RYDBERG ATOMS AND RADIATION IN A RESONANT CAVITY I, THEORY

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RYDBERG ATOMS AND RADIATION IN A RESONANT CAVITY I, THEORY

S. Haroche, C. Fabre, J. Raimond, P. Goy, M. Gross, L. Moi

To cite this version:

S. Haroche, C. Fabre, J. Raimond, P. Goy, M. Gross, et al.. RYDBERG ATOMS AND RADIATION IN A RESONANT CAVITY I, THEORY. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-265- C2-274. �10.1051/jphyscol:1982220�. �jpa-00221831�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-265

RYDBERG ATOMS AND RADIATION IN A RESONANT CAVITY I . THEORY S. Haroche, C. Fabre, J.M. Raimond, P. Goy, M. Gross and L. Moi

Laboratoire de Physique de l'Eoole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France

Résumé.- Nous analysons dans ce papier divers aspects de l ' i n t e r a c t i o n d'ato- mes de Rydberg avec une cavité électromagnétique. Nous montrons qu'un ensemble d'atomes couplés à un mode unique du champ à une température f i n i e se comporte comme un "objet quantique unique" manifestant des effets interprétables en terme de statistique de Bose ou de mouvement brownien. Nous décrivons l'ampli- f i c a t i o n t r a n s i t o i r e du champ thermique par un échantillon d'atomes de Rydberg et étudions l'évolution de la statistique des photons émis en fonction du temps. Nous montrons également qu'un atome de Rydberg isolé dans une cavité résonnante évolue suivant un régime oscillant ou amorti, suivant les valeurs respectives des constantes de temps du couplage atome-champ et champ-parois de la cavité. Nous analysons cet e f f e t comme une modification des propriétés d'émission spontanée de l'atome induite par la présence de la cavité. Des ex- périences destinées à v é r i f i e r ces effets sont en cours et seront décrites dans le papier suivant.

Abstract.- We analyze in this paper various aspects of the interaction of Rydberg atoms with resonant electromagnetic cavity. We show that an ensemble of Rydberg atoms coupled to a single field mode at a finite temperature beha- ves as a single guantum object exhibiting features of Bose-Einstein statistics and Brownian motion. We analyze the transient amplification of the blackbody field by a Rydberg sample and study the change of the photon emission statis- tics as a function of time. We also show that an isolated Rydberg atom in a cavity undergoes either an oscillating or a damped evolution according to the respective valuesof the atom to field coupling and cavity damping times and we analyze this effect in term of a modification of the vacuum field modes surrounding the atom. Experiments to check these effects are under progress and will be described in next paper.

1. Introduction.- Rydberg atom and cavity as a test system for the study of simple quantum electrod.ynamical effects

Theoretical models describing two-level atom systems interacting with a single mode of the electromagnetic field lie at the heart of modern quantum optics (1)(2) (3) C ) (5) - By considering various initial conditions for the atomic system and for the state of the field, such models are used to understand a variety of simple and important effects : optical nutation and Rabi oscillation of various kinds, statis- tics of laser and maser emission, superradiance ... Due to the intrinsically small atom to field coupling, most of these effects require very large atom or photon numbers to be observed in an actual experiment. When these numbers are reduced to small absolute values, the time constant of the atom-field evolution becomes indeed usually much longer than other characteristic times in the problem (atomic relaxa- tion or cavity mode damping time), so that models involving a single electromagnetic mode are not realistic to describe the experimental situation in these cases. For this reason, some simple effects predicted by the theory on very snail atomic system have not been observed so far. Among them, let us quote for example the modification of the spontaneous emission rate of a single atom in a resonant cavity (6) , the os- cillatory energy exchange between an isolated atom and the cavity mode (7) , the di- sappearance and quantum revival of optical nutation signals induced on a single

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982220

C2-266 JOURNAL DE PHYSIQUE

atom by a resonant f i e l d (8)(9).

