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HAL Id: jpa-00221736

https://hal.archives-ouvertes.fr/jpa-00221736

Submitted on 1 Jan 1981

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PROSPECTS FOR A MERCURY ION FREQUENCY STANDARD

M. Jardino, M. Desaintfuscien, F. Plumelle

To cite this version:

M. Jardino, M. Desaintfuscien, F. Plumelle. PROSPECTS FOR A MERCURY ION FRE- QUENCY STANDARD. Journal de Physique Colloques, 1981, 42 (C8), pp.C8-327-C8-338.

�10.1051/jphyscol:1981840�. �jpa-00221736�

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JOURNAL DE PHYSIQUE

CoZZoque C8, s u p p l h e n t au n012, Tome 42, de'cembre 1981 page

(28-327

P R O S P E C T S FOR A MERCURY ION FREQUENCY STANDARD

M. Jardino, M. D e s a i n t f u s c i e n and F. Plumelle

Laboratooire de ZrHorZoge ~ t o m i ~ u e * , Bat. 221, Uniuersite' Paris-Sud, 91405 Orsay, France

Abstract.- A mercury i o n frequency s t a n d a r d has been b u i l t and o p e r a t e d a t Orsay. I t i s b r i e f l y d e s c r i b e d and t h e r e s u l t s o b t a i n e d f o r i t s s t a b i l i t y a r e given. Measurement of v a r i o u s p h y s i c a l parameters and c a l c u l a t i o n of t h e i r e f f e c t o n t h e s h o r t and medium term s t a b i l i t y allow t o d i s c u s s p o s s i b l e i m - provements.

Systematic e f f e c t s a r e s t u d i e d and a method of measurerent i s given. It i s shown t h a t good s t a b i l i t y and accuracy can b e achieved.

I n t r o d u c t i o n . - The s p e c i f i c advantages o f the i o n s t o r a g e technique a p p l i e d t o frequency s t a n d a r d s a r e w e l l known. F i r s t , t h e i o n s can be c o n f i n e d d u r i n g very l o n g time w i t h o n l y very weak p e r t u r b a t i o n s due t o t h e c o n f i n i n g f i e l d s . Secondly, t h e p e r i o d i c c h a r a c t e r of t h e i r motion i n t h e s t o r a g e t r a p makes i t p o s s i b l e t o observe t r a n s i t i o n s f r e e from f i r s t o r d e r Doppler e f f e c t i f t h e i r wavelength i s l o n g e r t h a n t h e amplitude of the p e r i o d i c motion. T h i s i s t h e c a s e f o r h y p e r f i n e t r a n s i t i o n s .

On the o p p o s i t e , t h e second o r d e r Doppler e f f e c t cannot b e c a n c e l l e d , b u t one can reduce t h e s h i f t and broadening of t h e l i n e s due t o t h i s e f f e c t by lowering i t s mean k i n e t i c energy ( I ) ( 2 ) o r by choosing. f o r a given value of t h e temperature, an i o n whose atomic mass i s a s high a s p o s s i b l e . I n t h i s r e s p e c t , t h e mercury i o n i s a good choice. Furthermore, the h y p e r f i n e frequency of t h e mass 199 i s o t o p e of mercury i s h i g h (40.5 GHz), which a l l o w s one t o observe e a s i l y a very high Q h y p e r f i n e l i n e . For t h e s e r e a s o n s , lg9Hg+ i o n has been choosen by some a b o r a t o r i e s i n o r d e r t o rea- l i r e t h e f i r s t frequency s t a n d a r d based on i o n s t o r a g e 1)) (4) (5) ( 6 ) . I t s main fea- t u r e s a r e t h e following

:

- t h e h y p e r f i n e frequency of 4.05 10" H z a l l o w s Q f a c t o r s of 1 0 ' ~ o r b e t t e r

- t h e second o r d e r Doppler e f f e c t frequency s h i f t e q u a l s - 5 10-l2 f o r a mean k i - n e t i c energy of 1 eV

- a f o r t u i t o u s matching of lg9Hg+ and 2 0 2 ~ g + energy l e v e l s makes i t p o s s i b l e t o s e l e c t i v e l y pump 199Hg+ from t h e F

=

I t o t h e F = 0 h y p e r f i n e ground s t a t e ( f i g u r e l ) . I t i s t h e n p o s s i b l e to probe t h e h y p e r f i n e resonance by an o p t i c a l and microwave double resonance experiment and t o c o n t r o l t h e frequency of a q u a r t z c r y s t a l o s c i l l a t o r by t h e h y p e r f i n e frequencies.

Some promising r e s u l t s have been obtained s i n c e t h e p r e l i m i n a r y frequency s t a b i l i t y of t h e c o n t r o l l e d o s c i l l a t o r i n t h e range 10 s <

-c

< 3 500 s i o f the same o r d e r of magnitude a s t h a t of commercially a v a i l a b l e cesium c l o c k s a 7 ) . We d e s c r i b e t h i s experiment and d i s c u s s t h e l i m i t a t i o n s and f u t u r e developments of t h i s i o n frequency s t a n d a r d .

1.

