Laser without population inversion and coherent trapping

i-LILI

Volume 84, number 5,6

~ri_-Iti

OPTICS COMMUNICATIONS

F-~II_iL..:

l August 1991

F u l l length article

Laser without population inversion and coherent trapping O.A. K o c h a r o v s k a y a a, F. M a u r i b,c a n d E. A r i m o n d o c a Institute of Applied Physics, Academy of Sciences of the USSR, Nizhny Novgord, USSR b ScuolaNormaleSuperiore, 1-56100Pisa, Italy c Dipartimento di Fisica Universit,~ di Pisa, 1-56100 Pisa, Italy

Received 3 January 199l; revised manuscript received 21 March 1991

It is shown that in the basis of absorbing and nonabsorbingstates of coherent trapping, the atomic preparation in laser without inversion is described as a depopulation pumping. Optical BIochequations in that basis are derived for A and double-A configurations. Numerical results for realization of laser without inversion in a in STRbdouble-Aconfigurationare presented.

1. Introduction

In the atomic quantum-mechanical evolution, the coherences between atomic states play a very important role. The presence of atomic coherences in the interaction with radiation, demonstrated first in magnetic resonance experiments, has been widely exploited in laser coherent phenomena. An atomic coherence is connected to a well defined phase relation between the atomic states, and in most cases, a proper unitary transformation of the Hamiltonian greatly simplifies the mathematical and physical description of the phenomenon. In the "laser without population inversion", a phenomenon very recently theoretically investigated by several authors [ 1-7 ], stimulated emission of photons takes place between an upper atomic level 10), and a couple of lower levels I l ) and 12) as in fig. I a, in the case where an atomic coherence has been previously created between those lower levels. If laser photons induce simultaneously the two optical transitions between level 10) and levels I 1 ) and 12), owing to the presence of the coherence, an interference takes place between the two transition amplitudes. In this situation the absorption process does not take place, while stimulated emission is allowed. Thus, the laser emission is independent of the lower level populations and depends on the upper state population only. This phenomenon is strictly related to the coherent population trapping phenomenon [ 8 ] - - ~

IoU>

~

" IoU>

]A>

tNA>

7Y II>

12>

a)

IA>

[NA>

b)

11>

12>

c)

d)

Fig. 1. Energylevel schemes in the ltomic basis, (a) and (c), and in the absorbing/nonabsorbingbasis, (b) and (d), are presented for A and double-Aconfigurations. 0030-4018/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

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where, by the presence of a coherence between two lower atomic levels, absorption from those levels does not take place, and an atomic vapour appears transparent to laser radiation. As introduced by Gray et al. [ 8 ] and investigated in detail in ref. [ 9 ], coherent trapping may be intuitively analyzed within a proper atomic basis, i.e. by applying a unitary transformation to the optical interaction hamiltonian. In the basis of the so-called absorbing and nonabsorbing states, IA> and INA>, the optical Bloch equations, describing the density matrix time evolution in presence of the optical hamiltonian and the spontaneous emission process, assume a particularly simple form. Thus, by the creation of the atomic coherences through laser radiation acting on the two transitions in the A-configuration of fig. la and by the spontaneous emission processes, a depopulation pumping from the IA> state to the INA> one of fig. lb takes place. The aim of this article is to make use of that approach, and of the symmetries in the optical interaction of a three-level system [ l 0 ], to present the so called laser without inversion as a lasing from an upper state towards a lower empty IA ) state. Within this model the lasing condition, as well as the occurrence of laser instabilities derived in ref. [ 5 ], may be obtained in a very straightforward way. Moreover, the double A-configuration of fig. I c that has been proposed as a convenient scheme for both creating a coherence trapping and realizing a laser without inversion, may be simply analyzed as laser pumping in the four-level scheme of fig. ld. First we present equations for the propagation of a two-mode laser beam, interacting with a three-level system in the A-configuration, in the basis of the absorbing and nonabsorbing states, for the simplest case of a laser in resonance with the transition from an upper state to a lower degenerate pair of levels. Following this, the double A-configuration is investigated, and conditions for the realization of a laser without inversion are obtained. Numerical results of a four-level scheme in 87Rb atoms are presented. Finally the optical Bloch equations for the laser without inversion are examined for a laser not in resonance with any transition between upper and lower levels, and the population inversion versus a detuning parameter is numerically calculated.

