RADIATIVELY-AIDED INELASTIC COLLISIONS IN BOUND-STATE SYSTEMS

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RADIATIVELY-AIDED INELASTIC COLLISIONS IN BOUND-STATE SYSTEMS

P. Berman

To cite this version:

P. Berman. RADIATIVELY-AIDED INELASTIC COLLISIONS IN BOUND-STATE SYSTEMS.

Journal de Physique Colloques, 1985, 46 (C1), pp.C1-11-C1-24. �10.1051/jphyscol:1985102�. �jpa-

00224471�

JOURNAL DE PHYSIQUE

Colloque C I , supplement au n ° l , Tome 46, Janvier 1985 page C l - 1 1

RADIATIVELY-AIDED INELASTIC COLLISIONS IN BOUND-STATE SYSTEMS

P.R. Berman

Physics Department, New York University, 4 Washington Place, New York, New York 10003, U.S.A.

Résumé - Dans cet article, (1) on présente une brève revue de la théorie des collisions inélastiques assistées par le rayonnement (RAIC), (2) on décrit une méthode diagrammatique pour construire les opérateurs effectifs apparaissant dans la théorie, et (3) on propose une méthode pour créer des cohérences magnétiques et électroniques dans les réactions RAIC.

Abstract - In this paper, (1) a brief review of the theory of Radiatively- Aided Inelastic Collisions (RAIC) in bound-state systems is given, (2) a diagrammatic method for constructing the effective operators which appear in the theory is described, and (3) a method for producing magnetic and electronic state coherence in RAIC reactions is proposed.

I - INTRODUCTION

Fig. 1 - Energy level- diagram for the RAIC reaction A. + A'., + HO. •* A + Al,.

A Radiatlvely-Aided Inelastic Collision (RAIC) is most easily understood by direct reference to a prototype system. Two atoms, A and A', having the energy level diagrams shown in Fig. 1, undergo a collision. We wish to determine the cross section for a transition from an initial state having atom A in state i and A' in state i' to a final state having atom A in state f and A' in state f . In this discussion, it is assumed that atoms A and A' constitute part of an atomic vapor and that the energy difference between final and initial states greatly exceeds the mean thermal energy per atom in the sample. Consequently, the cross section

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985102

Cl-12 JOURNAL DE PHYSIQUE

f o r t h e i n e l a s t i c c o l l i s i o n from i n i t i a l t o f i n a l s t a t e w i l l b e s m a l l (of o r d e r 1 0 - ' ~ c m ~ o r l e s s ) . However, t h e c r o s s s e c t i o n c a n b e s i g n i f i c a n t l y enhanced i f a p u l s e d r a d i a t i o n f i e l d i s a p p l i e d whose f r e q u e n c y R i s c h o s e n t o match t h e t r a n s i - t i o n f r e q u e n c y from i n i t i a l t o f i n a l s t a t e . The combined r e a c t i o n ,

i n which two atoms c o l l i d e i n t h e p r e s e n c e o f a r a d i a t i o n f i e l d a n d a b s o r b o n e o r more p h o t o n s w h i l e c h a n g i n g t h e i r i n t e r n a l s t a t e s , i s what we r e f e r t o a s a

R a d i a t i v e l y - A i d e d I n e l a s t i c C o l l i s i o n (iUIC) / I / [ o t h e r names f o r t h i s r e a c t i o n i n - c l u d e R a d i a t i v e C o l l i s i o n (RC) / 2 / and Laser-Induced C o l l i s i o n a l E x c i t a t i o n

T r a n s f e r (LICET) 1 3 1 1 . The p u r p o s e of t h i s p a p e r i s t h r e e - f o l d : ( 1 ) t o p r e s e n t a b r i e f r e v i e w o f t h e t h e o r y of RAIL i n b o u n d - s t a t e s y s t e m s ; ( 2 ) t o o f f e r a diagram- m a t i c method f o r c o n s t r u c t i n g t h e e f f e c t i v e o p e r a t o r s which a p p e a r i n t h e o r i e s o f KAIC; a n d ( 3 ) t o d i s c u s s t h e manner i n which K A I C c a n b e u s e d t o g e n e r a t e b o t h m a g n e t i c and e l e c t r o n i c s t a t e c o h e r e n c e .

F o l l o w i n g t h e t h e o r e t i c a l p r e d i c t i o n / 2 / and e x p e r i m e n t a l o b s e r v a t i o n / 4 / of M I L , t h e r e was a f l o u r i s h of t h e o r e t i c a l / 5 , 6 / and e x p e r i m e n t a l 171 a c t i v e l y i n t h i s r e s e a r c h a r e a . I n t e r m s o f t h e o r e t i c a l development, b o t h a m o l e c u l a r - s t a t e ap- r o a c h / 6 / and a s e p a r a t e d - a t o m b a s i s a p p r o a c h / 2 , 5 / were employed t o c a l c u l a t e KAIC t r a n s i t i o n p r o b a b i l i t i e s . The m o l e c u l a r - b a s i s a p p r o a c h , i n which t h e c o l l i - d i n g atoms a r e t a k e n t o c o n s t i t u t e a s i n g l e m o l e c u l a r s y s t e m , i s t h e p r e f e r r e d method f o r o b t a i n i n g q u a n t i t a t i v e agreement between t h e o r y and e x p e r i m e n t , p r o v i d e d t h a t a c c u r a t e m o l e c u l a r - s t a t e p o t e n t i a l s c a n b e c a l c u l a t e d . On t h e o t h e r hand, t h e s e p a r a t e d - a t o m b a s i s a p p r o a c h a l l o w s one t o r e a c h g e n e r a l q u a l i t a t i v e c o n c l u - s i o n s r e g a r d i n g t h e dependence o f M I C t r a n s i t i o n r a t e s on s u c h p a r a m e t e r s a s l a s e r f i e l d s t r e n g t h , l a s e r f i e l d f r e q u e n c y , form o f t h e c o l l i s i o n a l i n t e r a c t i o n , e t c . , w i t h o u t e x t e n s i v e n u m e r i c a l a n a l y s i s . T h e s e p a r a t e d - a t o m b a s i s a p p r o a c h i s t h e one t h a t w i l l b e u s e d t h r o u g h o u t t h i s d i s c u s s i o n .

I n t h e s e p a r a t e d atom b a s i s , e a c h s t a t e of t h e AA' s y s t e m can b e w r i t t e n i n n com- p o s i t e b a s i s , i . r . ,

e t c . , where a c a p i t a l Roman l e t t e r i n d i c a t e s t h e c o m p o s i t e s t a t e . The atoms b e g i n i n some i n i t i a l s t a t e ] I > and e v o l v e t o s t a t e IF; a s a r e s u l t of t h e combined ac- t i o n o f t h e l a s e r f i e l d and t h e c o l l i s i o n . For s i m p l i c i t y , t h e atoms a r e assumed t o move on c l a s s i c a l l y d e s c r i b e d t r a j e c t o r i e s t h r o u g l l o ~ ~ t a c o l l i s i o n and t h e c o l l i - s i o n i s assumed t o o c c u r on a t i m e s c a l e s u f f i c i e n t l y s h o r t t o c o n s i d e r t h e l a s e r p u l s e a m p l i t u d e t o b e c o n s t a n t d u r i n t h e c o l l i s i o n . T h e r e a r e a d d i t i o n a l s t a t e s which a r e c o u p l e d t o s t a t e s 11, ^{and Ti;} v i a t h e v a r i o u s i n t e r a c t i o n s , b u t , owing

t o t i l e e n e r g y mismatch f o r t i l e v a r i o u s e x c i t a t i o n p r o c e s s e s , a l l s t a t e s o t h e r t h a n

11; a n d IF> e n t e r t h e c a l c u l a t i o n a s v i r t u a l s t a t e s o n l y . I t i s t h e n p o s s i b l e t o sum o v e r t h e s e v i r t u a l s t a t e s t o a r r i v e a t e f f e c t i v e o p e r a t o r s which c o u p l e tlle i n i t i a l and f i n a l s t a t e s .

