MODULATION EFFECTS IN PASSIVE Rb FREQUENCY STANDARDS

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MODULATION EFFECTS IN PASSIVE Rb FREQUENCY STANDARDS

P. Thomann, G. Busca

To cite this version:

P. Thomann, G. Busca. MODULATION EFFECTS IN PASSIVE Rb FREQUENCY STANDARDS.

Journal de Physique Colloques, 1981, 42 (C8), pp.C8-189-C8-197. �10.1051/jphyscol:1981822�. �jpa-

00221717�

JOURNAL DE P H Y S I Q U E

CoZZoque C8, suppZe'ment au n012, Tome 42,

de'cembre

1981 page C8-189

MODULATION EFFECTS I N P A S S I V E Rb FREQUENCY STANDARDS

P. Thomann and G. Busca

ASULAB S. A., 6, passage

M a x

Meuron, CH-2001 NeucMteZ, Switzer Zand

A b s t r a c t . - The l i n e s h a p e o f a microwave resonance i s s t u d i e d t h e o r e t i - c a l l y and e x p e r i m e n t a l l y i n t h e c a s e o f a phase-modulated RF-field and o p t i c a l d e t e c t i o n of t h e resonance. The c a l c u l a t i o n a l l o w s t o o p t i m i z e t h e e r r o r s i g n a l i n a Rb frequency s t a n d a r d with r e s p e c t t o t h e modula- t i o n parameters. Sine-wave and square-wave modulation a r e examined i n some d e t a i l .

1. I n t r o d u c t i o n . - The s h o r t term s t a b i l i t y o f a p a s s i v e Rb s t a n d a r d i s mainly li- mited by t h e s i g n a l t o n o i s e r a t i o o f t h e e r r o r s i g n a l needed t o lock t h e q u a r t z o s c i l l a t o r t o t h e atomic frequency. The e r r o r s i g n a l i s u s u a l l y o b t a i n e d by modu- l a t i n g t h e phase of t h e RF f i e l d and d e t e c t i n g t h e corresponding modulation i n t h e l i g h t i n t e n s i t y t r a n s m i t t e d by t h e resonance c e l l . I n t h i s paper we d e r i v e a n a l y t i - c a l r e s u l t s f o r t h e dependence of t h e e r r o r s i g n a l on modulation frequency, modula- t i o n depth and r e l a x a t i o n r a t e s , w i t h p a r t i c u l a r a t t e n t i o n t o t h e s p e c i f i c c a s e s of sine-wave and square-wave modulation.

Modulation e f f e c t s have been e x t e n s i v e l y s t u d i e d i n t h e p a s t ([I] - [8]). A l l au- t h o r s we a r e aware o f , however, c o n c e n t r a t e on t h e l i n e s h a p e o f t h e radio-frequency a b s o r p t i o n curve. Here we a r e concerned with o p t i c a l d e t e c t i o n of an RF resonance s i g n a l , which means t h a t t h e p o p u l a t i o n i n v e r s i o n between t h e two h y p e r f i n e l e v e l s , n o t t h e i r coherence, i s t h e r e l e v a n t parameter. Furthermore, most t r e a t m e n t s a r e r e s t r i c t e d a s t o t h e range of t h e modulation parameters (modulation frequency low [ 6

-

8 o r h i g h 1 3 , 43 compared t o t h e r e l a x a t i o n r a t e s , low modulation amplitu- d e s [2.

$, ^.

I n t h i s c a l c u l a t i o n we p u t no l i m i t a t i o n on t h e modulation amplitude and frequency, s o t h a t t h e e r r o r s i g n a l cannot be r e l a t e d i n g e n e r a l t o d e r i v a t i v e s of t h e s t a t i c l i n e s h a p e .

I n o r d e r t o s i m p l i f y t h e mathematical t r e a t m e n t and t o o b t a i n meaningful a n a l y t i c a l r e s u l t s , we make t h e two f o l l o w i n g assumptions:

1) Although t h e p o p u l a t i o n s o f a l l Zeeman s u b s t a t e s a r e coupled through t h e o p t i c a l pumping p r o c e s s , we assume t h a t t h e ( s m a l l ) p o p u l a t i o n changes induced by t h e RF f i e l d i n t h e two f i e l d - i n d e p e n d e n t s t a t e s have a n e g l i g i b l e e f f e c t on t h e o t h e r p o p u l a t i o n s . Thus we r e p l a c e t h e two Zeeman m u l t i p l e t s by two s i n g l e l e v e l s and assume t h a t t h e e f f e c t o f t h e c o u p l i n g w i t h i n m u l t i p l i c i t i e s can be c o n t a i n e d i n t h e r e l a x a t i o n r a t e s of t h e two-level system.

