PHYSICAL REVIEW A VOLUME 51, NUMBER 1 JANUARY 1995

Lasers without inversion in a Doppler-broadened medium

Andrzej KarawajczykFysik I, Kungl Tekniska Hogskolan, S I0-0$$ Stockholm, Sweden

Jakub ZakrzewskiInstytut Fizyki, Unimersytet Jagielloriski, ulica Reymonta g, 80 05-9 Krakow, Poland

and Iaboratoire Kastler Bro-ssel, Universite Paris VI, g place Jussieu, 75252 Paris Cedez OS, France(Received 14 April 1994)

Three-level models of lasers without inversion in V and A con6gurations are studied in a realisticDoppler-broadened gas medium. The intensity of a generated coherent radiation is numericallycalculated for a wide range of experimental parameters and the role of the laser cavity is fullytaken into account. Different geometrical arrangements of the experiment are compared. Significantdifferences in lasing properties of the studied models are shown and discussed. An amplificationmechanism based on the Doppler shift tuning to dressed-state inverted transitions is discussed.

PACS number(s): 42.65.Es, 42.65.Hw, 42.55.—f, 42.50.Hz

I. INTRODUCTION

The possible applications of systems exhibiting gainwithout population inversion between levels involved ina laser action have stimulated intensive studies in thepast few years. The process requires asymmetry be-tween absorption and emission properties of the activemedium. Theoretical models involving for this purposeautoionizing states [1,2] as well as an additional coher-ent driving field in two- [3,4], three- [5—8], and four-levelmodels [9—11] have been extensively studied theoretically.While a common term —lasing without inversion (LWI)—is used to describe the properties of all such models, theorigin of gain may be different in distinct cases.

Two main mechanisms of gain creation have beenfound. For most schemes, although the lasing appearsto be without inversion in the atomic basis, a clear pic-ture emerges once the system is represented in the so-called "dressed-state" basis obtained after diagonalizingthe Hamiltonian including an atom-coherent driving fieldinteraction. The gain is due to dressed-state inversion.The resulting inversion in the dressed-state basis for four-level schemes [10,11]may also be interpreted in the equiv-alent language of the population-trapping phenomenon[»l.

For other proposed models, in particular those involv-ing the autoionizing states [1,2] and some three-levelschemes in resonant conditions [5—8], no simple changeof basis may explain the amplification. The process isdue to coherence among different ionization channels inthe former and an induced coherence between dressedstates in the latter systems [13]. Alternatively, the am-plification may be explained in the language of coherentmultiphoton processes with the asymmetry leading to netgain creation introduced by the incoherent pumping [14].

While extensively explored theoretically, a practical re-alization of LWI seems to be a difficult experimental task.The dressed-state lasers have been shown to work [3], but

only partial success has been obtained with other lasingschemes, mainly using the population trapping mecha-nism [15,16]. Some experiments used a pulsed schemeand utilized not the steady state but a transient gain[16]. No success has been reported up to now for truly"inversionless" systems (of the second type describedabove), although a related phenomenon of self-inducedtransparency has been observed [17].

Most of the theoretical treatments have been restrictedto linear gain study with the notable exceptions [2,4,6,11]limited, however, to the simplest situations. The aimof this paper is to explore one possible source of ex-perimental difficulties in a realistic situation, namely,the inHuence of the inhomogeneous broadening due tothe Doppler effect. It produces detunings of the pumplaser beam and the resonant cavity &equencies &om theatomic transitions, which may affect the coherent recy-cling in atoms and damp the intensity of the laser radia-tion.

Despite its importance, the effect of inhomogeneous,Doppler-effect-induced broadening has received little at-tention in the present context. A notable exception avail-able in the literature is a detailed study of an enhance-ment of the index of refraction in the presence of van-ishing absorption in atomic models which are similar tothose studed here [18]. That work [18] considers a passiveoptical system; thus the inhomogeneous broadening maybe taken into account in quite a straightforward way byconvoluting the final results obtained for a Doppler-&eesystem with the assumed atomic velocity distribution.On the contrary, in the active. optical medium consideredhere, all atoms contribute to the dynamically evolving(and coupled to the cavity mode) macroscopic polariza-tion. The created field inQuences, in turn, the atomicevolution. Therefore, a full nonlinear laser theory mustinclude the effect of the inhomogeneous broadening fromthe very beginning, which makes the computation moredifficult. To our knowledge, such an analysis for LWImodels has not been presented up until now [19].

