Quantum theory of multiwave mixing. VI. Effects ofquantum noise on modulation spectroscopy
David A. Holm and Murray Sargent III
Optical Sciences Center, University of Arizona, Tucson, Arizona 85721
Received September 24, 1985; accepted January 2, 1986
Treating the weak-sideband fields quantum mechanically, we study the effects of quantum noise on amplitude-modulation and frequency-modulation spectroscopy of two-level media. Our analysis demonstrates that quantumnoise affects not only the intensities of the modulation sidebands but also their relative phase, which leads inparticular to a quantum limit for frequency-modulation spectroscopy.
INTRODUCTION
Modulation spectroscopy"4 in atomic and molecular mediais a branch of saturation spectroscopy in which a stronglaser-pump field is passed through an optical modulator,acquiring two sideband fields placed symmetrically in fre-quency about the laser frequency. Like the usual pump andprobe configurations of saturation spectroscopy,5 6 modula-tion spectroscopy can be used to measure the relaxationproperties of the medium, and it possesses certain advan-tages over two-wave saturation spectroscopy because only asingle laser source is required. The experimental configura-tions for modulation and two-field saturation spectroscopyare illustrated in Fig. 1. An important difference of modula-tion spectroscopy arises because the relative phase betweenthe sideband fields is an important quantity, playing a sig-nificant role in the wave-medium interaction. It is oftensufficient to measure only this phase modulation to findimportant features of the dynamics of the nonlinear medi-um.
Two special kinds of modulation are called amplitudemodulation and frequency modulation in analogy with radioterminology. Amplitude modulation implies that both side-bands have the same amplitude and phase, whereas frequen-cy modulation means that they have the same amplitude butare 180° out of phase. A semiclassical prediction of modula-tion spectroscopy is that for central tuning of the strongmode with respect to the atomic resonance, an amplitudemodulation or frequency modulation propagates through amedium unchanged, i.e., these modulations are eigenvectors.Frequency-modulation spectroscopy is based on null beat-note detection by square-law detectors and hence can besensitive to small effects that change the relative sidebandamplitudes or phase. It is currently of particular interestbecause of its potential application to optical memories.2
By passing a frequency-modulated wave through a mediumand measuring the output with a square-law detector, afrequency pattern in memory may be encoded and decoded.The production of an amplitude-modulation component inthe output field by spontaneous emission could detract fromthe accuracy of such a readout mechanism. Since large-scale memories require low power per bit to keep from over-heating, it is tempting to operate with sufficiently small
sideband amplitudes so that effects of spontaneous emissionmight become a limiting factor.
The quantum theory of multiwave mixing presented inthe previous papers 7-10 of this series can be applied to modu-lation spectroscopy. This theory shows how one strong clas-sical wave and one or two weak quantum-mechanical side-band waves interact in a two-level medium. Since we treatthe sideband fields quantum mechanically, we are able tostudy the effects of quantum noise on the fields and on theirmodulation. In the following sections we review the resultsof Refs. 7 and 8 and then derive the quantum counterpart tothe semiclassical coupled-mode equations. We demon-strate how spontaneous emission can alter the sideband am-plification obtained in amplitude-modulation spectroscopyand can yield a strongly spectrally dependent amplitude-modulation component in the transmitted field for an inci-dent frequency-modulation wave propagating through aspectrally symmetric medium.
SUMMARY OF BASIC EQUATIONS
In this section we summarize the theory developed in Refs. 7and 8 that forms the basis for this paper. Our Hamiltonian(in radians/second) is
3
H = (- 2)Z + E [( - V2)ajta; + (gajUjat + adjoint)].j=1
(1)
In this expression aj is the annihilation operator for the jthfield mode, U = Uj(r) is the corresponding spatial modefactor, o* and orz are the atomic spin-flip and probability-difference operators, o and vj are the atomic and field fre-quencies, and g is the atom-field coupling constant. Wetake mode 2 to be arbitrarily intense and treat it classicallyand as undepleted. Modes 1 and 3 are quantum fields treat-ed only to second order in amplitude and cannot by them-selves saturate the atomic response. This is an importantassumption and limits the applicability of the theory. Therotating-wave approximation has been made, and the Ham-iltonian is in an interaction picture rotating at the strong-field frequency 2. We define an atom-field density opera-
Fig. 1. (a) Two-field configuration for saturation spectroscopy.(b) Optical modulator puts frequency sidebands on a saturatingpump field for modulation spectroscopy.
