2406 J. Opt. Soc. Am. B/Vol. 10, No. 12/December 1993

Lasers without inversion: one-photon stimulated emissionin a three-level atom

Constantine Mavroyannis*

Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario KIA OR6, Canada

Received February 9, 1993; revised manuscript received June 14, 1993

I consider light amplification arising from the competition between absorption and stimulated emission of anexcitation, which is induced by a laser field at low intensities interacting with a three-level atom. The lifetimeof the induced excitation is 105-106 times longer than those of spontaneously emitted excitations. Conditionsare established under which the stimulated emission prevails over the induced absorption, and the net negativeintensity is 10-103 times greater than those of spontaneously emitted excitations, indicating that significantamplification occurs near the frequency of the laser field.

PACS numbers: 32.80-t, 42.50.-p, 42.50H 2

The possibility of obtaining amplification or even lasingwithout population inversion was recently proposed byHarris.' He considered the difference between the emis-sion and the absorption spectra that is due to Fano inter-ferences2 between two lifetime-broadened discrete levelsthat decay to the same continuum.'13 Many schemes forlasing without population inversion have been proposed,and the dependence of optical gain on various system pa-rameters has been examined.4'- 2

In this study I consider a three-level atom, shown inFig. 1, which is pumped by a laser field with frequency o,,operating in the 1) 12) and 1) 13) transitions. Theground (11)) and excited (12) and 13)) states are simulta-neously coupled by a weak perturbing electromagneticfield describing radiative transitions. In the low-intensitylimit of the laser field, the excitation spectra consist ofthree excitations describing two spontaneous processesfor the excited states 12) and 13) and an excitation inducedby the laser field; the lifetime of the induced excitation islong in comparison with those of excitations that are spon-taneously emitted by the excited states. When certainconditions prevail, the intensity (height) of the inducedpeak takes negative values, indicating that significantamplification occurs near the frequency of the laser field,oa. Thus, my amplification mechanism differs fromthose proposed in other studies,' 3 - 2 but it is analogous tothat recently suggested1 3 for a three-level atom in the cas-cade configuration driven by a bichromatic field, whereamplification occurs at the two-photon frequencies of thelaser fields. Since the processes under investigation oc-cur at finite frequencies, they are dynamic in origin, andtherefore I have used a suitable mathematical formalismthat was recently used in similar circumstances. 4"5

We may take

H 2,ca2ta2 + 31 a 3t a3

+ 2 g2[alta2 exp(-ioat) - 2 ta, exp(icat)]2

+ 2g93[altar3 exp(-iw0,) - ata, exp(icoat)]2

+ ckjkAtjkA + -iwE (k, A) kj]X, 2s~itcrjput kA[f ck

X (aiajjk - aj'aif,8)(1

to be the Hamiltonian for the atomic system depicted inFig. 1 in the electric dipole and rotating-wave approxima-tion, where ait and ai are the usual Fermi-Dirac operatorsdescribing the electron states i = 1, 2,3 and ni = itaiis the number operator. The functions fj/k,A) are theoscillator strengths for the atomic transitions Ii) - I),and op is the atomic plasma frequency; units with hi = 1are used throughout. The atom is pumped by the laserfield with frequency (0a, while 2 = 021 - Ha andA3 = 31 - toa are the detunings for the electronically al-lowed transitions 11) 12) and 11) 13), respectively, andg2 and g3 are the corresponding classical Rabi frequencies;(021 = )2 - )I and 31 = 3 - ol are the transition fre-quencies. The electronic transition 12) > 13) is electricdipole forbidden. The operators jkAt,jkA describe the vac-uum field, which is quantized with wave vector k, fre-quency ck, and transverse polarization A = 1,2. Inwriting Eq. (1) I have taken into consideration the relationn + 2 + n3 = 1.

The first two terms on the right-hand side (rhs) ofEq. (1) describe the free atomic fields, while the third andfourth terms denote the interaction of the two atomictransitions with the laser field, where the laser field hasbeen treated classically. The last two terms designate thefree vacuum (signal) field and its interaction with theatomic levels.

