Raman emission versus dopplerons in a two-level atomic cell

1336 J. Opt. Soc. Am. B/Vol. 11, No. 8/August 1994

Raman emission versus doppleronsin a two-level atomic cell

Andree Tallet

Laboratoire de Photophysique Mol6culaire du Centre National de la Recherche Scientifique,Batiment 213, Universit6 Paris-Sud, 91405 Orsay Cedex, France

Received September 30, 1993; revised manuscript received February 8, 1994

Dopplerons are the signature of the nonlinear response of moving atoms driven by a continuous standing-wave laser beam. They display maxima on the velocity-dependent atomic population difference at velocitiesnearly equal to ±(c/co)[(8(0

2 + 2f1 2 ) ' 2 /(2p + 1)], where fl is the Rabi frequency associated with an individualbeam field and 8o is the detuning between the driving frequency and the center of the Doppler linewidth.Probing the device with a weak signal yields gain for Doppler broadenings of the magnitude of the detuning,exclusively on the atomic-resonance-frequency side, with a maximum shifted from the driving frequency bythe generalized Rabi frequency associated with a standing wave, defined by (w2 + 4fl2)1/2. The existenceof dopplerons is demonstrated by peaks occurring at frequencies shifted from the driving frequency by evenharmonics of doppleron frequencies. The intensity of the peaks is shown to depend on the Doppler width,and the complexity of the gain curve is shown to increase as the pump intensity does.

1. INTRODUCTION

Dopplerons were originally introduced by Kyrold andStenholml to describe the energy quanta 1kv displayedby the successive emission and absorption processes froma Doppler-broadened atom or molecule with transitionangular frequency Wat; these processes are driven by astanding wave with wave number k. The absorption ofn + 1 photons from one traveling wave and the emissionof n photons into the opposite traveling wave give riseto resonances in the atomic population difference, W(v),when expanded in Fourier series,2 at the atomic veloc-ities v = (Wat - kc)/[(2n + 1)k]. The first resonance,v = ±[1/(3k)]& (0w = at - kc), was predicted primarilyby Hartmann and Haroche,3 who performed a perturba-tion treatment for the interaction of a moving two-levelatom with a strong pump field and a weak probe field ofthe same frequency but traveling in opposite directions.An experiment performed by Reid and Oka4 displayedthis resonance in a slightly different optical device: agas composed of CH3 molecules was driven by a standing-wave pump field and a weak traveling field with orthogo-nal polarization, which led to the situation of a three-levelmolecule with a ground state I1) and excited states 12) and13). The standing-wave pump beam saturated the tran-sition 1-2, and the molecule was probed by a travelingradio-frequency weak beam through the transition 1-3.The spectrum of the signal displayed not only Bennettholes,5 symmetrically located at ±I 1,2 - pumplk but alsodopplerons at +1/31(01,2 - pumpl. This experiment wasinterpreted with a three-level system model,6' 7 which re-produced the measured probe spectrum fairly well.

Continuous pump beams of very high intensity can nowbe achieved. It follows that many successive doppleronresonances can occur in the velocity-dependent atomicpopulation difference for atoms driven by an intensestanding-wave electric beam. As a matter of fact, probespectroscopy experiments should exhibit Raman emission

of light at frequencies shifted from the pump frequency bysome amount related to the doppleron frequency throughthe four-wave mixing processes. Dopplerons were pre-viously discussed for two-level atoms in the limit of highinput intensity.8 '10 In Ref. 9 a standing-wave inputbeam plus a probe beam were considered, leading toprobe gain at new frequencies. In the limit for verysmall intensity (with the input intensity being equalto 10-2 times the off-resonance saturation intensity) theprobe response displays only the Bennett holes, located atbat ± I wat - 10pump , but for a larger input intensity threenew lines occur: one of them appears at a frequencyshifted from the atomic frequency by the three-doppleronfrequency, (Wat - 1/31)at - pumpI.

The role of the atomic motion was also previously stud-ied in two-level atom phase-conjugate devices1 ' for largeintensities. Measures of the reflectivity factor indicateda maximum located at the driving frequency (Rayleighprocess) and secondary maxima, which were symmetri-cally located with respect to the driving frequency andshifted from a value equal to the detuning and to thegeneralized Rabi frequency, i.e., at copump ± aw andW0pump ± (w2 + 4fl

2 )1/2

, where fl is the Rabi angularfrequency associated with a single beam. These Ramanemission lines are generally expected in the standardsituation for nonmoving atoms and without foreign gas(T2 = 2T,). The two Raman lines, at ±6 w and ±(6e)w2 +4fl

2)

1/2, respectively, correspond to the extrema of the

z-dependent generalized Rabi frequency for the standing-wave device, [6w 2 + 4 cos2(kz)fl 2 ]" 2: resonances occurat these locations because there are many more atomsdriven by the maximum and the minimum intensities,where the slope is very small, than in the domain of in-tensity with the steepest slope.12 For Doppler-broadenedatoms these resonances survive because of the contribu-tion of slow atoms.

Actually, the two lines occurring on the resonanceside in Ref. 9, which were not identified as doppleron

0740-3224/94/081336-14$06.00 ©1994 Optical Society of America

Andree Tallet

Vol. 11, No. 8/August 1994/J. Opt. Soc. Am. B 1337

resonances, correspond to the Raman lines exactlyshifted from the driven frequency by the detuning andthe generalized Rabi frequency. These features areshown below.

It appears that the nonlinearities would generate, inthe case of Doppler-broadened two-level atoms, not onlymultidoppleron processes but also the Raman processesthat depend on the generalized Rabi frequency. Bothtypes of process are present in the standing wave-deviceand should be coupled through four-wave mixing.

