INTERACTION

BETWEEN

A TWO-LEVEL

A N D AN E L E C T R O M A G N E T I C QUASICLASSICAL V.

P.

FIELD

ATOM

IN T H E

APPROXIMATION

Yakovlev

UDC 539.186

An expansion in coherent states of the field is used to obtain a general equation that defines the energy s p e c t r u m of a s y s t e m consisting of a two-level atom and one mode of a quantized e l e c t r o magnetic field in the q u a s i c l a s s i c a l c a s e . Interaction with a low-frequency field under conditions of multiphoton r e s o n a n c e is investigated s e p a r a t e l y . w The c u r r e n t i n t e r e s t in the c l a s s i c a l quantum-mechanical problem of the interaction between an atom and an e l e c t r o m a g n e t i c field s t e m s p r i m a r i l y f r o m the investigation of multiquantum p r o c e s s e s in s t r o n g light fields that give r i s e to pronounced changes in the atomic s p e c t r u m [1]. T h e o r e t i c a l description of this situation r e q u i r e s that we go beyond the bounds of perturbation theory, and this entails m a j o r mathematical difficulties. The f e a t u r e s of the [mhavior of atoms in s t r o n g fields (primarily in the resonance case) can be investigated, at l e a s t qualitatively, by using the model of a two-level s y s t e m that interacts with one mode of aa e l e c t r o m a g n e t i c field. Within the confines of this model, the problem reduces to one of determining the s p e c t r u m of stationary states of a s y s t e m consisting of a two-level atom and a field. The effect of other atomic states and field modes (resulting in attenuation of the states in question) is a s s u m e d to be s m a l l and able to be allowed for by perturbation theory. This c i r c u m s t a n c e limits the field of application of the model in question. The stationary states

I ~ ( t ) ) = e x p ( - - iEt) {~1 [ rl ) + 72 I r2 > }

(1)

of a s y s t e m consisting of a two-level atom that interacts in dipole fashion with one mode of a quantized field a r e given by the equations

(~o (H.

- 0 1 h > + ~ - ~ ( c + - c) I h ) = 0 , (2) E + ~) I h > + 2 x ( c + - c )

I f, > = o.

Here ~01,2 and ~:e are the wave functions and energy levels of the atom; Hw = oJ(c+c + 1/2) is the Hamiltonian of the field; Ifl,2 > a r e state v e c t o r s that depend on the field v a r i a b l e s ; k = d~/2w/Vi~ and d is a dipole matrix element. These equations, and also the c o r r e s p o n d i n g equations for an atom in a c l a s s i c a l field, have been investigated in a number of studies [2-7, 12, 13]. In the r o t a t i n g - p h a s e approximation, paper [3] obtained a solution f o r the c a s e of one-photon r e s o n a n c e . Multiphoton r e s o n a n c e s were investigated in [4, 5]. Using the adiabatic approximation, Zaretskii and Kratuov [6] computed the probability of resonance multiquantum excitation of atomic levels in a low-frequency c l a s s i c a l field (oJ << a). Melikyan [12] employed the Hill method, f a m i l i a r in the theory of linear equations with periodic coefficients, to investigate the quasienergy s p e c t r u m of a two-level s y s t e m in a c l a s s i c a l field. Equations (2) can be solved exactly for degenerate (e = 0) levels [7]. The case of large frequency values (w >> e) was c o n s i d e r e d in [4, 13]. Moscow Engineering P h y s i c s Institute. T r a n s l a t e d f r o m Izvestiya Vysshikh Uchebnykh Zavedenii, Radio-fizika, Vol. 19, No. 12, pp. 1823-1832, D e c e m b e r , 1976. Original article submitted May 23, 1975. l IThis material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, N e w York. N. Y. 100! 1. NO part | o f this publication m a y be reproduced, stored in a retrieval system, or tr~nsmisted, in any form or by any megns, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publiM~er. A c o p y o f this article is avagable from the publisher for $ 7.50.

