POLARISATION EFFECTS IN ENERGY-TRANSFER COLLISIONS BETWEEN LASER-EXCITED ATOMS

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POLARISATION EFFECTS IN ENERGY-TRANSFER COLLISIONS BETWEEN LASER-EXCITED ATOMS

G. Nienhuis

To cite this version:

G. Nienhuis. POLARISATION EFFECTS IN ENERGY-TRANSFER COLLISIONS BETWEEN LASER-EXCITED ATOMS. Journal de Physique Colloques, 1985, 46 (C1), pp.C1-97-C1-106.

�10.1051/jphyscol:1985109�. �jpa-00224479�

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JOURNAL DE PHYSIQUE

Colloque CI, supplement au n°l, Tome 46, Janvier 1985 page Cl-97

POLARISATION EFFECTS IN ENERGY-TRANSFER COLLISIONS BETWEEN LASER- EXCITED ATOMS

G. Nienhuis

Fysisch Laboratorium, Rijksuniversiteit Utrecht, Postbus 80 000, 3508 TA Utrecht, The Netherlands

Résumé - Nous étudions l'équation pilote généralisée pour la matrice densité d'un atonie à deux niveaux dégénérés dans un champ laser ayant une polarisation arbitraire. Cette étude est utilisée pour l'analyse des collisions de transfert d'énergie entre deux atomes excités par laser, en termes des taux collisionnels pour les sous-états magnétiques. Les résultats sont appliqués à nos récentes mesures de l'ionisation associative entre deux atomes Na(3P) excités et polarisés par laser.

Abstract - The analysis of energy-transfer collisions between laser-excited atoms in terms of rate constants for separate magnetic substates requires knowledge of the excited-state density matrix for each applied laser polarisation. We study the generalised rate equation for this density matrix. Then we describe the relation between the collision signals and the M-dependent collision amplitudes. The results are applied to our recent measurements of associative ionisation of laser-excited Na(3P) atoms.

I - OBSERVABILITY OF POLARISATION EFFECTS

Polarisation effects in inelastic atomic collisions can be studied by two distinct methods, which are basically each other's time reverse. One may analyse the polarisation properties or the angular distribution of the radiation emitted by one or both of the atoms in this final excited state. Alternatively, one may prepare one or both of the atoms in the excited state by laser excitation prior to the collision.

Both techniques give information on partial cross sections for transitions to or from the magnetic substates |Me> of the excited state. The similarity and the difference between these two methods is aptly illustrated by considering the information content in terms of the scattering amplitudes f(Me). For notational con- venience we first* consider the case that only one of the collision partners takes part in the radiation process.

When f(Me) is the scattering amplitude for excitation of the magnetic substate |Me>, then the excited state emerging from the collision is described by the density matrix p, with matrix elements.

(1.1)

where the average is over the initial relative velocity and the substates of the (usually unpolarised) initial state, and an integration is performed over the direction a of the relative velocity after the collision. When one of the scattered atoms is detected in coincidence with the photon emitted after the collision, this integration extends over the angles selected by the aperture of the detector.

The intensity of the radiation with polarisation t emitted by spontaneous decay from the excited state to a lower state (usually the ground state) with magnetic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985109

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CI-98 JOURNAL DE PHYSIQUE

substates ]M > i s determined by t h e d e n s i t y m a t r i x p , according t o t h e proportiona- l i t y r e l a t 1 0 8 /1/

where

i s a m a t r i x element o f the component o f t h e atomic e l e c t r i c d i p o l e i n t h e p o l a r i - s a t i o n d i r e c t i o n , and where

serves as a d e t e c t i o n operator. Hence the p o l a r i s a t i o n d i r e c t i o n

b,

which i s selec- t e d by the p o l a r i s a t i o n analyser i n f r o n t o f the photodetector f u l l y determines t h e d e t e c t i o n operator, which a c t s as a sieve t h a t determines t h e piece o f i n f o r m a t i o n on p gained i n the measurement. I t i s the d e n s i t y m a t r i x p t h a t c o n t a i n s the i n f o r - mation on t h e s c a t t e r i n g process. I n the i d e a l s i t u a t i o n t h e p o l a r i s a t i o n a n a l y s i s and t h e angular d i s t r i b u t i o n o f t h e photons determine ^pcompletely. I n general t h i s i s o n l y t r u e when t h e angular momentum Je o f the e x c i t e d s t a t e i s n o t l a r g e r than 1, s i n c e

D

contains o n l y t h e m u l t i p o l a r ranks 0, 1 and 2 when expanded i n s p h e r i c a l tensors / 2 / . This r e s u l t s from t h e f a c t t h a t

D

i s a simple product o f the v e c t o r operators d and d i .

Very much t h e same i n f o r m a t i o n can be gained by s t u d y i n g t h e time reversed process, where t h e e x c i t e d s t a t e i s prepared by l a s e r e x c i t a t i o n as an i n i t i a l s t a t e t o the c o l l i s i o n . The corresponding s c a t t e r i n g amplitudes are denoted as g(M ) , and they a r e l i n k e d by time r e v e r s a l symmetry t o f ( M ) , provided o f course thag t h e i n i t i a l s t a t e i n t h e f i r s t process i s t h e same as tRe f i n a l s t a t e i n the second process.