tor Pa-f and obtain its time dependence from the standarddensity operator equation of motion
Pa-f = -i[H, Pa-f] + relaxation processes. (2)
In the present paper, we assume that the only relaxationprocesses are upper-to-lower level decay described by thedecay constant F (= 1/T 1 ) and the dipole decay described byy (=l/T 2 ). This is the usual experimental situation in laserspectroscopy. For pure spontaneous decay, y is equal toF/2. We calculate the reduced electric-field density opera-tor p that describes the time dependence of the two quan-tized fields by taking the trace of Pa-f over the atomic states.We assume all field amplitudes to vary little during atomicdecay times. This allows us to solve the atomic equations ofmotion in steady state and then to obtain the slowly varyingfield density operator equation of motion:
+ (same with 1 interchanged with 3) + adjoint, (3)
where Q, is the cavity quality for mode n and the coefficientsAl, B1, C1, and D1 are given by
A = 1 2 ) 2 21 + 2-C2 |2
I2 2Sg 2 _ - 02*(1 + r/iA)/2]_ 2 - , (4)
1 + I2gY7(YO1 + 0J3 )2
B - 1+ 1212 I 1 22
I2.-5r[(1 + I212/2)O1 + D2*(1 + r/iA)/21
2 +I~y2
C, = 92y), Ul*U 3 *1 +212
2T1V 22y{ 22 3* - 2*(1 + r/iA)/2]
X
1 + I25I-(0O1 + 1O3)2
(6)D - 1212 *X 2TV 2
25r[(l + I2.L2/2)3* + 02*(1 + /iA)/2]
1 + I2-Y(SO1 + D3)2
(7)
where, following the notation of Ref. 7, the complex Lorentz-ian denominators 0J, are given by
0 = /[y + i(w -vn)
the dimensionless Lorentzian 12 is
L2 = Y2/[2 + ( -v2)2],
the dimensionless intensity I2 is
I2 = 412 12TT 2 ,
and the dimensionless population pulsation term D is
= r/(r + iA),
and where cV2 = gU2 (n2 + 1)1/2, N is the total number ofinteracting atoms, and A = V2- i is the beat frequencybetween modes 1 and 2.
The equation of motion (3) for the field operator p and theexpressions for the coefficients Al through D1 are the funda-mental equations of our theory. As shown in Refs. 7 and 8,combinations of these coefficients yield physically meaning-ful quantities, some of which were derived previously byother methods. For example, Al + c.c. is the spectrum ofresonance fluorescence; B1 - Al is al, the semiclassical com-plex-absorption coefficient for a weak probe field in thepresence of a saturating field; and C1 - D1 is the semiclassi-cal complex-coupling coefficient -iK1* between the signaland conjugate fields in phase conjugation.
In Ref. 10 we showed how three-wave mixing in a cavityalters the spontaneous emission spectrum. In particular,the role of the C = C + C1* coefficient in reducing thecentral emission peak was pointed out. In this paper weshow how C enters into the propagation problem of modula-tion spectroscopy and how the effects of quantum noise areclosely connected to the C coefficient.
THE EQUATIONS OF MOTION
We may use Eq. (3) to obtain the equation of motion for anyoperator. This was also done in Refs. 7 and 10. For exam-ple, the annihilation operator (a1 ) corresponds to the classi-cal Fourier amplitude for the electric field of mode 1, 1, andits equation of motion is given by
E, Y2 I
-
D. A. Holm and M. Sargent III
734 J. Opt. Soc. Am. B/Vol. 3, No. 5/May 1986
d (a,) = d = (nlalln),.
= (Al - B, - v/2Q,)&, + (Cl - Dl)',3 (8)
where 63* = (a3t). Equation (8) is the semiclassical cou-pled-mode equation of motion for the field amplitude 61 andwould be valid inside a laser cavity. In modulation spectros-copy steady-state propagation of the fields is generally as-sumed and temporal variations are neglected. To treat thiscase the time derivative a/at is converted into the spatialderivative ca/az. This is valid classically from Maxwell'sequations and has been used in quantum-mechanical treat-ments of the fields by Yuen and Shapiro," Reid and Walls,' 2
and Levenson et al.3 We also make this assumption andrefer the reader to those papers for further discussion.