2) We assume t h a t t h e RF power i s low enough t o n e g l e c t s a t u r a t i o n e f f e c t s ; t h i s allows a p e r t u r b a t i v e t r e a t m e n t of t h e e q u a t i o n s of motion. I n t h i s r e s p e c t our c a l c u l a t i o n i s s i m i l a r t o t h e one by Karplus

[I],

b u t second-order, i n s t e a d of f i r s t o r d e r , r e s u l t s a r e n e c e s s a r y t o account f o r t h e o p t i c a l d e t e c t i o n of t h e RF-resonance.

2. D e r i v a t i o n of t h e resonance l i n e s h a p e . - The e v o l u t i o n of t h e d e n s i t y m a t r i x i s s p l i t i n t o t h r e e p a r t s corresponding t o o p t i c a l pumping, r e l a x a t i o n and i n t e r a c t i o n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981822

JOURNAL

DE

PHYSIQUE

with t h e RF f i e l d :

O P Re l R F

a , ~ = a , ~ + aP ,+ ~ , P

We t a k e a simple model f o r o p t i c a l pumping where l e v e l s 1 and 2 a r e d e p l e t e d by t h e pumping l i g h t a t r a t e s

rl

and

r2.

The pumping l i g h t i s assumed t o be weak and t o have a broad spectrum, i n which c a s e t h e pumping c y c l e i s a d e q u a t e l y d e s c r i b e d by r a t e e q u a t i o n s (no o p t i c a l c o h e r e n c e s ) . The l i g h t i s r e e m i t t e d spontaneously from a s i n g l e e x c i t e d l e v e l which decays t o l e v e l s 1 and 2 w i t h t h e same r a t e 4y, ( ~ > > r ) . We t h e n have t h e f o l l o w i n g e q u a t i o n s f o r t h e h y p e r f i n e 0-sublevels (we n e g l e c t t h e l i g h t - s h i f t f o r s i m p l i c i t y ) .

We assume t h a t a l l r e l a x a t i o n mechanisms ( c o l l i s i o n s w i t h o t h e r atoms o r with t h e w a l l s , magnetic f i e l d inhomogeneities) can b e d e s c r i b e d by a " l o n g i t u d i n a l " and a

" t r a n s v e r s e " r e l a x a t i o n time y i , and y;; t h e s t e a d y s t a t e d e n s i t y m a t r i x elements w i t h o u t o p t i c a l pumping a r e

011 = 022 = 4 and p l 2 = 0 , s o t h a t

a

Rel

,

^Pn

⁼ - ^Y;

^{(Pit - 1/21}

a

Rel

,

^{p22 = -}

^Y;

( P 2 2 - 1 4 )

a

Rel

,

^{p 1 2 =}

^-

^yiplz

F i n a l l y , t h e i n t e r a c t i o n w i t h t h e RF-field ( l i n e a r l y p o l a r i z e d p a r a l l e l t o t h e C- f i e l d ) i s given by

where p2 / ~ W O pe icp(t)

H - .

(

_Be-icp(t)

_-%

_{w o}

Uo i n c l u d e s t h e q u a d r a t i c Zeeman e f f e c t and t h e l i g h t s h i f t

p

= .-1 PBBRF i s t h e c o u p l i n g s t r e n g t h o f t h e 0-0 t r a n s i t i o n i n Rb-87 w i t h t h e RF f i e l d

Radiofrequency f i e l d , w i t h phase $ ( t ) t o b e B R F ( t ) = B R F COS cp(t)

s p e c i f i e d l a t e r

Defining A = p l l - p22, we o b t a i n t h e f o l l o w i n g e q u a t i o n s f o r A and p12 A = -

Yl

(A-AO)

+

4

p

Jm ( p12e - i V ( t )

biz=(-y2

+ i u o ) p,, -

,ae

^icp(tIA

where = y;

+

^{1 3 (}

r1 + r2

⁾

y , =

Yi

+ % ( r l + r 2 ) 0 - Yz(r

- r

A -&+b;r~+;:)

I n o b t a i n i n g e q u a t i o n s ( 4 ) we have a l s o made use o f t h e rotating-wave approximation, which i s j u s t i f i e d s i n c e i n p r a c t i c e b o t h

6

and t h e frequency

w,

o f phase modulation a r e much s m a l l e r than W o .