1

1264

In Sec. 2 of this p a p e r we employ an expansion in coherent field states to obtain and investigate general equation (11), which defines the e n e r g y s p e c t r u m of the s y s t e m for a field of a r b i t r a r y intensity and frequency in the q u a s i c l a s s i e a l case, when the m e a n number of photons is large. In p a r t i c u l a r , this equation includes the r e s u l t s obtained e a r l i e r in [3-5, 13]. In Sec. 3 we c o n s i d e r s e p a r a t e l y {for methodological p u r p o s e s ) i n t e r a c t i o n with a low-frequency field (w << e). We obtain the s p e c t r u m of s t a t i o n a r y s t a t e s of the s y s t e m and compute the populations of the atomic levels (32). In our opinion, the q u a s i c l a s s i c a l method of solving a s y s t e m of two s e c o n d - o r d e r differential equations that is p r o p o s e d in this section is of c e r t a i n methodological i n t e r e s t and can be used in investigating s i m i l a r equations in other physical p r o b l e m s . w

Let us now consider the functions ]f:~) = ( 1 / f ~ ( If~ } * If2)), which we expand in c o h e r e n t field states

[8]: c where

--i

z

(3)

~ I~>

I k > and integration is p e r f o r m e d over some contour C in the plane of cornk=0 ~ plex v a r i a b l e z. We substitute expansions (3) into s y s t e m (2), and, allowing for the effect of the field o p e r a t o r s on the c o h e r e n t s t a t e s ,

we integrate by p a r t s , t r a n s f e r r i n g the action of the d e r i v a t i v e s 0/3o~ to the functions u:~. We r e q u i r e that the integrated p a r t s vanish. F o r this we take C to be a closed contour and seek solutions u• that will r e t u r n to t h e i r initial v a l u e s a f t e r t r a v e r s i n g the contour C. M o r e o v e r , C should e n c o m p a s s singular points of u• since otherwise we would obtain the t r i v i a l solution I fl ) = [ f2 ) = 0. Then, equating the coefficients for I a ) in the integrand, we obtain the following s y s t e m of equations for u• --(l-z)

+

~+1--

u++~, u_=O,

(l-z)

(4) (l+z)~U - + [ ~+l --~-z

~~- ~ ( l + z ) ]

u_+--u+=O,

w h e r e p a r a m e t e r g is r e l a t e d to the s y s t e m e n e r g y by the e x p r e s s i o n t~ = (E/~o) + (~2/4W2) - 1/2. Equations (4) have t h r e e singular points: two r e g u l a r ones z = + 1with indices - 1 - g and --g, and one i r r e g u l a r one at infinity. L e t us now consider the s e c o n d - o r d e r differential equation for u+, which we will s e e k in the f o r m u+ = (1 --z)-1-~(1 + z)-,~.exp

---~2z w(z).

(5)

Then we obtain the following e x p r e s s i o n for w(z):

' I z.~ ) w ".+ [ l ( 1. 2 ~ ) z.~ ( l z 2 ) ] w ' [

"

-

--

~ ' ~ 2 k 2 t ~- '(- -i C05

2 0)2

--z)

] w=0.

(6)

We take C to be a closed loop that t r a v e r s e s singular points z = • in opposite directions. As a result, the function (1 - z)-1-/~(1 + z) -g r e t u r n s to its initial value. Then to obtain the requisite solution u+(z), we should s e e k r e g u l a r solutions w(z) of (6). Such solutions exist only for c e r t a i n values of t~. T h e s e eigenvalues /~ define the energy s p e c t r u m of the s y s t e m . We expand w(z) in gacobi polynomials [9]: Cm ~(z) = ~, ~AOv.,(z),

v~(z) = pc;,-~,-.(z),

v~(1) = r ( m - ~).

~=o

(7)

r (--~)ml

Using the differential equation, r e c u r s i o n r e l a t i o n s , and the orthogonality condition for Jacobi polynomials, we obtain the following t h r e e - t e r m r e c u r s i o n r e l a t i o n f o r the coefficients Cm: s' ~.' (

+"~

~+1/2

)

k2

4(m--~)' ] 4(m - - ~ ) 2 - - 1 j ~.' ( ~ + 1 1 2

m -- t~ + l/2 + 1 ( m + l - - p , ) c , , + l + ~ . - ~

m -- t~ -- ll2

1)(m--l--l~)Cm-~=O.

