INELASTIC COLLISIONS OF DRESSED ATOMS

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INELASTIC COLLISIONS OF DRESSED ATOMS

D. Pritchard, A. Migdall, R. Walkup

To cite this version:

D. Pritchard, A. Migdall, R. Walkup. INELASTIC COLLISIONS OF DRESSED ATOMS. Journal de Physique Colloques, 1985, 46 (C1), pp.C1-269-C1-274. �10.1051/jphyscol:1985126�. �jpa-00224499�

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INELASTIC COLLISIONS OF DRESSED ATOMS

D.E. P r i t c h a r d , A.L. Migdall and R.E. Walkup

Research Laboratory o f EZectronics and Department of Physics, Massachusetts I n s t i t u t e o f TechnoZogy, Cambridge, Massachusetts 02139, U.S.A.

R & ~ ~ ; - Des atomes B deux niveaux soumis 21 un rayonnement l a s e r monochro- matique peuvent Gtre d e c r i t s B l ' a i d e d l Q t a t s "habil16s1', q u i s o n t des s u p e r p o s i t i o n s c o h e r e n t e s , dependant du temps, de c e s deux niveaux. Nous e t a b l i s s o n s que l e s c o l l i s i o n s peuvent c o u p l e r d i r e c t e m e n t c e s atomes h a b i l l e s 2 un t r o i s i e m e niveau, b i e n s e p a r e en e n e r g i e des deux niveaux coupl6s p a r l e rayonnement, e t nous montrons comment exprimer l e taux de t e l l e s c o l l i s i o n s en f o n c t i o n de taux de c o l l i s i o n s r e l a t i f s aux niveaux de l'atome nu. Des donnees experimentales v e r i f i a n t ces p r e d i c t i o n s s o n t p r e s e n t e e s pour des c o l l i s i o n s Na-He.

A b s t r a c t - TWO le v e l atoms which a r e s u b j e c t t o monochromatic l a s e r r a d i a t i o n may be d e s c r i b e d u s i n g d r e s s e d s t a t e s , which a r e time-varying c o h e r e n t s u p e r p o s i t i o n s of t h e s e two l e v e l s . We argue t h a t c o l l i s i o n s can t a k e t h e s e d r e s s e d atoms d i r e c t l y t o a t h i r d l e v e l which i s w e l l s e p a r a t e d i n energy from t h e two r a d i a t i v e l y coupled l e v e l s and we show how t o e x p r e s s t h e r a t e f o r such c o l l i s i o n s i n terms of c o l l i s i o n r a t e s f o r t h e b a r e atomic l e v e l s . Data which v e r i f y t h e s e p r e d i c t i o n s a r e p r e s e n t e d f o r Na-He c o l l i s i o n s .

Most a t t e n d e e s a t t h i s meeting come from one o f t h r e e l a r g e l y d i s t i n c t s u b f i e l d s of Atomic and Molecular physics: atomic c o l l i s i o n s , l i n e broadening, and l a s e r spectroscopy. The t o p i c of t h i s meeting - r a d i a t i v e c o l l i s i o n s - n a t u r a l l y b r i n g s t h e s e t h r e e camps t o g e t h e r , because workers i n each camp have some r i g h t t o claim t h a t t h e s e c o l l i s i o n s a r e w i t h i n t h e i r s u b f i e l d . For example, t h e c l a s s i c r a d i a - t i v e c o l l i s i o n ,

which is o f t e n c a l l e d " l i g h t induced c o l l i s i o n a l energy t r a n s f e r " , h a s been s t u - d i e d l a r g e l y by l a s e r p e o p l e , belongs h i s t o r i c a l l y i n l i n e broadening where p r o c e s s e s i n v o l v i n g c o l l i s i o n induced a b s o r p t i o n have p r e v i o u s l y been s t u d i e d , and i s most e a s i l y understood u s i n g t h e language developed f o r c o l l i s i o n s i n v o l v i n g c r o s s i n g of i n t e r m o l e c u l a r p o t e n t i a l curves. C e r t a i n l y one of t h e c h i e f b e n e f i t s of t h i s meeting h a s been t o b r i n g d e v o t e e s of each s p e c i a l t y t o g e t h e r s o t h a t they can begin t o comprehend o t h e r ways of looking a t t h e same phenomena.

