1456 J. Opt. Soc. Am. B/Vol. 2, No. 9/September 1985

Quantum theory of multiwave mixing. IV. Effects ofcavities on the spectrum of resonance fluorescence

David A. Holm and Murray Sargent III

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

Stig Stenholm

Research Institute for Theoretical Physics, University of Helsinki, Helsinki, Finland

Received March 20, 1985; accepted May 29, 1985

In the presence of a high-finesse cavity, the resonance fluorescence spectrum can be altered by stimulated emissionand multiwave mixing processes. We show that under some conditions it is possible for spontaneous emission toinduce conjugate fields similar to semiclassical phase conjugation, requiring the use of the multiwave quantumtheory described in earlier papers in this series. The theory is applicable to laser spectroscopy and optical bistabil-ity. We present explicit spectra and suggest possible experimental tests. In particular we find a quantum limitfor FM spectroscopy that shows that even if the medium under study leaves the FM phase and amplitude relation-ships intact according to semiclassical theory, spontaneous emission can create an AM component detectable ina square-law detector.

1. INTRODUCTION

Resonance fluorescence describes the spontaneous emissionof a two-level system in the presence of a strong, monochro-matic field. It has been the subject of much study since thefirst correct analytical calculation of the scattered spectrumwas done by Mollow.' Since that landmark paper, extensionsof the original theory have been made and experiments havebeen performed.2 -5 Virtually all this research has assumedthat the spontaneously emitting atoms are in free space, andhence cavity effects have been neglected. In laser spectros-copy, Fabry-Perot cavities are often used to measure opticalfrequencies accurately.2 In addition, both lasers and opticalbistability require a high-quality cavity to provide the nec-essary feedback for operation. Instabilities in lasers andoptical bistability that grow from spontaneous emission areinfluenced by these cavities. It is of interest, therefore, toconsider the effects of cavities for these cases.

In this paper we show how collecting the spontaneousemission in a high-finesse cavity significantly alters theemergent radiation spectrum and can under some conditionslead to some new, quantum phenomena. Lugiato 6 and others 7

have studied fluorescent spectra in optical bistability, but they

were concerned with cooperative effects and did not considereffects of stimulated emission and phase conjugation on thespontaneous-emission spectrum.

In two recent papers8 '9 a quantum theory of multiwave in-teractions with a two-level medium is presented. The theoryis able to treat a number of topics in quantum optics involvingnonlinear interactions with several fields, including resonancefluorescence, saturation spectroscopy, laser and optical-bi-stability instabilities, and three- and four-wave mixing. Inthis paper we apply this theory to various experimental con-figurations involving resonance fluorescence in a cavity, suchas the work of Hartig et al.

2 We also find a quantum limit forthe three-wave mixing case known as FM spectroscopy.1 0

Section 2 summarizes the theoretical calculations presentedin Ref. 8. Section 3 unifies the phenomena of resonance flu-orescence and saturation spectroscopy by considering theeffects of stimulated emission and absorption on the fluo-rescent light captured in a cavity. Section 4 extends the re-sults of Section 3 to the case when the fluorescent light iscollinear or nearly collinear with the exciting laser light. Thepump and one-cavity mode driven by spontaneous emissioncan then interact in the medium to generate a phase-matchedconjugate field at a frequency symmetrically located on theother side of the pump frequency. This problem involvesthree-wave mixing, which considerably complicates thetheory. Section 5 studies the effects of a standing-wave pumpfield. Provided that the frequency of the induced conjugatepolarization coincides with a cavity mode, phase matching ofthe conjugate field is now possible for any direction of thefluorescent light. Averaging over the spatial holes burned bythe standing-wave pump must be done. Each section pre-sents explicit spectra and discusses the corresponding ex-perimental arrangements.