HE+

frequency s t a n d a r d a t L a b o r a t o i r e de 1'Horloge Atomique.- The experimental set-up has been d e s c r i b e elsewhere

( 3 )

Mercury i o n s a r e confined i n a c y l i n d r i c a l RF t r a p , whose diameter and h e i g h t a r e r e s p e c t i v e l y 38 m and 27 mm ( F i g u r e 2 ) . They a r e produced i n s i d e t h e t r a p by an e l e c t r o n beam which i o n i z e s a vapor of n e u t r a l

* ~ ~ u i ~ e de Recherche du C.N.R.S., a s s o c i 6 e B l ' U n i v e r s i t 6 Paris-Sud

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981840

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JOURNAL DE PHYSIQUE

Fig.

1

- S i m p l i f i e d energy diagram of t h e 1 9 9 ~ g + and 202Hg+ showing the l e v e l s involved i n t h e o p t i c a l pumping scheme.

lpMl

C o u n t ~ n g Photon S y s t e m

- -

F i l t e r

Fig. 2 - Scheme of t h e

experimental set-up.

mercur The pumping l i g h t , produced b y an e l e c t r o d e l e s s d i s c h a r g e lamp, f i l l e d with i s focused a t t h e c e n t e r of t h e t r a p through 2 h o l e s d r i l l e d i n the cy- l i n d e r w a l l . The f l u o r e s c e n c e l i g h t , e m i t t e d by t h e e x c i t e d i o n s , i s d e t e c t e d i n t h e d i r e c t i o n of the t r a p a x i s , p e r p e n d i c u l a r t o t h e d i r e c t i o n o f t h e pumping l i g h t . I t p a s s e s through t h e upper p l a t e of t h e trap,made of amesh. I t i s focused on t h e photo- cathode of a s o l a r b l i n d p h o t o m u l t i p l i e r . The microwave e x c i t a t i o n probing the hyper- f i n e resonance i s p r e s e n t l y introduced i n t h e t r a p by a horn l o c a t e d o u t s i d e of t h e q u a r t z vacuum envelope. The h y p e r f i n e resonance i s d e t e c t e d by t h e i n c r e a s e of t h e f l u o r e s c e n c e l i g h t which occurs when t h e frequency of the microwave e x c i t a t i o n e q u a l s the resonance frequency. Figure 3 shows a t y p i c a l resonance curve o b t a i n e d by coun- t i n g t h e p h o t o e l e c t r o n f l u x d e l i v e r e d by t h e p h o t o m u l t i p l i e r when t h e microwave f r e - quency i s swept through t h e resonance. The r e l a t i v e width of t h e l i n e i s about 2

10-Io

;

i t s h e i g h t i s about 10 % of t h e background l i g h t l e v e l .

The microwave e x c i t a t i o n i s produced b y frequency s y n t h e s i s from a q u a r t z

c r y s t a l o s c i l l a t o r whose frequency i s t o be c o n t r o l l e d by t h e hyperfine t r a n s i t i o n .

The c o n t r o l loop shown i n f i g u r e 4, i n c l u d e s a d i g i t a l i n t e g r a t o r and an analogue

f i l t e r i n o r d e r t o c o r r e c t any c o n s t a n t e r r o r o r l i n e a r d r i f t of the o s c i l l a t o r f r e -

quency ($1. D i g i t a l i n t e g r a t i o n of t h e frequency e r r o r r e s u l t s from up then down

counting t h e p h o t o e l e c t r o n s d e l i v e r e d by t h e p h o t o m u l t i p l i e r w h i l e t h e frequency

o f t h e microwave e x c i t a t i o n i s square wave modulated between two v a l u e s f l and f 2

on b o t h s i d e s of t h e resonance curve.

(4)

Counting Rate(s-') 57000

t

Fig. 3 - A t y p i c a l experimental resonance curve.

F i g . 4 - Scheme of t h e con- t r o l loop of t h e 5 MHz o s c i l - l a t o r .

The r e s u l t of an up then down counting i s zero when t h e mean frequency ( f ,

+ f 2)

1 2

of t h e e x c i t a t i o n e q u a l s t h e resonance frequency f . It i s added t o t h e p r e c e d i n g counting r e s u l t s , d i g i t a l t o analogue converted an2 f i l t e r e d b e f o r e b e i n g a p p l i e d t o t h e v a r a c t o r which c o n t r o l s t h e frequency of t h e q u a r t z c r y s t a l o s c i l l a t o r . As shown i n f i g u r e 5 , t h e measured frequency s t a b i l i t y of t h e c o n t r o l l e d o s c i l l a t o r i s comparable t o t h a t of commercially a v a i l a b l e cesium atomic c l o c k s . The square r o o t u (T) of t h e two sample v a r i a n c e of i t s f r a c t i o n a l frequency f l u c t u a t i o n s i s given gy ( 7 )

:

U ~ ( T )

= 3.6

( 1 )

i n t h e range 10 s < r < 3 500 s. These preliminary r e s u l t s can be used t o d i s c u s s

t h e f u t u r e improvements of t h e mercury i o n frequency s t a n d a r d .

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JOURNAL DE PHYSIQUE

Fig.

5 .

- Measured f r a c t i o n a l frequency stability of t h e c o n t r o l l e d o s c i l l a t o r i n t h e range 10 s < r < 3 500 s.

