Calculation of the forbidden electric tensor polarizabilities of free Cs atoms and of Cs atoms trapped in a solid ⁴He matrix

http://doc.rero.ch

Calculation of the forbidden electric tensor polarizabilities of free Cs atoms and of Cs atoms trapped in a solid

⁴

He matrix

A. Hofer,

*

P. Moroshkin, S. Ulzega,^†and A. Weis

Département de Physique, Université de Fribourg, Chemin du Musée 3, 1700 Fribourg, Switzerland^‡ 共Received 6 June 2007; revised manuscript received 25 October 2007; published 4 January 2008兲 We give a detailed account on our semiempirical calculations of the forbidden electric tensor polarizability

␣₂^共3兲 of the ground state of free Cs atoms and of Cs atoms implanted in a solid⁴He matrix. The results are compared with measurements of␣₂^共3兲in the free atoms关Europhys. Lett. 76, 1074共2006兲兴and in He-trapped atoms 关Phys. Rev. A 75, 042505 共2007兲兴. Features with respect to calculations by other authors are the inclusion of off-diagonal hyperfine interactions and an analysis of contributions from continuum states, which turn out to be negligible. For both samples the results of the calculations are in good agreement with the experimental values, thereby settling a long-standing discrepancy.

PACS number共s兲: 31.15.xp, 32.10.Dk, 32.60.⫹i, 67.25.D⫺

I. INTRODUCTION

The interaction of an atom with an external static electric field共Stark effect兲is one of the fundamental interactions in atomic physics. In atoms with degenerate orbital momentum states of opposite parity the change of level energies induced by the electric field is linear in the field strengthE, the hy- drogen atom being the most prominent example. In most atoms, however, the Stark shift is quadratic in the field strength, and the electric-field-induced shift of a magnetic hyperfine共hf兲sublevel兩␥典=兩n LJ,F,M典is commonly param- etrized in terms of the共electrostatic兲polarizability␣共␥兲 as

⌬E共␥兲= −1

2␣共␥兲E². 共1兲

In second order perturbation theory, the polarizability of statesnLJwithJⱖ1 can be written as the sum of anF- and M-independent scalar polarizability ␣₀^{共2兲 and an F- and} M-dependent tensor polarizability ␣₂^共2兲, while states with J

⬍1 have only a scalar polarizability. For spherically sym- metric states, such as the nS_1/2 and nP_1/2 states in alkali- metal atoms, the Stark effect is thus a purely scalar effect, meaning that all hyperfine sublevels兩nL_1/2,F,M典experience the same common shift. However, anF andM dependence of the Stark shifts in the 兩nS1/2典 alkali-metal ground states was already found experimentally in the late 1950’s and early 1960’s关1,2兴, and thus indicated the existence of a共for- bidden兲tensor polarizability of those states.

In 1967 Sandars关3兴could show that this forbidden polar- izability can be explained by expanding the perturbation treatment to third order, considering the Stark and the hyper- fine interactions as simultaneous perturbations. The corre- sponding third order polarizability can also be expressed in terms of an F-dependent third order scalar polarizability

␣₀^共3兲共F兲and anF- andM-dependent third order tensor polar-

izability␣₂^共3兲共F,M兲. The polarizability in Eq.共1兲can thus be written as

␣共␥兲=␣₀^共2兲共nLJ兲+␣₂^共2兲共nLJ,F,M兲+␣₀^共3兲共nLJ,F兲 +␣2共3兲共nL_J,F,M兲, 共2兲 where the superscripts refer to the order of perturbation, while the subscripts refer to the rotational symmetry共scalar or second rank tensor兲of the interaction.

Sandars’ expressions were evaluated numerically in关2,4兴 under simplifying assumptions, and yielded values of

␣₂^共3兲共F,M兲 for five alkali-metal isotopes whose 共absolute兲 calculated values were systematically larger than the corre- sponding experimental values 关4–7兴. Recently we have re- measured the third order tensor polarizability ␣₂^共3兲共F,M兲 of the Cs ground state in an all-optical atomic beam experiment 关7兴, yielding good agreement with the measurements from the 1960’s 关4,6兴. We have also measured ␣₂^共3兲共F,M兲 of ce- sium atoms implanted in the cubic phase of a ⁴He crystal, yielding a value whose modulus is⬇10% larger than in the free atom关8兴. In parallel we have reanalyzed and extended the theoretical expressions of the tensor polarizabilities 关5兴.

We have further extended the third order perturbation calcu- lation 关3兴 by including off-diagonal hyperfine interactions, not considered in the earlier calculations, as well as contri- butions from higher-lying bound and continuum states. This has yielded a theoretical value for␣₂^{共3兲of Cs}共6S_1/2兲, which is in good agreement with all existing experimental results.

As discussed in关5兴we have uncovered in an earlier cal- culation关3兴an error concerning the relative signs of the po- larizabilities of the two ground state hyperfine levels. Re- cently we have given an experimental verification of the sign of our calculation and discussed its implication for the dy- namic Stark shift of primary frequency standards by the blackbody radiation field关8兴.

In this paper we present more details of that calculation, whose results were already outlined in关5,8兴. In addition we present a calculation of␣₂^共3兲for cesium atoms in a cubic⁴He crystal by evaluating the matrix-induced alterations of the atomic energies and of the electric dipole and hyperfine ma-

*adrian.hofer@unifr.ch

†Present address: EPFL, Lausanne, Switzerland.

‡www.unifr.ch/physics/frap/

Published in "Physical Review A 77(1): 012502, 2008"

which should be cited to refer to this work.

http://doc.rero.ch

trix elements in the frame of the so-called standard bubble model 关9兴. Here too we obtain an excellent agreement with the experimental values.

II. THEORY

The Stark effect, i.e., the interaction of an alkali-metal atom with an external electric field E is described by the HamiltonianH_St= −dជ_·Eជ_{, whered}ជis the electric dipole opera- tor of the valence electron. Because of parity conservation, the Stark interaction vanishes in first order, and the effect of the electric field on the atomic level structure appears only in the next higher order共s兲, yielding shifts that are quadratic in the applied field strength.

