160 J. Opt. Soc. Am. B/Vol. 5, No. 1/January 1988

Theory of pump-probe spectroscopy

Galina Khitrova

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Paul R. Berman

Department of Physics, New York University, New York, New York 10003

Murray Sargent III

Optical Sciences Center, University of Arizona, Tucson, Arizona 85721

Received July 6, 1987; accepted September 29, 1987

We calculate the two-level pump-probe absorption coefficient including both upper-to-lower-level decay and leveldecays to a still lower-lying reservoir level. The probe-absorption profile can have arbitrary ratios of the naturallinewidth and detuning to the Doppler width. We observe and explain new line-shape features that occur when thetwo main levels decay to the reservoir level at different rates.

1. INTRODUCTION

Pump-probe spectroscopy is a technique that has been usedin high-resolution spectroscopy to obtain information aboutphysical properties of atomic systems. In a standard pump-probe experiment involving a single atomic transition, onemeasures the probe-absorption profile when an arbitrarilystrong pump field drives the same transition. In the past,calculations of probe-absorption profiles have been per-formed by several groups.'-' 0

A detailed theoretical investigation of the probe-absorp-tion line shape including coherent effects was undertaken byBaklanov and Chebotaevl and by Haroche and Hartman.2

The probe-absorption line shape can be calculated by evalu-ating the probe-field-induced dipole moment using masterequations in which the pump field is treated to all orders.Previous calculations have been performed for stationaryatoms 2' 4' 6'8 or for moving atoms in the so-called Dopplerlimit (Doppler >> decay rates)." 3

Absorption spectra for strongly driven stationary atomsexhibit the interesting phenomena of probe-field amplifica-tion2'4'6'8 at certain probe-field detunings from resonance.This negative absorption occurs with no population inver-sion and has been investigated by several authors. 4'7'8 "'Doppler-limit calculations for unidirectional' and opposite-ly3 traveling waves and moving atoms have been carried outby Baklanov and Chebotaev (no probe amplification waspredicted). Experimental curves obtained by Wu and co-workers 9 using an atomic beam agree with the stationary-atom theory for both zero4'8 and nonzero8 pump detunings.

The present paper extends these calculations to allow forarbitrary ratios of natural linewidth and detuning to theDoppler width. This theory accounts for contributionsfrom all velocity subclasses of atoms not only those that areDoppler shifted into resonance (Doppler limit). We alsoinclude both upper-to-lower-level decay and level decays toa reservoir level. By including them we are able to obtain

new line-shape features when the reservoir decay constantsdiffer. As in the references mentioned above, our treatmentis semiclassical, that is, the fields are treated classically, andthe atoms are treated quantum mechanically. The effectsof one or two quantized probe waves and a possible Lorentz-ian broadening have been studied by Sargent et al.12 andHolm et al.13 Their coefficients reduce to the present semi-classical absorption coefficient for stationary atoms and forcorunning waves in the Doppler limit.

Expanding the solution in powers of the pump-field inten-sity, one can predict the appearance of a narrow resonance(characterized by the lower-level width) for different combi-nations of level- and dipole-decay rates. This resonance isclosely related to the pressure-induced extra resonance infour-wave mixing (PIER4) resonance predicted by Bloem-bergen et al.

4 "15 in four-wave mixing and studied by a num-ber of groups.16,19-24

2. PROBE-WAVE ABSORPTION COEFFICIENT

We consider a medium subjected to a saturating wave offrequency 2 and study the transmission of a weak (nonsa-turating) probe wave of frequency P, as diagrammed in Fig.1. We assume that the saturating-wave intensity is constantthroughout the interaction region, and we ignore transversevariations. Our two-wave electric field has the form25

E(r, t) = 2 A,,(r)exp[i(Kn. r - Pat)] + c.c.,n=1

(1)

where the mode amplitudes 6n(r) are in general complex andKn are the wave-propagation vectors. This field induces thecomplex polarization

1 2P(r, t) = - E ?P(r)exp[i(Kn * r - vnt)] + c.c.,

n=l

0740-3224/88/010160-11$02.00 3 1988 Optical Society of America

(2)

Khitrova et al.

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. B 161

V2

V1

Fig. 1. Basic probe-saturatortion.

.IV1

saturation spectroscopy configura-

tions act as modulators (or like Raman shifters), puttingsidebands onto the medium's response to the v2 mode. Oneof these sidebands falls precisely at v1, yielding a contribu-tion to the probe-absorption coefficient. The other side-band would influence the absorption at the frequency V3.

In this section we derive the complete nonsaturatingprobe-absorption coefficient for the expanded two-levelscheme depicted in Fig. 3. This scheme is sufficiently gen-eral to include the standard excited-state configuration usedin typical laser media as well as the upper-to-ground-lower-level decay scheme often used in saturation spectroscopy.In particular it allows us to consider the effects of differingdecays yoa F4 'Yb, which lead to resonances analogous to thePIER resonances in the work of Bloembergen et al.1

4'15

The calculation is a semiclassical version of the corre-sponding two-mode quantized field case given in Refs. 12and 13, here including possible Doppler shifts due to movingatoms. The polarization P(r, t) of Eq. (2) is given in terms ofthe population matrix by

V2 V3

Fig. 2. Spectrum of the multiwave fields used in this paper. Theprobe wave has frequency i and is taken to be weak (nonsaturat-ing), while the 2 wave may have an arbitrarily large intensity.Pump-scattering off-population pulsations induced by the pump-probe beat frequency induces a polarization at the frequency Vs.