2. SUMMARY OF BASIC EQUATIONS

Our Hamiltonian (in radians/second) is

3H = ( - v2)o- + i [( - 2)ajta

j=1

+ (gajUjat + adjoint)]. (1)

In this expression aj is the annihilation operator for the jthfield mode, U = U(r) is the corresponding spatial modefactor, a and a, are the atomic spin-flip and probability-dif-ference operators, w and vj are the atomic and field frequen-cies, and g is the atom-field coupling constant. As in thesemiclassical theories, 1 ' we take mode 2 to be arbitrarily in-tense and treat it classically and undepleted. Modes 1 and

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Holm et al.

Vol. 2, No. 9/September 1985/J. Opt. Soc. Am. B 1457

3 are quantum fields treated only to second order in amplitudeand cannot by themselves saturate the atomic response. Thisis an important assumption and limits the applicability of thetheory. The rotating-wave approximation has been made,and the Hamiltonian is in an interaction picture rotating atthe strong field frequency v2. We define an atom-field den-sity operator Pa-f and obtain its time dependence from thestandard density operator equation of motion

Pa-f = -i [H, pa-f] + relaxation processes. (2)

Reference 8 carries out the calculations for a level scheme thatallows for interactions between two excited states as well asfor the usual situation in laser spectroscopy for which thelower level is the ground state. In this paper we restrict ourcalculations to the latter, which significantly simplifies ourexpressions. We assume that the only relaxation processesare upper-to-lower level decay, described by the decay con-stant r (= 1/T1 ), and the dipole decay, described by y (= 1/T2 ).

Our numerical examples consider pure spontaneous decay,for which y is equal to r/2. We calculate the reduced elec-tric-field density operator p that describes the time depen-dence of the two quantized fields by taking the trace of Pa-f

over the atomic states. We assume all field amplitudes to varylittle during atomic decay times. This allows us to solve theatomic equations of motion in steady state and then to obtainthe slowly varying field density operator equation of mo-tion:

= -Al(palait - aitpal) - (B1 + v/2Ql)(atap - alpait)+ Cl(altatp - a3tpait) + Dl(pa3tat - aitpa3t)+ (same with 1 interchanged with 3) + adjoint, (3)

where Qn is the cavity quality for mode n and the coefficientsA 1, B1 , C1 , and D1 are given by

the dimensionless intensity I2 is

I2 = 41 V 21 2 TlT 2 ,

the dimensionless "population pulsation" term 5 is

1= rr+iA

and where CY 2 = gU2(n2 + 1)1/2, N is the total number of in-teracting atoms, and A = V2- v is the beat frequency betweenmodes 1 and 2.

We use the density operator Eq. (3) primarily to derive theresonance-fluorescence spectrum. As shown in Refs. 8 and9, A1 + c.c. yields the spectrum of resonance fluorescence,while A 1 - B1 is the semiclassical complex gain/absorptioncoefficient a 1 for a weak probe field in the presence of a sat-urating field, and C1 - D1 is the semiclassical complex cou-pling coefficient iKl between the signal and the conjugatefields in phase conjugation.

3. SINGLE-SIDE-MODE INTERACTIONS

In this section we consider the interaction of the two-levelmedium with the strong mode 2 and a single quantized modeat frequency v1. This is the case when there is no phasematching or when the strong field is sufficiently detuned fromatomic resonance. The mode 3 coefficients and the couplingcoefficients C1 and D1 are all zero, and the operator Eq. (3)reduces to

= -Al(palalt - altpal)- (B1 + v/2Q)(altap - a1palt) + adj. (8)

We are mainly interested in the average photon number (n1 )

1l = g+2N 1 I2L 21 + 2L2 2

+ I2 212L2_

2

I2 r 2[I2 2D - 02*(1 + r/iA)/2f1

1 +I 2 ( 1 + SO3*) J2)

I2 29M[(1 + I2L2/2)Ol + 02*(1 - r/iA)/2]}

+ 25 (2 1 + 3*)

2TlT 22gJ[2 ys3 * - 2(1 + F/iA)]C, = - g2N Ul*U3*

+ 1I2 -C2 UU 1 + I2y (21 + 3*)

