Quantum interference of spontaneous decays: noise reduction in V lasers without inversion

15 February 1999

Optics Communications 160 Ž1999. 245–250

Quantum interference of spontaneous decays: noise reduction in V lasers without inversion Xiang-ming Hu, Jin-sheng Peng Department of Physics, Huazhong Normal UniÕersity, Wuhan 430079, China Received 27 July 1998; revised 24 September 1998; accepted 29 October 1998

Abstract For a four-level double-V laser without inversion we show that the interference of spontaneous decays leads to intensity noise squeezing. The optimum amount of noise reduction is 50%, which is the maximum squeezing in a resonant three-level inversionless L-laser with coherent driving and a resonant three-level Raman laser. By transforming into a combination state basis, the Mandel parameter and the output fluctuation spectrum are calculated and the mechanism is interpreted. q 1999 Elsevier Science B.V. All rights reserved. PACS: 42.50.Ar; 42.50.Dv; 42.50.Lc; 42.55.–f

1. Introduction Due to atomic coherence via external coherent driving, a laser without inversion w1–3x exhibits sub-Poissonian photon statistics. Gheri and Walls w4x, Manka et al. w5x, and Ritsch et al. w6x found that, under resonant condition, a closed three-level L inversionless system, an open threelevel L system, and a three-level Raman system can exhibit intensity noise reduction 50% below the shot noise. A common condition is that fast incoherent processes lead to a large population in the auxiliary level. For a closed L system, incoherent pumping from the lower to upper lasing level and a successive fast decay from the upper lasing level to the auxiliary level give rise to a large population in the auxiliary level. For a open system, the atoms are mainly populated in the auxiliary level before being injected into the cavity. For a Raman system, a direct decay process from the lower lasing level to the auxiliary level makes the laser electron swiftly recycle to the auxiliary level. Therefore, only when coherent driving is combined with fast incoherent processes can good squeezing be achieved. Although incoherent processes play an important role in noise reduction, the above mechanism is different

from dynamical noise reduction w7–10x. The differences are the following four. First, coherent driving is an externally imposed process while dynamical noise reduction is connected to the level structure of the laser medium itself. Second, coherent driving requires a dominant population in the auxiliary level but dynamical noise reduction demands rate matching. Third, for coherent driving, lasing can occur without bare-state population noninversion, but for dynamical noise reduction, lasing takes place only when population inversion is realized. Fourth, well above threshold, the intensity width is 4 k for coherent driving but 2 k for dynamical noise reduction where k is the cavity loss rate. When one combines coherent driving with dynamical noise reduction w10,11x or with other mechanisms w12,13x, this facilitates improvement of quantum noise reduction. As is well known, in addition to external coherent driving, quantum interference of some other processes creates atomic coherence. Such processes include autoionization w1,14–17x, spontaneous decay w18,19x, and incoherent pumping w20x. So far, for the effects of atomic coherence on the statistical properties, most publications have mainly focused on the case of coherent driving. In this paper we shall show that quantum interference of sponta-

0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 5 9 5 - 1

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X. Hu, J. Peng r Optics Communications 160 (1999) 245–250

neous decays can lead to about 50% intensity noise squeezing in a four-level double-V laser without inversion. Although Imamoglu et al. w16x and Fleischhauer et al. w19x showed that quantum interference can be realized in the dressed states of the coherently driven L or J threelevel system, this does not mean that the statistical properties of the corresponding systems are known. This occurs because interference can be only realized when the decay rate from the upper to the lower lasing level is dominant over the others, and when the incoherent pumping rates from the lower lasing level to both the upper lasing level and auxiliary level are equal Ži.e., symmetric pumping. w19x. Obviously, such incoherent processes deviate from the requirement for the inversionless L system w4x and the Raman system w6x in which incoherent processes result in a large population in the auxiliary level. As will be shown later, symmetric pumping in a three-level system limits the amount of squeezing to 20%. In this paper, our main purpose is to show that a four-level scheme, in which the excited doublet radiatively decays to two lower levels, can exhibit noise squeezing with an optimum amount of 50%, which is the maximum amount of squeezing in a resonant three-level inversionless L laser with coherent driving and in a resonant Raman laser w4–6x.

an appropriate rotating frame, the operator Langevin equations in the bare state basis ŽBSB. are derived as follows: a˙ s yka q g 1 Nsc a q g 2 Nsc b q Fa ,

s˙ca s y w

Ž1.

i Dca q g 1 q g 1X q L1 q L2

y Ž g 1g 2 .