Although a numerical s o l u t i o n o f e q s ( 4 ) can r e a d i l y be o b t a i n e d f o r any v a l u e o f t h e p a r a m e t e r s , we r e s t r i c t t h e range of RF amplitude i n o r d e r t o o b t a i n an approxi- mate a n a l y t i c a l s o l u t i o n . I f

6

<

yl, yz

we can expand A and i n a p e r t u r b a t i o n s e r i e s of powers o f

8.

This i s n o t a very r e s t r i c t i v e c o n d i t i o n : i n p r a c t i c e one a v o i d s t o s a t u r a t e t h e t r a n s i t i o n s i n c e s a t u r a t i o n a l s o means broadening o f t h e re- sonance curve and d e c r e a s e o f t h e d i s c r i m i n a t i n g s l o p e .

We t h e r e f o r e w r i t e

w i t h

d") - ^on

~2)- b n

I n s e r t i n g t h e e x p r e s s i o n s above i n t o e q s ( 4 ) and s o r t i n g o u t terms o f e q u a l powers of

B

y i e l d s t h e f o l l o w i n g o r d e r s of approximation f o r t h e permanent s o l u t i o n s : Order 0 : A(') = A0

:

= o

Order 1: A(') = 0

- i

p a 0

+iwt7 +iB@(t') .(-Y2 + iw,) ( t - t') d t' where we have used t h e following d e f i n i t i o n

64I(t) i s t h e phase modulation of t h e RF f i e l d ; we make use o f i t s p e r i o d i c i t y t o w r i t e it a s a F o u r i e r s e r i e s

-

we suppose f o r s i m p l i c i t y t h a t ak i s r e a l , which i s t h e c a s e i f 6 $ 1 ( t ) =

-

6 @ ( - t ) . Theref o r e

( l ) iwt a k e ikwmt

p I l ( t ) = - iPAoe 2

-a Y2+i(U +kW,) where Cf = W

-

^{U o} i s t h e RF-atom detuning.

Order 2 : p$) =

I n t e g r a t i n g eq. (9) we o b t a i n , f o r t h e l o w e s t (second) o r d e r i n

B,

t h e f o l l o w i n g e x p r e s s i o n f o r t h e permanent s o l u t i o n A ( ' ) ( t ) :

Replacing PI(:) ( t ' ) by i t s e x p l i c i t e x p r e s s i o n (eq. 8) and performing t h e i n t e g r a - t i o n , we g e t , a f t e r some m a n i p u l a t i o n s

A (2)(*) = -

2'

Cp COs pw, t

+ sp

sin pw,t

Y1 Y2

{

^P=o

JOURNAL DE PHYSIQUE

where A ( x ) = ( i

+

x Z ) - ' x k = ( a + k w , )

/

y, and

D ( x ) = ^{X . ( 1}

+

x 2 ) - I Y p = p W m / Y,

The i n v e r s i o n A ( ' ) shows o s c i l l a t i o n s a t a l l harmonics of t h e modulation frequency.

A s a f u n c t i o n of t h e d e t u n i n g a , t h e amplitude of t h e s e o s c i l l a t i o n s undergoes r e - sonances c e n t e r e d a t a = - kq,,. When % << y l , y~ t h e a d d i t i o n a l resonances (k

#

0 ) merely broaden t h e unmodulated l i n e shape whereas i f

q,,

> > y l , y2 t h e y appear a s r e s o l v e d sidebands. S i n c e t h e f i r s t harmonic ( p = 1) i s o f most i n t e r e s t h e r e we w i l l c o n c e n t r a t e on t h i s component and, i n o r d e r t o o b t a i n s p e c i f i c a n a l y t i c a l r e - s u l t s , we w i l l c o n s i d e r two s p e c i a l c a s e s of phase modulation, namely sine-wave and square-wave modulation.