This component couples weak modes in three- and four-wave mix-ing.

where P,(r) is a complex polarization coefficient that yieldsindex and absorption or gain characteristics for the probeand saturator waves. The polarization P(r, t) in general hasother components, but we are interested only in those givenby Eq. (2). For example, in homogeneously broadened me-dia, strong wave interactions induce components not only atthe frequencies and v2, but at k(v2 - v,) as well, wherek is an integer. To distill the components Pn(r) out of P (r,t), we can use the mode factors exp(iK, r), provided thatthey differ sufficiently from one another in distances forwhich the amplitudes vary noticeably. For nearly parallel(or parallel) waves, the mode functions do not vary suffi-ciently rapidly, and one must separate components by theirtemporal differences, for example, by heterodyne tech-niques. In this configuration, the conjugate wave at thefrequency 3 = 2 + ( - v,) is phase matched and builds up,involving a somewhat more complicated calculation thanthat given in this paper (Fig. 2).

The problem reduces to determining the probe's polariza-tion 91(r), from which the absorption coefficient is deter-mined from the equation

iK¶P,al=

0e,6

P(r, t) = p f dvW(V)pab(r, v, t) + c.c.,

where W(v) is the velocity-distribution function.tions of motion for the population matrix are

(-t + v V)Pab = -(iW + Y)Pab

+ ih 'Vab(Z t)(Paa Pbb),

at + V VPaa = AaPcc - (Ya + )Paa

- (Uh 'VabPba + c.c),

(a + bb

a

1b

I

(3)

One might guess that this absorption coefficient is simply aprobe Lorentzian multiplied by a population difference sat-urated by the saturator wave. However, an additional con-tribution enters because of population pulsations. Specifi-cally, the nonlinear populations respond to the superposi-tion of the modes to give pulsations at the beat frequency

A = V2 V1- (4)

Because we suppose the probe does not saturate, the pulsa-tions occur only at +A, a point proved below. These pulsa-

Fig. 3. Three-level atomic-energy-level scheme that treats purelyexcited-state interactions as well as upper-to-ground lower-stateinteractions in a uniform way.

(5)

The equa-

(6)

(7)

= AbPcc + "Paa - YbPbb

+ (Uh`lCVabPba + C.c.), (8)

Aa

-

Khitrova et al.

162 J. Opt. Soc. Am. B/Vol. 5, No. 1/January 1988

+ V V Pec -(Aa + Ab)Pcc + YbPbb + 'YaPa.

(9)

As shown in Fig. 3, Aa and Ab are pump constants from level cto levels a and b, respectively, zya and Yb are the correspond-ing decay constants, is the decay constant from level a tolevel b, and y is the dipole-decay constant given by (a + Yb

+ I)/2 + yph, where yph is a possibly nonzero contributiondue to phase-interrupting collisions. In addition, we havethe trace condition

Paa + Pbb + Pcc 1. (10)

Using Eq. (10) and the steady-state solution of Eq. (9), wecan eliminate Pcc from Eqs. (7) and (8) and eliminate paa fromEq. (8). For example, we have the relations

Pcc = 1 - Paa - [(A0 + Ab)P~c - YaPaa]/Yb

'Yb- Paa(Yb - Ya)

Yb + Aa + Ab

This reduces Eq. (7) to

The interaction energy-matrix element "Vab correspondingto Eq. (1) is given in the rotating-wave approximation by

2

CVab = 2- 2 6n(r)exp[i(Kn r - vnt)]. (17)n=1

To determine the response of the medium to this multi-mode field, we analyze, using a Fourier transform, the polar-ization component Pab of the population matrix as well as thepopulations themselves. We have

Pab = N exp[i(Kl.r - vt)]

X E Pm+l explim[(K 2 - K) r - At]), (18)m=-

where the unsaturated population difference

N a Xb (19)'Ya' Yb'

The population-matrix elements Paa have the correspondingFourier expansions

(da +v. Paa = Xa - Ya'Paa - (ihCVabPba + c-c),

where

'Yt = Y A + r + (Yb - Y.)Yb + A, + Ab

'Yb ± A0~ + Ab

Similarly for the lower-level population Pbb, we f

( + V * V)Pbb = b - Yb'Pbb + (ihcVabPba +

where

yb' = Yb +

Xb =

r(Yb + Aa + Ab) + Ab(Y. - Yb)

'Ya + Aa + Ab

AbYa + r(A0 + Ab)

-y + A0 + Ab

Hence the population-matrix equations of motiand (14) have the same form as those for gas-laseinclude a more general excitation-decay scherpaper we consider mainly two limiting cases oftion-decay scheme. In the upper-to-ground-lomcay configuration, Yo = Yb = 0, and the A, ancdences in Eqs. (12), (13), (15), and (16) drop out,-Yb' = r, \a = 0, and Xb = r. For the excitedconsidered in this paper, we can neglect A aicompared with the decay rates, since the reserNprobability is then assumed to be much largexcited-state probabilities p and Pbb. Thisgives yo' = Yo + r, xA = A,, Yb' = Yb(l+ r/-y), ar

F(A,, + Ab)/y0 . A nice feature of this three-le,that it allows one to move continuously among v,ing cases.

(11)Pac -N nk expl-ik[(K 2 - K) r - At], a = a, b.