A d i a g r a m m a t i c method f o r c o n s t r u c t i n g t h e e f f e c t i v e o p e r a t o r s i s g i v e n i n S e c . 11.

At t h i s p o i n t , however, i t is u s e f u l t o b r i e f l y d i s c u s s t h e g e n e r a l s t r u c t u r e of t h e o p e r a t o r s . I n d o i n g s o , i t i s assumed t h a t t h e i n i t i a l and f i n a l s t a t e s a r e e a c h n o n d e g e n e r a t e ( a n a s s u m p t i o n which w i l l be r e l a x e d i n Sec. 111, when RAIC-in- duced m a g n e t i c - s t a t e c o h e r e n c e i s d i s c u s s e d ) . The e f f e c t i v e o p f ; r a t o r s f a l l i n t o t h r e e c a t e g o r i e s : (1) F i r s t , t h e r e is t h e l i g h t s h i f t o p e r a t o r SL which v a r i e s a s

P.

t h e s q u a r e of t h e l a s e r f i e l d a m p l i t u d e . The o p e r a t o r SL r e s u l t s i n a s h i f t i n

t h e e n e r g y of tGe i n i t i a l and f i n a l l e v e l s . ( 2 ) Second, t h e r e i s t h e c o l l i s i o n a l

s h i f t o p e r a t o r S which i s q u a d r a t i c i n t h e c o l l i s i o n a l i n t e r a c t i o n p o t e n t i a l .

The o p e r a t o r 2 r e s u l t s i n a s h i f t o f t h e e n e r g y l e v e l s of t h e i n i t i a l a n d f i n a l s t a t e s . ( 3 ) F i n a l l y , t h e r e is t h e t r a n s i t i o n o p e r a t o r 7 w h i c h v a r i e s a s t h e p i o d u c t o f t h e l a s e r f i e l d a m p l i t u d e a n d t h e c o l l i s i o n a l i n t e r a c t i o n . T h e o p e r a t o r T i s r e s p o n s i b l e f o r t h e RAIC r e a c t i o n by c o u p l i n g t h e i n i t i a l s t a t e t o t h e f i n a l o n e ( t h e a d j o i z t o p e r a t o r T *+ c o u p l e s t h e f i n a l s t a t e t o t h e i n i t i a l o n e ) . B o t h t h e o p e r a t o r s S a n d T d e p e n d o n t h e c o l l i s i o n a l i r l t e r a c t i o n , w h i c h i t s e l f i s a f u n c - t i o n o f t h e c o l l i s i o n i m p a c t p a r a m e t e r b , t h e AA' r e l a t i v e s p e e d v a n d t h e t i m e t . G i v e n t h e a b o v e d e s c r i p t i o n o f t h e e f f e c t i v e o p e r a t o r s , i t is a p p a r e n t t h a t t h e p r o b a b i l i t y a m p l i t u d e s f o r s t a t e s I a n d F e v o l v e a s ( i n t h e i n t e r a c t i o n r e p r e - s e n t a t i o n )

w h e r e

a n d t h e d e t u n i n g A i s d e f i n e d by

T i e q u a n t i t y EG i s t h e e n e r g y o f s t a t e G a n d R is t h e a p p l i e d l a s e r f i e l d f r e - q u e n c y . W i t h t h e i n i t i a l c o n d i t i o n s a t t i m e t- b e f o r e a c o l l i s i o n g i v e n by

Eqs. ( 3 ) a r e s o l v e d t o o b t a i n t h e f i n a l s t a t e p r o b a b i l i t y / a F ( t + ) 1' a t t i m e t +

f o l l o w i n g t h e c o l l i s i o n . The f i n a l s t a t e p r o b a b i l i t y must b e a v e r a g e d o v e r i m p a c t p a r a m e t e r , r e l a t i v e s p e e d , a n d t i l e p o s i t i o n a n d t i m e d u r i n g t h e l a s e r p u l s e a t w h i c h t h e c o l l i s i o n o c c u r s i n o r d e r t o a r r i v e a t a n e x p r e s s i o n f o r t h e a v e r a g e RAIC c r o s s s e c t i o n u IF. I n m o s t c a s e s , a n u m e r i c a l s o l u t i o n o f Eqs. ( 3 ) i s re- q u i r e d .

The d e p e n d e n c e o f oIF o n s u c h p a r a m e t e r s a s t h e l a s e r f i e l d i n t e n s i t y , t h e de-

t u i t i n g A , a n d t h e n a t u r e o f t h e c o l l i s i o n a l i n t e r a c t i o n h a s b e e n d i s c u s s e d ex-

t e n s i v e l y i n t h e l i t e r a t u r e / 2 , 5 / . I:ithout e n t e r i n g i n t o a d e t a i l e d a n a l y s i s

from w h i c h s u c h g e n e r a l c o n c l u s i o n s c a n b e r e a c h e d , o n e c a n l i s t s e v e r a l i m p o r t a n t

r e s u l t s o f t i l e t h e o r y : ( 1 ) I n t h e " w e a k 1 ' - f i e l d l i m i t (power d e n s i t i e s <, 1 0 ~ 1 ! / c m ~ ) ,

t h e c r o s s s e c t i o n i s p r o p o r t i o r i a l t o t h e l a s e r f i e l d i n t e n s i t y a n d i s a n asymmet-

r i c f u n c t i o n o f d e t u n i n g A a b o u t A = 0 . The asymmetry a p p e a r s o n l y f o r ~ A ! T ~ 2 1

( T ~ = c o l l . i s i o n d u r a t i o n ) a n d i s r e l a t e d t o t h e f a c t t h a t , owing t o t h e c o l l i -

C1-14 JOURNAL DE PHYSIQUE

s i o n a l s h i f t o p e r a t o r Sc, t h e KAIC r e a c t i o n may be b r o u g h t i n t o i n s t a n t a n e o u s r e -

