NON-PERTURBATIVE METHODS APPLIED TO MULTIPHOTON IONIZATION

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NON-PERTURBATIVE METHODS APPLIED TO MULTIPHOTON IONIZATION

H. Brandi, L. Davidovich, N. Zagury

To cite this version:

H. Brandi, L. Davidovich, N. Zagury. NON-PERTURBATIVE METHODS APPLIED TO MUL- TIPHOTON IONIZATION. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-397-C2-405.

�10.1051/jphyscol:1982230�. �jpa-00221842�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 45, novembre 1982 page G2-397

NON-PERTURBATIVE METHODS APPLIED TO MULTIPHOTON IONIZATION H.S. Brandi , L. Davidovich* and N. Zagury*

*Departamento de Física, Pontificia Universidade Católica, Cx. P. 38071, Rio de Janeiro, Brasil

Departamento de Física, Universidade Federal de Pernambuco, SO. 000, Recife, PE, Brasil

Résumé : Nous discutons ici l'utilisation de méthodes non pertur- batives pour le traitement de l'ionisation atomique. Une atten- tion particulière est portée aux approches du type proposé par Keldysh, où l'ionisation multiphotonique et 1'autoionisation par effet tunnel ont lieu pour des champs très intenses. On montre que ces méthodes sont associées à une certaine catégorie de développements de la matrice T en puissances du potentiel intra- atomique. Ceci permet de suggérer un critère concernant le domai- ne de validité de ces méthodes non perturbatives. Une brève com- paraison des taux d'ionisation des atomes en présence d'une lu- mière polarisée rectilignement ou circulairement est présentée.

Abstract : We discuss the use of non-perturbative methods in the treatment of atomic ionization. Particular attention is given to schemes of the type proposed by Keldysh where multiphoton ioni- zation and tunnel auto-ionization occur for high intensity fields. These methods are shown to correspond to a certain type of expansion of the T-matrix in the intra-atomic potential, in this manner a criterium concerning the range of application of these non-perturbative scheme is suggested. A brief comparison between the ionization rate of atoms in the presence of linearly and circularly polarized light is presented.

1. Introduction - The existence of modern lasers available in frequen- cy ranges varying from the far infrared up to the vacuum-ultraviolet region, as well as focused pulsed laser beams with peak power densi- ties above gigawatts per cm2, greatly stimulated during the past deca- de the investigation of the physics of atoms, molecules and solids in the presence of strong coherent radiation fields. Particular attention has been given, both theoretically and experimentally, to the study of particle scattering, atomic excitation, atomic ionization, and mo- lecule dissociation in the presence of intense electromagnetic fields

(EMF). These subjects have been reviewed by several authors and we mention as reference the works of Delone [1], Lambropoulos [2], Bay- field [3], Ehlotzky [4]. For resonance phenomena, new effects occur at moderate laser intensities. For instance under adequate conditions the Weisskopk-Wigner Lorentzian lineshape gets altered for intensities of the order of lW/cm2, and two extra peaks show up in the spectra constituting the so-called dynamical Stark effect.

In this work we discuss atomic ionization in the presence of strong EMF when resonance processes are not involved. In this case, non-linear effects appear at high laser intensities, which are experi- mentally attained by C 02 and Neodimium lasers. Non-linearity implies that multiphoton processes become significant as compared to one pho- ton absorption; the theoretical description of the phenomena occuring under such condition lies outside the realm of lowest order perturba-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982230

C2-398 JOURNAL DE PHYSIQUE

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

a ) P e r t u r b a t i o n t h e o r y i n t h e a t o m - l a s e r c o u p l i n g H a m i l t o n i a n ( H i ) . The t r a n s i t i o n r a t e a s s o c i a t e d t o n-photon p r o c e s s e s , a r e ob- t a l n e d by e v a l u a t i n g t h e n - t h o r d e r c o n t r i b u t i o n t o t h e e v o l u t i o n o p e r a t o r ( o r e q u i v a l e n t l y t o t h e T - m a t r i x )

t 2

u ( ~ ) ( t ) = (i)-nj:dtnjtndtn-l..