1r2

qŽ

1r2 g 1X g 2X

.

x sc a

sc b

q g 1 a Ž sa a y sc c . q g 2 a s b a q Fc a ,

s˙cb s y w

i Dcb q g 2 q g 2X q L1 q L2

y Ž g 1g 2 .

1r2

x sc b

q Ž g 1X g 2X . 1r2 sc a

q g 2 a Ž s b b y sc c . q g 1 a s b†a q Fc b ,

s˙ b a s y w

i v b a q g 1 q g 2 q g 1X q g 2X

y Ž g 1g 2 .

1r2

q Ž g 1X g 2X .

1r2

s˙aa s y2 Ž y

1 2

Ž sa a q s b b . Ž4.

. sa a

Ž g 1g 2 . 1r2 q Ž g 1X g 2X . 1r2 Ž s b a q s b†a .

q 2 L1 sc c q 2 LX1 sd d y g 1 a†sc a y g 1 a sc†a q Fa a ,

2. Model and equation

s˙ bb s y2 Ž

g 2 q g 2X

y Ž g 1g 2 . Let us consider an ensemble of N atoms with the four-level double-V configuration sketched in Fig. 1. Such a system has been shown to have an enhanced refractive index without absorption or gain w21x. The levels of the doublet are closely spaced and carry the same quantum numbers J and m J . The radiative decays of the level doublet < a: and < b : to the lower states < c : and < d : lead to the build-up of coherence between the two upper levels. Lasing occurs on the quantum-beat transition a–c and b–c. Using the standard method w22x and transforming to

Fig. 1. The four-level double-V system in which quantum interference between the two upper levels is generated via radiative decay of two closely spaced upper levels a and b with the same J and m J quantum numbers to levels c and d.

Ž3.

x sb a

y g 1 a sc†b q g 2 a†sc a q Fb a ,

g 1 q g 1X

Ž2.

Ž5.

. sb b

1r2

q Ž g 1X g 2X .

1r2

Ž s b a q s b†a .

q 2 L 2 sc c q 2 LX2 sd d y g 2 a†sc b y g 2 a sc†b q Fb b ,

s˙cc s y2 Ž L1 q L2 . sc c q Ž g 1 g 2 .

Ž6. 1r2

Ž s b a q s b†a .

q 2g 1 sa a q 2g 2 s b b q g 1 a†sc a q g 1 a sc†a q g 2 a†sc b q g 2 a sc†b q Fc c , where g i and g iX and L i and LXi

Ž7.

Ž i s 1,2. are the spontaneous decay rates, Ž i s 1,2. are indirect incoherent pumping rates. Terms containing Žg 1g 2 .1r2 and Žg 1X g 2X .1r2 describe quantum interference effects which emerge from the radiative decays of the upper states to their common lower levels. As is well known, interference leads to lasing without inversion w1,15–20x. a and a† are the annihilation and creation operators of the cavity field which is coupled to the quantum-beat transitions with coupling constants g 1 and g 2 . si j s < i :² j <, Ž i, j s a,b,c,d . represents the atomic population operator when i s j and the dipole transition operator for i / j. The detunings are described by Dca s v ac y n and Dcb s v bc y n , in which v i j and n are the atomic transition frequency and the laser frequency, respectively. k is the cavity loss rate, and Fa , Fi j are the noise operators. Since the system is closed, we have saa q s bb q scc q sdd s 1. When g iX s 0 and LXi s 0, the present four-level system is reduced to a three-level system w19x. The steady state

X. Hu, J. Peng r Optics Communications 160 (1999) 245–250

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solution of the present four-level system is very complicated. For a similar system Ref. w17x used a perturbation method and only presented a semiclassical third order solution. In order to analyze the effects of the interference of spontaneous decays on the quantum statistical properties, we introduce a combination state basis ŽCSB. as <1: s Ž < a: y < b : . r'2 , <3: s < c : ,

<2: s Ž < a: q < b : . r'2 ,

<4: s < d :

Ž8.

and use the following reasonable assumptions,

g 1 s g 2 s gr2, g 1X s g 2X s g Xr2,

g 1 s g 2 s gr'2 ,

L1 s L2 s L ,

LX1 s LX2 s LX ,

Dc a s yDc b .