(S)

1265

T h e q u a s i c L a s s i c a l c a s e (large g), c o r r e s p o n d i n g to a t r a n s i t i o n to a c l a s s i c a l field, is of g r e a t i n t e r e s t . We i n t r o d u c e the new notation g = t~ + n, m = n + s, c m = Cn+s ~ Cs, w h e r e n is the n u m b e r of photons in the field in the a b s e n c e of i n t e r a c t i o n , this n u m b e r c o i n c i d i n g in the q u a s i c l a s s i c a l c a s e with the m e a n n u m b e r of photons in s t a t e (1)9 We a s s u m e that n >>1, n >> e/c0, and, m o r e o v e r , ~ - * 0 , but X~'-ff=dgo is a finite quantity ($0 is the f i e l d - s t r e n g t h amplitude of the c l a s s i c a l field). In this c a s e s t a t i o n a r y s t a t e (1) is equivalent to a s t a t e with a p a r t i c u l a r q u a s i e n e r g y [10, 11] in a c l a s s i c a l field. T h e n r e c u r s i o n r e l a t i o n (8) c a n be w r i t t e n as follows:

c~ + a ( s - ~)c~+, + a('~ - s)c~_~ = o,

a(x)=A ' (x+l)(x-1/2)

A'

(x~ - - "J[)(Xa -- v~)'

=

)~2(n+1/2) 4~ ,

{(_~

'q - v2 "=

_

1)~

}i/, + 2 A2

(9)

and we can a s s u m e that index s v a r i e s o v e r infinite limits (_oo < s < ~ ) . T h e e i g e n v a l u e s t7 a r e the r o o t s of the d e t e r m i n a n t of h o m o g e n e o u s s y s t e m (9): r

.... .

-

0

....

a(t~ +

i

0

a(~),

9 0

0

0

o

o

o

D(;)-- " 0

"

1 I a(--; --[ I

1)

- - 1)

1

0

9

9

9

9

9

9

9

'~ a ( 7 - -

1)

-

0

0

la(--~+l)

0

1

1

(10)

r

o -

0

a(--~)

1 9

0

I

9

.

,I

2) .

.

.

.

.

.

.

I

1 9

a(--~+2)

9

9

9

9

9

. 9

9

F u n c t i o n D(g~ p o s s e s s e s the fottowing p r o p e r t i e s : 1) D ( ~ is a m e r o m o r p h i c function with s i m p l e poles at points • 1 + k , • 2 + k (k = 0, +1, • . . . . ); 2) D(/~) is an e v e n p e r i o d i c function with p e r i o d 1; 3) D(~') ---1 a s ItTI --* ~o. T h e r e f o r e , D{t~) c a n be w r i t t e n a s f o l l o w s : D(~) = 1 + ~A, [ctg , : ( ~ -

v ~ ) - ctg ~ ( ~ - F ~)] + r.A~ [ctg~ (~ -- ~ . ) - ctg ~ ( ~ ' + v~)].

(lOa)

Coefficients A t and A 2 a r e independent of ~ and c a n be e x p r e s s e d in t e r m s of D(0) and D(1/2). (The c h o i c e of t h e s e g ' v a l u e s is convenient in that s o m e e l e m e n t s of the d e t e r m i n a n t vanish.) Since a (1/2) = 0, we have that D(1/2) is equal to the p r o d u c t of the t h r e e d e t e r m i n a n t s that a r e d i s t i n g u i s h e d by d a s h e d lines in (10). The c e n t e r d e t e r m i n a n t 1 - a 2 ( - 1 / 2 ) = 0, and h e n c e D(1/2) = 0. We note that t7 = 1 / 2 (and a l s o o t h e r h a l f - i n t e g r a l values) a r e not e i g e n v a l u e s of the p r o b l e m . T h e a p p e a r a n c e of t h e s e e x t r a r o o t s is a s s o c i a t e d with the fact that in going f r o m (8) to (9) we divided the r e c u r s i o n r e l a t i o n by the c o e f f i c i e n t f o r c m. T h e final equation f o r d e t e r m i n i n g the e i g e n v a l u e s of D(~) = 0 b e c o m e s sin 2 ~"~ = D(0) sin s ~'~l s In2 ~2, 1