O= g 2Nl 2T1lV 2291[(1 + I2C2/2)5J 3* + 0l)2(1-

~~~~~~~1 + I2L2 (1 + 03*)

where, following the notation of Ref. 8, the complex Lorent-zian denominators OJn are given by

O~n-Y + i(W - Vn)

the dimensionless Lorentzian L2 is£ 2 -y + ( 272 + -p)

(7)

= YnlnlPnini, where Pnini = (nillpnl). Projecting Eq. (8)onto the photon number states In1) and summing over n1, wefind the equation of motion:

dd (nl) = -A((n 2 ) + (ni)) - (B1 + v/2Q,)(ni2)

+ (B1.+ v/2Ql)((n, 2) -(nl)2)

+ Al((n,2) + 2(ni) + 1) + c.c.= (A-B-- v/Q)(n) + A, (9)

UeU4PWMC.0W

0nFso1-0

J (4)

(5)

(6)

Holm et al.

-

1458 J. Opt. Soc. Am. B/Vol. 2, No. 9/September 1985

rate depends on many factors, such as diffraction effects,mirror reflectivity, and nonsaturable absorptions. To relatethese two quantities uniformly we define ao as the value of themode 1 absorption when the strong field intensity I2 is zeroand when w = vl, i.e., the unsaturated centrally tuned ab-sorption rate

ao = (B - A)I,=o,i= =

The coefficients from Eqs. (4) to (7) can all be expressed interms of this quantity. We define the coefficients Al', B1 ',etc. as

Fig. 1. Experimental configuration of Ref. 2. The directions of theatomic beam, the intense laser field, and the Fabry-Perot axis are allmutually perpendicular.

A 1~B 1Al = Al, B'=l

COnt anoC1 ' = -, and Di' = l

1a2

(12)

where A -A + Al* and B B1 + B1 *. This equation is alsoderived in Refs. 8 and 9. Equation (9) gives the time differ-ential equation of motion for the average photon number ina cavity mode, where A - B is the gain coefficient and A is thesource term for spontaneous emission. In free space, thelosses typically greatly exceed A - B, preventing photonnumber buildup, i.e., (nj) = 0. Hence the derivative d/dt(nj) = A denotes the counting rate of photon detector ob-serving the resonance-fluorescence spectrum A. This was firstobtained by Mollow.1 The absorption coefficient, A - B,describes the absorption (or gain) of a weak probe field in thepresence of a strong field and, like the resonance-fluorescencecoefficient A, was also first obtained by Mollow.1 2 We nowstudy how these quantities are related.

We consider the experimental situation shown in Fig. 1. Anatomic beam passes through a high-finesse Fabry-Perot cavityand is radiated perpendicularly (or at a substantial angle) byan intense laser field. The fluorescent emission selected bya cavity-mode frequency is then measured by an externaldetector. This is essentially the configuration used in theexperiment of Ref. 2. Note that the cavity-mode separationmust be large compared to the overall width of the resonance-fluorescence spectrum. When the emission takes place in acavity, the photon number (nl) increases until a steady stateoccurs. Using Eq. (9) we solve for the steady-state value

These quantities are the dimensionless magnitudes of Al, B1,etc. in units of the unsaturated resonant absorption coeffi-cient. We also define the parameter : as the ratio of thecavity loss rate to ao0:

(13)a v/Q0

ao

Equation (10) can then be expressed as

A'(nl = Al(3+ (B - A') (14)

We now investigate how the scattered spectrum, given byEq. (14), depends on j3. Since we are interested in how theshape of the spectrum changes, we multiply Eq. (14) by : tonormalize our results. The limit 3 -X then recovers thefree-space answer. Figure 2 shows the spectrum of I2 = 50(corresponding to a Rabi flopping frequency of 5r) for: = 1.0,0.1, and 0.05. It can be seen that as 3 decreases (cavity finesseincreases), the sidebands increase in size, become somewhatsharper, and move slightly toward line center relative to thefree-space case. On the other hand, the central peak remainsunchanged. As 3 decreases further, the sideband intensitiescontinue to increase to infinity. At this point (the thresholdfor side-mode lasing), the denominator of Eq. (14) is zero andour linear theory has broken down. To obtain the spectrum