When u i s t h e d e n s i t y m a t r i x o f t h e i n i t i a l s t a t e , as prepared by l a s e r e x c i t a t i o n , then t h e p r o d u c t i o n r a t e R o f t h e f i n a l s t a t e obeys t h e p r o p o r t i o n a l i t y r e l a t i o n

R ^.^:

z

fdn [g(Me) <Me[o(Mel> g * ( M e 8 ) l a v = T r ue G

f ( 1 - 5 )

where G i s t h e d e t e c t i o n operator w i t h m a t r i x elements

The average i n (1.5) and (1.6) i s performed over the i n i t i a l d i s t r i b u t i o n o f r e l a - t i v e v e l o c i t i e s . An i n t e g r a t i o n i s performed over the d i r e c t i o n n o f the r e l a t i v e v e l o c i t i e s a f t e r t h e c o l l i s i o n , and t h e summation i s over the substates o f t h e f i n a l s t a t e . I n t h i s case i t i s the d e t e c t i o n o p e r a t o r G t h a t contains the informa- t i o n on t h e s c a t t e r i n g process, and the c h a r a c t e r i s t i c s o f t h e e x c i t i n g l a s e r determine t h e i n i t i a l d e n s i t y m a t r i x o , and thereby t h e p i e c e of i n f o r m a t i o n on G t h a t i s gained. A major advantage o f t h i s method o f s a t u r a t e d l a s e r e x c i t a t i o n over t h e p o l a r i s a t i o n a n a l y s i s o f spontaneous emission i s t h a t t h e m u l t i p o l e rank k o f t h e i n i t i a l d e n s i t y m a t r i x u i s n o t r e s t r i c t e d t o t h e values 0, 1 o r 2. By a proper choice o f t h e l a s e r p o l a r i s a t i o n d i r e c t i o n s i n successive measurements a f u l l determination o f G i s g e n e r a l l y f e a s i b l e .

The formal analogy between p o l a r i s a t i o n a n a l y s i s o f c o l l i s i o n a l l y e x c i t e d spontaneous emfssion and tine t i m e reverse orocess w i t h a l a s e r - e x c i t e d i n i t i a l s t a t e has been

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e l a b o r a t e d f o r e l e c t r o n - a t o m s c a t t e r i n g / 3 / . The a n a l o g y h o l d s equal l y w e l l f o r c o l l i s i o n processes i n which two atoms a r e e x c i t e d t o an e m i t t i n g e x c i t e d s t a t e i n a s i n g l e c o l l i s i o n /4/ as compared t o c o l ! i s i o n s between two i n i t i a l l y l a s e r - e x c i t e d atoms /5/. I n t h e p r e s e n t paper we d i s c u s s t h e d e s c r i p t i o n o f i n e l a s t i c c o l l i s i o n s between two l a s e r - e x c i t e d atoms, as a f u n c t i o n o f t h e p o l a r i s a t i o n o f t h e l a s e r . We s t u d y t h e v a l i d i t y o f g e n e r a l i s e d r a t e e q u a t i o n s i n terms o f E i n s t e i n c o e f f i c i e n t s as compared w i t h Bloch-type e q u a t i o n s i n d e t e r m i n i n g t h e s t a t i o n a r y d e n s i t y m a t r i x u of each one o f t h e atoms. The d e s c r i p t i o n i s a p p l i e d t o o u r c u r r e n t experiments on c o l l i s i o n s between two l a s e r - e x c i t e d Na atoms, b o t h i n a. s i n g l e atomic, beam. and i n two c o u n t e r r u n n i n g beams, l e a d i n g t o a s s o c i a t i v e ~ o n ~ s a t ~ o n . The r e l a t i o n w ~ t h o t h e r i n e l a s t i c processes, such as energy p o o l i n g , i s b r i e f l y discussed.

I 1

-

BLOCH EQUATIONS AND GENERALISED RATE EQUATIONS

An e s s e n t i a l i n g r e d i e n t i n t h e p o l a r i s a t i o n a n a l y s i s o f i n e l a s t i c c o l l i s i o n s w i t h l a s e r - e x c i t e d i n i t i a l s t a t e s i s t h e s p e c i f i c a t i o n o f t h e s t a t i o n a r y d e n s i t y m a t r i x ue o f t h i s e x c i t e d s t a t e . T h i s d e n s i t y m a t r i x i s a p a r t o f t h e combined d e n s i t y m a t r i x u o f t h e e x c i t e d s t a t e and t h e ground s t a t e , which a r e coupled by t h e r a d i a t i o n f i e l d . The l a s e r f i e l d i s d e s c r i b e d as a c l a s s i c a l o s c i l a t i n g f i e l d w i t h a f r e q u e n c y d i s t r i b u t i o n c e n t e r e d around uL. I n t h e r o t a t i n g - w a v e a p p r o x i m a t i o n t h e a t o m - f i e l d c o u p l i n g i s g i v e n by

w i t h d d e f i n e d as i n (1.3) i n terms o f t h e p o l a r i s a t i o n o f t h e e x c i t i n g r a d i a t i o n . The t i m e dependence o f E+ and