The spectral intensity of mode 1 is given by 61*61, whichin our quantum calculation is (altal). We denote this quan-tity by (nl) and the corresponding intensity for mode 3 by(n3). At the end of this paper we show how a real intensity,in watts per square centimeter, for instance, can easily berelated to these dimensionless intensities. Using Eq. (3), wefind the propagation equation for (n,) and (n3) to be
d (nl) = (Al + Al* - B, - B*)(nl) + Al + Al*
+ (C, - Dl)(a3talt) + (Cl* - D,*)(a3a,), (9)
dzdz(n,) = (A3 + A* -B - B,*)(n,) + A, + A,*
+ (C3 - D)(a3tat) + (C* - D3*)(a3a,), (10)
where the cavity loss terms v/Q have been dropped. Theterms (a3a,) and their complex conjugate (a3talt) in Eqs. (9)and (10) correspond to the semiclassical quantities 636, and63*61*. We obtain the equation for (a3al) in the samemanner and
d (aal) = (Al + A3 - B, - B)(a3a,) + C, + C3
+ (Cl - D,)(n,) + (C3 - D3)(nl). (11)
We call Eqs. (9)-(11) the quantum coupled-mode equations.To include the effects of spontaneous emission, these equa-tions contain source terms such as the resonance fluores-cence coefficient Al + Al* and C + C. Without thesesource terms, these equations follow from the correspondingsemiclassical coupled-mode equations
d6,dz-- 1 3 -
d_ = 3*63 + iK36,.dz
We see that the quantum coupled-mode equations recoverall the terms from the semiclassical theory. As stated earli-er, Al + Al* is the resonance fluorescence spectrum for mode1 and A3 + A3* is the fluorescence spectrum for mode 3.Both of these terms appear as source terms in the aboveequations of motion, which is quite reasonable. As the fieldspropagate through the medium, spontaneous emissions addto the intensity. This part of the quantum equations couldbe argued heuristically. What is not so obvious is that thecorresponding source for (a3al) is C, + C3. The quantity(a3al) contains the relative phase between the three modesand is analogous to Lamb's' 4 combination tone. Just as Al+ Al* is a quantum source for the intensity (nl), C, + C3 is aquantum source term for the coherence (a3al) and, by ex-tension, for the field modulation. All these quantitiesstrongly depend on frequency, as is well known for the reso-nance fluorescence spectrum. Figure 2 shows C, + C,* = Cversus AT2 for central tuning of the stong mode and theparameters I2= 50, T, = 2T2. For this large pump intensity,the C coefficient is similar to the resonance fluorescencecoefficient Al + Al* with a negative central peak. Thus thepredicted quantum effects depend on the frequencies of theside modes.
Equations (9)-(11) plus the complex conjugate of Eq. (11)are four coupled, linear differential equations. Becausethere are four equations instead of the semiclassical two,their solution is more difficult. In the present paper weassume central tuning of the strong mode, as in Ref. 10.This significantly simplifies the equations yet still allows fora variety of possible situations. When v2 = w, we have A3 =
Al*, C3 = Cl*, etc., so Eqs. (9)-(11) become
d (n1) = -a(nl) - iKl* (a3tait) + iK(a3a,) + A, (14)dz
d (n3) = -a(n3) + iK1 (atalt) - iK,*(a3a,) + A, (15)dz
where a = B + B*-Al + Al*, A = Al + Al*,-iK,* = C -Dl, and C = C, + C,*. Note that a, A, and C are real but Kl is
0. 61
C
(12)
(13)
Specifically, multiplying Eq. (12) by * and adding its com-plex conjugate, we obtain Eq. (9), provided that we identify(n,) with 61*, (a3al) with 6361, a withB, - A, and -iK,*with C, - D, and that we set the source term Al + Al* equalto zero. In a similar manner, we may relate Eq. (13) to Eq.(10) and Eq. (12) to Eq. (11). Alternatively, we could con-sider Eqs. (12) and (13) operator equations for the annihila-tion operator a and the creation operator at, respectively,as done by Reid and Walls.' 2
0.
-0.
Fig. 2. C = C + C versus AT2 for 2 = 50, T2 = 2T1 .