(n 1) = Av/Q - (A -B)

(10)

In the limit v/Ql >> B - A, corresponding to a poor cavity, Eq.(10) simplifies to become (nj) = QlA/v, and we recover thefree-space expression for the spectrum. Note that we are nowdefining the spectrum to be given by (n 1) as a function of v2- vj, where vl is given by a cavity resonance. This is consis-tent with other quantum-mechanical definitions of thespectrum. For further discussion see the recent review paperby Cresser.13 When v/Ql and B - A become comparable,however, the spectrum is altered appreciably.

(B - A) (nj) is the number of absorptions/second of themode 1, and (v/Q ) (n 1 ) is the cavity loss rate of mode 1. Theabsorption rate depends on such quantities as the atomicnumber density, the dipole matrix element between levels aand b, and the pump mode intensity I2. It is well known thatthis absorption can go negative, giving gain. The cavity loss

0.

0. 08

AT,

Fig. 2. The spectrum of (n1 ) given by Eq. (14) versus ( 2 - v)T= AT1 = A/r for f = 1.0, 0.1, and 0.05. 2 = 50 and T = 2T2.

>10

0C.)c4a

Q~

4)

U,U,

ATOMICBEAM

(11)

Holm et al.

DETECTOR

Vol. 2, No. 9/September 1985/J. Opt. Soc. Am. B 1459

0AT1

(a)

absorption on the resonance-fluorescence spectrum is signif-icant, but only for : < 0.1, that is, only when the cavity lossrate is much less than the unsaturated absorption. However,there is another way to observe this effect even when the cavitylosses are not so low. It is well known' that the resonance-fluorescence spectrum remains fully symmetric even whendetuned. However, the absorption B - A shows a pro-nounced asymmetry even for small detunings. Thus thespectrum (n,) given by Eq. (14) should show a noticeableasymmetry when detuned. Figure 4(a) shows this to be thecase for : = 1.0 and a detuning w - 2 of 1/Ti. For higherdetunings, the elastic portion of the spectrum (Rayleigh peak)becomes more dominant and the inelastic spectrum depictedhere decreases, but then the asymmetry can be seen for evenlarger values of 1. Figure 4(b) depicts such a case for: = 5and a detuning of 5F.

In the original experiments on resonance fluorescence,2-5

asymmetric spectra such as Fig. 4(a) were observed. Severalexplanations have been forwarded to account for this, in-cluding field irregularities and polarization effects. We donot suggest that the stimulated processes derived here are thecause of the observed asymmetry, but they are yet anothercause for such asymmetries and perhaps offer the best op-portunity to verify our calculations experimentally.

0.

<nl>

-8 0 8AT,

(b)

Fig. 3. (a) The free-space resonance-fluorescence spectrum A versusAT1 and (b) the semiclassical gain/absorption coefficient A - B =a1 of a weak field in the presence of a strong field versus AT1 . Posi-tive values of al correspond to gain. Same parameters as Fig. 2.

in this case it is necessary to include saturation of the sidemode in the theory. Because this theory neglects this, : mustnot be allowed to come too close to A' - B'. This saturationproblem does not occur in the semiclassical theory, since thecavity modes are below threshold. In a quantum field theory,such as that of Scully and Lamb,' 4 saturation is important justbelow as well as above threshold.