1050-2947/95/51(1)/830(5)/$06. 00 51 830 1995 The American Physical Society

51 LASERS WITHOUT INVERSION IN A DOPPLER-BROADENED MEDIUM 831

II. MODEL

We consider a sample of N three-level atoms in a closedcell placed inside a single-mode ring laser cavity of a pas-sive linewidth I', a resonant &equency u, and a wavevector of the cavity mode k . A thermal velocity distribu-tion of the atomic sample is a Maxwellian function char-acterized by a Doppler broadening D, which is relatedto a temperature of the sample T by D gl/T. Theformalism developed is also applicable to atomic beamarrangements; then D is related to the residual Dopplerwidth. We study two difFerent three-level models, shownin Fig. 1, i.e. , the A-level configuration [5] and the V-levelscheme [8]. In both cases, one of the optical transitionsis coupled by a strong, coherent pump beam of frequencyuz and wave vector kz (transition 2 ~ 3 in the A modeland 1 m 2 in the V model), while the laser action occurson the remaining transition (3 ~ 1 for both models)in the vicinity of which the cavity resonant frequency istuned. Additionally, an incoherent pumping on transi-tion 1 ~ 3 is assumed (with pump rate A). Each of theexcited states n decays radiatively to a ground state mwith a transition probability 2p . In our model we ne-glect processes leading to the depopulation of the groundstates such as collisions or escape out of the interactionregion.

The time evolution of the density matrix is describedby the Liouville —von Neuxnann equations (h = 1)

~ = -*[&,s]+ ~~~+ ~.s + ~~&

The Hamiltonian part of the evolution reads

N 3

p i=1

+ge '" "o+" ' ato."3+H.c. +(u a a, (2)

where j = 2 and i = 3 for the A-level model and j = 1and i = 2 for the V-level scheme. A pth atom moves

l3)

&&3+4I

I

I

II

I

I

A

l2&

l3&

+AI

I l(I

I

I

I

I

I

I

A

l3&

FIG. 1. Two-level schemes studied in our paper: (a) theA-level model, and (b) the V-level scheme. One of the opti-cal transitions is coupled by a strong pump beam (solid line,double arrow) and the lasing (dashed line) occurs on the re-maining transition to which the cavity resonant frequency istuned. The remaining arrows indicate pumping and sponta-neous decay.

To eliminate the time and spatial dependences &om theequations of motion we introduce a new set of variables.For the A laser

a = a exp (iu„t), 0; = cr;;,

012 ——O12 exp(i(k„—k, ) (ro + v"t) ),

0i3 —0]3 exp(i[~„t—k, (ro + v"t)])

023 —023 exp(i[u„t—k„'(ro + v"t)])(4)

(the transformation for the V model is obtained by anappropriate permutation of indices).

Since in our work we are interested in laser light in-tensity rather than in noise properties, we follow thesemiclassical approach in further calculations. Thus wedecorrelate the atomic and field variables and find a setof master equations in the rotating-wave approximationfor the quantum averages of atomic and field observables.

For the A model we obtain explicitly

N

a = —[I'+i (~, —~„)]a—ig) oi3P

oi =i(g oosi g is)+ ('Y + ) 3

0'2 = 'LO (0'32 0'23) + 2+23O3 l

03 —— i A (0'32 (723) i—(g'aosi —ga o'13)—2 (P23 + Pis + ~) os + 2«»

O12 —t[(kP kc) ~ V& —&21]O12 t~O13

+Zg aCF32 —AO l2 )

13 = t(&p + &13 —k~ v~)%13 —tOo12+ig'n (o3 —oi ) —(723+ /13+ 2~) o,"3,

23 ( P 32 c P) 23 + ( 3 2)—igaO21 —(P23 + +13 + A) 023

with a similar set of equations for the V model. Thevelocity-dependent terms describe the Doppler shifts ofthe pump beam &equency and the cavity frequency fromthe respective atomic transitions. The effect depends (seethe equation for 012 above) on the mutual orientation ofthe driving laser beam and the cavity axis.