.Io

dtlHI ( t n ) H I ( t n - l l . . .HI ( t

0

The t r a n s i t i o n r a t e f o r a n - t h o r d e r p r o c e s s i s p r o p o r t i o n a l t o E ; ~ , where Eo i s t h e EMF a m p l i t u d e , a s l o n g a s no i n t e r m e d i a t e r e s o n a n c e s o c c u r .

T h e r e f o r e h i g h e r o r d e r p e r t u r b a t i o n t h e o r y p r e d i c t s t h a t a l o g - l o g p l o t o f t h e i o n i z a t i o n r a t e ( o r i o n i z a t i o n p r o b a b i l i t y ) v e r s u s t h e i n t e n s i t y o f t h e l a s e r f i e l d i s a s t r a i g h t l i n e , w i t h s l o p e e q u a l t o t h e m u l t i p l i c i t y o f t h e p r o c e s s ( n ) . I n f a c t t h i s h a s b e e n v e r i f i e d by Lompre e t a 1 [ S ] f o r f i e l d i n t e n s i t i e s up t o 10'' W/cmZ. On t h e o t h e r h a n d , r e c e n t l y , Boreham e t a 1 [ 6 ] have r e p o r t e d d e v i a t i o n s from t h e p r e d i c t i o n s o f p e r t u r b a t i o n t h e o r y f o r f i e l d i n t e n s i t i e s o f t h e o r d e r o f 1016 W/cmz. For v e r y i n t e n s e f i e l d s , p e r t u r b a t i o n t h e o r y becomes q u e s t i o n a b l e due t o c o n v e r g e n c e problems. A l t e r n a t i v e a p p r o a - c h e s t o p e r t u r b a t i o n t h e o r y have been s u g g e s t e d by s e v e r a l a u t h o r s and t h e y l y e i n t h e c l a s s o f t h e u s u a l l y c a l l e d " n o n - p e r t u r b a t i v e " methods.

b ) " N o n - p e r t u r b a t i v e " m e t h o d s , g e n e r a l l y d e n o m i n a t e t h e a p p r o a - c h e s which a t t e m p t t o i n c l u d e t h e EMF t o a l l o r d e r s i n a n a p p r o x i m a t e d manner.

The p r e s e n t work c o n c e r n s a p a r t i c u l a r t y p e o f " n o n - p e r t u r b a t i v e "

t r e a t m e n t which o r i g i n a t e d w i t h t h e p i o n e e r work o f Keldysh [ 7 ] , almost two d e c a d e s a g o , and whose e x p r e s s i o n f o r t h e i o n i z a t i o n p r o b a b i l i t y f i t s t h e e x p e r i m e n t a l r e s u l t s o f Boreham e t a 1 [ 6 ] and Baldwin and Boreham [ 8 ] . Many have been t h e works b a s e d i n a p p r o a c h e s s i m i l a r t o K e l d y s h ' s , making u s e o f t h e f a c t t h a t S c h r B d i n g e r e q u a t i o n f o r an e l e c t r o n i n t h e p r e s e n c e o f a n o s c i l a t i n g , monochromatic EMF c a n be s o l v e d e x a c t l y ( o r i g i n a t i n g t h e s o - c a l l e d Volkov s o l u t i o n s [ 9 1 ) . These a p p r o x i m a t i o n s we denominated h i g h i n t e n s i t y a p p r o x i m a t i o n s . Without b e i n g e x a u s t i v e , we m e n t i o n t h e works o f Perelomov e t a 1 [ l o ] , t h e r e l a t i v i s t i c a p p r o a c h f o r s p i n z e r o p a r t i c l e s o f N i k i s h o v and R i - t u s [ I l l , t h e works c l o s e l y r e l a t e d t o t h e s e two o f Berson [ 1 2 ] , Ma- nakov and Rapoport [ 1 3 ] , Geltman and Teague [ 1 4 ] , G e r s t e n and M i t t l e - man [ I S ] and o f P e r t [ 1 6 ] ; t h e s p a c e t r a n s l a t i o n a p p r o x i m a t i o n (STA)