D. A. Holm and M. Sargent III
Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. B 735
complex. We solve these equations by using the Laplacetransform. Let [(nl)] denote the Laplace transform of (n1 ),[(n3 )] that of (n3 ), and [a3ai] that of (a3a,). Taking thetransform of Eq. (16) and solving for [a3a,], we find that
(a 3al) 0 - iKl*[(n3)] + iKj[(nl)] + C/s[a3a1] =(17)+ a
where (a 3al)o is the value of (a 3al) at z = 0 and s is thetransform parameter. We substitute Eq. (17) into thetransforms of Eqs. (14) and (15), recalling that (a3 talt) isthe complex conjugate of (a3a,). This gives two equationsfor the unknowns [(nl)] and [(n3 )]. We write the solutionfor these as
and where q = s + a. The expression for [(n3)] can beobtained by interchanging 1 and 3. Equations (6) and (7)for v2 = w show that this implies that Ki and K1* should also beswitched by this interchange.
To obtain (n1) as a function of z, we calculate the inverseof Eq. (18). Equation (18) is sufficiently complicated thatexpressions for the inverse Laplace transform are difficult toobtain from a table. Therefore we calculate the inversefrom the definition of the Laplace transform:
1 i+of+(z) * FL(s)eszds,
where f+(z) is f(z) for z > 0 and where To is large enough toinclude all the poles of FL(S). The integral is most easilyperformed using the calculus of residues in the complex splane. The denominator term of Eq. (19) can be factored toobtain
d=(s+a-K)(s+a+ K)(S+a-K)(S+a+K'), (20)
where K and K are twice the real and imaginary parts of K1.
All the residues of the integrand are first-order poles, andthe evaluation of the inverse transform is straightforward.For example, for the (a3 ai)o term in Eq. (18) we have
(a3aR)sKResls +a =-K] +Res(s +aK)= *2
(e-KZ eKZ)
= -32 -a sinh KZ.
(21)
Repeating this procedure for the other terms in Eq. (18), wefinally obtain the solution for (nl):
Interchanging the subscripts 1 and 3 yields the solution for(n3):
(n3) = [(n)o + (n3) 2 Aa -C h()
((a3al)o + (a3talt)o + AK -Ca)sinh(Kz)
- (nl)o - (n3)0) (,)-V 2 CSKZ
((aa)o -(a3 talt)o
2)sin(Kz)]e-az + Aa-CK
(23)
We determine the solution to Eq. (16) for (a3al) by substi-tuting Eqs. (22) and (23) into Eq. (16). The equation canthen be readily integrated and
(a3al) - (a3 ab)o + (a 3talt)o AK-Ca cosh(KZ)
- ( ( ' 1 ) 0 2 2- A2 c K ) sh()
/(nl)o + (n30 AC- CK\ .ih(2 ae2 K2 Sh(Z
((ala)o - (a3talt)o)+ 2(~a) (3a )COS(K'Z)
-(nl)o-(n3)0 1 AK-CCa+ 2 )sin(K'z) eaz -
2 K2
(24)
Equations (22)-(24) satisfy the original Eqs. (14)-(16) aswell as the appropriate boundary conditions. The termscontaining the A and C coefficients represent the effects ofquantum noise. When these are small compared with (n1 )oor (n3)0, they reproduce the semiclassical solutions. It ap-pears that when a = +K the sum of these quantum termsdiverges because of their common denominator a2 - K2.
However, the sum of the corresponding numerators alsovanishes in this limit, as we can show by using l'Hospital'srule. In fact, differentiating the numerator and denomina-tor with respect to K or -K, we find for a = ±K that
)sin(Kz)]e-z + Aa-CK
(22)
D. A. Holm and M. Sargent III
736 J. Opt. Soc. Am. B/Vol. 3, No. 5/May 1986
(n) = [((nl)o + (n3)0 -F-)cosh(z)
- (a3 a)o + (a 3ta1 t)o A .sinh(oz)
+ ((fl1 2 2a ) c
+ (nl)o -(n3)0\ CO(K2 CSKz
+ ((a3al)o -(atalt)o ) * , 12 )mKz2-
+ az(A :F C) : C (25)2a
(n 3 ) is given by the same expression with the subscripts 1and 3 interchanged, and (a3al) is given by a similar expres-sion.
It is well known that it is possible for the semiclassicalprobe absorption coefficient a to become negative, givinggain instead of attenuation in the e-az term. Therefore Eqs.(22)-(25) can, for some frequencies, increase without boundas z increases. Eventually, of course, side-mode saturationwould prevent this, but this model does not consider this.Except for large-input photon numbers, the above equationsare valid below the gain threshold and give the variation ofthe spectrum and modulation with distance in that case.