A. Second order perturbation theory

The second order energy perturbation of the magnetic hy- perfine sublevel 兩␤典=兩6S_1/2,F,M典 of the ground state is given by

⌬E^共2兲共␤兲=

兺

_␥ ^兩具␤_E^兩H_␤−^St兩_E^␥_␥^{典兩2, 共3兲}

where, according to the selection rule ⌬L= ± 1 imposed by the Stark operator, the sum is to be taken over all excitedP states兩␥典=兩nPJ,f,m典, including continuum states, and where E_␥ are the unperturbed energies of those states. Following 关10兴one can define an effective second order Stark operator

2H_eff=

共

^dជ·Eជ

兲

^2␭

共

^dជ·Eជ

兲

^{, 共4兲}

in which the scalar projection operator₂␭ is defined as

2␭=

兺

_{␥ E}^兩␥_␤^典具₋^␥_E^兩_␥^{. 共5兲}

With this definition of ₂H_eff the second order energy shift

⌬E^共2兲 of the state兩␤典 is given by the expectation value

⌬E^共2兲共␤兲=具␤兩₂H_eff兩␤典, 共6兲 similar to the expression from first order perturbation theory.

Note that the superscripts on the right of⌬E^共n兲and␣^共n兲and the subscripts on the left of_nH_effand_n␭refer to the order of perturbation.

The interaction Hamiltonian can be factorized关10兴into an electric-field-dependent part and a part which depends only on atomic properties by defining the components of two rank-Ktensor operators as

EQ共K兲=关E丢E兴Q共K兲=

兺

q,q⬘具11qq⬘兩KQ典Eq共1兲Eq^共1兲⬘, 共7兲 DQ共K兲=关d^共1兲丢₂␭丢d^共1兲兴Q共K兲=

兺

q,q⬘具11qq⬘兩KQ典d_q^共1兲₂␭d_q^共1兲_⬘, 共8兲 where the具11qq⬘^兩KQ典 are Clebsch-Gordan coefficients and whered_q^共1兲 and E_q^共1兲 are the spherical components of the di- pole operator and the electric field, respectively. The effec-

tive Stark operator can then be written as the sum

2H_eff=

兺

K=0 2

共− 1兲^K2H_eff^共K兲=₂H_eff^共0兲+₂H_eff^共2兲 共9兲

of its multipole components

2H_eff^共K兲=

兺

_Q ^{共− 1兲QEQ共K兲D−Q共K兲. 共10兲}

The vector contribution vanishes since E^共1兲⬀关E丢E兴^共1兲⬀Eជ

⫻Eជ. The scalar termE^共0兲⬀关E丢E兴^共0兲⬀Eជ_·Eជ_=E²depends only on the magnitude of the field, while the rank-2 tensor term

2H_eff^共2兲depends on its orientation, as can be seen, e.g., from its Q= 0 component E₀^共2兲⬀3Ez2−E²=E²共3 cos²␪− 1兲. With this notation, the second order Stark effect⌬E^共2兲共␤兲can be writ- ten as the expectation value

⌬E^共2兲共␤兲=具␤兩2H_eff^共0兲+₂H_eff^共2兲兩␤典. 共11兲 The Wigner-Eckart theorem implies that the matrix ele- ment具nLJ兩₂H_eff^共K兲兩nLJ典vanishes unless 0ⱕKⱕ2J, so that the tensor part共K= 2兲 of the interaction vanishes,

具␤兩₂H_eff^共2兲兩␤典⬀具6S_1/2储₂H_eff^共2兲储6S_1/2典 ⬅0 共12兲 for the spherically symmetric 6S_1/2state. As a consequence, the Stark effect in the alkali-metal ground state treated in second order perturbation theory is a purely scalar effect which is parametrized in terms of theF- andM-independent second order scalar polarizability␣₀^{共2兲 as}

⌬E^共2兲共␤兲= −1

2␣0共2兲E². 共13兲 This energy perturbation results in an overall shift of the ground state sublevels. The experimental value of

␣₀^共2兲共6S_1/2兲 is 9.98共2兲⫻10³Hz/共kV/cm兲² 关11兴, and is theo- retically well understood at a level of 10⁻³ 关12,13兴. The F- and M-dependent shifts of the ground state sublevels dis- cussed below are approximately 5 and 7 orders of magnitude smaller than this global scalar shift.

B. Third order perturbation theory

As already mentioned, Sandars has shown 关3兴 that the forbidden electric-field-induced lifting of the Zeeman degen- eracy can be explained by extending the perturbation theory to third order, including simultaneously the Stark interaction and the hyperfine interactions in terms of the perturbation operatorW=H_St+H_hf^Fc+H_hf^dd+H_hf^q, which consists of the Stark 共H_St兲, the hyperfine Fermi contact 共H_hf^Fc兲, the hyperfine dipole-dipole共H_hf^dd兲, and the hyperfine quadrupole 共H_hf^q兲 op- erators. In关5兴we have given explicit expressions of the hy- perfine operators in terms of irreducible tensor operators.

Following the general rules of perturbation theory the third order energy perturbation of the ground state level兩␤典 is given by

http://doc.rero.ch

⌬E^共3兲共␤兲=_␥⫽␤,

兺

_␦⫽␤^{具␤兩W兩}_共E␤^␥₋^典具_E^␥_␥^兩W兩_兲共E^␦_␤^典具₋^␦兩W兩_E␦兲^␤典

−具␤兩W兩␤典_␥⫽␤

兺

^兩具_共E^␥_␤^兩W₋^兩_E^␤_␥^典兩_兲^{22, 共14兲}

whereE_␥andE_␦are the unperturbed state energies.

The two terms of Eq.共14兲are trilinear forms of the per- turbation operatorW, of which only terms proportional toE² give nonzero contributions to the Stark interaction. As dis- cussed by Ulzegaet al.关5兴the ground state hyperfine inter- actionE_hf共6S兲 is factored out from the sum over pure Stark matrix elements in the second term of Eq.共14兲. This leads to M-independent, but F-dependent shifts of the ground state levels which can be parametrized, according to Eqs.共1兲and 共2兲, by a third order scalar polarizability␣₀^共3兲共F兲. The second term thus does not contribute to the tensor polarizability, but gives the leading contribution共type A interactions in the no- tation of关5兴兲to the Stark shift of the hyperfine 共clock tran- sition兲frequency关14,15兴. The contribution from this term is thus suppressed by a factor on the order of E_hf共6S兲/

⌬E6P-6S⬇10⁻⁵with respect to the second order scalar polar- izability␣₀^共2兲.