2. Absorbing and nonabsorbing states We will investigate the evolution of the three-level density matrix of an atomic system interacting with resonant laser radiation in the degenerate case, i.e. when in the scheme of fig. 1a the I 1 ) and 12 ) levels have the same energy and when the two laser fields interacting with the I l ) -, I0 ) and 12 ) --, I0 ) transitions have the same frequency t2 equal to the atomic frequency O)o. For this particular case the mathematics becomes simple to handle and the main physics remains essentially the same. Moreover, in the atomic configuration we are planning to consider, the dipole matrix elements between levels 11 ) -, I0 ) and 12 ) --, I0 ) are supposed to be equal. This case corresponds to the He configuration investigated in ref. [ 9 ] and to the configuration to be considered in sec. 3. To analyze the propagation equation or the lasing condition in the case of electric fields interacting with the three-level system, the equations for the slowly varying amplitude of the electromagnetic field and of the atomic density matrix have to be examined. The wave equations for the slowly varying complex amplitudes of a two-mode degenerate field: E = ½[E, e x p ( - i g 2 t + i k z ) +E2 e x p ( - i f 2 t + i k z ) ] + c . c . ,

( 1)

interacting with the 0--, 1 and 0 ~ 2 transitions of fig. la result in BE,

1 8El

4~tiNI2p.o ~

6---~ + c - - ~ - + x , E , - -

-

PO,

~E2

1 ~E2

4niN~2/to ~

--~z + c - - - ~ + x 2 E 2 - -

-

PO2

(2)

where/~o, and/~o2 are the slowly varying off-diagonal elements of the density matrix: Po, =/~o, exp(--i£2t) , Po2 =rio2 e x p ( - i , Q t ) .

(3)

In eqs. (2), N is the atomic density,/to is the dipole matrix element between level 0 and either level 1 or level 394

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2, and r~ and x2 are the damping rates for the electric fields. In terms of a Rabi frequency for each laser wave, a = ItoE~/2h and r = ltoE2/2h, the propagating equations become 8a 1 8or a--z + c - ~ - + x , a =

2rtiN12 g~ ~ hc Po,,

~fl 1 8fl 2rtiN12#~ a---z+ c--~ +K2fl= hc POE"

(4)

The equation for the time evolution of the density matrix p subject to an internal hamiltonian and an interaction with the laser fields is written as i h d p / d t = [p, ~at + ~ i . , l ,

(5)

where ~t=E,

I I ) ( I I +Ez 12) (21 +Eo 10) ( 0 l ,

(6)

E~ ( i = 0 , 1, 2) being the energy of the atomic state Ii), with E, = E 2 = 0 and Eo =hogo. Within the RWA the interaction hamiltonian is ~int = - h a l 0 ) ( 11e x p ( - i12t) - h f l l O ) ( 2 [exp(-i12t) +c.c..

(7)

The hamiltonian evolution of eq. (5) is completed by relaxation terms specified below. The equations for the density matrix components in the basis of the Ii) (i=0, 1, 2) states have been previously written down several times, for instance in ref. [ 5 ], and will not be reported here. Instead the density matrix equations will be written in the basis of the absorbing and nonabsorbing states defined by IA)=(1/W)[a*ll)+fl*12)],

(8)

INA)=(1/W)[flI1)-a[2)],

with W= ( l a [ 2 + Ifll 2) ~/z the total intensity of the incident laser beams. In this new basis the interaction hamiltonian of eq. (7) becomes ~in~ = - h W [ IA) (01 exp(i12t) + 10) ( A l e x p ( - i 1 2 t ) ] .