ANALYSIS OF SOLUTIONS FOR WHICH (nl)aAND/OR (n3 )0 VANISH
We 'now study Eqs. (22)-(25) in greater detail. We mustnormalize our parameters to scaled units. As in Ref. 10, wedefine a0 to be the centrally tuned, weak-field absorptioncoefficient for I2 = 0. In terms of the wave vector K of thefield, the number density N of the interacting atoms, and thedipole moment s, ao is given by
a0 = K A2 N/hecoy. (26)
We express z, the propagation distance, in units of ao- 1, andthe quantities A, C, a, K, and K' in units of a 0.
Although we are concerned in this paper primarily withquantum effects on modulation spectroscopy in which twononzero input fields are directed into the medium, the solu-tions given by Eqs. (22)-(25) are also valid for the situationsof no sideband input fields or just one. In this section weconsider these cases.
In semiclassical theory there is no emission from the vacu-um, and so if (n1)o and (n3 )0 are zero, (n 1) and (n 3) alwaysremain zero. For the quantum-mechanical solutions we seefrom Eqs. (22)-(25) that this is not the case. The sourceterms A and C cause (n1 ) and (n 3 ) to increase, even if bothare zero initially. In this case the side modes grow fromspontaneous emission and undergo three-wave mixing in themedium. Figure 3 plots the spectrum of mode 1 for threepropagation distances into the medium, 20a0-1 , 40a 0-1, and50ao-1 . A strong pump field of I2 = 50 and pure radiativedecay are assumed. We see from Fig. 3 that for short dis-tances the spectrum resembles the resonance fluorescencespectrum, although the sidebands are somewhat higher rela-
tive to the central peak. For longer distances the sidebandssharpen and increase, whereas the increase of the centralpeak tends to saturate. This agrees with the results of Ref.10 for three-wave mixing in a cavity. For even longer dis-tances than those shown in Fig. 3, the sidebands continue toincrease, giving a two-peaked spectrum. However, satura-tion of the side-mode emission enters, and our theory cannotbe considered valid at this point.
In many media of interest, collisional dephasing of thecoherence between the atomic levels is important, and thiscauses T1 to be much different from T2, i.e., T 1 >> T2. Figure4 repeats the plot of Fig. 3 with the same parameters, exceptthat now T2 =0.01T1 . In contrast to Fig. 3, we note that thesidebands have moved in considerably. This result can bereadily explained by the fact that in this case the Rabifrequency is much smaller, i.e., equal to 1/A/2T2 for I2 = 50.
We consider next the case for which (n1 )o is nonzero but(n3 )0 is zero. In this case the semiclassical theory predicts,because of three-wave mixing, a nonzero (n 3)(z) as well.
3
A
V
Fig. 3. (n1 ) versus AT2 for the propagation distances z = 20ao-1,40a 0-1 , and 50ao-1, for (nl)o = (n 3 )0 = 0, I2 = 50, and T2 = 2T1.
30-
A
V
Fig. 4. Same as Fig. 3, except that T2 = 0.01T 1 .
D. A. Holm and M. Sargent III
Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. B 737
3
A
cv
20AT2
Fig. 5. (n,) versus AT2 from the quantum (Q) solution, Eq. (21),and the corresponding semiclassical (S) solution. (n,)0 = 1, (n3)0 =0, z = 50ao-', I2 = 50, and T2 = 2T,.
Fig. 6. (n,) versus AT2 from the quantum (Q) and semiclassical (S)solutions for (nl)o = 20. Other parameters are the same as those inFig. 5.
Figure 5 plots the spectrum (nl) for the quantum calcula-tion of Eq. (22) and the corresponding semiclassical resultfor an initial photon number (nl)o = 1 and a propagationdistance 50ao-1. We note the similarity of the quantumcurve to the 50ao01 curve of Fig. 3. This assumes that wetake the same (nl)o for all A. In the next section we trans-form these scaled units into observable quantities, but forthe time being we note that, for the units that we are nowusing, terms such as A/a have roughly a maximum value of10 (recall that a B - A and is the difference between twonearly equal quantities). Thus the quantum terms in Eq.(22) dominate the spectrum, and we see it is quite differentfrom the semiclassical. If we allow (n,)o to increase so thatit is slightly larger than the quantum terms, the differencebetween the quantum and semiclassical curves decreases.This is shown in Fig. 6. We see that, even for (nl)o = 20photons, which is not much larger than the quantum terms,
the semiclassical theory provides an excellent representa-tion of the spectrum. The largest discrepancy occurs justwithin the Rabi sidebands. As it must, the semiclassicalcurve has the same shape as the double-side-mode gain/absorption coefficient. The quantum curve lacks some ofthe semiclassical features, but it is not too different. Forstill larger initial fields, the difference narrows even more, asexpected.