E-

allow: f o r a m p l i t u d e and f r e q u e n c y m o d u l a t i o n , and t h e r e b y a f i n i t e l a s e r bandwidth. We i n t r o d u c e t h e d i m e n s i o n l e s s o p e r a t o r s Qu by s p e c i f y i n g t h e i r m a t r i x elements as a Clebsch-Gordan c o e f f i c i e n t , a c c o r d i n g t o

f o r u =

-

^1, ^0, ^1,w i t h Fe and F t h e a n g u l a r momenta o f t h e e x c i t e d s t a t e and t h e ground s t a t e . I n a r o t a t i n g frams, t h e L i o u v i l l e e q u a t i o n f o r t h e complete d e n s i t y m a t r i x a t a k e s t h e f a m i l i a r f o r m o f g e n e r a l i s e d B l o c h e q u a t i o n s /6/, w h i c h r e a d

Here w i s t h e t r a n s i t i o n frequency between t h e s t a t e s e and g. Spontaneous e m i s s i o n i s meagured b y t h e E i n s t e i n c o e f f i c i e n t A, and g i v e s a g a i n t e r m i n t h e e q u a t i o n f o r

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Cl-100 JOURNAL DE PHYSIQUE

u ( t ) , and a l o s s t e r m i n t h e o t h e r e q u a t i o n s . I n monochromatic r a d i a t i o n , t h e p 8 s i t i v e - and n e g a t i v e - f r e q u e n c y p a r t s E+ and E- o f t h e f i e l d a r e c o n s t a n t s . The B l o c h equations (2.3) can be used t o d e r i v e g e n e r a l i s e d r a t e e q u a t i o n s f o r t h e p o p u l a t i o n s o f t h e e x c i t e d s t a t e and t h e ground s t a t e b y e l i m i n a t i n g oeg and age, .which i s p o s s i b l e a f t e r some s i m p l i f y i n g assumptions. t h a t a r e v a l i d i n s p e c i a l cases / 7 / . The e q u a t i o n s (2.3) f o r oeg and oge can be f o r m a l l y w r i t t e n as an i n t e g r a l o v e r a, and og i n t h e p a s t , and we f i n d

t

ueg ( t ) =

f

d t ' e x p [ ( i ( u L

-

u0)

-

$A) ( t - t ' ) ]

03

-[E h i

+

( t ' ) do ( t ' ) 9

-

o e ( t l ) d E + ( t l ) ] (2.4)

and an analogous e q u a t i o n f o r

age.

S u b s t i t u t i n g these e q u a t i o n s i n t h e B l o c h e q u a t i o n s f o r ae and a y i e l d s r a t e e q u a t i o n s o f t h e M a s t e r t y p e f o r oe and og, p r o v i d e d t h a t we nay r g p l a c e o e ( t l ) and o ( t o ) by u e ( t ) and u g ( t ) . T h i s ^IS j u s t i f i e d o n l y i f oe and u change n e g l i g ? b l y d u r i n g t h e t i m e o f i n t e r e s t i n t h e i n t e g r a n d . T h i s t i m e o f i n a e r e s t i s r e s t r i c t e d b y t h e spontaneous decay term $A, and p o s s i b l y by t h e f i n i t e bandwidth o f t h e l a s e r r a d i a t i o n , which a r i s e s a f t e r s u b s t i t u t i n g

f o r t h e average o v e r t h e l a s e r f l u c t u a t i o n s , w i t h I ( u ) t h e s p e c t r a l i n t e n s i t y o f t h e l a s e r . For a f i n i t e bandwidth A, t h e r e l e v a n t v a l u e s o f t-t' a r e r e s t r i c t e d t o a t most A-1. When t h e s t i m u l a t e d - t r a n s i t i o n r a t e i s s m a l l e r than t h e maximum o f A and A , we can r e p l a c e o e ( t l ) and o g ( t ' ) by o e ( t ) and o g ( t ) i n t h e i n t e g r a n d , and we o b t a i n r a t e e q u a t i o n s f o r oe and a

,

which do n o t c o n t a i n oeg

These r a t e e q u a t i o n s a r e o b t a i n e d i n a d i & n s i o n l e s s f o r m i f we intro:Ecdeo!~; r e a l parameters a and b b y t h e d e f i n i t i o n

The f a c t o r i n f r o n t o f t h e i n t e g r a l i s s i m p l y t h e r a t i o B/A o f t h e E i n s t e i n c o e f f i - c i e n t f o r s t i m u l a t e d emission and spontaneous emission, and t h e r e a l p a r t o f t h e i n t e g r a l i n (2.6) i s t h e o v e r l a p o f t h e s p e c t r a l i n t e n s i t y o f t h e l a s e r and t h e n a t u r a l p r o f i l e . Hence a i s t h e s a t u r a t i o n parameter d e f i n e d as t h e r a t i o o f t h e s t i m u l a t e d t o t h e spontaneous e m i s s i o n r a t e , and b i s a c o r r e s p o n d i n g l i g h t - s h i f t parameter. Furthermore we i n t r o d u c e t h e dimensionless t r a n s i t i o n o p e r a t o r

w i t h

Go

t h e s p h e r i c a l u n i t v e c t o r s

One n o t i c e s t h a t Q becomes equal t o Q d e f i n e d i n (2.2) when t h e l a s e r p o l a r i s a t i o n i s equal t o t h e s p h e r i c a l u n i t vectorol?

