THE ROLE OF DISPERSIVE FORCES IN THE ONSET OF SUPERRADIANT BEATING

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THE ROLE OF DISPERSIVE FORCES IN THE ONSET OF SUPERRADIANT BEATING

F. Brugè, C. Leonardi, A. Vaglica

To cite this version:

F. Brugè, C. Leonardi, A. Vaglica. THE ROLE OF DISPERSIVE FORCES IN THE ONSET OF SUPERRADIANT BEATING. Journal de Physique Colloques, 1986, 47 (C6), pp.C6-321-C6-326.

�10.1051/jphyscol:1986639�. �jpa-00225882�

THE ROLE OF DISPERSIVE FORCES IN THE ONSET OF SUPERRADIANT BEATING

F. B R U G ~ , C. LEONARDI* and A. VAGLICA*

Gruppo CRRNSM, Istituto di Fisica delllUniversita, via Archirafi 36, I-90123 Palermo, Italy

*Gruppo Nazionale CNR e Centro Interuniversitario MPI di Struttura della Materia, Istituto di Fisica delllUniversita, Via Archirafi 36, 1-90123 Palermo, Italy

ABSTRACT - Superradiance is a stochastic process which shows macroscopic shot to shot fluctuations in some of the properties of the emitted pulse.

These fluctuations may have microscopic and quantum-mechanical origin.The aim of this paper is to discuss a simple and hopefully intriguing example of one of the approaches used to deal with the statistical properties of this process.

The recovery of equilibrium by a gas of atoms initially pumped to an excited le- vel is usually described as a chaotic process,in which the time evolutions of the mi- croscopic electric dipoles associated with the transition spontaneously performed by the atoms are independent from each other. No sizable polarization would develop in the gas,and the process follows the single-atom exponential decay law. On the contra- ry,in coherent spontaneous emission (superradiance,superfluorescence) the motion of the dipoles is correlated and the decay becomes a cooperative process,since electric dipoles pile up to give a macroscopic polarization able to produce a coherent field.

A simple theoretical model able to show such a behaviour is the celebrated Dicke mo- del for superradiance /l/,in which the atoms are described as two-level sources con- tained in a volume whose linear dimensions are small compared to the resonant wave- length. The aim of this assumption is to obtain a situation in which all atoms expe- rience the same radiation field during the whole decay and,consequently,evolve in ti- me identically. However,in these conditions dispersive forces among the emitting di- poles are present and,according to the geometry of the active region,may lead to de- cohering or recohering effects on the evolution of the atomic system 121. We investi- gate here the statistical properties of the early stages of the decay process in an idealistic recohering case and we show that the r6le of dipole-dipole interaction is particularly transparent when two slightly different radiation frequencies are pre- sent,as in the case of superradiant beating /3/,/4/.

Before going to details of this specific problem,let us comment shortly on the statistical description of superradiance. Coarsely speaking,the time evolution of a superradiant gas can be divided into three parts which are associated respectively to the linear quantum beginning of the decay,the subsequent linear classical evolu- tion and the non linear classical growing up of the superradiant intensity. The li- neariry refers to the dependence of this intensity on the number N of the sources.

In the case of transition frequencies sufficiently high or low enough temperatures,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986639

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the first part of the decay is dominated by single-source spontaneous emission and is characterized by quantum fluctuations. The second and the third parts differ from each other because of the number of the emitted photons which is larger than unity but small compared whith N in the second,and comparable with N in the third. In both of them superradiant decay shows sizable shot to shot fluctuations in the time evo- lution of the emitted pulse,which have their origin in the microscopic fluctuations at the beginning of the process. Differently from the non linear regime,the first and the second parts of the decay can be easely described by analytical means and we shall limit our discussion to the statistical properties of the time evolution of the atomic system during the linear regime. A very useful tool to combine quantum and classical aspects,both present during this regime,is provided by quasi-probabi- lity distribution functions,often used in quantum optics and laser theory 151. This kind of approach has been used by Glauber and Haake / 6 / and Polder et al. 171. Their analysis are different from each other because of a different choice of the order in which electric field operators appear in the equations of motion for the atoms 181.

Actually,according to whether the chosen order is norma1,antinormal or symmetric,the quantum onset of superradiant decay can be attributed to the fluctuations o$ the e- lectric dipoles associated to the atoms / 6 / , / 9 / , to the vacuum field fluctuations 171, or to a superposition of them /lO/,respectively. Therefore,according to the order o- ne obtains different physical pictures of the same quantum process,which are never- theless all correct and selfconsistent. In the antinormal ordering picture here ado- pted,the time evolution of the atomic system can be described by quantum Langevin e- quations in which stochastic forces appear which simulate the action of the vacuum e.m. field on the atoms and supply the necessary push to start the decay of the ato- mic system from its fully excited configuration. In the following we shall refer to this as Langevin picture.