MODULATION SPECTROSCOPY
We now consider nonzero values of both (nl)o and (n3 )0 andhence of (a3a,)O. This is the case of primary interest in thispaper, namely, that of modulation spectroscopy. It is possi-ble to calculate the propagation spectrum for an arbitraryinitial phase difference between the side modes. As wementioned in the Introduction, the special cases of primaryinterest are amplitude modulation and frequency modula-tion, and we restrict our discussion to these. From semiclas-sical theory,1"15 in amplitude modulation the two side modesare in phase and they tend to work together. In particular,for central saturator wave tuning ( 2 = W), the populationpulsations induced by the saturator with each side modeadd, effectively doubling the coherent effects. For example,this leads to side-mode gain for substantially smaller satura-tor intensities. For a population difference lifetime T, >>than the dipole lifetime T2, amplitude-modulation spectros-copy provides a sensitive way"15 of measuring T1, as demon-strated by Keilmann16 and Hillman et al.
6 The T T2case is important in the theory of laser and optical bistabilityinstabilities.'7
In contrast, with frequency modulation the two sidemodes oscillate 1800 out of phase and thereby induce out-of-phase population pulsations that cancel out, yielding nocoherent effects. In essence, the nonlinear atomic mediumacts much like a square-law detector, incapable of respond-ing to frequency-modulation beat-frequency contributions.What remains is the incoherent part of the absorption,which has a standard Lorentzian shape. Semiclassically,side-mode gain cannot occur for frequency modulation whenthe central mode is tuned to line center and the medium hasa symmetric absorption spectrum.
Consider first the frequency-modulation case. Figure 7again plots the spectrum of (nl) from Eq. (22), but now with(ni)o = (n3 )0 = 40 photons and (a3al)o = -40, correspond-ing to a relative phase NI of 180°. Note the dramatic differ-ence between the quantum and semiclassical curves, eventhough the initial photon numbers are twice those of Fig. 6.The semiclassical curve has the usual Lorentzian shape, and(nl)o is always below the initial value of 40 photons becauseof incoherent absorption. The quantum curve, in contrast,shows strong gain just within the Rabi sidebands and has aless extreme dip for AT2 = 0. Although the relative propor-tions of the approximate Lorentzian features are different, ithas a similar shape to that of the C coefficient of Fig. 2. Thequantum curve, even for initial frequency modulation, clear-ly reveals the presence of coherent effects. Thus the insen-sitivity of frequency modulation predicted semiclassically isovercome by quantum multiwave mixing processes.
The above results are also valid in the limit of a shortdipole lifetime. Figure 8 plots the spectrum (ni) versus AT2for the semiclassical and quantum solutions for the sameparameters as Fig. 7, except that T 2 = 0.01T,. Again we see
D. A. Holm and M. Sargent III
738 J. Opt. Soc. Am. B/Vol. 3, No. 5/May 1986
70-
A
V 0
-20 0 20AT2
Fig. 7. (nj) versus AT2 from the quantum (Q) and semiclassical (S)solutions for the frequency-modulation case of (n1 ) = (n3 )0 =
-(a 3 a,)o = 40, z = 50ao', I2 = 50, and P2 = 2T1.
7
A
CV
0 0AT2
Fig. 8. Same as Fig. 7, except that T2 = 0.01T,.
a big difference between the two. The semiclassical solutionis a usual incoherent Lorentzian with width 2/T2. The co-herent dip of amplitude-modulation spectroscopy has beencanceled by the out-of-phase side modes. The quantum6urve has the peaks inside the much-reduced Rabi frequen-cies, as in Fig. 4, and we also see that there is no absorptionfar out into the wings.
The. difference between the quantum and semiclassicalpredictions of Figs. and 8 is so large that this ought to beobservable experimentally. The measurement of gain fromfrequency modulation is unexpected semiclassically andcould confirm the theory presented here. We show in Fig. 9that significant differences exist between the quantum andsemiclassical theories for frequency modulation for evenlarger input fields. Figure 9 (nl) plots (nl) for frequencymodulation with initial photon numbers (nl)o = (n3)0 = 500,which is much larger than the quantum terms. We still haveside-mode gain, only now it is substantially smaller than inFig. 7. We see that the quantum curve approaches the
semiclassical for increasing photon numbers, as one wouldexpect.
