Table Of ContentHyperfine-mediated electric quadrupole shifts in Al+ and In+ ion clocks
K. Beloy,1 D. R. Leibrandt,1,2 and W. M. Itano1
1National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305, USA
2Department of Physics, University of Colorado, 440 UCB, Colorado 80309, USA
(Dated: February 1, 2017)
Weevaluatetheelectricquadrupolemomentsofthe1S and3P clockstatesof27Al+ and115In+.
0 0
To capture all dominant contributions, our analysis extends through third order of perturbation
theory and includes hyperfine coupling of the electrons to both the magnetic dipole and electric
7
quadrupole moments of the nucleus. For 27Al+, a fortuitous cancellation leads to a suppressed
1
frequencyshift. Thisshouldallowforcontinuedimprovementoftheclockwithoutspecialtechniques
0
to control or cancel the shift, such as the averaging schemes that are critical to other optical ion
2
clocks.
n
a
J I. INTRODUCTION analyzed for ground-state alkali-metal atoms, with po-
1 tentialrelevancetomicrowaveatomicclocks[12]. Asone
3 critical difference to that work, here we find it necessary
State-of-the-art optical ion clocks owe their low sys-
] tematic uncertainties, in part, to the external electric to proceed through third order of perturbation theory
h fields that provide strong confinement to the ion being (first order in the external field, second order in the hy-
p perfine interaction) to capture all dominant effects.
probed. Electricfieldgradientsareaninevitablebyprod-
- Gaussian electromagnetic expressions are employed
m uct of this trapping and, when coupled with the electric
throughout. We let e and a denote the elementary
quadrupole(E2)momentsoftheclockstates,inducesub- B
o charge and the Bohr radius, respectively.
stantial frequency shifts in clocks based, for example, on
t
a Hg+ [1], Sr+ [2, 3], and Yb+ [4, 5]. Exploiting the ro-
.
s tational symmetry of the E2 interaction allows cancel-
c lation of the shift by averaging over different magnetic II. THE E2 ENERGY SHIFT
i
s substates or bias magnetic field orientations [6, 7]. The
y
additionaloperationalburdenoftheseaveragingschemes HerewepresentformulasfortheE2energyshiftforan
h
is accompanied by distinct technical challenges, such as atom in a hyperfine state nFM , where n identifies the
p | i
[ maintainingstringentcontroloverthebiasmagneticfield electronicstate(inclusiveofJ),nuclearspinI isimplicit,
direction. and F and M specify the total angular momentum and
1 In contrast to the clocks above, the Al+ clock [1, 8] its projection onto the z-axis, respectively. To clarify,
v
here nFM represents an eigenstate of the full atomic
6 employs two J = 0 clock states, where J specifies the | i
4 total electronic angular momentum. Given the spherical Hamiltonian, inclusive of nuclear spin and the hyperfine
1 symmetry implied by J = 0, the electrons nominally do interaction. Generally, we are interested in a scenario
9 not contribute to an E2 moment. Rather, the E2 mo- in which static and radiofrequency electric fields simul-
0 ments of the clock states reduce to the negligibly small, taneously perturb the atom. Here we limit our concern
. to effects first order in the field, in which case only the
1 state-independentnuclearE2moment. To-date,thishas
0 allowed the Al+ clock to operate without special regard staticcomponentneedstobeconsidered. Describingthis
7 for electric field gradients, including those attributed to fieldwiththeelectricpotentialΦ,theE2interactioncan
1 the co-trapped Be+ or Mg+ logic ion. be written as the scalar product
:
v Undercloserscrutiny,theJ =0symmetryoftheclock
Xi states is weakly broken by the coupling of the electrons VE2 =E2·Q2, (1)
to the electromagnetic moments of the nucleus, i.e. the
r where is shorthand for the field-gradient tensor
2
a hyperfine interaction. This gives rise to state-dependent E
“hyperfine-mediated” corrections to the E2 moments, 1
which may readily overshadow the bare nuclear E2 mo- E2 ≡ √6{∇⊗∇}2Φ,
ment. Estimates of these effects have given justification
for their prior neglect [9]. However, given the continued understood to be evaluated at the center-of-mass posi-
improvement of the Al+ clock [10], a more refined anal- tion of the atom. Here ∇ ∇ represents the sec-
ysis is now warranted. In this paper, we evaluate the E2 ond rank tensor formed b{y co⊗upli}n2g ∇ with itself [13].