.

However t h e d e f i n i t i o n (2.7) a l l o w s f o r an a r b i t r a r y p o l a r i s a t i o n , t h a t i s n o t R e c e s s a r i l y c i r c u l a r o r 1 in e a r . Obviously, t h e d i p o l e component d i s p r o p o r t i o n a l t o t h e o p e r a t o r , due t o t h e Wigner-Eckart

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theorem /2/.

The generalised equations f o r oe and a take t h e form 9

These equations a r e given i n a dimensionless form, and they c o n t a i n o n l y m u l t i - p l i c a t i o n s w i t h t h e operators Q and Q

',

which a r e d e f i n e d i n (2.2). For l i n e a r o r c i r c u l a r p o l a r i s a t i o n , Q i s gqual

t g

one o f t h e Q when t h e q u a n t i s a t i o n a x i s i s p r o p e r l y chosen. One immediately checks t h a t i n t R a t case t h e d e n s i t y m a t r i c e s ag and ae a r e diagonal i n t h e s t a t e s

I M

> and [Me>, provided t h a t they were d i a - gonal a t some i n i t i a l time. Hence i n t h g case o f l i n e a r o r c i r c u l a r p o l a r i s a t i o n t h e s t a t e o f t h e atom i s e n t i r e l y described i n terms o f p o p u l a t i o n s o f magnetic substates, and coherences do n o t occur. The Master equation f o r these p o p u l a t i o n s i s d i r e c t l y e x t r a c t e d from (2.9) i n t h i s case, and we f i n d t h e usual equations / 3/

-f -+

f o r E = u

,

where N denotes t h e p o p u l a t i o n s o f t h e e x c i t e d substates, and n t h e ground s t s t e . One n o t i c e s t h a t t h e f i r s t terms on t h e right-hand s i d e o f (2.9) do n o t c o n t r i b u t e i n t h i s case, s i n c e t h e r e l e v a n t operators a r e a l l diagonal and the commutators disappear.

I n t h e case o f e l l i p t i c a l p o l a r i s a t i o n i t i s n o t g e n e r a l l y p o s s i b l e t o choose a b a s i s f o r which a and og a r e diagonal a t a l l times, and we have t o use t h e o p e r a t o r equations 72.9), r a t h e r then t h e equations (2.10) f o r populations. This i s an i m p o r t a n t remark, since, as we s h a l l see, t h e use o f e l l i p t i c a l l y p o l a r i s e d l i g h t can be necessary f o r a f u l l determination of t h e d e t e c t i o n o p e r a t o r G i n t h e case o f c o l l i s i o n experiments w i t h l a s e r - e x c i t e d i n i t i a l s t a t e s . The commutators i n (2.9) r e f l e c t energy s h i f t s , which g i v e r i s e t o an e f f e c t i v e Hamiltonian f o r t h e e x c i t e d s t a t e and t h e ground s t a t e . The l i g h t - s h i f t parameter b vanishes when the l a s e r spectrum i s symmetrically d i s t r i b u t e d around the resonance frequency w

0 '

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C1-102 JOURNAL DE PHYSIQUE

We demonstrated t h a t the g e n e r a l i s e d r a t e equations (2.9) a r e v a l i d when the s t i m u l a t e d - t r a n s i t i o n r a t e s a r e small e i t h e r compared w i t h t h e n a t u r a l decay r a t e o r w i t h t h e l a s e r bandwidth. T h i s i s t r u e when t h e s a t u r a t i o n parameter a i s small compared w i t h 1 o r w i t h A. I n many cases o f p r a c t i c a l i n t e r e s t , where t h e atoms a r e e x c i t e d by an i n t e n s e narrow-band l a s e r source, t h i s c o n d i t i o n i s n o t f u l f i l l e d . Nevertheless, eqs. (2.9) can s t i l l be used t o evaluate t h e s t a t i o n a r y l i m i t o f t h e d e n s i t y m a t r i c e s oe and o

.

This can be j u s t i f i e d by i n s p e c t i n g eq. (2.4) and t h e analogous equation f8r a g e ( t ) . I n t h e s t a t i o n a r y l i m i t , t h e time dependence o f o and o e i n the i n t e g r a n d o f (2.4) can be omitted. S u b s t i t u t i o n o f t h e r e s u l t i n f h e f i r s t two equations (2.3) then leads t o t h e same s t a t i o n a r y values o f oe and o as t h e r a t e equations (2.9).

Hence t h e Bloch equations (2.3), and t h e generafised r a t e equations (2.9) y i e l d the same d e n s i t y m a t r i c e s oe and G i n t h e s t a t i o n a r y s t a t e , even though t h e time-dependent approach o f t h e sta?ionary s t a t e can be q u i t e d i f f e r e n t according t o these two s e t s o f equations. This conclusion i s o f g r e a t importance t o o u r present discussion, s i n c e t h e atoms u s u a l l y have s u f f i c i e n t t i m e t o reach a steady s t a t e i n the l a s e r f i e l d b e f o r e t h e beginning o f t h e c o l l i s i o n . This steady s t a t e i s c o r r e c t l y p r e d i c t e d by eqs. (2.9) even f o r an i n t e n s e monochromatic l a s e r .