As discussed by Ulzegaet al.关5兴the first term of Eq.共14兲 has contributions from both diagonal and off-diagonal hyper- fine matrix elements. The diagonal contributions共typeBin- teractions in the notation of 关5兴兲 dominate共see Fig. 1兲 and are suppressed by a factor on the order ofE_hf共6P兲/⌬E_6P-6S

⬇10⁻⁷with respect to the second order scalar shift␣₀^{共2兲. The} Zeeman splitting of the ground state levels by the electric field is thus approximately 100 times smaller than the shift of the clock transition frequency.

The third order perturbation can again be expressed in terms of an effective Hamiltonian

3H_eff=共dជ_·Eជ_兲

3␭共dជ_·Eជ_{兲, 共15兲}

in which the projection operator₃␭ is given by

3␭=

兺

_␥ ^兩␥典具_共E^␥_␤^兩H₋^hf_E^兩␥_␥兲^{典具2␥兩. 共16兲}

The effective Hamiltonian can be expressed as the sum of a scalar and a rank-2 tensor, yielding the third order energy perturbation

⌬E^共3兲共␤兲=具␤兩₂H_eff^共0兲+₂H_eff^共2兲兩␤典, 共17兲 in a form equivalent to Eq.共11兲. The scalar part turns out to have the sameFdependence as the third order scalar polar- izability␣₀^共3兲共F兲from the second term of Eq.共14兲and gives a correction to the latter on the order of 1%.

By applying the Wigner-Eckart theorem, the contribution of the second rank tensor part can be written as

具␤兩₂H_eff^共2兲兩␤典⬀关3M²−F共F+ 1兲兴共3Ez2−E²兲⫻具F储关d^共1兲丢₃␭

丢d^共1兲兴^共2兲储F典. 共18兲 The electronic and nuclear angular momenta in this equation cannot be decoupled since the operator 关d^共1兲丢₃␭丢d^共1兲兴^共2兲 depends explicitly on the hyperfine interaction termJ·Iand one can state in general that 具F储关d^共1兲_丢3␭丢d^共1兲兴^共2兲储F典⫽0 for states withFⱖ1. The tensor part has therefore an explicit FandM dependence, which can be parametrized in terms of a third order tensor polarizability␣₂^共3兲共6S1/2,F,M兲.

We can summarize the above results by parametrizing the two terms of the third order interaction关Eq.共14兲兴by a total third order polarizability as

⌬E^共3兲= −1

2␣^共3兲共F,M兲E², 共19兲

with

␣^共3兲共F,M兲=␣₀^共3兲共F兲+␣₂^共3兲共F兲3M²−F共F+ 1兲

2I共2I+ 1兲 f共␪兲. 共20兲 The function f共␪兲= 3 cos²␪− 1 expresses the dependence of

␣^共3兲 on the angle␪ between the electric field and the quanti- zation axis.

III. THIRD ORDER POLARIZABILITY OF THE FREE CESIUM ATOM

A. Earlier calculation revisited

The leading term in the perturbation sum共14兲is given by

⌬E^共3兲共6S_1/2,F,M兲=

兺

n,J,F,m

具nP_J,F兩H_hf兩nP_J,F典

⫻ 兩具nPJ,F,m兩HSt兩6S1/2,F,M典兩² 共E_nP

J,F,m−E_6S

1/2,F,M兲² , 共21兲 where we have used the fact thatH_hfdoes not mixFvalues, and where we have allowed forM mixing byH_Stin case the electric field is not along the quantization axis. This term is diagrammatically represented as diagram B in 关5兴, and we shall refer to it below as typeBinteraction. The correspond- ing tensor polarizability of theF= 4 state of Cs was evalu- FIG. 1.共Color online兲Dependence of the different contributions

to the perturbation sum共14兲leading to the tensor polarizability on n_max, the maximum principal quantum number considered in the sum. Note that previous calculations took only diagonal 共type B兲 hyperfine interactions into account.

http://doc.rero.ch

ated in 关4兴 considering—among other simplifying assump- tions discussed in 关5兴—only diagonal hyperfine matrix elements in the 6P_1/2 and 6P_3/2 states. The result of that calculation is shown as point共f兲in Fig.4and is in disagree- ment with the experimental results. We have redone this cal- culation after dropping all simplifying assumptions of 关4兴, while still considering only diagonal matrix elements. We also used recent more precise experimental values关16兴 for the reduced dipole matrix elements 具6S_1/2储d储6PJ典, which yield the leading contribution. Point共g兲in Fig.4represents the result of this reanalysis. As a result, the gap between theoretical and experimental values of the Cs tensor polariz- ability increases. Extending the perturbation sum to nP_J states withn⬎6 does not affect the discrepancy at a signifi- cant level共lineBin Table 5 and Fig. 1兲.

B. Inclusion of off-diagonal hf matrix elements All previous third order calculations have considered only diagonal hyperfine matrix elements共diagram Bin Fig. 1 of 关5兴兲. However, the first term of the general third order ex- pression关Eq.共14兲兴contains also off-diagonal hyperfine ma- trix elements. In关5兴we have identified five different types of off-diagonal hyperfine mixing contributions:

共1兲Type 1:n mixing ofnS_1/2states.

共2兲Type 2:n mixing ofnP_Jstates with givenJ.

共3兲Type 3:Jmixing ofnP_Jstates with givenn.

共4兲Type 4:n andJmixing ofnPJstates.

共5兲Type 5: mixing ofmD_3/2and 6S_1/2states.

Note that the type 1 interactions have an F, but no M dependence and contribute thus only to the clock frequency shift. The different mixing types are represented as diagrams in Fig. 1 of关5兴and the labeling used here follows the nota- tion of that figure. The energy shifts due to interactions of types 2, 3, and 4 are given by sums of terms with the general structure

具␤兩HSt兩␥1典具␥1兩Hhf兩␥2典具␥2兩HSt兩␤典 共E_␤−E_␥

1兲共E_␤−E_␥

2兲 , 共22兲

with 兩␤典=兩nS1/2,F,M典, 兩␥2典=兩nPJ,f,m典, and 兩␥1典

=兩n⬘^PJ_⬘^,f,m典. The contributions to the interaction of type 5 have the form

具␤兩HSt兩␦典具␦兩HSt兩␥典具␥兩Hhf兩␤典

共E_␤−E_␦兲共E_␤−E_␥兲 , 共23兲 where兩␥典=兩mD3/2,F,M典 and兩␦典=兩nPJ,f,m典.