For further discussion we consider two geometrical re-alizations of the experiment: (i) the collinear configura-tion (CC), in which the pump beam is sent along theoptical axis of the cavity, and (ii) the orthogonal config-uration (OC), in which the pump beam illuminates thecell &om the side Ic„Jk .

Note that the CC requires the application of wave-

with a velocity v& and its position at t is ro + vt"t. Theconstant g represents a coupling strength between theatomic transition and the ring-cavity mode a and 0 isthe Rabi &equency of the driving field.

The I iouville operators Z~, Z~, and Z~ describe thecavity damping, the spontaneous emission, and the in-coherent pumping, respectively, and take the standardform [20], e.g. ,

C~p = I' (2apat —atap —pata) .

832 ANDRZEJ KARAWAJCZYK AND JAKUB ZAKRZEWSKI

length selective mirrors that are transparent for the pumpbeam wavelength and highly reQective for the generatedcoherent radiation. However, for the energy splitting ofthe two optical transitions larger than several nanome-ters, such mirrors are commercially available.

III. NUMER. ICAL RESULTS

To facilitate the numerical integration of the prob-lem we divide the velocity distribution into discretegroups along the relevant direction. The population ofthose groups is numerically calculated using a Gaussianquadratic procedure and all atoms belonging to a givengroup are assumed to move with the same mean groupvelocity.

For the CC (k„~~k, ) the problem is one dimensionalsince the equations become dependent only on the ve-locity component parallel to the cavity optical axis. On

+

the other hand, the OC (k„Jk, ) requires integrationover ~he two-dimensional velocity space. That signifi-cantly increases the number of equations. In calculationsthe number of velocity groups is typically 30 for the CCand 64 for the OC, corresponding to 272 and 578 realequations, respectively.

We assume natural initial conditions for atomic vari-ables (atoms in the ground state with vanishing initialatomic coherences). As usual in semiclassical laser the-ory, a = 0 gives a trivial solution. We take, therefore, asmall nonzero value for the initial field a(0). Equationsare numerically integrated using the standard predictor-corrector method. The system of equations reaches astable solution typically after the time of the order of300—500 lifetimes of the excited level.

We have studied numerically the behavior of both theV and A models in a wide range of parameters [21]. Forsimplicity let us consider here only the resonant case[~, = ~sq and ~„=cu, ~. , in the index convention ofEq. (2)]. Then the analytic solution for the laser intensityis available for the limiting homogeneous case (D = 0)both for the A model [6] and for the V system [21]. Bothmodels lead then to LWI in any basis.

The question we now pose is whether this character ofthe lasing is preserved in the presence of the inhomoge-neous Doppler-effect-induced broadening (D ) 0). Foratoms at rest the gain is peaked around the two-photonresonance (w„—w, = u;z —wqs). The Doppler effectsmears out the resonance so that the intensity decreasewith increasing D may be expected. The effect is due tothe (k, —k„) v~ term in Eqs. (5). Note that the influ-ence of the inhornogeneous broadening is sensitive to themutual orientation of k and A:„. We discuss the ring-cavity arrangement in which the driving field and thegenerated field are copropagating. For counterpropagat-ing fi.elds the Doppler effect will be stronger due to theminus sign above. Let us also note that had we consid-ered the ladder atomic level scheme, the transformationto time-independent kame would have resulted in the(k, + k„) v" term in Eqs. (5) and the counterpropagat-ing configuration would have been favored.