a s p r o p o s e d by F a i s a l 1171 f o r i o n i z a t i o n , e t c . . . R e c e n t l y B r a n d i and D a v i d o v i c h [18] and B r a n d i e t a 1 [ I 9 1 h a v e shown t h a t some o f t h e must commonly used h i g h - i n t e n s i t y a p p r o x i m a t i o n s , namely t h e Keldysh a p p r o - x i m a t i o n [ 7 ] , Perelomov e t a l . a p p r o a c h [ l o ] , t h e G e r s t e n and N i t t l e - man f o r m u l a t i o n 1151 and t h e STA [17] c o r r e s p o n d t o t h e same k i n d of e x p a n s i o n , o f t h e G r e e n ' s f u n c t i o n , i n t h e i n t r a a t o m i c p o t e n t i a l and d i f f e r a t most by an u n i t a r y o r by a gauge t r a n s f o r m a t i o n . Although many of t h e p r e v i o u s mentioned a p p r o a c h e s s h e d l i g h t i 3 t h e u n d e r s t a n d i n g o f t h e r e l a t i o n between m u l t i p h o t o n and t u n n e l i n g p r o c e s s e s a s mechanisms f o r a t o m i c i o n i z a t i o n , i t s t i l l remained t h e q u e s t i o n con-

c e r n i n g t h e i r r a n g e of v a l i d i t y . T h i s i s one o f t h e q u e s t i o n we d i s c u s s i n t h e f o l l o w i n g s e c t i o n s , t o g e t h e r w i t h t h e i o n i z a t i o n r a t e o f atoms by s t r o n g l i n e a r l y and c i r c u l a r l y p o l a r i z e d l a s e r l i g h t .

11. I o n i z a t i o n i n an a l t e r n a t i n g e l e c t r i c f i e l d and t h e a d i a b a t i c a p p r o x i m a t i o n .

I n o r d e r t o have b e t t e r u n d e r s t a n d i n g o f t h e i o n i z a t i o n p r o c e s s

o c u r r i n g i n t h e p r e s e n c e o f a n a l t e r n a t i n g EMF we f i r s t l y d i s c u s s a s i m p l e r p r o b l e m ; a t o m i c i o n i z a t i o n i n t h e p r e s e n c e o f a s t a t i c e l e t r i c f i e l d = E z . I n t h i s c a s e t h e K a m i l t o n i a n i s g i v e n by ( a t o m i c u n i t s a r e u s e d t h r o u g h o u t t h i s work)

where

~ ( f )

i s t h e i n t r a a t o m i c p o t e n t i a l . The e l e c t r o n i n i t i a l l y bound w i t h e n e r g y - I o c a n i o n i z e by t u n n e l l i n g . The i o n i z a t i o n r a t e , f o r f i e l d s n o t v e r y s t r o n g , may be c a l c u l a t e by t h e WKB method [ 2 0 ] , and f o r t h e h y d r o g e n atom i n i t i a l l y i n t h e ground s t a t e i t y i e l d s

where Eo i s t h e a m p l i t u d e o f t h e e l e c t r i f i e l d due t o t h e a t o m i c n u c l e u s i n t h e f i r s t Bohr o r b i t E0(21 )'"? 5 x l g 9 V/cm. Eq. ( 2 ) i s v a l i d

f o r E<cEo. ⁰

A s i m i l a r s i t u a t i o n o c c u r s f o r an a l t e r n a t i n g e l e c t r i c f i e l d i f t h e p e r i o d o f t h e f i e l d T F =

2

i s much l a r g e r t h a n t h e i o n i z a t i o n t i m e T I , which c a n be e s t i m a t e d from E q . ( 2 ) a s

T h e r e f o r e

where we have i n t r o d u c e d t h e q u a n t i t i e s y : Z ( U / I ~ ) ( E ~ / E ) 1 and F (X) = X Z exp (=) 2

.

