequencyunits; a0 istheBohrradiusandǫ0
6s6p3P0 49451 41719 52320 50288 −16 5.8 1.7 is the electric constant.
6s6p3P1 52394 44743 55114 53060 −15 5.2 1.3 While we do not use the sum-over-state approach in
6s6p3P2 61728 61728 65044 62669 −14 5.4 1.5 the calculation of the polarizabilities, it is useful to es-
6s6p1P1 75663 75663 76866 76145 −7 1.6 0.6 tablishwhichlevelsgivethedominantcontributions. We
6s7p3P0 119361 119361 122299 120155 −11 2.5 0.7 evaluate several leading contributions to polarizabilities
6s7p3P1 119576 119576 122602 120472 −11 2.5 0.8 by combining our values of the E1 matrix elements and
6s7p3P2 122209 122029 124873 122675 −11 2.3 0.5 energiesaccordingtothesum-over-statesformulaforthe
6s7p1P1 122379 122379 126014 124019 −9 3.0 1.3 valence polarizability [4]:
6s5f 3F2 136216 136216 138873 136600 −10 2.0 0.3
6s5f 3F3 136115 136115 138868 136577 −10 2.0 0.4 2 |hvkDkni|2
66ss55ff 13FF34 113366226330 113366226330 113388989770 113366755965 −−1100 21..09 00..43 αv0 = 3(2J +1)Xn En−Ev (3)
where J is the total angular momentum of state v, D is
the electric dipole operator, and E is the energy of the
i
TABLE II: Comparison between experimental [13, 14] and state i.
CI+all-order transition energies in cm−1. Therelative differ- The breakdown of the contributions to the 6s2 1S0
ences are given in the last column in percent. and 6s6p 3P0 polarizabilities α0 of Tl+ in a.u. is given
in Table III. Absolute values of the corresponding re-
Transition Expt. CI+all-order Dif. (%)
6s6p3P0−6s7s3S1 55778 55739 0.07% duced electric-dipole matrix elements are listed in col-
6s6p3P0−6s6d3D1 66701 66569 0.20% umn labeled “D” in a0e. To demonstrate the size of the
6s6p3P0−6p2 3P1 75887 76152 −0.35% correlation corrections, we list valence results obtained
6s6p3P0−6s8s3S1 84117 83899 0.26% intheCI,CI+MBPT,andCI+all-orderapproximations.
The contribution of the other terms listed in the row
“Other” is obtained by subtracting the sum of the con-
tributions that are calculated separately from the total
valence electrons. The ionic core polarizability is evalu- valence polarizability result obtained by the direct solu-
ated in the random-phase approximation (RPA), an ap- tionoftheEq.(2). Withtheexceptionofthelastcolumn
proach that is expected to provide core values accurate labeledCI+allB,weusethetheoreticalenergiesobtained
to better than 5% [4]. We approximate the vc term by in the respective approximations. To obtain data listed
addingvccontributionsfromtheindividualelectrons,i.e. in the last column, we combine CI+all-order E1 matrix
α (6s2)=2α (6s),andα (6s6p)=α (6s)+α (6p). elementsandexperimentalenergies. Thepolarizabilityof
vc vc vc vc vc
For consistency, this term is also calculated in RPA. We the ground state changes by 0.5% as expected from the
note that αvc contributions are small, but their contri- accuracyofthe6s21S0−6s6p1P1transitionenergylisted
bution to the ∆α(3P0− 1S0) polarizability difference is inTableI.Thepolarizabilityoftheexcited6s6p3P0state
significant,15%, due to severecancelationof the valence remains the same to four significant figures. Such re-
polarizabilitiesofthese twostates. The valence polariz- markable agreementis due to excellent, 0.07%, accuracy
4
TABLE III: Contributions to the 6s2 1S0 and 6s6p 3P0 polarizabilities in a.u. The dominant contributions to the valence
polarizabilities are listed separately with the corresponding E1 matrix elements given in columns labeled D. The remaining
valencecontributionisgiveninrowOther. Thecontributionfrom thecoreandvctermsaregivenbyαc andαvc,respectively.