Ž9. Fig. 2. The equivalent system in the combination state basis.

Choosing normal ordering, †

a,

† s 21 ,

† s 32 ,

† s 31 ,

† s 14 ,

† s 24 ,

† s 34 ,

s 11 , s 33 ,

s44 , s 34 , s 24 , s 14 , s 31 , s 32 , s 21 , a,

Ž 10.

and defining a correspondence between c numbers and operators as

a l a,

a † l a† ,

† Õ†32 l s 32 , † Õi4 l si4† ,

Õ1 l s 21 ,

Õ 31 l s 31 , z j l sj j

† Õ†1 l s 21 ,

† Õ†31 l s 31 ,

Õ 2 l s 32 ,

Õi4 l si4 ,

Ž i s 1 y y 3; j s 1 y y 4 . , Ž 11 .

we transform BBS operator equations Ž1. – Ž7. into a set of c-number Langevin equations in CSB as follows,

a˙ s yk a q gNÕ 2 q Fa ,

Ž 12. †

Õ˙ 1 s yg 12 Õ 1 q V Ž z 1 y z 2 . y g a Õ 3 q FÕ 1 ,

Ž 13.

Õ˙ 2 s yg 23 Õ 2 q g a Ž z 2 y z 3 . q V Õ 3 q FÕ 2 ,

Ž 14.

Õ˙ 3 s yg 13 Õ 3 y V Õ 2 q g a Õ1 q FÕ 3 ,

Ž 15 .

X

z˙1 s 2 L z 3 q 2 L z 4 y V Õ1 y V Õ†1 q Fz 1 ,

Ž 16 .

z˙3 s y4 L z 3 q 2g z 2 q g a † Õ 2 q g a Õ†2 q Fz 3 ,

Ž 17.

X

X

z˙4 s y4 L z 4 q 2g z 2 q Fz 4 .

Ž 18 . X

with z 1 q z 2 q z 3 q z 4 s 1, g 12 s g q g , g 23 s g q g X q 2 L and g 13 s 2 L. It is obvious that the set of Langevin equations in CSB Ž12. – Ž18. has a simplified form and is equivalent to the corresponding equations of a resonant cascade system as shown in Fig. 2. The level splitting in BSB corresponds to coherent driving Ži.e., transition 1–2. with Rabi frequency

V s 12 v b a .

Ž 19 .

The V quantum-beat laser transition is equivalent to the 2–3 transition with coupling g. g and g X are the rates of spontaneous decays 2 ™ 3 and 2 ™ 4, respectively. L Ž i s 1,2. are the rates of the incoherent pumpings 3 ™ 1 and 3 ™ 2, respectively. LX Ž i s 1,2. are the rates of the incoherent pumpings 4 ™ 1 and 4 ™ 2, respectively. It should be noted that in contrast to the three-level L inversionless system w3,4,6x and the three-level Raman system w5,8x, the incoherent pumpings 3 ™ 1 and 3 ™ 2 with rate L are symmetric, and so are the incoherent