2. Improvements t h a t can be e a s i l y o b t a i n e d f o r the s h o r t and medium term s t a b i l i t y . The s h o r t and medium term s t a b i l i t y of the c o n t r o l l e d o s c i l l a t o r can be r e l a t e d t o some p h y s i c a l parameters of t h e experiment, assuming t h a t i t i s l i m i t e d by t h e pho- t o n s h o t n o i s e which p e r t u r b s t h e measurement of t h e f l u o r e s c e n c e l i g h t and t h a t t h e c o n t r o l loop i n c l u d e s a d i g i t a l i n t e g r a t o r of t h e k i n d d e s c r i b e d above. I t can be shown t h a t , f o r F o u r i e r f r e q u e n c i e s s m a l l e r than the cut-off frequency of t h e c l o s e d loop t h e one-sided power s p e c t r a l d e n s i t y S of the f r a c t i o n a l frequency f l u c t u a t i o n s o f t h e o n t r o l l e d o s c i l l a t o r i s r e l a t g d t o t h e mean v a l u e of the photo- e l e c t r o n f l u x by t 5 )

:

I

a 1

where

(=)

i s the slope of t h e resonance curve, g i v i n g t h e p h o t o e l e c t r o n f l u x I

c I1

,2

a s a f u n c t i o n of t h e microwave frequency f , f o r the v a l u e s f and f 2 of t h e micro- wave frequency. The square r o o t of t h e two-sample v a r i a n c e o i t h e r e l a t i v e frequsn- cy f l u c t u a t i o n s i s t h e n

:

e

f

182

The f o l l o w i n g d i s c u s s i o n r e s u l t s from t h i s e x p r e s s i o n

:

a ) E_ffect-of-~kodu1at_io-f!~~idf.h-o-n-the_~st~b_i1if:~

a 1

The s l o p e

(-)

i s maximal f o r t h e f o l l o w i n g v a l u e s of f and

f

(5) .

af

I

2

f 1 , 2

f

= f o +_ 0.29 6 f

192 ( 4 )

where 6 f i s t h e f u l l width h a l f maximum of t h e l i n e . I n t h e f o l l o w i n g , we assume t h a t t h i s c o n d i t i o n i s s a t i s f i e d .

b) E f f est-of -th_e_-microw_a!!eeex_cit_atio_nn~owe-r

For the v a l u e s o f f and f given above, we have (9) .

2

(6)

where I R i s the amplitude of t h e resonance l i n e , given by

(') :

"

Y r l

+

3Y

I , = ; K n ' r l r l + 2 y P -

I + S

s

I n t h i s r e l a t i o n , K i s t h e t o t a l e f f i c i e n c y o f t h e d e t e c t i o n of t h e f l u o r e s c e n c e l i g h t ,

y

i s t h e pumping r a t e of the i o n s , d e f i n e d a s t h e r a t e a t which t h e F

=

0,

% = 0 & a t e p o p u l a t i o n i n c r e a s e s , by o p t i c a l pumping d i v i d e d by t h e t o t a l popula- t l o n of the o t h e r l e v e l s o f t h e ground s t a t e , r l i s t h e t o t a l l o n g i t u d i n a l relaxa- t i o n r a t e , n the number of i o n s and s t h e s a t u r a t i o n f a c t o r g i v e n by

:

b2 T 1 + 2 Y s = -

r1r2 r 1

+

3Yp

( 7 )

I n t h i s e x p r e s s i o n , r 2 i s t h e t o t a l t r a n s v e r s e r e l a x a t i o n r a t e and b the Rabi f r e - quency r e l a t e d t o t h e microwave magnetic f i e l d i n t h e t r a The l o n g i t u d i n a l r 1

and t r a n s v e r s e T2 t o t a l r e l a x a t i o n r a t e s a r e given by (101':

where y l and y2 a r e the l o n g i t u d i n a l and t r a r s v e r s e h y p e r f i n e r e l a x a t i o n r a t e s o f t h e i o n s and y i s t h e i n v e r s e o f t h e i r s t o r a g e time.

The mean p h o t o s l e c t r o n f l u x Im i s t h e sum of t h r e e terms

:

I m = I B + I + - I 3

0

4 R (10)

where I and I. a r e t h e f l u x e s due r e s p e c t i v e l y t o t h e background l i g h t and t o t h e par! of t h e f l u o r e s c e n c e l i g h t which does n o t depend on t h e microwave e x c i t a - t i o n . FosI_the v a l u e s f l and f 2 of t h e e x c i t a t i o n frequency, t h e resonance s i g n a l e q u a l s (--i--).

K

Assuming t h a t t h e main s o u r c e of s t r a y l i g h t i s t h e d i f f u s i o n of pho- tons from t h e pumping beam we w r i t e

:

IB = K1y

P (11)

where K1 i s a dimensionless c o e f f i c i e n t which c h a r a c t e r i z e s t h e experimental s e t - u p . Using t h e e x p r e s s i o n s f o r I. and IR given i n r e f e r e n c e ( 9 ) , we o b t a i n

:

When t h e s a t u r a t i o n f a c t o r s v a r i e s , a

(PC)

i s minimal f o r a v a l u e s which depends on t h e r e l a x a t i o n r a t e s and on t h e r a t l o K1/Kn. We have

: 0

where a i s given by :

It i s easy t o v e r i f y t h a t s v a r i e s between 1 and 2 when a v a r i e s between

0

and i n f i n i t y . The f i g u r e 6 shows t h e v a r i a t i o n s of

0

80

( T )

a s a f u n c t i o n of s , assu- ming y,,

=

y , f o r the t y p i c a l v a l u e s o f o u r e x p e r i m z n t a l set-up

:

n

=

4 105, K

=

8

yp : 2.6 s - ' , K 1 = I I Kn and f o r v a r i o u s values of t h e r a t i o (y1+ys)/y .