For the CC the value of the net detuning is small and

10 I

6 i ~ ~

I

I

'i

'l

l

l

0 I

0 40

4

2

30D

50

FIG. 2. A-level scheme. Intensity of the generated radi-ation versus the Doppler width D. N = 10, 0 = 30,I' = 10, g = 10, A = 10, A/3 —0.5, and p23 ——1.The squares connected by the solid line correspond to the or-thogonal con6guration. The circles connected by the dashedline represent the lasing intensity in the collinear configura-tion. I asing without inversion appears for the CC on theshoulder of lasing, to the right of thin vertical line.

may vanish when both transitions are degenerate in fre-quency. Yet one would prefer a large difference betweenboth transition frequencies to be able to generate radia-tion in a new frequency range. We assume ~k,

~

= 1.5~k„~so the generated frequency is significantly blueshiftedwith respect to the driving frequency. For the OC thetwo-photon resonance condition is not satisfied for mostof the active atoms, which should strongly inBuence las-ing capabilities.

The intensity of the A laser as a function of the Dopplerwidth D is presented in Fig. 2. All frequencies and decayrates are expressed in units of p23. The laser intensityis dimensionless, measured by the corresponding photonnumber I = ~a~ . Indeed the CC setup works better;lasing is suppressed for smaller inhomogeneous broaden-ing for the OC. However, even for the CC the Dopplerwidth permitting lasing is roughly one order of magni-tude smaller than the typical inhomogeneous broadeningin the atomic cells at the realistic temperatures. On theother hand, much smaller residual widths are possible inthe atomic beam experiment. Then, however, the degreeof beam collimation is strongly related to the maximumatomic density in the active region. We have verifiedthat for the realistic residual width D = 8 the thresholdatomic densities for the CC are four orders of magnitudesmaller than for the OC.

The parameters chosen for the A system correspond toLWI in any atomic basis in the linear gain theory [5,13].As already noted in Ref. [6], for a homogeneously broad-ened model, a nonlinear theory yields negative inversiononly very close to the threshold. Once lasing is estab-lished, inversion in the A model is created via redistri-bution of the population due to the generated light. Asimilar situation occurs for our Doppler-broadened sys-tem. By integrating the steady-state populations overthe velocity profile we have found that the steady-statelaser action is accompanied by atomic inversion (exceptfor the largest D, just at the border of lasing). A lack of

LASERS WITHOUT INVERSION IN A DOPPLER-BROADENED MEDIUM 833

10 I

4

2I

gN&-4f 0

II

l

1I

1Ii1

I

I1II

0 11 ~ I

100I

200 300D

400

FIG. 3. Same as Fig. 2, but for the V scheme. N = 10,A = 0.75, and the remaining parameters are the same as inFig. 2 (but in units of pi2). Lasing without inversion appearsin the CC for D ( 105, to the left of the thin vertical line.

inversion occurs for smaller N = 5 x 10; then the lasingD interval is much smaller than in Fig. 2.

Figure 3 presents the calculated laser intensity as afunction of the Doppler width for the V model. We keepthe values of atomic, cavity, and pump field parametersthe same as before, but the incoherent pumping rate Avalue must be significantly larger to achieve lasing. Thedifference between both considered geometries is evenmore striking. For the OC the lasing occurs only forvery small inhomogeneous broadening. For the CC thesurprising growth of the lasing intensity with Dopplerwidth is observed. Strong lasing persists for inhomoge-neous broadening values corresponding to typical, realis-tic temperatures of the gas sample.

To explain the striking behavior of the V-laser modelwe note, following Zhu [8], that for the driving beamsufIiciently detuned from the resonance the gain appearsdue to the population inversion in the dressed-state ba-sis. Moreover, the gain profile is peaked around theshifted frequency value corresponding to the transitionfrom state 3 towards one of the dressed states. Thus, toachieve an efIicient amplification, u should be tuned tothat frequency. Such is the situation in a homogeneouslybroadened system.