pumpings 4 ™ 1 and 4 ™ 2 with rate LX. Since the states <1: and <2: are combination states of the states < a: and < b :, the symmetrical pumpings c ™ a, c ™ b and d ™ a, d ™ b remain unchanged, which are listed as 3 ™ 1, 3 ™ 2, 4 ™ 1, and 4 ™ 2, respectively, in CBS. We would like to emphasize that, due to quantum interference, the case for the decays is completely different from that for incoherent pumpings. The interference terms in Eqs. Ž2. – Ž7. lead to the vanishing decays 1 ™ 3 and 1 ™ 4. Thus, only the decay processes 2 ™ 3 and 2 ™ 4 are present in the equivalent system. Certainly, such an equivalent system is different from a real bare-state atomic system. As will be seen later, the interference effects are the ones that lead to noise reduction. On the other hand, if the interference effects are absent, the present system is reduced to an ordinary laser system with excited level splitting and lasing without inversion is impossible. For the simplified set of equations, the analytic solutions can be relatively easily obtained, and are listed in the Appendix. Then the steady state solutions in BBS are easily obtained via relations Ž8.. Since there is no coherent coupling between level <4: and < i : Ž i s 1–3., the polarizations Õi4 Ž i s 1,2,3. are always zero in the steady state and have no effect on the steady state laser intensities. So among the atomic operators associated with the level <4:, only the population operator s44 contributes to the steady state laser intensities. We also see from Eqs. Ž12. – Ž18. that Õi4 Ž i s 1,2,3. are uncorrelated with the field operators and other atomic operators. According to the dissipative theorem w22x, under these conditions Ži.e., g , g X 4 k ., the noise in Õi4 is not coupled to the laser field. So, we may neglect the equations and noise correlations involved with the noise terms FÕ i4 . The noise correlations are generally written as ² Fx Ž t . Fy Ž tX . : s 2² Dx y :s Ž t y tX . , x , y s a † , Õ†1 , Õ†2 ,Õ†3 , z 1 , z 3 , z 4 , Õ 3 , Õ 2 , Õ1 , a .

Ž 20.

Applying the dissipative theorem w22x we may obtain the

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nonzero diffusion coefficients 2² Dx y : for the c-number variances as 2² DÕ Õ : s 2 V Õ1 , 2² DÕ Õ : s 2 g a Õ 2 , 1 1

2

2

2² DÕ†1 Õ1 : s 2 Ž g q g X . z 1 q 2 L z 3 q 2 LX z 4 , 2² DÕ†2 Õ 2 : s 4 L z 2 q 2 L z 3 q 2 LX z 4 ,

SŽ v . s Q

2² DÕ†3 Õ 3 : s 4 L z 1 q 2 L z 3 q 2 L z 4 , 2² DÕ 2 Õ1 : s yV Õ 2 q 2 Ž g q g X . Õ 3 , 2² DÕ †2 Õ 3 : s 4 L Õ1 ,

2² Dz1 Õ 2 : s V Õ 3 y 2 L Õ 2 , 2² Dz 3 Õ 1 : s y2g Õ1 , 2² Dz 3 Õ 3 : s 4 L Õ 3 ,

2² Dz 1 Õ 3 : s y2 L Õ 3 ,

2² Dz 4 Õ 1 : s y2g X Õ1 ,

2² Dz1 z1 : s yV Õ†1 y V Õ1 q 2 L z 3 q 2 LX z 4 , 2² Dz 3 z 3 : s yg a Õ†2 y g †a Õ 2 q 4 L z 3 q 2g z 2 , X

2² Dz 4 z1 : s y2 L z 4 ,

2² Dz 3 z 1 : s y2 L z 3 ,

2² DÕ 1 a : s ygÕ 3 . 2

Ž 21 .

From Eqs. Ž12. – Ž18., and Ž21. we can analyze the statistical properties in the steady state. 3. Mandel Q parameter and output spectrum Assuming the good-cavity limit Ži.e., g , g X 4 k . greatly simplifies the calculation of the quantum statistical properties of the emitted laser light, since the atoms can then be eliminated adiabatically. After eliminating adiabatically the atomic variables, we can obtain the Langevin equation of the laser intensity I s a †a as dI s d I q FI , Ž 22. dt where d I and FI are the drift coefficient and Langevin force, respectively. Thus the fluctuation width l and the noise correlation ² FI Ž t . FI Ž tX .: are obtained as E dI ls , ² FI Ž t . FI Ž tX . : s 2² DII :d Ž t y tX . , Ž 23. EI where 2² DII : is the diffusion coefficient. The Mandel Q parameter in the steady state is obtained as Qs

²Ž D I .2 :

y1s

ž /

l2

4. Results and conclusion

2² Dz 3 Õ 2 : s 4 L Õ 2 ,

2² Dz 4 z 4 : s 4 LX z 4 q 2g X z 2 ,

4k

. Ž 26. l l2 q v 2 SŽ v . s 0, y1 - SŽ v . - 0, and SŽ v . s y1 represents the Poissonian statistics, sub-Poissonian statistics, and perfect squeezing, respectively.