We s e e t h a t u

( T )

v a r i e s slowly when s i s i n the range

1 4

s

4

2. P

We w i l l choose i n Y t h e f o l l o w i n g s

=

2.

(7)

C8-332

JOURNAL DE

PHYSIQUE

Fig. 6 - V a r i a t i o n s of fi u

( T )

a s a func- t i o n of t h e s a t u r a t i o n f a c t o r Y s f o r va- r i o u s v a l u e s of t h e r e l a x a t i o n r a t e s and f o r t y p i c a l v a l u e s of t h e o t h e r parame- t e r s

:

n

=

4 l o 6 , K

=

8 loq4, y

=

2.6s - 1

P

F i g . 7 - V a r i a t i o n s of fi a

(T)

a s a Y

f u n c t i o n of t h e pumping r a t e y , assuming

P

y , = y2 ; s

=

2, K

= 8

; n

=

4 lo6 ; K'

=

1 1 Kn and f o r v a r i o u s v a l u e s of Y I

+

Y,.

F i g u r e 7 shows t h e v a r i a t i o n s of fi a

( T )

a s a f u n c t i o n of t h e pumping r a t e y , as-

suming

y, =

y2, s = 2 and our typicalYvalues K

=

8 10-4, n

=

4 106, K ' - 1 1 &.

(8)

The c u r v e s a r e drawn f o r v a r i o u s v a l u e s of t h e sum y + y which c h a r a c t e - r i z e s t h e r e l a x a t i o n p r o c e s s e s a p a r t from t h e o p t i c a l pumping. fie t y p i c a l v a l u e s of t h e r e l a x a t i o n r a t e s i n our experiment a r e t h e following

:

y = 2.6 s - l , y 1

=

1

s - l , y2

=

3.9 s-1, y = 2.3 s-1. These curves show t h a t t h e r e P i s an optimal v a l u e of t h e pumping r a t e , w k c h d e c r e a s e s when the sum y + ys decreases. The b e s t s t a - b i l i t y , achieved f o r t h e lowest value of t h e r e l a x a h o n r a t e s needs only low pum- ping r a t e s . For i n s t a n c e , assuming t h a t y + ys =

'1,

we should have :

J

which i s 30 times b e t t e r than our experimental r e s u l t . This would be obtained f o r Y = 0.25 s-l only, which corresponds t o a pumping l i g h t i n t e n s i t y t e n times s m a l l e r tRan our t y p i c a l v a l u e g i v e n p r e v i o u s l y . Consequently, i n o r d e r t o improve t h e s t a b i l i t y , we must improve t h e s t o r a g e time and t h e h y p e r f i n e r e l a x a t i o n r a t e s r a t h e r than produce a more e f f i c i e n t pumping l i g h t . This c o n c l u s i o n i s s t i l l v a l i d when no s t r a y l i g h t i s d i f f u s e d from t h e pumping beam, a s i t is shown i n f i g u r e 8. I n t h i s c a s e , assuming a g a i n y , + ys . 1, we should have

:

-1 Y

w i t h y = .32 s . This shows t h a t reducing d r a s t i c a l l y t h e background l i g h t l e v e l i s mucR l e s s e f f i c i e n t , r e g a r d i n g t o frequency s t a b i l i t y , t h a n improving t h e r e l a - x a t i o n r a t e s .

Fig. 8 - V a r i a t i o n s o f d? u

(T)

a s a f u n c t i o n of t h e pumping r a t e Y y , assuming

-4 6

y 1 = y 2 , s = 2 , K = 8 1 0 , n = 4 1 0 and K '

=

0 (no s t r a y l i g h t ) f o r v a r i o u s v a l u e s of y 1

+

ye..

One must keep i n mind t h a t the r e s u l t s given h e r e a r e v a l i d o n l y i n t h e l i m i t where t h e background l i g h t i s p r o p o r t i o n a l t o t h e pumping r a t e . I n our experiment, t h e v a r i o u s c o n t r i b u t i o n s t o t h e background p h o t o e l e c t r o n s f l u x a r e t y p i c a l l y t h e f o l - lowing

:

n o i s e of t h e p h o t o m u l t i p l i e r 40 s-I l i g h t e m i t t e d by t h e e l e c t r o n gun 150 s-1 l i g h t due t o t h e e x c i t a t i o n of atoms by

t h e e l e c t r o n beam 7 000 s-I

d i f f u s e d l i g h t from t h e pumping

beam 50 000 s-I

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C8-334 JOURNAL DE PHYSIQUE

3. Accuracy and l o n g term s t a b i l i t y of t h e mercury i o n frequency s t a n d a r d . - The long term s t a b i l i t y and accuracy a r e l i m i t e d by s y s t e m a t i c e f f e c t s which per- t u r b t h e confined i o n and a r e n o t c o n t r o l l e d s u f f i c i e n t l y . The magnetic s h i f t , f i r s t and second o r d e r Doppler e f f e c t , l i g h t s h i f t and S t a r k e f f e c t must be e v a l u a t e d i n o r d e r t o d i s c u s s the long term p r o p e r t i e s of t h e mercury s t a n d a r d .