When t h e p o l a r i s a t i o n

-:

i s equal t o one o f the s p h e r i c a l u n i t vectors

&,

ⁱ.e.

f o r l i n e a r o r c i r c u l a r p o l a r i s a t i o n , the steady s t a t e i s diagonal, and the s t a t i o n a r y p o p u l a t i o n s a r e determined by eqs. (2.10) by s e t t i n g t h e r i g h t - h a n d sides equal t o zero. The s t a t i o n a r y d i s t r i b u t i o n o f t h e e x c i t e d atoms over t h e magnetic substates i s independent o f the i n t e n s i t y i n t h i s case, as has been shown b e f o r e /3,7/. When F = F

-

^1,no s t a t i o n a r y p o p u l a t i o n o f t h e e x c i t e d s t a t e i s p o s s i b l e when .Gi?? ?e= fdhen Fe = F a c i r c u l a r p o l a r i s a t i o n pumps a1 1 t h e azoms t o t h e ground s t a t e w i t h M =

+ ?:

^{f o r}^G⁼

+

1. L i n e a r p o l a r i s a t i o n ( z = u,) leads t o an i s o t r o p i c s t a t i 8 n a r y d i s t r i b u t i o n over t h e e x c i t e d sub- s t a t e s when Fe = F i s a h a l f i n t e g e r , and no s t a t i o n a r y e x c i t e d atoms a r e p o s s i b l e when Fe = g ~ i s i n t e g e r /8/. F i n a l l y , when Fe = Fg + 1, c i r c u l a r pola- r i s a t i o n pumps t h e aQoms t o t h e s t a t e s M =

+

F

,

M =

+

Fe f o r u =

+

1. L i n e a r p o l a r i s a t i o n produces a s t a t i o n a r y e x c i t z d d i s t ? i b u ? i o n - f o r Fe = Fg ? 1 where / 3/

The t o t a l e x c i t a t i o n f r a c t i o n depends on t h e s a t u r a t i o n parameter a, b u t the r e l a t i v e d i s t r i b u t i o n i s f u l l y s p e c i f i e d by t h e p o l a r i s a t i o n

z

i n these cases.

The use o f e l 1 ip t i c a l p o l a r i s a t i o n provides an a d d i t i o n a l p o s s i b i 1 i t y f o r modi- f y i n g t h e e x c i t e d s t a t e , as does t h e use o f p a r t i a l l y p o l a r i s e d r a d i a t i o n .

I 1 1

-

POLARTSATION-DEPENDENT RATE OF INELASTIC COLLISIONS

We consider t h e case o f i n e l a s t i c c o l l i s i o n s between two l a s e r - e x c i t e d atoms A and

B y

which we s h a l l t a k e t o be i d e n t i c a l . The magnetic substates o f t h e e x c i t e d s t a t e s a r e denoted as ]MA M p . The p r o d u c t i o n r a t e o f f i n a l - s t a t e products can be expressed as i n eq. (1.6), and we o b t a i n t h e p r o p o r t i o n a l i t y r e - l a t i o n

w i t h t h e d e t e c t i o n operator

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We assume t h a t t h e p o s s i b l e substates o f t h e f i n a l s t a t e are n o t d i s c r i m i n a t e d , and a summation over these substates i s i m p l i e d i n (3.2). When b o t h atoms a r e e x c i t e d by t h e same l a s e r beam, the two d e n s i t y m a t r i c e s UA and ag a r e i d e n t i c a l . F i n a l l y we consider the case t h a t t h e d i r e c t i o n o f t h e s c a t t e r e d p a r t i c l e s a f t e r t h e c o l l i s i o n remains unobserved, so t h a t the i n t e g r a t i o n over t h e d i r e c t i o n

n

o f t h e f i n a l r e l a t i v e v e l o c i t y extends over a l l d i r e c t i o n s . Then t h e symmetry p r o p e r t i e s o f the d e t e c t i o n o p e r a t o r G simply r e f l e c t t h e symmetry o f t h e d i s t r i - b u t i o n o f i n i t i a l r e l a t i v e v e l o c i t i e s . I n t h e case o f a s i n g l e atomic beam o r two counterrunning beams, G has an a x i a l symmetry. Then i t obeys t h e symmetry r e l a - t i o n s

i M M _{A B}l G I M A 1 MB1> = 0 f o r MA

+

MB

#

MA'

+

MB1

,

_(3.3)

where we assumed t h a t t h e q u a n t i s a t i o n a x i s f o r the magnetic substates i s taken along t h e beam a x i s . From r e f l e c t i o n symmetry about a plane through t h i s a x i s we o b t a i n the r e l a t i o n s

Since t h e c o l l i d i n g atoms a r e i d e n t i c a l , G obeys t h e r e l a t i o n s

F i n a l l y , t h e h e r m i t i c i t y of G y i e l d s the e q u a l i t i e s