Thedominantcontributionstoα0 listedincolumnsCI+allA andCI+allB arecalculatedwithCI+all-orderenergiesandNIST
[13, 14] energies, respectively. The differences of the 3P0 and 1S0 polarizabilities calculated in different approximations are
given in thelast row.
CI CI+MBPT CI+allA CI+allB
State Contribution
D α0 D α0 D α0 α0
6s2 1S0 6s2 1S0−6s6p3P1 0.424 0.589 0.658 1.149 0.597 0.984 0.997
6s2 1S0−6s6p1P1 2.789 16.131 2.619 13.057 2.646 13.450 13.535
Other 0.269 0.143 0.155 0.155
αc 4.983 4.983 4.983 4.983
αvc −0.071 −0.071 −0.071 −0.071
Total 21.901 19.261 19.501 19.599
6s6p3P0 6s6p3P0−6s7s3S1 1.044 3.113 0.975 2.499 0.980 2.519 2.517
6s6p3P0−6s6d3D1 2.007 9.563 1.893 7.860 1.897 7.912 7.897
6s6p3P0−6p2 3P1 1.616 5.219 1.557 4.603 1.562 4.690 4.706
Other 1.782 1.630 1.660 1.660
αc 4.983 4.983 4.983 4.983
αvc −0.338 −0.338 −0.338 −0.338
Total 24.322 21.236 21.426 21.425
∆α0(3P0− 1S0) 2.421 1.975 1.925 1.826
where yn = ωnv/T; α0 is the static dipole polarizabil-
TABLE IV: Contributions to dynamic corrections η for
ity of the state v, and J the total angular momentum
6s2 1S0 and 6s6p3P0 states. of the state v. We list the dominant contributions to η
State Transition yn η of the clock states calculated using CI+all-order E1 ma-
trixelementsandexperimentalenergiesinTableIV.The
η(6s2 1S0) 6s2 1S0−6s6p3P1 363 0.000099 sumin the expressionfor η above convergesveryrapidly
6s2 1S0−6s6p1P1 251 0.000015 making all other contributions negligible. The values of
0.000114 η forthe6s1S0 and6s6p3P0 statearealmostequal,and
η(6s6p3P0) 6s6p3P0−6s7s3S1 268 0.000031 their difference listed in the last row gives only 0.0016%
6s6p3P0−6s6d3D1 320 0.000068 contribution to the BBR shift.
6s6p3P0−6p2 3P1 364 0.000031
0.000130
IV. EVALUATION OF THE UNCERTAINTY
∆η(3P0− 1S0) 0.000016 AND CONCLUSION
We use Table III to evaluate the uncertainty to the
oftheCI+all-order6s6p3P0−6s7s3S1 transitionenergy BBRshiftdue tothe core-valencecorrelationcorrections
andoppositesignsofthedifferencebetweenCI+all-order
by comparing the CI, CI+MBPT, and CI+all-order re-
6s6p3P0−6s6d3D1and6s6p3P0−6p2 3P1transitionen- sultsfor∆α0(3P0−1S0)listedinthelastrowofTableIII.
ergiesand experiment(see Table II). We note that while
The difference between the CI and CI+MBPT results is
thechangeinthegroundstatepolarizabilityisonly0.5%,
23%, which is expected owing to poor agreement of CI
the corresponding change in the final polarizability dif-
energies with experiment. The difference between the
ference ∆α(3P0−1S0) is 5%. CI+MBPT and CI+all-order results is only 3%. As we
We have also calculated the dynamic correction η
notedabove,theuseoftheexperimentalenergieschanges
of both clock states. The total dynamic correction η
CI+all-order value by 5%.