3. F i r s t harmonic of t h e p o p u l a t i o n i n v e r s i o n , s i n e - and square-wave phase modulation.-

The f i r s t harmonic component o f t h e i n v e r s i o n i s given by e q . . l l :

where C1 and S 1 a r e given by e q s ( l l b , c ) a ) sine-wave modulation

The phase e x c u r s i o n i s t h e n

8 g , ( t ) = m s i n w m t ; t h e F o u r i e r development of ei6'(t) r e a d s

i m s i n % t +o

= J~ ( m ) e i k w m t k=-w

and we have ak = J k ( m ) , where Jk(m) i s t h e s t a n d a r d B e s s e l f u n c t i o n of o r d e r k . b ) square-wave modulation

-

i f k = O

= [ i s I i n m i f i f k k i s i s even odd

,.,,

Table 1 g i v e s a summary o f a n a l y t i c a l r e s u l t s which a r e v a l i d i n some l i m i t i n g c a s e s of i n t e r e s t . For small modulation f r e q u e n c i e s t h e l i n e shape i s , a s expected, e q u a l t o t h e d e r i v a t i v e o f t h e s t a t i c L o r e n t z i a n l i n e shape (eq. 1 5 ) . A t high modu- l a t i o n f r e q u e n c i e s , however, t h e l i n e shape becomes a s t a n d a r d d i s p e r s i o n curve (eq. 1 3 ) .

Comparison between t h e s e two l i m i t i n g c a s e s ( e q s 13 and 1 5 , t a b l e 1) i n d i c a t e s a simple method t o determine Y 2 f r o m t h e width of t h e experimental curves and y l from t h e i r amplitude. A t h i g h modulation f r e q u e n c i e s , f o r example,the d i s p e r s i o n curve peaks a t a / y Z = I , which g i v e s an immediate measurement of '(2. Once y2 i s known, y l can b e determined i n t h e f o l l o w i n g way. Keeping t h e phase e x c u r s i o n m and t h e de- t u n i n g a f i x e d , t h e amplitude C 1 o f t h e e r r o r s i g n a l i s measured a t two frequencies:

(&, y2 and Um2

>>

y2. I n s e r t i n g t h e e x p e r i m e n t a l v a l u e s C l ( % l ) and C 2 ( ( & 2 ) in t o eqs 1 3 and 15 y i e l d s

ffl ffl

"

w .rl

0 rn h

4 0

g 8 I"

rn

.

8

.i 0

3 "

B 8

A

r-

=!

0

-

" z 5 2

~ I L

^II

e

V V

s s

CI

0 I

- N Y - I

II II II a

-

4J

- ^s

8

-

Y ,

8

;

A-

- $

U h .rl

::

2 2

+ A

" " _{; 31;} ^V _G

v 2 g

^I

31;

'I ^?I

0

-

"I U rn

- - S

^N

*

N ^II

A A A A

N m ln W

=! - ^{5 "} - ⁵

T' 2

al

m

"

ffl

B

u

"

A

8 I

.rl C rn

N

A

*

N

$ 2 ₊

.rl _rn rn

-

+

g ₈

U

-

I

3 ²

4 3

I b -

II II E 4

-

₀

-

^II

^V

2? #

^2:

e

a m

I al .r( rn

4

4

* : :

_rn

⁸¹⁹

2

^{+ N}

u rn

-

^N

,I ,I

0

-

4-1 4

b

-

_+J II

-

N

-

4

u

9

A

I 8

2

^{2: I}

u ^(I

8 G

rn ⁺

N

-

..

";,

^N

-

JOURNAL DE PHYSIQUE

Once y2 and

yl

a r e de'termined, t h e modulation frequency and t h e modulation index can b e chosen s o a s t o o p t i m i z e t h e s l o p e o f A1 ( x ) =

(c: + ^s:)%

n e a r o = 0 , u s i n g t h e c u r v e s no 3 , 4. I t should b e n o t e d t h a t Al, t h e maximum amplitude o f t h e s i g n a l , r a t h e r t h a n C1 o r S1, i s t h e r e l e v a n t parameter f o r determining t h e d i s c r i m i n a t i n g s l o p e s i n c e C1 and S 1 a r e b o t h e q u a l t o z e r o a t o = 0 and a r e p r o p o r t i o n a l t o each o t h e r n e a r

a

= 0. I n o t h e r words t h e phase s e t t i n g o f t h e phase s e n s i t i v e d e t e c t o r i n a servo-loop u s i n g A ( 2 ) ( t ) a s an e r r o r s i g n a l s e r v e s o n l y t o maximize t h e s i g n a l b u t i n t r o d u c e s no s h i f t i n t h e s t a b i l i z e d frequency. That i s t r u e f o r pure s i n e - wave a n d . p u r e square-wave modulation, b u t it may n o t h o l d f o r an a r b i t r a r y modula- t i o n s i g n a l .