/Z=-(

(20)

It is further convenient to define the population difference(12) D(r, t) with the expansion

D(r, t) Paa(r, t) - Pbb(r, t)

(13) = N i dk expl-ik[(K 2 - K) r - At]), (21)

find where dk -nah-nbk. We now substitute these expansions

into the population-matrix equations of motion and identifyc.c.), (14) coefficients of common exponential-frequency factors. We

suppose that 61 does not saturate, that is, it appears onlyonce. We show that in this approximation that only Pi, P2,and p3 occur in the polarization expansion [Eq. (18)] and

(15) that only do and d+1 appear in the population-differenceexpansion [Eq. (21)]. Physically this simplification occursbecause once a product of 61 and 62 creates the pulsations

(16) d +l, then only C2 can interact. One obtains the polarizationsidebands of v2 at frequencies v and V3, which subsequentlycombine with v2 only to give back d±1 components.

on (6), (11), We calculate the coefficient of exp(iK 2 r - iv2t) for ther theory but saturator wave by neglecting the nonsaturating probe field.[ne. In this We findthis excita-ier-level de- i(v K2 - 2)p2 = -(iW + Y)P2 - i(8P 2/2h)do,I Ab depen-,iving ,,' = that is,-state cases.id Ab when'oir-level PeerEr than theassumptionid b Ab +

el model is

arious limit-

P2 = -i(pl2h)eA2do,

where the complex denominator 0J 2 is the n = 2 case of

= 1~I

On= + i(w + v Kn-vn) y + i(A + v K)

where the miode detuning

(22)

(23)

An = -v 0 (24)

Khitrova et al.

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. B 163

The coefficient of exp(iK1 r - ivit) for the probe waveincludes an extra term, 62di,

i(v K1 -)pl = -(iw + -y)p1 - i(p/2h)[61do + C2d],

giving

P1 = -i(p/2h)%)1[&do + 62d1]. (25)

where A' is given by

A' = A - v- (K2 - K1) (34)

and where we have used the fact that d-1 = d*. For co-running waves, A' A A, since K 2v n K1v. Similarly nbl isgiven by nal by changing sign and replacing Sat by Ob', givenby

'Db= Ya + Aa + Ab + iA(_Yb + tA')('Ya + iA') + AU(-Yb + iA') + Ab(-Ya + i A') + (Yb + A, + Ab + iA')r

The extra 62d1 term gives the scattering of '2 into the 61mode by the population-pulsation component d. po re-mains zero when only do and d 1 are nonzero, since it isproportional to 61dj, involving at least two 61's.

The component p3 does have a value, namely,

Solving for d1 = na1 - nb1, we have

[°a(A') + b(A\ )](12/2h) 616'2*(°V1 + 0 2*)do1 + 1602/2hj2[2La(A') + Db(A/)](°l + D3*)

P3 = i(p2/2h)' 36'2d_, (26)

in which K3 = 2K2 - K1, while Pj>3 vanishes since d(h>l)would be involved.

We calculate the dc-population-difference Fourier com-ponent do = n- nb0 saturated by the saturator wave 62alone. Substituting Eq. (20) into Eq. (11), we have

0 = Xa/N - ya'naO + [i(jPC2/2h)P 2* + C.C.],

yielding with Eq. (22)

na0 = X/Nya' - (2,ya'y) 1ItP62/hI1Z 2dO. (27)

Here £2 is the dimensionless Lorentzian

L2 = Y2/[_Y2 + (w + v - K 2 - v 2)2]. (28)

The 61 contributions are ignored, since we assume that 61does not saturate. The dpopulation component nbO isgiven by Eq. (27) with a - b and a change of sign. Thisgives the population-difference component

do= -1 - 12 2do,

that is,

(pl)26d 12*TjT2Y,(//) 2 (AZ) + yD2 *)2

do, (36)

1 + 25((') Y ()1 + O3*)

where the dimensionless complex population-pulsation fac-tor

5(A') = (2T1 ) [a(A') + Zb(Ai)]- (37)

This factor approaches unity as A' - 0.Our calculation is self-consistent, since only do and d±1 can

obtain nonzero values from Pi, P2, P3, and vice versa. Com-bining the pulsation component [Eq. (36)] with the polariza-tion component [Eq. (25)], setting P = 2Npl, and usingEq. (3), we find the complex absorption coefficient

a = ao dvW(v) 1 Y If 1+12132

X -

I2JY(A')2 (')1 + 1)2*)2 +

1 + I2gI(A') 'Y (D 1 + 03)) 2 _j

(38)

(29)do = 1/(1 + 122),

where the dimensionless intensity

12 = I6' 2/hIT 1T2 , (30)T2 1/-y is the dipole lifetime, and T1 is the population-difference lifetime:

T1 = I (1 + 1 ). (31)

Proceeding with the population-pulsation terms nal, nbl,and d1, we have

iA'fnal = -,ynal + ip/2h)[e1 P2* + &2P3 - 2*P1]

giving

nal = -,a(A/)(p/2h) 2[161&2*(Dl + 02~*)do

where

+ 1 212 (i + b 3*)d,],

-v + An + A, + iA'

where the homogeneous-broadening linear absorption coef-ficient a 0 is given by

K 1Nsp2

a0 - heoy (

Equation (38) has the same form as previous derivations butapplies to the more general level scheme of Fig. 3 and can beused to calculate probe absorption by Doppler-broadenedmedia with a probe wave propagating in any direction withrespect to the saturator wave.

The general formula [Eq. (31)] for the population-differ-ence decay time T1 and the corresponding population-pulsa-tion factor W(A') simplify in the limits of the pure two-levelsystem (Pcc = 0), and excited-state systems such that Pcc 1,while Paa, Pbb << 1. For the latter, we note that since thepump and decay probability flows have the same generalsize, ya and Yb >> A, and Ab. Hence Eq. (31) reduces to

a + . --a + A .0 _ A)a/ (-yb + iAt)(-ya + i/) + Aa('Yb + t') + Ab(-Ya + t') + (Yb Aa + Ab + i )r' 33

Khitrova et al.

d =

(39)

164 J. Opt. Soc. Am. B/Vol. 5, No. 1/January 1988

1 Ya\

2 Ya + F ( +Yb (40)and Eq. (37) reduces to

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Ya ± iA '

2T1 Ya+ + i + A Yb + iA') (41)

The limiting value of T1 for a closed two-level system isrecovered by setting Ya, Yb - 0, such that Ya/Yb - 1- Ingeneral, if Ya = Yb, Eq. (41) reduces to

J( = 1Tl(-ya + F + iA')

a simple resonant behavior studied in detail in the past.However if ya #d Yb, Eq. (41) shows that 5c has an extra beat-frequency dependence because the level populations decayat different rates as far as the population difference is con-cerned. This fact leads to a new kind of special resonancetreated in this paper. Grynberg 23'24 and Pinard 2 3 also con-sidered an extra resonance in two-wave mixing due to radia-tion decay as well as to collision broadening.