D(O) = D~('% ~), D('q,'*~) ---- a ( - - 2 ) 0 9

o

~

(i!)

a(1)

0

1 a(--3)

a(2) 1

9

.

9

~

.

,

0

. 9 .

0 . . . a(3)... ,

,

,

~

o

,

This equation defines the e n e r g y s p e c t r u m of the s y s t e m in the g e n e r a l c a s e f o r a r b i t r a r y field f r e q u e n c y and intensity. T h e d e t e r m i n a n t D(vl, v 2) c a n be c o m p u t e d in finite f o r m in s o m e l i m i t i n g c a s e s 9 If the field f r e q u e n c y is Large as c o m p a r e d to the d i s t a n c e b e t w e e n a t o m i c l e v e l s , e/w << 1, then v1 ~1/4 + 2A 2, v2 ~ (1/2vl)(e/w) << 1, and to d e t e r m i n e the e i g e n v a l u e s t~ it s u f f i c e s to c o m p u t e the d e t e r m i n a n t D(vl, 0), which c a n be e x p r e s s e d in t e r m s of a B e s s e l function in this c a s e : D(,., 0)-- 2"----L-tJ . ( 4 A ) ,

h =

+ 2A'. 1/~ f m

Thus, the eigenvalues g b e c o m e [4, 13] ~=•

A)=_+

8

2

9

w

For a weak field A 2 << 1 (and arbitrary e/w values) w e can limit ourselves to a few terms in the series of D(vi, v2) in p o w e r s of A2:

expansion

1266

a, ctg ~ + ,x a, ctg v~ 4 ( ~ - - ~)~ '

D (v~, v2) ~ 1 - - A'"

a~ -- [(1-b'~2)'-- "~] [(1--,~,)'-- v~] \ 4 ' ~ -- 1 + 3,~g + , ~ - - 4 . ,

(12)

F o r the r e s o n a n c e s i t u a t i o n (of g r e a t e s t i n t e r e s t in the c a s e of a weak field), when e/w ~ k + 1/2, the e x p a n sion t e r m s we have w r i t t e n a r e s u f f i c i e n t if k = 0 and k = 1. In p a r t i c u l a r , f o r t h r e e - p h o t o n r e s o n a n c e (e/w 3/2, v 2 ~ 1/2, v~ ~ 3/2) we obtain the following e x p r e s s i o n f o r the s p e c t r u m : -V - __+ ( 1

,,--3)'+

1

m~--t----~-s. 3A~ t ,

3[

which c o r r e c t l y y i e l d s the r e s o n a n c e p o s i t i o n with allowance f o r Stark shift. F o r r e s o n a n c e s of higher m u l t i plicity, which can be c o n s i d e r e d s i m i l a r l y , we need to allow f o r the s u b s e q u e n t t e r m s in the e x p a n s i o n of D(vt, v2) in p o w e r s of A 2, s o m e t h i n g that r e q u i r e s e x t r e m e l y l a b o r i o u s c o m p u t a t i o n s . We should note that our r e s u l t s d i f f e r f r o m those of [12]; evidently, this r e s u l t s f r o m the inadequacy (close to multiphoton r e s o n a n c e , in p a r t i c u l a r ) of those e x p a n s i o n s in the i n t e r a c t i o n p a r a m e t e r that w e r e u s e d in [12]. w

It is c o n v e n i e n t to i n v e s t i g a t e the c a s e of a l o w - f r e q u e n c y field (e/w >> 1) f o r l a r g e v. v a l u e s (p >> but Xg~-is finite) d i r e c t l y using the q u a s i c l a s s i c a l solution of s y s t e m (2).

e/w, X ~ O,

C h a n g i n g to c a n o n i c a l v a r i a b l e s ~ = p and ~ = i(d/dp) u s i n g the c u s t o m a r y f o r m u l a s c = (1/4-2)(~ + i~), c + = (1/~2)(~ - i~), we c a n w r i t e s y s t e m (2) f o r functions f~,2(p) in the f o r m (

1 d~

1

i

-~)[z +

1 ?