The c o l l i s i o n process can be assumed t o be u n a f f e c t e d by t h e s t a t e o f the n u c l e a r spins, and i t i s s u f f i c i e n t t o take f o r a and a the p a r t i a l t r a c e over t h e n u c l e a r spins. The magnetic quantum numbers MA an% MB de!ine t h e o r i e n t a t i o n o f t h e elecL t r o n i c angular momentum J o f t h e atoms. Since t h e n u c l e a r s p i n has s u f f i c i e n t time t o couple t o t h e e l e c t r o n i c angular momentum J d u r i n g t h e o p t i c a l pumping, i t i s e s s e n t i a l t o describe t h e l a s e r e x c i t a t i o n i n terms o f the h y p e r f i n e l e v e l s . I n fact, i t i s p r a c t i c a l t o tune t h e e x c i t i n g l a s e r t o a s p e c i f i c h y p e r f i n e t r a n s i t i o n , which leads t o a p o l a r i s e d p o p u l a t i o n o f t h e h y p e r f i n e l e v e l o f t h e upper s t a t e as des- c r i b e d i n t h e previous s e c t i o n . The d e n s i t y m a t r i x ^OA= u then f o l l o w s by t a k i n g t h e p a r t i a l t r a c e o f t h e d e n s i t y m a t r i x f o r the sublevel

pe.

The number o f independent parameters determining G f o r a given value o f t h e e l e c t r o - n i c angular momentum J can be c a l c u l a t e d from t h e symmetry r e l a t i o n s (3.3)

-

(3.6).

This number amounts t o (J+1)(4J2+5J+3) / 3 f o r i n t e g e r J-values, and t o (2J+1) (8J2+14J+9)/12 f o r ha1 f - i nteger J - v a l ues. Thi's determines t h e number o f independent measurements f o r a f u l l determination o f G. Both t h e p o l a r i s a t i o n and the beam d i r e c t i ' o n o f t h e e x c i t i n g l a s e r can be v a r i e d t o produce a d i f f e r e n t value of t h e d e n s i t y matrices. A d e t a i l e d a n a l y s i s o f t h e i n f o r m a t i o n e x t r a c t e d by measurement o f t h e production r a t e o f t h e f i n a l s t a t e w i t h s p e c i f i c p o l a r i s a t i o n charac-

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C1-104 JOURNAL DE PHYSIQUE

t e r i s t i c s o f t h e e x c i t i n g l i g h t i s g i v e n i n a r e c e n t paper /8/, where we focused on the case o f a s s o c i a t i v e i o n i s a t i o n . The same d e s c r i p t i o n i s a p p l i c a b l e t o any i n e l a s t i c process, provided t h a t t h e t o t a l production r a t e o f f i n a l - s t a t e p a r t i c l e s can be measured.

A f i r s t measurement i n o u r l a b o r a t o r y on p o l a r i s a t i o n e f f e c t s on a s s o c i a t i v e i o n i s a - t i o n o f e x c i t e d sodium atoms was r e c e n t l y published 151. I n t h e n e x t s e c t i o n we w i l l discuss some more extensive measurements.

I V

-

MEASUREMENTS ON ASSOCIATIVE IONISATION

We have performed measurements on t h e p o l a r i s a t i o n e f f e c t o f a s s o c i a t i v e i o n i s a t i o n o f l a s e r - e x c i t e d sodium atoms, according t o t h e r e a c t i o n equation

Our e a r l i e r measurements were performed i n a s i n g l e atomic beam 151, where c o l l i - sions a r e due t o t h e spread i n t h e atomic v e l o c i t i e s i n t h e beam. A remarkably s t r o n g p o l a r i s a t i o n e f f e c t was observed, when we detected t h e i o n production r a t e as a f u n c t i o n o f t h e angle e between t h e d i r e c t i o n o f t h e l i n e a r p o l a r i s a t i o n and the atomic-beam a x i s : t h e i o n p r o d u c t i o n was maximal f o r e = 0, and minimal f o r 0 = 90°, w i t h a r a t i o 1.5. The d e t e c t i o n o p e r a t o r G contains 16 independent parameters f o r the P312-state i n t h e a x i a l l y symmetric s i t u a t i o n . I f t h e r e a c t i o n (4.1)

i s i n s e n s i t i v e t o t h e e l e c t r o n spin, then e f f e c t i v e l y G a c t s o n l y on t h e o r b i t a l angular momentum L = 1 o f the e x c i t e d s t a t e , which would s t i l l y i e l d 8 independent parameters. The i o n i s a t i o n s i g n a l as a f u n c t i o n o f e has t h e form /5,8/

R(e) = Ro

+

R1 cos 20

+

R2 cos 4e

,

(4.2)

w i t h Ro, R1 and R2 ? i n e a r combinations o f m a t r i x elements o f G. We adopted t h e s i m p l i f y i n g assumption t h a t t h e o f f - d i a g o n a l elements <MA MBIGIMA'M~'> w i t h M