Physically, one can interpret this resonance intuitively bythinking of the populations as forced, damped, anharmonicoscillators with a zero-resonance frequency. Because theyare nonlinear oscillators, they respond to the superpositionof two nondegenerate fields by pulsating at the field beatfrequency. If the applied fields propagate in different di-rections, the nonlinear population pulsations are accompa-nied by a spatial variation, which is typically called spatialhole burning or an induced grating. More precisely, thepopulations respond to the walking plane-wave field fringewith the dependence cos[(K2 - K1) r - (v2 - )t]. Since,unlike the induced dipole, the populations have a zero-reso-nance frequency, they inevitably lag behind any nonzeroforcing frequency such as the possibly Doppler-shiftedprobe saturator beat note A' of Eq. (34). The faster thepopulations decay, the wider their frequency response andhence the smaller the lag. The ' contribution to the leveldecays affects both populations in the same way and henceleads to identical induced gratings for any A'. However, if'Ya Fd Yb, the populations lag different amounts as functionsof A' [compare Eqs. (33) and (35)], thereby leading to aresonance behavior in the population difference.

In addition, the two-level analogs to the PIER resonancesfollow from Eq. (38) for ya = Yb when evaluated to secondorder in the pump amplitude. In this approximation, Eq.(38) reduces to

al = o f dvW(v),yD 1 [1 - '2-2 - I21(A')'Y(D 1 + D2*)/21.

(42)

Combining the complex denominators D1 + °2*, we find

al = ao0 dvW(v)y0 1 [1 - 12-32 - 2x2D lD2*

X~i + Y:+A') (43)r + hYa + i'' (3

where e = 2 y - F - a and x = IpC2/2hl is one half of theRabi flopping frequency. Here we see that the A' depen-dence of the dipole resonance sum D1 + DJ2* cancels out theA' dependence given by the population-difference factor

7(A') if r = 2y and Ya = Yb = 0, which is valid for pureradiative decay. This cancellation is imperfect either ifpressure is introduced, causing 2 y to exceed r (pressure-induced extra resonance), or if the pump is too intense topermit the use of second-order perturbation theory. Inthese cases, we expect to see a beat-frequency resonance.

We refer to the system as closed if there are no populationtransfers to external reservoirs (Aa = Ab = Ya = Yb = 0) andthere are no collisions. This case occurs for pure radiativedecay from the upper to the ground lower level. This casesatisfies the condition r = 2 and hence exhibits no A'dependence up to third order.

As for the ya #4 Yb induced resonance, we can gain someunderstanding of the r 2 induced resonance in terms offorced, damped, anharmonic oscillators. Specifically, twokinds of second-order population pulsation contribute to theW term in Eq. (42), one starting with a first-order dipole Pabcomponent induced by the probe field and one with a Pbacomponent induced by pump. The sum of these contribu-tions produces the complex Lorentzian sum Yi + °2 *. Thissum has a phase shift equal to the sum of the individualphase shifts minus that for the beat note A' with a decayconstant 2 y. The population-difference phase shift alsoresults from the beat note A', but with a decay rate r.Hence, if r equals 2 ,y, the two phase shifts add to zero. Ineffect the population-pulsation response lags as usual be-cause of the finite temporal response of the population dif-ference, but this phase lag is canceled by a correspondingphase lead that is due to the interference between the probe-and pump-induced dipoles. As soon as these phase shiftsdiffer, the A'. dependence fails to cancel out. Furthermore,in higher order, the saturation denominator containing inEq. (38) yields a A' dependence even when those in thenumerator cancel one another.

Similar results occur in three-wave mixing of two weakfields and a pump. This mixing includes amplitude- andfrequency-modulation spectroscopy. In our notation, thecoupling coefficient xi appearing in the three-wave coupled-mode equation (neglecting phase matching)

(44)da,dz -a 16, + X163*

has the value

-ao| dv W(v) 1+XI = f ~ 1 2132

X

(PG2/h)2Tly-lI(A') (2 +°3*)(45)

1 + I2y(A') ( + 03*)2

where the conjugate-wave frequency 3 = 2 + 2 - v. Thisresult follows by carrying out the derivation above withthree fields in Eq. (1). It is straightforward to see that Eq.(45) has the same A' dependences as the absorption coeffi-cient a1 of Eq. (38). For four-wave mixing with counterprop-agating pump waves, the formulas become more complex.Homogeneously broadened treatments have been given inRefs. 18, 26, and 27, and Doppler-broadened treatmentshave been considered in Refs. 28 and 29. In particular,averages over the pump spatial holes have to be carried out.

Khitrova et al.

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. B 165

However, the suppression A' dependences discussed abovesurvive this average.

The observations on population-difference resonances arebased directly on the absorption formula [Eq. (38)] and itsthird-order approximation, which is quite general. We nowconsider a number of special cases, starting with a homoge-neously broadened medium.

stimulated Rayleigh and Raman resonances given by Eqs.(46)-(48).