There i s now a r e a l p o s s i b i l i t y o f observing a l l these e f f e c t s w i t h the deve- lopment o f Rydberg atom experiments i n resonant electromagnetic c a v i t i e s . There are a t l e a s t t h r e e f e a t u r e s o f Rydberg systems which make them p a r t i c u l a r l y a t t r a c t i v e f o r t h i s k i n d o f s t u d i e s :

( i ) Rydberg atoms are much more s t r o n g l y coupled t o r a d i a t i o n than o r d i n a r y atoms. The e l e c t r i c d i p o l e m a t r i x elements between neighbour l e v e l s are, f o r p r i n c i -

p a l quantum numbers n around 30, about t h r e e o r d e r s o f magnitude l a r g e r than i n ground o r low e x c i t e d s t a t e s . This makes t h e i n t r i n s i c c o u p l i n g o f a s i n g l e atom t o a c a v i t y mode very strong.

( i i ) Rydberg atoms are resonant f o r m i l l i m e t e r wave t r a n s i t i o n s , which corres- pond t o reasonably l a r g e low-order-mode electromagnetic c a v i t i e s , i n s u r i n g a r a t h e r l o n g i n t e r a c t i o n time - i n the us domain- f o r an atom propagating a t thermal v e l o c i t y across t h e c a v i t y .

(iii) Rydberg atoms have r e l a t i v e l y l o n g spontaneous emission times, which means t h a t t h e i r c o u p l i n g t o t h e o t h e r modes o f the f i e l d can be neglected d u r i n g t h e i n - t e r a c t i o n time w i t h the c a v i t y - s e l e c t e d s i n g l e mode.

For these reasons, Rydberg atoms and c a v i t i e s c o n s t i t u t e q u i t e new r a d i a t i v e systems, i n which a t o m - f i e l d c o u p l i n g reveals i t s e l f a t an unusual scale. The u l t i - mate goal of experiments i n t h i s area w i l l be t o study the spectacular s i n g l e atom e f f e c t s mentioned above. Even before t h i s goal i s reached, t h e r e i s already a whole

range o f new i n t e r e s t i n g e f f e c t s which have been observed and analyzed : superra- d i a n t behaviour has been obtained w i t h atomic systems made o f very small number o f atoms (I0), Rabi n u t a t i o n e f f e c t s have been induced by the presence o f v e r y small number o f photons i n t h e c a v i t y (I1), thermal r a d i a t i o n e f f e c t s have been detected i n which the atomic system behaves as a Bose gas exchanginq a very small amount o f energy w i t h the f i e l d ( I 2 )

. . .

2. O u t l i n e o f the quantum d e s c r i p t i o n f o r an N-two l e v e l atom system coupled t o a s i ng l e electromagnetic mode

We consider an ensemble of two l e v e l atoms i n t e r a c t i n g w i t h a s i n g l e f i e l d mo- de ( p r a c t i c a l l y an eigenmode o f a Fabry-Perot c a v i t y resonant w i t h t h e atomic t r a n - s i t i o n ) . The atoms a r e coupled t o t h e f i e l d v i a the well-known e l e c t r i c d i p o l e i n - t e r a c t i o n :

a and a a r e t h e a n n i h i l a t i o n and c r e a t i o n operators f o r the f i e l d , Di'

+

which i s t h e Rabi n u t a t i o n frequency associated t o the f i e l d o f a s i n g l e photon i n

t h e mode o f frequency w ( d : e l e c t r i c d i p o l e m a t r i x element o f t h e t r a n s i t i o n ; '?Y : e f f e c t i v e volume o f t h e c a v i t y ) . For sake of s i m p l i c i t y , we assume i n t h e f o l l o w i n g t h a t a l l atoms see the same f i e l d [ f ( r i ) = 1 f o r a l l r i 1s 1

.

The basic f e a t u r e o f t h e i n t r i n s i c a l l y symmetrical i n t e r a c t i o n V i s t o couple c o l l e c t i v e l y and coherently t h e atoms t o t h e f i e l d ( 4 ) ( 5 ) . When a photon i s absorbed, t h e e x c i t a t i o n i s shared by t h e whole atomic system considered as a s i n g l e quantum o b j e c t . F i g u r e 1 shows t h e r e l e v a n t atom and f i e l d l e v e l s : t h e harmonic o s c i l l a t o r - l i k e l e v e l s o f t h e f i e l d , l a b e l l e d 1, 2,

...

...

...

...