:~+~

2 dP d~ +Ip~__~__.~+

h=o, (14)

:,=0.

In what follows we will c o n c e r n o u r s e l v e s with the c a s e of a field that is not too l a r g e , so that

x V;<<

1, but - - ~

I, ! ~ } 1 .

$o)

T h e n we c a n u s e p e r t u r b a t i o n t h e o r y to s o l v e (14). in t e r m s of fl by u s i n g the second equation in (14),

(D

We will s e e k a solution such that f2 << fl-

We e x p r e s s f2

(15) and s u b s t i t u t e it into the f i r s t .

As a r e s u l t we obtain the following equation f o r fl:

2 dp~+

2s(o]

(16)

2

whose solution is an o s c i l l a t o r y wave function, f, (p) = q, (p i), ~. (z)

1

( 1--2er ) )2

f=

exp (

@z=)H,,(z),

V2.n,V 114 ~

1

(17)

: 8~r

and whose eigenval, aes g have the f o r m ~(t)=

1

~ +~

2

~

n+

~n

(18)

~

4~

F u n c t i o n ~bn(p~) d e s c r i b e s a s t a t e with a c e r t a i n n u m b e r of photons of f r e q u e n c y ~ = ~2w. ff we expand ~bn(p~) in functions ~bk(P), the s q u a r e s of the e x p a n s i o n c o e f f i c i e n t s a~ yield the s t a t e p r o b a b i l i t i e s with a c e r tain n u m b e r of "old" photons (with f r e q u e n c y w) in s t a t e I q,(t)) (1) (function f2 can be d i s r e g a r d e d ) . In the g e n e r a l c a s e t h e s e q u a n t i t i e s c a n be e x p r e s s e d in t e r m s of J a c o b i p o l y n o m i a l s , while in the q u a s i c l a s s i e a l

1267

l i m i t (n >>

e/w

>> 1) t h e y c a n b e e x p r e s s e d i n t e r m s

of B e s s e l f u n c t i o n s : a2n+~s = J2s(X2n/8ew) ( s = 0, e l , +2, .. . ).

H e r e the m e a n n u m b e r of photons k = n, the fluctuation ~/(Ak)2 = ( 1 / ( 2 ) [X2(n + 1/2)/4ew], while the r e l a t i v e fluctuation 6 ~ d2/sV is i n v e r s e l y p r o p o r t i o n a l to the v o l u m e V o c c u p i e d by the field. F o r a field with ~0 ~ 108 V / c m and w a v e l e n g t h l = 5 . l03 .~ in a v o l u m e V ~/3, n ~ 3.109, while the r e l a t i v e fluctuation (for a d i s tance b e t w e e n l e v e l s 2e ~ 10 eV) is negligible, 6 ~ 10 -1~ T h e o t h e r l i n e a r l y independent solution has the f o r m 9

~l+--

(:9) p.(~)_~ 1 . . ] _ 2

-z -i- ~-~(m -1- 1/'2) ~.~ m -t- ~--l- ))(m § 1/2) o) :,) 4 s~o

T h e r e s u l t a n t solutions (17)-(19) b e c o m e i n c o r r e c t f o r the c a s e of multiphoton r e s o n a n c e , when the eigenvalues of (18) and (19) a r e c l o s e to one a n o t h e r [~(i) ~ g(2) n - m = 2k + 1, k >> 1]. In this c a s e ft ~ f2 and the c o r r e c t s o l u t i o n c a n be given as a l i n e a r c o m b i n a t i o n of solutions (18) and (19). We will seek the solution in the f o r m 0 0 f, = cos -:- z, + sin V z,, (20) 0