(9)

For the components of the Bloch vector defined using the states of eqs. (8), the resulting optical Bloch equations are /~o.o = - 2 W Sy A , jbA,A=2WSyA,

SXA = A S y A ,

2~yA=--ASXA+ W (po.o--PA,A) ,

/~NA,NA= 0 ,

CxA.

= w syN

,

6XA,NA = -- W S x N A ,

dXN. =aSy

A+ WCy..

A,

£'yN. =

WCxA,

, (10)

where we have defined 3=12-O9o and CXA.NA= ( A I p [ N A ) / 2 + c . c . ,

CYA,NA=i(AIPINA)/2+c.c.,

SXA = (01PlA) exp(i12t)/2+c.c.,

SyA = i ( 0 1 P l A ) exp(i12t)/2+c.c.,

together with similar definitions for SXNA and SyNA. In eqs. (10) we have grouped on a line a set of equations coupled to each other by the interaction hamiltonian. In the basis of eq. (8) the constants of motion investigated by Hioe [ 10 ] and Mallesh and Ramachandran [ 11 ] for the purpose of analyzing the dynamic symmetries in a three-level system in absence of relaxation mechanisms, assume a particular simple shape. For instance, from eqs. (10) it is straightforward to derive that the population ffNA,NAin the nonabsorbing state and KI =APo.o + 2 WSXA are constant. The basis of absorbing and nonabsorbing states provides a simple interpretation for the equation for propagating of the laser intensity, W, through the medium 395

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~W6__z_+ -1 ~W 4- x~ q-x2 W + x~--x2 [ a l 2 - Ifll 2 _ __2nNK2#2SyA.

c-~-

--Y--

2

w

(11)

~c

Thus, at x, = x2 the optical polarization SyA corresponding to an absorbing state only controls the laser propagation. Eqs. (10) connect the Sy A polarization to the population difference P0.0--PA,A and decouple that polarization from the low frequency coherences C X A , N A and CyA,NA. At this point, in order to analyze the stationary state, the influence of the relaxation processes has to be considered. If we suppose a spontaneous emission decay from the upper 10 > state with a rate constant 1"/2 to each I 1 > and 12 > level, the complete equations for the optical polarizations are

£XA = A SyA -- (1"/2 ) SXA , SyA = -- d SXA + W (Po,o - PA.A) -- ( F / 2 ) SYA , SXNA =ASyNA + W CyAINA -- ( F / Z ) SXNA,

( 12 )

SyNA = --ASXNA -- WCXA,NA -- (F/Z) SyNA.

Owing to the fast spontaneous emission rate, compared to the optical pumping and ground state relaxation rates, we apply an adiabatic elimination of the optical polarizations to obtain

SXA --iSyk = [ - i W / ( F / 2 - i A )

] (PA,A --Po,o),

SXNA --iSYNA = [ - i W / ( I ' / 2 - i A )

] (CXA,NA--iCyA,NA) •

Substitution into the steady state (5/St = 0) propagation eqs. (4) leads to coupled equations for the wave components in terms of the new basis: ~Ol/~Z "3I- 1( l OL= g [ ot (Poo - P A A ) -- fl* ( C X A , N A

--

(14a)

iCYA, NA ) ] ,

(14b)

5fl/5z + x2fl= g [fl(Poo --PAA) + Or*(CXA,NA -- iCYA,NA) ] , where we have introduced

g= ( 4 n N O u U h c r / 2 ) [ ( r / 2 ) / ( r / 2 - ~ )

(14c)

].

The most rigorous and physical way to analyze the local and linear properties for the field propagation is to derive the normal waves. For normal waves characterized by components or, fl* ~ exp(ixz) we obtain from eqs. (14) the following solutions of a characteristic equation: Zl`2=i~/2+{-(~/2)2+~g~2(~C~A~NA[2+~CyA~NA~2)+[g(pA~A-p~)-xl][g(pA~A-p~)-~2]~

1/2 ,

(15a)

where ) ~ = - 2 R e ( g ) (Po.o--PA,A)+Xl +X2.