s o n a n c e d u r i n g t h e c o l l i s i o n f o r one s i g n of t h e d ~ t u n i n g , b u t n o t t h e o t h e r . ( 2 ) F o r s t r o n g l a s e r f i e l d s , t h e l i g h t s h i f t o p e r a t o r SL s h i f t s t h e c e n t r a l r e s o n a n c e f r e q u e n c y of t h e RAIC r e a c t i o n . The c r o s s s e c t i o n becomes a symmetric f u n c t i o n of A a b o u t t h e l i g h t - s h i f t e d c e n t r a l f r e q u e n c y , h a v i n g a w i d t h t h a t d e c r e a s e s w i t h i n c r e a s i n g f i e l d s t r e n g t h . Moreover, t h e c r o s s s e c t i o n no l o n g e r i s p r o p o r t i o n a l t o t h e l a s e r f i e l d i n t e n s i t y , b u t v a r i e s a s t h e i n t e n s i t y t o some power l e s s t h a n o r e q u a l t o o n e - h a l f , depending on t h e n a t u r e of t h e c o l l i s i o n a l i n t e r a c t i o n . (3) F o r weak f i e l d s , t h e dependence o f t h e c r o s s s e c t i o n o n A f o r l a r g e ] A / c a n b e r e l a t e d s i m p l y t o t h e n a t u r e o f t h e c o l l i s i o n a l i n t e r a c t i o n ( i . e . f o r a n a t t r a c - t i v e d i p o l e - d i p o l e i n t e r a c t i o n , t h e "red" wing f a l l s o f f a s / A / - ~ ' ' ) . The domi- n a n t c o n t r i b u t i o n t o t h e c o l l i s i o n i n t e r a c t i o n u s u a l l y comes from t h e l o w e s t o r d e r n o n v a n i s h i n g t e r m i n t h e m u l t i p o l e e x p a n s i o n f o r t h e e l e c t r o s t a t i c i n t e r a c t i o n between A and A' atoms. (The p a r i t y of t h e c o n t r i b u t i n g t e r m must b e odd i f s t a t e s I and F h a v e t h e same p a r i t y and e v e n i f t h e y have o p p o s i t e p a r i t y ) . ( 4 ) I n weak f i e l d s , t h e v a l u e o f A l e a d i n g t o a maximu2 i n t h e RAIC c r o s s s e c t i o n i s d e t e r - mined by t h e r e l a t i v e s t r e n g t h s o f $- and T. I f t h e major c o n t r i b u t i o n from t h e ?

C

..

o p e r a t o r is d e r i v e d from c o l l i s i o n s h a v i n g i m p a c t p a r a m e t e r s where S is p r o d u c i n g a s t r o n g e f f e c t , t h e maximum i s s h i f t e d from A = 0.

The e x p e r i m e n t a l r e s u l t s /4,7-111 a r e i n q u a l i t a t i v e agreement w i t h p r e d i c t i o n s o f t h e t h e o r y b a s e d on Eqs. ( 3 ) . One i m p o r t a n t e x c e p t i o n t o t h e agreement between t h e o r y and e x p e r i m e n t c a n b e found i n h i g h r e s o l u t i o n s t u d i e s of RAIC p r o f i l e s . I n e x p e r i m e n t s i n v o l v i n g Sr-Eu / 8 , 9 / and Na-Ca 1101 R A I C r e a c t i o n s , t h e RAIC c r o s s s e c t i o n was found t o v a r y a s y m p t o t i c a l l y a s l ~ l - ~ ' ~ , whereas t h e t h e o r e t i c a l p r e - d i c t i o n b a s e d on Eqs. (3) i s J A J - ~ ' ~ , assuming t h a t t h e dominant c o n t r i b u t i o n t o t h e AA' i n t e r a t o m i c p o t e n t i a l comes from t h e d i p o l e - d i p o l e term i n t h e e l e c t r o - s t a t i c m u l t i p o l e e x p a n s i o n . I t i s p o s s i b l e t h a t a m o l e c u l a r - b a s i s c a l c u l a t i o n c o u l d r e p r o d u c e a / A ] -

dependence, b u t s u c h a c a l c u l a t i o n i s y e t t o be performed f o r t h e s e s y s t e m s . On t h e o t h e r hand, t h e RAIC p r o f i l e f o r a Sr-Li r e a c t i o n /11/

was c o n s i s t e n t w i t h t h e / A 1 -"' p r e d i c t i o n .

11 - EFFECTIVE RAIC OPERATORS

The RAIC r e a c t i o n c a n b e d e s c r i b e d i n t e r m s of t h e o p e r a t o r s ZL, and T i n t r o - * duced i n t h e p r e v i o u s s e c t i o n . Once t h e s e o p e r a t o r s have been s p e c i f i e d , i t is a r e l a t i v e l y e a s y t a s k t o s o l v e Eqs. ( 3 ) and t o o b t a i n t h e c o r r e s p o n d i n g K A I C c r o s s s e c t i o n . I n t h i s s e c t i o n , a diagrammatic method f o r c o n s t r u c t i n g t h e e f f e c t i v e RAIC o p e r a t o r s i s d i s c u s s e d .

F o r m a l l y , t h e e f f e c t i v e RAIC o p e r a t o r s a r e found by i t e r a t i n g t h e e q u a t i o n s of motion f o r t h e s t a t e a m p l i t u d e s s o a s t o e l i m i n a t e a l l t e r m s i n v o l v i n g v i r t u a l s t a t e a m p l i t u d e s . I n t h i s manner, one a r r i v e s a t e q u a t i o n s , s i m i l a r t o Eqs. ( 3 ) , i n which t h e i n i t i a l and f i n a l s t a t e s a r e c o u p l e d by e f f e c t i v e o p e r a t o r s , which t h e m s e l v e s i n v o l v e a sum o v e r a l l p o s s i b l e v i r t u a l s t a t e s p a r t i c i p a t i n g i n t h e RAIC r e a c t i o n .

To c o n s t r u c t t h e RAIC o p e r a t o r s d i a g r a n m a t i c a l l y , t h e a t o m - f i e l d i n t e r a c t i o n which t a k e s t h e AA' s y s t e m from some s t a t e G t o s t a t e B i s r e p r e s e n t e d by t h e v e r t e x

3 ^, w h i l e t h e c o l l i s i o n a l i n t e r a c t i o n which t a k e s t h e atoms from s t a t e G t o

G H I

H i s r e p r e s e n t e d by t h e v e r t e x I . To o b t a i n t h e a p p r o p r i a t e diagrammatic G 1:

r e p r e s e n t a t i o n s of t h e M I C o p e r a t o r s , one combines t h e s e v e r t i c e s i n s u c h a

manner t h a t all i n t e r m e d i a t e s t a t e s ( i . e . t h o s e a p p e a r i n g between v e r t i c e s ) a r e

v i r t u a l s t a t e s of t h e s y s t e m w h i l e t h e "end" s t a t e s ( i . e . t h e s t a t e a p p e a r i n g t o

t h e l e f t o f t h e f i r s t v e r t e x and t h e r i g h t of t h e l a s t v e r t e x ) a r e e i t h e r i n i t i a l

o r f i n a l s t a t e s of t h e RAIC r e a c t i o n . I n g e n e r a l , t h e dominant c o n t r i b u t i o n t o

the effective KAIC operators comes from the diagrams having the fewest number of vertices (but see below).

The light-shift operator has the diagrammatic representation

while the collisional shift operator can be represented as

The operator * S L involves the excitation of either atom A or A' to a virtual state and the subsequent deexcitation of that atom to its initial state; the operator 2

involves the virtual excitation of both atoms A and A ' to a virtual state and the subsequent deexcitation of tiese atoms to their initial states. ASince there are two atom-field vertices for SL and two collisional vertices for Sc, it is immediately obvious that

ST - varies as the square of the laser field amplitude and that S varies as the square of the collisional interaction strength.

The diagram for the transition operator associated with the RAIC reaction of Fig. 1

where it is assumed that the overall energy defect,

is less than the thermal energy kT. It is apparent from the diagram that IF,^)

depends on the product of the laser field amplitude and collisional interaction strength; consequently it vanishes in the absence of either the applied laser field or the collision. In other words, the RAIC reaction is a combined laser field-collisional interaction, as stated in the introduction.