momentsoftheAl+ clockstatesusingmethodsofmany- is a rank-2, even-parity operator that acts on both
2
body atomic structure theory and discuss implications Qelectronic and nuclear coordinates. It is given by =
2
for the Al+ clock going forward. We further extend our q r2C (ˆr ), with q the charge and r the positiQon of
analysis to In+, motivated by the pursuit of a many-ion Ptheii-ithi el2ectironornucileon(ri =riˆri)anidwithC2(ˆr)be-
clock based on this species in Braunschweig [11]. ing the conventionalrank-2 C-tensor (normalized spher-
Hyperfine-mediated E2 moments have recently been ical harmonic).
2
The first order E2 energy shift to the state nFM attention to J = 0 atomic states, with the immediate
| i
reads consequence that F = I (I 1 assumed throughout).
≥
To arrive at practical expressions for evaluating the E2
δE = nFM V nFM = nFM nFM ,
h | E2| i E20h |Q20| i moments, we treat the E2 interaction and the hyperfine
where,followingfromelementaryangularselectionrules, interaction on equal footing as perturbations,
the matrix element nFM nFM is non-zero only
h |Q20| i V = (e)+ (n),
for F 1. Generally, VE2 mixes states of different M, E2 E2·Q2 E2·Q2 (2)
≥
suggestingtheneedtodiagonalizea(2F+1)-dimensional V =µ(n) (e)+ (n) (e).
matrix. Here contributions off-diagonal in M are as- hfi 1 ·T1 Q2 ·T2
sumed to be negligible. In practice this is ensured by Subscripts appearing on the right-hand-side of the ex-
application of a sufficiently large bias magnetic field, pressions specify tensor rank, while superscripts identify
which defines the z-axis and lifts the degeneracy in the operators acting exclusively on electronic (e) or nuclear
M states. This permits us to take M as a “good” quan- (n)coordinates,inclusiveofspin. Notethattheoperator
tum number and has the consequence of isolating the 2oftheprecedingsectionhasbeenpartitionedintoelec-
Q
zero-component of the scalar product in Eq. (1). The tronicandnuclearparts. Vhfiaccountsforcouplingofthe
zero-component of is electrons to the magnetic dipole and electric quadrupole
2
E moments of the nucleus. We will refer to the respective
1 ∂2 1 ∂2 1 ∂2 1∂2Φ
terms as the M1-hyperfine interaction and E2-hyperfine
= + Φ= ,
E20 (cid:18)−6∂x2 − 6∂y2 3∂z2(cid:19) 2 ∂z2 interaction, where the latter is not to be confused with
the E2 interaction with the external field, V . In the
where we invoked 2Φ = 0 to arrive at the last expres- E2
∇ limit of a point nucleus, (e) = e (1/r2)(α ˆr )
sion. T1 − i i i× i
We proceed to define the E2 moment Θ as the expec- and (e) = e (1/r3)C (ˆr ), whePre α is composed
T2 − i i 2 i
tation value of 20 for the “stretched” state, of the conventionPal Dirac matrices (αx,αy,αz) and the
Q
summations run over all electrons. We neglect coupling
Θ nFF nFF .
≡h |Q20| i of the electrons to higher-multipolarmoments of the nu-
Meanwhile, from the Wigner-Eckart theorem, cleus (M3-hyperfine, E4-hyperfine, etc.).
The E2 moment Θ is attributed to energy shifts first
F 2 F
order in V and all orders in V . In practice, however,
E2 hfi
nFM nFM =( 1)F−M(cid:18)−M 0 M (cid:19) we only retain the leading contributions in Vhfi. The no-
h |Q20| i − F 2 F tation Θ(1+m) is used to denote the (1 + m)-th order
(cid:18) F 0 F (cid:19) contribution, with m being the number of hyperfine in-
−
teractions involved. Applying conventional perturbation
nFF nFF .