4. Experimental.- I n o r d e r t o check t h e r e s u l t s e s t a b l i s h e d i n t h e p r e c e d i n g sec- t i o n , we have used a s e t u p v e r y s i m i l a r t o t h e one used normally i n Rb frequency s t a n d a r d s : a Rb-87 lamp, followed by a Rb-85 f i l t e r , p r o v i d e s t h e pumping l i g h t f o r a Rb-87 c o a t e d c e l l w i t h o u t b u f f e r - g a s . The RF f i e l d i s produced from t h e 5 MHz o u t p u t of a Cs-clock, phase modulated, m u l t f p l i e d by 1368 and mixed w i t h t h e o u t - p u t o f a swept frequency s y n t h e t i z e r . I n o r d e r t o avoid any coupling o f t h e RF sidebands w i t h t h e m a g n e t i c - f i e l d dependent Zeeman s u b s t a t e s a t t h e h i g h e s t modu- l a t i o n f r e q u e n c i e s ( - 10 kHz), a r a t h e r l a r g e DC magnetic h i e l d o f 0.5 Gauss was used t o s e p a r a t e t h e Zeeman l e v e l s by

-

^{350 kHz.}

Fig. 1 Amplitude A1 o f t h e f i r s t harmonic o f t h e i n v e r s i o n ( t ) v e r s u s RF-atom d e t u n i n g ( t h e o r y and experiment).

Y P

= 2050 s - l , h+,, = 1 4 . 4 . 1 0 ~ s - l , sine-wave phase modulation, m = 1.4

Fig. 1 shows t h e shape of t h e resonance (amplitude A1 of t h e f i r s t harmonic of

A ( ' )

( t ) a s a f u n c t i o n o f t h e RE'-atom d e t u n i n g

a ) .

I n t h i s exemple t h e modulation frequency i s much l a r g e r t h a n t h e r e l a x a t i o n r a t e s , s o t h a t w e l l r e s o l v e d sidebands appear a t i n t e g e r v a l u e s o f t h e r a t i o

o / ~ .

The q u a n t i t y o f i n t e r e s t f o r a frequen- cy s t a n d a r d i s t h e s l o p e of t h e curve n e a r

a

= 0 , which we d e f i n e a s

P I = aA_

a c a / r ~ ) (18)

Fig. 2a Slope of t h e d i s c r i m i n a t i n g s i g n a l v e r s u s modulation frequency f o r

y1/Y2

= 0.3; 0.7; 1.5; 3.0, sine-wave phase modulation ( m = 2 ) s o l i d l i n e s : eq. 11; o: experiment

Fig. 2 b Slope of t h e d i s c r i m i n a t i n g s i g n a l v e r s u s modulation frequency for

Y1/Y2 = 0 - 3 ; 0 - 7 ; 1.5; 3.0, square-wave modulation m = ~ / 4 ; s o l i d l i n e s : eq. 17 ( t a b l e 1) ; o: experiment

C8- 196 JOURNAL DE PHYSIQUE

The d i s c r i m i n a t i n g s l o p e [ ~ o l t s / ~ z ] i s t h e n given by

where V i s t h e DC v o l t a g e change g e n e r a t e d a t t h e photo-detector o u t p u t when t h e RF power l e v e l i s switched from

B

= 0 t o

B

>> y l y 2 ( s a t u r a t i o n ) , t h u s inducing an atomic i n v e r s i o n v a r i a t i o n e q u a l t o

A'

(eq. 5 ) .