My aim in this paper is to treat exactly the nonlinear in-teraction of moving two-level atoms with opposite strongtraveling pumps together with the linear interaction witha signal and to discuss the effect of these dopplerons onthe gain curve of the signal as a function of the variousparameters. I start from the solutions, originally givenby Stenholm and Lamb,2 for the stationary regime of amoving two-level atom. The atomic concentration in thecell is assumed to be small enough to permit neglectingthe depletion of the pumps but large enough to generatea phase variation of the field amplitudes during propaga-tion. The input probe, sent, for instance, in the forwarddirection, generates both an idler field in the same direc-tion and signal + idler beams in the opposite directionbecause of the propagation back and forth in the cell oflength 1. Linear analysis provides a propagation equa-tion that takes the form of a four-dimension vector, thetreatment of which is described in detail in a previouspublication.'3 This equation is numerically solved, lead-ing to the probe spectrum at the exit of the cell. Hereour results will be compared with the Kyrbld-Stenholm 9

results, for which propagation back and forth in the cellwas neglected.

are, as usual,

*ga() dt] = 1)2W iT, -grd(W) +-i=-_(W+ 1) - ..(pfl* - cc)lT1~~v.~a dtj2

T2[v grad(P) +d = -(1 + i) - QWT2, (4)

where fl(Z) stands for the z-dependent angular Rabi fre-quency:

(f(z) = fl{exp[ikz + i(z - 1/2)] + c.c.},

with

fl= fi Eo(1

(5a)

(5b)

and with A = (at - w)T2 = 8wT2. The scalar productv grad reduces to v2 (d/dz) when transverse effects areneglected. Then, with the definition

P = C + is (6)

and by expanding in Fourier series any Bloch componentX (X = W, C, S),

X = x exp{in[kz + i(z - 1/2)]} (n = -, +o),

(7)

we can follow the same procedure originally proposedby Stenholm and Lamb2 for the semiclassical theoryof lasers. When definitions (7) are introduced into theBloch equations, the stationary solutions wn, Sn, and cnobey the relations

(inkvT2 + 1)cn = ASn, (8a)

2. STATIONARY SOLUTIONS

A. EquationsLet E0 be the amplitudes of the forward and the backwardplane-wave pump beams at the entrance, z = 0, and theexit, z = , of the cell, respectively. When there is noabsorption of the pump intensities through the cell, thepositive-frequency part of the electric field at location zin the cell can be written generally as

E(z, t) = E(z)exp(-iwt), (1)

(inkvT2 + )sn = -ACn - T21FWn-l + f1Bwn+1), (8b)(inkvT + )wn = -n,O + Tl(flBSn+l + OFSn-1) (8c)

with v = v and F,B = (li)IEFBI. There are severalways to solve the infinite set of Eqs. (8). The simplestone consists in defining a vector X = {xn} of infinite di-mensions with components 2n = W2n and X2n+l = S2n+l

for n = -oo, +oo, i.e.,

anXn + (qnxn+l + qnxn-1) = an, (9)

withwhere w is the pump angular frequency and E(z) is thecomplex amplitude (the time dependence is omitted be-cause we are dealing with only the stationary solutions):

an= -(inkvT, + 1), qn = flT, for n even(lOa)

E(z) = Eo exp(i;){exp[ikz + i(z - 1/2)] + c.c.}

= CEF(z)exp(ikz) + EB(z)exp(-ikz). (2)

Within the no-pump-depletion assumption the field-atominteraction cell creates the nonlinear phase ± cz for thefield amplitudes, where ; stands for the phase in midcell,

= e1/2. Therefore the Bloch equations for the popula-tion difference, W(z, v, t), and the polarization amplitude,P(z, v, t), for an atom with velocity vector v at location zin the cell and time t, with the definition

P(z, v, t) = P(z, v, t)exp(i;)exp(-iwt),

or

an = nkvT 2 + 1 + A2inkvT2 + 1qn = QT2

for n odd, (lOb)

where Ql stands for F,B because of the framework of theno-pump-depletion approximation. The set of equationsis equivalent to

BX = Xo, (11)

with X0 = (. .. , 0, ... , n,O, ... .) and B a tridiagonal(3) matrix such that bnsn = an, bn,n-1 = qn and bn+lsn = qn-

Andree Tallet


Equation (9) or (11) can be easily solved because of theparticular form of X0. The method consists in definingthe quantities 2

En = -n+Xnxn

Xn =An-Xn+1

for n 2 0,

for n < 0

and using the recurrence law to get the homogeneoussolution. n and Xn can be expressed as follows:

= + (13a)1 + (qn/an+1)6n+l

Xn qn/ (13b)1 + Xn+1

B. Multidoppleron QuantaThe lowest-order term in power of 2 , i.e., I1,2 of Eq. (16a),displays the resonances kvT 2 = ±A that corresponds tothe rate-equation solution. Keeping the term I2,3 but ne-glecting the next term of the continuous fraction gives riseto the resonance obtained by Haroche and Hartmann, 3

kvT2 = +1/3A, but only in the limit AT2 << A. When thenext terms in the continuous fraction are taken into ac-count, nonlinearities" 7 shift the lowest-order resonancekvT2 = ±'/3A and introduce new resonances, which at thelowest order are located at ±[1/(2n + 1)]A.

When all the successive terms of continuous fraction(16a) are taken into account, the angular frequency, in-volved for characterizing the Doppleron resonances, isfound to be close to

(s = (w 2 + 22)112 (19)

Then Eq. (11), with the right-hand member, can be solved:the vector component x0 corresponding to the Fouriercomponent wo, which satisfies [from Eq. (9)]

wo - f11T(s+, + s.,) = +1, (14)

becomes, from Eqs. (13),

ao 1 + (l/ao)(qoeo + qox-) (15)

Equation (15) can be expressed in the form of a continuousfraction as

ao= = I1,2 123 (16a)

1- 13,4

with the definition

Ii,(v) (12722 (16b)ai(v)aj(v)

and with

1 ((1 fl * (16c)a0 a0o

It is easy to verify that, for atoms at rest, wo is nothingbut the well-known expression {1/[1 + 4Io ,(0)]}l/2, whereIo,(O) equal to TT22(12(1 + A2), is the input intensityscaled to the off-resonance saturation intensity: for v =0, a2n = -1 and a2n+l = 1 + A2. Then we get, fromEq. (13b),

T of = Iobi(O) (17)

which has the obvious solution

TflQeo = 1/2{1 - (I + 4 10o,(0)]1/2}. (18)

Equation (18) leads, when Eq. (15) is used with ao = -1and eo = X-l, to the value that we were looking for.

