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INSTABILITIES IN THERMAL BATH : PHONON ENHANCEMENT AND SATURATION IN A
THREE-LEVEL SYSTEM
B. Fain
To cite this version:
B. Fain. INSTABILITIES IN THERMAL BATH : PHONON ENHANCEMENT AND SATURATION IN A THREE-LEVEL SYSTEM. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-540-C6-542.
�10.1051/jphyscol:19816158�. �jpa-00221235�
JOURNAL DE PHYSIQUE
CoZZoque C6, supple'ment au n o 12, Tome 42, dgcembre 1981 page C6-540
INSTABILITIES IN THERMAL BATH : PHONON ENHANCEMENT AND SATURATION IN A THREE-LEVEL SYSTEM
B . Fain
Chemistry Department, TeZ-Aviv University, R m t - A v i v , TeZ-Aviv, 69978, I s r a e l .
Abstract - The equations, determining temporal behavior of coupled e l e c t r o n i c m p h o n o n subsystems i n t h e electromagnetic f i e l d , a r e analyzed. The steady s t a t e solution describing saturation of the three level i n such a system i s achieved. I t i s shown t h a t laser-l ike sel f-excitation of phonons cannot be obtained though essential phonon enhancement may be gained provided c e r t a i n conditions are f u l f i l led. Field induced transparency is predicted between f i r s t two levels when t h e f i e l d s a t u r a t e s ground and highest l e v e l s , and the concentration of ions i s high.
1. Introduction.- Usually, i n various non-equil i brium processes, i t is t a c i t l y assumed t h a t phonons play a passive role. They form a thermal bath t h a t i s assumed t o be preserved a l l t h e time i n i t s s t a t e o f thermal equilibrium. B u t as i t has been s t r e s s e d i n paper [l] by t h e author, t h i s i s not always the case. Under c e r t a i n con- di tions o r a t c e r t a i n stages of t h e non-equi l i bri um process, i n s t a b i l i t i e s may a r i s e i n t h e phonon system. These i n s t a b i l i t i e s correspond t o stimulated emission of phonons. In the papers of Bron and Grill [Z], observations have been reported of a stimulated phonon emission achieved by t h e inversion of a three-level e l e c t r o n i c system of v4+ ions i n A1203. The theoretical analysis of the three-level system influenced by the strong coherent electromagnetic f i e l d , under circumstances when phonons are not necessarily i n t h e equilibrium, i s the aim of the present paper.
2. Basic equations and steady s t a t e of t h e ensemble of inhomogeneously broadened three-level systems interacting with monochromatic f i e l d and phonons.- The above mentioned paper [ l ] of t h e author was devoted t o the derivation of the equations describing temporal behavior of t h e e l e c t r o n i c subsystems coupling w i t h phonons.
The s p e c i f i c example of the e l e c t r o n i c system consisted of two-level subsystems was explored. I t i s easy t o perform generalization t o the case of many-level e l e c t r o n i c subsystems interacting with t h e electromagnetic f i e l d and phonons both. The corres- pondi ng equations have the form:
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19816158
Here omn - t h e density matrix of c e r t a i n ion (defect) i n t e r a c t i n g w i t h phonons of t h e s o l i d , E - i t s Hamiltonian, V(t) - the energy of the interaction of t h i s ion with the electromagnetic f i e l d ; n - mean values of the phonon numbers with t h e
frequency U and wave-vector q , W q - t h e relaxation r a t e of n t o i t s equilibrium
4 q q
value n i . Aq, - matrix elements of interaction operator between e l e c t r o n i c system and phonons. The sumnation i n (3) over (mk) means t h e summation over various ions with various umk, i . e . the inhomogeneous broadening i s assumed. The equations
(1)-(3) are the system of coupled equations f o r umn and n both.
We will assume t h a t each ion has three levels En<Ek<Em ( o r j u s t these levels q are e s s e n t i a l f o r the problem). These l e v e l s a r e inhomogeneously d i s t r i b u t e d and grouped with c e r t a i n bandwidths . We suppose t h a t t h e monochromatic electromagnetic f i e l d
-(,.l?"=- (4)
kd ;f. Q!, ,,
.
i s imposed on the c r y s t a l . Then t h e steady s t a t e solution may be looked f o r i n the
- - i w o t
form amn=omne ; onm=~nmeiuot where Zmn, F nm do not depend on time as well as umm, unn, ukk and n F r m t h e equation (3) we obtain the steady s t a t e equation f o r nq w i t h the solution 9'
o mm' ukk are the mean values of t h e p r o b a b i l i t i e s of the s t a t e s m and k and we assume t h a t
Gm qk + CO = 1
From equation (2)' we obtain f o r the t r a n s i t i o n p r o b a b i l i t i e s Wmk, Wkm the expressions
where
I t i s easy t o see from Eqs.(3) ,(6) t h a t t h e threshold condition of the self-excita- t i o n of phonons has t h e form (see a l s o [ l ] )
C6-542 JOURNAL DE PHYSIQUE
T h i s condition being s a t i s f i e d , the steady s t a t e solution does not e x i s t . I t can be seen from Eqs. (5) , ( 8 ) , since the q u a n t i t i e s n Wkm could not be negative.