Consider next an open system with arbitrarily large pumpintensity for which Yb << TYa, ' < A2, and consider beatfrequencies A of the order of Yb. For simplicity, we furtherneglect F, which just modifies the saturation intensity inthese approximations. Then T, of Eq. (40) reduces to 1/27b,W of Eq. (41) reduces to TYb/(TYb + iA), and Eq. (38) reduces to

3. HOMOGENEOUSLY BROADENEDOPERATION

For homogeneously broadened media, we can drop the vdependence in Eq. (38), including that in the O's, in -2, andin A'. Our discussion focuses on large detunings, but similarfeatures occur for detunings less than the natural linewidth.We are interested in some simple analytic approximations tothe absorption coefficient [Eq. (38)] that reveal simple reso-nant features. We then illustrate these features numerical-ly.

Consider first the real part of the second-order complexprobe-absorption coefficient given by Eq. (42). For Tla = TYb= 0, pump detunings I1A21 >> oy, and beat frequencies A of theorder of y, we can approximate the 0Z, of Eq. (23) by -i/A 2 .Hence the real part of Eq. (43) reduces to

Rel, I ° ,o - r+a (46)A2

3 F,2 + A2

which has a dispersive line shape. Such dispersionlike ab-sorption resonances are well known in two-level saturationspectroscopy. They are dispersionlike instead of Lorentz-ian because the dipole is being driven way off its resonanceleading to a phase lag or lead of 7r/2. This resonance vanish-es if e vanishes, that is, if r = 2-y.

It no longer vanishes if we expand Eq. (38) to the nextorder in the pump intensity and assume the resonance con-dition F = 2. For similar detuning values and A of theorder of y, we find the fifth-order contribution (first order inprobe, fourth order in pump)

Rela 1 Retal(5)} -ao 2 F2 A2- (47)

In addition in this order, there is a resonance in the vicini-ty of A3 0 from the 03* term [and hence from p3 of Eq.(26)] in the denominator of Eq. (38). This resonance occursfor A A2. This corresponding contribution is

R 4 7 2Relalln- Relal( 5 )1 - -ao -

A24 -Y2 +A 3

2 (48)

and originates from three-photon processes that induce apolarization 3 at the conjugate-wave frequency 3. Morespecifically, the pump-wave scattering off the populationpulsations contributes to two side modes, one being theprobe itself and the other being the conjugate wave. Thecontribution in Eq. (48) is sometimes called a Raman reso-nance and reveals probe amplification at the expense of thepump field. Several authors 2' 4' 6' 11' 17 "8 have predicted am-plification of a weak probe field by a strongly saturatedresonance medium in spite of an uninverted population.This amplification was observed by Wu et al.9 It resultsfrom constructive scattering of the pump off populationpulsations into the probe wave. Figure 4 illustrates the

al = -ia 0 1 + 4x2 [/22b 1-4x 2

-y/AlA 2 1Tb(l + 4x2 Y/ybAA 2) + iA

(49)

where x = lp&2 /2hl is one half of the Rabi flopping frequen-cy. Dropping the remaining differences between the A,, wefind the real part of a1 to be

Re~,1} a0° 1 + 2 2Y/YbA22

4Ax2/A2 1Tb2 (1 + 4x 2'Y/'YbA2

2 )2 + A2 ](50)

To second order in the pump field, we neglect 4x2

-Y/ybA22

compared to 1 in the denominators. This decision gives thestill simpler formula

Re~al} ~' ao A2-

2 / 2 ]- (51)

The first term is the linear probe absorption. The secondterm has a dispersive line shape as a function of the beatfrequency A and is due to scattering of the pump wave off thepopulation pulsations. In these limits, amplification (nega-tive absorption) occurs when

4A(x 2 /A 2 ) > 1. (52)

Tb2 + A2

In particular, for A 'Yb we have gain for 2x2 /ybA2 > 1.Since our perturbation expansion is valid only for X

2Y/'YbA2

2

212Xz0

C',

Lin

a-

-c

C I

$) 220Lr(b)OF

K- , - T

l I- 30 -10 10 80 100 120 220 240

A,

Fig. 4. Probe absorption line shape as a function of probe detuningfor the stationary atom limit and a nearly closed system (a = 'yb =0.001, r = 1, -y = 0.501, x = 40, A2 = 100 pump detuning). Allfrequency parameters in this and subsequent figures are in units ofthe total upper-level spontaneous decay rate ya + , which is setequal to unity. Linear absorption, Rayleigh (a), and Raman (b)resonances are identifiable. The same dimensionless units are usedfor relative probe absorption in all the following figures, and Al = - , A2 = 2 - w, which differs from the text by a minus sign.

I I

Khitrova et al.

166 J. Opt. Soc. Am. B/Vol. 5, No. 1/January 1988

4 --t -~ l

where u is the average atomic speed. The absorption coeffi-cient [Eq. (38)] can be written in the form

ai = -io dvW(v)f Ku + Al1 -

X [1-(Kv + A2 + iy')(Kv + A2 - i-')]

I2 (A)y2 -y + iA2 + iKv- Cl0 2 J dvW(v) 2 + (A2 + Kv)2

8 9 10 I 1 12

Fig. 5. Rayleigh resonance for stationary atoms in the perturba-tion limit for an open system Yb = 0.101, -y = 1.001, r = O, Y = 0.551,x = 2, and A2 = 10. No such resonance (to order x2) occurs for aclosed system (see text).

2y+iA -y-iA3 -iKv32 + (A 2 + Kv) 2 -y + iAl + iKv

where

f 2 = ( + i)[y + iA + _YI2 5(A)]

Y' Y 1\.X0

,_ a-8CL0:C)

m B

mLiio= 4

!. 2LiL

0 a 9 10 11 12

A,

Fig. 6. Same as Fig. 5 but for a weaker pump field of x = 0.5.Perturbation theory is valid. There is a resonance structure but noamplification.

