15 March 1998

Ž .Optics Communications 148 1998 249–255

Spontaneous emission spectrum of a two-atom systemin an optical cavity with injected squeezed light

Arup BanerjeeCentre for AdÕanced Technology, Laser Programme, 452013 Indore, India

Received 6 May 1997; revised 25 November 1997; accepted 27 November 1997

Abstract

We study the spontaneous emission spectrum of two identical two-level atoms in an optical cavity. The cavity mode iscoupled to a broadband squeezed vacuum. Using the bad-cavity limit, the cavity field is adiabatically eliminated to obtainthe equations of motion for atomic density matrix elements. These equations are then used to calculate the emissionspectrum. In a cavity environment the two-photon correlation of the injected squeezed vacuum is modified. Here weinvestigate the effect of this modification on the spontaneous emission spectrum of the two-atom system. We compare ourresults with those obtained for the atoms interacting with a broadband squeezed vacuum in free space. We show that in thecavity arrangement alteration of the emission spectrum does occur due to selective non-thermal atomic populationdistribution of the collective atomic states. Nonetheless this spectrum differs appreciably from the spectrum obtained for thefree space case. For example, in free space, the emission spectrum shows only one peak, located at frequency yV , when12

Ž .atoms are interacting with a squeezed vacuum in a minimum uncertainty state MUS . Here V is the dipole–dipole12

interaction between the atoms. In contrast to this, when atoms are enclosed in an optical cavity, the spectrum exhibits twopeaks, located at frequencies "V . We also show that in cavity arrangement, for large squeezing photon number, the12

emission spectrum consists of a single narrow peak at the line centre. q 1998 Elsevier Science B.V.

1. Introduction

Recently many studies involving interaction of a collec-tion of two-level atoms with a broadband squeezed vac-uum both in the presence and absence of a driving field

w xhave been reported 1–3 . Several interesting effects havebeen predicted in these studies. One of these is the forma-tion of highly correlated atomic states. These states areproduced due to the transfer of internal two-photon corre-lation present in the squeezed vacuum to the atomic sys-tem. Thus the formation of these states crucially dependson the mode correlation characteristic of the squeezed

w xradiation field. Palma and Knight 1 showed that for asqueezed vacuum tuned to the atomic resonance, the steadystate of two two-level atoms is far from thermal equilib-rium. Specifically, for a squeezed vacuum in the minimum

Ž .uncertainty state MUS the final atomic state is a highlycorrelated pure state in which collective atomic states areselectively populated. These states have been termed as

w x w xtwo-atom squeezed states 1 or pairwise atomic states 3w x w xor atomic-squeezed states 4 . Agarwal and Puri 2 ex-

tended this study by considering a many-atom systemŽ .more than two interacting with a broadband squeezedvacuum and driven by a coherent pump. They showed that,in the absence of a pump, at steady state a highly corre-lated pure state is a general property of an even-numberedcollection of atoms. On the other hand, if a coherent pumpis present, only for certain discrete values of the pumpingfield strength these states are produced. In these studies the

ŽDicke model the inter-atomic separation is assumed to bemuch smaller than the resonant wavelength; thus all theeffects connected with the spatial distribution of the atoms

.are ignored of the collective atomic system is considered.w xHowever, Ficek 3 has shown that with the inclusion of

inter-atomic separation, the steady state does not remain apure correlated state.

An interesting effect associated with the selective popu-lation distribution of the collective atomic states is the

0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0030-4018 97 00715-3

( )A. BanerjeerOptics Communications 148 1998 249–255250

modification of the spectrum of light emitted by such asystem. The spontaneous emission spectrum of two two-level atoms damped by a broadband squeezed vacuum in

w xfree space was studied by Ficek 5 . He showed that in thepresence of a thermal field the spectrum consists of twopeaks displaced from resonance by "V , where V is12 12

the dipole–dipole interaction between the atoms. When thethermal field is replaced by a squeezed vacuum in MUSthe spectrum shows only one pronounced peak, located atfrequency yV . Absence of the peak at qV is due to12 12