To obtain a better understanding of the curves of Fig. 2,Figs. 3(a) and 3(b) depict the free-space resonance-fluores-cence coefficient A and the gain coefficient A - B for the sameintensity and decay constants as Fig. 2 versus the beat fre-quency AT,. It is well known that for frequencies v1 withinthe Rabi sidebands the weak field 1 can be increased ratherthan absorbed.12"5 Thus spontaneously emitted light inthose frequencies is amplified. Also, since the maximum gainoccurs at a point near but within the Rabi sidebands, thefluorescent sideband peaks tend to move in. Notice that thereis near total bleaching of the gain when vj = v2, and so thecentral peak remains unchanged. For frequencies outside theRabi frequencies there is a small absorption, and so the lightemitted at those frequencies is attenuated. These conclusionsare consistent with the curves of Fig. 2.

As seen from Fig. 2, the effect of stimulated emission and

0.

0.

<n,>

0. 01-10 0 10

AT,(b)

Fig. 4. Effects of 1 on the detuned spectrum of Eq. (14) versus AT 1 .I2 = 50 and T1 = 2T2. In (a) we have ,B = 1 and a detuning of w -2

=r. In(b),=5andco-v2=5r.

0.

A

0.

0.

A-B

0.

e,x

CI):0

MM0

0cc

O00

-0.

AT1(a)

8

Holm et al.

1460 J. Opt. Soc. Am. B/Vol. 2, No. 9/September 1985

4. THREE-WAVE MIXING IN A CAVITY

The results of the previous section are not valid for directionsof mode 1 nearly collinear to the intense mode 2. This is be-cause for spontaneous emissions in this direction the two fieldsinduce a grating that scatters the strong field, producing aphase-matched or nearly phased-matched third field. Thisis the origin of resonant three-wave mixing. This is relevantfor lasers and optical bistability, since this now correspondsto a double side-mode case. To treat this it now becomesnecessary to use the complete reduced density operatorequation of motion, Eq..(3). This means that there are nowfour times as many terms to consider.

One way to achieve the desired mode-frequency arrange-ment is to use the typical unidirectional optical-bistabilityconfiguration for which a strong mode is injected at a cavityresonance, as shown in Fig. 5. Side-mode fluorescence is thenmeasured as a function of mode spacing. In this way thecavity selects pairs of side modes for study. The detectorheterodyne electronics measures the amplitudes of the beatfrequencies of interest. This kind of experiment is directlypertinent to laser and optical-bistability instabilities, as wellas to modulation spectroscopy. For the single side-mode caseof Fig. 1, it is possible to determine the entire emission spec-trum by tuning the cavity to the frequencies of Fig. 2. In thedouble side-mode case, however, the strong mode frequencyP2 has to be retuned to be resonant with a new cavity modewhenever the side-mode frequencies v, and v3 are changed.A complete measurement of the spectrum is therefore moredifficult although still possible in principle.

We proceed as before, multiplying the p by the numberoperators altal and a 3 ta 3 , projecting onto the basis ketsJn1n3), and summing over n1 and n3. We then find that

dt (n+) = (Al + Al* -B ) -B*)(n) + A+ A,* + (C - D,)(a3talt)

+ (C* -Di*) (a3ai),

d (n 3) = (A3 + A3* - B3 - B3*)(n 3 ) + A3

+ A3 * + (C3 - D3)(a3tait)+ (C3* -D3*)(a3a ). (15)

A(C - D) + C(PIQ + B -A)(aita3t) = (aia3O = ()2C--(C - D)(Y/Q + B-A)2 (-) (20)

where we assume that Q, = Q3= Q. We rewrite Eq. (19) usingthe 13 parameter of Eq. (13) to obtain

(n,) =A'( + B' - A') + C'(C' - D')

(1 + B'- A')2 - (C- D')2 (21)

As in Section 3 we investigate the dependence of the spec-trum (nl) on the cavity quality by varying 1. We againnormalize Eq. (21) so that for large 13 we recover the free-spaceresult. Figure 6 shows the resonance-fluorescence coefficientversus AT, for I2 = 50, r = 2y, and 1 1.0, 0.2, and 0.1.These curves are seen to be very similar to those of Fig. 2. The13 values, and hence the cavity loss rate, are now twice as largefor the 13 = 0.2 and 0.1 curves. This can readily be explainedas due to the fact that there are now two quantized modesparticipating in the interaction, and hence the effects appearfor larger 1 values and poorer cavities. One noteworthy dif-ference between Fig. 2 and Fig. 6 is that now the central peakdrops as 13 decreases, whereas before it remained unchanged.Equation (21) is quite complicated and difficult to interpretanalytically, but it may be easily investigated graphically.