This quantity is nothing but the generalized Rabi fre-quency for atoms, the population difference of whichwould be sensitive only to a net intensity equal to thesum of the forward and backward pump intensities, thatis, the mean intensity obtained after the full intensityfor a standing-wave is averaged over a half-wavelength,which cancels the crossed term exp(2ikz)EFEB* + c.c.That behavior for a mean atom is totally different fromthe behavior of an atom at rest, located at some z inthe cell, which is characterized by the z-dependent gen-eralized Rabi frequency [6w2 + fl(Z)2 ]1/2 . Whereas thedopplerons are primarily a result of the crossed termsexp(2ikz)EFEB* + c.c., the frequency that character-izes them is related to the net intensity. Finally, thedoppleron resonance frequencies are dressed frequencies,

ken = + 1 (,-2n + 1 (20)

when the nonlinearities are accurately taken into account.Let us now examine the numerical results that have

lead me to propose Eq. (20). For low enough input in-tensity, the truncature of the continuous fraction is valid,as is shown in Fig. 1(a), where the Fourier componentwo(v) is plotted as a function of kvT2 for (1T2 = 32,T2 = 2T, (Io = (12T2T = 500), and A = 100, giving rise toT2(s = 109.5. For these parameters a strong doppleronresonance occurs at kvT 2 = +37, slightly shifted from

1/3A, and well approximated by the dressed frequency

±1/3T2S = 36.5; a small resonance also occurs at kvT 2 =

+22, in agreement with ±1/(1ST2, while the outer max-ima, corresponding to the Bennett holes at kvT 2 = + 102.5,undergo almost no nonlinear effect. As the input inten-sity increases, higher-order doppleron resonances occur:these resonances are depicted in Fig. 1(b) for (1T2 = 82.The peaks are located at kvT 2 = +51, 30.5, 21.5, ...instead of at the bare frequencies, 33, 20, 14, ... but ingood agreement with the dressed frequencies 51, 30.7, 22.Nonlinearities shift the Bennett holes by -20%. For avery large input intensity, (127T2T, = 104, the velocity-dependent atomic Fourier components also conform to thedressed frequencies predicted by Eqs. (19) and (20). Thedressed doppleron frequencies, Eq. (20), are consequentlyvalid for any value of the input intensity.

These results show that the fundamental frequencythat determines the resonances is related to the sum

Andree Tallet

Andr6e Tallet Vol. 11, No. 8/August 1994/J. Opt. Soc. Am. B 1339

kvT2

(a)

0

zWz

C.,

Ck

Ca

C3

~J0

-250 -200 -ISO -too -so o 1 50 tO IO o0 250kvT2

(b)

0

Z

,0

CL

C(

Cd)

Ot U,3

C)oE-

-350 -30 -20 -20 -150 -100 -50 0 50O 100 1SO 200 250 300 350kvT2

(C)Fig. 1. Bloch velocity-dependent stationary Fourier component wo(v) for various input intensities: (a) Io = 500, (b) Io 3400. Themaxima exhibit doppleron-frequency resonances, except for the external maxima, which correspond to the so-called Bennett holes.Other parameters are A = 100, T 2 = 2T1.


of the individual intensities. Nevertheless, neglectinga priori the intensity crossed terms will forbid the oc-currence of the oscillations in the velocity-dependent sta-tionary atomic Fourier-components, i.e., the generation ofdopplerons. These crossed terms induce a z-dependentgrating, which can explain the generation of new fre-quencies through four-wave mixing when a probe field isadded."1

In conclusion, the effect of the nonlinearities in thepresence of moving atoms is subtle: as a matter of factthe dopplerons result from the z-dependent Rabi fre-quency for an atom at location z in the cell, fi(z), butthey display resonances at velocities that depend on themean intensity.

The higher-order Fourier components of the Bloch vec-tor, W2n and S2n+1, can be expressed in terms of wo by useof Eqs. (12), giving rise to

W2n = fe162 ... 62n}WO ,

W-2n = (2n)*,

S2n+1 = {f162 ... 2n+ 1}Wo

S-2n-1 = (2n+)* - (21a)

If Eq. (8a), which relates cn to Sn, is used, then the Fouriercomponents for the polarization amplitude Pn = Sn + intake the form

P2n+1 = 1 + i(2n + 1)kvT2 + 2n+1

P-2n-1 = (P2n+1)* - (21b)

Like wo all these components exhibit multidopplerons,with weights that decrease as n increases.

Before dealing with the four-wave mixing that shouldprovide a way for displaying the multidopplerons, let usrecall the reduced Maxwell equations for forward andbackward plane-wave amplitudes13"4 :

dF 2 ia J dy/u exp[-(V/u) 2 ]p,(V, z)

= f dy/u exp[-(V/U)2] a+if 2(1 + iA + ikv722)

X (WoEF + W1EB),

dEB(z) _i

dz = 2 aoJ dv/u exp[-(V/u) 2]p_,(v, z)

= f dv/u exp[-(V/U)2 ] a+J 2(1 + iA + ikvT22)

X (woEB + W-1EF), (22)

where ao is the on-resonance absorption coefficient and uthe atomic mean velocity, equal to [8kT/(-rM)]V' 2 . Whenthe depletion of the pump is negligible, then Eqs. (22) giverise to the nonlinear phase qz introduced at the beginningof this section in Eq. (1).

3. LINEAR ANALYSIS FOR THEGAIN OF A SIGNAL

To probe the stationary atomic system, a weak plane-wavesignal is sent into the cell at z = 0 in the forward direction,

with its angular frequency shifted by 77 from the drivingone and with the inclination 0 from the optical axis,

Es(z, r, t) = Eof+,o(z)exp[-i(co + 7)t + ikz + ikOr],(23)

with the relation f+,o(z = 0)I << 1. The probe's propaga-tion provides four-wave mixing in the atomic cell drivenby the strong standing-wave pump laser, so that the totalforward electric field takes the form

EF,B(z, r, t) = EF,B(Z) + 85 EF,B(Z, r, t), (24a)

with

.5EF(Z, r, t) = Eo exp(i; + ik'z)[f+,O(z)exp(-i-7t + ik~r)+ f_,o(z)exp(+i-7t + ik~r)+ f+,-o(z)exp(-i-qt - ikor)+ f_,_o(z)exp(+iy7t - ik~r)], (24b)

and the total backward electric field is

5EB(z, r, t) = Eo exp(i - ik'z)[b+,o(z)exp(-i-qt + ik~r)+ b-,O(z)exp(+i?7t + ikor)X b+,0 (z)exp(-i?7t - ik~r)+ b-,_o(z)exp(+iq7t - ik~r)],

with the boundary conditions

f+,o(z = 0) 0 0,

f+,-o(z = 0) = 0,f-,+o(z = 0) = 0,b.+,o(z = 1) = 0.