Some of the off-diagonal hyperfine matrix elements were considered earlier in calculations关14,15,17兴of the Stark shift of the兩6S_1/2,F= 3 ,M= 0典→兩6S_1/2,F= 4 ,M= 0典 clock transi- tion, but were never considered in a calculation of the tensor polarizability. The off-diagonal共type 1兲matrix elements of the Fermi contact interaction betweenS_1/2 states contribute only to the clock frequency shift and are of no relevance here.

The共diagonal兲 matrix elements of the hyperfine quadru- pole interaction H_hf^q are only relevant for states with L,J

⬎1/2 and their numerical values are two orders of magni- tude smaller than all other hyperfine matrix elements. The

matrix elements of the quadrupole part of the hyperfine Hamiltonian can be decomposed in a similar way. However, since their numerical evaluation yields only a relative contri- bution of 10⁻³ to the tensor polarizability we do not repro- duce the corresponding algebraic expressions here. The only hyperfine operator contributing to the off-diagonal matrix el- ements is the dipole-dipole operator H_hf^dd, which can be ex- pressed in terms of irreducible tensor operators as

H_hf^dd=a共r兲共L^共1兲−

冑

10关C^共2兲丢S^共1兲兴^共1兲兲·I^共1兲, 共24兲 whereL^共1兲,S^共1兲, andI^共1兲 are the irreducible vector operators associated with the orbital angular momentum, the electronic spin, and the nuclear spin, respectively, andC^共k兲=

冑

_2k+1^4␲ Y^共k兲 are the normalized spherical harmonic operators of rank k.

The radial dependence of the hyperfine operator is given by a共r兲. The two terms in Eq. 共24兲 correspond to the dipolar magnetic interaction of the nuclear spin I with the orbital 共⬀L^共1兲兲 and the electronic spin 共⬀C^共2兲⫻S^共1兲兲, respectively.

The two interactions obey the selection rules ⌬L= 0 and

⌬L= 0 , ± 2, respectively. The relevant ⌬L= 0 interactions 共types 2,3,4兲 have nonvanishing off-diagonal matrix ele- ments of the form具␥1兩H_hf^dd兩␥2典where兩␥i典=兩n_iP_J

i,f,m典. The explicit reduction of those matrix elements leads to

具␥1兩H_hf^dd兩␥2典=共− 1兲^f+L1+J2+13

冑

7

冑

共2J1+ 1兲共2J2+ 1兲

⫻

再

^{7/2J1 7J/22 1f}

冎

^{共⑀orbital+⑀spin兲具a共r兲典,}

共25兲 where we have separated the contribution of the orbital mag- netic dipole interaction

⑀^orbital=共− 1兲^J22

冑

3

再

^{LJ12 LJ12 11/2}

冎

^{␦L1,L2, 共26兲}

from that of the spin dipolar interaction

⑀^spin=共− 1兲^7/23

冑

10

冑

共2L1+ 1兲共2L2+ 1兲

冉

^L0^{1 2}0 ^L0²

冊

⫻

冦

^{1/2 1/2 1LJ11 LJ22 21}

冧

^{. 共27兲}

The coupling constant is given by

具a共r兲典=

冕

0

⬁⌿n₁P_J

1

* a共r兲⌿n₂P_J

2

r²dr

=2gI

h

␮0

4␲␮b 2

冕

0

⬁

⌿n₁P_J

1

* 1

r³⌿n₂P_J

2

r²dr. 共28兲

Because of the second rank tensor character of the spheri- cal harmonic operatorC^共2兲in Eq.共24兲, the spin dipolar term can also couple states with⌬L= ± 2 and can thus contribute to the third order Stark effect with off-diagonal matrix ele- ments of the form具␥兩H_hf^dd兩␤典, given by Eq.共23兲, where兩␥典

=兩mD_3/2,F,M典withmⱖ5 and 兩␤典=兩6S_1/2,F,M典. In the lat- ter case⑀^orbital vanishes and Eq.共25兲reduces to

http://doc.rero.ch

具␥兩H_hf^dd兩␤典=

冑

5 −F

冑

F− 2

冑

F+ 3

冑

F+ 6

4 具a共r兲典, 共29兲

where the corresponding coupling constant is given by 具a共r兲典=2gI

h

␮0

4␲␮b 2

冕

0

⬁

⌿mD_3/2

* 1

r³⌿6S_1/2r²dr. 共30兲

C. Electric dipole matrix elements

Besides the hyperfine matrix elements the expressions in- volve matrix elements of the Stark interactionH_St, i.e., of the electric dipole operatordbetweenS andPstates. The latter can be reduced by applying the Wigner-Eckart theorem and the standard angular momentum decoupling rules, leading to the reduced matrix elements

具nS1/2储d储mPJ典=共− 1兲^{J−共1/2兲}具mPJ储d储nS1/2典

= −

冑

^2J3^{+ 1}D_nS,mP

J, 共31兲

in which the radial integral is given by

D_nS,mP

J=e

冕

0

⬁

RmP_J共r兲r³RnS_1/2共r兲dr, 共32兲 whereR_nL

Jare the radial wave functions.

We note that the phases 共signs兲 of the reduced matrix elements 关Eq. 共31兲兴 are irrelevant for evaluating contribu- tions involving diagonal hyperfine interactions, but are of fundamental importance for the contributions involving off- diagonal matrix elements.

Although dipole matrix elements between low-lying states in the Cs atoms can be calculated quite accurately using rela- tivistic Hartree-Fock calculations共see, e.g.,关19,21兴兲we have decided to rather use 共more precise兲 experimental values, whenever they were available. TableI lists共in bold兲 the re- duced dipole matrix elements used in the present calculation.