#

MAi, MB-# M

'

a r e small compared t o the diagonal ones, as a r e s u l t o f t h e

i f f t e g r a t i o n I n (g.2) over products o f amplitudes w i t h independent phases. The diagonal elements have t h e s i g n i f i c a n c e o f M-dependent i o n i s a t i o n r a t e s

which obeys t h e symmetry r e l a t i o n s

Neglecting t h e o f f - d i a g o n a l terms gives r i s e t o a d i r e c t determination o f t h r e e independent combinations o f r a t e s from t h e measured values o f Roy R1 and R2. These r e l a t i o n s f o r t h e P3/2 s t a t e a r e /5,8/

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w i t h t h e a b b r e v i a t i o n s

I n t h e absence o f p o l a r i s a t i o n e f f e c t s , we would have K 1 = K2 = K

,

and R and R

should have been zero. I n f a c t , our f i r s t measurements i n d i c a t e d ghat

~ ~ / k ~

^%^1.g.

whereas K2 was s i g n i f i c a n t l y smaller than e i t h e r K1 o r K3 IVa

-

Counterrunning beams

Recently these measurements have been extended i n two respects. F i r s t o f a l l , the same measurement was performed w i t h two counterrunning atomic beams, so t h a t t h e a x i a l symmetry o f t h e c o l l i s i o n system was preserved. By s u b t r a c t i n g t h e i o n i s a t i o n s i g n a l w i t h o n l y one o f t h e two beams present, i t i s p o s s i b l e t o i s o l a t e t h e r a t e s K f o r a s s o c i a t i v e i o n i s a t i o n between two atoms, one from each o f t h e two counterrunning beams. Hence t h e average r e l a t i v e v e l o c i t y was much l a r g e r than i n t h e case o f a s i n g l e beam. I t t u r n s o u t t h a t t h e r a t i o K1/K3 becomes 2.0, which i s s i g - n i f i c a n t l y l a r g e r than t h e value 1.6 i n t h e case o f a s i n g l e beam. D e t a i l s o f t h e measurements w i l l be published elsewhere. Here we emphasise t h e remarkable f a c t t h a t t h e p o l a r i s a t i o n e f f e c t s seem t o g e t more pronounced when t h e r e l a t i v e v e l o c i t y i s increased. A t f i r s t s i g h t t h i s i s s u r p r i s i n g . One expects p o l a r i s a t i o n e f f e c t s t o be present when t h e various molecular s t a t e s a r e unequally populated, and when t h e process o f a s s o c i a t i v e i o n i s a t i o n has unequal p r o b a b i l i t i e s f o r t h e d i f f e r e n t quasi-molecul a r s t a t e s o f t h e c o l 1 i s i o n complex. The non-uni forni p o p u l a t i o n d i s t r i

-

b u t i o n must be preserved d u r i n g t h e c o l l i s i o n i n o r d e r t o produce p o l a r i s a t i o n e f f e c t s i n t h e i o n production, and t h i s p r e s e r v a t i o n o f a n i s o t r o p y i s favoured by a d i a b a t i c behaviour o f t h e molecular s t a t e . However, a d i a b a t i c i t y must be assumed t o decrease f o r i n c r e a s i n g r e l a t i v e v e l o c i t y , whereas t h e p o l a r i s a t i o n e f f e c t i s observed t o increase. On t h e o t h e r hand, a t h i g n e r r e l a t i v e v e l o c i t y t h e e l e c t r o n spins can be expected t o decouple from t h e i n t e r n u c l e a r a x i s , even when t h e o r b i t a l angular momentum o f the two valence e l e c t r o n s i s s t i l l coupled a d i a b a t i c a l l y t o t h e a x i s , and t h e question a r i s e s whether t h e decoupling o f the s p i n can e x p l a i n the increase o f t h e p o l a r i s a t i o n e f f e c t . T h i s may be t h e case when t h e i o n i s a t i o n process occurs t o a l a r g e e x t e n t by t h e molecular s t a t e s which do have non-zero spin, i .e. t h e t r i p l e t s t a t e s .

IVb

-

C i r c u l a r p o l a r i s a t i o n

Another extension o f t h e experiment was t h e use o f c i r c u l a r l y p o l a r i s e d l a s e r l i g h t t o e x c i t e t h e atoms. The l a s e r beam remained d i r e c t e d orthogonal t o t h e atomic beam.

I n t h e s t a t i o n a r y s t a t e , t h e atoms a r e i n the pure s t a t e lJM> = 13 3>, when t h e l a s e r beam i s s e l e c t e d as q u a n t i s a t i o n a x i s . I f we take the z - a x i ? g l o n g t h e atomic beam, the steady s t a t e can be represented as t h e pure s t a t e

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C1-106 JOURNAL DE PHYSIQUE

since t h i s i s t h e e i g e n s t a t e o f J w i t h eiqenvalue 7. 3 Hence t h e p o p u l a t i o n s o f t h e

x 3 1 1 3

f o u r n a g n e t i c substates w i t h M =

-, , , - - ,

have t h e r a t i o ' s 1 : 3 : 3 : 1.