Figure 7 depicts the C coefficient for the parameters of Fig.6. Note that the sidebands of C are similar to the resonance-fluorescence coefficient A but that the central peak of C is

-Two-level medium

1/

\ ~~~~~~~~~~~/Fig. 5. Unidirectional optical-bistability cavity producing three-wavemixing.

(16) 0.

We also must determine the equations of motion for (a 3ta 1t)and (ala 3 ). We have

dt (a3tat) = (A* + A3* - B * - B3*)(a3tat)+ (C,* -D*) (na)+ (C3* - D3*)(n,) + C,* + C3 *,

d (a 3ai) = (A, + A 3 - B,- B3)(a 3ai) + (C1 - Di)(n 3)dt

+ (C3 - D3 )(n) + C1+ C3.

<nI>

(17)

(18)

To simplify these equations we assume central tuning of thestrong mode, v2 = . This implies that A, = A3*,B1= B3*,etc. WeletA=A,+A,*,B=B,+B,*,C=C+C,*,andD = D, + D,* and solve each equation in steady state. It isstraightforward to show that

(n,) = (n3) = ( /QQ+ B -A) (C -D) (19)(v/Q +B - A) 2 - (C -D)2

0.-8 0 8

AT,

Fig. 6. (n 1) versus AT, given by Eq. (21) with the three-wavecoefficients Eqs. (27), (28), (33), and (34). The three-wave mixingspectra are shown for ,B = 1.0, 0.2, and 0.1, with 12 = 50 and T, =2T 2.

;a0(A0

U,

,^

k

Holm et al.

Vol. 2, No. 9/September 1985/J. Opt. Soc. Am. B 1461

0. 5

c

0.

-n

0I

U. -I-8 0 8

AT,

Fig. 7. The quantum coupling source coefficient C versus AT, for12 = 50, T, = 2T2 -

inverted. From Eq. (22), we see that the term containing Cis in the numerator with A. When 1 becomes small enoughso that A'(1 + B' - A') is comparable with C'(C' - D'), thisterm pulls the central peak down, in contrast to the singleside-mode case of Fig. 2. This is the origin of the drop of thecentral peak in the curve of Fig. 6.

Three of the quantities appearing in Eq. (21), the resonance-fluorescence spectrum A, the semiclassical side-mode ab-sorption coefficient B - A, and the semiclassical phase-con-jugation coupling coefficient C - D, are well known and havepreviously been derived in many ways. The fourth term, theC coefficient, is new, having first been derived in earlier papersin this series.8 As seen from Eqs. (17) and (18), it is a quan-tum-mechanical source term for the couplings (alta 3 t) and(ala3 ). Just as A leads to the buildup of the side modes (nl)

and (n3), C increases the field coherences. In the problemconsidered here, C is seen to alter the emission spectrum by

reducing the central peak. This represents a quantum effectthat can be accounted for only by a quantum multiwave theorysuch as that used here.

Consider the important case for which v/Q dominates A -B and C - D. Equations (19) and (20) then reduce to

(ni = (n3 ) = A-,

C =(aita 3t) = (aia3) = -Q

(22)

5. FOUR-WAVE MIXING

For the three-wave mixing considered in the previous section,it was pointed out that owing to phase-matching consider-ations, all three fields must be nearly collinear. To discrim-inate against the intense pump field, heterodyne methods aregenerally necessary. In this section we consider the case offour-wave mixing in which the strong pump field consists oftwo oppositely directed running waves, forming a standingwave. In this case a given pump wave interferes with thesignal wave to induce a grating that scatters the other pumpwave into the direction opposite that of the signal wave. Thisprovides the source for the conjugate wave. The conjugateis phase matched for all directions. Because of the stand-ing-wave pump pattern in the medium, atoms in differentlocations in the medium experience a varying amount of sat-uration that is due to the spatial hole burning of the upper andlower population difference. We account for this by averagingthe coefficients over the spatial hole burning for one wave-length.

