Equation of motion for the photon distribution function derived from the laser masterequation

Equation of motion for the photon distribution function derived from the laser masterequation

Volume 27A. number 3 EQUATION PHYSICS LETTERS 17 June 1968 OF MOTION FOR THE PHOTON DISTRIBUTION DERIVED FROM THE LASER MASTEREQUATION FUNCTION ...

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Volume 27A. number 3

EQUATION

PHYSICS LETTERS

17 June 1968

OF MOTION FOR THE PHOTON DISTRIBUTION DERIVED FROM THE LASER MASTEREQUATION

FUNCTION

U. GNUTZMANN and W. WEIDLICH

Institut f~r Theoretische Physik der Universitiit Stuttgart Received 8 March 1968

From the complete laser masterequation for atoms and field, an equation of motion for the field distribution function is derived, which differs from the semiclassical Fokker-Planck equation [ 1.2] by second order time derivatives. The solutions of both equations are compared.

Recently, equations of motion for d i s t r i b u t i o n functions of the field and a t o m i c v a r i a b l e s in a l a s e r s y s t e m have been d e r i v e d by different methods [3-5] f r o m the m a s t e r e q u a t i o n of the density m a t r i x . Usually t h e s e equations have a c o m p l i c a t e d s t r u c t u r e which can not i m m e d i a t e l y be c o m p a r e d with equations of the F o k k e r - P l a n c k type. In our method of t r e a t i n g the m a s t e r e q u a t i o n we finally obtain an equation of motion for the field d i s t r i b u t i o n function which may easily be solved and c o m p a r e d with the F o k k e r - P l a n c k equations d e r i v e d by s e m i c l a s s i c a l methods [1,2]. We s t a r t f r o m the m e a s t e r e q u a t i o n and the G l a u b e r P - r e p r e s e n t a t i o n of the density m a t r i x p a s given in ref. 3: p

-- ~fd2/3 --~-

I/3) P(fl,/3*,

..

")(/31

N

P = ~ Pv v=l

(i)

Pv = [PE(fl,/3*)lv +P(r(/3, P I~ l v alv - COy aov)+ + p.(/3,~*)a~vaov+P(/3 ' [3*)aovalv] + P(/3,/3",... ) explicitly depends on the atomic o p e r a t o r s . The a n s a t z (1) a c c o u n t s for the i n fluence of the a t o m i c f l u c t u a t i o n s on the field mode. The m a s t e r e q u a t i o n for p leads to a coupled s y s t e m of n o n l i n e a r p a r t i a l d i f f e r e n t i a l equations for the four t i m e dependent functions PC, Pc, p* and p [3]. T h e s e equations have now been solved i n the t i m e dependent case. In v e r y good a p p r o x i m a t i o n it i s p o s s i b l e to e l i m i n a t e the functions Pc, P*, P, and to obtain the following equation of motion for the field d i s t r i b u t i o n

function F(fl, fl*) = Tratom(P) = 2NpNE(fl,fl*): _

2

z-z

(b~+(K+v~)p)

+

g 2 N 4 2(Z m - Z)

(2)

1 +2[u(2F lu - ( a - u ) F ) ] l u = 0 + uFi~o[~p /3=

~--Y-/(Zm-~

a=

~ 2

The c o n s t a n t s Z, Zm, 2, ~, a r e d e f i n e d ~ y Z = =Nv, , /4K, Z rn = ½(ao+1), 2 = Z((ro - i f ) ; Z = 2 for 2 > 0, Z = 0 for 2 < 0. a o m e a n s the i n v e r s ion without l a s e r action, ff = VlK/ff~is the c r i t i cal i n v e r s i o n , ~±, 7 H, K a r e r e l a x a ~ o n constants of the l a s e r s y s t e m , and g is the coupling c o n stant between field and a t o m s , as defined in ref. 3. In d e r i v i n g eq. (2) no r e s t r i c t i o n s with r e s p e c t to the photon n u m b e r a r e n e c e s s a r y as w e r e made in d e r i v i n g the F o k k e r - P l a n c k equation. We r e m a r k that in the t r a n s i e n t state the e l i m i nation of Pc, P*, P, leads to additional t i m e d e pendent coefficients in eq. (2). This c a s e will be t r e a t e d explicitly in ref. 6. F o r the calculation of s t a t i o n a r y field c o r r e l a tion functions we need the G r e e n function solution F(fl,/3*,~;/31,(3~,O) of eq. (2), defined by the i n i tial condition F(/3, /3", 0 ; /31,131,0) * ~ 52(fl-/31) and