Consider a single nonzero velocity group. For the mov-ing atoms both cuz and u are detuned from the resonancein the same direction. The moving atom "sees" the cav-ity shifted in the vicinity of the inverted transition inthe (velocity-dependent) dressed-state basis and acts asan amplifier. Note that the effect is not dependent onthe velocity sign. A change of the velocity sign changesthe sign of the Doppler shift, i.e. , the sign of the drivingGeld detuning from the resonance with a given velocitygroup. Then the relative populations of dressed-statesalso change [22]. But as the cavity-mode Doppler shift isalso dependent on the velocity sign, the cavity remainsclose to resonance with the inverted transition. The de-tuning from the inverted dressed-state transition is de-pendent, via Doppler shifts, on the velocity value for agiven velocity group and on a relative magnitude of driv-ing and generated waves' wave vectors. Thus some ofthe velocity groups contribute via this mechanism much

more strongly to the observed gain than the other groups.At the same time, low-velocity groups contribute also togain due to the LWI mechanism of Ref [8]. That explainsthe effect observed.

An amplification mechanism (and lasing) based onan inverted dressed-state transition has been studied indetail both theoretically and experimentally [3,4, 10,11]for homogeneously broadened systems. An interestingfeature of the inhomogeneously broadened system con-sidered here is that the required driving laser detun-ings appear as Doppler shifts. The discovered velocity-dependent amplification enhancement mechanism dis-cussed above is very sensitive to the mutual directions ofthe driving coherent field and the ring-cavity mode. Con-sider that we discuss here the case when kz and k areparallel (copropagating pumping and generated beams).For antiparallel orientation (counterpropagating beams)uz and u, would be shifted in opposite directions in theatomic frame (defined for each velocity group separately).Then the similar dressed-state analysis shows that thedressed atoms would act as strong absorbers (since thecavity would be "tuned" to the noninverted dressed-statetransition). In a similar way one can convince oneselfthat for the ladder level scheme the counterpropagatingconfiguration would be favored.

For the OC a small fraction of atoms only is effectivelytuned in the vicinity of the dressed transition. The num-ber of atoms contributing to gain becomes very low forhigher Doppler widths and the lasing does not occur.

By monitoring the final populations of dressed statesand integrating them over the velocity profile we havefound that lasing without inversion in the dressed basisoccurs for D & 100 for the parameters in the figures,while dressed-state inversion dominates lasing for highervalues of the Doppler width.

IV. CONCLUSIONS

To summarize, we have presented a theoretical anal-ysis of lasers without inversion in the realistic inhomo-geneously broadened medium. We have found that thepossibility of observing the lasing depends strongly onthe mutual orientation of the pumping laser beam andthe ring-cavity axis. Although our theory is not valid forstanding-wave cavities, one may expect intuitively thatthe effect of the inhomogeneous broadening in such cav-ities will be rather similar to the lasing observed here inthe less favorable orthogonal configuration. The wavespropagating in both directions have to be considered insuch a cavity and the two-photon resonance conditionwill be strongly suppressed.

Although the V-laser model [8] requires much strongerincoherent pumping in order to lase as compared to theA scheme [5], the lasing in the former model may actuallybe enhanced by the inhomogeneous broadening. This isdue to the strong amplification generated in the Doppler-broadened medium. The origin of this mechanism hasbeen traced to gain on inverted dressed-state transitionswhere the unequal populations of dressed states are dueto effective detunings caused by Doppler shifts. For suffi-

834 ANDRZEJ KARAWAJCZYK AND JAKUB ZAKRZEWSKI

ciently strong incoherent pumping and large atomic den-sities both models may lase efBciently with atomic cellarrangement in a collinear geometry only. As far as ex-perimental verification of LWI (in any basis) is concerned,

it seems that well collimated atomic beams are necessary(and also the CC) since small Doppler widths are re-quired. For the A model, LWI appears only just abovethe lasing threshold (similarly to the D = 0 case [6]).

[1] V. G. Arkhipkin and Y'u. I. Heller, Phys. Lett. 98A, 12(1983); S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989);A. Imamoglu, Phys. Rev. A 40, 2835 (1989). A. Lyras,X. Tang, P. Lambropoulos, and J. Zhang, ibid. 40, 4131(1989); G. S. Agarwal, S. Ravi, and J. Cooper, ibid. 41,4721, 4727 (1990).

