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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�
JOURNAL DE PHYSIQUE
CoZZoque C8, s u p p l h e n t au n012, Tome 42, de'cembre 1981 page
(28-327P 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
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 rFig. 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.
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 .
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
I2
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
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 + Ss
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 ,
yi 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--).
KAssuming 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
: 0where 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
0and 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
080
( 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 4s
42. P
We w i l l choose i n Y t h e f o l l o w i n g s
=2.
C8-332
JOURNAL DE
PHYSIQUEFig. 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 &.
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
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
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
AVDi 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
yc 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 ft 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 . These112 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)
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
8v 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
Ep 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
: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 improvementof 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
f3P F
20 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
WPI12, 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 thermale 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
= 1ground 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
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
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