The spectral functions describing the excitation spectrafor the excited states 12) and 13) are determined' bythe imaginary parts of the Green functions G2,2( O) =((a2ia2t)) and G3,3()) = ((a 3ia 3 t)), respectively. Using theHamiltonian (1), one may derive the equations of motionfor G2,2((0):

d2 G2, 2 (0)) = - i g2Gl2(( - iKG3,2()Q,

i ~~~~~id i.Ga, 2((0) = 2G 2,2 (G) + -g3G 3,2()X),

2 2

d G3,2(O) = -- 3Gl,2Go) -iK22Q

(2)

(3)

(4)

where

0740-3224/93/122406-04$06.00 © 1993 Optical Society of America

Constantine Mavroyannis

Vol. 10, No. 12/December 1993/J. Opt. Soc. Am. B 2407

12>

72 0)a 73

11>

Fig. 1. Energy-level diagram of a three-level atom or ion.

G2,2() = 2 1 + 7F23 - 2if 23 \27r d 2 dy/2Ak2

g2(P23 + iR 2 3)G 2 (0)) = 21rdy 2A 2

= y(L - 2iM)4irKdA 2A 3

where

d = (- (a + i/2,

Gl.,2((t)) = ((a, exp(ioat); a 2 t)),

d2= W - (021 + i2/2,

di. = (0 - (021 + A2,

d3= - 31 + iy3/2,

2K = (727Y3) X/,

72 = (4/3)(0 2 1I/c)3 1P1212 ,

G3,2(c)) = ((a3 ;a2l)),F23 =f23 =

. P23 =R23 =

L =M =

3 = (4/3)() 31/c) 3 IP13 12 .

Here P,2 and P,3 are the transition dipole moments. Thefunctions 72 and 73 denote the spontaneous-emission prob-abilities for the radiative decays 12) -> 11) and 13) -- 11), re-spectively, while 2/Y2 and 2/73 define the radiativelifetimes of the corresponding electron states 12) and 13).Solution of Eqs. (2)-(4) yields

G2,2(0)) = (d 0d 3 - g 32/4)/27rD, (5)

Gla,2(0) = (ig2d3 + Kg3)/4-TD, (6)

G3,2(cW) = (g 2g3 - 4iKdla)/8IrD, (7)

where

D = dla(d2d3 + K2) - g -d3 -g d2 + Kg2 g94 4 2

The Green function G2,2(() describes the excitation spec-tra of an electron in the excited state 12), while Ga,2((c)represents the induced process in which a laser photon isabsorbed by the ground state 11) of the atom. The Greenfunction G3,2(o)) describes a Raman-type spectra arisingfrom the interference between the two excited states 12)and 13), where the indirect coupling is due to the productg2g3 of the two Rabi frequencies as well as to the product Kof the two raditive decay rates 72 and y3.

Considering the symmetric form of the Hamiltonian,Eq. (1), as far as the interactions of the laser and vacuumfields with the excited states 12) and 13) are concerned, onecan easily prove that the expressions for the Green func-tions G3,3(cw), Ga 3 ((t), and G2,3 (G), which describe thespectra of the excited state 13), can be derived from thecorresponding expressions (5)-(7) through the inter-change of the indices 2 3.

The low-intensity limit occurs when Y22>> g22 and

732 >> g32, which imply that both transitions are not satu-

rated. At this limit an expansion of Eqs. (5)-(7) intopower series of g2

2/Y22 << 1 and g32/73

2 << 1 yields

-1 + 4 2A/Y,_A - 722 + MF3/A3 ,1 -VA3A"2,

-712 + M/A 3 /.12,

V[(r~r3)112/y -1] + 4mA/y,M(r2r3)112/ - m - Av/y,

F2 = 22/Y2,

'q2 = 2/Y2,212 =~~~

A2 = 1 + 4222,sA = 02I3 + MI0,SY = ( 3 F2

1/2 + 212 r31/2)2 ,

0= 213 F2 + 21213,

m = 712 + 213,

v= 1 - 4q2213,

, = F2/F3 ,F3 = 32/-/3

713 = A3 /713 ,

A3 = 1 + 432,

S= m2 + 4222232,44 = (r2 1/2 - r3l"2)2 .

