SHIFT OF 2Sl/2 HYPERFINE SPLITTINGS DUE TO BLACKBODY RADIATION AND ITS INFLUENCE ON FREQUENCY STANDARDS

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SHIFT OF 2Sl/2 HYPERFINE SPLITTINGS DUE TO BLACKBODY RADIATION AND ITS INFLUENCE

ON FREQUENCY STANDARDS

W. Itano, L. Lewis, D. Wineland

To cite this version:

W. Itano, L. Lewis, D. Wineland. SHIFT OF 2Sl/2 HYPERFINE SPLITTINGS DUE TO BLACK-

BODY RADIATION AND ITS INFLUENCE ON FREQUENCY STANDARDS. Journal de Physique

Colloques, 1981, 42 (C8), pp.C8-283-C8-287. �10.1051/jphyscol:1981835�. �jpa-00221731�

CoZZoque C8, suppZ&ent au n012, Tome 42, ddeembre 1981 page C8-283

SHIFT OF 'slI2 HYPERFINE SPLITTINGS DUE TO BLACKBODY RADIATION AND ITS INFLUENCE ON FREQUENCY STANDARDS

W.M. I t a n o , L.L. Lewis and D.J. Wineland

Frequency and Time Standards Group, Time and Frequency Divieion, National Bureau of Standards, BouZder, CoZorado 80303, U. S. A .

Abstract. - Frequency shifts of hyperfine splittings of 2Sq states due to the blackbody electric field are calculated. It is shown that they can be estimated from the dc hyperfine Stark shifts, which have pre- viously been measured in the ground states of hydrogen and the alkali atoms. The shifts scale as T4. The fractional shift for Cs at 300 K is -1.7 x 10-14, which is large enough to be significant in primary frequency standards, and should be measurable. A simple method of calculating the hyperfine Stark shifts is described, which is based on the Bates-Damgaard method for determining radial matrix elements and the Fermi-Segrg formula for determining the contact hyperf i ne matrix elements. It agrees with the experiment to within 12% for the entire a1 kali series. It is applied to ~ a + and Hg+, for which no experimen- tal data are yet available, and which are currently of interest for frequency standards. At 300 K, the fractional shifts are -9.9 x 10-I and -2.4 x 10-l5 for Hg+ and ~ , ' a respectively. The shift due to the blackbody magnetic field is -1.3 x 10-l7 [T(K)/300I2 for any 2S4 state.

Introduction. - The most accurate and stable atomic frequency standards are based on hyperfine transition frequencies in 25% ground states, such as in 133Cs,

lH,

and 87Rb. In this communication, we estimate the temperature-dependent shift of 25% hyperfine splittings due to the blackbody radiation field. We note that this effect is large enough to be observable in a Cs atomic beam apparatus. The shift of the Cs hyperfine splitting at T = 300 K from the unperturbed

(T =

0 K) value causes a frequency off set which i s significant for primary frequency standards.

However, since the temperature of these standards is kept very stable (for other reasons), the correction can be made very precisely.

Blackbody Shifts. - According to the Planck radiation law,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981835

C8-284 JOURNAL DE PHYSIQUE

where E2(w)dw [B2(w)dw] is the squared amplitude of the blackbody electric (mag- netic) field in a bandwidth dw around w. [Atomic units (a.u.) are used unless otherwise specified

(3 =

me

=

e

=

I)]. The mean-squared fields are

=

(8.319 ~ / c m ) ~

x

[T(K)/~oo]~' and

Gallagher and Cooke [I] pointed out that these fields induce temperature- dependent shifts of transition frequencies in atoms and molecules through the ac Zeeman and Stark effects [2,3] They estimated the fractional blackbody ac Zeeman shift of the ground-state hyperfine splitting in H or Cs to be about

10-l6

at

T =

300 K.

We have derived the following expression for this shift in any 25% ground state, which is valid at zero dc magnetic field and at temperatures such that the peak of the blackbody spectrum is at a much higher frequency than the hyperfine frequency. We find

where gJ and gI are the electronic and nuclear g factors, respectively. In the last 1 ine, we have assumed that gJ

=

2 and that

I

gI/gJ I

<< 1.

At laboratory temperatures, the blackbody ac Stark shift of the hyperfine splitting, which has previously been neglected, is generally larger than the ac Zeeman shift. The ac hyperfine Stark shift due to an electric field of fre- quency, w , is approximately equal to the dc hyperfine Stark shift due to a static field with the same rms value, if w

<<

wres, where w res is the lowest allowed

2

electric dipole transition frequency. The correction is of order (w/wres) . ^For

the frequency corresponding t o the peak o f the blackbody spectrum a t 300 K.

Therefore, a t 300 K, the blackbody ac hyperfine s h i f t i s approximately equal t o the s h i f t caused by a dc f i e l d o f 8.3 V/cm.

The dc hyperfine Stark s h i f t was f i r s t observed i n Cs by Haun and Zacharias [4]. Later, i t was observed i n H (see Ref. 5) and other a l k a l i atoms [6]. These

1 1

experiments measured t h e Stark s h i f t o f the

( F

= I +

2,

MF

=

0) c-,

(F

= I

- 2,

MF = 0) t r a n s i t i o n . They can be considered t o be measurements o f the scalar hyperfine p o l a r i z a b i l i t i e s , which are independent o f

MF

and the o r i e n t a t i o n o f the e l e c t r i c f i e l d , since the c o n t r i b u t i o n s from the tensor p o l a r i z a b i l i t i e s can be estimated and are less than the experimental u n c e r t a i n t i e s

173.