For f i e l d s s u c h t h a t T I / T F < < l , t h e i o n i z a t i o n r a t e may be ob- t a i n e d by a v e r a g i n g Eq. ( 2 ) o v e r one c y c l e o f t h e a l t e r n a t i n g f i e l d

( o f c o u r s e f o r t h e WKB be v a l i d E < < E o ) , y i e l d i n g [ l o ] w ~ ~ = ( ~ E / . ~ ~ E ~ ) ~ / ~ w ~ ~ From E q . ( 4 ) i t i s c l e a r t h a t T I / T F < < l i s s a t i s f i e d i f y < < l ( i n t h i s

c o n n e c t i o n y h a s b e e n c a l l e d t h e a d i a b a t i c i t y p a r a m e t e i ) . However t h i s i s n o t a s u f f i c i e n t c o n d i t i o n t o h a v e T I / T F < < ~ , s i n c e t h e r a t i o E/Eo i s a l s o p r e s e n t i n E q . ( 4 ) . For t y p i c a l v a l u e s o f , Io=lOeV (%0.37 a - u . ) w=leV (s0.037 a . u . )

,

one s h o u l d h a v e E2Eo/10) i n o r d e r t o s a t i s f y T 1 / T F < < l . But f o r t h e s e v a l u e s o f t h e f i e l d , t h e a p l i c a t i o n o f t h e WKB method may b e q u e s t i o n a b l e .

111. The h i g h i n t e n s i t y a p p r o x i m a t i o n .

An a l t e r n a t i v e a p p r o a c h t o t h e one d i s c u s s e d i n S e c . 1 1 h a s been p r o p o s e d by Keldysh [ 7 ] and i t c o n t a i n s e s s e n t i a l l y t h e whole i d e a b e h i n d t h e h i g h - i n t e n s i t y a p p r o x i m a t i o n s . S t a r t i n g from t h e t r a n s i - t i o n a m p l i t u d e (Keldysh [ 7 ] , STA [ 1 7 ] )

t:

T

-

⁼ d t ' <

B f ,

HI

mo>

f 1

JOURNAL DE PHYSIQUE

w h e r e , <,> d e n o t e s i n t e g r a t i o n o v e r s p a c e c o o r d i n a t e s , $ 5 i s a Volkov s t a t e [ 9 ] , +o i s t h e a t o m i c bound s t a t e i n t h e a b s e n c e o f EMF and HI i s t h e l a s e r atom c o u p l i n g H a m i l t o n i a n , B r a n d i and D a v i d o v i c h [ 1 8 ] have shown t h a t Eq. ( 5 ) c a n be e x p r e s s e d a s t h e f i r s t t e r m s o f a n e x - p a n s i o n i n powers o f t h e i n t r a a t o m i c p o t e n t i a l v ( ( % ( ) . I n f a c t , l e t u s i n t r o d u c e t h e G r e e n ' s f u n c t i o n s GA and G d e f i n e d by

and

The Volkov s o l u t i o n s c a n be w r i t t e n i n t e r m s o f t h e G r e e n ' s f u n c t i o n GA d e f i n e d by E q . ( 6 ) a s

where + f i s a p l a n e wave.