The h i s t o r i c a l o r i g i n s of our work i l l u s t r a t e t h e i n t e r p l a y of t h e s e t h r e e sub- f i e l d s , and a l s o s e r v e t o i n t r o d u c e and motivate our p r e s e n t a t i o n . It began w i t h our experiments t o use t h e Doppler s h i f t t o s e l e c t t h e v e l o c i t y of e x c i t e d atoms undergoing i n e l a s t i c f i n e s t r u c t u r e c o l l i s i o n s /I/, and our d e s i r e t o push t h i s technique a s f a r a s p o s s i b l e i n t o t h e wing of t h e l i n e i n o r d e r t o e x c i t e atoms with a s g r e a t a v e l o c i t y a s p o s s i b l e . A s t h e detuning from l i n e c e n t e r i s

i n c r e a s e d , atoms with l a r g e r and l a r g e r v e l o c i t i e s a r e r e s o n a n t l y e x c i t e d , b u t t h e number of atoms moving with t h e n e c e s s a r y speed d e c r e a s e s extremely r a p i d l y , a s r e q u i r e d by t h e Maxwell-Boltzmann d i s t r i b u t i o n . Beyond a r a t h e r sudden t r a n s i t i o n p o i n t , t h e number of atoms f a s t enough t o be i n resonance h a s decreased s o f a r

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985126

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C1-270 JOURNAL DE PHYSIQUE

t h a t most of t h e l i g h t is absorbed by t h e l a r g e number of slow atoms n e a r t h e c e n t e r of t h e v e l o c i t y d i s t r i b u t i o n . These slow atoms a r e n o t r e s o n a n t l y e x c i t e d , of course, b u t t h e l i g h t which t h e y s c a t t e r ( i n t h e tail of t h e i r Lorentzian resonance e x c i t a t i o n f u n c t i o n ) i s e s s e n t i a l l y i n d i s t i n g u i s h a b l e from r e s o n a n t l y s c a t - t e r e d l i g h t . The i n t e n s i t y of t h i s l i g h t f a l l s o f f only a s t h e s q u a r e of t h e detuning, much slower t h a n t h e l i g h t from atoms i n t h e v e l o c i t y s e l e c t e d p o r t i o n of t h e Maxwell-Boltzmann d i s t r i b u t i o n , and hence it becomes dominant a t d e t u n i n g s beyond 4 o r 5 Doppler l i n e w i d t h s . These atoms may be r e f e r r e d t o a s being v i r t u - a l l y e x c i t e d , and our i n t e r e s t i n t h e i r c o n t r i b u t i o n t o t h e i n e l a s t i c c o l l i s i o n r a t e f i r s t a r o s e i n t h i s c o n t e x t .

I n s o f a r a s Doppler v e l o c i t y s e l e c t i o n experiments a r e concerned, we now understand t h a t t h i s p r o c e s s imposes t h e u l t i m a t e l i m i t t o t h e t h e o r e t i c a l l y a t t a i n a b l e velo- c i t y ; however, a s a r e s u l t of t h e work r e p o r t e d h e r e , we know how t o c o r r e c t f o r its e f f e c t s a t most v e l o c i t i e s which may be a t t a i n e d i n p r a c t i c a l experiments /2/.

A t r e a t m e n t of i n e l a s t i c c o l l i s i o n s of v i r t u a l l y e x c i t e d atoms which i s p a r t i c u - l a r l y germaine t o t h e v e l o c i t y s e l e c t i o n problem has been p r e s e n t e d elsewhere /3/.

I n t h e p r e s e n t a t i o n h e r e , we use a d r e s s e d atom approach /4/ which makes t h e col- l i s i o n p h y s i c s c l e a r e r a t t h e expense of being i n v a l i d f o r small detunings.