For the matrix elements involving higher-lying states neither experimental nor theoretical values were available. In that case we have evaluated the corresponding radial integrals using nonrelativistic wave functions obtained by solving the Schrödinger equation as described in the next paragraph. The values of Table I show that this approach reproduces the matrix elements to better than 10%. Assuming that the quoted accuracy also holds for excited states, and consider- ing that the excited states give only a minor contribution to the final result共cf. TableI兲, we are confident that there is no significant uncertainty introduced by using our theoretical values for excited state matrix elements.

D. Wave functions of the free cesium atom

The nonrelativistic wave functions of the states兩nL,J典can be separated into radialR_nL,J共r兲and angularY_L,m共␪^,␾兲parts

⌿nL_J共r兲=具r兩nL,J兩典=R_nL,J共r兲Y_L,m共␪^,␾兲. 共33兲 The radial wave functions are found as solutions of the radial Schrödinger equation foru共r兲=rR_n,J共r兲,

−1 2

d²u共r兲

dr +

冋

^{VCsfree共r兲+L共L2r+ 1兲2}

册

^{u共r兲=Eu共r兲. 共34兲}

For the potentialV_Cs^free共r兲we have used a scaled Thomas- Fermi model potentialV_TF共r,␭兲 共with a scaling parameter␭兲 including dipolar and quadrupolar core-polarization correc- tions V_pol共r兲 and the spin-orbit interaction V_spin-orbit共r兲 that depends on the angular momentumL,

V_Cs^free共r兲=V_TF共r,␭兲+V_pol共r兲+V_spin-orbit共r兲. 共35兲 This model follows the work of Gombas关22兴and Norcross 关23兴and was explained in detail in关9兴. Using this approach we have calculated the electronic wave functions for nSJ, nPJ, and nDJstates up ton= 200.

E. Terms with diagonal hf matrix elements

As stated above terms involving diagonal hyperfine ma- trix elements共hyperfine coupling constants兲in thenPJstates give the leading contribution to the third order tensor polar- izability. In the numerical evaluation of the terms we use again experimental values, when they are available. In Table IIwe list the hyperfine coupling constants used共in bold兲. A comparison of experimental coupling constants of low-lying states with constants calculated using our Schrödinger wave functions is also made in order to illustrate the accuracy of the theoretical approach. As for the radial integrals there are TABLE I. Radial integralsD_nS1/2,mP_Jin atomic units. The val- ues marked in bold were used in the calculation of the third order tensor polarizability. The theoretical values calculated from Schrödinger wave functions were used for matrix elements involv- ing states withn⬎7. The theoretical values of the states involving n= 6 , 7 were not used in the calculation but are shown merely to illustrate the accuracy of our calculation.

nS_1/2,mP_J Experiments

Theory

共this work兲 Deviation 6S_1/2, 6P_1/2 ±5.5087共75兲^a −5.3982 2%

6S_1/2, 6P_3/2 ±5.4829共62兲^a −5.3009 3%

6S_1/2, 7P_1/2 ±0.3377共24兲^b −0.3092 8%

6S_1/2, 7P_3/2 ±0.5071共43兲^b −0.4617 9%

7S_1/2, 6P_1/2 ±5.184共27兲^c +5.245 1%

7S_1/2, 6P_3/2 ±5.611共27兲^c +5.696 2%

7S_1/2, 7P_1/2 ±12.625^d −13.298 5%

7S_1/2, 7P_3/2 ±12.401^d −13.015 5%

6S_1/2, 8P_1/2 −0.078

6S_1/2, 8P_3/2 −0.130

6S_1/2, 9P_1/2 −0.045

6S_1/2, 9P_3/2 −0.078

6S_1/2, 10P_1/2 −0.030

6S_1/2, 10P_3/2 −0.054

aExperimental values from Rafacet al.关16兴.

bExperimental values from Vasilyevet al.关18兴.

cExperimental value quoted in关19兴.

dBennettet al.关20兴, with signs determined from Eq.共31兲.

http://doc.rero.ch

no experimental data for the high-lying states and we used our theoretical values, and, again, our accuracy is sufficient since it affects only the共very small兲 contributions from ex- cited states.

It can be easily seen that the off-diagonal contributions are sensitive to the sign of the radial matrix elements. When- ever we extracted such matrix elements from experimental data—which all determined squared matrix elements—we have used the sign of the dipole matrix elements given by Eq.共31兲.

F. Terms with off-diagonal hf matrix elements S-mixing off-diagonal hyperfine matrix elements have played an important role in the measurement of parity viola-

tion in the Cs atom and they were measured and calculated with a high accuracy. However, such matrix elements inter- vene only in type 1 interactions, and do thus not contribute to the present calculation, in which the relevant terms come fromnPJ−n⬘^PJ_⬘^mixing 共types 2, 3, and 4兲and 6S_1/2-mD_3/2 mixing共type 5兲. We have evaluated all relevant diagonal and off-diagonal hyperfine matrix elements using Schrödinger wave functions and the expressions given above. A selection of numerical values of those matrix elements for the lowest- lying states is shown in TablesIIIandIV.

G. Contribution of continuum states

In principle, the summation in Eq.共14兲has to be carried out over continuum states as well as over bound states. It was shown in 关14,15兴 that the continuum states contribute significantly 共⬃15%兲 to type 1 interactions, which them- selves do not contribute to the tensor polarizability. In order to estimate the influence of the continuum on the tensor po- larizability we calculated the 共positive energy兲 continuum wave functions using our Schrödinger approach and used them to evaluate bound-continuum and continuum- continuum matrix elements of the electric dipole and hyper- fine interactions.

For the numerical evaluation of transition matrix elements 具p,L_J⬘_⬘兩W兩nLJ典 between bound 兩n,LJ典 and continuum states 兩p,L_J⬘_⬘典 as well as between continuum states we use the ex- pressions共32兲and共28兲. We extended the perturbation sums 共integrals兲 of all relevant interactions 共types B, 2–5兲 over bound and continuum states.