To simplify the calculations we w i l l assume t h a t the temperature of the crystal q' equals zero ( o r i t i s very low) so t h a t n 0 =O; Wnk=Wnm=O. As t o W and Wkm, we take
q ?'k
i n t o account t h e i r dependence on U,,,,,, and ukk. (We consider the s i t u a t i o n when umn-akk can be p o s i t i v e and may lead t o t h e e s s e n t i a l enhancement of n and Ukm,
'4
Wmk, and a t the same time ukk-unn and umm-unn being negative cannot e s s e n t i a l l y influence on t h e q u a n t i t i e s W n k , Wkn, W,,, W,,). After transformations we obtain from (1)-(9) t h e following equations:
f 2 [ ( i - y ? ~ ( i - 2 ? ) - ~ + ~ i d I ( t - g ~ - ~ ( ) = ? - f 0 - ~ ) C I C *&-g?- dlj'(r-dq}
Here f C ~ V 2/li2~&; g=rq/Wq ; e= Gk/Wmn; d= hk/Wkn;
The necessary condition f o r t h e inversion of t h e populations umm and ukk (W-0) i s Wkn>rmk o r d < l . Solving equation (11) and finding n as a function of dimension- l e s s f i e l d strength f we can find a l l t h e populations unn, ukk and U,,,, from (12).
I t can be shown t h a t g is proportional to t h e concentrations of ions (defects)[l].
In t h e case of low concentrations (g&) we obtain f o r n=umm-okk and f o r unn-U,,,,,, the usual expression describing t h e saturation i n three-level system (see e.g. [3]).
3. Conclusions. - According t o (3) ,(5) ,(10) the thresh01 d f i e l d corresponding t o s e l f - e x c i t a t i o n o f phonons could e x i s t provided l-gas0 i s t h e solution of the equation (11). B u t t h e analysis shows t h a t such f i e l d does not e x i s t . The f i r s t conclusion i s t h a t i n three-level system, the self-excitation of phonons cannot be achieved. When the difference of populations n=umm-ukk begins t o increase w i t h the increasing of the f i e l d f the number o f phonons (5) and Wkm,Wmk a l s o increase.
This leads t o t h e saturation of l e v e l s m and k which i n i t s turn prevents the ful- f i l lment of the s e l f-exci t a t i o n condition (10).
The second conclusion i s t h a t , s t i l l , e s s e n t i a l enhancement of the phonon numbers may be achieved f o r g>l : n =L >>l ( f o r g > > l ) ; n = 9 >>l (d-g-2).
q 3d 9 3
The t h i r d conclusion i s t h a t a t large enough concentrations (when g > > l ) should a n t i c i p a t e t h e quenching of the l a s e r e f f e c t a t t h e levels n,k since the difference of (inverted) popul ations n,=uk-uik tends t o zero when t h e saturation e f f e c t between levels m and n holds (f2>>1): ~ l - = ( l - d ) / g - 3 d ( l - d ) / ~ ( g > > l ) and n,=g7'-d(l+g@(g-2) (d<<g-2).
And l a s t , but not l e a s t , we p r e d i c t f i e l d induced transparency i n the f i r s t two l e v e l s n and k of three level system, when f i e l d s a t u r a t e s levels n and m and g > > l . In this case u ~ ~ - u ~ ~ = ~ , + O when g- while t h e absorption of the probe l i g h t signal ( a t t h e frequency wkn) i s proportional t o unn-ukk.
R3ferences
,.
l . B. Fain, Phys. Rev. ( i n press.).
2. W.E. Bron and W. G r i l l , Phys.Rev. Lett. 40,1459(1978); W. G r i l l , i n Phonon Scattering i n Condensed Matter ed. by Humphrey J . Maris, Plenum Press ,N.Y. (1980).
3. B. Fain, Y .I. Khani n,Quantum Electronics ,V01 . I , Pergamon Press, Oxford (1969).