<< 1, we find that -Y/A2 must be <<1, which is consistent withour original approximations. This result shows that alreadyin third-order perturbation theory, the scattering off popu-lation pulsations is larger than the linear absorption at thisdetuning. Gain in the absence of population inversion iswell known for larger pump intensities. 2'4'6

Corresponding results of a numerical evaluation of Eq.(38) are shown in Figs. 5 and 6 and are in good agreementwith our analytic approximations. Note that alternativelywe could have taken r + Y, << Yb. The same formulas [Eqs.(49)-(51)] result, with Yb replaced by r + ya. In the firstcase, the population difference lifetime T1 is given by 1/7b,

and in the second case it is given by 1/(r + 'ye). The reso-nance in both cases is characterized by the corresponding T1limit. This discussion has centered on cases for large pumpdetuning, but similar resonances also occur within the homo-geneous linewidth.

4. MOVING ATOMS

We now calculate the probe-absorption coefficient [Eq. (38)]for an atomic medium assuming a Maxwellian distributionof atomic velocities given by

W( = 1 exp(-v 2 /u2 ), (53)uX7

(54)

(55)

(56)

The approximation made for obtaining Eq. (54) is the ne-glect of the term (K2 - K1 ) v compared to y. This action isjustified for unidirectional waves, since (K2 - K1 )v is typi-cally of the order of 10-6-y. In actual experiments, the pumpand probe typically propagate in slightly different direc-tions, leading to a residual Doppler width of the order 0.1y.We neglect corrections of this sort in the following. Thisrepresents a major difference from the counterpropagating-wave case for which the Doppler shifts add, rather thansubtract, and the (K2 - K1) v terms play a decisive role.

We make use of the plasma-dispersion function defined as

Z(4) =-| dx 1 e (5_* A±u i

with Iml/A4 > 0. When we use the method of partial frac-tions, the velocity integral of Eq. (54) can be reduced to asum involving functions Z 2, Z3, and Z4, which, as defined inAppendix A, can each be written as a sum of plasma-disper-sion functions defined by Eq. (57).

The general result as a sum of Z, Z2, Z3, and Z4 functions is

aloY a0J2 7 3

a,* = -i Ku Z(G1) + 2 2(Ku) [Z2(01 6; -1) + Z2(y1415; 1)]

I2'Y2(2'y - iA)lI*/- a0 4f ( Ku) (Z2 ('tlI 4; -1) + Z2(/i1 13 ; 1)

+ [Z3 (AlA4A2 ; -1 - 1) + Z3(0u1 A31 A2; 1 -1)]

2-Ku Z 3(0 1 6 14; -1 - 1) + Z3(AlIuA5i4; 1 -1)

+ Z 3(01 i5A3; 1 1) + Z3(0V 6113; -1 +1)

+ A 1Z4GylA642; -1 -1 -1) + Z4(/1Al5 A3 /A2; 1 -1 -1)

+ Z4(A1t 6A31A2; -1 1 -1)

+ Z4 (A1l 5 93 92; 1 1 -1)}), (58)

C

4

.1

C,Q,

0a-

Lii

. . . . .

Khitrova et al.

(57)

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. B 167

$2 = (Y - A2)/Ku,

$4 = (i - A2)/Ku,

$6 = (iY' - A2)/Ku. . (59)

The general form of our result allows us to consider a num-ber of limiting cases., First, we can reproduce a result previ-ously obtained in the Doppler limit.

5. DOPPLER LIMIT

The Doppler limit is achieved when all decay constants (Yb,-Ya, r, y), detunings (dAJI, Al), and the Rabi frequency 2x aremuch smaller than the Doppler width Ku. Using the Dopp-ler-limit expressions given in Appendix A for Z2, Z3, and Z4and after long but straightforward calculations, we obtain

a1 = ao' exp[-(A/Ku) 2 ]

I 2 12 _YI2 (2y + iA)9(A){ ,(-y + + iA) 2 y2 -32

( +,Y)(Y'YiA) (3+,Y)(f3-Y-iA)]1x [( Y (, + .y+ A ) r ( +Y i ) JL '(I + z+ ii, 00 + 1 + iW j(60)

where the inhomogeneous broadening linear-absorption co-efficient ao' is defined by

(63)]. In each case this leads to a resonance at (A = A, i.e., A= 0). The A = 0 resonance comes from satisfying velocity-selection rules [Eqs. (63)], and this result is in agreementwith the analytical expression of Eq. (62) (see Fig. 7).

For Rabi frequencies satisfying Ku > x > Yb, the perturba-tion limit is no longer valid, but the Doppler limit still holds.The velocity-selection conditions, obtained from Eq. (54) inthis limit are

Ku = 2 i 2 _ X2,Ku=A2+C A-4Ky A,

Kv = A2.

CI)

.WLi

Lii

-160 -120 -80 -40I I I I

(64)

0 40 80 120 160

A,

Fig. 7. Probe-absorption line shape for moving atoms in the Dopp-ler limit (Ku = 100, x = 0.2, A2 = 5), a closed system (Yb = 0.001, Ya =0.001, r = 1, y = 0.501), and the perturbation-theory limit [x << Y].The dip near A2 = A results from velocity selected atoms. Allsubsequent figures are for the closed systems.

a0 = J7aoy/Ku. (61)

This result is the same as that of Baklanov and Chebotaev,1

except that r is included in AT, fl, and T, and hence in thedimensionless intensity I2. Including the r accounts forspontaneous emission from level a to level b and allows us toinvestigate open and closed systems (no population loss tothe external reservoir and no collisions present). The Dopp-ler-limit results can also be obtained by direct integration ofEq. (38) using the residue method in the complex plane.'. Inthe Doppler limit, velocity-selected atoms provide the majorcontributions to the line shape, and the slowly varying atom-ic-velocity distribution functions can be evaluated at theselected velocity.