[2] D. Kupiszewska, P. Masiak, and K. Rzg, zewski, Opt.Commun. 92, 266 (1992).

[3] A. Lezama, Y. Zhu, M. Kanskar, and T. W. Mossberg,Phys. Rev. A 41, 1576 (1990); M. Lewenstein, Y. Zhu,and T. W. Mossberg, Phys. Rev. Lett. 64, 3131 (1990).

[4] J. Zakrzewski, M. Lewenstein, and T. W. Mossberg,Phys. Rev. A 44, 7717 (1991);44, 7732 (1991);44, 7746(1991);J. Zakrzewski and M. Lewenstein, ibid. 45, 2057(1992).

[5] A. Imamoglu, J. E. Field, and S. E. Harris, Phys. Rev.Lett. 66, 1154 (1991).

[6] K. M. Gheri and D. F. Walls, Phys. Rev. A 45, 6675(1992).

[7] G. B. Prasad and G. S. Agarwal, Opt. Commun. 86, 409(1991).

[8] Y. Zhu, Phys. Rev. A 45, R6149 (1992).[9] O. Kocharovskaya and Y'u. I. Khanin, Pis'ma Zh. Eksp.

Teor. Fiz. 48, 581 (1988) [JETP Lett. 48, 630 (1988)];O. Kocharovskaya and P. Mandel, Phys. Rev. A 42, 523(1990).

[10] O. A. Kocharovskaya, F. Mauri, and E. Arimondo, Opt.Commun. 84, 393 (1991);L. M. Narducci et ai. , ibid 81, .379 (1991);86, 324 (1991);C. H. Keitel et al. , Phys. Rev.A 48, 3196 (1993).

[11] A. Karawajczyk, J. Zakrzewski, and W. Gawlik, Phys.Rev. A 45, 420 (1992); A. Karawajczyk and J. Za-krzewski, Opt. Commun. 107, 145 (1994).

[12]

[13][14]

[i5]

[16]

[i7]

[»][»]

[20]

[21][22]

E. Arimondo and G. Orriols, Nuovo Cimento Lett. 17,333 (1978); H. R. Gray, R. M. Whitley, and C. R. Stroud,Jr. , Opt. Lett. 3, 218 (1978).G. S. Agarwal, Phys. Rev. A 44, R28 (1991).C. Cohen-Tannoudji, B. Zambon, and E. Arimondo, J.Opt. Soc. Am. 10, 2107 (1993).J. Gao et al. , Opt. Commun. 93, 323 (1992); E. S. Fry etal , Phy. s. Rev. Lett. 70, 3235 (1993).A. Nottelmann, C. Peters, and W. Lange, Phys. Rev.Lett. 70, 1783 (1993);W. E. van der Verr et aL, ibid. 70,3243 (1993).K.-J. Boiler, A. Imamoglu, and S. E. Harris, Phys. Rev.Lett. 66, 2593 (1991).M. Fleischhauer et aL, Phys. Rev. A 46, 1468 (1992).Since this paper was submitted we have become awareof the work of K. J. Schernthanner and H. Ritsch, Phys.Rev. A 49, 4126 (1994), who discussed the infiuence ofDoppler shifts on the quantum-noise reduction in Ra-man lasers. The authors find the Mandel q parameteraveraging over the rectangular velocity distribution. Bycontrast, in this work we take a realistic atomic velocitydistribution and consider the lasing intensity rather thanquantum properties of the emitted radiation. The atomicmodel is also diferent since we consider lasing withoutinversion schemes.W. H. Louisell, Quantum Statistical Properties of Radiotion (Wiley, New York, 1973).A. Karawajczyk and J. Zakrzewski (unpublished).For a review of dressed-state properties and their appli-cations to quantum optics problems see, e.g. , C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions: Basic Processes and App/ications(Wiley, New York, 1992).