#(/3,/3", o ;/31,/3~, o) = o. The g e n e r a l solution of eq. (2) r e a d s

F(u,¢,t) = n~m AnmFnm(U)exp(-in¢)Tnm(t ) w h e r e Fnm(U) s a t i s f i e s the equation n2 i.i + + X.m)F.m

(-

(3)

: 0 179

Volume27A. number 3

PHYSICS LETTERS

with e i g e n v a l u e s Xnm , and Trim(t) h a s the f o r m O l n+Vn

Tnm(t)

exp ( - ~ m t)

=

- Ol n pVl

m a t e l y Tnm(t ) ~ exp ( - a n m t ) and (~nm ~ ?~nm" In

+

exp ( - ~ n m t)

+

O/n~Fl

17 June 1968

- olnm

this approximation, our solution completely agrees with the exact solution of the FokkerPlanck equation [1,2], which holds only in the threshold region.

K +y± Oln~

-

i

1

O- (~+~1)~-

z-~

Equation (2) a g r e e s with the F o k k e r - P l a n c k e q u a tion d e r i v e d by R i s k e n [1] and Lax [2] except f o r the s e c o n d t i m e d e r i v a t i v e P which a p p e a r s in eq. (2). C o r r e s p o n d i n g l y , the e i g e n v a l u e s ;~nm a g r e e with t h o s e d e r i v e d in r e f s . 1 and 2, while the function Trim(t) in eq. (3) contains two e x ponential t e r m s in c o n t r a s t to the solution of the F o k k e r - P l a n c k equation. Only in the t h r e s h o l d r e g i o n , w h e r e ~ n+m >> ~ n-m , we have a p p r o x i -

THEORY

OF

We wish to thank P r o f . H. Haken f o r s t i m u l a t ing d i s c u s s i o n s and the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t for f i n a n c i a l support.

Refewence8 1. H.Risken. Z. Physik 186 (1965) 85; 191 (1966) 302. 2. R.O. Hempstead and M. Lax. Phys. Rev. 161 (1967) 350. 3. W. Weidlich. H.Risken and H.Haken. Z. Physik 201 (1967) 396. 4. H.Haken. H.Risken and W.Weidlich. Z. Physik 206 (1967) 355. 5. J.P.Gordon. Phys. Rev. 161 (1967) 367. 6. U.Gnutzmann. to be published.

R A Y L E I G H S C A T T E R I N G IN I S O T R O P I C CRYSTALS R. H. ENNS

Simon Fraser University, Vancouver, Canada Received 5 March 1968

A calculation of the spectral distribution to be expected for light scattering by spontaneous temperature fluctuations in an isotropic dielectric crystal with three branches of acoustical phonons is described.

We r e p o r t h e r e on a c a l c u l a t i o n of the s p e c t r a l d i s t r i b u t i o n to be e x p e c t e d when a m o n o c h r o m a t i c light b e a m i s s c a t t e r e d by spontaneous t e m p e r a t u r e f l u c t u a t i o n s in an i s o t r o p i c d i e l e c t r i c solid with t h r e e b r a n c h e s of a c o u s t i c a l phonons. We s t a r t by s et t i n g up a t e m p e r a t u r e fluctuation at t i m e t = 0 (a " s p o n t a n e o u s fluctuation") in a gas of a c o u s t i c a l phonons and studying how it p r o p a g a t e s with t i m e . I m a g i n e that at t = 0, a p o r t i o n of the phonon gas is c o m p r e s s e d b e t w e e n two i n finite p a r a l l e l p l a t e s at - Z o and +Zo, say, in such a way that the phonon d i s t r i b u t i o n b e c o m e s N(P)(z, 0) = N(Op) + N~I)(Z , O) where

180

NtP) ( z, O) = No(P)¢1 +No(P)). ]~S(p)g. 6T(0)

KT °

0

To , - Z o < Z < Z o , otherwise

and the t e m p e r a t u r e T = T O + ST(0) f o r - Z o < Z < < Z o and T = T o , o t h e r w i s e . H e r e the i n d ex (p) l a b e l s the b r a n c h (two t r a n s v e r s e , one l o n g i t u d i nal), s the o r d i n a r y sound v e l o c i t y a n d N o the e q u i l i b r i u m a c o u s t i c a l phonon d i s t r i b u t i o n . At t i m e t = 0, the phonon gas h as no d r i f t v e l o c i t y . At a l a t e r t i m e , the phonon gas will be d e s c r i b e d l by a l o c a l t e m p e r a t u r e T ( z , t) = T o + Tl(Z, t) and i a l o cal d r i f t v e l o c i t y U(z, t). As usual, we take the l i n e a r a p p r o x i m a t i o n f o r s m a l l v a r i a t i o n s i