Atomic collective states

3 -

-

2- 1-

.O-

Field states

F i g u r e 1 : Schematic d e s c r i p t i o n o f t h e N-atom-single f i e l d mode system : t h e two energy scales represent r e s p e c t i v e l y t h e atomic and f i e l d s t a t e s , cohe- r e n t l y coupled by an i n t e r a c t i o n p r o p o r t i o n a l t o a. The f i e l d i s damped i n t o a thermal r e s e r v o i r w i t h a r a t e w/Q.

For a more r e a l i s t i c d e s c r i p t i o n o f t h i s e v o l u t i o n , i t i s necessary t o take a l s o i n t o account d i s s i p a t i v e processes. I n Rydberg atom experiments, t h e atoms r e - l a x a t i o n can g e n e r a l l y be neglected d u r i n g t h e c a v i t y atom i n t e r a c t i o n time, so t h a t i t i s a good approximation t o i n c l u d e i n t h e p i c t u r e t h e r e l a x a t i o n o f t h e f i e l d alone : t h e harmonic o s c ! l l a t o r r e p r e s e n t i n g t h e fie!d i s supposed t o be cou- p l e d t o a thermal b a t h reservoir a t temperature T ( d e s c r i b i n g f o r example t h e c a v i t y w a l l s ) . The e f f e c t o f t h i s c o u p l i n g a c t i n g alone i s t o b r i n g t h e f i e l d mode t o t h e r - mal e q u i l i b r i u m w i t h i n a c h a r a c t e r i s t i c time Q/w ( Q : c a v i t y q u a l i t y f a c t o r ) . T h i s

process i s described by a master equation f o r t h e d e n s i t y m a t r i x o f t h e f i e l d , whose e q u i l i b r i u m value i s the well-known Bose-Einstein d i s t r i b u t i o n . The N-atom f i e l d evo- l u t i o n r e s u l t s from a c o m p e t i t i o n between t h e coherent atom-single mode c o u p l i n g ( c h a r a c t e r i s t i c time 1/a) and i r r e v e r s i b l e f i e l d damping ( c h a r a c t e r i s t i c t i m e Q/w).

T h i s e v o l u t i o n i s a l s o described b a master equation f o r t h e combined system densi- t y m a t r i x . We w i l l n o t g i v e here X i s equation, b u t o n l y d7scuss r a p i d l y v a r i o u s ef- f e c t s p r e d i c t e d by i t s r e s o l u t i o n f o r d i f f e r e n t i n i t i a l c o n d i t i o n s .

3. S i n g l e atom e f f e c t s

L e t us f i r s t consider t h e case o f a s i n g l e atom (N = 1 ) . The energy l e v e l s of t h e combined atom and f i e l d system when t h e c o u p l i n g V i s n e g l i g i b l e (n = 0) are sketched on F i g u r e 2-a, w h i l e F i g u r e 2-b presents t h e eigenenergies o f t h e system t a k i n g V i n t o account. These "dressed" s t a t e s ( I 3 ; c o n s i s t of a ground l e v e l

lg,O

representing t h e atom i n i t s lower s t a t e lg > w i t no photon, and o f a succession o f two l e v e l manifolds separated from each o t h e r by t h e energy Mw o f a photon. The

( 2 - 2 6 8 JOURNAL DE PHYSIQUE

nth m a n i f o l d i s made o f two dressed e i g e n s t a t e s

\+

-

t i o n s of t h e unperturbed le,n > and ] g, n + l > l e v e l s ( e : e x c i t e d atomic s t a t e ) :

-

+ n > = - [ [ e , n > 2 lg, n + l > ] ( 3 )

I-

The energy separation between these two l e v e l s i s 2 b R m . The atomic system evolu- t i o n can be very simply described i n t h i s dressed atom b a s i s :

F i g u r e 2 : S i n g l e two l e v e l atom-single f i e l d mcde system a t resonance (wo = w ) a) Energy l e v e l s i n t h e uncoupled case ( a = 0)

b) Dressed energy l e v e l s t a k i n g t h e c o u p l i n g 0 i n t o account.

3.1. Spontaneous e f f e c t s

---

L e t us assume t h a t the atom i s prepared a t time t = 0 i n l e v e l l e > i n an empty c a v i t y mode ( n = 0, T = 0 K

.

