Physics Letters A 307 (2003) 8–12

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Modifying spontaneous emission via interferences from incoherentpump fields

David Bullock, Jörg Evers, Christoph H. Keitel∗

Theoretische Quantendynamik, Fakultät für Mathematik und Physik, University of Freiburg, Hermann-Herder Str. 3, 79104 Freiburg, Germany

Received 10 July 2002; received in revised form 9 September 2002; accepted 13 September 2002

Communicated by P.R. Holland

Abstract

The spontaneous emission spectrum is evaluated from the doublet of a 4-level atomic system. Excitation from the ground stateof the closed system is supplied solely by two incoherent pump fields with one involving the doublet states; i.e., no laser fieldsare involved at any stage of the problem. We find that the interference which arises due to the incoherent pump field inducessubstantial modifications of the spontaneous emission spectrum. In particular, the occurrence of a narrow spectral feature isshown to be quite sensitive to the incoherent pump-induced interference. 2002 Elsevier Science B.V. All rights reserved.

PACS: 42.50.Lc; 42.50.Ct; 42.50.Hz

Laser field induced quantum interferences havebeen proven to be seminal in controlling the op-tical properties of an atomic medium. Both stimu-lated processes such as light absorption and refrac-tion [1] and also spontaneous processes were shownto be altered almost at will due to laser-induced, i.e.,stimulated extra indistinguishable quantum mechani-cal pathways [2–4]. In addition incoherent processessuch as spontaneous emission [3,4] and incoherentpumping [5] turned out to induce interferences in par-ticular circumstances and thus to modify stimulatedprocesses like absorption and refraction. What thusremains to be shown is that incoherently induced in-terferences are in the position to modify spontaneous

* Corresponding author.E-mail address: keitel@uni-freiburg.de (C.H. Keitel).

processes as well. In this Letter we consider sponta-neous emission from doublet states of an incoherentlypumped 4-level system and point out that interferencesdue to the incoherent field may indeed notably modifythe spontaneous emission spectrum without any laserfield. Special emphasis will be placed on the role of theincoherently induced interferences on narrow spectralfeatures.

The 4-level system of interest involves aΛ config-uration driven by a linearly polarised incoherent laserfield, with a spectral width which may be greater thanthe separation of the two closely lying doublet states.The level configuration and notation, including that ofthe various spontaneous emission and pumping ratesmay be viewed from Fig. 1.

The spontaneous emission spectrum of the doubletstates shall be calculated in dependence in particularof the involved pumping processes (we have assumed

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D. Bullock et al. / Physics Letters A 307 (2003) 8–12 9

Fig. 1. The 4-level system of interest with the corresponding levelnotation and showing all employed decay ratesγ1 and γ2 andindirect pumpr and direct pump processesr1 andr2. Emphasis isplaced here on the role of the interferences arising from direct pumpprocessesr1 andr2 on the spontaneous emission from the doubletwith level spacing 2δ.

the decay rates of level|c〉 to be negligible forsimplicity). The indirect pump processr via someunspecified auxiliary state is employed to supplysteady state population in the excited states. The directpumping however is known to induce interferences[5]. Following [3–6], the equations of motion for therelevant matrix elements of the density operatorρ canbe cast in the form:

ρaa = −(γ1 + r1)ρaa + r1ρcc

− 1

2

(Ks

√γ1γ2 + Kp

√r1r2

)(ρa′a + ρaa′),

ρa′a′ = −(γ2 + r2)ρa′a′ + r2ρcc

− 1

2

(Ks

√γ1γ2 + Kp

√r1r2

)(ρa′a + ρaa′),

ρbb = −rρbb + γ1ρaa + γ2ρa′a′

+ Ks√

γ1γ2 (ρa′a + ρaa′),

ρcc = −(r1 + r2)ρcc + r1ρaa + r2ρa′a′ + rρbb

+ Kp

√r1r2 (ρa′a + ρaa′),

ρab = −(

iωab + 1

2(γ1 + r1 + r)

)ρab

− 1

2

(Ks

√γ1γ2 + Kp

√r1r2

)ρa′b,

ρa′b = −(

iωa′b + 1

2(γ2 + r2 + r)

)ρa′b

− 1

2

(Ks

√γ1γ2 + Kp

√r1r2

)ρab,

ρa′a = −(

iωa′a + 1

2(γ1 + γ2 + r1 + r2)

)ρa′a

− 1

2

(Ks

√γ1γ2 + Kp

√r1r2

)(ρaa + ρa′a′)

(1)+ Kp

√r1r2 ρcc.