Of c o u r s e t h e H a m i l t o n i a n HI d e p e n d s on t h e c h o i c e o f g a u g e . I f we s u b s t i t u t e Eq. ( 8 ) i n t o ( 5 ) we may r e w r i t e i t s y m b o l i c a l l y a s

where t h e symbol

I >

s t a n d s f o r i n t e g r a t i o n o v e r a l l t i m e and s p a c e c o o r d i n a t e s . T h i s e x p r e s s i o n i s t h e same u s e d by G e r s t e n and M i t t l e m a n

[ I 5 1 e x c e p t t h a t t h e y n e g l e c t t e r m s o r i g i n a t i n g from A2/2c2 i n t h e c o u p l i n g H a m i l t o n i a n . E q . ( 5 - a ) may b e compared w i t h t h e e x a c t t r a n s i - t i o n m a t r i x w h i c h i s e x p r e s s e d i n t e r m s o f t h e t o t a l G r e e n ' s f u n c t i o n ( 7 ) a s

i f one u s e s t h e r e l a t i o n

E q . ( 5 - a ) may be r e g a r d e d a s t h e f i r s t two t e r m s o f a n e x p a n s i o n o f t h e t r a n s i t i o n m a t r i x i n t e r m s o f t h e i n t r a a t o m i c p o t e n t i a l V , which i s e q u i v a l e n t t o r e p l a c e i n t h e e x a c t T m a t r i x ( E q . ( 9 ) ) G by G A . T h i s a p p r o x i m a t i o n i s e x a c t i n t h e l i m i t i n g s i t u a t i o n s where V+O o r E-, and i t i s a l s o e v i d e n t from t h e f i r s t t e r m i n t h e r i g h t hand s i d e o f E q . ( 5 - a ) t h a t f i r s t o r d e r p e r t u r b a t i o n t h e o r y i s r e c o v e r e d by t h i s t y p e o f a p p r o x i m a t i o n .

A c r i t e r i u m c o n c e r n i n g t h e r a n g e o f v a l i d i t y o f t h e h i g h - i n t e n - s i t y a p p r o x i m a t i o n s c a n b e e s t a b l i s h e d by c o n s i d e r i n g t h e f i r s t t e r m o f t h e T m a t r i x d e p e n d i n g i n t h e i n t r a a t o m i c p o t e n t i a l ( V ) , and t o impose a c o n d i t i o n f o r f a s t c o n v e r g e n c e o f t h e s e r i e s e x p a n s i o n [ 1 8 ] i . e .

An e x a c t e v a l u a t i o n o f t h e n u m e r a t o r o f i n e q u a l i t y ( 1 1 ) i s c o m p l i c a t e d , b u t a p p r o x i m a t e c a l c u l a t i o n s h a v e shown t h a t t h e above c o n d i t i o n i s s a t i s f i e d i f t h e a d i a b a t i c i t y p a r a m e t e r i s s m a l l a s compared t o u n i t y

( i n f a c t w i t h i n t h e a p p r o x i m a t i o n s c o n s i d e r e d i n e q u a l i t y ( 1 1 ) i s s a t i s f i e d f o r y 6 < < 1 w i t h E < Eo)

To e v a l u a t e ( 6 ) we c o n s i d e r f o r c o n v e n i e n c e [ 1 9 ] t h e gauge $=O, d i v l = 0 , s o t h a t

1 + 1

HI ( t ) =

-

7 p s X ( t ) + -? 2 c A2 ( t ) . ( 1 2 ) For l i n e a r l y p o l a r i z e d f i e l d s w i t h

x ( t ) = -

CB

c o s w t ;

2

⁼ E x

w ( 1 3 )

i t c a n b e shown [ 1 9 ] t h a t t h e m u l t i p h o t o n t r a n s i & i o n r a t e from t h e i n i t i a l s t a t e t o a f i n a l s t a t e w i t h momentum p i s g i v e n by

w h e r e & n _" ($) i s t h e F o u r i e r t r a n s f o r m o f Qn

- (XI.