Given the diagrammatic representation above, it is a simple matter to establish rules 1121 for constructing the operators from the diagrams. For an applied laser field of the form

+ -+

and a collisional interaction potential V(t), one assigns a value -

-+ -?* 4 pT*E

(or - ^{% t ; p )} to each atom-field interaction vertex and a value V(t) to each

1 -+

- + - +

collisional interaction vertex (the quantity p, = p + ^p' is the sum of the dipole

moment operators p and * for atoms A and A', :espectively). In addition, an

appropriate energy denominator (or product of energy denominators if more than two

vertices appear in an individual diagram) must be included to account for the

energy defect between the intermediate virtual states and the original state (i.e.,

the state appearing to the left of the first vertex) in the diagram. For any

atom-field interaction vertex other than the last vertex occurring in a diagram,

one must include the photon energy 5 t i G in the energy denominators since this

Cl-16 JOURNAL DE PHYSIQUE

p h o t o n e n e r g y i s v i r t u a l l y a b s o r b e d o r e m i t t e d by t h e AA' s y s t e m . Using t h e s e r u l e s , one f i n d s t h a t t h e l i g h t - s h i f t o p e r a t o r i s g i v e n by

t h e c o l l i s i o n a l s h i f t o p e r a t o r by

and t h e t r a n s i t i o n o p e r a t o r by

I n o r d e r t o g a i n a d d i t i o n a l i n s i g h t i n t o t h e s t r u c t u r e o f t h e t r a n s i t i o n o p e r a t o r , i t i s c o n v e n i e n t t o r e w r i t e e a c h o f t h e d i a g r a m s i n E q . ( 9 ) a s t h e sum of two d i s - t i n c t d i a g r a m s i n which t h e l a s e r f i e l d a c t s o n l y on atom A o r o n l y on atom A'.

The f o u r d i a g r a m s f o r t h e t r a n s i t i o n o p e r a t o r a l o n g w i t h t h e c o r r e s p o n d i n g e n e r g y l e v e l d i a g r a m s a r e shown i n F i g . 2. They c o n t a i n a l l t h e p o s s i b l e o r d e r i n g o f t h e l a s e r f i e l d and c o l l i s i o n a l i n t e r a c t i o n s i n d r i v i n g t h e Ah' s y s t e m from i n i t i a l s t a t e I t o f i n a l s t a t e F. I n F i g . 2 ( a ) , t h e c o l l i s i o n a c t s t o v i r t u a l l y e x c i t e t h e atoms from s t a t e l i i ' > t o s t a t e h f l > and t h e f i e l d t h e n a c t s on atom A t o com- p l e t e t h e e x c i t a t i o n t o f i n a l s t a t e I f f

I n P i g . Z(c), t h e f i e l d a c t s o n atom A t o e x c i t e t h e v i r t u a l s t a t e \ h i ' > and t h e c o l l i s i o n c o m p l e t e s tlie e x c i t a t i o n . The a n a l y s i s o f F i g s . 2 ( b ) and 2 ( d ) i s s i m i l a r t o t h a t f o r F i g s . 2 ( a ) and 2(c), r e - s p e c t i v e l y , b u t tlie f i e l d a c t s on atom A ' r a t h e r t h a n atom A.

F i g u r e 2 i s h e l p f u l f o r u n c o v e r i n g s i t u a t i o n s i n which =-resonant enhancement of t h e KAIC c r o s s s e c t i o n c a n b e a c h i e v e d . I f one o f t h e v i r t u a l e n e r g y l e v e l s i n F i g . 2 n e a r l y c o i n c i d e s w i t h a r e a l e n e r g y l e v e l of atom A o r A ' , t h e r e can b e a d r a m a t i c enhancement o f t h e RAIC c r o s s s e c t i o n ( i t i s s t i l l assumed, however, t h a t t h e e n e r g y d i f f e r e n c e between t h e r e a l a n d v i r t u a l e n e r g y l e v e l i s g r e a t e r t h a n kT t o a s s u r e t h a t t h e v i r t u a l l e v e l r e m a i n s v i r t u a l ) . I n s u c h a c a s e , t h e dominant c o n t r i b u t i o n t o t h e sum i n Eq. ( 1 4 ) w i l l come from t h e n e a r l y r e s o n a n t i n t e r m e d i a t e s t a t e . T h i s t y p e o f n e a r - r e s o n a n t enhancement h a s b e e n e x p l o i t e d i n many of t h e e x p e r i m e n t s i n v o l v i n g K A I C r e a c t i o n s .

The d i a g r a m m a t i c t e c h n i q u e c a n b e e x t e n d e d t o much more complex s i t u a t i o n s . I n

p a r t i c u l a r , i t c a n b e used t o a n a l y z e RAIC r e a c t i o n s , s u c h a s t h o s e proposed by

Smirnov and C h a p l i k 1 1 3 1 and Nayfeh e t . a l . 1141, i n which t h e r e a r e c o n t r i b u t i o n s

t o t i l e KAIC o p e r a t o r s from h i g h e r o r d e r t h a n i s i m p l i e d by t h e d i a g r a m s c o n t a i n i n g

t h e f e w e s t number of v e r t i c e s . Such c o n t r i b u t i o n s c a n become i m p o r t a n t i f t h e r e

i s n e a r - r e s o n a n t enhancement f o r t h e s e h i g h e r o r d e r p r o c e s s e s .

i i' hf' f f'

i i' fh' f f '

i i' h i ' f f '

i i' ih' f f '

Fig. 2 - Schematic representation of the four types of contributions to the transi-

tion operator. Wavy lines with arrows represent the atom-field interaction and

solid lines wit11 arrows represent the collisional interaction.

CI-18 J O U R N A L DE PHYSIQUE

111 - MAGNETIC-STATE: COHERENCE

I n t h e p r e v i o u s s e c t i o n s , a method f o r o b t a i n i n g RAIC t r a n s i t i o n p r o b a b i l i t i e s was r e v i e w e d . By e x a m i n i n g t h e RAIC e x c i t a t i o n c r o s s s e c t i o n a s a f u n c t i o n o f de- t u n i n g A , o n e may b e a b l e t o e x t r a c t some i n f o r m a t i o n a b o u t t h e AA' i n t e r a t o m i c p o t e n t i a l . The l i n e s h a p e becomes a s e n s i t i v e p r o b e o f t h e c o l l i s i o n a l i n t e r - a c t i o n o n l y f o r d e t u n i n g s o u t s i d e t h e s o - c a l l e d " i m p a c t c o r e " o f t h e l i n e ; t h a t i s , f o r d e t u n i n g s A s a t i s f y i n g I A ~ T ^{> I} w h e r e T i s t h e c o l l i s i o n d u r a t i o n . S i n c e t h e s i g n a l s t r e n g t h i s w e a k e s t f o r s u c h d e t u n i n g s , i t may b e d i f f i c u l t t o o b t a i n l a r g e s i g n a l - t o - n o i s e r a t i o s i n p r e c i s e l y t h a t p a r t o f t h e RAIC e x c i t a t i o n p r o f i l e t h a t p r o b e s t h e c o l l i s i o n a l i n t e r a c t i o n . One m i g h t a s k w h e t h e r t h e r e i s a n o t h e r f e a - t u r e o f t h e RAIC e x c i t a t i o n p r o c e s s t h a t c a n p r o v i d e i n f o r m a t i o n on t h e c o l l i - s i o n a l i n t e r a c t i o n i n t h e i m p a c t c o r e o f t h e l i n e ( I A / T < I ) , w h e r e s i g n a l s t r e n g t h i s a maximum.