1

d i s s i p a t i o n - f r e e system evo u t i o n corresponds t o a quantum mechanical oscillation between t h e unperturbed combined s t a t e s l e y 0 > and lg, 1 >. I n t h e dressed atom p i c - t u r e , t h i s o s c i l l a t i o n appears as the e v o l u t i o n o f a coherent s u p e r p o s i t i o n o f sub- s t a t e s i n t h e n = 0 manifold. The i n i t i a l s t a t e [e, 0 > i s indeed a s u p e r p o s i t i o n o f t h e l e v e l s

1-

0 TIME 0 TIME

F i g u r e 3 : S i n g l e two-level atom i n a resonant c a v i t y . a) o s c i l l a t i o n o f t h e proba- b i l i t y P,(t) i n case o f weak c a v i t y damping (w/? = 0.2 R; T = 0 K) b) i r r e v e r s i b l e decay o f P e ( t ) i n case of s t r o n g c a v i t y damping (w/Q = 5n; T = 0 K)

Q A 3 r where r = expression [Equ. 2 ] , one can w r i t e indeed

r,,,

-

4a2 W SP SP 8.rr2

- - d2 i s t h e spontaneous r a t e o f emission between l e v e l s l e > and l g > i n

free

k o h 3

space. The above r e l a t i o n between r ca v _and r S P had been p r e d i c t e d long ago by P u r c e l l ( 6 ) who had n o t i c e d t h a t t h e number o f o s c i l l a t o r modes per u n i t frequency i n t e r v a l

8.rr2 v 2

a v a i l a b l e f o r spontaneous emission i s equal t o -

- ^V

3 - 3

Q/v i n a resonant c a v i t y , t h e r a t i o o f these two q u a n t i t i e s accountin! ~ r e c i s e l y L f o r the f a c t o r rcav /rsp > 1. The f a s t damping o f P e ( t ) i n t h e c a v i t y i s thus n o t h i n g b u t the m o d i f i c a t i o n of f r e e space spontaneous emission induced by the presence o f t h e c a v i t y w a l l s which changesthe boundary c o n d i t i o n s f o r t h e vacuum f i e 1 d surround- i n g the atoms. Note t h a t t h e opposite e f f e c t (suppression o f spontaneous emission i s a non-resonant c a v i t y ) has r e c e n t l y been discussed by 0. Kleppner i n t h e c o n t e x t o f Rydberg atom experiments ( I 4 ) .

3.2. Blackbody r a d i a t i o n induced e f f e c t s

...

A t a f i n i t e temperature T # 0 K, the above e f f e c t s a r e complicated by t h e oc- curence o f absorption and induced emission o f thermal photons s t o r e d i n t h e c a v i t y . The atomic system evolves according t o an o s c i l l a t o r y o r t o an i r r e v e r s i b l y damped t r a n s i e n t regime towards t h e f i n a l s t a t e d i s t r i b u t i o n corresponding t o thermal equi- l i b r i u m a t temperature T. The nature o f t h i s t r a n s i e n t regime depends upon t h e ave-

Iw/ kBT -1

rage number ?r = [ e

-

I n t h e case o f weak c a v i t y damping (Q >> w/Q), t h i s t r a n s i e n t regime can be expanded along a sum o f elementary Rabi o s c i l l a t i o n terms. The modulated p a r t o f P e ( t ) can be w r i t t e n as ( 1 5 ) ( 1 6 ) :

I n t h i s expression, r n ( t ) i s a damping parameter p r o p o r t i o n a l t o w/Q and depending upon n. Equation (4) merely expresses t h e f a c t t h a t t h e atomic system o s c i l l a t e s i n a f i e l d i n which t h e number o f photons i s a random q u a n t i t y obeying t o the Bose- E i n s t e i n s t a t i s t i c s . The t i m e dependance o f Equation ( 4 has been s t u d i e d i n d e t a i l s i n t h e l i m i t i n g case o f a p e r f e c t l y r e f l e c t i n g c a v i t y ( )

4

4. C o l l e c t i v e behaviour o f N-atoms i n t h e c a v i t y a t T ⁼ 0 K

The s i n g l e atom e f f e c t s described above a r e e a s i l y g e n e r a l i z e d t o N atom sys- tems. L e t us consider f i r s t t h e e v o l u t i o n o f an ensemble o f N two l e v e l atoms s t a r t - i n g a t t i m e t = 0 from t h e upper s t a t e ( ~ ~ ~ ( 0 ) > = l e , e, e,

. . .