Kp and Ks are measures of the interferences be-tween the two direct incoherent pump processes andthe two processes of spontaneous emission, respec-tively. In both casesKp/s = 0 corresponds to no in-terference whileKp/s = 1 means maximum interfer-ence. These parameters were introduced to comparethe situations with and without interference. Usingρaa + ρa′a′ + ρbb + ρcc = 1 for a closed system, theequations of motion can be transformed into matrixform: ρ = B ρ + A. Here ρ = 〈σ 〉 with 〈 〉 denot-ing the quantum mechanical expectation value andσ = (|a〉〈a|, |a′〉〈a′|, |b〉〈b|, |a〉〈b|, |b〉〈a|, |a′〉〈b|,|b〉〈a′|, |a′〉〈a|, |a〉〈a′|)T. The precise form of the timedependent matrixB and of the constant vectorAfollows from Eqs. (1). In order to obtain the spon-taneous emission spectrum, we need calculate theFourier transform of the two-time correlation func-tion, 〈D(+)(t)D(−)(t ′)〉 whereD(+)(t) = µabσ4(t) +µa′bσ6(t) and D(−)(t) = (D(+)(t))†. For the dipolemoments on the transitionsa ↔ b anda′ ↔ b we have|µab|2 ∝ γ1, |µa′b|2 ∝ γ2 and µabµa′b ∝ Ks

√γ1γ2

andσi is theith elements ofσ for i ∈ {1, . . . ,9}. Thuswe obtain

S = Re(K5

(γ1 R4 + Ks

√γ1γ2 R6

)(2)+ K7

(γ2 R6 + Ks

√γ1γ2 R4

)),

where Re denotes the real part. HereK = (iωI− B)−1

with Ki being itsith row. Furthermore, we employedRi = 〈δσ (0)δσi(0)〉ss with σi = 〈σi〉 + δσi for i ∈{1, . . . ,9}, which can be transformed to readRi =〈σσi〉ss − 〈σ 〉ss〈σi〉ss. The superindices “ss” refer tothe steady state values of the corresponding variable.By integration we then obtain the total intensity

(3)

Itotal = π(γ1ρ

ssaa + γ2ρ

ssa′a′ + Ks

√γ1γ2

(ρss

a′a + ρssaa′

)).

In what follows we evaluate and discuss the spectrumS as a function of the frequencyω − ω0, whereω0 isdefined to be the energy gap between level|b〉 and themiddle between the states|a〉 and|a′〉.

The general features of a typical spectrum in Fig. 2are as expected. There are two Lorentzian peaks

10 D. Bullock et al. / Physics Letters A 307 (2003) 8–12

Fig. 2. Inelastic spectral intensityS via Eq. (2) as a functionof frequencyω. For all four linesγ1 = γ2 = 1, r1 = r2 = 0.2,r = 0.01 andδ = 0.7. The dotted and dot-dashed lines denote thesituation with no interferences due to spontaneous emissionKs = 0,with rather small dependence onKp (dotted:Kp = 1, dot-dashed:Kp = 0). For the dashed line:Ks = Kp = 1 and for the solid line:Ks = 1,Kp = 0. The vertical grey solid lines indicate the frequencyof the transitions|a〉 ↔ |b〉 and|a′〉 ↔ |b〉.

centred on the transition frequencies from levelsa anda′ to the ground levelb, because we are calculatingthe light emitted due to the spontaneous emission fromthe doublet. When the levels are brought closer (2δ <

γ1, γ2), the peaks are found to merge into one.In Fig. 2 the role of interferences due to incoherent

pumping is found to be small without the interferencevia spontaneous emission; i.e., forKs = 0 settingKp

to 0 or 1 is found to make little difference on the scaleof the figure. Thus we find for these parameters thatpump interference terms only show a significant effectwhen we also have decay interference. Mathematicallythis can be associated with the fact that many of theinterference terms inS in Eq. (2) appear as a functionof the product ofKs and Kp . In the case of allinterference terms being present(Ks = 1, Kp = 1),we find destructive interference, predominantly atthe centre, whereas in the case of there only beingdecay interference we have higher peaks and no largetrough in the middle of the spectrum. When wecompare the maxima of the peaks to the transitionfrequencies shown in the figure, we see that onlythe plot with just the decay interference terms hasits maxima off these frequencies, actually nearer thecentre. Thus, there is clear evidence that interferencedue to incoherent pumping substantially changes thespontaneous emission spectrum.