in Eq. (1) is the difference of individual corrections,
We studied the effect of the Breit interaction by re-
∆η(3P0 −1 S0) = η(3P0) − η(1S0). The dynamic cor- peating the CI+all-order calculation with the one-body
rectionη ofthe state v is evaluatedusing the formula [5]
part of the Breit interaction incorporated into the DF
equations and construction of the basis set on the same
(80/63)π2|hvkDkni|2 21π2 336π4
η = 1+ + , footing with the Coulomb interaction. We find that the
Xn α0T (2J +1)yn3 (cid:18) 5yn2 11yn4 (cid:19) Breitinteractionaffectsboth1S0 and3P0 polarizabilities
5
TABLEV: BBRshiftsatT =300K inB+,Al+,In+,andTl+. B+,Al+,andIn+valuesaretakenfromRef.[6]. Polarizabilities
α0 and theirdifferences∆α0 aregiven in a.u.; clock frequenciesν0 andtheBBR shifts|∆νBBR|are givenin Hz. Uncertainties
in thevalues of ∆νBBR/ν0 are given in column labeled “Uncertainty”.
Ion α0(1S0) α0(3P0) ∆α0 ∆νBBR (Hz) ν0 (Hz) |∆νBBR/ν0| Uncertainty
B+ 9.624 7.772 −1.85(19) 0.0159(16) 1.119×1015 1.42×10−17 1×10−18
Al+ 24.048 24.543 0.495(50) −0.00426(43) 1.121×1015 3.8×10−18 4×10−19
In+ 24.01 26.02 2.01(20) −0.0173(17) 1.267×1015 1.36×10−17 1×10−18
Tl+ 19.60 21.43 1.83(18) −0.0157(16) 1.483×1015 1.06×10−17 1×10−18
by approximatelythe same amount, −0.5%. As a result, and the uncertainty in the relative BBR shift of Tl+.
the correctionto the BBR shift due to the Breit interac- The Tl+ values are compared with the results obtained
tion is negligible (0.6%) at the present level of accuracy. for B+, Al+, and In+ ions in Ref. [6]. The results listed
To evaluate the uncertainty in the α contribution to in Table V demonstrate that near cancelation of the
vc
the polarizability,wecalculate this terminboth DF and 1S0 and 3P0 state polarizabilities in divalent B+, Al+,
RPA approximations. The difference between these re- In+ ions of group IIIB [6] continues for much heavier
sults is taken to be the uncertainty. We find that the Tl+, leading to anomalously small BBR shift for this
uncertainty of the vc term contributes 2.4% to the un- system. This calculation demonstrates that the BBR
certainty in the BBR shift. The ionic core polarizability contribution to the fractional frequency uncertainty of
α is thesameforbothstatesanddoesnotcontributeto the Tl+ frequency standard at 300 K is 1×10−18. We
c
the BBR shift. find that Tl+ has the smallest fractional BBR shift
Based on the comparison of the CI, CI+MBPT, and among all present or proposed frequency standards with
CI+all-order data, estimated of the accuracy of the α the exception of Al+.
vc
terms, and estimated effect of the Breit interaction,
we place an upper bound on the uncertainty of our
∆α0(3P0 −1 S0) polarizability difference and the corre-
sponding BBR shift of Tl+ at 10%.
Our final result for the BBR shift of the
ACKNOWLEDGEMENTS
6s2 1S0 − 6s6p 3P0 transition in Tl+ is ∆νBBR =
−0.0157(16) Hz at 300 K. The corresponding relative
BBR shift at 300 K is |∆νBBR/ν0| = 1.1(1) × 10−17. We thank Hugh Klein for bring our attention to the
Our final results are summarized in Table V, where we problem of BBR shift in Tl+ frequency standard. This
list the clock state polarizabilities, their difference ∆α0, workwassupportedinpartbyUSNSFGrantsNo.PHY-
BBR shift at T = 300 K, 1S0 −3 P0 clock frequencies 1068699 and No. PHY-0758088. The work of MGK was
ν0, absolute value of the relative BBR shift |∆νBBR/ν0|, supported in part by RFBR grant No. 11-02-00943.
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