I n view of t h e t r i p a r t a t e n a t u r e of t h i s audience noted above, we now g i v e our p r e s e n t a t i o n a t i t l e s u i t a b l e t o each group. For t h e l a s e r s p e c t r o s c o p i s t s ,

"Transfer from d r e s s e d atom t o bare l e v e l " ; f o r t h e c o l l i s i o n a l i s t s , "Fine s t r u c - t u r e changing c o l l i s i o n s of v i r t u a l l y e x c i t e d atoms"; and f o r t h e l i n e broadeners,

" R e d i s t r i b u t i o n of weak o f f - r e s o n a n t r a d i a t i o n by i n e l a s t i c c o l l i s i o n s " . By what- e v e r name you c a l l it, t h e p r o c e s s is t h e same: l i g h t from t h e p l e v e l of Na ( ~ 2 l i n e ) i s monitored a s a l a s e r i s tuned n e a r t o t h e p level3(& l i n e ) ; D2 l i g h t which i n c r e a s e s l i n e a r l y w i t h t h e p r e s s u r e of t h e J d g t u r b e r g a s s i g n i f i e s a d i r e c t c o l l i s i o n a l t r a n s f e r from t h e v i r t u a l l y e x c i t e d p l e v e l t o t h e - p

l e v e l . 1 /2 312

F i g 1.Level scheme of Na and a schematic r e p r e s e n t a t i o n of t h e i n t e n s i t y of f l u o r e s c e n c e i n o r n e a r t h e D2 l i n e a s a f u n c t i o n of t h e t u n i n g o f t h e e x c i t i n g l a s e r .

The most s a l i e n t f e a t u r e i n t h e D2 l i g h t shown i n F i g 1 i s t h e l a r g e L o r e n t z i a n

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resonance frequency / 5 / , t h e f i r s t being independent of p e r t u r b e r p r e s s u r e and t h e l a t t e r being d i r e c t l y p r o p o r t i o n a l t o t h e product of t h i s p r e s s u r e and t h e quasi- e l a s t i c c o l l i s i o n r a t e f o r t h e p l e v e l . ( o u r d e t e c t o r cannot r e s o l v e t h e s l i g h t frequency d i f f e r e n c e betw& t h e s e , but i s capable of complete r e j e c t i o n of a l l l i g h t whose frequency i s n e a r t h e Dl l i n e . )

The key i s s u e i n our s t u d y was "IS THERE AN INCREASE I N LIGHT AT D2 AS THE LASER TUNES THROUGH Dl DUE TO SINGLE COLLISIONS NITH THE PERTURBER?" The p e r t u r b e r p r e s s u r e dependence of t h i s l i g h t i s c r u c i a l : t h e r e i s c e r t a i n l y no l i g h t a t D2 from Dl e x c i t a t i o n i n t h e absence of c o l l i s i o n s , and t h e r e i s d e f i n i t e l y a two s t e p c o l l i s i o n p r o c e s s ( q u a s i - e l a s t i c t o p and subsequent f i n e s t r u c t u r e chang- i n g t o p ) which w i l l n e c e s s a r i l y have a l & a d r a t i c p r e s s u r e dependence. The l i n e a r p?-dgsure dependence i s t h e s i g n a t u r e of t h e d i r e c t c o l l i s i o n a l t r a n s f e r p r o c e s s under s t u d y here.

THEORY

The theory f o r our s t u d y i s a s t r a i g h t f o r w a r d combination of t h e i n e l a s t i c c o l l i - s i o n s with 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 d r e s s e d atom. We c o n s i d e r t h a t t h e atom has t h r e e l e v e l s : t h e ground s t a t e , g , and two e x c i t e d s t a t e s , e , and e l . For Na t h e s e a r e t h e 3s, 3p s t a t e s r e s p e c t i v e l y . (These " s t a t e s " a r e

s u b s t r u u r e ; we i g n o r e t h i s i n t h e f o l l o w i n g d i s - a c t u a l l y l e v e l s with magnetID2 and 3P242

c u s s i o n and have taken p a i n s e x p e r i m e n t a l l y t o measure f l u o r e s c e n c e a t magic a n g l e s , and i n t h e a n a l y s i s t o use c r o s s s e c t i o n s w i t h proper a v e r a g i n g o v e r t h e s u b l e v e l s s o t h a t t h e r e a r e no o b s e r v a t i o n a l consequences of t h i s f a c t -- ^/3/).