(15b)

The condition of amplification I m (Z~,2) < 0, for equal losses on the two laser modes, x l = x2 --x, assumes a very simple expression, )~< 0, that is expressed as Po,o --PA.A > x / R e (g) .

(16)

This result implies that an amplification may be obtained if a population inversion exists between the I0 > and IA > states. Moreover, the gain Re (g) (Po,o-PA,A) must exceed the wave losses x. From the definitions of eq. (8) it appears that population inversion is realized through the presence of coherence between the atomic I 1 > and 12 > states. However, the states IA > and INA > are determined by the a and fl fields through eqs. ( 8 ). Thus, self-consistent relations for a and ~ fields are obtained from eqs. ( 14)., All properties of normal waves found in terms of the new basis coincide with properties of normal waves found in terms of the usual atomic basis. Normal wave amplitudes depend on stationary values for the atomic density matrix, and specific hypotheses for source terms in the optical Bloch equations have to be considered in order to obtain stationary values. In 396

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the simplest case of an experiment in an atomic beam, as in ref. [ 9 ], where spontaneous emission is the only relaxation mechanism, the stationary values of density matrix elements can be derived in a straightforward way. The population of the ground ]NA> state, which is a constant of motion for the atom-laser interaction of eqs. (1 0), is modified by the spontaneous emission decay: emission decay from l0 > state feeds the ]NA > state population. As a final result all atomic population is trapped in the INA > state, PNA.NA= 1, and all other populations, optical polarizations and low-frequency coherence vanish. This process represents a depopulation pumping to prepare atoms in the ]NA) state decoupled from laser radiation in the scheme of fig. lb.

3. Double-A configuration We investigate now the case where the double-A configuration is applied, as in refs. [ 4,6 ], in order to prepare a coherent trapping superposition of the ground states and where an incoherent pumping to an upper state is applied to realize an inversion. We consider this arrangement as the most convenient for experimental realization. In the double-A configuration, shown in fig. lc, the I l ) and 12) ground states are connected in two A-schemes to excite states I0 U) and I0 ) . Two laser beams, El and E2, resonant with the transition I 1 ) --, I0) and 12) -, [0 ) may be used to transfer, through the depopulation pumping presented above, atoms from the absorbing IA ) state into the nonabsorbing superposition INA). In the simplest case where El~E2 =E~/EU2 it results fA ) = IAU). Thus pumping from the ground states to the upper 10 U) level may create a population inversion between I0 U) and IN A ) = INA u ) states, in the scheme of fig. 1d. As a consequence lasing from the upper level I0 U}, on a two-mode electric field with amplitudes E~u and E2v, may take place. As shown in eq. (8), the IA ) state itself is determined by the E, and E2 fields, and is modified by propagation through the medium of the lasing electric fields E~ and E2u. Thus, in general, self-consistent relations for the propagations of El, E2, E~ and E2u fields are obtained. Here considering the El and E2 electric field strong enough to be not affected by the propagation and t o create constant PA.A and PNA,NA,we calculate the conditions for realizing inversion between 10U) state and the INA) ground state. The equation describing the atom-laser coupling for the double-A configuration are obtained easily by writing eqs. (10) for both A configurations and adding both terms. The feasibility of laser without inversion depends on the relaxation rates of populations and coherences. The general treatment of relaxation processes presented in ref. [ 5 ], when converted into the absorbing and nonabsorhing basis, leads to expressions that are quite awkward. A simpler case, that describes with a good approximation several experimental configurations, is here analysed. We will suppose that the excited I0 ) state decays towards both lower states with spontaneous emission rate'F/2 to each of them, and that spontaneous emission decay takes place for the 10tJ) state with total decay rates F U and equal probability to both ground states. For the ground states we will introduce a rate to describe the decay of the population difference and of the coherence towards a zero value, as in the case of a negligible Boltzmann difference between those levels. In order to describe the population inversion mechanism, we will introduce a pumping rate 2 from the lower states to the upper state I0 U). Thus the relaxation terms are written: /~0U0]rel = -- FUp0U0 "~ ~ (Pll "~-P22) ,