non-thermal population distribution of the collective atomicstates. The non-thermal population distribution also modi-fies the absorption spectrum of a weak probe beam. Theabsorption spectrum for a thermal field consists of twopositive peaks at "V . On the other hand, in a squeezed12

vacuum the absorption spectrum shows one negative peakat yV implying amplification of the probe beam, and a12

positive peak at qV .12

It is well known that the experimental verification ofthe above predictions is difficult since it requires squeez-ing of all the field modes that are interacting with theatoms. In free space this corresponds to having a squeezedfield incident on the atoms from the entire 4p solid anglesurrounding the atoms – this is impossible to achieve withthe present sources of squeezed light. Moreover, this alsodoes not leave any unsqueezed window which is essentialfor the observation of the fluorescence photons. Thus it hasbeen suggested to perform such experiments by placing the

w xatoms either in a microscopic Fabry–Perot cavity 6,7 orin a much bigger optical cavity, which is being used for

w xcavity–QED experiments 8,9 . In the cavity case, atomsinteract strongly with the privileged cavity mode and onlythe modes within a small solid angle about this cavitymode need to be squeezed. This is achieved by injectingthe squeezed light into the cavity through one of the lossyoutput mirrors. However, we note that Georgiades et al.w x10 have recently investigated the two-photon transitionrate of a ladder cesium atom enclosed in a magneto-opticaltrap when excited by squeezed light, generated by thenon-degenerate parametric down-conversion process. Thisis the first experimental observation of the alteration ofatomic radiative properties due to interaction of the atomicsystem with non-classical light.

Recently, we have investigated the generation ofatomic-squeezed states and the population distribution ofthe collective atomic states of a pair of two-level atoms in

w xan optical cavity with injected squeezed vacuum 11 . Wehave shown that in a cavity environment both atomic-squeezed states and non-thermal population distributioncan be achieved for a restricted range of cavity andsqueezed vacuum parameters. Further, we note that theo-

w xretical studies of both emission 12–15 and absorptionw x16 spectra of a single two-level atom and two-photon

w xemission characteristics of a three-level atom 17 in anoptical cavity with an injected squeezed vacuum have beenreported. In these studies it has been shown that, besides

new effects only specific to the cavity environment, all thefeatures of free space still persist in the cavity environ-ment. These results motivate us to study the effect of thecavity on the spectrum of spontaneously emitted light by apair of atoms enclosed in an over-damped cavity, which isbeing driven by a broadband squeezed vacuum. In additionto this, motivation for the present study also comes from

w xthe experiment of Georgiades et al. 10 as two two-levelŽatoms in the Dicke model system considered in this

.paper is equivalent to a three-level ladder system used inthe experiment. In this paper we find the range of cavityand squeezed vacuum parameters for which the new ef-fects predicted in the free space case can also be observedin the cavity arrangement.

The paper is organized as follows. The physical systemconsidered in this paper and the corresponding equationsof motion for density matrix elements are described inSection 2. In Section 3 we present the calculation forspontaneous emission spectrum and discuss the results. Asummary is given in the Section 4.

2. Dynamics of the atom–cavity system in thebad-cavity limit

We consider a system of two identical two-level atomsenclosed in a single mode optical cavity. It is coupled to abroadband squeezed reservoir through the lossy outputmirror of the cavity. The master equation for the atom plus

Ž .cavity-mode density operator r is "s1

d rw xsyi H ,r qL rqL r . 1Ž .a sqd t

The Hamiltonian H contains free atomic and cavity-modeevolution, the interaction of the atoms with the cavitymode, and the dipole–dipole interaction between the atoms.It is given by