Our graphical analysis has so far been concerned with theemerging intensities of the probes. Another possible experi-mental test would be to measure the modulation of thepropagating waves. We now briefly review the semiclassicaltheory for the experimental situation of Fig. 1(b). By defi-nition, a square-law detector measures the squared modulusof the sum of the electric fields El + E2 + E3. This is givenby Sargent et al.15 as
I = '/4ce0[E22 + 2EjE2 CoS(AZ/C + 02 -A)
+ 2E2E3 COS(AZ/C + 03-02)] (27)
where terms of order E12 have been dropped and kk is thephase of Ek. Since we assume central tuning of the E2 field,E= E3 for all z, provided thatE,(0) = E3(0). Equation (27)then reduces to
6000
S
A
CV
01 __ -20 0 20
AT2
Fig. 9. Same as Fig. 7, except that (nl)o = (n3)0 =-(ala 3 )o = 500.
10
C0
-4)am3
U). T
AT2
Fig. 10. Beat-frequency term of Eq. (28) versus AT2 .(n3 ) = -(ala 3 ) = 20, z = 50ao&', 2 = 50, and T2 = 2T,.
20n-
D. A. Holm and M. Sargent III
n
. .I
Vol. 3, No. 5/May 1986/J. Opt. Soc. Am. B 739
O. O0
0. CI/
-UJ. WJ' -5 0 5
AT2
Fig. 11. Effective double-sideband absorption coefficient a + K
versus AT2 in the short-dipole-lifetime limit T2 = 0.01T1 . 2 = 50,and w = 2.
100.
A
cv
-5 0 5AT2
Fig. 12. Quantum (Q) and semiclassical (S) transmitted intensitiesfor waves passed through a medium with the effective double-side-band absorption coefficient of Fig.11. (n1)0= (n3)0 = (a3al)o = 10,T2 = 0.01T 1, I2 = 50 and z = 50a0o 1.
where = 202 - 'k - 03 is the relative phase between thefields. The beat-frequency term is thus doubled for ampli-tude modulation ( = 0°) and vanishes for frequency modu-lation ( = 1800).
In Fig. 10 we plot the amplitude of the beat-frequencyterm of Eq. (28) for (nl)o = (n3)0 = -(ala 3 )O = 20, using ourquantum-mechanical solutions of Eqs. (22)-(25) for the Elfield and the relative phase . According to Eq. (28) thisshould be zero for all frequencies, since T = 1800. We see,however, that instead we obtain nonzero contributions for anarrow range of frequencies just within each Rabi sideband.Although one might expect spontaneous emission to inter-fere with the modulation in general, we predict that this
effect will turn out to be strongly frequency dependent.Detection of a signal at the frequencies of Fig. 10 providesanother means to test this theory.
The effects of spontaneous emission on amplitude-modu-lation spectroscopy are less dramatic since they only modifythe shape of the transmitted spectrum rather than suddenlyprovide a strongly frequency-dependent beat-frequency sig-nal. Nevertheless, for large saturation intensities we findsubstantial differences between the spectra for the quantumand semiclassical amplitude-modulation transmission in-tensities. Figure 11 shows the effective double-sidebandabsorption coefficient a + K for central saturator tuning.Figure 12 shows the corresponding quantum and semiclassi-cal transmitted intensities for waves passed through such amedium. Whereas semiclassically amplification is restrict-ed to small beat frequencies, quantum mechanically, ampli-fication occurs at all frequencies plotted. Such variationsmight be observable in precise amplitude-modulation ex-periments such as those of Hillman et al.