Expression (8) represents the excitation spectra of anelectron in the excited state 12), where the two terms de-scribe a spontaneous and an induced excitation whose life-times are equal to 2/y2 and 2/7, respectively. Expressions(9) and (10) describe the absorption of a laser photon by,the ground state 11) and the interference effect betweenthe two excited states 12) and 13), respectively. The lastterm on the rhs of Eq. (8) and Eqs. (9) and (10) describecontributions to an excitation that is induced by the laserfield at the frequency (0 = woa and has a long lifetime, 2/y,in comparison with the lifetimes 2/y2 and 2/73, respec-tively, that are short in duration; that is, 2/7 >> 2/72 and2/y >> 2/73. In deriving Eqs. (8)-(10), I have discardedterms describing spontaneous processes whose amplitudesof occurrence are of the order of g2

2/Y22 << 1 as well as an

induced frequency shift A, since A << «A 2 and A << A3.Taking the imaginary parts of Eqs. (8)-(10), we

may write the total spectral function for an electron inthe excited state 12) as J2 J2(co) = I2 + Ila,2 + I3,2,where I2 = -2ir Im G2,2 W,), Ila,2 = -2ir Im Gla,2(0), I3,2 =-2'r Im G3,2(co), and

2 r 722/4 1 W2 32/4 + (W-a)Y 23 1

J2 (= -(021)2 + 722/4 A2 ((0 -(0a)2 + 722/4 |(11)

with W 23 = F23 + g2 P23/7 - LY 2 /2KA 3 , Y = f23 - S23and 2S23 = g2R23 + YMY2 /1KA3. Similarly, the total spec-tral function J3 J3 ()) for an electron in the excitedstate 13) turns out to be

2 732/4 1 W32Y2/4 + (__ - W32_3 = ( _03)2 + 732/4 +A Wa _ 4+)2 + Y2/4 '

(12)

where the functions W32 and Y32 can be obtained from thecorresponding functions W23 and Y23 through an inter-

(8)

(9)

(10)

Constantine Mavroyannis

2408 J. Opt. Soc. Am. B/Vol. 10, No. 12/December 1993

z

I-'

z>

LCl

-2001 1 a ir-24 -16 -8 0 8 1 6 24

105 ( Ca)/Y3Fig. 2. Stimulated-emission spectra when g 2 /g3 = 72/73.

change of the indices 2 3. The first terms on the rhsof Eqs. (11) and (12) describe the spontaneous-emissionprocesses arising from the excited states 12) and 13), re-spectively, while the last terms designate contributions ofthe induced peak whose spectral line is denoted by anasymmetric Lorentzian line peaked at o) = &)a and whichhas a spectral width equal to 7/2. We may write Eqs. (11)and (12) as J = J2 + J3 = I2'P + I3'P + Iind, where I2P andI3'P represent the first terms on the rhs of Eqs. (11) and(12) for the spontaneous processes, respectively, whileIind Iind(&0) denotes the induced process, that is,

WY2/4 + ( - a)YI'md = 2 (_ (0)2 + 72/4

comes a function of p:

= - [(1 pl/2 )2 + 4 2 1/2 (16)

which is negative for any value of p. In Eq. (16), 3W isroughly proportional to p'12 . The induced spectra in thiscase are shown in Fig. 2, where the relative intensityy3Ind() and the function W computed from Eqs. (13) and

4000

3000 X=0.27

2000 a 0.25 0

g0.23

1000

z 0 -I- P= O

Z

> -1000 93 = 01Y3

La.I -7 72= 5

1 O (a W)/ 3Fig. 3. Stimulated-emission and induced-absorption spectrawhen g2 /g3 #6 72/73-

(13)

and functions W and Y are defined as W = W23/y2A2 +

W32/y3A3 and Y = Y23/Y2A2 + Y32/y3A3, respectively. Thespectral function (13) indicates that at the frequencya = wa the maximum intensity (height) of the inducedpeak is equal to 2W, while its spectral width is of the orderof 7/2.

We may begin to examine the form of the functions Wand Y by considering the case when m = 212 + 13 = 0,which implies that A2 = A3 = v, while Wand Y become

(1 W = -2)

2 i- 42132

A3 A3(1 - pl 2 )2

X [y (1 - p/ 2 )(1 - p"/2) + (1 - Op)21,

I-

n

zWL

IL

(14)

2yy= A 3 -( 0P)[ 93 (1- /2)(j - 0.p1/2) - X s] (15)

where p = y3/y2, C- = g2 /g3 , and g3 /y3 < 1. When op =1, that is, when g2 /g3 = Y2/Y3, then Y = 0, while W be-