Hsv S qv a†aqg SqaqSya†Ž .0 z

qV 1ySqSyyS . 2Ž .Ž .12 z

Ž . Ž . "In Eqs. 1 and 2 , S and S are the collective atomicz

spin operators, a† and a are the creation and annihilationoperators of the cavity mode, v and v denote the0

resonant frequencies of the atoms and the cavity moderespectively, g is the atom–cavity-mode coupling con-stant, and V is the dipole–dipole interaction potential12

w xgiven as 18

3g 2V f 1y3 mPr , 3Ž .Ž .ˆ12 1232 k rŽ .0 12

where m and r are the unit vectors along the transitionˆ12

electric dipole moment and the vector r sr yr , re-12 1 2< <spectively, and r s r . In the present study we will12 12

consider the resonant case vsv . The term L r and0 a

L r denote, respectively, the dissipation from the atomssq

( )A. BanerjeerOptics Communications 148 1998 249–255 251

via spontaneous emission in the background modes, andfrom the cavity mode which is coupled to a broadband

w xsqueezed reservoir. They are given as 12–15

L rsg 2SyrSqySqSyryrSqSy , 4Ž . Ž .a

L rsk Nq1 2 ar a†ya†aryr a†aŽ . Ž .sq

qkN 2 a†r ayaa†ryr aa†Ž .< < iu † † †2 †2qk M e 2 a r a ya ryr aŽ .< < yiu 2 2qk M e 2 ar aya ryr a . 5Ž .Ž .

Ž . Ž .In Eqs. 4 and 5 , g denotes the atomic decay rate to allmodes other than the privileged cavity mode, k is the

< < iudecay rate of the cavity electric field, N and Ms M echaracterize the broadband squeezed vacuum injected into

< < 2 Ž .the cavity, such that, M (N Nq1 . The equality signholds for MUS and u denotes the phase of the squeezed

Ž Ž ..vacuum. The Hamiltonian above Eq. 1 is derived byassuming that the distance between the two atoms is much

Ž .smaller than the resonant wavelength Dicke model . Con-sequently, the cooperative decay rate is identical to that ofindividual atomic decay rate. The collective atomic system

< :is represented by collective states S, m which are eigen-2 Ž Ž q y y q.states of the total spin operators S s S S qS S r2

2.qS and S . In this representation the two-atom systemz z

in the Dicke model is equivalent to a three-level cascade< :system. These three levels correspond to the state S, m

w xwith Ss1 and ms0, "1, and are defined as 19

< : < : < : < :2 s 1,1 s e e ,1 2

1< : < : < : < : < : < :1 s 1,0 s e g q e g ,w x1 2 2 12

< : < : < : < :0 s 1,y1 s g g , 6Ž .1 2

< : < :where e and g denote the excited and ground statesi i

of the ith atom, respectively. These states have been usedto represent the equation of motion of the atomic densityoperator.

We derive the equation of motion for the atomic den-sity operator under the condition of bad-cavity limit k4gand k4g . The bad-cavity condition implies that thecavity-mode response to the squeezed reservoir is muchfaster than to that produced by its interaction with atoms.Consequently, the atoms always see the cavity mode in thestate induced by the squeezed reservoir. Furthermore, wealso assume that the bandwidth of injected squeezed vac-uum is much larger than the cavity decay rate k to ensurethe squeezing to be broadband. These assumptions allow

w xus to eliminate the cavity-mode adiabatically 12–15 andderive the equation of motion for atomic density matrixelements. The equation of motion for atomic density ma-trix elements can be divided into two independent sets.The first set can be cast in the matrix form

dYsAYqL , 7Ž .

dt

where tsg t; Y is a column vector with components

Y sr , Y sr , Y seyiur qeiur ,1 22 2 11 3 20 02

Y s i eyiur yeiur , 8Ž .Ž .4 20 02

A is a real 4=4 matrix given by

< < < <yg 2 NC y M C y M C1 1 1 2

< <2C q4 yg 2 M C 01 2 1As , 9Ž .