0

f.~ = - - sln ~- 7.: + cos 2 Z2, where tanO = (1/~2)(Xp/e) and cos 0 = 1/~/1 + (X2p2/2e~). Substituting (20) into (14) and neglecting s m a l l t e r m s containing f o r Xi and )/2:

dO/dp, we

obtain independent equations

d ~ZL__.___~2 -l- ~,'2 (P) ZL 2 = O,

df qh.2----

2~-t-l--P

(21) ~+2~

1+'~-2

9

In the q u a s i c l a s s i c a l a p p r o x i m a t i o n , the solutions of these equations a r e p

u(p)-----1/.
~,dp),

u*(p), {22)

v (p) = ~ r ~ The c l a s s i c a l t u r n i n g points at which 4,1,2 v a n i s h will be denoted by +Pl and +P2, r e s p e c t i v e l y . Aside f r o m t h e s e , functions 4,1,2 have two m o r e c o m p l e x b r a n c h i n g points ~ip0 = • (g'2-g/X) at which ,/1 + (X2p2/2e 2) v a n i s h e s . When the b r a n c h i n g points • a r e t r a v e r s e d in the c o m p l e x plane p, the e x p r e s s i o n ~/1 + (X2p2/2e2) c h a n g e s sign and the functions 4,1 and 4,2 go o v e r into one a n o t h e r (u ~ v, r e s p e c t i v e l y ) . M o r e o v e r , in this c a s e c o s (0/2) --* sin (0/2) and sin (0/2) ~ - c o s (0/2}. We take the solution u(p) in (22) f o r p < 0. When this solution is a n a l y t i c a l l y continued into the r e g i o n p > 0 along the r e a l axis

u (o < o)-~ ~(p > 0). In the c a s e of analytic continuation a c r o s s the u p p e r h a f t - p l a n e [in the u p p e r h a f t - p l a n e u(p) d e c r e a s e s e x p o n e n tially] with t r a v e r s i n g of the point ip 0 (this b e i n g clone at a d i s t a n c e f r o m ip 0)

(p < o) -~ R~ (p > o), w h e r e tl is an exponentially s m a l l a b o v e - b a r r i e r r e f l e c t i o n c o e f f i c i e n t [6]: R ---- exp { - -

[r

(ip) - -

r (ip)] dp} ~ exp -- -~-

1

,,

0

T h e quantity in the exponent c a n be e x p r e s s e d in t e r m s of a c o m p l e t e elliptic i n t e g r a l . too l a r g e , R has the f o r m [6]

1268

F o r fields that a r e not

R=exp

--+

e

2~tO

)~ V'~ + 1/2 -~

jln 4

)~z(F + 1/2~ 2-~ / (R <(1).

(23a)

Thus, the relationship between the solutions in regions p < 0 and p > 0 is as follows:

cos ~0 u (p) -,- cos ~2 u (p) + R sin 2

7./ (p).

(24)

Similarly,

0

0

0

sin ~ v (p) -,- sin - : v (p) -- R cos -x u (p).

(24a)

Z

Z

T h e e x p r e s s i o n s f o r u* and v* a r e obtained by the c o m p l e x c o n j u g a t e s of (24) and (24a). T h e n we take the function • in the f o r m of a l i n e a r c o m b i n a t i o n of functions u(p) and u*{p) such that the solution d e c r e a s e s exponentially f o r p < - P l [for )/2 we m u s t c o r r e s p o n d i n g l y c h o o s e a l i n e a r c o m b i n a t i o n of v(p) and v*(p) that d e c r e a s e s exponentially f o r p < -P2]- T h e n fl(P) has the following f o r m f o r p < 0:

f, = o, c o s ~

0

{e ~~ (s'-~)u(p)'

~ (s,- ~-)

+c.c. }+c~sln~{e

-~

v(p)+

c.c.