a) Magnetic s h i f t

I n the low f i e l d l i m i t , t h e frequency of t h e AmF = 0 t r a n s i t i o n used i n the mercury i o n frequency s t a n d a r d depends on t h e a x i a l magnetic f i e l d B according t o the second o r d e r law

where B i s given i n T e s l a and Af i n Hz. The magnetic f i e l d i n s i d e t h e t r a p can be c o n t r o l l e d w i t h h i g h accuracy. T E ~ r e l a t i v e frequency f l u c t u a t i o n s r e l a t e d t o magnetic f l u c t u a t i o n s 6B a r e g i v e n by

:

i f B = T. FluctuationLso6B s m a l l e r than T a r e e a s i l y a c h i e v a b l e and t h e magnetic s h i f t would consequently not l i m i t t h e s t a b i l i t y and accuracy u n t i l the

range.

b) Eo_~~ler-ef f e _ c ~

I f t h e h y p e r f i n e resonance i s probed by a microwave with a t r a v e l l i n g component, t h e f i r s t o r d e r Doppler e f f e c t e x a c t l y c a n c e l s o n l y when t h e mean displacement of t h e i o n s d u r i n g t h e time of t h e i r i n t e r a c t i o n w i t h the e l e c t r o m a g n e t i c f i e l d i s zero. I n t h e worst c a s e , t h i s displacement e q u a l s the t r a p r a d i u s r . T h i s w o r s t case o c c u r s when t h e i o n s , c r e a t e d on t h e a x i s of t h e t r a p a r e l o c a t e d a t t h e d i s - tance r of t h i s a x i s a t t h e end of t h e r e i n t e r a c t i o n w i t h the e l e c t r o m a g n e t i c f i e l d . Bue t o t h e symmetry of the t r a p , t h i s e f f e c t produces a broadening of t h e l i n e r a t h e r than a n e t s h i f t . I t s o r d e r of magnitude i n o u r experiment can be a s h i g h a s 2 10-lO (8 Hz). This e f f e c t can b e reduced by using a s t a n d i n g wave ( t h e t r a p would then b e a microwave c a v i t y ) o r by i n c r e a s i n g s t r o n g l y t h e r e l a x a t i o n time and d e c r e a s i n g t h e t r a p volume.

On the o t h e r hand, t h e second o r d e r Doppler s h i f t Af can b e reduced only by lowering the mean k i n e t i c energy of t h e i o n s . For a meanDZinetic energy

Q C >

we have

:

A c

It h a s been shown ( 1 1 ) ( 1 2 ) ( 1 3 ) th a t t h e mean k i n e t i c energy of i o n s confined i n an RF t r a p i s a c o n s t a n t f r a c t i o n of the pseudopotential w e l l depth f o r given experi- mental c o n d i t i o n s . I t i s consequently p o s s i b l e t o measure the second o r d e r Doppler s h i f t by v a r y i n g t h i s w e l l d e p t h , p r o p o r t i o n a l t o t h e r a t i o v2/$ ( V i s t h e ampli- tude of the v o l t a g e a p p l i e d t o t h e t r a p e l e c t r o d e s and R / W t B f r e q 8 e n c y ) , and e x t r a p o l a t i n g t o zero. The e f f e c t of t h e l i g h t s h i f t must be taken i n t o account i n

t h i s measurement a s shown i n t h e n e x t c h a p t e r .

Therefore, f l u c t u a t i o n s of t h e background p r e s s u r e and of t h e pseudo poten- t i a l w e l l depth would limit t h e frequency s t a b i l i t y . For a t y p i c a l v a l u e <E >

=

2 eV, t h e r e l a t i v e f l u c t u a t i o n s of t h i s energy must b e l e s s than 10-3 i n o r s e r t o achieve a s t a b i l i t y b e t t e r t h a n 10-l4. This means t h a t t h e amplitude and frequency of t h e c o n f i n i n g v o l t a g e must be s t a b l e t o b e t t e r t h a n 5 10-4.

I t must be n o t i c e d t h a t t h e second o r d e r Doppler s h i f t can b e s t r o n g l y re- duced by c o o l i n g t h e i o n s . Laser c o o l i n g , whi h allows k i n e t i c temperature s m a l l e s t than 1 P b r i n g s t h i s e f f e c t i n the range. B u f f e r gas c o o l i n g i s much l e s s e f f i c i e n t b u t c a n a l s o b e used.

c ) L C g k t - s h i f ~

W e assume i n t h i s d i s c u s s i o n t h a t t h e o p t i c a l pumping of t h e i o n s i s performed by

the l i g h t e m i t t e d by a lamp f i l l e d w i t h 2 0 2 ~ g . T e nergy diagram of mass 199 and

mass 202 i s o t o p e s of i o n i z e d mercury ( f i g u r e 1) 7147 shows t h a t t h e wavelength of

(10)

t h e pumping l i g h t i s a p p r o x i m a t e l y t h e mean v a l u e o f t h a t of t h e l i n e s 2 S I I 2 F

=

I++

1/2F= 1-2P

.