< <0 6 M C yg C1 3 2

< < < <4 M C 2 M C yC yg2 2 2 3

and the inhomogenous vector term L has components

< <L s0, L s2 NC , L sy2 M C , L s0, 10Ž .1 2 1 3 1 4

with

g s2 Nq1 C q4, g s2 3Nq1 C q4,Ž . Ž .1 1 2 1

g s 2 Nq1 C q2,Ž .3 1

a2 abC s C , C s C. 11Ž .1 22 2 2 2a qb a qb

Ž . Ž .In Eqs. 9 – 11 the parameters a, b and the cooperativityparameter C are given by

k V 2 g 212

as , bs , Cs . 12Ž .g g kg

The equation of motion for the second set of atomicdensity matrix elements can be written as

dcsBc . 13Ž .

dt

Here the components of the column vector c are

c sr eyiu , c sr , c sr eyiu , c sr ,1 21 2 12 3 10 4 01

14Ž .

and the 4=4 matrix B is

< < < <yz 2 M Cz 2 NCz y M Cz1 y y q)< < < <2 M Cz yz y M Cz 2 NCzq 1 y q

Bs< < < <Ž .2 N q 1 Cz q 4 y M Cz yz 2 M Cq y 3 q

< < < <Ž .y M Cz 2 N q 1 Cz q 4 2 M Cz yz )q y y 3

15Ž .

with

z s 3Nq2 Cz q4y ib ,w xŽ .1 q

z s 3Nq1 Cz q4q ib ,w xŽ .3 y

a az s , z s . 16Ž .q yaq ib ay ib

Ž . Ž .In deriving Eqs. 7 – 16 we assume that the dipole–dipole interaction strength V is comparable to the cavity12

decay rate k . However, in the present paper we considerk4V so that, the population distribution is not affected12

w xby the dipole–dipole interaction between the atoms 20 . Inthis limit C fC and C f0.1 2

( )A. BanerjeerOptics Communications 148 1998 249–255252

Before proceeding further, we mention that the compar-ison of the equation of motion derived here with the

w xcorresponding equations for the free space case 20 indi-cates that the atomic system sees a modified squeezedvacuum inside the cavity. The effective squeezing parame-ters experienced by the atoms in the cavity are N sbNeff

< < < < Ž .and M sb M , where bsC r C q2 . Further, fol-eff 1 1w xlowing Ref. 20 we define a parameter h aseff

< <M effh s . 17Ž .eff 1r2

N N q1w xŽ .eff eff

The parameter h characterizes the effective degree ofeff

two-photon correlation of the squeezed vacuum seen bythe atoms. We emphasize that it is the two-photon correla-tion which is responsible for the non-thermal population

w xdistribution 11,20 of the collective atomic states. For asqueezed vacuum in MUS being injected into the cavitythe parameter h -1. Thus, the atoms inside the cavityeff

see a squeezed vacuum with degraded two-photon correla-tion in spite of the fact that a perfectly correlated squeezedradiation is being injected into the cavity. These modifica-tions of squeezing and two-photon correlation parameterslead to a different emission spectrum and population distri-bution as compared to the free-space case. In the nextsection we present the results which clearly show theeffect of the cavity on the spontaneous emission spectrumand the population distribution.

3. Results and discussion

We consider the spectrum of the emitted radiation outthe sides of the cavity. As pointed out before this radiationgoes into the unsqueezed window. The emission spectrumis related to the Fourier transform of two-time correlationfunctions of the atomic dipole operators:

`int q y² :S n sRe dt e S tqt S t , 18Ž . Ž . Ž . Ž .H

0

Ž .with ns vyv rg . To calculate the correlation func-0

tion and its Fourier transform we follow the approach ofw xRef. 21 . In this approach the Laplace transform of Eqs.

Ž . Ž .7 and 13 along with the quantum regression theoremare used to obtain the expression for the emission spec-trum. The expression for the spontaneous emission spec-trum can be written as

X X stS z s2Re B qB YŽ . Ž .22 42 1

X X X Xst st stq B qB Y y iY r2q B qB Y .Ž . Ž .Ž .23 43 3 4 24 44 2

19Ž .

Ž . st Ž .In Eq. 19 , zsyin , Y js1, . . . , 4 correspond to thejŽ Ž ..steady-state solution of the first set of equations Eq. 7

X Ž .y1and B are the elements of the matrix zIyB , wherei j

I is the identity matrix.