I t t u r n s o u t t h a t , i n c e r t a i n c a s e s , s u c h i n f o r m a t i o n c a n b e p r o v i d e d by s t u d y i n g t h e p o l a r i z a t i o n p r o p e r t i e s o f RAIC e x c i t a t i o n / 1 2 , 1 5 , 1 6 / . I t i s n o t d i f f i c u l t t o u n d e r s t a n d b o t h t h e o r i g i n o f RAIC i n d u c e d p o l a r i z a t i o n a n d i t s s e n s i t i v i t y t o t h e f o r m o f t h e i n t e r a t o m i c p o t e n t i a l . To d o s o , one m u s t g e n e r a l i z e t h e d i s c u s s i o n o f t h e p r e v i o u s s e c t i o n s by a s s u m i n g t h a t e a c h o f t h e a t o m s ' e n e r g y l e v e l s ( s e e F i g s . 1 a n d 2), a c t u a l l y c o n s i s t s o f a number o f m a g n e t i c a l l y d e g e n e r a t e s u b l e v e l s . I f t h e i n i t i a l s t a t e i s u n p o l a r i z e d , t h e f i n a l s t a t e m a g n e t i c c o h e r e n c e p r o d u c e d i n t h e RAIC r e a c t i o n r e s u l t s f r o m t h e combined e f f e c t o f ( I ) t h e p o l a r i z a t i o n a n d d i r e c t i o n a l i t y o f t h e a p p l i e d r a d i a t i o n f i e l d w h i c h t e n d s t o c r e a t e a m a g n e t i c - s t a t e c o h e r e n c e a n d ( 2 ) t h e c o l l i s i o n a l i n t e r a c t i o n w h i c h t e n d s t o d e s t r o y i t . Thus, a n y m a g n e t i c - s t a t e c o h e r e n c e p r o d u c e d i n RAIC r e a c t i o n s c a n , i n p r i n c i p l e , b e u s e d t o e x t r a c t i n f o r m a t i o n a b o u t t h e M' i n t e r a t o m i c p o t e n t i a l . Cy m o n i t o r i n g b o t h t h e RAIC i n d u c e d f i n a l s t a t e p o p u l a t i o n p o l a r i z a t i o n a s a f u n c t i o n o f de- t u n i n g A , o n e g a i n s a more c o m p l e t e p i c t u r e o f t h e RAIC r e a c t i o n t h a n t h a t p r o - v i d e d by p o p u l a t i o n m e a s u r e m e n t s a l o n e . T h i s i d e a h a s b e e n e x p l o i t e d i n t h e n o t u n r e l a t e d p r o b l e m o f p o l a r i z a t i o n e f f e c t s i n ' f c o l l i s i o n a l r e d i s t r i b u t i o n o f r a d i a - t i o n " / 1 6 , 1 7 / , a l t h o u g h s u c h m e a s u r e m e n t s , by t h e i r v e r y n a t u r e , a r e r e s t r i c t e d t o l a r g e d e t u n i n g s , I A I T , ² 1. I n t h e c a s e o f RAIC-generated m a g n e t i c s t a t e co- h e r e n c e , b o t h t h e i m p a c t c o r e a n d l a r g e d e t u n i n g r e g i o n s o f t h e RAIC p r o f i l e c a n b e u s e d t o e x t r a c t i n f o r m a t i o n r e l a t e d t o t h e c o l l i s i o n a l i n t e r a c t i o n .

The RAIC r e a c t i o n d e p i c t e d i n F i g . 1 c a n s e r v e t o i l l u s t r a t e t h e b a s i c f e a t u r e s o f RAIC-generated m a g n e t i c - s t a t e c o h e r e n c e . I n o r d e r t o s i m p l i f y t h e d i s c u s s i o n , i t i s assumed t h a t ( 1 ) t h e l a s e r f i e l d i s s u f f i c i e r l t l y weak t h a t e f f e c t s p r o d u c e d by t h e l i g h t - s h i f t o p e r a t o r c a n b e n e g l e c t e d , ( 2 ) t h e l a s e r f i e l d i s s u f f i c i e n t l y weah t o treat t h e t r a n s i t i o n o p e r a t o r t o l o w e s t o r d e r i n p e r t u r b a t i o n t h e o r y , a n d (3) t h e l a s e r f r e q u e n c y is c h o s e n t o b r i n g t h e RAIC r e a c t i o n i n t o e x a c t r e s o n a n c e , A = 0 . (The c o n s e q u e n c e s o f r e l a x i n g t h e s e a s s u m p t i o n s i s d i s c u s s e d b r i e f l y a t t h e e n d o f t h i s s e c t i o n ) . we c o n t i n u e t o f o l l o w t h e s e p a r a t e d - a t o m b a s i s ap- p r o a c h , a l t h o u g l i a m o l e c u l a r b a s i s a p p r o a c h c a n a l s o b e u s e d e f f e c t i v e l y / 1 6 / . The i n t e r a t o m i c p o t e n t i a l V ( t ) c a n b e w r i t t e n q u i t e g e n e r a l l y a s

k , k t V ( t ) = A::: ( t ) T

T

k , k ' , q , q l

w h e r e Tk a n d 'It:: a r e i r r e d u c i b l e t e n s o r o p e r a t o r s which a c t on t h e e l e c t r o n i c c o o r d i n a t e s o f a t o m s A and A ' , r e s p e c t i v e l y . As s u c h , t h e c o l l i s i o n a l i n t e r - a c t i o n c a n b e v i e w e d a s p r o d u c i n g t h e same e f f e c t a s two c o r r e l a t e d m u l t i p o l a r f i e l d s ( e a c h f i e l d i n t e r a c t i n g w i t h o n l y o n e o f t h e a t o m s ) , w h e r e t h e c o r r e l a t i o n

k k '

i s d e t e r m i n e d by t h e c o u p l i n g p a r a m c t e r s A q q , ( t ) . Moreover, t h e i n c i d e n t m u l t i -

. .

p o l a r f i e l d s s h o u l d h e t a k e n l o be u n p o l a r i z e d a n d i n c i d e n t f r o m a l l d i r e c t i o n s

i n o r d e r t o s i m u l a t e a n a v e r a g e o v e r a l l c o l l i s i o n o r i e n t a t i o n s . Thus, t h e c o l l i - s i o n i n t e r a c t i o n can b e r e p l a c e d by two c o r r e l a t e d , u n p o l a r i z e d m u l t i p o l a r f i e l d s i n c i d e n t from a l l d i r e c t i o n s on t h e atoms, p r o v i d e d t h a t e a c h f i e l d is r e s t r i c t e d t o i n t e r a c t w i t h o n l y one of t h e atoms / 1 5 / . I t i s t h e n p o s s i b l e t o i n t e r p r e t t h e c r e a t i o n o f a f i n a l s t a t e R A I C c o h e r e n c e a s t h e r e s u l t o f t h e a c t i o n o f three

f i e l d s , t h e two m u l t i p o l a r f i e l d s p l u s t h e i n c i d e n t l a s e r f i e l d . T h i s p i c t u r e en- a b l e s oile t o u n d e r s t a n d many of t h e f e a t u r e s o f t h e RAIC g e n e r a t e d m a g n e t i c - s t a t e c o h e r e n c e .