F i g u r e s 2a and 2b show t h e dependence o f P l on the modulation frequency f o r seve- r a l v a l u e s of t h e r a t i o y1/y2. For low v a l u e s of %, P1 i n c r e a s e s w i t h modulation frequency b u t i s independent of Y1. However, t h e modulation frequency a t which P1 r e a c h e s a maximum and t h e maximum v a l u e i t s e l f both depend on y l . A measurement o f P I a s a f u n c t i o n o f

y,

a l l o w s , a s d e s c r i b e d b e f o r e , a simple experimental determi- n a t i o n o f y1/Y2. I n t h e c o n d i t i o n s of o u r experiment, t h e d a t a f i t t h e t h e o r e t i c a l curve corresponding t o Y1/y2 = 0.7 ( f i g .

g:

square-wave modulation w i t h m = T/4;

f i g .

&:

sine-wave modulation w i t h m = 2 ) . The r e l a x a t i o n r a t e s y; and y; ( s e e e q s 3, 5 ) c o u l d be measured by r e p e a t i n g t h e s e measurements a t v a r i o u s pumping r a t e s and e x t r a p o l a t i n g t o z e r o l i g h t i n t e n s i t y .

The dependence of PI on (sine-wave) modulation frequency i s shown i n f i g . 3 f o r v a r i o u s modulation depths. For low v a l u e s of t h e modulation index m ( m 2)

,

t h e optimum modulation frequency s t a y s c o n s t a n t b u t t h e s l o p e PI i n c r e a s e s with m. For m

5

2 , t h e maximum s l o p e s t a y s c o n s t a n t b u t t h e optimum modulation frequency de- c r e a s e s with i n c r e a s i n g m: t h e l a r g e s t s i g n a l i s o b t a i n e d when t h e frequency excur- s i o n

mm

i s roughly e q u a l t o t h e width y 2 o f t h e resonance.

Fig. 3 Slope o f t h e d i s c r i m i n a t i n g s i g n a l v e r s u s modulation frequency (sine-wave modulation) f o r v a r i o u s modulation d e p t h s ~ 1 / Y 2 = 0.7.

S o l i d l i n e s : eq. 11; experimental d a t a :

+

^(m = 0 . 5 ) ; 0 (m = 1 ) ; o ( m = 2 ) ; A ( m = 5 )

F i g . 4 shows how t h e s l o p e P1 o f t h e e r r o r s i g n a l depends on t h e phase e x c u r s i o n m f o r b o t h sine-wave and square-wave modulation. I n t h e l a t t e r c a s e , P I i s pro- p o r t i o n a l t o sin2m. The maximum s l o p e i s t h u s o b t a i n e d when m = T/4, r e g a r d l e s s o f a l l o t h e r parameters. I n sine-wave modulation PI depends i n a more complicated way on both t h e modulation index and t h e modulation frequency b u t t h e optimum s l o p e

can be up t o 50% l a r g e r t h a n i n square-wave modulation.

Fig. 4 Slope of t h e d i s c r i m i n a t i n g s i g n a l v e r s u s phase e x c u r s i o n m f o r sine-wave phase modulation and square-wave phase modulation

(wm/y2 = 0 . 7 7 ) . S o l i d l i n e s : sine-wave (eq. 11); square-wave (eq. 1 7 ) experiment: o sine-wave phase modulation;

+

square-wave phase

modulation, ( s l o p e i n a r b i t r a r y u n i t s )

5. Conclusions.- A dynamical c a l c u l a t i o n of t h e double-resonance l i n e s h a p e s i n t h e presence of phase-modulation o f t h e RF f i e l d h a s been performed. T h i s calcu- l a t i o n i s v a l i d f o r any combination of modulation frequency and depth b u t i s r e s - t r i c t e d t o u n s a t u r a t e d resonances. I t allows t o p r e d i c t t h e modulation parameters t h a t w i l l maximize t h e d i s c r i m i n a t o r s l o p e once t h e r e l a x a t i o n r a t e s o f t h e system a r e known. The r e l a x a t i o n r a t e s can b o t h be measured, u s i n g t h e same t h e o r e t i c a l r e s u l t s , by comparing t h e amplitude o f t h e s i g n a l a t low and h i g h modulation f r e - quencies. Experimental r e s u l t s a r e i n good agreement with t h e o r e t i c a l p r e d i c t i o n s and show t h a t t h e assumptions underlying t h e c a l c u l a t i o n s a r e j u s t i f i e d , p a r t i c u - l a r l y concerning t h e n e g l e c t of Zeeman pumping.

Acknowledgements

We thank D r . H. Brandenberger f o r having p o i n t e d o u t t h a t problem t o u s and f o r h e l p f u l d i s c u s s i o n s .

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