In the weak-pump-field limit, the probe-absorption pro-file consists of a broad Gaussian of width Ku containing aDoppler-free hole with width r + Ya (or 2 y or Yb). Keepingonly first-order terms in I2, one can obtain for a closedsystem

(r + iA) (62)

To explain this perturbation-theory result, we return to Eq.(54) and note that atoms having velocities

Kv = A2 ,

Kv = A 2 i A,

Kv = Al (63)

are resonant with the applied fields. The contribution tothe probe absorption comes from atoms with velocities thatsimultaneously satisfy at least two of the equalities [Eqs.

C'14at

W

CD0) I'

U2

m tC)

Q

ic-JWi

I I I I I I I

-120 -80 -40 0 40 80 120

A,

Fig. 8. Probe-absorption line shape in the Doppler limit but for amoderately strong field (x = 20, Ku = 100, A, = 0). A dead zone canbe seen centered around A, = A2 = 0 of width 4x. In this detuningrange no atoms can satisfy the velocity-selection criteria to be reso-nant with the Stark-shifted transition frequencies.

Co IEx2 C)14

a-

U) IC

Lii

ac:

-Jm

W

I I I I I I-120 -80 -40 0

A140 80 120

Fig. 9. Same as Fig. 8 but for a pump detuning of A 2 = 20, whichshifts the center of the dead zone.

where

Y = (iy + Al)/Ku,

A3 = (i + A2)/Ku,

$5 = (iy' + A2 )/Ku,

us '

F1

Khitrova et al.

168 J. Opt. Soc. Am. B/Vol. 5, No. 1/January 1988

If one first considers zero detuning, the condition of Kv =(A12 - 4X2)1/2 can be satisfied only for A1 > 2x or A1 < -2x

for which no resonant atoms will exist. This means that inthe region of detuning 1A11 < 2x the absorption coefficient isequal to zero. As can be seen in Fig. 8, there is a symmetricdead zone in the Gaussian profile. For arbitrary pumpdetuning A2 the velocity-selection condition requires that A2> 2x and A2 < -2x for absorption to occur. Consequently,the dead zone, is shifted from the center by A2, as can be seenin Fig. 9. The size of the dead zone depends on the fieldstrength; by changing x and keeping A fixed, one can actual-ly move the borders of dead zone, as illustrated in Fig. 9.

For the strong-pump field, the Doppler-limit results donot contain any probe amplification (gain in the profile) andshow no narrow Doppler-free resonances related to theground- or excited-state width.

6. ABSORPTION LINE SHAPE ANDDETUNING DEPENDENCE

We are now in a position to consider cases that were notobtainable in previous calculations, namely, those in whichthe detuning A2 is of order Ku and the pump field is quitestrong, 2x Ku. The line shape consists of the contribu-tions from atoms with all possible velocities, not only thosefrom atoms whose velocities Doppler shift their frequenciesinto resonance with the fields. As can be seen in Fig. 10,there is a Doppler-limit contribution to the line shape; how-ever, internal structure can also be seen in a place where thedead zone used to be. To be able to provide a physicalexplanation of the line shape, we consider a perturbativedevelopment of the profile for a relatively weak pump field.

Expanding the general solution in powers of x2, we obtaina linear absorption term (x2)0, which is Doppler broadenedsince the resonance condition - = -Kv can be satisfiedfor a range of detunings 1 - v Ku. The nonlinear (x2)1

term gives a Rayleigh resonance about A - that remainsDoppler free, since both waves are Doppler shifted nearlyequal amounts. The condition V - Kv = 2 - Kv impliesthat the beat frequency A - - . This is true only forcopropagating fields. If pump and probe fields are propa-gating in opposite directions, then this resonance conditiongives lv -v 21 <2Ku, that is, a broad resonance. For the nextnonlinear term, (x2)2, the Raman resonance becomes veloci-ty broadened, since the resonance condition can be satisfiedwhen ( 2 - Kv) - ( - Kv) + (V2 - Kv) = w, that is, when (2v2- V1 - c) <Ku.

It is now possible to see how the total line shape has beenformed. With zero detuning, the line shape for (x = 50, Ku= 25) is shown in Fig. 11. This intermediate case was impos-sible to obtain with the previous limits and shows the transi-tion from homogeneous broadening to inhomogeneousbroadening. As can be seen in Fig. 11, a Doppler-limitcontribution does occur and is similar to that of Fig. 8.However, we can see additional structure in a place wherethe dead zone used to be. Our line shape accounts for thenegative probe absorption, which occurs in the (-2x < 1 <2x) region. This is definitely a non-Doppler-limit contribu-tion. It can be traced to a contribution from all atoms thatsatisfy the velocity-independent stimulated Rayleigh-reso-nance condition whenever A 0.

The amplification occurs in the dead zone (-2x < A1 <2x), as in the homogeneously broadened case in Fig. 4. Thus

the dead zone provides openings in the inhomogeneouslybroadened profile that allow us to see the Doppler-free con-tributions. The disadvantage of the zero-detuning case isthat it is impossible to distinguish contributions from differ-ent resonances (they all are at the same place, A1 = A = 0)and to obtain any information about ground- and excited-state widths.

Consider now the large-detuning case (Ku 2A2). TheDoppler-limit contribution occurs only for (A > 2x) and (A <-2x) (the position of the dead zone is shifted from the center

ca:

0

gC)W

LuIm0D

a-_Mi

U-

X 1o312-

10 8

8 -4 - X lo,

6 ~~ ~~ ~~0 0 18 26

6 - / l 106 °1-4

4 -8 -3

-30 -18 -6 6 - 18 30 42A,

Fig. 10. Probe-absorption line shape when the Doppler limit is nolonger valid, with x = 10, Ku = 20, A2 = 10. Structure is now seen inthe dead zone (Rayleigh and Raman resonances), although theDoppler-limit contribution is still evident.