-3000 L

Fig. 4. As in Fig. 3P = 73/72-

-4 -2 0 2 4 6

1 0 5 (CO w a)/ Y3but for different values of a = ,2/g3 and

Constantine Mavroyannis

-

Vol. 10, No. 12/December 1993/J. Opt. Soc. Am. B 2409

(16), respectively, are plotted versus the relative frequency(0 - a)/IY3 for up = 1, 213 = -2 = 5 g3 = 0.173, and forp = 10, 20, 100. Figure 2 indicates that the relative in-tensity (height) of the induced peak increases negativelyand, consequently, the amplification increases as the valueof p = 73/72 increases. For instance, for p equal to 10, 20,and 100 and at the frequency X = W.a, Iind(cW) takes the val-ues of -62.7/73, -88.8/73, and -199.6/73, respectively.Comparison of these values with those of 2/y2 and 2/y3 forthe intensities of the spontaneous emission that occurfrom the states 12) and 13) at = 021 = Ota + A&2 anda = 031 = 0). + A3, respectively, indicates that signifi-cant amplification is likely to occur at the frequencylo = Ma of the laser field.

At frequencies ) F wa, and when -p # 1, the asym-metry of the spectral lines described by Eq. (13) dependsstrongly on the value of Y Since the ratio /73 is of theorder of 10-s_10-6, the function Y takes large positive ornegative values at # c&oa. Therefore there will beabrupt changes in the intensities of the spectral linesfrom negative values to positive ones at o.# co wheno-p 0 1. This behavior is demonstrated in Figs. 3 and 4,where the functions W and Y have been computed fromEqs. (14) and (15), respectively. For p = 10, 20, 100, Wbecomes negative for - ' 0.2726, 0.18, 0.0656, respec-tively. Figure 3 depicts the computed spectra for p = 10and for o- = 0.27, 0.25, 0.23. It is shown that for a givenvalue of p the maximum negative and positive intensities(heights) of the induced peaks that occur when ( -(0a)/73 > 0 and ( - (a)/73 < 0, respectively, decrease asthe value of a decreases. Figure 4 depicts the spectra forvarious values of a- and p, where the intensities of the in-duced peaks are smaller than those in Fig. 3. Inspectionof Figs. 2-4 implies that light amplification occurs at

= a in Fig. 2 for a-p = 1, while in Figs. 3 and 4 it oc-

curs at c > co,, for -p =# 1, which is due to the asymmetryof the spectral lines.

In conclusion, we have shown that our amplificationmechanism is a characteristic feature of the low-intensityfield 3 and is likely to occur near the frequency w = a ofthe laser field provided that the favorable conditions indi-cated in Figs. 2-4 are satisfied.

*Permanent address, 1945 Fairmeadow Crescent,Ottawa, Ontario K1H 7B8, Canada.

REFERENCES

1. S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989).2. U. Fano, Phys. Rev. A 124, 1866 (1961).3. V G. Arkhipkin and Y. I. Heller, Phys. Lett. 98A, 12 (1983).4. A. Imamoglu, Phys. Rev. A 40, 2835 (1989).5. Z. Lyras, Z. Tang, R Lambropoulos, and J. Zhang, Phys. Rev.

A 40, 4131 (1989).6. S. E. Harris and J. J. Macklin, Phys. Rev. A 40, 4135 (1989).7. M. 0. Scully, S.-Y. Zhu, and A. Gavrielides, Phys. Rev. Lett.

62, 2813 (1989).8. G. S. Agarwal, S. Ravi, and J. Cooper, Phys. Rev. A 41, 4721,

4727 (1990).9. E. E. Fill, M. 0. Scully, and S.-Y. Zhu, Opt. Commun. 77, 36

(1990).10. S. Basile and P. Lambropoulos, Opt. Commun. 78, 163 (1990).11. 0. A. Kocharovskaya and Ya. I. Khanin, Pis'ma Zh. Eksp.

Teor. Fiz. 48, 581 (1988) [JETP Lett. 48, 630 (1988)]; 0.Kocharovskaya and P. Mandel, Phys. Rev. A 42, 523 (1990).

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

13. C. Mavroyannis, Phys. Rev. A 46, R6785 (1992).14. C. Mavroyannis, Can. J. Phys. 68, 321, 411, 1389 (1990); 70,

1, 427 (1992).15. C. Mavroyannis, Quantum Opt. 2, 237 (1990); 4, 109, L145

(1992).16. D. N. Zubarev, Soy. Phys. Usp. 3, 320 (1960); Nonequilibrium

Statistical Thermodynamics (Plenum, New York, 1976),Chap. 16.

Constantine Mavroyannis