...

The exact shape o f t h e o s c i l l a t i o n depends upon w i t h an a p p a r e n t l y complicated b e a t i n g p a t t e r n f o r r e l a t i v e l y small N values ( ) (see F i g u r e 4). I n t h e presence o f c a v i t y r e l a x a t i o n , t h i s o s c i l l a t i o n i s damped. It can be observed as l o n g as 0. ^V% > w/Q. On t h e c o n t r a r y , f o r s t r o n g damping o r small atom numbers (Q fl < w/Q)

C2-270 JOURNAL DE PHYSIQUE

t h e atomic system energy i r r e v e r s i b l y decays according t o a monotonous law (17). The decay process i s governed by a c h a r a c t e r i s t i c c o l 1 e c t i ve decay r a t e T R - ~ = 4 . Q 2 ~ ~ / w which i s equal t o N times t h e s i n g l e atom c a v i t y enhanced decay r a t e rcav. I n t h e same way as t h e s i n g l e emission r a t e i s enhanced by t h e presence o f t h e c a v i t y , t h e c o l l e c t i v e superradiance r a t e of t h e N atom system, which i s equal t o Nrsp i n f r e e space, i s increased by t h e f a c t o r (3/47r2) Q X ~ / Z P and becomes equal t o NTcav i n t h e c a v i t y (see s e c t i o n 6).

pelt] F i u r e 4 : N = 5 atoms i n a resonant c a v i t y w i t h -

-%r

5. C o l l e c t i v e a b s o r p t i o n o f blackbody photons i n t h e c a v i t y a t T # 0 K ( 1 2 ) -

L e t us consider now t h e problem o f blackbody r a d i a t i o n absorption by t h e N-atoms i n t h e c a v i t y . The atomic system i s i n i t i a l l y i n i t s lower s t a t e (g, g,

...

-

ylwl kBT

-

m = E =

This c o m e s ~ o m t h e f a c t t h a t t h e atoms are n o t absorbing independently from each o t h e r t h e r a d i a t i o n i n t h e c a v i t y , b u t r a t h e r c o l l e c t i v e l y . The existence o f c o l l e c - t i v e energy l e v e l s f o r t h e atomic system does indeed i m p l y s t r o n g d i p o l e - d i p o l e cor- r e l a t i o n between t h e atoms. These c o r r e l a t i o n s , which a r e p h y s i c a l l y produced b y t h e s p a t i a l coherence o f t h e thermal f i e l d i n t h e c a v i t y mode, make the,system behave as a s i n g l e quantum o b j e c t made o f i n d i s c e r n a b l e p a r t i c l e s . I t thus obeys, as does t h e f i e l d , a Bose-Einstein s t a t i s t i c s . I t i s w e l l known t h a t a gas o f bosons reaches thermal e q u i l i b r i u m by absorbing much l e s s energy than a gas o f independant atoms. T h i s e x p l a i n s t h e s a t u r a t i o n of t h e a b s o r p t i o n process t o a v e r y small l i m i t corresponding t o an e x c i t a t i o n p r e c i s e l y equal t o t h e number of black- body photons i n t h e c a v i t y mode

...

6. O u t l i n e o f t h e Bloch v e c t o r d e s c r i p t i o n f o r an ensemble of N-atom coupled t o a s i n g l e c a v i t y mode

The f u l l quantum mechanical d e s c r i p t i o n i n terms o f a s t a t e v e c t o r o r a d e n s i t y operator f o r t h e compound atom

+

system e v o l u t i o n can then be a l t e r n a t i v e l y described i n a c l a s s i c a l way, using t h e concept o f Bloch v e c t o r ( l a ) . We present here q u a l i t a t i v e l y t h i s p o i n t o f view, and 1 im i t ourselves - f o r sake o f s i m p l i c i t y - t o t h e case o f a p u r e l y symmetrical cou- p l i n g t o t h e f i e l d ( f ( r i ) = 1 f o r a l l rils )

.