-

2 P l I 2 F'= 0 and 2 s

112 F'= 1 o f t h e s t o r e d i o n s . Consequently, b o t h t r a n - s i t i o n s c o n t r i b u t e t o t h e pumping scheme s i n c e t h e Doppler w i d t h o f t h e a b s o r p t i o n l i n e s a r e much l a r g e r t h a n t h e m i s t u n i n g .

Three t r a n s i t i o n s c o n n e c t t h e F

=

1 ,

2

=

0 l e v e l of t h e ground s t a t e t o t h e l e v e l s of t h e 2P l 2 s t a t e , w i t h e q u a l p r o b a i l i t i e s ( f i g u r e 9 ) Two o f them a r e mistuned by abou! - 3.5 GHz and t h e t h i r d by a b o u t

+

3.5 GHz ( i 4 ) . Each o f t h e s e t r a n s i t i o n s produces a l i g h t s h i f t w i t h e q u a l amplitude b u t w i t h o p p o s i t e s i g n .

2k-77 /\ I and LS,,,, s t a t e s o f " ' H ~ ' .

L e t us c a l l 6 t h e a m p l i t u d e of t h i s a n g u l a r f r e q u e n c y s h i f t . We have (15)

*

R

where vL i s t h e f r e q u e n c y o f t h e pumping r a d i a t i o n , v t h e resonance f r e q u e n c y and AV t h e Doppler w i d t h o f t h e a b s o r p t i o n l i n e . Av i s o r e l a t e d t o t h e mean k i n e t i c

D D

e n e r g y o f t h e i o n s <E >. With o u r t y p i c a l v a l u e s we have

where <Ec> i s g i v e n i n eV. T h i s r e s u l t , v a l i d i n t h e l i m i t where

AVD

i s l a r g e r t h a n t h e m i s t u n i n g , shows t h a t , f o r our t y p i c a l v a l u e of

y

, t h e pumping l i g h t i n t e n s i t y must be s t a b l e t o b e t t e r t h a n 1

%

i n o r d e r t o r e a c h tRe 10-14 s t a b i l i t y r a n g e and t h a t i t would be o f i n t e r e s t t o o p e r a t e t h e t r a p i n c o n d i t i o n s s u c h t h a t

y

c o u l d be reduced, a s d e s c r i b e d p r e v i o u s l y . The l i g h t s h i f t , which depends on thePmean e n e r g y of t h e i o n s , must be t a k e n i n t o account when measuring t h e second o r d e r Doppler s h i f t and must b e measured i n d e p e n d e n t l y . The l i g h t s h i f t c a n be measured and c a n c e l l e d , u s i n g a p o l a r i z e d pumping l i g h t ( I 5 ) . AS a conse uence of l i n e a r p o l a r i z a t i o n of t h e b e m , t h e t h r e e t r a n s i t i o n s c o n n e c t i n g t h e j S F

=

1 ,

1

/ 2 m~ = O

l e v e l t o t h e e x c i t e d s t a t e have no l o n g e r e q u a l p r o b a b i l i t i e s

: t h e p r o b a b i l i t y o f

t h e 2 S l I 2 F

=

1 , % = O*->2P F '

=

0 , m' = 0 a t r a n s i t i o n i s d i f f e r e n t from t h a t of t h e

U+

and u t r a n s i t i b b l 'S F

=

T, mF

=

& p I I 2 F'

=

I ;

1 . These

112 2

p r o b a b i l i t i e s , an and a a r e p r o p o r t i o n a l r e s p e c t i v e l y t o c o s 0 and s l n 8, 0 b e i n g t h e a n g l e between t h e d i a l magnetic f i e l d and t h e p o l a r i z a t i o n v e c t o r . The n e t l i g h t s h i f t i s t h e n p r o p o r t i o n a l t o

:

2 2 2

GR(cos 0 - 2 s i n 0)

=

( 3 c o s 0 - 2) 6 R (22)

(11)

JOURNAL DE PHYSIQUE

F i g u r e 10 shows t h e v a r i a t i o n s of t h i s e f f e c t when

8

v a r i e s between 0 and 90". The t o t a l v a r i a t i o n of t h e s h i f t e q u a l s t h r e e times i t s v a l u e f o r

9 =

0 , which can consequently be measured. Furthermore, t h e l i g h t s h i f t e q u a l s zero f o r a v a l u e of 8 c l o s e t o 35'. It can consequently be s t r o n g l y reduced.

,

LIGHT SHIFT ILRBITRRRI UNIT,

F i g . 10 - V a r i a t i o n of t h e l i g h t s h i f t when t h e i o n s a r e o p t i c a l l y pumped by a p o l a r i z e d beam, a s a f u n c t i o n of t h e a n g l e between t h e

TNLTR

p o l a r i z a t i o n v e c t o r and t h e magnetic f i e l d .

d) Stark-e_f f ect

I n an RF t r a p , t h e c o n f i n i n g e l e c t r i c f i e l d

E

p e r t u r b s the i o n s , which modifies t h e h y p e r f i n e frequency ( S t a r k e f f e c t ) . The r e l a t i v e frequency s h i f t i s approximately given by

(17) :

where W6 0, W6, I , W7,0 a r e r e s p e c t i v e l y t h e e n e r g i e s of t h e l e v e l s 6 s , 6p and 7s of t h e l b n .