Recently,this approach has been used to give a statistical description of the ge- neration of superradiant beats Ill/. A simple model able to produce these beats is provided by a modified version of the Dicke model /l/,in which the N two-level atoms are divided into two groups. Each V (v=1,2) group has N/2 atoms which are different from those belonging to the other group by the energy spacing between the two levels, only. The two emitted superradiant fields would have different frequencies and their superposition leads to modulation of the intensity of the pulse,or b~ating. In this mode1,each group of atoms can be described by a collective operator RV whose compo- nents RV (q=+,z) obey angular momentum commutation relations and whose length is RV = N/4. As promised above,we shall describe the behaviour of the atomic system by ; looking at the time evolution of the quasi-probability distribution function

where ,and nv are numerical variables associated to RV;- ,Rv;+ ,Rv;, ,respec- tively,and the superscript (a) means that atomic operators in (1) are antinormally ordered 1121. The quantum - classical correspondence /13/ connected with the use of quasi-probability distribution functions can be drawn by considering the equation

obtained by ordering R,,;? operators appearing in A according to the chosen order, and replacing them by their associated numerical variables. It has been shown 1111 that in the linear region the quasi-probability distribution has the form

where we have introduced the correlation functions

and it is

Q = R (41 ) R ( 2 , 1 )

-

The meaning of Eq.(3) is particularly transparent in the case of large values of the detuning parameter

A = la,-w,l% (5)

where wv is the resonant frequency of the atoms in the v group and TR is the super- radiant characteristic time for zero detuning. In these conditions,R(v,q) and R(v,v) for v#q vanishes and

~ " ~ t l =

?"'(p,,&,

P ' " ' V ~ , P ~ ; ~ )

factorizes into the distributions describing the motion of the two groups. This e- quation is valid for t>(2.rr/lw,-w2 1 ) ^{and it is}

pWq4

)p:

r-p4~:/~(v,v){

According to these results,the transverse components of the two macroscopic vectors

'Av under the action of the vacuum field fluctuating force execute brownian motions 1141 which are statistically independent from each other. On the contrary,for smal- ler detunings ( A%2a),the two motions are correlated since in these conditions the effects on each group of radiation field emitted by the other are no longer negligi- ble. A suitable physical quantity to describe such a behaviour is the phase cp of the intensity modulation,which corresponds in our model to the angle between the two macroscopic emitting dipoles which are proportional to the transverse parts of 3 , and 8, . The distribution for V ,P(cp;t),can be derived from ~ ( ~ ) ( t ) taking BV = pv x

exp(icpv) and integrating over all the atomic variables exceptcp=cp,- cp,

.

The results discussed so far have been obtained neglecting dispersive dipolar forces /15/,/16/ among the atoms and taking into account dissipative forces,only. We wish now to describe the main modifications of the statistical properties of the e- volution of the two groups of atoms in the particular case in which dipole-dipole in- teraction is effective only between detuned atoms and is identical for any couple of them. In these conditions,one can consider dipolar interaction as occurring between

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the transverse components of 1, and 3 , . One could expect that during the linear re- gime the effects of this interaction are more evident in the beating case than for A+O. In fact,the main effect of recohering interaction in this last case amounts to a time dependent frequency shift of the emitted radiation (chirping),which is rela- ted to the variation of the population of the atomic levels during the decay 141. It is likely that these effects are negligible in the linear part of the decay,during which the variation of the level populations is vanishing small. On the other hand when,as in the case here considered,two frequencies are present,the effects of re- cohering interaction between detuned atoms can be conspicuous even in the linear re- gion. In fact,dipolar forces compete with the detuning which try to disalign tran- sverse dipole components belonging to unlike atoms. A parameter suitable to describe such a competition is

6 = ( N y T R / k / \ ) ($1

where y is the dipole-dipole coupling between any couple of detuned atoms. In the Langevin picture,the time behaviour of can be summarized as follows. For y=O the phase distribution,which is initially very sharp around the value q=O,moves because of the free precession toward positive values (U, >w,)and quickly broadens under the influence of the stochastic vacuum e.m. field. As shown in Fig.l(a),this diffu- sion process occurs during the quantum part of the decay (t<TR) and whether the di-

stribution attains its uniform limit or not depends on the detuning. In fact,for A>1 and y=(t/TR)>l,P(qI;t) can be written in the form 1111