L L L L

When t h e o f f - d i a g o n a l terms o f G a r e ignored, we can p r e d i c t the i o n s i g n a l by using t h e r a t e s K1, K2 and K3 as determined from t h e case w i t h l i n e a r l y p o l a r i s e d l a s e r l i g h t . Both i n t h e case o f a s i n g l e atom beam and two counterrunning beams a s i g n i f i c a n t d e v i a t i o n was observed: t h e measured r a t e w i t h e x c i t a t i o n w i t h c i r - c u l a r l y p o l a r i s e d l i g h t was l a r g e r than t h e c a l c u l a t e d value based on t h e observa- t i o n w i t h l i n e a r l y p o l a r i s e d l i g h t . I n f a c t , t h e i o n s i g n a l w i t h c i r c u l a r l y pola- r i s e d l i g h t was s i g n i f i c a n t l y l a r g e r than t h e maximum value ( f o r 8 = 00) o f t h e i o n s i g n a l w i t h l i n e a r l y p o l a r i s e d l i g h t , where t h e s i g n a l s were normalised t o the t o t a l observed e x c i t e d - s t a t e p o p u l a t i o n .

This r e s u l t seems t o i n d i c a t e t h a t our n e g l e c t i n g the o f f - d i a g o n a l terms o f G i s n o t j u s t i f i e d . This would mean t h a t the coherences between magnetic substates con- t r i b u t e s i g n i f i c a n t l y t o t h e i o n p r o d u c t i o n . Note t h a t t h e coherences between these s t a t e s a r e maximal f o r e x c i t a t i o n w i t h c i r c u l a r l y p o l a r i s e d l i g h t , s i n c e then the e x c i t e d s t a t e i s a pure s t a t e .

I n t h e case o f c i r c u l a r p o l a r i s a t i o n , t h e two e l e c t r o n s p i n s i n t h e i n i t i a l s t a t e are b o t h p a r a l l e l t o t h e l a s e r beam, so t h a t o n l y molecular t r i p l e t terms a r e produced. The h i g h i o n p r o d u c t i o n i n t h i s case suggests t h a t t h e t r i p l e t s t a t e s p l a y an i m p o r t a n t p a r t i n the process o f a s s o c i a t i v e i o n i s a t i o n . T h i s i s n o t q u i t e what one would expect, s i n c e t h e t r i p l e t s t a t e s have an antisymmetric wave f u n c t i o n o f t h e two valence e l e c t r o n s , which tends t o d i m i n i s h the Coulomb i n t e r a c t i o n needed f o r t h e r e a c t i 6 n (4.1).

An e x p l a n a t i o n o f these r e s u l t s i n terms o f the dynamics o f the c o l l i s i o n i s n o t easy, s i n c e t h e p o t e n t i a l curves o f t h e system o f two e x c i t e d Na-atoms are n o t known i n any d e t a i l

.

I n f a c t , t h e r e a r e 12 quasimolecular terms ending up i n t h e se a r a t e d atoms Na(3P) _3.ZU,

P

_I.Z:, _'nu, _{3 ~ g ,}

+

Na(3P), w i t h t h e designations jaU. These terms produce 21 pote8;ial

Ic+ lei,

curves. ~ a e

lc;,

I n

,

Inu, s t r o n g lag, p o l a r i s a t i o n e f f e c t s suggest t h a t o n l y a few o f these p a r t i c i p a t e i n t h e i o n i s a t i o n process, probably s i n c e o n l y a few cross w i t h t h e ground-state p o t e n t i a l curve o f the molecular i o n ~a;.

The importance o f t h e o f f - d i a g o n a l terms o f G can be d e r i v e d from a complete e x p e r i - ment, u s i n g e l l i p t i c a l l y p o l a r i s e d l i g h t . The a n a l y s i s o f t h i s experiment r e q u i r e s t h e c a r e f u l e v a l u a t i o n o f the e x c i t e d - s t a t e d e n s i t y m a t r i x , along t h e l i n e s o f s e c t i o n 11. This we i n t e n d t o do i n t h e near f u t u r e . For t h e moment we may conclude from t h e observations t h a t t h e p o l a r i s a t i o n e f f e c t s increase when the v e l o c i t y i s increased from subthermal t o thermal values, and t h a t t h e o f f - d i a g o n a l terms o f G cannot be ignored. Since t h e same p o t e n t i a l curves g i v e r i s e t o o t h e r energy-trans- f e r processes, such as energy pooling, we can expect t h a t s i m i l a r conclusions h o l d f o r these cases.

REFERENCES

/I/ FAN0 U., MACEK J.H., Rev.Mod.Phys. 45 (1973) 553.

/2/ FAN0 U., RACAH G., I r r e d u c i b l e T e n s o r i a l Sets (Academic, New York, 1959).

/3/ MACEK J., HERTEL I . V . , J.Phys. 67 (1974) 2173.

/4/ DE VLIEGER G.J.N.E., HEIDEMAN H.G.M., VAN ECK J., NIENHUIS G., J-Phys. 815 (1982) L345.

/5/ KIRCZ J.G., MORGENSTERN R., NIENHUIS G . , Phys.Rev.Lett. 48 (1982) 610.

/6/ COHEN-TANNOUDJI C., i n : F r o n t i e r s i n Laser Spectroscopy, R. Balian, S. Haroche, S. Liberman, eds. (North Holland, Amsterdam, 1977) 3.

/7/ DUCLOY M., Phys.Rev. A8 (1973) 1844.

/8/ NIENHUIS G., Phys.Rev. A26 (1982) 3137.