The i o n i z a t i o n t h r e s h o l d i s I o + E Z / 4 w Z c Z = I, ( 1 + 1 / 2 y 2 ) , i n s t e a d o f I o . The e x t r a t e r m I o / 2 y 2 h a s b e e n d i s c u s s e d by K i b b l e [ Z l ] , who h a s shown t h a t i t c a n be i n t e r p r e t e d a s a p o t e n t i a l e n e r g y a s s o c i a t e d w i t h t h e p o n d e r o m o t i v e f o r c e , when t h e m o t i o n o f t h e e l e c t r o n i s a v e r a g e d o v e r many p e r i o d s o f t h e a p p l i e d f i e l d . A s t h e e l e c t r o n l e a - v e s t h e f o c a l r e g i o n o f t h e l a s e r beam, t h i s p o t e n t i a l e n e r g y i s t r a n s f o r m e d i n t o e l e c t r o n i c t r a n s l a t i o n e n e r g y . F o r a Neodimium l a s e r

( a

=

1 . 1 7 eV), one h a s E 2 / 4 w Z c 2

=

1 0 - l 3 IeV, where t h e i n t e n s i t y I i s g i v e n i n W/cmZ. T h i s shows t h a t f o r i n t e n s i t i e s a r o u n d 1 0 1 3 W/cmz t h i s t e r m s t a r t s t o become i m p o r t a n t .

The i n f i n i t e sum o v e r B e s s e l f u n c t i o n s , i n E q . ( l 4 j , c a n be e v a l u a t e d u s i n g t h e i n t e g r a l r e p r e s e n t a t i o n

D e f i n i n g t h e t o t a l m u l t i p h o t o n t r a q s i t i o n r a t e by i n t e g r a t i n g E q . ( 1 4 ) o v e r t h e e l e c t r o n f i n a l momenta p . One h a s

C2-402 JOURNAL DE PHYSIQUE

2 1 / 2

- fi

s i n e + ¹ s i n 2 0 1 1

0 Y 4 Y

The i n t e g r a l i n E q . ( 1 5 ) c a n be e v a l u a t e d by t h e s a d d l e p o i n t method [ 7 , 1 0 , 1 1 , 1 9 ] , u n d e r t h e c o n d i t i o n I o / w > > l , and i n t h e r e g i o n where y+O t h e a m p l i t u d e o f t h e EMF (E) must b e much s m a l l e r t h a n QEo [ 1 9 ] . T h i s l a s t c o n d i t i o n imposes a h i g h i n t e n s i t y l i m i t on t h e r e g i o n o f a p p l i c a b i l i t y o f Keldysh f o r m u l a . I n t h e s e c o n d i t i o n s one h a s

and

The t o t a l t r a n s i t i o n r a t e i s t h e n e v a l u a t e d by summing o v e r a l l m u l t i p h o t o n P r o c e s s e s W . =

1

w @ ) , y i e l d i n g a t u n n e l i n g l i k e e x p r e s - s i o n f o r y < < l f 1 N

Except f o r t h e p r e - e x p o n e n t i a l f a c t o r E q . ( 1 7 ) r e d u c e s t o t h e s t a t i c f i e l d s c a s e f o r y+O. The d i s c r e p a n c y i n t h e p r e - e x p o n e n t i a l f a c t o r may be due t o t h e f a c t t h a t t h e l o n g r a n g e n a t u r e o f t h e Cou- lomb f o r c e h a s n o t been p r o p e r l y t a k e n i n t o a c c o u n t . The s p e c i f i c form o f f ( E / E o ) depends on t h e bound s t a t e wave f u n c t i o n and f o r s h o r t r a n g e p o t e n t i a l s Eq. ( 1 8 ) r e p r o d u c e s e x a c t l y ( f o r y+O) t h e a d i a b a t i c r e s u l t o f Sec.11 [ 1 0 , 1 1 ] .