(24c)

(25)

These weak fields induce changes in the Bloch vectorcomponents,

P(z, r, v, t) = P(z, v) + p(z, r, v, t), (26a)

W(z, r, v, t) = W(z, v) + [W(z, r, v, t) + c.c.], (26b)

or, when the Fourier expansion is introduced,

P(z, r, v, t) = Y-exp[i(2n + 1)k'z][p 2 .+l(z, V)

+ exp(-i77t + ik0r)6p 2 n+1(z, V)

+ exp(i77t - ik0r)3p2 n+1'(z, v) + ... ],(27a)

with k' = k + 0 z and pn = Cn + isn as defined in Eqs. (1),(6), and (7). In the same way

W(z, r, v, t) = Y exp(i2nk'z)[w2 n(z, v)

+ exp(-i77t + ik0r)8w 2n(z, v)

+ exp(ii7t + ikr)...]. (27b)

In general W(z, r, v, t) is a complex number; thisproperty makes the expansions 8P(z, r, v, t) =8C(z, r, v, t) + iS(z, r, v, t), such as C(z, r, v, t) =C(z, v) + 5C(z, r, v, t) and S(z, r, v, t) = S(z, v) +

Andree Tallet


8S(z, r, v, t), useless, because both C(z, r, v, t) and8S(z, r, v, t) are also complex numbers. Consequentlylet us consider the complex Fourier components P2n+1and 8

W2n, where for the sake of simplicity the dependencewith respect to the atomic velocity v and the penetrationinto the cell, z, have been omitted and we are lookingfor the Fourier components that are involved in thereduced Maxwell equations for the forward and thebackward probes and idlers. If we introduce relations(24) for the electric field and relations (26) and (27) forthe Bloch components into Eqs. (4), we get, after somestraightforward calculations, recurrence relations for theFourier components W2n and 5 P2n+1. Then we maydefine the vector Y with components

36Y2n = W2n = {3W-2n} ,

' 5Y2n+1 = P2n+1 - 8P-(2n+1) } (28)

where, from definition (27a), {8P-(2n+1)}* is associatedwith the same frequency shift ,/ and the same 0 as 6P2n+1-The vector Y obeys the equation

SBSY = R.

for n = 0, + 1, 2, ... , depending on the strength of thepump. On the other hand, it follows for Eqs. (32b) thatthe components r2p+j exhibit maxima for

+ - 77T2 + (2p + 1)kvT2 = 0. (34)

These two relations give rise to resonances, either am-plification or absorption, for the probes and idlers at theangular frequency 77 such that

77T2 = +A - 2p + sT22n + 1 (35)

for any positive or negative integers n and p. In thesame way the even components r2p may exhibit reso-nances for

77 - 2pkv, = 0, (36)

giving rise to(29)

where B is an infinite-dimensional tridiagonal matrixand R is an infinite-dimensional vector with componentsrn, generally all being nonvanishing. The matrix ele-ments of 3B are

8bnn = 1,

8bnn-1 = 6bnn+l, (30)

with

5ib2.2n- = i IT 7 +

1 - i72, + ~

L 1 + iA - i77T2 + i(2n + 1)8

+ 11)3 (31)1 - iA - i77T2 + i(2n + 1)8

where and ' stand for kT 2 and kvT,, respectively.The right-hand components r depend on the stationaryFourier components, which are given by Eq. (19):

77 = - 2p (1s 2n + (37)

for any positive or negative n and p. Actually thesepredictions for the positions of peaks of emission or ab-sorption are qualitative, because a possible nonlinearshift caused by continuous-fraction recombinations hasnot been considered.

These resonances may induce a positive or a negativegain at frequencies shifted with respect to the pump fre-quency by an amount equal to an even harmonic of themultidoppleron frequencies, as implied by Eq. (34), or re-lated to an odd harmonic, as implied by Eq. (35).

The Bennett resonances are deduced from kv ±A,leading to 77 0 O ±2A.

The gain of the signal is the result of a four-wave mixingprocess1 generated by propagation in the cell, where theleft-hand side of the Maxwell equation is proportional to

J dv exp[-(v/u) 2 ]8p±1(V, z).

r i 1 - i77 T, + in' (f2*p. + b-*P2.-,

- f+p2n-l* -b+P2n+l*)X

= -i(2 2 1 + i - i77T2 + i(2n + 1)8

X (f+W2n -b+W2n+2)X

-m 1- l121u2 1 - i - inT2 + i(2n + 1)8

X (f*w2n+2* - b*W2n*)-

All stationary Fourier components 2n and P2n+1 thatoccur in Eqs. (32) for the vector R components may exhibitmultiple resonances at velocities such that

kv = 2 +lS (33)

Then knowledge of 8p±,(v, z) is required for solving theMaxwell equations for the probe and idlers. This time

(32a) it is not possible to get the solutions for the componentsof the vector Y by using the same method as for thestationary solutions because of the right-hand member ofEq. (29). Nevertheless this equation can be solved by theso-called LU method, for which B becomes the productof two matrices15

B = LU,(32b)

(38)

where the lower matrix L has only diagonal elements,nn, and off-diagonal elements, n+1,f,, different from

zero, whereas the upper matrix U has un,n and Unn+lthat are not zero. (There are other methods such asthe continuous-fraction Green-function method 6 used inRef. 9 or the straightforward matrix inversion used for

Andree Tallet


the study of two-copropagating pump beams.17) With

Eqs. (28), Eq. (29) becomes

LU8Y = R (39a)

4. GAIN OF THE SIGNAL

The forward perturbation f+(1) at the exit of the atomiccell, associated with the driving angular frequency (w +)7), can be expressed as

or f+(l) =()GF+(0),

LA = R (39b)

where A is the vector defined as A = U8Y. The compo-nents of the vector, An, obey the recurrence law

lA 3 = R 1 ,

1jj_iAj- + j,jAj = Rj, (40)

where GI, is the matrix element of the 2 X 2 gain matrixG defined in our previous papers,12"13"58 which is a functionof the frequency shift 77.