"down" p o s i t i o n s correspond t o t h e f u l l y e x c i t e d and deexcited atomic s t a t e s respec- t i v e l y . During t h e system e v o l u t i o n , t h e angle 8 made by t h e Bloch v e c t o r w i t h t h e

"up" d i r e c t i o n changes i n t i m e and t h e system e v o l u t i o n i s determined by t h e 8 ( t ) law. The p o p u l a t i o n d i f f e r e n c e between t h e e x c i t e d and deexcited s t a t e s i s equal t o N cose whereas t h e g l o b a l atomic e l e c t r i c d i p o l e i s p r o p o r t i o n a l t o N sine

.

l u t i o n o f 8 can be d e r i v e d r i g o r o u s l y by s t a r t i n g from t h e Heisenberg equations o f motion f o r t h e operators a, a+, Di' and by going t o a c l a s s i c a l l i m i t v a l i d f o r l a r g e N's and amounting t o n e g l e c t i n g t h e commutators o f these operators. T h i s equa-

t i o n can a l s o be i n f e r r e d by i n t u i t i v e arguments p a r t l y based on energy conserva- t i o n requirements (I9). A t T = 0 K, one f i n d s f o r 8 ( t ) t h e f o l l o w i n g equation :

d28 w d8 w

- + -

- - -

d t * 2Q d t 4QTR

- 1

where TR has been d e f i n e d above. This equation describes t h e e v o l u t i o n o f a f i c t i - t i o u s pendulum damped by a viscous drag term

1 !&

I

For T = 0 K, t h e departure from e q u i l i b r i u m comes from the f l u c t u a t i o n s i n h e r e n t t o spontaneous emission. I t can be shown ^{( I 9 )} t h a t these f l u c t u a t i o n s a r e c o r r e c t l y described by assuming a small randon non zero i n i t i a l value f o r e, obeying a Gaussian s t a t i s t i c s . The e v o l u t i o n o f the system down from t h i s i n i t i a l s t a t e depends upon t h e r e s p e c t i v e value o f w/Q and TR (i.e. ^w/Q and Q fl). I f w/Q < T ~ , -t h e decay ~ occurs according t o an o s c i l l a t i n g regime. I f on t h e c o n t r a r y w/Q > T ~ - ' , one can n e g l e c t i n Equation (5.a) t h e d28/dt2 term and t h i s equation becomes :

1

de

-

T h i s equation describes t h e i r r e v e r s i b l e e v o l u t i o n o f an overdamped pendulum which monotonically decays from 0 t o IT w i t h o u t undergoing any o s c i l l a t i o n . One thus r e - t r i e v e s i n t h i s c l a s s i c a l model t h e general conclusions o f t h e f u l l quantum des- c r i p t i o n ( s e c t i o n 4).

7. The Brownian motion o f t h e atomic p o l a r i z a t i o n on t h e Bloch sphere f o r T f 0 K I n t h e overdamped regime a t T # 0 K, Equation (5-b) has t o be completed by an e x t r a term accounting f o r t h e e f f e c t o f a small c h a o t i c r a d i a t i o n f i e l d a c t i n g on t h e atomic p o l a r i z a t i o n . I f Ti << N (much l e s s blackbody photons than atoms i n the c a v i t y ) , t h e a c t i o n o f t h i s f i e l d i s o n l y important i n t h e case where 8 i s c l o s e t o 0 o r a so t h a t s i n 8 i n Equation (5-b) can be replaced by 8 o r a - 8 (as soon as 8 departs from these values, t h e s i n 8/2 TR term, which describes t h e e f f e c t of t h e s e l f r a d i a t e d f i e l d i n t h e system, s t a r t s t o d r i v e t h e atomic e v o l u t i o n and t h e thermal f i e l d can be neglected). One can thus r e s t r i c t t h e d e s c r i p t i o n o f t h e e f f e c t o f thermal r a d i a t i o n on l i n e a r i z e d equation o b t a i n e d from Equation (5-b) by r e p l a - c i n g s i n 8 by 8 o r a

-

C2-272 JOURNAL DE PHYSIQUE

This amounts t o d e f i n i n g two independant p a r t i a l Bloch angles ex and By. I f t h e Bloch v e c t o r i s i n i t i a l l y i n t h e up p o s i t i o n ( f i g u r e 6-b), 8 i obey t h e f o l l o w i n g l i n e a r i z e d equation ( i = X,Y) :

doi ei - d

ggi

d t

Pr

2 T ~

i n which gB i ( t ) a r e two o u t o f phase random components o f t h e blackbody f i e l d i n t h e c a v i t y . I f , on t h e contrary, t h e Bloch v e c t o r i s i n i t i a l l y deexcited ( f i g . 6-c), one d e f i n e s a i = r - Bi and t h e a i obey t h e equation :