X

2² DÕ 3 Õ1 : s V Õ 3 ,

flux Žintensity. operator. When the atoms can be adiabatically eliminated and when the laser is operated in the regime where one can linearize the equations for the intensity fluctuations, one finds

² DII :

. Ž 24. I lI Q s 0 represents the Poissonian statistics, y1 - Q - 0 denotes the sub-Poissonian statistics, and Q s y1 means perfect squeezing. A further experimentally interesting quantity is the normally ordered part SŽ v . of the output field intensity fluctuation spectrum, which is given by ² :i Ž t q t . ,i Ž t . :: ` S Ž v . s 2 dt cos Ž vt . , Ž 25. ² i Ž t .: 0

H

with iŽ t . s 2 ka† Ž t . aŽ t . corresponding to the output photon

Let us first consider the three-level system, i.e., g X s 0 and LX s 0. The Mandel Q parameter and output intensity fluctuation spectrum at zero frequency versus cooperativity C0 s g 2 NrG k Žwhere G is the decay rate. are plotted in Fig. 3 for different incoherent pumping rates Lrg s 5r10, 6r10, 8r10, 1, which correspond to the solid, dashed, dotted and dot-dashed curves, respectively. The parameters are: g s 0.25 G , v b a s 0.5 G , k s 0.01 G . Compared to the resonantly driven three-level system w4–6x, the noise reduction is only moderate, and less than 20%. For the four-level system, the Mandel Q parameter and output intensity fluctuation spectrum are shown in Fig. 4 for g Xrg s 1, 2, 4, 8, which correspond to the solid, dashed, dotted and dot-dashed curves, respectively. The parameters are chosen as: Lrg s LXrg s 2r5, g s 0.25 G , v b a s 0.5 G , k s 0.01 G . It can be seen that as the ratio g Xrg increases, the Mandel Q parameter is down close to the optimum value of Q s y0.5. At the same time, the best achievable output spectrum is up to SŽ0. s y0.5. Obviously, compared with the above three-level system, squeezing is greatly enhanced. In order to get an idea of the physical cause of squeezing, one can compare the cases with and without quantum interference effects. First, let us look at the case of absence of quantum interference Ži.e., the interference terms proportional to Žg 1g 2 .1r2 and Žg 1X g 2X .1r2 in Eqs. Ž2. – Ž7. are absent.. In this case, the three-level system Ži.e., g X s 0, LX s 0. is reduced to an ordinary laser system with the upper level doublet. Under the condition of noninversion Ž L F g , as shown in Fig. 3., neither squeezing nor lasing is possible. Without quantum interference, the four-level system degenerates into an ordinary laser system with upper level doublet and additional level <4:. For the noninversion condition Ž L F g , LX F g X , as shown in Fig. 4., no lasing occurs, needless to say squeezing. Next, we analyze the quantum interference effects, which are more easily understood in CBS. The basic mechanism of squeezing is that recycling of the two-photon process and the incoherent processes lead to regularization of the laser electron w4,6x. In CBS, the decays 1 ™ 3 and 1 ™ 4 vanish due to quantum interference. For the three-level system, due to the vanishing decay 1 ™ 3, the pumping 3 ™ 1 and the two-photon process make the laser

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g X 4 g , L, LX , such recycling greatly improves the noise statistics. In another words, quantum interference plays a crucial role in both lasing operation and noise squeezing. Furthermore, it should be noted that the Rabi frequency V amounts to half of the bare-state level splitting wi.e., Eq. Ž19.x and plays an important role in both lasing and noise reduction. Otherwise, if v b a s 0, we have z 1 s 1, z 2 s z 3 s z 4 s 0, I2 s 0. That means that the atoms are trapped in level <1: and no lasing occurs. Therefore, only when quantum interference is combined with the appropriate level of splitting, can both lasing operation without population inversion and good squeezing be achieved. Ideally, for the above optimum conditions, the Q parameter has an optimum value of y1. However, we only obtain Q s y0.5. This result can be further understood in terms of ac-Stark effect. The strong laser field a leads to ac-Stark splitting of the lasing transition, which is proportional to the Rabi frequency g a . Due to the effects of the level shifts the coherent driving is detuned from the atomic dressed-state transition 1–2. For such a case, recycling of the active laser electron is moderately regularized. On the other hand, since the present four-level system is equivalent to a driven system, the intensity fluctuation width is