×h |Q20| i theory through third order, we arrive at the following
Evaluating the 3j-symbols and combining the preceding expressions specific to J =0 states,
expressions, we arrive at the energy shift expression
1
Θ(1+0) = Q,
1∂2Φ3M2 F(F +1) 2
δE = − Θ.
2 ∂z2 F(2F −1) Θ(1+1) = 51Q[Q2,T2]1,
Note that the M-dependence appears explicitly in this
2
last expression, with Θ itself being independent of M. Θ(1+2) = µ2A [ , , ]
11;I 2 1 1
√15 Q T T
The definition of the E2 moment Θ is chosen to be
1
consistent with previous works for J 1 atomic states, + µ2A [ , , ]
≥ 11;I 1 2 1
where nuclear structure and hyperfine effects are typ- 3 T Q T
ically neglected. Expressions in the following section 1
+ µQA [ , , ]
make explicit reference to the nuclear E2 moment Q. 5 12;I Q2 T1 T2
For this quantity, we adhere to the conventional nuclear 1
+ µQA [ , , ]
physics definition, which incorporates an additional fac- √15 12;I Q2 T2 T1
tor of two. Specifically, in a complete absence of elec- 1
trons, Θ equates to Q/2. In any case, our formulas are µQA12;I[ 1, 2, 2]
− √15 T Q T
self-consistent and unambiguously specify the physical
1
meaning of Θ. + Q2A [ , , ]
22;I 2 2 2
10 Q T T
1
+ Q2A [ , , ]
III. HYPERFINE-MEDIATED E2 MOMENTS 20 22;I T2 Q2 T2
FOR J =0 STATES 1
+ µ2QB [ , ]
6 1;I T1 T1 2
The expressions in the previous section are valid for + 1 Q3B [ , ] ,
generic hyperfine states. In the remainder, we limit our 40 2;I T2 T2 2
3
where µ and Q are the conventionalnuclear M1 and E2 corresponding terms in the sum-over-states expressions
moments. The I-dependent angular factors A and entering with small energy denominators. From the pre-
k1k2;I
B are given in terms of 3j- and 6j-symbols by cedingarguments,weidentifyonesecondordertermand
k;I
two third order terms to be of principal interest,
I 2 I k k 2
1 2 1 Q
(cid:18) I 0 I (cid:19)(cid:26) I I I (cid:27) 3P 3P 3P 3P , (3)
A =( 1)2I − , −5∆ h 0||Q2|| 2ih 2||T2|| 0i
k1k2;I − I k I I k I 20
1 2 8√2 µ2
(cid:18) I 0 I (cid:19)(cid:18) I 0 I (cid:19) 3P 3P 3P 3P 3P 3P ,
− − 75 ∆ ∆ h 0||Q2|| 2ih 2||T1|| 1ih 1||T1|| 0i
20 10
I I k I I k
(4)
(cid:26)I I 2 (cid:27)−(cid:26)I I 0 (cid:27)
Bk;I =(−1)2I I k I 2 . 145r125∆µ22 h3P0||T1||3P1ih3P1||Q2||3P1ih3P1||T1||3P0i,
(cid:18) I 0 I (cid:19) 10
− (5)
Tcohmepsatcrtinngootaftoiopnerfaotrotrhseappupreeareilnegctirnonsiqcufaarcetobrrsackets is tw∆oi2t0Ih≡fi=ne(5E/s23tPr2u(2c−7tAuErl3e+P)0s.)p.liFttoTirnergomsths∆e(r140)I,≡atnh(dEes3(eP51)t−earmrEes3Ps0sp)heocauinfilddc
′ ′
n X n n Y n
[X ,Y ] = h || k|| ih || k|| i, be multiplied by an additional factor (5/8)(2I 1)/I.
k k r Xn′ (En−En′)r For the sake of completeness, below we evaluate−all first
through third order contributions to the E2 moments of
[X ,Y ,Z ]=
k1 k2 k3 both clock states. Terms (3)–(5) are found to be compa-
′ ′ ′′ ′′
n X n n Y n n Z n
h || k1|| ih || k2|| ih || k3|| i, rable to one another and dominate significantly over all
nX′n′′ (En−En′)(En−En′′) other contributions. Given that the fine structure split-
tings and nuclear moments are known, the critical task
where we have dropped the superscript (e) on the oper- reducestothe accuratedeterminationofthe fivedistinct
ators involved. Here n and E , with arbitrary number matrix elements appearing in these three terms.