As a s p e c i f i c example, we c o n s i d e r t h e c a s e where t h e dominant c o n t r i b u t i o n t o t h e MIC r e a c t i o n of F i g . 1 comes from e i t h e r o f t h e r e s o n a n t l y enhanced c h a n n e l s shown i n F i g s . 3 ( a ) a n d 3 ( b ) , r e s p e c t i v e l y . Tile development of S e c s . I and I1 is e a s i l y g e n e r a l i z e d t o i n c o r p o r a t e t h e f a c t t h a t e a c h l e v e l not? c o n s i s t s o f a number of m a g n e t i c a l l y d e g e n e r a t e s u b s t a t e s . The c o l l i s i o n a l o p e r a t o r

Sc(G) (G = I o r F) now c o u p l e s any two s u b s t a t e s w i t h i n l e v e l G ; i n o t h e r words SF

r e s u l t s i n b o t h a s h i f t o f t h e e n e r g y o f t h e i n i t i a l and f i n a l s t a t e s a s w e l l a s a -

c o u p l i n g between t h e m a g n e t i c s u b s t a t e s w i t h i n e a c h of t h o s e l e v e l s . The t r a n s i - t i o n o p e r a t o r c o u p l e s a s u b s t a t e of l e v e l I t o one o f l e v e l F. S i n c e K A I C e x c i t a - t i o n o c c u r s f o r l a r g e impact p a r a m e t e r c o l l i s i o _ n s and s i n c e t h e t r a n s i t i o n o p e r a t o r v a r i e s a s t h e c o l l i s i o n i n t e r a c t i o n V ( t ) w h i l e Sr v a r i e s a s t h e s q u a r e o f V ( t ) , i t

-

i s assumed t h a t t h e t r a n s i t i o n o p e r a t o r p r o v i d e s t h e dominant c o n t r i b u t i o n t o t h e f i n a l s t a t e p o l a r i z a t i o n .

F i g . 3 - An example o f e n e r g y l e v e l d i a g r a m s f o r which n e a r - r e s o n a n t enhancement of t h e RAIC r e a c t i o n , Ai + ^A;, + hi2 ^-+ Af + A;,, i s p o s s i b l e .

I n T i g . 3 ( a ) , t h e m u l t i p o l a r f i e l d s p r o d u c e t h e v i r t u a l s t a t e ( f r l > . The l a s e r

f i e l d c o m p l e t e s t h e RAIC r e a c t i o n by t a k i n g A ' from ( v i r t u a l ) s t a t e r ' t o f i n a l

s t a t e f ' . Thus, t h e f i n a l s t a t e p o l a r i z a t i o n of s t a t e f ' s h o u l d b e e q u a l t o t h a t

produced i n a two-photon t r a n s i t i o n from s t a t e i f t o s t a t e f ' , i n which t h e f i r s t

f i e l d (an u n p o l a r i z e d f i e l d , i n c i d e n t from a l l d i r e c t i o n s ) d r i v e s t h e i ' t o r '

v i r t u a l t r a n s i t i o n and t h e s e c o n d f i e l d ( t h e l a s e r f i e l d ) c o m p l e t e s t h e t r a n s i t i o n

from r ' t o f ' . The same c o n c l u s i o n i s r e a c h e d i n a f o r m a l t r e a t m e n t of t h e prob-

lem 1151 and h a s been v e r i f i e d e x p e r i m e n t a l l y 1 9 1 ( t h e f i n a l s t a t e c o h e r e n c e i s

measured by m o n i t o r i n g t h e p o l a r i z a t i o n o f f l u o r e s c e n c e from l e v e l f ' ) . I f t h e

i n i t i a l s t a t e is u n p o l a r i z e d (always assumed t o be t h e c a s e i n c h i s d i s c u s s i o n ) ,

t h e f i n a l s t a t e c o h e r e n c e c a n n o t depend o n t h e n a t u r e o f t h e c o l l i s i o n a l i n t e r -

a c t i o n , owing t o t h e f a c t t h a t t h e c o l l i s i o n e f f e c t i v e l y p r o d u c e s a n u n p o l a r i z e d

v i r t u a l s t a t e r ' ; t h e f i n a l s t a t e p o l a r i z a t i o n i s t h a t produced by t h e l a s e r f i e l d

d r i v i n g a t r a n s i t i o n from l e v e l r ' ( u n p o l a r i z e d ) t o l e v e l f ' . C o n s e q u e n t l y , a t

l i n e c e n t e r , - t h e f i n a l s t a t e p o l a r i z a t i o n f o r s u c h a s y s t e m does n o t p r o v i d e a

p a r t i c u l a r l y e f f e c t i v e p r o b e o f t h e c o l l i s i o n a l i n t e r a c t i o n .

C1-20 JOURNAL DE PHYSIQUE

The s i t u a t i o n changes somewhat when one c o n s i d e r s t h e l e v e l scheme of F i g . 3 ( b ) . The l a s e r f i e l d t a k e s atom A from s t a t e i t o ( v i r t u a l ) s t a t e r and t h e c o l l i s i o n c o m p l e t e s t h e r e a c t i o n t o s t a t e I f f ' , . I t would a p p e a r a t f i r s t g l a n c e t h a t s t a t e f ' i s u n p o l a r i z e d s i n c e i t i s e x c i t e d by a n u n p o l a r i z e d m u l t i p o l a r f i e l d i n c i d e n t from a l l d i r e c t i o n s ; however, t h i s need n o t b e t h e c a s e . Imagine t h a t t h e l a s e r f i e l d i s l i n e a r l y p o l a r i z e d a l o n g z and t h a t s t a t e s i, r , and f have j v a l u e s of 0, 1 and 0, r e s p e c t i v e l y . Only t h e m = 0 component of l e v e l r i s e x c i t e d by t h e f i e l d ; c o n s e q u e n t l y , o n l y t h a t p a r t of t h e m u l t i p o l a r o p e r a t o r is used which c a n c a u s e a Am = 0 t r a n s i t i o n between s t a t e s r and f . Owing t o t h e c o r r e l a t i o n between t h e m u l t i p o l a r f i e l d s [ s e e Eq. ( 1 5 ) ] a c t i n g on e a c h atom, o n l y p a r t of t h e m u l t i p o l a r f i e l d a c t i n g on atom A' i s used i n t h e i' -+ f ' e x c i t a t i o n ; t h a t is, i t is no l o n g e r a s i f l e v e l f ' i s e x c i t e d by an u n p o l a r i z e d f i e l d i n c i d e n t from a l l d i r e c - t i o n s . As s u c h , l e v e l f ' c a n be p o l a r i z e d w i t h a p o l a r i z a t i o n t h a t w i l l depend on t h e n a t u r e of t h e c o l l i s i o n a l i n t e r a c t i o n ; f o r m a l c a l c u l a t i o n s 1151 c o n f i r m t h i s p r e d i c t i o n . Thus, f o r t h e l e v e l scheme of F i g . 3 ( b ) t h e RAIC-generated f i n a l s t a t e p o l a r i z a t i o n a t l i n e c e n t e r (A=O) c a n p r o v i d e i n f o r m a t i o n a b o u t t h e i n t e r - a t o m i c p o t e n t i a l which is n o t c o n t a i n e d i n measurements of RAIC-generated popu- l a t i o n s .