The experimental configuration is very similar to that ofFig. 1, except that the atomic beam passes through a stand-ing-wave pump formed by two counterpropagating wavesinstead of through a single running-wave pump. Thestanding-wave pump field is given by

E2 = 2 {A2 exp[i(K2 -r - v 2t)]2

+ A2 exp[-i(K 2 r + v2t)] + cc.}.

We define the spatially dependent intensity

52r = 4l~v~l2Tl = 14p2IA2I2 os2(K2 r)TiT 2= 41V2 12TT 2 = h2 C

= 4I2 COS 2 (K 2 -.

(24)

(25)

where ,u is the dipole moment and I2 is the dimensionless,spatially independent intensity for each wave. For simplicitywe take the wave vector K2 to be along the z axis, so K2 - r =K2 z = 27rz/X2. We integrate each coefficient along the z di-rection for one wavelength. For example,

(A,)SHB =- dzAi(z).X2 o

(26)

The standing-wave averages that we need have been calcu-lated in a previous paper in this series.17 The results are

(23) (A1)SHB = g1 O -b + C/L2 -b + c/d(A,)SHB 2 - d [(1 + 412L2)1/2 (1 + 4I2d)1/2

This shows that while A is the source term for the fluorescencespectrum, C is the source term for the field coherence spec-

trum. In particular C determines the relative phase betweenthe side modes. As illustrated in Fig. 7, for beat frequenciesA less than the Rabi flopping frequency, C is negative, leadingto an FM phase relationship, while for larger beat frequencies,C is usually positive, leading to an AM phase relationship.This has the consequence for FM spectroscopy discussedelsewhere in this issue1O,16 that for some beat frequencies, FMspectroscopy should detect an AM component owing tospontaneous emission. This effect places a quantum limit onFM spectroscopy while at the same time providing an addi-tional test of our theory.

+ c(- - I (27)

g 2 Ul*U 3 *01 [ -b 3 + c/-L 2 -b 3 + c/d\iI5HB = £ -2 - d [(1 + 4I212)1/2 (1 + 4I2d)1/2

(28)

where

b L2 + 'YD2* gr + r/w,2 4

b3 = - -D2 gr+ F/wA,4

(29)

(30)

C

M0uM_.

e0uaMO0V

if

Holm et al.

I - 1+ C

�d -CJ I

1462 J. Opt. Soc. Am. B/Vol. 2, No. 9/September 1985

0.

O. --15 0

Fig. 8. Resonance-fluorescence intensity A of Eq. (27) versus AT,in a standing wave. Each running wave has an intensity I2 = 50, T,= 2T 2.

0.

'I' SH8

0. 0--15 0 15

AT,

Fig. 9. (nl)sHB versus AT1 given by Eq. (21) with the four-wavecoefficients Eqs. (27), (28), (33), and (34) for 3 = 1.0 and 0.1. Sameparameters as in Fig. 8.

0.

<B-A

0. 0

-0. 01-15 0 15

AT,

Fig. 10. Semiclassical Wbak probe absorption coefficientB - A av-eraged over a standing-wave fringe. Same parameters as in Fig. 8.

'yL2 YJ 3*4

d = 2 5r(Dl1 + 03*)-

(31)

(32)

We need the standing-wave averages for the semiclassicalquantities A1 -B 1 and C1 - D1. These are also done in Ref.17 with the results

(Al - B1)SHB = - + 4I2L2)1/(1 + 4I212 )1/2

I 2I2-;Y(2 Ol + 02*)1 + 4 2d + [(1 + 4I2-L2)(1 + 4I2d)]1/2 , (33)

(C1- D1)SHB

2g2Ui*U3*Y)iA 2 5IY(D1J2 + .3*)

(1 + 4I21 2)1/211 + 412d + [(1 + 4I2.L 2)(1 + 412d)]l/2 j

(34)Equation (27) plus its complex conjugate is the formula for

resonance fluorescence from a standing-wave pump field.Figure 8 plots the resonance-fluorescence spectrum of Eq. (27)for T2 = 2T 1 and an intensity I2 of 50 for each pump field.Note that the central peak is much sharper and higher thanfor the running-wave formula and that the sidebands are in-distinct. Because the fluorescence is emitted by atoms indifferent locations in the standing-wave fringe, a distributionof Rabi sidebands contributes, tending to wash out the side-mode Lorentzians characteristic of the unidirectional case.However, all parts of the fringe contribute to the central fre-quency, reinforcing the central peak.