n
| i
of primes attachedto the n, denote electronic states and
energies. That is, these are eigensolutions to the atomic
IV. NUMERICAL ANALYSIS AND RESULTS
Hamiltonian in absence of nuclear spin and the hyper-
fine interaction. Reduced matrix elements and energies
are independent of the magnetic substates, and it is un- To evaluate electronic properties, we employ the
derstood that the sum-over-states expressions above ex- methodofconfigurationinteractionplusmany-bodyper-
clude summation over magnetic quantum numbers. The turbationtheory(CI+MBPT)[14,15]. Inbrief,we start
unprimed n designates the J =0 state of concern, while withaDirac-Hartree-Fockdescriptionoftheatomiccore
′ ′′ ′
n and n run over all other electronic states (having J (nuclear charge plus core electrons). The CI procedure
′′ ′
and J , respectively). Selection rules insist that J = k treats the strongly-interacting valence electrons in the
′ ′′
for the top expression and J = k and J = k for the presence of the “frozen” core. The MBPT extension ac-
1 3
bottom expression. Moreover,since only even-parity op- counts for additional perturbative effects of the valence
eratorsplayarole,allstatesinvolvedmusthaveidentical electrons on the core. Further details of our CI+MBPT
parity. implementation, with only minor modification, can be
The first order Θ(1+0) represents the E2 moment in found in Ref. [16].
absence of the hyperfine interaction. In accordance with The CI+MBPT method has been applied to numer-
the Introduction, this reduces to the bare nuclear value, ous divalent systems, including heavy systems such as
with a factor of 1/2 bridging the atomic physics and nu- Ra [17]. Al+ is comparatively simple. To best gauge
clear physics definitions of the E2 moment. At second the accuracy of our calculations, in particular the ma-
order,a correctionΘ(1+1) arisesdue to the E2-hyperfine trix elements appearing in terms (3)–(5), we first iden-
interaction. Selection rules preclude the M1-hyperfine tify relevant data in the literature for comparison. Hy-
interaction from having effect at this order, prompting perfine intervals for the 3P state of 27Al+ [9] and the
1
further progression to third order. The expression for 3P states of 25Mg [18] have been measured. Mg has
1,2
Θ(1+2) includes several terms. The first two terms in- similar electronic structure to Al+, justifying its inclu-
volve two M1-hyperfine interactions, while the remain- sion here. Neglecting higher order effects, the hyperfine
ingtermscanberegardedashigher-orderininteractions intervals may be combined with nuclear moments [19]
that already contribute to Θ(1+0) and Θ(1+1). For the to infer diagonal M1-hyperfine and E2-hyperfine ma-
specific problem at hand, we further note that the ex- trix elements, 3P 3P and 3P 3P , for the
J 1 J J 2 J
cited clock state is part of a 3P fine structure man- respective statehs. |T|The||se riesultshare|p|Tre|s|enteid in Ta-
J
ifold. The hyperfine-mediated effects are consequently ble I with the label “inferred, uncorrected.” To these
expectedtobedominatedbyhyperfine-mixingofthe3P values, we add second order corrections attributed to
0
clock state withthe neighboring3P and3P states,with hyperfine mixing between the neighboring 3P states.
1 2 J
4
Thecorrectionsareevaluatedusingoff-diagonalab initio three orders of magnitude larger than the bare nuclear
CI+MBPThyperfine matrixelements, togetherwiththe value of 2.62 10−9ea2. Remaining second and third
× B
nuclear moments and fine structure intervals [20]. The ordercontributionswereevaluatedusingCI+MBPTma-
results are presented in Table I with the label “inferred, trix elements and energies, with experimental nuclear
corrected.” Finally, these values are comparedto the re- moments. For both clock states, these contributions
spective diagonal ab initio CI+MBPT matrix elements, amount to 1 10−8ea2, verifying the dominance of
− × B
labeled “CI+MBPT.” the terms(3)–(5). Table III providesabreakdownofthe
First, we examine the M1-hyperfine matrix elements contributions to the E2 moments of both clock states.