From t h e above examples, i t can b e s e e n t h a t s t u d i e s of R4IC-generated p o l a r i z a - t i o n can p r o v i d e s u p p l e m e n t a r y i n f o r m a t i o n f o r a n a l y z i n g RAIC r e a c t i o n s . I n addi- t i o n , t h e g e n e r a t i o n of a m a g n e t i c - s t a t e c o h e r e n c e i n a M I C r e a c t i o n p r o v i d e s a n i n t e r e s t i n g example o f t h e i n t e r p l a y between t h e a t o m - f i e l d and c o l l i s i o n a l i n t e r - a c t i o n s . The r e p l a c e m e n t of t h e c o l l i s i o n a l i n t e r a c t i o n by a n e f f e c t i v e m u l t i - p o l a r f i e l d a i d s i n t h e i n t e r p r e t a t i o n o f t h e RAIC-generated m a g n e t i c - s t a t e co- h e r e n c e i n t h e weak f i e l d , impact c o r e l i m i t . T h i s a p p r o a c h s h o u l d a l s o prove u s e f u l i n t h e l i m i t where t h e t r a n s i t i o n o p e r a t o r c a n no l o n g e r b e t r e a t e d u s i n g p e r t u r b a t i o n t h e o r y . The RAIC t r a n s i t i o n o p e r a t o r a c t s a s a n e f f e c t i v e t h r e e - p h o t o n o p e r a t o r f o r t h e system; two p h o t o n s coming from u n p o l a r i z e d m u l t i p o l a r f i e l d s i n c i d e n t from a l l d i r e c t i o n s and one photon from t h e l a s e r f i e l d . The pre- s e n c e of a s t r o n g l a s e r f i e l d i s e q u i v a l e n t t o s a t u r a t i o n of t h e "three-photon"

t r a n s i t i o n and t h e f i n a l s t a t e p o l a r i z a t i o n i s t h e same a s t h a t g e n e r a t e d i n a s a t u r a t e d t h r e e - p h o t o n t r a n s i t i o n .

On t h e o t h e r hand, i t i s q u e s t i o n a b l e w h e t h e r t h e e f f e c t i v e m u l t i p o l a r f i e l d a p p r o a c h i s p a r t i c u l a r l y u s e f u l f o r l a r g e d e t u n i n g s , I A ~ T , ^> 1. I n t h i s l i m i t , t h e c o l l i s i o n a l s h i f t o p e r a t o r becomes i m p o r t a n t s i n c e i t may b r i n g t h e i n i t i a l and f i n a l s t a t e s of t h e RAIC r e a c t i o n i n t o i n s t a n t a n e o u s r e s o n a n c e a t c e r t a i n t i m e s d u r i n g t h e c o l l i s i o n . At t h e t i m e s of i n s t a n t a n e o u s r e s o n a n c e , t h e t r a n s i - t i o n o p e r a t o r r e s u l t s i n a n e x c i t a t i o n ( o r d e e x c i t a t i o n ) between t h e i n i t i a l and f i n a l s t a t e s . I n a d d i t i o n , f o l l o w i n g t h e f i r s t " i n s t a n t a n e o u s resonance", t h e c o l l i s i o n o p e r a t o r S (F) mixes t h e m a g n e t i c s u b s t a t e s which were e x c i t e d by t h e t r a n s i t i o n o p e r a t o r . Owing t o t h e c o m p l i c a t e d n a t u r e of t h e e x c i t a t i o n p r o c e s s , i t i s n o t o b v i o u s t h a t t h e r e p l a c e m e n t of t h e c o l l i s i o n a l i n t e r a c t i o n by m u l t i - p o l a r f i e l d s a i d s i n t h e i n t e r p r e t a t i o n of t h e r e s u l t s . Measurements of t h e f i n a l s t a t e magnetic c o h e r e n c e r e s u l t i n g from t h e a c t i o n of b o t h t h e t r a n s i t i o n and c o l l i s i o n a l s h i f t o p e r a t o r s c a n b e combined w i t h measurements of f i n a l s t a t e p o p u l a t i o n s t o p r o v i d e a s e n s i t i v e t e s t of i n t e r a c t i o n p o t e n t i a l s 116,171. To d a t e , t h e r e h a s been o n l y one e x p e r i m e n t / 9 / i n which RAIC-generated magnetic- s t a t e c o h e r e n c e was o b s e r v e d .

I V - ELECTRONIC-STATE COHERENCE

When one o r more l a s e r f i e l d s a r e i n c i d e n t on a n a t o m i c vapor i t i s p o s s i b l e t o

c r e a t e a c o h e r e n c e between d i f f e r e n t e l e c t r o n i c s t a t e s , p r o v i d e d t h a t t h e e x c i t a -

t i o n mechanism c o n s e r v e s b o t h e n e r g y and momentum ("phase-matchingg'). A neces-

s a r y c o n d i t i o n f o r t h e e x c i t a t i o n of a n e l e c t r o n i c s t a t e c o h e r e n c e i s t h a t t h e

f r e q u e n c y ( o r sums o r d i f f e r e n c e s of f r e q u e n c y ) of t h e a p p l i e d f i e l d s b e e q u a l t o

a n e l e c t r o n i c l e v e l s p a c i n g i n t h e atoms. I t i s a p p a r e n t t h a t t h e RAIC e x c i t a -

t i o n scheme of F i g . 1 c a n n o t b e used t o g e n e r a t e a n e l e c t r o n i c s t a t e c o h e r e n c e

a n d t h a t , i n g e n e r a l , KtlIC r e a c t i o n s i n v o l v i n g t h e a b s o r p t i o n o r e m i s s i o n of a s i n g l e p h o t o n do n o t s a t i s f y t h e c r i t e r i o n f o r t h e g e n e r a t i o n o f a n e l e c t r o n i c s t a t e c o h e r e n c e . I n o r d e r t o p r o d u c e e l e c t r o n i c s t a t e c o h e r e n c e i n RAIC r e a c t i o n s , o n e must a b s o r b ( o r e m i t ) a t l e a s t two p h o t o n s f r o m o n e o r more r a d i a t i o n f i e l d s . A RAIC r e a c t i o n i n w h i c h e l e c t r o n i c s t a t e c o h e r e n c e c a n be g e n e r a t e d i s i l l u s t r a - t e d i n F i g . 4.

F i g . 4 - RAIC r e a c t i o n s i n w h i c h i t i s p o s s i b l e t o c r e a t e a c o h e r e n c e b e t w e e n l e v e l s 2 ' a n d 3 ' .

Two p u l s e d l a s e r f i e l d s o f a m p l i t u d e E2 a n d E3, h a v i n g f r e q u e n c i e s G2 a n d R3, re- s p e c t i v e l y , a r e s i m u l t a n e o u s l y i n c i d e n t o n t h e c o l l i s i o n p a r t n e r s A and A' whose e n e r g y l e v e l d i a g r a m s a r e shown i n F i g . 4. The RAIC r e a c t i o n may b e w r i t t e n

A2 + ^A;, +%11112 + h R + A 1 + A t ( 2 ' , 3 ' ) , ( 1 6 ) 3

w h e r e t h e n o t a t i o n A' ( 2 ' , 3 ' ) i n d i c a t e s a f i n a l s t a t e c o h e r e n c e b e t w e e n s t a t e s 2 ' a n d 3 ' . I n F i g . 4 ( a ) t h e f i n a l s t a t e c o h e r e n c e o s c i l l a t e s a t f r e q u e n c y ( 0 -Q )

3 ' 2 w h i l e i n F i g . 4 ( b ) i t o s c i l l a t e s a t f r e q u e n c y (R3+n2). The f i n a l s t a t e c o h e r e n c e b e t w e e n l e v e l s 2' a n d 3 ' d e p e n d s o n t i l e p r o d u c t o f t h e a m p l i t u d e s f o r t h e i n d i v i d - u a i RAIC r e a c t i o n s A2+A;,+hR2 + A1+A;, a n d A +A' ,+iiR3

A +A' r e s p e c t i v e l y . The

- _{2 1 1} 3"

r e l a t i v e s p a t i a l a n d t e m p o r a l p h a s e o f t h e two a p p l i e d p u l s e d l a s e r f i e l d s m u s t b e s u f f i c i e n t l y s t a b l e t o i n s u r e t h a t t h e p h a s e o f t h e c o h e r e n c e c r e a t e d a t d i f f e r e n t t i m e s a n d p o s i t i o n s i n t h e l a s e r f i e l d s h a s t h e c o r r e c t s p a t i a l a n d t e m p o r a l de- p e n d e n c e .