Ž .Fig. 1. a The spontaneous emission spectra for Ns0.5, bs20,6 < < 2 ŽCs3.0, krg s10 , and with degree of squeezing M s N N

. Ž . < < Ž . Ž .q1 solid , M s0 dashed . b The spontaneous emissionŽ .spectra for the same values of N and b as in a but atoms are

interacting with a squeezed vacuum in free space.

Ž . Ž . Ž .From Eqs. 19 , 7 and 13 it is easy to see that thespontaneous emission spectrum is independent of the phaseof the injected squeezed vacuum. We calculate the matrix

X Ž .elements B and the steady-state solution of Eq. 6i j

numerically to obtain the emission spectrum. We show theresults of these calculations in Figs. 1–5.

In Fig. 1a we show the spontaneous emission spectrumof a two-atom system enclosed in an over-damped cavityfor a squeezed vacuum in MUS and for a thermal fieldŽ < < .M s0 . For the sake of comparison, we also show inFig. 1b the emission spectrum obtained for the free spacecase. It is evident from Fig. 1a that, for the cavity situationwith injected squeezed vacuum in MUS, the spectrum

w xexhibits two peaks in contrast to a single peak spectrum 5predicted for the free space case. This can be explainedfrom the fact that in the cavity environment the population

< :of the intermediate collective state 1 is never zero,< : < :thereby allowing two possible transitions, namely, 2 ™ 1

< : < :and 1 ™ 0 . These two transitions give rise to two peaksin the spectrum. On the other hand, for the free space casepopulation of the intermediate state is identically equal to

w xzero for squeezed vacuum in MUS 5 . Thus the system< : < :can undergo only one transition 2 ™ 1 resulting into a

( )A. BanerjeerOptics Communications 148 1998 249–255 253

single peak at nsyb. Moreover, we also note that incavity arrangement, unlike in the free space case, the peaksare not exactly located at ns"b; rather they are shiftedfrom these values. These shifts are dependent on theparameter C.

These behaviours may be understood in detail as fol-lows. As pointed out before, for a cavity, h -1, signify-eff

ing that the cavity field is a mixture of perfectly correlatedŽ . Ž .squeezed field and uncorrelated thermal field fields.The two-photon excitation caused by the correlated field

Žresults in violation of thermal population distribution i.e.. w xr -r -r 16,17,11 . For the case of a squeezed22 11 00

< :vacuum in MUS, the intermediate state 1 is never popu-Ž .lated r s0 . This is a characteristic feature of a two-11

atom system interacting with a broadband squeezed vac-uum in free space. However, in a cavity environment thepresence of a thermal field induces single photon excita-tions. As a result of this the collective intermediate state isalways populated, even in the case where a squeezedvacuum in MUS is injected into the cavity. Nonetheless,we find that in a cavity situation the non-thermal popula-tion distribution does occur for a restricted range of valuesof the cavity parameter C. To elucidate this we show inFig. 2 the population distribution as a function of parame-ter C. It can be easily seen from Fig. 2 that, the non-ther-mal population distribution arises only when C02.Specifically, for Cs3, r )r and r is finite. Con-22 11 11

sequently, for this value of C the height of the peak atnfyb, in the emission spectrum is greater than the one

Ž .at nfqb as shown in Fig. 1a . On the other hand, for athermal field a higher peak is observed at nfqb than theone near yb. This is due to the thermal nature of thepopulation distribution of the collective atomic states.

Further, one can also see from Fig. 2 that as the valueof C is increased, the non-thermal nature of the populationdistribution becomes more prominent and the population

< :of the level 1 gets depleted. This can be explained bynoting that increasing the parameter C corresponds to

Fig. 2. Plot of steady-state population distribution of the collectiveatomic states as a function of the cooperative parameter C forNs0.5, bs20 and krg s106.