in Table I. For all three Mg and Al+ states, the sec- To estimate uncertainty of Θ for the excited clock
ond order corrections are small, with the resulting “in- state, we ascribe a 3% uncertainty to each of the five
ferred,corrected”valuesbeingaccuratetowithinthedis- distinct matrix elements entering terms (3)–(5). We as-
played digits. The corresponding CI+MBPT results ex- sume uncorrelated error in these matrix elements and
hibitagreementintherange0.7–1.6%. Thisisindicative propagateuncertaintyaccordingly,renderingafinalvalue
of the CI+MBPT accuracy for the M1-hyperfine matrix Θ=−1.7(6)×10−6ea2B.
elements between the different 3P states. Next, we ex- In Ref. [9], the E2 moment of the excited clock state
aminetheE2-hyperfinematrixeleJmentsforthethreeMg was given as Θ ≈ −1.2×10−5ea2B, this being an order-
and Al+ states. Here, non-negligible uncertainty in the of-magnitude larger than the present evaluation. After
“inferred, uncorrected”values stems from uncertainty in inspecting notes from that work, a sign error was dis-
the nuclear E2 moments. At the same time, the sec- covered for the term (5) contribution. A large relative
ondordercorrectionsaremoreprominent. Thesecorrec- error in Θ resulted from an absence of the cancellation
tionsaredominatedbyterms involvingoff-diagonalM1- discussed above. Correcting the sign error, the previous
hyperfine matrix elements. Our preceding assessment of result is brought into agreement with the present result.
the CI+MBPT performance for these matrix elements
allows us to ascribe a fair uncertainty to the corrections.
Thisisonlyofsignificanceforthe3P stateofAl+,where V. IMPLICATIONS FOR THE Al+ ION CLOCK
1
the correctionleads to a doubling ofthe uncertainty. Fi-
nally,for the 3P states ofMg,the diagonalCI+MBPT Al+ ion clocks to-date have operated with a single
1,2
matrix elements are found to be within 1σ ( 1.5%) of Al+ ion and a single logic ion (Be+ or Mg+) simulta-
the “inferred, corrected” values. Meanwhile, ∼for the 3P neously confined in a linear RF Paul trap [23]. For this
1
state of Al+, the CI+MBPT resultis found to be within configuration, there are two contributions to the electric
3σ ( 5%) of the “inferred, corrected” value. This pro- quadrupole field atthe positionof the Al+ ion: that due
∼
vides us with a measure of the CI+MBPT accuracy for to the static axial confining trap potential and that due
the E2-hyperfine matrix elements between the different to the logic ion. Following Ref. [24], we write the total
3P states. Unfortunately, literature data is lacking to trap electric potential as
J
directlyassessCI+MBPTperformanceforE2matrixel-
ewmeehnatvse(nQo2reoapseornattoore)xbpeetcwtetehnetEh2em3PaJtrsixtaetleesm. eHnotswteovebre, ΦT = V20 cos(Ωt)X2R−2Y2 +U0Z2−αX2d−2(1−α)Y2,
anylessaccuratethantheM1-hyperfineorE2-hyperfine
where V /2 and Ω are the amplitude and angular fre-
0
matrix elements between these states. We will briefly
quency of the voltage applied to the rf electrodes, U
0
return to the discussion of accuracy below.
is the voltage applied to the endcap electrodes, R and d
Table II presents our CI+MBPT results for the five arecharacteristicradialandaxialdimensionsofthetrap,
distinct matrixelements appearinginterms(3)–(5). To- and α parameterizes the radial asymmetry of the static
gether with nuclear moments [19] and fine structure field. Above, X, Y, and Z are spatial coordinates in the
splittings [20], these matrix elements yield the results trap coordinate frame, while t denotes time. Dropping
1.08 10−5ea2 for term (3), 3.92 10−6ea2 for term the rf term, transforming to the quantization coordinate
(−4), an×d 5.13 B10−6ea2 for term×(5). TheBthird or- frame, and taking the second derivative with respect to
× B
der M1-hyperfine-mediated terms (4) and (5) are seen z, we get
to be comparable in magnitude and add constructively.