The l a s e r i s detuned from t h e Dl l i n e by only 1% of t h e f i n e s t r u c t u r e s e p a r a t i o n s o only t h e 3s and 3p s t a t e s a r e d r e s s e d by t h e l a s e r - t h e p3r(12 s t a t e i s e s s e n t i a l l y "barett. only type of i n e l a s t i c c o l l i s i o n s which u s t be con- s i d e r e d a r e f i n e s t r u c t u r e c o l l i s i o n s which connect t h e two 3p s t a t e s .

We r e p r e s e n t t h e e f f e c t s of a c o l l i s i o n w i t h t h e p e r t u r b e r g a s by t h e S-matrix, which i s a 3 by 3 m a t r i x f o r t h e l i m i t e d s e t of s t a t e s which we a r e c o n s i d e r i n g .

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To i n c o r p o r a t e t h e presence of t h e l a s e r f i e l d a t frequency u , ^{we use}f i r s t o r d e r p e r t u r b a t i o n theory f o r t h e d r e s s e d s t a t e which c o r r e l a t e s wiEh t h e ground s t a t e a s UR+O,

This e x p r e s s i o n i s v a l i d when t h e d e t u n i n g

i s much l a r g e r t h e n t h e Rabi frequency

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

qe+, which c o r r e l a t e s with l e v e l e i s n e g l i g i b l e . We s t r e s s t h a t @g, (and Jte*

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a l s o ) i s a s i n g l e s t a t e , a c o h e r e n t s u p e r p o s i t i o n of ground and e x c i t e d s t a t e s with coherence which o s c i l l a t e s a t t h e l a s e r frequency. it i s n o t p r o p e r t o r e g a r d t h e second term of Eq. 2 s e p a r a t e l y from t h e f i r s t , a s a v i r t u a l l e v e l i n t o and o u t of which c o l l i s i o n s t r a n s f e r population. Terms i n v o l v i n g $,+. have been omitted from this e x p r e s s i o n because t h e detuning i s ,,-I00 t i m e s large* f o r t h a t f i n e s t r u c t u r e component, and i t s admixture i s n e g l i g i b l e .

The e f f e c t s of c o l l i s i o n s on t h e d r e s s e d system may now be determined. The s t a t e of t h e atom a f t e r a c o l l i s i o n which o c c u r s a t tc i s

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Jlf = 3g*

O R -hLtC -&, L t c

= S g g ~ g + b e s e e $ e +

T

^~ ^~ ¹ ^'^~ ⁰ ^~ ^s

The first two terms, when p r o j e c t e d onto $e, d e s c r i b e t h e u s u a l q u a s i - e l a s t i c l i n e broadening p r o c e s s which o c c u r s w i t h r a t e c o n s t a n t k , ,. (Note t h a t t h e r e i s no longer any coherence between J, and y e f o r t h e ensem%lg because t i s d i f f e r e n t f o r each atom). g

The t h i r d term of Eq. 5 i s t h e p r o c e s s under study. It d e s c r i b e s a d i r e c t t r a n s f e r from 9 , ^{t o}^J, ^,which o c c u r s w i t h t h e u s u a l f i n e s t r u c t u r e changing amplitude reducgd i n mzgnitude by t h e f a c t o r O R / A . Thus t h e t r e a t m e n t p r e d i c t s t r a n s f e r t o e ' w i t h a r a t e c o n s t a n t

where k , ^{i s}t h e u s u a l f i n e s t r u c t u r e t r a n s f e r r a t e c o n s t a n t from e t o e ' (which i s prop&eional t o I S e t 2). T h i s may be c o n t r a s t e d w i t h t h e q u a s i - e l a s t i c l i n e broadening r a t e

where kb is t h e u s u a l q u a s i - e l a s t i c r a t e c o n s t a n t . EXPERIMENT