The n a t u r e of t h e f i n a l s t a t e c o h e r e n c e d e p e n d s c r i t i c a l l y o n w h e t h e r o r n o t t h e f i n a l s t a t e s h a v e t h e same o r o p p o s i t e p a r i t y . F o r f i n a l s t a t e s h a v i n g t h e same p a r i t y , o n e c a n e s t i m a t e / 1 8 , 1 9 / t h a t t h e c r o s s s e c t i o n f o r t h e c r e a t i o n o f f i n a l s t a t e c o h e r e n c e i s c o m p a r a b l e t o t h a t f o r t h e c r e a t i o n o f f i n a l s t a t e p o p u l a t i o n s (' 1 0 - l 6 cm2 p e r biW/cm2 of l a s e r i n t e n s i t y ) . T h i s f i n a l s t a t e c o h e r e n c e c a n b e m o n i t o r e d by a p p l y i n g a p r o b e p u l s e t o a t o m 11' a t some t i m e f o l l o w i n g t h e p a s s a g e of t h e M I C e x c i t a t i o n p u l s e s ( e . g . , i f t h e p r o b e p u l s e h a s f r e q u e n c y Q2, a s i g n a l a t f r e q u e n c y . S13 c a n be g e n e r a t e d ) .

F o r f i n a l s t a t e s h a v i n g o p p o s i t e p a r i t y , t h e s i t u a t i o n c h a n g e s d r a m a t i c a l l y . The

C1-22 JOURNAL DE PHYSIQUE

e x c i t a t i o n schemes of F i g . 4 a p p e a r t o r e p r e s e n t a n o v e l means f o r sum o r d i f f e r - ency f r e q u e n c y g e n e r a t i o n i n v a p o r s . I t i s w e l l known, however, t h a t i t i s i m p o s s i - b l e t o g e n e r a t e t h e sum o r d!-fferency f r e q u e n c y o f two l a s e r f i e l d s i n a c e n t r o -

- symmetric medium ( i . e . , t h e s e c o n d o r d e r e l e c t r i c s u s c e p t i b i l i t y v a n i s h e s f o r s u c h a medium). When a c o m p l e t e c a l c u l a t i o n i s c a r r i e d o u t 1191, i n c l u d i n g e f f e c t s o f l e v e l d e g e n e r a c y a n d a n a v e r a g e o v e r a l l c o l l i s i o n o r i e n t a t i o n s , one d o e s f i n d t h a t t h e f i n a l s t a t e c o h e r e n c e a v e r a g e s t o z e r o , l e a v i n g u s w i t h o u t any sum o r d i f f e r e n c y f r e q u e n c y r a d i a t e d by t h e atoms. A l l i s n o t l o s t , however. By v e l o c i t y - s e l e c t i n g e i t h e r t h e A o r A' atoms ( o r b o t h ) , i t i s p o s s i b l e t o b r e a k t h e c e n t r o - symmetric s t r u c t u r e o f t h e v a p o r and t o c r e a t e a n o n v a n i s h i n g f i n a l s t a t e coher- e n c e t h a t c a n r a d i a t e sum o r d i f f e r e n c e f r e q u e n c y . Even though t h e v e l o c i t y - s e l e c t e d s y s t e m i s n o t c e n t r o s y m m e t r i c , i t t u r n s o u t 1191 t h a t t h e c r o s s s e c t i o n s t i l l v a n i s h e s u n l e s s t h e l a s e r s a r e d e t u n e d from e x a c t r e s o n a n c e f o r t h e RAIC r e - a c t i o n s . A s a c o n s e q u e n c e , c r o s s s e c t i o n s f o r RAIC e x c i t a t i o n o f e l e c t r o n i c s t a t e c o h e r e n c e between f i n a l s t a t e s h a v i n g o p p o s i t e p a r i t y i s s m a l l e r by a f a c t o r o f a t l e a s t 100 from t h o s e f o r f i n a l s t a t e s h a v i n g t h e same p a r i t y .

A l t e r n a t i v e l e v e l schemes f o r c r e a t i o n o f RAIC-induced e l e c t r o n i c s t a t e c o h e r e n c e s h a v e b e e n p r o p o s e d / 1 8 , 1 9 / a n d t h e r e a r e many a d d i t i o n a l p o s s i b i l i t i e s , n o t d i s - c u s s e d i n t h o s e r e f e r e n c e s , which c a n be u s e f u l . A RAIC r e a c t i o n which may h a v e some r e l e v a n c e t o s e c o n d harmonic g e n e r a t i o n i n a l k a l i v a p o r s / 2 0 / i s shown s c h e m a t i c a l l y i n F i g . 5.

Fig. 5 - A l e v e l scheme f o r s e c o n d harmonic g e n e r a t i o n u s i n g a RAIC r e a c t i o n ,

L e v e l 2 ' h a s p a r i t y o p p o s i t e t o t h a t of l e v e l 1' s o d i r e c t two-photon e x c i t a t i o n of l e v e l 2 ' i s f o r b i d d e n . However, by two-photon e x c i t a t i o n v i a t h e RAIC r e a c t i o n

one c r e a t e s a pathway f o r sum-frequency g e n e r a t i o n i n atom A' n e a r t h e 1'-2' t r a n - s i t i o n f r e q u e n c y . For r e a s o n s d i s c u s s e d above, t h e f i n a l s t a t e c o h e r e n c e v a n i s h e s u n l e s s some mechanism i s p r e s e n t t o c r e a t e a n a n i s o t r o p i c c o l l i s i o n d i s t r i b u t i o n . One must v e r i f y ( a s i n a l l s u c h RAIC r e a c t i o n s ) t h a t phase-matching c o n d i t i o n s f o r t h e g e n e r a t i o n o f a s i g n a l f i e l d a t f r e q u e n c y 2i2 c a n b e met. A s y e t , t h e r e i s no e x p e r i m e n t a l e v i d e n c e f o r t h e e x i s t e n c e o f MIC-induced e l e c t r o n i c - s t a t e c o h e r e n c e . I t would seem t h a t RAIC-generated c o h e r e n c e r e p r e s e n t s a r i c h a r e a f o r f u t u r e t h e o - r e t i c a l and e x p e r i m e n t a l development.

ACKNOWLEDGMENTS

I s h o u l d l i k e t o t h a n k P r o f s . E. G i a c o b i n o , E. Robinson and S. H a r r i s f o r many

h e l p f u l comments. P r o f e s s o r G i a c o b i n o s u g g e s t e d t h e l e v e l scheme shown i n F i g . 5.

T h i s r e s e a r c h i s s u p p o r t e d by t h e G.S. O f f i c e o f N a v a l R e s e a r c h a n d , i n p a r t , by t h e N a t i o n a l S c i e n c e F o u n d a t i o n , G r a n t No. INT 79-21530.

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