Fig. 3, involving smaller values of the pump rates,displays the effect of the pump root terms, i.e., the

pump interference terms, more clearly. Only the caseof Ks = 1 has been drawn to accentuate the differencebetween the two plots, showing the effect of the pumproot terms. As in Fig. 2, the solid graph has peaksnearer the centre than the plot with both root terms. Wesee in the solid graph that when varying the doubletseparation, at some point we observe a high peakat the centre of the spectrum, presumably an effectof constructive interference between the decay paths,namely, spectral narrowing [7–10]. We could expectthis feature to be seen somewhere in the spectra, but itis interesting that when the pump interference termsare included (dashed plot), the narrowing effect islost. At the extreme cases of large and small doubletseparation we see that the interference effects play alessening role. The plots merge into each other.

Finally, in Fig. 4 we note that interference effectscan also be seen for larger values of the pump rates.All four possibilities of including or excluding the in-terference terms of the pump and decay processes areshown and are well distinguishable. There are no spec-tral narrowing effects here to be seen. The only largevariation is the height of the peaks and the minimumat the centre of the spectrum. In both diagrams thedot-dashed plot, corresponding to neither root termsbeing included, is highest in the central region andthe dashed plot, corresponding to all the terms be-ing present, has both the lowest peaks and minimum.Thus the action of the terms in this example is destruc-tive interference. Fig. 4(b) shows the spectrum for theasymmetric case of one of the pump rates being larger.Noticeable is that the plot of no interference terms iseven more clearly above the others. The main result ofthis figure though is, that interference due to incoher-ent pumping substantially changes spontaneous emis-sion when there is no interference due to spontaneousemission.

A qualitative understanding of pump interferencesand coherences arises from the fact that the width ofthe pump field covers both transitions from the aux-iliary level c to the doublet states. This way we ob-tain a cross coupling of the coherences to the groundstate and a substantial coherence between the doubletstates themselves. The pump interference terms thusmodify the equations of motion similarly to the wellstudied decay interference terms. As a consequencethe pump interference terms modify the doublet popu-lations and all coherences involving the doublet states

D. Bullock et al. / Physics Letters A 307 (2003) 8–12 11

Fig. 3. Inelastic spectral intensitiesS via Eq. (2) for varyingδ. For both graphs in all figures,γ1 = γ2 = 1, r1 = r2 = 0.01, r = 0.1 andKs = 1.In (a) δ = 1.5, (b) δ = 0.7, (c) δ = 0.1, (d)δ = 0.03. The dashed line considers the situationKp = 1 and the solid lineKp = 0, i.e., we note thesignificance of interferences via incoherent pumping.

Fig. 4. Inelastic spectral intensityS via Eq. (2) for large pump rates. For all lines in both figures,γ1 = γ2 = δ = r = 1 and furthermore in(a) r1 = r2 = 1 and in (b)r1 = 1, r2 = 3. The solid line corresponds toKs = 1, Kp = 0, the dashed toKs = Kp = 1, the dotted toKs = 0,Kp = 1 and the dot-dashed toKs = Kp = 0. Interference due to incoherent pumping is significant even without the presence of interferencevia spontaneous emission.

and thus also the spontaneous emission spectrum. Themost sensitive area in the spectra with respect to pumpinterferences is as usual the central frequency betweenthe transitions from each of the doublet states. We notefurther that quantitative interpretations are demandingdue to the complexity of the system and the equation

for the spectrum in Eq. (2). We find in this equationthat two of the four terms fully disappear without de-cay interference and decay and pump interferences of-ten appear in sums in the equations of motion (1).Since the various matrix elements in the dynamicalequations are coupled we expect the spectral response

12 D. Bullock et al. / Physics Letters A 307 (2003) 8–12

to involve higher order mixed expressions of decayand pump interference terms. Thus it is understand-able that pump interferences may strongly depend onthe existence of decay interferences such as in Fig. 2.

Concluding, interferences via fully incoherentpumping processes have been proven to have a sub-stantial effect on the spontaneous emission spectrum,even though to a somewhat lesser extent than via co-herent fields. In particular, narrow spectral features inthe area half way between two transitions turned outto be remarkably sensitive to this incoherently inducedinterference.

Acknowledgements

Funding by Erasmus Exchange Program and Deut-sche Forschungsgemeinschaft (Nachwuchsgruppewithin SFB 276) is gratefully acknowledged. We thankMartin Haas for valuable advice.

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