The e s s e n t i a l measurement of t h e experiment i s t h e r a t i o of f l u o r e s c e n c e i n t h e D2 l i n e t o t h a t from t h e D l l i n e . To make c o n t a c t with t h e preceding t h e o r e t i c a l d i s c u s s i o n we d e s i g n a t e t h e s e I , f o r t h e l e v e l populated by i n e l a s t i c c o l l i s i o n s and I, f o r f l u o r e s c e n c e o r i g i n a % i n g from t h e neighborhood of e. P o p u l a t i o n i s g e n e r a t e d i n e 1 by t h e d i r e c t t r a n s f e r from g* ( t h e p r o c e s s we a r e s t u d y i n g ) a s w e l l a s by i n e l a s t i c c o l l i s i o n s from l e v e l e ; hence

where n is t h e d e n s i t y of p e r t u r b e r s and f i s t h e s t e a d y - s t a t e f r a c t i o n of popu- l a t i o n !?n e. Fluorescence from t h e neighbo$hood of e i n c l u d e s both a Rayleigh component l(OR/L1)2rN] and t h e q u a s i - e l a s t i c broadening term mentioned p r e v i o u s l y ,

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Eq. 9 d e s c r i b e s t h e observed f l u o r e s c e n c e r a t i o i n terms of k There a r e two ways i n which we have checked whether our p r e d i c t i o n f o r k Eq. 6) i s c o r r e c t :

( 1 ) t h e term i n b r a c k e t s i n d i c a t e s t h a t t h e r a t i o w i l l n o t be a l i n e a r f u n c t i o n of n u n l e s s ^Eq.6 is t r u e (because of

f e a"P). P

( 2 ) I f Eq. 6 holds, t h e n t h e p r e d i c t e d r a t i o becomes simply np k e l e / r N .

PRESSURE _tTorr)

F i g 2. Data f o r I , / I f o r v a r i o u s p r e s s u r e s of He g a s , + I s a r e f o r v i r t u a l o f f - e e

r e s o n a n t e x c i t a t i o n s and 0 ' s a r e f o r on-resonant e x c i t a t i o n s .

We n o t e t h a t t h e d a t a shown i n F i g 2 confirm both checks: The p o i n t s (+) f o r off-resonant e x c i t a t i o n show a n extremely l i n e a r dependence on n ( n o t e t h a t f ( A / ~ J ~ ) ~ = 1 a t a p r e s s u r e of "'0.5 t o r r ) . I n a d d i t i o n t h e d a t a fgr 1,,/1 madee u s i n g on-resonant e x c i t a t i o n show e x c e l l e n t agreement w i t h d a t a u s i n g off-resonant

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excitation at all values of n . This confirms that I ,/Ie is equal to n k ,e/rN.

(The on-resonant measurementsPwere made with detuningesufficient to assuze %hat the collision pairs had an average relative kinetic energy of 3kT/2, see /3/ for details). These data are for A = 5 GHz. This work was supported by the National Science Foundation.

References

1. W.D. Phillips and D.E. Pritchard, Phys. Rev. Lett. z, 1254 (1974).

J. Apt. and D.E. Pritchard, J. Phys. B 12, 83 (1979).

2. N. Smith, T.A. Brunner, and D.E. Pritchard, J. Chem. Phys. 3, 467 (1981).

3 . R. Walkup, A. Migdall and D.E. Pritchard, Phys. Rev. A 29, 2651 (1984).

4. C. Cohen-Tannoudji, in Frontiers in Laser Spectroscopy, Session 27 of the Les Houches Summer School, 1975, edited by R. Balian, S. Haroche, and S. Lieberman

(~orth-~olland, Amsterdam, 1976); C. Cohen-Tannoudji and S. Reynaud, J. Phys. B 10, 345 (1977).

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5. J.L. Carlsten, A. Szoke, and M.G. Raymer, Phys. Rev. A 2, 1029 (1977).