P00 ]rel -m"-- F~oo,

])ll ]rel ~-~(F/2)Poo+ (Fu/2)P~o - 7 (fill --P22 ) --I~01 l ,

P22 ]tel = (F/2)Poo + (Fu/2)pUoo--7(p22--Pll ) --2P22, /~12]r¢l= --7Pl2 --2p12 •

(17)

Rewriting these equations in the [A) and ]NA) basis, the total expressions for the density matrix elements are obtained. Populations PA.A and PNA.NA and coherences CXA,r~A and CyA,NA may be derived from the stationary solution of these total equations and substituted in eqs. (14) to determine the propagation equation for the electric field amplitudes. We have examined the numerical solution of the optical Bloch equations in the IA ) and IN A ) basis for the case of the double-A configuration in a S7Rb atomic beam composed of the following levels" 397

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[OU)=I62p3/2F=I, mF=O) ; IO)=152p3/2F=l, mF=O) ; I1)=152SI/2F=l, mF=l) ; 12)=152SI/2F=I, m F = - - l ) . The transitions from ground levels to these excited levels are not cycling transitions and in order to allow an efficient depopulation pumping with creation o f coherent trapping in the ground state, a repumping laser from the F = 2 ground state hyperfine levels may be required. The modes o f e q . ( 1 ) correspond to a - and a + polarizations of electric field. Atomic parameters corresponding to those transitions at wavelengths 420.2 nm and 780.0 n m respectively, are F v = 8.93 × 106 s - t and F = 3.77 × 107 s - ~ respectively. A numerical calculation was performed for Rabi frequencies ot = fl= 0.5F, a ground state relaxation rate 7= 5 × 104 s - t, a pumping rate 2 = 0.5 X 106 s - i, and considering the transition as a cycling one. If the spontaneous decay to the hyperfine F = 2 ground state level and an incoherent pumping from that level are included in the numerical calculation, similar results are obtained. In fig. 2 we have reported the population inversion between levels P~o -PAp, as function of the interaction time O with the depopulating laser beams. For a pumping time O = 4F/ot z, a P~o--PAA population inversion around 0.016 is obtained, i.e. for a 10 ~2 cm -3 atomic density a 1.6× 101° cm -3 population inversion. It may be noticed that the density matrix occupation difference between upper and absorbing states is large enough to reach the 7/F~ threshold value for instability determined in ref. [ 5 ].

4. Detuned lasers The unitary transformation o f eq. (8) may be applied to analyze the interaction and propagation o f a twomode laser field also when the ground levels are not degenerate and the laser frequencies are not in resonance with atomic transitions. In the general case we denote hogij as the energy separation between states I i) and IJ ) and £21 and ~r~2 a s the laser frequencies of the laser fields in eq. ( 1 ), and take the reference energy o f I 1 ) state as zero. Thus the total atomic hamiltonian in the interaction representation may be written as

~at +~int = h

(o -°p J

-

,

(18)

-zll/

where we have introduced zll = f21 - COo~ and J = CO21- 121 +/22. If S denotes the unitary transformation mod-

P~o- PA A 0.~ 0.0 -0,1 -0,2 -0,3

S

0, P~o- PA*

t

•

- 0 . 1 ~ 0 ". .0. . . . . . . . . . . . . . . . . . . . . . . .

-0.4 -0.5 -0.6

i

i

i

i

10

20

30

40

O"

50

-2

-1

'

0

'

1

Ho

F

Fig. 2. Population inversion pUoo--PAA in a S7Rb double-A degenerate configuration versus pumping time O for pumping rate 2 = 0.5× 106 s -1 and other parameters specified in the text. The pumping time is measured in units of the spontaneous emission lifetime for the [0 ) level involved in the depopulation pumping, the maximum value of the population difference p~--PAA is 0.016.

398

-0.2

Fig. 3. Population inversion p ~ --pAA in a STRb double-A notdegenerate configuration versus the applied external magnetic field Ho in gauss a t pumping time O=20/F, pumping rate 2 = 2 × 106 s- i and other parameters as in the text.