With t h e same approximation, t h e p o l a r i z a b i l i t y a is defined by

:

2 <6,11x16,0> 2

a = 2q (24)

W6,1-W6,0

where q i s the e l e c t r o n charge and x the p o s i t i o n o p e r a t o r . a can be deduced from the experimental value of t h e o s c i l l a t o r s t r e n g t h of t h e 6p<->6s t r a n s i t i o n f

s i n c e we have

:

SP

where m i s t h e e l e c t r o n mass, h t h e wavelength of t h e t r a n s i t i o n and h t h e Planck c o n s t a n t .

The experimental value of f i s

(18)

SP

SP = 0.15 which g i v e s a

=

2.9 S . I . and

The e l e c t r i c f i e l d i n t h e t r a p v a r i e s w i t h t h e c o o r d i n a t e s r and z according t o 2

l ~ ( r , z , t ) l 2

=

9 ( r 2 + 4z2) (27)

r

V(t) i s t h e v o l t a g e a p p l i e d t o t h e t r a p g l e c t r o d e s . I n t h e following, we s h a l l ne-

g l e c t t h e DC component of t h i s v o l t a g e . On t h e o t h e r hand, t h e p o t e n t i a l energy of

t h e i o n s i n t h e p s e u d o p o t e n t i a l w e l l i s

:

(12)

Consequently, t h e mean v a l u e of the e l e c t r i c f i e l d a p p l i e d t o t h e i o n s can be r e l a - t e d t o t h e i r mean p o t e n t i a l and k i n e t i c energy. We f i n d

:

Since <E > i s p r o p o r t i o n a l t o t h e p s e u d o p o t e n t i a l w e l l depth ( 1 1 ) ( 1 2 ) ( 1 3 ) , we f i n d k being €he p r o p o r t i o n a l i t y c o e f f i c i e n t

:

2 V 2

< [ E [ >

=

k (9) (30)

0

I n s t a n d a r d c o n d i t i o n s , we have

:

and

Af s e

- = -

3 10-l4 (32)

0

The measurement of t h i s e f f e c t can b e performed by v a r y i n g t h e v o l t a g e V , t a k i n g c a r e t o m a i n t a i n the mean k i n e t i c energy of t h e i o n s by a d j u s t i n g simult8neously t h e frequency n/2n i n o r d e r t o keep t h e second o r d e r s h i f t c o n s t a n t .

conclusion

: c o o l i n g t h e mercury ions.- I t h a s been shown t h a t i m p o r t a n t improvement

of t h e s h o r t and medium term s t a b i l i t y of t h e mercury i o n frequency s t a n d a r d c a n b e achieve by lowering t h e r e l a x a t i o n r a t e s of the i o n s i n t h e t r a p . As a consequence, lower pumping r a t e s a r e n e c e s s a r y . The r e l a x a t i o n r a t e s a r e s t r o n g l y dependent upon the n e u t r a l mercury background p r e s s u r e which must be c o n t r o l l e d more p r e c i s e l y t h a n i t i s p r e s e n t l y .

The r e s i d u a l f i r s t o r d e r Doppler e f f e c t can produce l a r g e broadening of t h e l i n e which can be c a n c e l l e d by using a s t a n d i n g wave t o probe t h e h y p e r f i n e reso- nance. For t h a t purpose, the t r a p must be a microwave c a v i t y , tuned t o t h e resonance

frequency. The l i g h t s h i f t and i t s f l u c t u a t i o n s w i l l b e reduced by lowering t h e pumping r a t e , furthermore t h e use o f a p o l a r i z e d beam can n e a r l y cancel t h i s e f f e c t .

The second o r d e r Doppler s h i f t remains t h e main l i m i t a t i o n t o t h e accuracy and long term s t a b i l i t y s i n c e i n our t y p i c a l c o n d i t i o n s , where t h e i o n s a r e n o t cooled, i t s n e t e f f e c t on t h e frequency s h i f t i s a s l a r g e a s 10-11 and v a r i e s w i t h the n e u t r a l mercury background p r e s s u r e and t h e v o l t a g e a p p l i e d t o t h e t r a p , which b o t h modify the mean k i n e t i c energy of t h e i o n s . The only way t o overcome t h i s d i f f i c u l t y i s t o cool t h e i o n s . A p o s s i b l e scheme f o r c o o l i n g lg9Hg+ i o n s and pro- bing t h e i r hyperfine resonance i n an RF t r a p i s t h e following. The mercury i o n s can be cooled by Coulomb i n t e r a c t i o n (19) w i t h o t h e r i o n s such a s M ~ + o r ~ a + f o r example which w i l l be cooled. The hyp5rfine t r a n s i t i o n w i l l t h e n be probed by a l a s e r beam

tuned to the 2 ~ 1 1 2 ,

F =

I

f3

P F

2

0 t r a n s i t i o n whose upper s t a t e can b e con- 112'

nected t o t h e 2

2 ~ 1 1 2 , F

=

1 s t a t e only. The wing of t h e 2 ~ 1 1 2 , F

=

I

W

PI12, F

=

I t r a n s i t i o n w i l l be e x c i t e d by the l a s e r beam ( 2 0 ) , which consequently w i l l pump t h e i o n s i n t h e 2~ F

=

0 state, a s w e l l as any r e l a x a t i o n process ( t h e energy

112'

d i f f e r e n c e between t h e F

=

1 and F

=

0 l e v e l s of t h e i o n s e q u a l s 1.9 K

: a t thermal

e q u i l i b r i u m , t h e p o p u l a t i o n of t h e F

=

0 s t a t e i s consequently l a r g e r than t h a t of t h e F

=

1 s t a t e , i f t h e temperature i s of t h e o r d e r o r lower t h a n 1 K ) .