The situation in which the atoms are at rest has beenstudied extensively.12"13,1

8-20 In this case the signal may

exhibit Rayleigh or Raman resonances located in the de-tuning region 3w and at the generalized Rabi frequency,which for a standing wave can be written as

where the integer j increases from zero to N, whichis assumed to be infinitely large. The elements of theL matrix are calculated in Appendix A. This forward-recurrence law gives rise to the general expression for

Aj = E [l)j n Om (41)1n,n m m,m

where the summation over index n goes from 1 to j andthe product over index m goes from n to j. In Eq. (41) thequantity Oj stands for jj_1, introduced in Appendix A.To determine the components of the vector 8Y, let us solvethe backward-recurrence law, deduced from the definitionA = U8Y,

AyN = AN, 3 yI + ul,j+,yl+l = Al, (42)

where the integer decreases from N to 0; in Eqs. (42) therelation Unn = 1 has been taken into account, and the off-diagonal elements of matrix U are expressed in Eqs. (A3)below. In the same way as for Eq. (41), we get

8yk = Ak + E [(-1)n-kAn H ] (43)

where the summation over index n goes from k + 1 toN and the product over index m goes from k to n - 1.Knowledge of Aptl, as is required by the right-hand sideof the Maxwell equation,

= i(2 2 1 ± iA - i7T 2 ± ikvT2

x [(3wO + 3w±2) + (f+wo,- 2 + b+w2vo)], (44)

implies that we must perform the calculation of {Syk} onlyfrom k = N to (N/2) + 3, because we do not need any8W-2n for n > 1.

Finally, after integrating over the atomic velocities, wehave to solve a propagation equation that formally has thesame form as Eq. (2.20) of Ref. 13 but with the boundaryconditions given above by Eqs. (25) for the present casewithout any external feedback.

flT = 1 [A 2 + 4((1T2)2]1V2.T2

(46)

These gains generally vary exponentially with propaga-tion inside the cell,' 8 and they are generally positive(negative) and on the opposite (same) side of the detun-ing with respect to the pump frequency. (This is true aslong as the propagation does not mix the waves of oppo-site frequencies too much.) Here, finite positive (nega-tive) gains are expected at angular frequencies relatedto the doppleron resonances as a result of amplification(absorption) in the cell, whereas the gain at the Rabi fre-quency ±(1T is questionable because of the atomic motion.

Another gain may be also expected if the condition

det(G-') = 0 (47)

is fulfilled for some particular values of the set of theparameters of the problem and a given value of thefrequency shift, 77th; in this case the gain is infinite be-cause no probe is sent into the cell; this situation cor-responds to the instability-threshold or Hopf bifurcation,where new frequencies (wpump ± 77th) are generated by theincrease in nonlinearities. The present study was under-taken to explain the role of atomic motion in the Hopf-bifurcation characteristics for the device considered here.Indeed, experiments2 1-2 3 exhibit results that cannot beexplained within the nonmoving-atom assumption. Theaffect of atomic motion on the instability threshold is notpresented in detail here.

As was pointed out in Section 3, resonances may occurfor frequencies (pump + 77) close to even harmonics ofthe multidoppleron frequencies or odd harmonics shiftedby the detuning. However, there is no direct evidenceof frequencies that might be generated for the four-wavemixing process. We can only conjecture that the Dopplerlinewidth has to be large enough that atoms with veloc-ities near a doppleron resonance may be involved in thenonlinear interaction process.

For moving two-level atoms interacting with two-strongcounterpropagating pump fields plus a probe, this is, toour knowledge, the first attempt at getting numericalresults from the full treatment of the linear analysis forvarious Doppler linewidths. The main difficulty lies inthe numerical integration on the Doppler profile of someunknown function, which may have several resonances.Actually, the Gauss quadrature method2 4 fails because it

(45)

Andree Tallet


introduces resonances at the n zeros, n = 1, 2,... k, of thekth k Hermite polynomial that is used for the expansion.With such a method the locations of the resonances arefound to vary as the order of the Hermite polynomialis changed for integration over the Doppler profile andalso as the Doppler linewidth is changed. This methodworks only in the limit of a very small Doppler linewidthWD compared with the homogeneous linewidth, i.e.,WDT2 << 1, when the atomic motion is negligible. Here,with large Doppler broadening, WDT2 >> 1, a variable-step Runge-Kutta integration method is used, whichappears to be reliable only for Doppler linewidths of themagnitude of or larger than a few units of the homoge-neous linewidth. In between, for 1 << WDT2 c 10, noreliable procedure has been found.

In Subsection 4.A the role of Doppler broadening isstudied in the simplest case, in which the stationaryregime exhibits only the three-doppleron resonance, i.e.,for very small intensity scaled to the off-resonance satu-ration intensity, (127T2T1/(1 + A 2) << 1 (Figs. 2 and 3).Then in Subsection 4.B the input intensity is increasedand the gain of the signal is discussed as function of boththe input intensity and the Doppler broadening (Figs. 4and 5).

The on-resonance absorption coefficient is chosen tobe small enough that the net absorption of the pumpsis negligible, as is assumed for solving the propagationequations. In all numerical calculations fulfillment ofthat requirement was verified a posteriori. With aol =200, the requirement is satisfied for the sets of parametersstudied in this paper.

Note that the horizontal axes in Figs. 2-5, which givethe variation of the frequency shift of the probe, are la-beled with /3 equal to - -. Therefore resonance occurs onthe negative side, at /3T2 = -A. The zero corresponds tothe frequency of the pump, as is usual in four-wave mix-ing.

A. Role of Doppler BroadeningThe parameters chosen in this subsection are those ofFig. 1(a), with A = 100, (12T1T2 = 500, T2 = 2T1, whichexhibit Bennett holes and doppleron resonance at kvT2equal to ±102.5 and ±36.5, respectively. When theatoms are at rest, the signal displays a large Ramangain peaked at an angular frequency slightly shiftedfrom the Rabi frequency flT, only on the opposite sideof resonance, 8 2 0 and also some Rayleigh gain. As themean velocity of the atoms increases, the gain of theprobe is modified: for DT2 = 10, [Fig. 2(a)], the Ra-man gain on the opposite side of resonance still survives,slightly shifted but broadened, as occurs for one-way andsingle-pass propagation,2 5 except that here the gain isshifted toward the high frequencies. On the resonanceside the maximum occurs near -ITT2 (curve a), withfITT2 = 118. Smaller peaks occur near -IST2 (curve b),with fIST2 = 109, and near -A (curve c). Symmetricallylocated peaks are also visible. No emission or absorptionis obtained near a doppleron resonance as predicted byEqs. (35) and (37) because of the small Doppler broaden-ing, 0

WD < 1s/ 3 . As the Doppler linewidth is increasingfurther [coDT2 = 100 in Fig. 2(b)], the gain on the sideopposite resonance, at flT, is vanishing, while the maxi-mum gain still occurs in the vicinity of -T (curve a).