}

(25)

,

Z

where Pl, 2

$1,~ = 2 ~ r

~ ~(~ +

1/2) +_ S,

o

tO

0LJ

(26)

V

~2

By u s i n g (24) and (24a) we continue solution (25) into the r e g i o n p > O. The r e s u l t a n t solution should d e c r e a s e exponentially to the r i g h t of the t u r n i n g points Pi and P2- T h i s leads to a h o m o g e n e o u s s y s t e m of equations for the c o e f f i c i e n t s cl and c2: 1 c, cos S, - - c2 R cos ~ (S, + S~) = 0,

1 c~ R cos ~- (S, + S2) + c2 cos S~

(27) =

0.

E q u a t i n g the d e t e r m i n a n t of this s y s t e m to z e r o , We obtain sln~ ~:(F + 1/2) = cos2S + R ~.

(28)

N e a r r e s o n a n c e the p h a s e S is c l o s e to ,r(k + 1/2); i . e . , 1S~.~_~ +~(F-k

~

1 / 2 ) ~ k + I/2,

(29)

4~

while the e i g e n v a l u e s ~ a r e c l o s e to h a l f - i n t e g e r s :

~=n+-~+A~,

iV[ S - ~ (k+

~q---,:

+R ~ (30)

(n >>/~ >> 1). N e a r r e s o n a n c e , the c o e f f i c i e n t s c i and c 2 have the f o r m c~

:,2=~ 1 [ 1 ~

S--~:(kq-1/2)]

,

2 c 1 c ~ = (-- 1)~+l

R .

(31)

F o r m u l a s (28)-(31) c o r r e s p o n d to the r e s u l t s of [6]. T h e p r o b a b i l i t i e s that an a t o m will be in the "low" and "high" s t a t e s a r e , r e s p e c t i v e l y , w, ~ c~ + ~.t(~ + 1/2) [ ci

8~'

~-

c~~

1~, w ~ c ~ +

).2(F + 1/2) {

8,'

' ~-cl)

(32)

N e a r r e s o n a n c e e 2 ~ c 2 ~ 1, and t h e r e f o r e we c a n d i s r e g a r d the s e c o n d t e r m s in these e x p r e s s i o n s (since k2t~/e 2 << 1). At the e x a c t r e s o n a n c e , w 1 = w 2 = 1 / 2 . At a d i s t a n c e f r o m r e s o n a n c e , one of the coefficients

1269

is s m a l l (e. g., c 2 ~ R << },2/~/e2 << 1), while the other is close to 1. Then w 1 = 1, w 2 ~ A.2(/~ + 1 / 2 ) / 8 e 2 = 1 ( d [ o / s ) 2, this coinciding with the perturbation=theory r e s u l t s (15). The r e s u l t s c o r r e s p o n d to a s a t u r a t i o n m o d e for a t w o - l e v e l s y s t e m in a l o w - f r e q u e n c y c l a s s i c a l field with adiabatic s t a r t - u p of the interaction. In conclusion, the author wishes to thank V. P. Krainov for c r i t i c i s m and for a n u m b e r of useful r e m a r k s . LITERATURE

CITED

N. B. Delone, P r e p r i u t No. 146, Fiz. Inst. Akad. Nauk SSSR i m . P. N. Lebedeva, Moscow (1974). S. Stenholm, P h y s . Rep., 6__CC(1973). E. T. J a y n e s and F. W. Cummings, P r o c . IEEE, 51, 89 (1963). J. H. Shirley, P h y s . Rev., 138B, No. 4, 979 (1965). S. Stenholm, J. P h y s . B: Atom. Molec. Phys., 5, 878, 890 (1972). D. F. Z a r e t s k i i and V. P. Krainov, Zh. ]~ksp. T e o r . Fiz., 6_./7,No. 2, 537 (1974); 6_/7, No. 10, 1301 (1974). N. Polonsky and C. Cohen-Taunoudji, J. Phys., 26, 409 (1965). R. J. Glauber, P h y s . Rev., 131, 2766 (1963). H. B a t e m a n and A. E r d e l y i , Higher T r a n s c e n d e n t a l Functions, M c G r a w - H i l l (1953). Ya. B. Zel'dovich, Zh. l~ksp. T e o r . F i z . , 51, 1492 (1966). V. I. Ritus, Zh. Eksp. T e o r . Fiz., 51, 1544 (1966). A. O. Melikyan, Zh. ]~ksp. T e o r . Fiz., 6..88, No. 4, 1228 (1975). V. P. Yakovlev, Zh. Eksp. T e o r . Fiz., 67, No. 8, 921 (1974).