T h e r e i s s t i l l a n o t h e r p o i n t t h a t must b e commented which con- c e r n s t h e p r o p e r t u n n e l i n g l i k e e x p o n e n t i a l b e h a v i o r o f t h e i o n i z a t i o n r a t e ( E q . 1 8 ) . The WKB r e s u l t , which i s r e c o v e r e d from Eq. ( 1 6 ) , i n t h e l i m i t y+O o r w+O

,

i s v a l i d f o r f i e l d a m p l i t u d e s s a t i s f y i n g t h e con- d i t i o n E<<Eo. T h i s b r i n g s t h e q u e s t i o n o f t h e r e g i o n o f a p p l i c a b i l i t y o f t h e s t e e p e s t d e s c e n t method, which h a s been u s e d t o o b t a i n E q . ( 1 6 ) and c o n s e q u e n t l y t h e a d i a b a t i c l i m i t o f E q . ( 1 8 ) . I n h a s b e e n shown [ 1 9 ] t h a t f o r y?O and a r b i t r a r y f i e l d i n t e n s i t i e s t h e i n f i n i t e sum- m a t i o n o f Eq. (14') c a n n o t be e v a l u a t e d by t h e u s u a l s a d d l e p o i n t method, a s a p p l i e d by Keldysh [ 7 ] , due t o t h e p r e s e n c e o f c o a l e s c i n g s a d d l e p o i n t s . I n f a c t f o r y+O, t h e r e g i o n o f a p p l i c a b i l i t y o f

Keldysh t r e a t m e n t i s r e s t r i c t e d t o f i e l d a m p l i t u d e s s u c h t h a t E<<4Eo [191.

Of c o u r s e t h i s i s a l s o t h e r e g i o n o f v a l i d i t y o f E q . ( 1 6 ) i n t h e r e g i m e where y+O, which r e s t r i c t s i t s a p p l i c a t i o n t o t h e s e f i e l d i n -

t e n s i t i e s . The f a c t t h a t an a p p r o x i m a t i o n e x a c t f o r E + m , r e p r o d u c e s r e s u l t s i n t h e weak f i e l d r e g i o n i s r e l a t e d w i t h t h e e x p a n s i o n p a r a - m e t e r o f i n e q u a l i t y ( 1 1 ) . The p r e s e n t r e s u l t s t e n d t o c o n f i r m t h a t t h i s p a r a m e t e r must be i n some way c o n n e c t e d w i t h t h e s m a l l n e s s o f t h e p a r a m e t e r y = w m , which r e p r o d u c e s p r o p e r l y t h e a d i a b a t i c

(w+O), t h e h i g h i n t e n s i t y (E+m) and t h e weak p o t e n t i a l (Io+O) l i m i t i e s . T h i s q u e s t i o n may be s a t i s f a c t o r i l y answered by p r o p e r l y e v a l u a t i n g t h e e x p r e s s i o n ( 1 1 ) .

F i n a l l y we would l i k e t o d i s c u s s v e r y b r i e f l y t h e r e s u l t s con- c e r n i n g c i r c u l a r p o l a r i z e d l a s e r l i g h t . The t r a n s i t i o n m a t r i x ( 5 ) c a n be e v a l u a t e d c o n s i d e r i n g t h e same i n t e r a c t i o n H a m i l t o n i a n o f E q . ( 1 2 ) , w i t h t h e v e c t o r p o t e n t i a l

i ( t ) = E ( ? coswt +

7

s i n w t ) ( 1 9

The t r a n s i t i o n m a t r i x ( 5 ) c a n be e v a l u a t e d and f o r t h i s c a s e y i e l d s a t r a n s i t i o n r a t e a s s o c i a t e d t o N-photon p r o c e s s e s g i v e n by

Pij 2

~ 2 : '

⁼

^{-1 Go}

^{( p i )}

¹

^{( I o + p N}

' 1

^{JR (} sinO)sinOdO (20)

where from e n e r g y c o n s e r v a t i o n

To o b t a i n E q . ( 2 0 ) we have assumed, f o r t h e s a k e o f s i m p l i c i t y , t h a t t h e i n i t i a l bound s t a t e i s s p h e r i c a l l y s i m m e t r i c a l ( S - t y p e ) . A g e n e r a l e x p r e s s i o n c a n a l s o be o b t a i n e d f o r o t h e r c a s e s .