Together they are close in magnitude to the second or- ∂2ΦT = 2U0 3cos2θ−1 α 1 sin2θcos(2φ) ,
derE2-hyperfine-mediatedterm(3)butofoppositesign. ∂z2 d2 (cid:20) 2 −(cid:18) − 2(cid:19) (cid:21)
Thisresultsinsignificantcancellation,withasuppressed
cumulative result of 1.7 10−6ea2 for these three where θ and φ are the polar and azimuthal angles of the
terms. Clearly,itwoul−dhave×beenerroBneoustolimitour bias magnetic field as referenced from the trap frame.
analysis to second order perturbation theory or to ac- This result is independent of the position of the Al+ ion
count for hyperfine-mediated effects using only the M1- in the trap. The electric potential due to the other ion
hyperfine interaction. can be written as
Despite the cancellation, the terms (3)–(5) provide a e
Φ = ,
I
correction to the E2 moment of the excited clock state (X X )2+(Y Y )2+(Z Z )2
i i i
− − −
p
5
TABLE I. Diagonal M1-hyperfineand E2-hyperfinematrix elements for 3PJ states of Mg, Al+, and In+. Uncorrected matrix
elements are inferred from hyperfine intervals [9, 18, 21] and nuclear moments [19] found in the literature, neglecting higher
order effects of the hyperfine interaction. These results are subsequently corrected for second order effects using off-diagonal
ab initio CI+MBPT hyperfine matrix elements between the 3PJ states, together with nuclear moments and fine structure
intervals [20]. In many cases, thecorrections do not or barely change theexpressed result. The corrected matrix elements are
then compared to ab initioCI+MBPT hyperfinematrix elements. Allvalues are in atomic units[22].
Mg, 3s3p3PJ Al+, 3s3p3PJ In+, 5s5p3PJ
J =1 J =2 J =1 J =1 J =2
inferred, uncorrected 0.07938(<1) 0.1573(<1) 0.1723(<1) 1.059(<1) 1.733(4)
h3PJ||T1||3PJi inferred, corrected 0.07936(<1) 0.1572(<1) 0.1723(<1) 1.059(<1) 1.733(4)
CI+MBPT 0.07847 0.1561 0.1696 1.150 1.841
inferred, uncorrected −0.482(7) 0.70(1) −1.72(1) −6.9(6) 7(2)
h3PJ||T2||3PJi inferred, corrected −0.466(7) 0.71(1) −1.26(2) −6.4(6) 7(2)
CI+MBPT −0.4621 0.7023 −1.199 −6.175 8.582
(0,0, (ed2/8U )1/3), we get
TABLE II. Ab initio CI+MBPT results for the five distinct ± 0
matrixelementsappearinginterms(3)–(5),evaluatedforAl+
∂2Φ 2U 3cos2θ 1
and In+. State labels refer to the 3s3p3PJ and 5s5p3PJ fine I = 0 − .
structure manifolds of the respective ion. All values are in ∂z2 d2 2
atomic units[22].
The trap dc potential satisfies
Al+ In+
h3P0||Q2||3P2i −6.271 −6.932 U0 = m1ω2 µ ,
h3P1||Q2||3P1i −5.428 −5.825 d2 2e 1+µ 1 µ+µ2
h3P ||T ||3P i 0.1195 0.7556 − −
0 1 1 p
h3P1||T1||3P2i −0.1545 −0.7973 wherem1 andm2 =µm1 arethe massesofthe logicand
h3P ||T ||3P i −1.382 −7.114
0 2 2 clockionsandωistheangularfrequencyoftheaxialcom-
mon secular mode. Given that ω is a readily-accessible
experimental parameter, U /d2 can be immediately de-
0
termined from the expression above.
TABLE III.Contributions to theE2 moments of the ground
For the Al+ clock reported in Ref. [8], with ω/2π =
and excited clock states in 27Al+ and 115In+. All values are
◦
in unitsof ea2. The notation x[y]indicates x×10y. 3.00 MHz, α = 1.65, and θ = φ = 45 , we calculate
B an E2 clock shift of 28 µHz, or 2.5 10−20 fraction-
27Al+ 115In+ ally. The magnetic fi−eld orientatio−n wi×th respect to the
bare nucleus trap axes was not characterized with high accuracy in
Θ(1+0) 2.62[−9] 15[−9] thisexperiment,butduetogeometricconstraintswecan
ground clock state bound the uncertaintyof θ andφ to be better than 5◦.