2

0

1--

o

Liim

Cra-

;i

JLi

o 20 60 100-2-6

-10

I I-104 -100

A

I I100 104

Fig. 11. Probe-absorption line shape for the zero-pump-detuningcase with Ku = 25, x = 50. Owing to contributions from atoms withall velocities, negative absorption occurs in the area that is a deadzone in the Doppler limit.

20

0m0c

C)

LI

-lJ.I

I1 I I I I-30 -10 10

A,30 50

Fig. 12. Line shape for Ku = 10, x = 10, A2 = 5. Contributionsfrom atoms with all velocities can be seen, including a Rayleighresonance at A = A2 and Raman dip in the area where the dead zoneused to be.

-

Khitrova et al.

. , |

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. B 169

6 _

-0oX

;Zto 4_

cr .

CO

<0:

C>, Wo \Q.

>U

l'1:1

.2 0 0 0 0 0 2 2

20 60 100 140 180 220 260 300

Al

Fig. 13. Line shape for Ku = 100, x = 40, A2 = 100. The substruc-ture occurring in the area of the dead zone is shown in detail. Onecan see that the Doppler-limit contribution dominates the Ramandip.

-W0>< 20C10co-- I

LU

W

r -2

LJco

Cr

Q_

ft

I I I I100 140 180 220 260 300

A,

Fig. 14. Same as Fig. 13 except for the increased field strength, x =50. The Raman dip dominates the Doppler-limit contribution.

We have seen that, by changing the field strength, we canmove the border of the dead zone. At the same time theshape of the Raman dip itself is only slightly affected, sinceit is Doppler broadened. For the case x = 40, A2 = Ku = 100(Fig. 13), the Doppler contribution dominates over the Ra-man dip, and there is only a small gain before a large Dopp-ler-limit contribution, which appears exactly at (Al = 2x +A2 180) (Fig. 13). Comparing with the stationary atomline shape (Fig. 4), we can see that the stimulated Rayleighresonance remains the same but that the Raman resonanceis changed substantially. If we now increase the fieldstrength (all other parameters remaining the same), the lineshape changes dramatically, the border of the dead zonemoves (x = 50, A1 = 200), and the Raman gain dominates theprofile (see Fig. 14). There is a narrow dip just before thedead-zone border, then a small peak occurring exactly at theborder, and, finally, additional gain that occurs in the detun-ing range of the Doppler-limit contribution. Consequently,by properly choosing the field strength, we can divide theRaman gain into two parts. Thus we can vary the pumpintensity to produce Doppler-free structures in the openingsof a Doppler-limit profile. The line profiles displayed inFigs. 7-14 indicate that both velocity-selected and non-ve-locity-selected atoms can make important contributions tothe probe absorption for different values of the detunings,decay rates, and pump-field strength.

APPENDIX A

This appendix defines the higher-order functions Z,, used inEq. (58) and evaluates them in terms of the plasma-disper-sion function Z(,u) of Eq. (57). They are

/ JA ; e2 W(v)dv32 /1M283 ~El) J-E (l + ElV/u)(12 + e2V/u)

Z ` 2`31C \ = W(v)dvt k 2 El E (1 + ElV/U)(1 2 + E2V/U)(J13 + E3VIU)

Z4 i E12134 C2 3 4 = W(v)dv1k2' e E el (Al + E1v/u)(u 2 + E2 V/U) Cu3 + E3 V/U)(. 4 + E4 V/U)

by A2). For (Al < -2x + A2), there is a Doppler-limitcontribution that just adds to the velocity-broadened linearabsorption peak. An interesting feature is the Rayleighresonance (vl = 2), which is Doppler free and can be seen inthe dead-zone region. In the strong-field limit, the Rayleighresonance exists for both open and closed systems (Fig. 12).

Another interesting structure that occurs in the dead-zoneregion is a broad dip that terminates at exactly the dead-zone border (see Fig. 12). The origin of this dip can betraced to the velocity-broadened Raman resonance dip,which is competing with the Doppler-limit contribution(Fig. 12). As for purely homogeneous broadening, it is athree-photon process that is responsible for the probe ampli-fication, but it is possible to observe this gain in the profileonly because it occurs in the dead-zone region. One actuallycan change the shape of the gain by considering that thelocation of the Doppler-limit contribution depends on thefield strength (Al > A2 + 2x).

In terms of Eq. (58), they are given by

- ~Z(A2) -Z(Ad)Z 2(pl/t 2 ; e) = )

-2 CEtl

Z 3(tlA 2 A3 ; C') =

Z 4 (A1 21'3 A4 ; ee/'") =

Z 2 (91 A3 ; e - Z2(L 2 A3; e/e')

(A4)

I (A5)/'2 -e)eZ3 (,t 1lt 3 p4; C'C") - Z3 (AL2/L3 / 4; C/C' C""/E)

"2 - (Al

(A6)

ACKNOWLEDGMENTS

This research was supported in part by the U.S. Office ofNaval Research, in part by the U.S. Army Research Office,in part by the U.S. Air Force Office of Scientific Research,

(Al)

(A2)

(A3)

01 - - 1 .

Khitrova et al.

170 J. Opt. Soc. Am. B/Vol. 5, No. 1/January 1988

and in part by the National Science Foundation under grantPHY-8415781. It is based in part on the Ph.D. dissertationof G. Khitrova (New York University, New York, 1986).

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Khitrova et al.