6
RELATIONSHIP TO OBSERVABLE QUANTITIES
In the previous section our results were scaled to the weak-field absorption coefficient a0. In this section we indicatehow these are related to measurable parameters. We wishto relate (n1) to the dimensional spectral intensity I,1 Wedo this by multiplying (n,) by hv1 to obtain an energy, thenby N, the number density of the interacting atoms, to obtainan energy density, and by c, the speed of light, to convert theenergy density into an intensity. Finally, we multiply byAQ, the solid angle subtended by the detector as viewed fromthe interaction region. Thus in the above expressions for(nl) and (n3 ) we substitute
(nl) - hvNcAQ(nl) = I. (29)
In a similar manner we note that A, C, a, K, and K all have thesame units and that in Eqs. (22)-(25) they always appear asdimensionless ratios of one another or multiplied by thepropagation distance z. In the first case we apply the sametransformation to them as to (nl) and (n3) in Eq. (29), so,for example,
Aa - CK A - CK2
K2 hvNcA/Q 2 2a - K2 a 2- K2
(30)
Expressions such as az can easily be dealt with by express-ing a in units of a 0, and z in units of ao-1 , where a0 is given byEq. (26). Thus all the results of this section may be ex-pressed in real, dimensional quantities. In Eqs. (22)-(25)we replace (nl) and (n 3 ) by I and I,3 (a 3 ai) byJlil 631ei,and the quantum terms involving A and C by the right-handside of formula (30). We saw in the previous section thatquantum effects are usually relevant only when (n1) 10or, equivalently, when (n1 ) is of the same order of magnitudeas the expression of formula (30). Whether this is thecase for a given experiment can be determined from theabove equations.
To summarize, in this paper we solve and analyze thequantum-mechanical coupled-mode equations predictedfrom the quantum theory of multiwave mixing 7-10 for severalclasses of side-mode input photon numbers. As expected,we find that quantum noise is usually relevant only for smallside-mode intensities. Three important cases influenced by
D. A. Holm and M. Sargent III
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740 J. Opt. Soc. Am. B/Vol. 3, No. 5/May 1986
quantum noise (1) amplified spontaneous emission, whichthe semiclassical theory does not consider, (2) amplitude-modulation spectroscopy, for which substantially widerside-mode gain regions can occur than those predicted semi-classically, and (3) frequency-modulation spectroscopy, inwhich, in contradiction to the semiclassical theory, ampli-tude-modulation components in the spectrum can appear atcertain frequencies near the Rabi sidebands. This thirdphenomenon places an important, potentially observable,limit on weak-sideband frequency-modulation spectrosco-py, such as might be used to read out frequency-domainoptical memories.
ACKNOWLEDGMENT
This research was supported in part by the U.S. Office ofNaval Research under contract N00014-81-K-0754.
REFERENCES
1. M. Sargent III, Phys. Rep. 43C, 223 (1978).2. G. Bjorkland, Opt. Lett. 5, 15 (1980).
3. D. E. Cooper and T. F. Gallagher, Opt. Lett. 9,451 (1984). Forapplications of frequency-modulation spectroscopic techniquesinvolving passive cavities, see R. W. P. Drewer, J. L. Hall, F. W.Kowalski, J. Hough, G. M. Ford, A. G. Manley, and H. Wood,Appl. Phys. B 31, 97 (1981).
4. M. A. Kramer, R. W. Boyd, L. W. Hillman, and C. R. Stroud, Jr.,J. Opt. Soc. Am. B 2, Sept. (1985).
5. F. Y. Wu, S. Ezekial, M. Ducloy, and B. R. Mollow, Phys. Rev.Lett. 38, 1077 (1977).
6. L. W. Hillman, R. W. Boyd, J. Kransinski, and C. R. Stroud, Jr.,Opt. Commun. 45, 416 (1983).
7. M. Sargent III, D. A. Holm, and M. S. Zubairy, Phys. Rev. A 31,3112 (1985).
8. S. Stenholm, D. A. Holm, and M. Sargent III, Phys. Rev. A 31,3124 (1985).
9. D. A. Holm, M. Sargent III, and L. M. Hoffer, Phys. Rev. A 32,963 (1985).
10. D. A. Holm, M. Sargent III, and S. Stenholm, J. Opt. Soc. Am. B2, 1456 (1985).
11. H. Yuen and J. Shapiro, Opt. Lett. 4, 334 (1979).12. M. Reid and D. F. Walls, Phys. Rev. A 31, 1622 (1985).13. M. D. Levenson, R. M. Shelby, A. Aspect, M. Reid, and D. F.
Walls, Phys. Rev. A 32, 1550 (1985).14. W. E. Lamb, Jr., Phys. Rev. 134, A1429 (1964).15. M. Sargent III, P. E. Toschek, and H. G. Danielmeyer, Appl.
Phys. 11, 55 (1976).16. F. Keilmann, Appl. Phys. 14, 29 (1977).17. For a review, see S. T. Hendow and M. Sargent III, J. Opt. Soc.