Figure 1. - Time variation of the normalized phase uncertainty f o r A = Z n and foc three different values of the parameter U. The curves refer to o=O (a),

- 5 (b), 1 (c). (AV): = 71'13 is the uncertainty for uniform distribution

between - n and a .

of o as in Fig.1

.

which shows that for large detuning the diffusive motion randomize the phase comple- tely. In the intermediate case,a trace of the drift motion of the phase due to pre- cession is still present and,depending on time,some values of are more likely than others. Dipole-dipole interaction modifies both the diffusive and the drift motion of the phase. For o#O,as shown in Fig.1, (b) and (c) ,phase uncertainty is less than for o=O and decreases with time. This means that the spreading of the phase proba- bility due to the stochastic forces during the quantum regime is compensated at classical times (t>TR) by a narrowing process due to dipolar forces. Moreover,as shown in Fig.Z,(b) and (c),the probability of having q=O increases gradually with time,indicating that the two transverse dipoles are going to remain aligned and the drift motion due to precession tends to be suppressed. In conclusion we wish to re- call that the results here reported hold in the case in which dipolar forces have recohering effects. however,it is likely that the kind of statistical approach here adopted can be extended to describe the r6le of dipole-dipole interaction in the on- set of superradiant decay in more general cases.

REFERENCES

2. R.FRIEDBERG,S.R.HARTMANN,3.T.MANASSAH,Phys.Lett .A%,365 (1972) ;Phys .Rep.l, 102 (1973) ; R. FRIEDBERG,S .R.HARTMANN,Phys .Rev.AE, 1728 (19741 ; R. FRIEDBERG,B.COF- FEY ,Phys .Rev.A13,1645 (1976) ; B.COFFEY ,R.FRIEDBERG,Phys.Rev.A~,1033 (1978) ; B.COFFEY, ~ h ~ s ~ ~ e v . ~ ~ , 6 3 3 (1984).

C6-326 JOURNAL DE PHYSIQUE

3. Q.H.F.VREHEN and H.M.GIBBS,in Dissipative Systems in Quantum Optics-,ed. R.Boni- facio (Springer-Verlag,Berlin 1982)p.l and references quoted therein.

4. M-GROSS and S.HAROCHE,Phys.Rep.z,302 (1982)

5. R.GLAUBER,in Quantum Optics and Electronics,C.deWitt and C.Cohen Tannoudji ed.s (qordon and Breach 1965) p.65 ; M.LAX,in Brandeis University Summer Institute in Theoretical Physics,(Gordon and Breach 1968)p.269 ; H.HAKEN,Laser theory.,En- cyclopedia of Physics vol.XXV/2c ; W.H.LOUISELL,Quantum Statistical Properties of Radiation,(3ohn Wiley,1973) ; L.A.LUGIATO,F.CASAGRANDE,L.PIZZUTO,Phys:Rev.

A26,3438 (1982) ; M.HILLERY,R.F.O1CONNELL,M.O.SCULLY,E.P.WIGNER,Phys.Rep.~, 121 (1984).

6. R-GLAUBER and F.HAAKE,Phys.Lett.A68,29 (1978) ; see also V.DE GIORGI0,Optics Comm.2,362 - (1971) ,and R.BONIFACIO,P.SCHWENDIMANN,F.HAAKE,Phys.Rev.A~, 302,854(1971) 7. D. POLDER ,M. F. H. SCHUURMANS and Q. H. F. VREHEN, Phys

.

8. L. ALLEN, 3 .H.EBERLY ,Optical Resonance and Two-Level Atoms, (3 .Wiley, 1975) 9. F.HAAKE,H.KING,G.SCHRODER,LHAUS and R.GLAUBER,Phys.Rev.A20,2047 (1979) 10. 3. DALIBARD, 3. DUPONT-ROC, C. COHEN TANNOUD31,3. Phys .S, ^{637 ( 1984)}

11. C.LEONARDI,A.VAGLICA,Optics Comm.x,340 (1985)

12. F.T.ARECCHI,E.COURTENS,R.GILMORE,H.THOMAS,in Fundamental and Applied Laser Phy- sics,A.3avan,N.A.Kurnit,M.S.Feld ed.s (3.Wiley 1973),and Phys.Rev.A6,2211 (1972) 13. W.H.LOUISELL,in Quantum Optics,S.M.Kay ,A.Maitland ed.s (Academic Press 1970) 14. Q. H. F .VREHEN,M. F .H.SCHUURMANS and D .POLDER ,Nature =,70 (1980)

15. R.R.MCLONE and E.A.POWER,Mathematika g , 9 1 (1964)