Only the scalar p o l a r i z a b i l i t y contributes t o the blackbody ac Stark s h i f t , because o f the i s o t r o p y o f the blackbody r a d i a t i o n . The f r a c t i o n a l ac Stark s h i f t o f t h e Cs hyperfine s p l i t t i n g can be estimated from the measured dc hyperf i ne p o l a r i z - a b i l i t y (see Table 1 o f Ref. 6) and Eq. (2) t o be -1.69(4) x 10-l4 [T(K)/300I4.

This s h i f t i s large enough t o a f f e c t the c a l i b r a t i o n o f primary Cs frequency standards and therefore should be taken i n t o account. For example, the f r a c t i o n - a l uncertainty o f one primary Cs frequency standard (CS 1 o f the Physikalisch- Technische Bundesanstalt) i s stated t o be 6.5 x 10-Is, b u t has n o t been corrected f o r the blackbody s h i f t [8]. For H and Rb, the s h i f t s are too small t o be experi- mentally s i g n i f i c a n t a t present.

The theory of the dc hyperfine Stark s h i f t o f 2S4 ground states i s q u i t e w e l l developed, and the c a l c u l a t i o n s are i n good agreement w i t h the experiments.

For hydrogenic atoms and ions, an a n a l y t i c s o l u t i o n has been obtained C9,10].

Numerical c a l c u l a t i o n s have been made f o r the neutral a1 kal i atoms [ll-151.

We w r i t e the t h i r d - o r d e r p e r t u r b a t i o n expression f o r the scalar f r a c t i o n a l dc hyperfine Stark s h i f t o f the ns 2Sq s t a t e o f an a l k a l i - l i k e atom o r i o n i n the f o l l o w i n g form, which i s independent o f the s p i n and magnetic moment o f the nucleus:

where

2

(n 'Sq

IIr1I

^{n'' 'Pj)}

= 6[W(n1'PJ)-W(nS)J2 n" 3

and

C8-286 JOURNAL DE PHYSIQUE

The reduced m a t r i x elements o f

;,

the p o s i t i o n operator o f the valence electron, are defined w i t h t h e conventions o f Edmonds [16]. We choose the phases o f the r a d i a l wave functions so t h a t they are r e a l . W(nUPJ) and W(nS) are the energies (not i n c l u d i n g the hyperfine i n t e r a c t i o n ) o f the n"p 2PJ and the ns 2S states,

4

respectively, and $nS(0) i s the value o f the ns 2S wave f u n c t i o n a t the o r i g i n .

4

The dc magnetic f i e l d i s assumed t o be so small t h a t the Zeeman s p l i t t i n g i s much l e s s than the hyperfine spl i t t i n g .

We have developed a simple method o f approximately evaluating Eqs. (6a) and (6b). We c a l c u l a t e the r a d i a l m a t r i x elements using the Coulomb (Bates- Damgaard) approximation [17] and the values o f the s-state wave functions a t the o r i g i n using the Fermi-Segrg formula 1181. We have used t h i s method t o c a l c u l a t e t h e scalar f r a c t i o n a l ground-state hyperfine p o l a r i z a b i l i t i e s o f L i , Na, K, Rb, and Cs and have obtained agreement w i t h experiment t o w i t h i n 12% o r b e t t e r i n a l l cases. The lowest three p states and the lowest f i v e s states were included i n the basis.

This method can be used f o r other atoms, f o r which no experimental data o r c a l c u l a t i o n s have y e t been published, such as the s i n g l y ionized a l k a l i n e earths. We have c a r r i e d out the c a l c u l a t i o n s f o r the ground states o f Hg

+

and

~ a + , which are c u r r e n t l y o f i n t e r e s t f o r applications i n stored-ion frequency standards [19,20]. I n atomic u n i t s , k = 37.9 f o r Hg+ and k = 902 f o r Ba

+ .

^The

conversion between atomic and laboratory u n i t s o f E2 i s given by

A t 300 K, the f r a c t i o n a l blackbody ac hyperfine Stark S h i f t s are -9.9 x 1 0 - l 7 and -2.4 x 1 0 - l 5 f o r Hg+ and ~a', respectively. I n rf t r a p experiments, the ac Stark s h i f t due t o the trapping f i e l d s may be l a r g e r .

The ground-state hyperfine s h i f t due t o an ac e l e c t r i c f i e l d o f magnitude E(t)

=

E(w) cos wt can be obtained by the same method t h a t was used t o derive Eqs. (6a) and (6b), except t h a t the formula f o r the ac Stark s h i f t [2,31 i s used.

For Cs a t 300 K, the blackbody ac Stark s h i f t i s 1.4% greater than the s h i f t due t o a dc f i e l d o f the same rms value.

The blackbody s h i f t could be observed i n a Cs frequency standard which was modified so t h a t the temperature o f a tube surrounding the atoms i n the resonance region could be varied. I f the temperature were changed, f o r example, from 300 K t o 400 K, the f r a c t i o n a l frequency s h i f t would be 3.7 x 10-14. I f the frequency standard had the same frequency s t a b i l i t y as NBS-6, the primary frequency standard o f the United States, [21] t h i s s h i f t could be determined t o 30% o r b e t t e r i n an averaging time o f several hours.

This work was supported i n p a r t by the A i r Force O f f i c e o f S c i e n t i f i c Research and the O f f ic e o f Naval Research.

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