Secondary maxima occur (curves b and c), as in Fig. 2(a),and an expected maximum occurs at -2fs/ 3 (curved), i.e., at twice the largest dressed doppleron quan-tum, in agreement with prediction (37). Curve d cor-responds to line discussed in Ref. 9. Let us alsopoint out that the Rabi frequency (1 for an individ-ual pump beam presents a negative gain symmetri-cally on both sides of the driving frequency (hole e for-fl). For greater Doppler broadening, DT2 = 103in Fig. 2(c), gains arise approximately at the same lo-cations, except the one located close to f1s, which ishidden in the tail of the broadened peak located nearQIT. I also point out that the maximum of the Ramangain begins to decrease when the Doppler linewidth be-comes larger than the detuning, although it is approxi-mately constant for smaller linewidths. The Rayleighfeatures correspond to the location of one of the so-calledBennett holes. Note that there is no feature near theother Bennett hole, at -2A. The absence of the secondBennett hole near -2A is not surprising with the presentparameters; indeed, the atomic system is expected to betransparent for frequencies far from the Rabi-frequencydomain.

Curves a, c, and d are also seen in Ref. 9, whereas onlycurve d was identified by the authors. Curve b is notpresent, presumably for the same reason as in Fig. 2(c).

As a matter of fact the strong amplification on the reso-nance side, for a frequency close to the Rabi frequency,flT, is the one of the main features of these numericalresults. Such an amplification, related to the Rabi fre-quency, was indeed foreseen, but in a different setup,with a traveling pump beam and a weak signal in theopposite-traveling direction.3 In this case Haroche andHartmann 3 predicted a possible enhancement at a fre-quency shifted from the driving frequency by plus orminus the Rabi frequency.

For the present device with two counterpropagatingpump beams, four-wave mixing provides a positive ornegative gain that involves atoms with a very small ve-locity about the center of the Doppler profile. The con-tribution of the system to such a resonance at the Rabifrequency can be shown to result from the central compo-nent r of the vector R in Eq. (29). The approximation

R = (. . 0, 0, 0, r, 0, 0, 0, .) (48)

gives the Fourier component wo the same form as thestationary component wo, i.e., a continuous fraction likeEq. (16). Then, with it assumed that only the atomswith a negligible velocity are involved in the gain process,the summation gives rise to solutions for wo, p±l,....proportional to [see Eq. (A15) of Ref. 13]

1

J flTT 1 + i8 2 112+ 4(272722[(1 + i72 2)

2+ A2 ](1 + i372)

1 (49)

which displays a maximum at / = +QT and a smallerpeak at 83 = ±Qs. Exact integration on the Doppler pro-file gives rise to [3wo(v, /T32)]mean, the real part of whichexhibits symmetrical bumps at , = ±QT but dispersion-like behavior with a large amplitude near /3 = 0; the

Andree Tallet


ID

7- a

cc

zcm

Cd00..I

0

C

a

b_ d

-150.0 -120.0 -90.0 -60.0

vs('4

('A

za:

0D

-50.0 0.0

beLoT2

(b)

30.0 60.0 90.0 120.0 ISO.0

ISO.0

Fig. 2. Gain curve of the signal for various Doppler linewidths.T2 = 2T1, aol = 200, Io = 500, and (a) &JDT2 = 10, (b) &)DT2

beLoT2

(c)The x-axis origin is the driving frequency. Parameters are A = 100,

= 100, (c) WDT2 = 1000.

Andree Tallet

bet.oT2

(a)

z

2:CD

a

S10

9- -

-


C

a

CM0

CD

a: CD d0 I

-130 -12S -120 -I Is -11bets

(a)

IRn'

U,

0

-130 -12S -120 -itS -Ibeti

(b)Fig. 3. The Rabi gain originates in the component ro as displayedare those of Fig. 2(b).

imaginary part has features at the same locations. Ac-tually the quantity, depending on 8S, that is involved forsignal enhancement is proportional to

[1 + iA + iT2 + iV T2 mean 32) (0

which is found to present, only on the resonance side, twomaxima located near -T and -8w. Figure 3(a) showsthe probe gain when it is assumed that the right-handside of Eq. (29) is given by Eq. (48). The figure agreeswith the prediction of two peaks as indicated by Eq. (50)at -8cw and at -T. The results of Fig. 3(a) can becompared with the exact gain in Fig. 3(b) [an enlargementof Fig. 2(b)].

The enhancement at 2/3fls that is the signature of adoppleron is small because the frequency does not belongto the Rabi domain - 6, - TI, where three-photon pro-cesses are favored.

b

3T2

b c

in (a) and compared with the actual gain in (b). Parameters

B. Effect of Higher IntensityFor a larger pump intensity the main results are un-changed; the shape of the probe gain tends to becomemore complex, just because the number of doppleronresonances in the stationary atomic Fourier componentsis increasing as shown in Figs. 1, giving rise to newresonances in the probe gain.

Figure 4 shows gains for Io = 3400 and DT2 =

25, 50, 100. The various resonances related to adoppleron frequency (18 /(2n + 1) may exhibit largeenough peaks only if the Doppleron frequency is smallerthan D. For this reason, for DT2 = 25, the reso-nance at -4fs/5 (curve r, BT2 = -124) appears with asmall maximum, while the resonance at -8fs/ 7 (curves, 3T2 = -161) exhibits the greatest maximum of thedoppleron lines. For T2w(OD = 50 in Fig. 4(b) the situationis reversed: curve r has the highest peak at -4fls/5,and the next peak by decreasing height corresponds tocurve s. The enhancement at -4(1s/5 (line r) becomesthe maximum probe gain for T2wD = 100. In Fig. 4(b) a

Andree Tallet


za:es

CL

II

E

cc

CD

w

Cd

n

a:CD

CdIn_0-

-2s0.0

a

St

r

betoT2

(a)

200.0 2S0.0

a

betaT2

(b)

200.0 250.0

r

-50.0 0.0beLoT2

a

(C)

Fig. 4. Gain curve of the signal for different Doppler linewidths. The(a) &JDT2 = 25, (b) WDT2 = 50, (c) WDT2 = 100.

parameters are the same as in Fig. 2 except that Io = 3400:

Andree Tallet


a

-35.0 -280.0 -210.0 -140.0 -70.0 0.0betcT2

70.0 140.0 210.0 280.0 350.0

Fig. 5. Gain curve of the signal for Io = 103. Other parameters are those of Fig. 2 with wDT2 = 100.

smaller peak, located at pT 2 = -144, curve t, cannot beinterpreted simply with the help of relation (35) or (37).It seems that new resonances occur through the buildingof the vector Y, which may be odd harmonics of theDoppleron frequency shifted from the Rabi frequency,tfIT- Indeed, the peak, at 3T2 = 144, agrees withflT - ls/3. Other peaks can be also attributed tosome QT - (2k + l)f1S/(2p + 1), but there is not yetmathematical proof of such resonances. Finally, notethat the Rabi curve a broadens, decreases, is shiftedtoward the center, and is split in Fig. 4(c) for largeDoppler broadening (D = aw) and that the minima attfl still exist and are deep enough when WD becomes ofthe magnitude of or larger than [Q. Peak r is magnifiedin Fig. 4(c) because the outer resonance in the stationaryFourier components (associated with the Bennett holes)displays a frequency kvout, slightly larger than aw andclose to 4fls/5: therefore the velocities near kvout havea nonnegligible weight when WD 8w, leading to a globalresonance of all terms occurring in the products such asin Eq. (43). For larger Doppler broadening this peakstill exists, but its height decreases.

For a very large input intensity, Io = 104, i.e., foran input intensity equal to the off-resonance saturationintensity equal to unity, results are depicted in Fig. 5with T2WD = 100 and other parameters taken unchanged.Unexpectedly, the maximum gain does not occur in theRaman frequency range but in the Rayleigh one. TheRabi gain still survives, but curve a at -QT is broadened,while a noisy behavior characterizes the process in the re-maining Rabi domain below -8w. This blurring behav-ior may be a consequence of all the doppleron resonancesthat might occur between -8w and -QT. The vanishingof curve a can be understood as a result of the saturationeffect and of the many doppleron oscillations imposed onthe stationary velocity-dependent Fourier components inthe vicinity of the center of the Doppler profile. It fol-lows that the conjugate action of the atomic motion andof saturation tends to cancel the Raman gain in the limitof input intensity of the magnitude of or larger than theoff-resonance saturation intensity. In this limit only theRayleigh process survives.

5. CONCLUSIONS

In this paper we have discussed the effect of atomic mo-tion on the nonlinearities in a device made of a two-levelcell driven by counterpropagating laser beams and probedby a weak signal. The model that is developed is validwhatever the pump intensity might be, compared with thesaturation intensity, whereas the interaction with the sig-nal is treated in the standard linear analysis that assumesthe generation of four weak beams through propagation,two each of counterpropagating probes and idlers. Thelinear analysis includes no simplification with respect tothe nonlinear pump power.

The main results are the following:

(a) The doppleron angular frequencies are close to±[1/(2n + l)]fs, where s is the generalized Rabi fre-quency,

fls = (w 2 + 2 2)"2.

The meaning of the factor 2 in the above equation hasbeen discussed in Subsection 2.B.

(b) Except for the Rayleigh domain that correspondsto one of the Bennett holes, the frequency domain forwhich the signal can be amplified is bounded by theextrema of the generalized z-dependent Rabi frequency,W ± [80 2 + 4f1 2 cos2(kz)]2 2. The maximum of the gainin the Raman-frequency range occurs on the resonanceside, at + (w 2 + 4 2)1/2. The atomic motion beginsto affect this gain when the Doppler linewidth increasesfurther above the value of the detuning. This gain is alsovanishing because of the combining effects of saturationand multidoppleron occurrence in the limit in which theinput intensity is larger than the off-resonance saturationintensity. Other peaks in the Raman range support theexistence of dopplerons; this conclusion is demonstratedby the occurrence of signal enhancement at a frequencyshifted from the driving frequency by even harmonicsof the doppleron frequencies (8w2 + 2 2 )v2/(2p + 1), inagreement with prediction (37). Doppleron enhancementrequires that the Doppler broadening be larger than the

-0-n-

aa

coC3

| T , . . .

Andree Tallet


doppleron frequency,

(802 + 2 2)"2

2p + 1

Furthermore, large enhancement is favored if a frequencyshift associated with the p doppleron belongs to the Rabidomain:

2 2n 8w • 2+ (8w2 + 2(12)12 • (8w2 + 4fI2)I2~2p + 1

The conjecture is realized for the parameters of the Fig. 4that display beautiful doppleron lines. These resultsallow us to be optimistic about the possibility of observ-ing doppleron lines from two-level atoms, in contradictionto the conclusions of Ref. 9.

All these results show without any ambiguity thatthe atomic motion can be neither neglected nor quali-tatively introduced when the Doppler linewidth becomeslarger than the homogeneous linewidth. The occurrenceof dopplerons in the velocity-dependent atomic Fouriercomponents caused by the opposite-traveling beams hasdrastic effects on the gain of a probe, making the gainoccur on the resonance side instead of on the side oppo-site the resonance, as is often expected in four-wave mix-ing studies in the limit CODT2 << 1. Actually the usualapproximation, which makes the polarization follow theelectric field adiabatically, fails when the system is in-homogeneously broadened. Even in the limit of strongexcitation, when the saturation and the blurring effectscancel the Raman processes (see Fig. 5), i.e., when the lon-gitudinal grating seems definitely to vanish, accurate re-sults require an exact treatment of the nonlinearities plusatomic motion. Indeed, previous calculations neglectingthe longitudinal grating but taking account of the Dopplerbroadening did predict the enhancement of the signal atfrequencies shifted from the driving frequency by approxi-mately ±fHs. Neglecting the crossed terms in the inten-sity leads to some treatment that looks like the treatmentfor a unidirectional beam setup; within this approxima-tion the treatment of the motion of the atoms becomesqualitatively the same for counterpropagating beams asfor unidirectional beams, which implies a shift of theresonance frequency toward the center and a broadeningof the line. Therefore the building of dopplerons is totallyignored. The actual treatment gives rise to a complex be-havior of the system that depends on the input intensityand the Doppler broadening, which would be never ac-curately described without the possibility for an atom toabsorb or emit photons from counterpropagating waves.