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

DEVELOPMENT TRAVELING N.

B.

OF

LOCAL

PERTURBATIONS

IN SELF-FOCUSING Baranova

and

B.

IN A WAVE

MEDIA Ya.

Zel'dovich

UDC 538.56

In a l i n e a r i z e d approximation, the space - t i m e evolution of local amplitude and phase p e r t u r b a tions in a plane m o n o c h r o m a t i c wave has been investigated for a nonlinear d i s p e r s i v e m e d i u m . Asymptotic e x p r e s s i o n s have been d e r i v e d for the G r e e n ' s functions which c h a r a c t e r i z e the r e spouse to m o n o c h r o m a t i c p e r t u r b a t i o n s as well as to p e r t u r b a t i o n s which a r e t i m e - l o c a l i z e d and which a r e o n e - d i m e n s i o n a l or t w o - d i m e n s i o n a l along the t r a n s v e r s e c o o r d i n a t e s . A r e a s of wave instability w e r e found in the s p a c e of t r a n s v e r s e - w a v e v e c t o r s , p e r t u r b a t i o n - f r e q u e n c y shifts (Iqt, ~2), and t r a n s v e r s e coordinates and time ( Ir l , T) for s e l f - f o c u s i n g and s e l f - d e f o c u s i n g media with different signs of d i s p e r s i o n . 1.

INTRODUCTION

Wave instability in a nonlinear r e a c t i v e m e d i u m was f i r s t investigated by Bespalov and Talanov in [1, 2]. Recently, the question of s m a l l - p e r t u r b a t i o n buildup in the amplitude of a powerful light wave in s e l f - f o c u s i n g media acquired g r e a t p r a c t i c a l i m p o r t a n c e in connection with the c r e a t i o n of powerful s o l i d - s t a t e l a s e r s [3, 4]. Most studies investigate the development of sinusoidal p e r t u r b a t i o n s . This i m m e d i a t e l y allows one to find the e x p r e s s i o n for the i n c r e m e n t . The s o u r c e s of wave field p e r t u r b a t i o n s a r e usually the m i c r o i m p u r i t i e s of the m e d i u m or the b o u n d a r i e s (diaphragms) of the optical s y s t e m . In the p r e s e n t work, within the limits of the l i n e a r i z e d theory, the b e h a v i o r of the G r e e n ' s functions (i. e., of r e s p o n s e s to localized p e r t u r b a t i o n s ) has b e e n analyzed for a s e r i e s of p r o b l e m s a r i s i n g f r o m the propagation of a powerful light wave in a nonlinear r e a c t i v e aud, generally speaking, d i s p e r s i v e m e d i u m . The s t r u c t u r e of these functions (the location of m a x i m a and minima) enables us to d e t e r m i n e the distribution of the p e r t u r b a t i o n buildup in r e l a t i o n to the original inhomogeneity. F o r the sake of c o m p l e t e n e s s of exposition, r e s u l t s a r e p r e s e n t e d both for l i n e a r P. N. Lebedev Physics Institute, Academy of Sciences of the USSR, Moscow. T r a n s l a t e d f r o m Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 19, No. 12, pp. 1833-1840, D e c e m b e r , 1976. Original a r t i c l e s u b m i t t e d July 15, 1975. This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, N e w York, N . Y . l OOl l. N o part 1 o f this publication m a y be reproduced, stored in a retrieval system, or transmitted, in any f o r m or by any means, electronic, mechanical, photocopying, | microfilming, recording or otherwise, w i t h o u t written permission o f the publisher. A c o p y o f this article is available f r o m the publisher for $7.50. ]

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