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ifying only the ground states given in eq. (8), the resulting hamiltonian in the IA ) , IN A ) and 10) basis is h f 51#12 -- 5a*fl* ~A'NA=s*(~1+~i"t)S= W ~ _W 3

-aot13 51otl 2 0

-

3 / 7 --.41W2/"

(19)

It may be noticed that the S transformation leaves the INA) state decoupled from the excited state: however, for 5 ~ 0 the state INA) is coupled to IA) state, so that the INA) state population does not remain unperturbed. For the Bloch vector components defined on the basis of the states of eq. (8), the following optical Bloch equations are obtained: Po.o = - 2 W S y A ,

/~NA,NA= 5

PAA = 2 WSyA - 8 a f t i- Wa*fl* ot*fl* CyA,NA, 2 CXA,NA- - 8 aft+W2

~13-a*~* ~¢+a*b'* iW: CXA'NA"11-5 W2 CYA'NA'

I 2 SyA _a ap-o~*13* Sx,, = (,4, + a__13wl__¢) SXN~ --5 a13+ ~ a*#* SYNA 2i W 2

( '"3

SyA = -- Z~,+ 5--W--r SXA + W (Po.o--PA,A) +a a#+ a * ~ SXNA --a ORB--0~*~* Sy~A ' 2W2 2i W 2 CXANA ,

=

WSyNA"I-5 ~ 1 32 -i ~W* 2~

CYA,NA-------

SXNA = SYNA=--

(PA,A--PNA,NA)+ 5 lal2

2 (PA,A--PNA,NA) - 5 WSXNAdl-50[fl;WOl*fflt

z~1 OI-5

dl +8

SyNA + 8 O~13-OL*j~ Sx A --5

2iW 2

SXNA +8

1ill2 CYA,NA '

W 2

~

SXA + 8

]Otl2--Ifl[ 2 CXA,NA' W2

2W 2

,o.,.

2iW2

@A'aft W CyA.NA

'

SyA-- WCXA.NA.

(20)

The same set of equations can be derived by applying the unitary S transformation to the density matrix in the atomic basis. This transformation allows us to write in the new basis the relaxation terms introduced by eqs. (17). The resulting relaxation terms will be not reported here. Eqs. (20), coinciding with eqs. (10) for 5=0, represent optical Bloch equations in the absorbing-nonabsorbing basis even when o921S0. Thus results obtained before for the degenerate case remain valid also for the o921# 0 nondegenerate case. We have applied the transformation to the IA ), IN A ) basis to evaluate the population inversion in a doubleA configuration with lasers detuned from the atomic transition as a modification of the scheme investigated in the previous section. If a magnetic field Ho is applied to the STRb atoms, a magnetic splitting takes place in the 2 :Sl/2F= 1 ground state. The atomic energy separations, referred to the I 1 ) = IF= I, m e = + 1 ) energy, result: o921= 2 IgF= 11 ltBHo/h, O9ol =C.Oo+ Ig~= ,I lZBHo/h with gr= t the negative Land~ factor of the F = 1 hyperfine level. For a degenerate pumping laser field at frequency I2, varying the applied Ho magnetic field, a double-A scheme with detuned lasers is obtained. We have calculated the population inversion versus the magnetic field for the parameters given in previous section, Rabi frequencies ot = 13= 0.51, supposing the pumping laser frequency £2 resonant with the atomic transitions at Ho = 0. The result, presented in fig. 3, is interpreted as a coherent trapping, observed on the population inversion, with its width determined by the ground state relaxation rate. The difference pUo --PAA may provide amplification on generated E u = E U electric fields.