I f a microwave e x c i t a t i o n a t t h e frequency of t h e h y p e r f i n e t r a n s i t i o n i s a p p l i e d ,

every i o n e x c i t e d i n t h e F

= 1

ground s t a t e w i l l emit a l a r g e f l u o r e s c e n c e photon

f l u x due t o t h e s h o r t l i f e t i m e of t h e 2 ~ 1 / 2 s t a t e . The microwave t r a n s i t i o n

(13)

C8-338 JOURNAL qE PHYSIQUE

w i l l c o n s e q u e n t l y be d e t e c t e d w i t h a good e f f i c i e n c y . The 2 ~ i , 2 F . ltsZP F 0 t r a n s i t i o n c a n a l s o b e e x c i t e d b y a l i g h t beam e m i t t e d b y a amp a s i n o u r 'k2periment.

Reducing t h e k i n e t i c t e m p e r a t u r e o f t h e i o n s t o 1 K w i l l r e d u c e t h e s e c o n d o r - d e r Doppler s h i f t by more t h a n f o u r o r d e r s o f magnitude. By t h e same way, t h e s t o -

r a g e time o f t h e i o n s i n t h e t r a p w i l l be s t r o n g l y enhanced, a l l o w i n g t o o p e r a t e i t w i t h t h e e l e c t r o n beam o f f and w i t h a v e r y low mercury vapor p r e s s u r e , r e d u c i n g c o n s e q u e n t l y t h e r e l a x a t i o n r a t e s . Any r e s i d u a l f i r s t o r d e r Doppler e f f e c t and t h e S t a r k e f f e c t w i l l a l s o b e reduced.

T h i s work was s p o n s o r e d by D.R.E.T.

R e f e r e n c e s

(

1 ) NEWAUSER W., HOHENSTATT M. and TOSCHEK P.E., Phys. Rev. A 2

(

1980) 1 137

(

2 ) WINELAND D.J. and ITANO W.M., Phys. L e t t . A 8 2 (1981) 75

(

3) MAJOR F.G. and WERTH G., Phys. Rev. L e t t . 2 (1973) 1155

(

4 ) Mc GUIRE M.D., PETSCH R. and WERTH G., Phys. Rev. A 17 (1978) 1999

(

5 ) JARDINO M., DESAINTFUSCIEN M., BARILLET R., VIENNET J., PETIT P. and AUDOIN C., Proc. of t h e 34th Annual Frequency C o n t r o l Symposium ( P h i l a d e l p h i a , Pennsyl- v a n i a ) (1980) 353

(

6) M c GUIRE M.D., B u l l .

Am.

Phys. Soc. (1981) 615

(

7) JARDINO M., DESAINTFUSCIEN M., BARILLET R., VIENNET J., PETIT P. and AUDOIN C., Appl. Phys. 2 (1981) 107

(

8) VIENNET J . , JARDIN0 M., BARILLET R. and DESAINTFUSCIEN M., s u b m i t t e d t o IEEE Trans. on I n s t r . and Meas.

(

9) JARDINO M. and DESAINTFUSCIEN M., IEEE T r a n s . on I n s t r . and Meas. IM-29 (1980) 163

(10) J A R D I N O M., Ph. D. D i s s e r t a t i o n , U n i v e r s i t y Paris-Sud (1981) (CNRS r e g i s t r a - t i o n number 2403)

(1 1) IF'FL~DER R. and WERTH G., M e t r o l o g i a 2

(

1977) 167 (12) KNIGHT R.D. and PRIOR M.H., J. Appl. Phys. 50 (1979) 3044 (13) SMAF

H . ,

SCHMELING U. and WERTH G., Appl. Phys. 2 (1981) 249

(

14) GUERN

Y . ,

BIDEAU-MEHU A., ABJEAN R. and JOHANNIN-GILLES A., Phys . S c r i p t a 2

(1977) 273

(15) COHEN-TANNOUDJI C., Metrologia 2

(

1977) 16

1

(16) CAGNAC B., P r i v a t e communication

(17) ANDERSON L.W., Nuovo Cirnento 2 (196 1) 936

(

18) BRENETEAU A.M. , ICOLE A.M., ROUILLE C., POQUERUSSE A. and DOUCET H. J . , Phys.

L e t t . 46 A (1974) 309

(19) D R U U I ~ R R.E., WINELAND D . J . and BERGQUIST J.C., Appl. Phys. 2 (1980) 365 (20) WINELAND D. J. , BERGQUIST J. C., ITANO W .M. and DRULLINGER R. E . , Opt. L e t t . 5

(1980) 245

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