X X Fig. 3. For the three-level system Ži.e., g s 0 and L s 0., the Mandel Q parameter Ža. and output intensity fluctuation spectrum Žb. at zero frequency versus cooperativity C0 s g 2 Nr G k Žwhere G is the decay rate. for different incoherent pumping rates L r G s 5r10, 6r10, 8r10, 1, which correspond to the solid, dashed, dotted, dot-dashed curves, respectively. The parameters are: g s 0.25 G , v b a s 0.5 G .

electron undergo unidirectional recycling, which regularizes the laser electron. It can be seen that the vanishing decay 1 ™ 3 plays an important role in noise reduction. Ideally, a resonant three-level system produces 50% squeezing. However, because there is not a successive incoherent process to the pumping 3 ™ 2 and then the population in the level <2: cannot be depleted, the amount of squeezing is limited. In another words, quantum interference leads to noise squeezing but the symmetrical pumpings limit the amount of squeezing. For the four-level system, quantum interference leads to the vanishing decays 1 ™ 3 and 1 ™ 4. In spite of the symmetric pumpings 3 ™ 1 and 3 ™ 2 and the symmetric pumpings 4 ™ 1 and 4 ™ 2, the pumping 4 ™ 2 is negligible compared to the fast decay 2 ™ 4 when g X 4 LX. The decay 2 ™ 4 and pumping 4 ™ 1 are successive incoherent processes to pumping 3 ™ 2. When the laser is operated well above threshold, the population in the level <2: is depleted and the population in the auxiliary level is greatly increased. Thus the multi-step incoherent pathways 3 ™ 2 ™ 4 ™ 1 together with pumping 3 ™ 1 and the two-photon process make the electron recycle unidirectionally. When

Fig. 4. For the four-level system, the Mandel Q parameter Ža. and X output intensity fluctuation spectrum Žb. for g rg s1, 2, 4, 8, which correspond to the solid, dashed, dotted, dot-dashed curves, X respectively. The parameters are chosen as: L rg s L rg s 2r5, g s 0.25 G , v b a s 0.5 G , k s 0.01 G .

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l s 4 k, which limits the best achievable noise reduction in the output field to 50%. In summary, by transforming into CBS we have studied the statistical properties of a laser without inversion in four-level configuration in which the excited state doublet radiately decays to two lower levels. It has been shown that when the decay rates from the excited doublet to the auxiliary level are much larger than those to the lower lasing level, the optimum amount of noise reduction in the intracavity and output fields is 50%, which corresponds to that of a resonant three-level L inversionless laser with coherent driving and a resonant three-level Raman laser. As a special case, the three-level V system only exhibits 20% noise squeezing. We have also explained the physical origin of squeezing in terms of CBS. Quantum interference results in vanishing decays in CBS. When some spontaneous decays are absent, unidirectional recycling of the laser electron occurs and thus noise squeezing is achieved. In the four-level system, all incoherent processes and the two-photon process facilitate unidirectional recycling. In the three-level system, however, there is one incoherent process which deviates the recycling and so the amount of squeezing is reduced.

Acknowledgements This work is supported by the National Natural Science Foundation of China and the Natural Science Foundation of Hubei Province. One of the authors ŽJ.S. Peng. would like to thank the International Atomic Energy Agency and UNESCO for hospitality at the Abdus Salam International Center for Theoretical Physics, Trieste, Italy.