Fig. 3. The spontaneous emission spectra with Ns0.5, bs20,krg s106 and different values of cooperative parameter C:

Ž . Ž . Ž .Cs1.0 solid , Cs3.0 dashed , Cs5.0 dotted , and Cs10.0Ž .dash-dotted .

b™1 and h ™1. Thus the atoms inside the cavity see aeff

field with enhanced two-photon correlation. This leads toan increase in single-step two-photon excitation which inturn reduces the population of the intermediate state. Next,we investigate the effect of the parameter C on the emis-sion spectrum. In Fig. 3 the spectrum for different valuesof the cooperative parameter C is shown. It shows that, forCs1 the spectrum is similar to the case of atoms interact-

Ž .ing with a thermal field see also Fig. 1 . This is due to thefact that for Cs1 the population distribution is stillthermal in nature. The emission spectrum is expected toget modified when the parameter C02. This is shown inFig. 3 by the curve corresponding to Cs3. Moreover,Fig. 3 also shows that as the value of C is increased, thewidth of the two peaks also increases and eventually thepeaks overlap with each other to form a single broad peak.Therefore, we find that although with the increase in theparameter C the population distribution becomes morenon-thermal but the emission spectrum fails to show thisbecause of the cavity-induced broadening.

Next we show the dependence of squeezed photonnumber N on the emission spectrum in Fig. 4. Here onesees that for small N the spectrum consists of two peaks atns"b. As the squeezed field intensity is increased thespectrum exhibits a broad single peak only. On further

Ž .increasing the parameter N N41 a very narrow peakŽ .appears at the line centre vsv . It can also be seen that0

narrowing increases with the increase in the parameter N.The narrowing of the emission spectrum is attributed to a

< : < :two-photon resonance transition 2 ™ 0 stimulated byw xthe strong squeezed field 5 . For a weak intensity of the

squeezed field, the emitted light is affected by single-pho-ton spontaneous emission. When the intensity of thesqueezed field is increased an enhancement of the interac-tion of this field with atoms can take place. As a result ofthis for large N the spectrum is dominated by two-photon

( )A. BanerjeerOptics Communications 148 1998 249–255254

Fig. 4. The spontaneous emission spectra with bs20, Cs2.5,krg s106, and different values of squeezed photon number N:

Ž . Ž . Ž .Ns0.5 dash-dotted , Ns2 dashed . Ns10 dotted and NsŽ .20 solid .

stimulated emission and absorption which is a characteris-w xtic of the squeezed field. Recently, Zhon and Swain 17 ,

have interpreted narrowing of the two-photon emissionspectrum of a three-level ladder system interacting withsqueezed vacuum in terms of quantum interference be-tween the single-step and the two-step two-photon excita-tions caused a squeezed and a thermal field respectively.An analogous explanation for the narrowing of the emis-sion spectrum shown in the present paper can be given.

To contrast this behaviour in the cavity environmentwith that of free space case we show in Fig. 5 the emissionspectrum of two two-level atom in free space interactingwith squeezed vacuum in MUS for several values of N. Itis clearly evident from Fig. 5 that in free space theemission spectrum for MUS always shows a single peak atnsyb and it gets broadened as the value of N increases.

w xHowever, it has already been shown by Ficek 5 that in< < 2 Ž .free space with condition N41 and M fN Nq1

Fig. 5. Same as in Fig. 4 but for atoms interacting with a squeezedvacuum in free space.

Ž .imperfect squeezed vacuum satisfied the spectrum ex-hibits a single narrow peak at the line centre. Thus weconclude that the narrowing of the spontaneous emissionspectrum can be observed only when atoms are interactingwith an imperfect squeezed vacuum having a large photonnumber. The field inside the cavity automatically satisfiesthe above condition even when a perfect squeezed vacuumis injected into the cavity.