TABLE II. Hyperfine coupling constantsA_hf共nL_J兲in MHz. The values marked in bold were used in the calculation of the third order tensor polarizability, where they contribute as diagonal matrix ele- ments ofH_hfto type Binteractions. The theoretical values calcu- lated from Schrödinger wave functions were used for states with n⬎8. The theoretical values of the statesn= 6 , . . . , 8 were not used in the calculation but are shown to illustrate the accuracy of our calculation. The hyperfine constantsA_nL

Jare related to the coupling constantsa共r兲defined in the text by the relation具a共r兲典=_L^J共_共^J+1_L+1^兲_兲A_nL

J.

nL_J Experiments Theory Deviation

6P_1/2 291.920共19兲^b 317.9 8%

6P_3/2 50.275共3兲^c 48.3 4%

7P_1/2 94.35共4兲^a 99.9 6%

7P_3/2 16.605共6兲^a 15.4 7%

8P_1/2 42.97共10兲^a 45.0 5%

8P_3/2 7.58共1兲^a 7.0 7%

9P_1/2 24.2

9P_3/2 3.8

10P_1/2 14.5

10P_3/2 2.3

11P_1/2 9.4

11P_3/2 1.5

aExperimental value taken from Arimondoet al.关24兴.

bExperimental value taken from Rafacet al.关25兴.

cExperimental value taken from Tanneret al.关26兴.

TABLE III. Off-diagonal hyperfine coupling constants具a共r兲典 共in MHz兲representing interactions of types 2, 3, and 4. The numerical evaluation is based on Schrödinger wave functions as described in the text.

6P_1/2 7P_1/2 8P_1/2 9P_1/2 6P_3/2 7P_3/2 8P_3/2 9P_3/2

6P_1/2 785.3 827.1 386.4 187.7 106.1 71.5 52.5

7P_1/2 422.2 309.4 105.2 85.0 57.3 42.1

8P_1/2 290.7 70.6 57.1 53.8 39.5

9P_1/2 51.8 41.8 39.4 41.3

6P_3/2 187.7 105.2 70.6 51.8 14.3 9.6 7.1

7P_3/2 106.1 85.0 57.1 41.8 7.8 5.7

8P_3/2 71.5 57.3 53.8 39.4 5.4

9P_3/2 52.5 42.1 39.5 41.3

TABLE IV. Off-diagonal hyperfine coupling constants具a共r兲典 共in MHz兲of type 5 interactions. The numerical evaluation is based on Schrödinger wave functions as described in the text.

n 具mD_3/2兩a共r兲兩6S_1/2典

5 74.4

6 42.9

7 28.4

8 21.2

9 16.4

http://doc.rero.ch

The calculation poses no convergence problems, except for the type 5 interaction, which is the only one that involves dipole matrix elements between continuum states. Such ma- trix elements also occur, e.g., in the calculation of brems- strahlung transitions, and pose some technical difficulty. For their evaluation we have used the method of exterior com- plex scaling introduced in 关27,28兴 with a numerical imple- mentation on a grid. The regular and outgoing waves in the complex plane were obtained through a step-by-step propa- gation using recursive relations 关27兴. As a result we find a total contribution from continuum states to the final value of

␣₂^共3兲 on the order of 0.2%.

H. Numerical evaluation of the third order tensor polarizability for the free Cs atom

We have done a numerical evaluation of all diagonal共type B兲 and off-diagonal 共types 2–5兲 contributions to the tensor polarizability summing over bound states up ton= 200. For theF= 4 state we find

␣2共3兲共F= 4兲= − 3.72共25兲⫻10⁻²Hz/共kV/cm兲², 共36兲 in which the contribution from the quadrupole hyperfine in- teraction is 2⫻10⁻⁵Hz/共kV/cm兲². This result is shown as a dotted line in the left part of Fig.4, together with previous theoretical and experimental results, described in the figure caption.

TableV shows the relative contributions of the different diagonal共typeB兲and off-diagonal共types 2–5兲hyperfine in- teractions to the third order tensor polarizability. One notes that the diagonal共typeB兲contributions which were the only ones used in previous calculations overestimate the modulus of the third order tensor polarizability by approximately 50%. Of all the off-diagonal contributions the interactions of type 2 and 5 共n-mixing of nPJ states with given J and 6S_1/2-nD_3/2 mixing兲 give large contributions with opposite signs which cancel each other to a large extent.

We estimate the uncertainty of the final result to be⬇7%.

This estimation is based on the precision 共2 – 9 %兲 with which our wave functions reproduce measured hyperfine constants共TableII兲and dipole matrix elements共TableI兲, and considers that we have used共more precise兲experimental val- ues for the leading terms. As shown in the previous para- graph the relative contribution of the continuum states to the tensor polarizability is on the order of 10⁻³ and is thus neg-

ligible in the final result. Figure1 shows the relative contri- butions from the different interactions in a cumulative way.

Our theoretical value for␣₂^共3兲共F= 4兲of the free Cs atom is in good agreement with all experimental values共Fig.4兲.

IV. THIRD ORDER POLARIZABILITY OF CESIUM IN SOLID HELIUM

A. Experiment

In connection with a proposed search for an electric di- pole moment of Cs atoms in solid helium we were led to study the tensor Stark shifts in that unusual sample. The exceptionally long longitudinal and transverse spin relax- ation times of Cs in the body-centered cubic共bcc兲phase of

4He crystals form the basis for high resolution magnetic resonance experiments. Recently we have measured the Stark shift of magnetic resonance lines in the ground state of Cs atoms implanted in bcc ⁴He by two different tech- niques 关8兴. The experimental values of the corresponding tensor polarizabilities −4.07共20兲⫻10⁻²Hz/共kV/cm兲² and

−4.36共28兲⫻10⁻²Hz/共kV/cm兲² for ␣₂^共3兲共F= 4兲 are shown as points共d兲 and共e兲 in Fig.4.