Equation (6-b) i s f o r m a l l y i d e n t i c a l t o t h e Langevin equation o f a Brownian p a r t i c l e o f v e l o c i t y cai , and mass m = ~ f i w / 2 c ~ s u b j e c t e d t o a r e s t o r i n g f o r c e ( - a i / 2TR) and t o a f l u c t u a t i n g f o r c e Fi such t h a t Fi ( t ) / mc = d gB i ( t ) / 1

.

2

according t o e q u i p a r t i t i o n theorem, towards kRT. As a r e s u l t , t h e number o f e x c i t e d atoms i s l i m i t e d by kBT/flw and we f i n d again t h e r e s u l t o f t h e quantum model (sec- t i o n 5 ) . The absorption o f t h e blackbody f i e l d i n t h e c a v i t y can thus be p i c t u r e d as t h e Brownian motion o f t h e atomic p o l a r i z a t i o n immersed i n a b a t h o f thermal photons

...

Moreover, t h e f l u c t u a t i o n s o f e2 around ?? reproduce i d e n t i c a l l y t h e f l u c t u a t i o n s o f t h e number o f photons n around F, so t h a t the s t a t i s t i c s o f the Bloch angle i s given by a Gaussian law :

V

w i t h an average value F2 i n c r e a s i n g e x p o n e n t i a l l y w i t h t i m e (see Equation 7 ) . This r e s u l t holds o n l y d u r i n g t h e e a r l y stage o f the e v o l u t i o n , when B i s small.

8. S t a t i s t i c s o f superradiant emission t r i g g e r e d by t h e blackbody f i e l d

We conclude t h i s a n a l y z i s o f t h e Bloch v e c t o r model by a d e r i v a t i o n o f t h e Bloch v e c t o r average e v o l u t i o n and f l u c t u a t i o n s i n a superradiant process down t h e 8 = 0 i n i t i a l p o s i t i o n . I t i s convenient t o d i v i d e t h e system e v o l u t i o n i n t o two stages ( 2 0 ) ( 1 6 ) : an i n i t i a l l i n e a r phase near 8 = 0, described as a Brownian d i f f u - sion, and a subsequent non l i n e a r phase, d u r i n g which t h e main p a r t o f t h e emission occurs and t h e e f f e c t o f the blackbody f i e l d can be neglected. These two steps a r e connected a t an a r b i t r a r y time to = qTR ( q = a few u n i t s ) . L e t us consider a p a r t i - c u l a r r e a l i z a t i o n o f t h e process, corresponding t o a s p e c i f i c value 8, o f 8 a t time to. The subsequent e v o l u t i o n o f t h e system from time to on i s described by the non l i n e a r equation (5-b) w i t h t h e i n i t i a l c o n d i t i o n 8 ( t o ) = 8,. A f t e r a s t r a i g h t f o r w a r d i n t e g r a t i o n , one f i n d s f o r t h i s case :

[ t - ;y;o)

-

with t D ( e o ) = -2TR L0g(e0/2) (10)

This "hyperbolic-tangent solution" shows t h a t the atomic energy i s i r r e v e r s i - bly damped a f t e r a delay of the order of t D ( e o ) within a time of a few T R . Due t o the 8, f l u c t u a t i o n , t h i s delay i t s e l f fluctuates from one r e a l i z a t i o n of the expe- riment t o the next, the "average" system evolution being obtained by replacing in Equation (9) O0 by i t s average mean square root value

$

=---

.

n + l

The fluctuations of 8 around i t s average a t time t are c l e a r l y related t o the one of t h e number of photons emitted by t h e system between times 0 and t , which is equal t o N (1 - cos 8 ) . Let us c a l l P(n,t) the probability f o r t h i s number t o be equal t o

k

1 N - t / T R - t / T R

P ( n , t ) =-(-- _{(ii+l) N-n} )' e ^{x exp}

EJ-

1

Figure 5 represents the P ( n , t ) d i s t r i b u t i o n a t various times, e i t h e r s h o r t e r o r longer than tD. A t the beginning of the emission, the photon s t a t i s t i c s c l e a r l y obeys a Bose-Einstein law c h a r a c t e r i s t i c of an amplified thermal f i e l d , whereas a t l a t e r times, the non-linear character of the system evolution equations leads t o a probability d i s t r i b u t i o n with a maximum peaked a t a n f 0 value (Poisson-like s t a - t i s t i c s typical of a coherent f i e l d ) .