Θ(1+1) −10[−9] ±
Taking this field-gradient uncertainty together with un-
Θ(1+2) −0.1[−12]
certainty in Θ, we find an uncertainty in the clock shift
Θ total −8[−9] of19µHz, or1.7 10−20 fractionally. As expected, both
excited clock state ×
the shift andits uncertaintyarenegligible incomparison
Θ(1+1), term (3) −10.8[−6] −18.5[−6]
with the total clock uncertainty of Ref. [8]. Even as the
Θ(1+1), all other 9[−9]
Al+ clock continues to improve [10], this shift will likely
Θ(1+2), term (4)a 3.9[−6] 1.1[−6]
remain small with respect to the uncertainty budget for
Θ(1+2), term (5)a 5.1[−6] 1.7[−6]
the immediate future.
Θ(1+2), all other −19[−9]
Θ total −1.7[−6] −15.7[−6]
a For115In+,terms(4)and(5)aremultipliedby10/9,as VI. ESTIMATES FOR AN In+ MANY-ION
appropriateforthenuclearspinI=9/2. CLOCK
Following Refs. [25] and [26], we consider a many-ion
where (X ,Y ,Z ) is the position of the logic ion. Trans- clock based on 115In+. We evaluate terms (3)–(5), with
i i i
forming to the quantization coordinate frame, tak- appropriatescaling for the nuclear spin I =9/2,andne-
ing the second derivative with respect to z, and sub- glectallothercontributions. Resultsaretabulatedalong-
stituting the equilibrium ion positions (X,Y,Z) = side 27Al+ results in Tables I–III above. Unlike 27Al+,
6
thereisnotsignificantcancellationamongtheterms(3)– corresponding clock shift in optical ion clocks based on
(5). The second order E2-hyperfine-mediated term (3) 27Al+ and 115In+. Interestingly, for 27Al+, the second
dominates, with a final value of Θ = 1.6 10−5ea2. orderE2-hyperfine-mediatedcontributionisnearlyequal
− × B
Through inspection of Table I, this result is expected to but opposite to the third order M1-hyperfine-mediated
be accurate to within 20%. contributions,leadingto asubstantialsuppressionofthe
∼
To estimate the clock shift, we assume a linear crystal already small shift.
of eight In+ clock ions and two Yb+ sympathetic cool- While the magnitudes of the E2 shifts for the clocks
ing ions, with the Yb+ ions residing in the center. We considered here are small with respect to current to-
supposethatthe trapstrengthissuchthatthe axialsec- tal systematic uncertainties, the E2 shift may be more
ularfrequencyofasingle172Yb+ ionis330kHzandthat significant for future clocks with lower total system-
◦
α 1/2 and θ = 25 . By generalizing the above formu- atic uncertainties, clocks with stronger confinement, or
≈
las to the many-ion case, we calculate the mean of the clocks based on other atomic species. For these cases,
E2 shift of eachof the eightIn+ ions to be 490µHz,or in addition to the established techniques of cancelling
3.9 10−19 fractionally. The full width o−f the associ- the shift by averaging over different magnetic substates
− ×
ated inhomogeneous broadening is 530µHz, which given or bias magnetic field orientations, it is worth noting
the lifetime limited transition linewidth of 820 mHz [27] that the shift can also be suppressed by setting θ =
willnotlimitthespectroscopylinewidth. Wenotethatat arccos(1/√3) 54.7◦ and φ = 45◦. Finally, while the
≈
somelevelitmightbe necessarytotake intoaccountan- quadrupole moments are rather small, it may be possi-
harmonicity in the trapping potential over the extended ble to do a direct experimental measurement using the
sizeoftheioncrystalforamoreaccurateestimateofthe technique of Ref. [28].
E2 shift.
ACKNOWLEDGMENTS
VII. CONCLUSION
TheauthorsthankS.BrewerandD.Nicolodifortheir
Here we have derived expressions for hyperfine- careful reading of the manuscript. This work is the con-
mediated electric quadrupole moments of J = 0 atomic tributionof NIST,anagency ofthe USgovernment,and
states and have used these expressions to evaluate the is not subject to US copyright.
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