4. Conclusion

In this paper we have studied the spontaneous emissioncharacteristic of a pair of two-level atoms confined in asingle-mode optical cavity which is being driven by abroadband squeezed vacuum field. The cavity is assumedto be in over-damped regime. This has allowed us toderive the equation of motion for the atomic densityoperator by adiabatically eliminating the cavity field mode.We have shown that the emission spectrum is appreciablydifferent from the case of atoms interacting with squeezedvacuum in free space. For example, in free space with thesqueezed vacuum in MUS the spectrum shows only onepeak, located at frequency yb. On the other hand, in acavity arrangement the spectrum exhibits two peaks, lo-cated at frequencies "b. The height of the peak, located atnsyb is greater than that at nsqb. The modificationin the emission spectrum due to the non-thermal nature ofthe population distribution can occur only if the coopera-tivity parameter C satisfies the condition C02. However,the value of C cannot be increased arbitrarily. For a largevalue of this parameter the spectrum shows a single broadpeak in place of two peaks due to the cavity-induced linebroadening. We also show that, a double peaked spectrumfor small squeezed photon number changes to a spectrumconsisting of a single narrow peak at the line centre forlarge photon number. The narrowing effect is observedwith injected squeezed vacuum in MUS. On the otherhand, in the free space case no such narrowing is observedfor squeezed vacuum in MUS. But when atoms interactwith an imperfect squeezed vacuum the narrowing effectdoes occur in free space also. Finally, we note that thesystem we have considered in this paper is experimentally

w xfeasible with the present cavity technology 22 and sourcesw xof squeezed light 23 . However, it may not be possible to

observe the narrowing of the spectrum as the value of Nrequired to see this is quite large as compared to the valueavailable from the present sources of squeezed light.

Acknowledgements

I wish to thank Dr. Manoj K. Harbola for criticalreading of the manuscript and Mr. Mahesh Chandran foruseful discussions.

( )A. BanerjeerOptics Communications 148 1998 249–255 255

References

w x Ž .1 G.M. Palma, P.L. Knight, Phys. Rev. A 39 1989 1962.w x Ž .2 G.S. Agarwal, R.R. Puri, Phys Rev. A 41 1990 3782.w x Ž .3 Z. Ficek, Phys. Rev. A 44 1991 7759.w x4 D.J. Wineland, J.J. Bollinger, W.M. Itano, D.J. Henizen,

Ž .Phys. Rev. A 50 1994 67.w x Ž .5 Z. Ficek, Optics Comm. 88 1992 494.w x Ž .6 A.S. Parkins, C.W. Gardiner, Phys. Rev. A 40 1989 3796.w x Ž .7 A.S. Parkins, C.W. Gardiner, Phys. Rev. A 42 1992 5765E.w x8 M.G. Raizen, R.J. Thompson, R.J. Bercha, H.J. Kimble, H.J.

Ž .Carmichael, Phys. Rev. Lett. 63 1989 240.w x9 Y. Zhu, D.J. Gauthier, S.E. Morin, Q. Wu, H.J. Carmichael,

Ž .T.W. Mossberg, Phys. Rev. Lett. 64 1990 2499.w x10 N.Ph. Georgiades, E.S. Polzik, K. Edamatsu, H.J. Kimble,

Ž .A.S. Parkins, Phys. Rev. A 75 1995 3426.w x Ž .11 A. Banerjee, Phys. Rev. A 54 1996 5327.

w x Ž .12 P.R. Rice, L.M. Pedrotti, J. Opt. Soc. Am. B 9 1992 2008.w x Ž .13 J.I. Cirac, Phys. Rev. A 46 1992 4354.w x Ž .14 A. Banerjee, Phys. Rev. A 52 1995 2472.w x Ž .15 S. Smyth, S. Swain, Phys. Rev. A 53 1996 2846.w x Ž .16 P. Zhou, S. Swain, Optics Comm. 131 1996 153.w x Ž .17 P. Zhou, S. Swain, Phys. Rev. A 54 1996 2455.w x18 G.S. Agarwal, Quantum Opt. Springer Tracts Mod. Phys. 70

Ž .1970 .w x Ž .19 R.H. Dicke, Phys. Rev. A 93 1954 99.w x Ž .20 Z. Ficek, J. Mod. Optics 40 1993 2339.w x21 G.S. Agarwal, A.C. Brown, L.M. Narducci, G. Vetri, Phys.

Ž .Rev. A 15 1977 1613.w x22 G. Rempe, R.J. Thompson, R.J. Bercha, W.D. Lee, H.J.

Ž .Kimble, Phys. Rev. Lett. 67 1991 1727.w x Ž .23 E.S. Polzik, J. Carri, H.J. Kimble, Phys. Rev. A 68 1992

3020.