Atomic motion should probably be introduced into thetheory for treating optical instabilities in gaseous and in-homogeneously broadened media driven by counterpropa-gating beams. That modification might explain someexperimental results of spatiotemporal instabilities thathave not received any convincing interpretation, such asthose that occur for a detuning of the magnitude of theDoppler linewidth. 22 23

APPENDIX A

Let us recall that the lower matrix L has only diagonalelements ly and off-diagonal elements 1 n+1,n differentfrom zero, whereas the upper matrix U has Un,n and Un,n+lthat are not zero. The elements of L and U can be de-duced from the B elements given by Eqs. (28) with asimple algorithm:

dbn=n 1 = nn-Un-1,n + nsnUngn

5bn-lgn = lnnUnn+l = On,

,bn+lbn = n+lsnUnxn = On

(Al)

(A2)

where n goes from zero to infinity. If setting uj,j = 1,then lj+1,j = Oj for any j and 11,1 = 1, U1, 2 = 01, equa-tions (Al) and (A2) allow us to determine jj and uj+,,j.Actually an expansion as continuous fractions emerges:

12,2 = 1 - 0102,

12,2 = 1 - 02031 - 0102

02U2,3 = -

03U3,4 - 0203

- 1 - 0102

(A3)

This algorithm is used for performing the calculation ofthe Fourier components of the Bloch vector for each classof atomic velocity, both for the stationary part Y and theperturbation part Y,

BY = Yo,

MB8Y = R. (A4)

as given in Eqs. (11) and (27)-(30).

REFERENCES AND NOTES1. E. Kyr6ld and S. Stenholm, "Velocity tuned resonances as

multidoppleron processes," Opt. Commun. 22, 123 (1977).2. S. Stenholm and W. E. Lamb, Jr., "Semi-classical theory of

a high-intensity laser," Phys. Rev. 181, 181 (1989).3. S. Haroche and F. Hartmann, "Theory of saturated-

absorption line shapes," Phys. Rev. A 6, 1280 (1972).4. J. Reid and T. Oka, "Direct observation of velocity-tuned

multiphoton processes in the laser cavity," Phys. Rev. Lett.38, 67 (1977).

5. W. R. Bennett, Jr., "Hole burning effects in a He-Ne opticalmaser," Phys. Rev. 126, 580 (1962).

6. R. Corbalan, G. Orriols, L. Roso, R. Vilaseca, and E.Arimondo, "New phenomena in doppleron resonances," Opt.Commun. 28, 113 (1981).

7. L. Roso, R. Corbalan, G. Oriols, and R. Vilaseca, Dressed-atom approach for probe spectroscopy in Doppler-broadenedthree-level systems with standing-wave saturator," Appl.Phys. B 31, 115 (1983).

8. J. Ziegler and P. R. Berman, "High-intensity single-modelaser theory," Phys. Rev. A 16, 681 (1977).

9. E. Kyrdld and S. Stenholm, "Probe-spectroscopy of multi-Doppleron processes," Opt. Commun. 30, 37 (1979).

10. H. Friedmann and A. D. Wilson-Gordon, "Doppleron-inducedstimulated reflection," Opt. Lett. 14, 737 (1989).

11. Robert A. Fisher, ed., Optical Phase Conjugation (Academic,London, 1983), pp. 234-251.

12. M. Le Berre, E. Ressayre, and A. Tallet, "Gain and reflec-tivity characteristics of self-oscillations in self-feedback anddelayed feedback devices," Opt. Commun. 87, 358 (1992).

13. M. Le Berre, E. Ressayre, and A. Tallet, "Self-oscillations ofthe mirrorlike sodium vapor driven by counterpropagatinglight beams," Phys. Rev. A 43, 6345 (1991).

Andree Tallet


14. Propagation was treated for moving atoms driven by coun-terpropagating pump beams of unequal input intensities byJ. H. Shirley, "Semi-classical theory of saturated absorptionin gases," Phys. Rev. A 8, 34 (1973).

15. P. Manneville, Dissipative Structures and Weak Turbulence(Academic, London, 1990).

16. C. G. Aminoff and S. Stenholm, "Recoil effects in quantumelectronics II. Saturated absorber spectroscopy," J. Phys.B 9, 1039 (1976).

17. Z. Ficek and H. S. Freedhoff, "Resonance-fluorescence andabsorption spectra of a two-level atom driven by a strongbichromatic field," Phys. Rev. A. 48, 3092 (1993).

18. I. Bar-Joseph and Y. Silberberg, "Instability of counter-propagating beams in a two-level-atom medium," Phys. Rev.A 36, 1731 (1987).

19. A. J. van Wonderen and L. G. Suttorp, "Instability for ab-sorptive optical bistability in a nonideal Fabry-Perot cav-ity," Phys. Rev. A 40, 7104 (1989).

20. M. Le Berre, E. Ressayre, and A. Tallet, "Physics in coun-terpropagating light-beam devices: phase-conjugation andgain concepts in multiwave mixing," Phys. Rev. A 44, 5958(1991).

21. D. Grandclement, G. Grynberg, and M. Pinard, "Observationof continuous-wave self-oscillation due to pressure-inducedtwo-wave mixing in sodium," Phys. Rev. Lett. 59, 40 (1987).

22. G. Giusfredi, J. F. Valley, R. Pon, G. Khitrova, and H. M.Gibbs, Optical instabilities in sodium vapor," J. Opt. Soc.Am. B 5, 1181 (1988). The theoretical model developed inRef. 13 partially disagrees with the experimental measure-ments in the limit of small detuning, presumably because ofthe Doppler effect.

23. G. Khitrova, J. F. Valley, and H. M. Gibbs, "Gain-feedbackapproach to optical instabilities in sodium vapor," Phys. Rev.Lett. 60, 1126 (1988). In the case of the setup withoutfeedback mirrors, the observation of an instability thresh-old frequency of the magnitude of the radiative linewidthdisagrees with the theoretical prediction of a Raman insta-bility Ref. 18. It appears that the atomic motion may beresponsible for this Rayleigh instability as indicated by pre-liminary numerical simulations.

24. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T.Vetteling, eds., Numerical Recipes (Cambridge U. Press,New York, 1987), pp. 121-126.

25. G. Khitrova, "Theory of pump-probe and four-wave mix-ing spectroscopy," Ph.D. dissertation (New York University,New York, 1986); see also M. Gruneisen, K. R. MacDonald,and R. W. Boyd, J. Opt. Soc. Am. B 5, 123 (1988).

Andree Tallet

Raman emission versus dopplerons in a two-level atomic cell

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