Appendix A The steady state solutions are listed here. Neglecting the noise and setting the derivatives to zero in Eqs. Ž12. – Ž18., we may obtain the steady state solutions. The laser intensity defined as I2 s g 2a )a is I2 s g 2 I s

(

yB q B 2 q 4 A Ž C1 y C2 . 2A

,

where I s a )a is the steady state number of photons in the cavity. The parameters A, B, C are A s L q g Xr2, B s Ž TL q L q g q g Xr2 . I1 q Ž L q g Xr2 . g 13g 12 q L Ž g q g X . g 23 , C1 s Ž g 2 Nrk . w Ž 2 L y g . g 13 q L Ž g q g X . x I1 , C2 s Ž 2TL q 2 L q g . I12 q w Ž 2TL q 2 L q g . g 13g 23 qL Ž g q g X . g 12 x I1 q L Ž g q g X . g 12 g 23g 13 ,

with I1 s V 2 , T s 1 q g Xr2 LX. The threshold condition is C1 ) C 2 . The steady state populations and polarizations are obtained as z 1 s  2 L I12 q Ž L q g Xr2 . I22 q Ž g y L . I1 I2 q2 w Lg 13g 23 q L Ž g q g X . g 12 x I1 q w Ž L q g Xr2 . g 13g 12 q L Ž g q g X . g 12 x I2 qL Ž g q g X . g 12 g 23g 13 4 rW , z 2 s 2 L I12 q L I1 I2 q 2 Lg 13g 23 I1 rW , z 3 s g I12 q Ž 2 L q g Xr2 . I1 I2 q 2gg 13g 23 I1 rW , z 4 s 1 y z1 y z 2 y z 3 , Õ1 s V w L Ž g q g X . I1 q Lg 13 I2 qL Ž g q g X . g 23g 13 x rW , Õ 2 s g a w Ž 2 L y g . g 13 q L Ž g q g X . g 12 I1 x rW , Õ 3 s g aV w Ž g y 2 L . I1 q L I2 q L Ž g q g X . g 23 x rW , where W s AI22 q BI2 q C2 . References w1x S.E. Harris, Phys. Rev. Lett. 62 Ž1989. 1033. w2x M.O. Scully, S.Y. Zhu, A. Garielides, Phys. Rev. Lett. 62 Ž1989. 2813. w3x A. Imamoglu, J.E. Field, S.E. Harris, Phys. Rev. Lett. 64 Ž1991. 1154. w4x K.M. Gheri and D.F. Walls, Phys. Rev. Lett. 68 Ž1992. 3428; Phys. Rev. A 45 Ž1992. 6675. w5x A.S. Manka, C.H. Keitel, S.Y. Zhu et al., Optics Comm. 94 Ž1992. 174. w6x H. Ritsch, M.A.M. Marte, P. Zoller, Europhys. Lett. 19 Ž1993. 7. w7x A.M. Khazanov, G.A. Koganov, Phys. Rev. A 42 Ž1990. 3065. w8x H. Ritsch, P. Zoller, C.W. Gardiner, D.F. Walls, Phys. Rev. A 44 Ž1991. 3361. w9x T.C. Ralph, C.M. Savage, Opt. Lett. 16 Ž1991. 1113. w10x T.C. Ralph, C.M. Savage, Phys. Rev. 44 Ž1991. 7809. w11x H. Ritsch, M.A.M. Marte, Phys. Rev. A 47 Ž1993. 2354. w12x C. Saavedra, J.C. Retamal, H. Keitel, Phys. Rev. A 55 Ž1997. 3802. w13x K.J. Schernthanner, H. Ritsch, Phys. Rev. A 49 Ž1994. 4126. w14x A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40 Ž1989. 4131. w15x S.E. Harris, J.J. Macklin, Phys. Rev. A 40 Ž1989. 4135. w16x A. Imamoglu, S.E. Harris, Opt. Lett. 14 Ž1989. 1344. w17x S.Y. Zhu, E.E. Fill, Phys. Rev. A 42 Ž1990. 5864. w18x A. Imamoglu, Phys. Rev. A 40 Ž1989. 2835. w19x M. Fleischhauer, C.H. Keitel, L.M. Narducci, M.O. Scully, S.Y. Zhu, M.S. Zubairy, Optics Comm. 94 Ž1992. 599. w20x M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, Optics Comm. 87 Ž1992. 109. w21x M. Fleischhauer, C.H. Keitel, M.O. Scully, C. Su, B.T. Ulrich, S.Y. Zhu, Phys. Rev. A 46 Ž1992. 1468. w22x M. Sargent III, M.O. Scully, W.E. Lamb Jr., Laser Physics, Addison-Wesley, Reading, MA, 1974.