399

~litL I ++I l ~ i l l l '~HII~L t

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5. Conclusions We have a p p l i e d the scheme o f the absorbing a n d n o n a b s o r b i n g states to investigate the p h e n o m e n a taking place in the A or d o u b l e - A configurations interacting with two laser modes. This a p p r o a c h provides a simpler interpretation and a m o r e straightforward calculation o f the n o r m a l modes, p r o p a g a t i o n equations a n d threshold conditions for the laser without inversion. The whole process is interpreted as a d e p o p u l a t i o n p u m p i n g o f a t o m s from the absorbing level to the n o n a b s o r b i n g one, a n d as a four-level lasing scheme with inversion realized between an u p p e r level a n d an e m p t y absorbing lower one. The change o f basis point o f view was briefly discussed in ref. [ 7 ], where attention was focused mainly on the preparation o f the lower state coherence through a microwave field. W h e n d e p o p u l a t i o n p u m p i n g is the source for coherence preparation, our absorbing and nonabsorbing states, even if d e p e n d i n g on the laser electric field a m p l i t u d e s a n d not energy eigenstates for nondegenerate lower levels, are the convenient basis to describe the phenomenon. This statement implies that when density m a t r i x equations are written in this basis, the laser without inversion m a y be treated as a two-level problem, apart m i n o r perturbations. In the present investigation we have not included the role o f the atomic velocity on the coherent trapping o f the ground states. H o w e v e r the results o f previous investigations on the A-configuration [8,9] let's to derive that for the case o f copropagating laser fields acting on the two branches o f the A-configuration, all the a t o m s contribute to the coherent t r a p p i n g p h e n o m e n o n . O n the contrary for the case o f counterpropagating laser fields only the a t o m s with a t o m i c m o m e n t u m equal _+hk contribute to the coherent t r a p p i n g p h e n o m e n o n . Thus, for the realization o f the laser without inversion, the copropagating scheme is required in o r d e r to have the largest population inversion. Moreover, for counterpropagating laser beams in the double-A configuration, where different p h o t o n m o m e n t a are involved in the lower a n d upper transitions, the atomic velocity classes contributing to coherent t r a p p i n g on one transition are not a p p r o p r i a t e for laser without inversion on the second one.

Acknowledgements The authors wish to t h a n k R. B e n h e i m for useful c o m m e n t s on the manuscript.

References [ 1] O.A. Kocharovskaya and Ya.I. Khanin, Pis'ma Zh. Eksp. Teor. Fiz. 48 (1988) 581; JETP Len. 48 (1988) 630; J. Opt. Soc. Amer. B7 (1990) 2016. [2 ] M.O. Scully, in: Noise and Chaos in Nonlinear Dynamical Systems, eds. F. Moss, L.A. Lugiato and W. Schleich (Cambridge Univ. Press, Cambridge, 1990) p. 93. [3] M.O. Scully, S.Y. Zhu and A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813. [4] E.E. Fill, M.O. Scully and S.Y. Zhu, Optics Comm. 77 (1990) 36. [5 ] O.A. KOcharovskaya and P. Mandel, Phys. Rev. 42A (1990) 523. [6] O.A. Kocharovskaya, R.D. Liand P. Mandel, Optics Comm. 77 (1990) 215. [7] S.Y. Zhu, M.O. Scully and E.E. Fill, in: Coherence and Quantum Optics, vI, eds. J.H. Eberly, L. Mandel and E. Wolf (Plenum Press, New York, 1990) p. 1285. [8 ] For early descriptions see E. Arimondo and G. Orriols, Lett. Nuovo Cim. 17 (1976) 33; H.M. Gray, R.M. Whitley and C.R. Stroud Jr., Optics Lett. 3 (1978) 218; Fora review see'E. Arimondo, in: Interaction of radiation with matter, eds. E. Arimondo, G. Alzetta, F. Bassani and L.A. Radicati (Scuola Normale Superiore, Pisa 1987 ) p. 343. [9] A, Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste and C. Cohen-Tannoudji, Phys. Rev. Lett. 61 (1988) 826; J. Opt. Soc. Am. B6(1989) 2112. [ 10] F.T. Hioe, Phys. Rev. A28 (1983) 879. [ l 1] K.S. Mallesh and G. Ramachaandran, J. Phys. B: At. Mol. Opt. Phys. 22 (1989) 2311. 400