B. Wave function of Cs in solid⁴He

For calculating the tensor polarizability of Cs in solid he- lium one has to include the influence of the He matrix on the atomic energies and wave functions. Because of the Pauli repulsion Cs atoms form spherical bubbles, which are de- scribed by a spherically symmetric continuous He density distribution

␳共R,R0,⑀兲=

再

^{0,␳0兵1 −关1 +⑀共R−R0兲兴e−⑀共R−R0兲其 RR,⬍ⱖRR00,}

共37兲 whereR₀ is the bubble radius.⑀ describes the steepness of the interface共density changes from zero to the bulk density兲 and␳0is the bulk density which depends on the He tempera- ture and pressure. In Eq. 共37兲 we have assumed that solid helium is an incompressible fluid, an assumption well justi- fied by the quantum nature of condensed ⁴He. Our calcula- tion of the Cs wave functions relies on an extension of the so-called bubble model关9兴, which was shown in the past to be well suited for describing energies, absorption or emission TABLE V. Relative contributions共in %兲to␣₂^共3兲共F= 4兲from diagonal, typeB, Eq.共21兲and off-diagonal,

types 2–5, Eqs.共22兲and共23兲, hyperfine interactions of excited statesnL_J. n

Type 6 7 8 9 10–200 Total

B 146.8 ⬇0 ⬇0 ⬇0 ⬇0 146.8

2 73.4 10.2 4.3 6.4 94.3

3 −37.0 −0.1 ⬇0 ⬇0 ⬇0 −37.1

4 −2.1 −0.4 ⬇0 −0.2 −2.7

5 −116.7 −3.7 12.6 2.2 4.3 −101.3

Total −6.9 67.5 22.4 6.5 10.5 100.0

http://doc.rero.ch

line shapes, hyperfine structure, and lifetimes of alkali metals in solid helium关9,29兴.

The total interaction potential experienced by the Cs va- lence electron is given by

V_Cs^bub共r,R0,⑀兲=V_Cs^free共r兲+

冕

^{d3R␳共R,R0,⑀兲}

⫻关VHe共r,R兲+V_cross共r,R兲+V_cc共R兲兴, 共38兲 whereV_Cs^free共r兲is the potential of the valence electron with the Cs⁺ core introduced in Eq.共35兲. The ionic core and the He atoms are assumed to have fixed spatial positions 共Born- Oppenheimer approximation兲. In Eq. 共38兲 V_He共r,R兲 and V_cc共R兲represent the interactions of the valence electron and the Cs⁺ ion with a He atom. The potential V_cross共r,R兲 de- scribes the three-body interaction resulting in the simulta- neous polarization of the He atom by the Cs core and the valence electron.randRpoint from the core to the electron and to each He atom, respectively. Explicit forms for all the potentials as well as numerical parameter values are given in 关9兴. The potentials seen by the valence electron in the free Cs atomV_Cs^free and in the Cs atom trapped in solid He V_Cs^bub are shown, together with the energies of the lowest states in Fig.

3.

The energy needed to from a bubble consists of a pressure volume term, a surface energy共with surface tension param- eter ␴兲, and the kinetic energyE_kin, which arises from the localization of the He atoms at the bubble interface

E_bub共R_b,⑀兲=4

3␲^Rb3p+ 4␲^Rb2␴^+Ekin, 共39兲 whereRb=f共R0兲is the center of gravity of the interface. The total energy of the bubble defect is thus V_tot^bub共r,R₀,⑀兲

=V_Cs^bub共r,R₀,⑀兲+E_bub共R0,⑀兲, and the average bubble radiusRb

is found by numerically minimizing the total energy with respect to the two parameters R₀ and ⑀. Using the known bubble parameters and the interaction potential one can then solve the radial Schrödinger equation as for the valence elec- tron of the free Cs atom. In order to illustrate the effect of the He bubble we compare in Fig. 2 the wave function of the 9P_1/2state of the free Cs atom with the corresponding wave function in the bubble. As expected, the wave function in the bubble is compressed due to the repulsive interaction with the surrounding helium atoms.

Using the solutions共wave functions兲one can then evalu- ate the matrix elements required for the numerical calcula- tion of the third order polarizability, in analogy to the free atomic case.

C. Numerical evaluation of the third order tensor polarizability of Cs in solid He

The numerical evaluation of the tensor polarizability␣₂^共3兲 for Cs in solid He was done in analogy to the case of the free atom, with wave functions and energy levels calculated using the bubble model introduced in Sec. IV B. While in the case of the free Cs atom we have considered bound states up to

the principal quantum numbern= 200 we need only to con- sider states up to 9P in the He matrix, since higher-lying states are not bound共see Fig.3兲. Moreover, the energy levels of the higher-lying states are strongly shifted to higher ener- gies. For example, the energies of the 8P_1/2and 9P_1/2 state are displaced by 5580 cm⁻¹ 共22%兲 and 5830 cm⁻¹ 共21%兲, respectively, with respect to the free atom. Since the pertur- bation sum involves squared energy denominators these ex- cited states give smaller contributions than in the free atomic case. In TableVIwe analyze the dependence of the relative contributions from the different diagrams on the number of states included in the perturbation sum, which qualitatively reflects the same features as in the free atomic case. Our final theoretical value of the tensor polarizability of Cs in solid

4He is

␣₂^共3兲共F= 4兲= − 4.11⫻10⁻² Hz/共kV/cm兲². 共40兲

On the right of Fig.4we compare this theoretical value with the experimental values of ␣₂^共3兲 of Cs atoms trapped in a body-centered cubic solid⁴He matrix关8兴and find an excel- lent agreement.

r (a.u.) (a.u.)

energy(a.u.)

FIG. 3. 共Color online兲 Energies of the lowestnS_1/2 共left兲 and nP_1/2共right兲states of the free Cs atom共dashed lines兲and of Cs in the bcc solid⁴He共solid lines,R₀= 10.2,⑀= 2.45 atomic units兲. The corresponding potentials for the valence electron of the free Cs atom关Eq.共35兲兴and of the Cs atom in solid He关Eq.共38兲兴are shown as dotted and solid lines, respectively. Then= 9 states are the high- est bound states in the bubble. The bump in the potentialV_Cs^bubis due to the repulsion by the bubble interface. The arrows between the graphs indicate the ionization limits.

r (a.u.)

FIG. 2. 共Color online兲Comparison of the 9P_1/2wave function of the free Cs atom共black solid line兲with the same wave function of a Cs atom in a spherical helium bubble共red dotted line兲. The bubble parameters correspond to the equilibrium bubble shape of a ground state Cs atom共R₀= 10.2, indicated by the vertical dotted line and⑀= 2.45 in atomic units兲.

http://doc.rero.ch

V. SUMMARY

We have presented calculational details of our semiempir- ical evaluation of the cesium ground state tensor polarizabil- ity␣₂^{共3兲. ThisF- and}M-dependent polarizability is forbidden in second order perturbation theory and arises only when considering the Stark effect and hyperfine interactions in a third order perturbation calculation. The tensor polarizability is suppressed by seven orders of magnitude with respect to the usual scalar polarizability of the atom.