The i n t e g r a l a p p e a r i n g i n Eq.(ZO) c a n be e v a l u a t e d w i t h t h e h e l p of a t a b l e of i n t e g r a l s i n v o l v i n g B e s s e l f u n c t i o n s 1221

Using t h e power s e r i e s d e f i n i t i o n o f t h e g e n e r a l i z e d h i p e r g e o - m e t r i c f u n c t i o n , 1221

i t i s e a s i l y shown t h a t t h i s f u n c t i o n c a n be w r i t t e n i n t e r m s o f an i n t e g r a l o v e r a B e s s e l f u n c t i o n , i . e .

JOURNAL DE PHYSIQUE

T h i s l a s t e q u a t i o n shows t h a t , e x c e p t f o r t h e c h o i c e o f g a u g e , t h i s r e s u l t i s e q u i v a l e n t t o t h e one o b t a i n e d by Perelomov e t a 1 [ l o ] . The i n t e g r a l o n t h e r i g h t hand s i d e o f e q . ( 2 4 ) may be e v a l u a t e d i n a s i m i l a r manner a s done i n [ l o ] . I f o n e i n t r o d u c e s t h e v a r i a b l e s ,

v ( I o / w ) ( 1 + l / y z ) and t E ( 2 v c / N ) - 1 ,

C

Eq. ( 2 0 ) c a n b e r e w r i t t e n a s

For I o > > w we h a v e N > > 1 , and t h e B e s s e l f u n c t i o n o f e q . ( 2 5 ) h a s l a r g e v a l u e s o f t h e argument and o f t h e i n d e x . An a s s y m p t o t i c e x p r e s - s i o n f o r t h i s c a s e c a n b e u s e d [221 and E q . ( 2 5 ) c a n b e t r a n s f o r m e d

1101 i n

x e x p C - 4 v c $ ( t , y ) l ( 2 6 )

where [ l o ]

F o l l o w i n g Perelomov e t a 1 [ l o ] , i t i s e a s y t o show t h a t t h e t o - t a l t r a n s i t i o n r a t e c a n be w r i t t e n i n t h e f o r m

where f o r y+O, g c ( y ) + l - y 2 / 1 5 , r e p r o d u c i n g a l s o a t u n n e l l i n g l i k e be- h a v i o r . For a r u b y l a s e r , l i n e a r l y p o l a r i z e d l i g h t s h o u l d i o n i z e atoms more s t r o n g l y t h a n c i r c u l a r l y p o l a r i z e d l i g h t [ l o ] , b u t f o r C02 o r Neodimium l a s e r s t h e p r o b a b i l i t i e s a r e b a s i c a l l y t h e same.

I n c o n c l u s i o n we must m e n t i o n t h a t t h e t y p e o f n o n - p e r t u r b a t i v e a p p r o a c h p r e s e n t e d h e r e seems t o be c o n v e n i e n t t o d e s c r i b e s e v e r a l d i f f e r e n t r e g i o n s o f a p p r o x i m a t i o n s , i n c l u d i n g t h e s t r o n g f i e l d l i m i t . The e x p e r i m e n t a l r e s u l t s o f Boreham e t a 1 [ 6 ] and Baldwin and Boreham

[ 8 ] , p e r f o r m e d i n t h e r e g i o n O.lhyLl and E=Eo, shows d e v i a t i o n s from t h e p r e d i c t i o n s o f p e r t u r b a t i o n t h e o r y and a t t h e moment t e n d s t o s u p p o r t t h e Keldysh t y p e o f a p p r o a c h d i s c u s s e d i n t h i s work.

One o f t h e a u t h o r s ( H . S . B . ) w i s h e s t o acknowledge h e l p f u l l d i s c u s s i o n s w i t h D r . O.L.!~lalta.

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