Of course, a1 1 the properties described i n t h i s paper are not a priori r e s t r i c t e d t o Rydberg atoms. Clearly, t h e orders of magnitude of the atom t o f i e l d coup1 ing make them however impossible o r very d i f f i c u l t t o observe on ordinary atomic systems. In the next communica- t i o n , we describe experiment performed on Rydberg atom in which we have veri- f i e d several aspects of the theory outlined above. Researches a r e i n progress t o check o t h e r predictions of the theory, in p a r t i c u l a r those concerning the behaviour of single atoms in c a v i t i e s .

Figure 5 : Probability P ( n , t ) of emission of n photons between times 0 and t as a function of n f o r d i f - f e r e n t t values, equal respectively

t o : a ) 0.58 tD; b) 0.65 tD;

0

C ) 0.71 t ~ ; d) 0.88 t ~ ; e ) 0.95 tD; f ) 1.22 tD.

C2-274 JOURNAL DE PHYSIQUE

References

ALLEN L. and EBERLY J.H., O p t i c a l Resonance and Two Level Atoms (Wiley, New York, 1975)

JAYNES E.T. and CUMMINGS F.W., Proc. I n s t . E l e c t . Eng.

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2

SCHARF G., Helv. Phys. Acta,

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TAVIS M. and CUMMINGS F.W., Phys. Rev.

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HAROCHE S., i n Atomic Physics 7, D. Kleppner and F.M. P i p k i n e d i t o r s (Plenum Press, New York), 1980

MEYSTRE P., GENEUX E., QUATTROPANI A. and FAIST A., Nuovo Cimento, 625 (1975), 521; EBERLY J .M., NAROZHNY N.B. and SANCHEZ-MONDRAGON J. J., Phys. R e C L e t t . 44 (1980), 1323

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(9) KNIGHT P.L. and RADMORE P.M., Phys. L e t t e r s ( t o be published)

( l o ) GROSS M., GOY P., FABRE C., HAROCHE S. and RAIMOND J.M., Phys. Rev. L e t t e r s 43 (1979), 343 and r e f . ( 7 ) above

-

( I 1 ) HAROCHE S., GOY P., RAIMOND J.M., FABRE C. and GROSS M., Proceedings o f t h e

Royal Society, t o be p u b l i s h e d

( I 2 ) RAIMOND J.M., GOY P., GROSS M., FABRE C. and HAROCHE S., Phys. Rev. L e t t e r s 49 (1982), 117

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( l 3 ) HAROCHE S. , Annales de Physique,

5

( I 5 ) CUMMINGS F.W., Phys. Rev.

140

8

(1975), 95

('6) HAROCHE S., Proceedings o f Les Houches Summer School, New Trends i n Atomic Physics (1982 session; t o be published by North Holland)

(17) For a quantum mechanical d e s c r i p t i o n o f N-atom system damping i n a c a v i t y , see BONIFACIO R., SCHWENDIMANN P. and HAAKE F., Phys. Rev. A

4

( I 8 ) For t h e connexion between t h e quantum mechanical and t h e Bloch v e c t o r approach, see f o r example BONIFACIO R., KIM D.M. and SCULLY M.O., Phys. Rev.

187

( I 9 ) M O I L., GOY P., GROSS M., RAIMOND J.M., FABRE C., HAROCHE S., Phys. Rev. A, t o be published (1982)

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(21) A s i m i l a r s t a t i s t i c s o f photon emission i s found a t T ⁼ 0 K (spontaneous emis- s i o n t r i g g e r i n g ) . See f o r example DE GIORGIO V. and GHIELMETTI F., Phys. Rev.

A 4 (1971), 2415. The s t a t i s t i c s o f i n i t i a l f l u c t u a t i o n s being Gaussian i n both cases (T = 0 K o r T # 0 K), i t i s n o t s u r p r i s i n g t o f i n d t h e same law f o r P(n,t).