We have evaluated␣₂^共3兲both for the free cesium atom and for Cs embedded in a cubic solid⁴He matrix using solutions of the Schrödinger equation with appropriate potentials for evaluating the relevant matrix elements in both cases. We have found that off-diagonal hyperfine matrix elements, which were not considered in previous treatments give sub- stantial contributions共of different signs兲to␣₂^共3兲. As a result we obtain theoretical values for the tensor polarizabilities that are in excellent agreement with previous and recent

measurements. The modulus of the experimental tensor po- larizability of Cs in the He matrix is found to be 8 – 10 % larger than the one of the free Cs atom.

It thus seems that the 40 year old discrepancy between experimental and theoretical tensor polarizabilities has now found a satisfying final solution.

ACKNOWLEDGMENT

This work was supported by the Grant No. 200020- 103864 of the Schweizerischer Nationalfonds.

关1兴R. D. Haun and J. R. Zacharias, Phys. Rev. 107, 107共1957兲. 关2兴E. Lipworth and P. G. H. Sandars, Phys. Rev. Lett. 13, 716

共1964兲.

关3兴P. G. H. Sandars, Proc. Phys. Soc. London 92, 857共1967兲. 关4兴H. Gould, E. Lipworth, and M. C. Weisskopf, Phys. Rev. 188,

24共1969兲.

关5兴S. Ulzega, A. Hofer, P. Moroshkin, and A. Weis, Europhys.

Lett. 76, 1074共2006兲.

关6兴J. P. Carrico, A. Adler, M. R. Baker, S. Legowski, E. Lipworth, P. G. H. Sandars, T. S. Stein, and C. Weisskopf, Phys. Rev.

170, 64共1968兲.

关7兴C. Ospelkaus, U. Rasbach, and A. Weis, Phys. Rev. A 67, 011402共R兲 共2003兲.

关8兴S. Ulzega, A. Hofer, P. Moroshkin, R. Müller-Siebert, D. Net- tels, and A. Weis, Phys. Rev. A 75, 042505共2007兲.

关9兴A. Hofer, P. Moroshkin, S. Ulzega, D. Nettels, R. Müller- Siebert, and A. Weis, Phys. Rev. A 76, 022502共2007兲. 关10兴J. R. P. Angel and P. G. H. Sandars, Proc. R. Soc. London, Ser.

A 305, 125共1968兲.

关11兴J. M. Amini and H. Gould, Phys. Rev. Lett. 91, 153001 共2003兲.

关12兴A. Derevianko and S. G. Porsev, Phys. Rev. A 65, 053403

共2002兲.

关13兴H. L. Zhou and D. W. Norcross, Phys. Rev. A 40, 5048 共1989兲.

关14兴K. Beloy, U. I. Safronova, and A. Derevianko, Phys. Rev. Lett.

97, 040801共2006兲.

关15兴E. J. Angstmann, V. A. Dzuba, and V. V. Flambaum, Phys. Rev.

Lett. 97, 040802共2006兲.

关16兴R. J. Rafac, C. E. Tanner, A. E. Livingston, K. W. Kukla, H. G.

Berry, and C. A. Kurtz, Phys. Rev. A 50, R1976共1994兲. 关17兴J. D. Feichtner, M. E. Hoover, and M. Mizushima, Phys. Rev.

137, A702共1965兲.

关18兴A. A. Vasilyev, I. M. Savukov, M. S. Safronova, and H. G.

Berry, Phys. Rev. A 66, 020101共R兲 共2002兲.

关19兴M. S. Safronova, W. R. Johnson, and A. Derevianko, Phys.

Rev. A 60, 4476共1999兲.

关20兴S. C. Bennett, J. L. Roberts, and C. E. Wieman, Phys. Rev. A 59, R16共1999兲.

关21兴V. A. Dzuba, V. V. Flambaum, A. Y. Kraftmakher, and O. P.

Sushkov, Phys. Lett. A 142, 373共1989兲.

关22兴P. Gombas, Pseudopotentiale 共Springer-Verlag, Berlin- Gottingen-Heidelberg, 1956兲.

关23兴D. W. Norcross, Phys. Rev. A 7, 606共1973兲. TABLE VI. Relative contributions共in %兲to the tensor polariz-

ability for Cs trapped in solid He. Note that Cs in solid He has no bound states withn⬎9.

n= 6 , 7 n= 8 n= 9 Total

B 143.4 ⬇0 ⬇0 143.4

2 21.1 13.4 8.5 43.0

3 −26.8 ⬇0 ⬇0 −26.8

4 −0.2 −0.1 −0.1 −0.4

5 −65.4 4.1 2.1 −59.2

Total 72.1 17.4 10.5 100.0

e atomic beam

solid He⁴

(g) (f)

FIG. 4. The Cs tensor polarizability ␣₂^共3兲共F= 4兲. Atomic beam measurements of共a兲Carricoet al.关6兴,共b兲Gouldet al.关4兴, and共c兲 Ospelkauset al. 关7兴. Points 共d兲 and 共e兲 represent recent measure- ments in solid helium关8兴. The circles are the theoretical values共f兲 from Gouldet al.关4兴and 共g兲 from Ulzegaet al.关5兴. The dashed lines represent the results of our new calculations for the free atom and for Cs in a solid helium matrix, together with their uncertainties 共shaded bands兲.

http://doc.rero.ch

关24兴E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys.

49, 31共1977兲.

关25兴R. J. Rafac and C. E. Tanner, Phys. Rev. A 56, 1027共1997兲. 关26兴C. E. Tanner and C. Wieman, Phys. Rev. A 38, 1616共1988兲.

关27兴M. Chrysos and M. Fumeron, J. Phys. B 32, 3117共1999兲. 关28兴M. Chrysos, J. Phys. B 33, 2875共2000兲.

关29兴P. Moroshkin, A. Hofer, S. Ulzega, and A. Weis, Fiz. Nizk.

Temp. 32, 1297共2006兲 关Low Temp. Phys. 32, 981共2006兲兴.