We once again restrict our numerical examples to centraltuning. Equation (21) is still valid, provided that the coeffi-cients are replaced with their averages over the spatial holeburning, Eqs. (27), (28), (33), and (34). Figure 9 depicts thespectrum predicted by the averaged Eq. (21) for : values of1.0 and 0.1. Note that, in contrast to the previous cases, thesidebands are not increased and actually drop a little.However, we see that the central peak again decreases.

Figure 9 is, unlike Figs. 2 and 6, due to the averages over thespatial hole burning. Three of the quantities that are in Eq.(21), A', C', and C' - D', do not change appreciably when av-eraged according to Eq. (26). As shown in Fig. 1d, thestanding-wave averaged absorption (B' - A')SHB differsconsiderably from the undirectional case. Specifically, (B'- A')SHB is only slightly negative and, instead of nearlyvanishing when V = 2, it is a maximum. Because of this,there is little amplification. This is due to the fact that spatialhole burning leaves substantial amounts of unsaturated ab-sorbing medium, averaging out the gain. Hence the sidebandsof Fig. 9 do not increase as they did in Figs. 2 and 6. Because(C')SHB and (C' - D')SHB are similar to their respectiverunning-wave coefficients, the central peak again decreasesas : decreases.

6. CONCLUSION

We hav~e applied the mnultiwave quantum theory of Refs. 8 and9 to the problem of the spectrum of resonance fluorescencein various configurations in a high-finesse cavity. In the caseof a single side mode, we predict an increase and a narrowing

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Vol. 2, No. 9/September 1985/J. Opt. Soc. Am. B 1463

of the sidebands when the cavity loss rate is less than the un-saturated absorption rate for a centrally tuned intense field.When this is detuned, we predict an asymmetric spectrumowing to the asymmetry of the absorption and stimulatedemission.

When it is possible for the strong classical field and the weakquantized field to induce another weak quantized field, thetheory is complicated by three- and four-wave mixing. In thecollinear, three-wave mixing case, the results are similar to thesingle side-mode solution, but the central peak is reduced.We predict that this is due to a new quantum-mechanicalcoefficient, C, that can be described as a source for field cou-pling terms. This effect is also seen to be the case in thefour-wave mixing case, but higher-finesse cavities are requiredowing to the averaging by the spatial hole burning.

For FM spectroscopy, we find that spontaneous emissioncan produce an AM signal detectable in a square-law detector.This effect places a quantum limit on FM spectroscopy whileat the same time providing an additional test of our theory.We are currently studying this quantum limit in detail in-cluding effects of propagation. Preliminary investigationsalso show that the height and the width of the coherent dip"found using AM spectroscopy18 can be significantly alteredby spontaneous emission.

ACKNOWLEDGMENTS

It is a pleasure to thank Matthew Derstine and DarwinSanders for their indispensable assistance in developing theC-language software able to graph all equations in this paperon an IBM-PC. You might say that we use C to investigateC!

The work of David A. Holm and Murray Sargent III was

supported in part by the U.S. Office of Naval Research undercontract N00014-81-K-0754.

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