Enhanced frequency conversion

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Optics Communications 277 (2007) 186–195

Enhanced frequency conversion of nonadiabatic pulses in a doubleK system driven by two pumps with and without carrier beams

A. Eilam, A.D. Wilson-Gordon *, H. Friedmann

Department of Chemistry, Bar-Ilan University, Ramat Gan 52900, Israel

Received 29 January 2007; received in revised form 29 March 2007; accepted 20 April 2007

Abstract

We show that nonadiabatic, resonant amplitude- and phase-modulated pulses can be frequency converted with greater efficiency thanadiabatic resonant pulses in a double K system, interacting with two strong cw beams on one side of the system, and a weak pulsed probeon the other. Indeed, in this double EIT (electromagnetically induced transparency) configuration, conversion efficiencies close to unity,similar to those achieved using highly detuned pulses, can be obtained using highly nonadiabatic resonant pulses. The distance at whichthe maximum conversion occurs is shorter than in a coherently-prepared K system. This counteracts the increased absorption that occursin the double EIT configuration, so that both produce similar conversion efficiencies. The absorption experienced by matched nonadi-abatic pulses in the double EIT system, at all propagation distances, can be overcome by superimposing the nonadiabatic pulses as ampli-tude modulations on carrier fields. Thus we demonstrate the formation of adiabatons in the double EIT system, and of diabatons in boththe coherently-prepared K system and the double EIT system. Both the diabatons and adiabatons satisfy pulse-matching conditions. Inaddition, the asymptotic amplitude of the complementary amplitude modulations is proportional to the ratio of the pump to probe car-rier Rabi frequencies, and is the same in each of the configurations.� 2007 Elsevier B.V. All rights reserved.

PACS: 42.65.Jx; 42.65.Ky

1. Introduction

Recently, we compared the frequency conversion of adi-abatic and nonadiabatic pulses in the coherently-prepareddouble K system, shown in Fig. 1a [1]. The lower K system,consisting of the states j1i, j2i, and j3i, interacts with twocw resonant fields with Rabi frequencies V31,32 that prepareit in a coherent, nonabsorbing superposition of the lowerstates [2–9]. A weak probe pulse which may be either adia-batic or nonadiabatic, with initial Rabi frequency V42(t,0),then interacts with one of the transition of the upper K sys-tem, consisting of the states j1i, j2i, and j4i, and is con-verted via four-wave mixing (FWM) to the resonancefrequency of the other transition. When the incident pulsesatisfies the adiabaticity condition [10]

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.04.052

* Corresponding author. Fax: +972 35351250.E-mail address: [email protected] (A.D. Wilson-Gordon).

j _V 42ðt; 0Þ=V 42ðt; 0Þmaxj � jD42 þ iC42j; ð1Þwhere D42 and C42 are the one-photon detuning and thetransverse decay rate for the j2i to state j4i transition, themaximum conversion that can be obtained when C41 =C42 and the two-photon detuning D21 = D41 � D42 = 0, isCmax = jV41(t 0,z)/V42(t 0, 0)jmax ’ jq21j (with t 0 = t � z/c)for a resonant pulse, and Cmax ’ 2jq21j when the pulse ishighly detuned so that D42� C42. Thus when V31 = V32,so that the two-photon coherence q21 = �1/2, the maxi-mum conversion that can be achieved is 50% for a resonantpulse and 100% for a highly detuned pulse. However, thepropagation distance at which the maximum conversion isobtained is greater for the case of a detuned pulse thanfor a resonant pulse by a factor of approximately D42/C42.We showed in [1] that when the incident pulse is nonadia-batic, much higher values of Cmax can be obtained, evenfor resonant pulses, at distances that are much shorter thanthose required for detuned adiabatic pulses.

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1

4

2

3

1

4

2

3

ω41

ω31

ω32

ω42

ω41

ω31

ω32

ω42

ω41

ω31

ω32

ω42

ω41

ω31

ω32

ω42

1

4

2

3

1

4

2

3

ω41

ω31

ω32

ω42

ω41

ω31

ω32

ω42

41ω

31ω

32ω

42

ω41

ω31

ω32

ω42

a

b

Fig. 1. Double K systems. (a) Two strong fields of frequency x31,32

interact with the lower K system, establishing two-photon coherence. Aweak field of frequency x41 is converted, via FWM, to one with frequencyx42. (b) Two strong fields of frequency x31,41 interact with one side of thedouble K system. A weak field of frequency x32 is converted, via FWM, toone with frequency x42.

A. Eilam et al. / Optics Communications 277 (2007) 186–195 187

We also extended the concept of ‘‘matched pulses’’ [7,11]to the case of nonadiabatic pulses. Specifically, we derivedthe result that a pair of pulses will propagate unchangedprovided they are initially matched according to the expres-sion V42(t 0, 0)/V41(t 0, 0) = V32/V31. In addition, if only oneof the pulses is initially present, the incident and generatedpulses will be matched asymptotically according toV42(t 0,z)/V41(t 0,z)! V32/V31.

In this paper, we study an alternative version of the dou-ble K system (for a recent review, see [12]), shown inFig. 1b, and compare the results with those for the systemof Fig. 1a. In this version, two strong cw fields interact withthe j1i ! j3i and j1i ! j4i transitions. A nonadiabaticweak probe pulse that interacts resonantly with thej2i ! j3i transition is converted via FWM to the resonancefrequency of the j2i ! j4i transition. This system has beendiscussed previously in the context of pulse matching andslow group velocity [13–15], controlled light storage [16–18], adiabatic frequency conversion [19], and phase-con-trolled light switching [20]. For convenience, we will referto the system as the double EIT (electromagneticallyinduced transparency) system since, in the adiabatic limit,the probe and converted pulses do not experience furtherabsorption once pulse-matching conditions are achieved.However, as we will show, nonadiabatic pulses experiencecontinuous absorption at all propagation lengths.

In order to overcome the problem of absorption of thenonadiabatic weak probe pulse and the pulse generated

by FWM, we consider a modified strategy. We apply theprobe and FWM pulses as complementary, weak, nonadi-abatic amplitude modulations, superimposed on constantcarrier beams. This creates a matched stable coherent con-figuration for the carrier waves in the double K configura-tions of Fig. 1. Once pulse matching of all four modulatedfields is achieved, they continue to propagate without fur-ther absorption. These nonadiabatic modulated carrierwaves are the nonadiabatic analogs of the adiabatons[15,21–23] , discussed by Cerboneschi and Arimondo [15]in the coherently-prepared double K system. We will there-fore refer to them as ‘‘diabatons’’ in this paper. The exis-tence of adiabatons was demonstrated both analytically,using an approach based on dressed states, and numerically[15]. We will show that adiabatons and diabatons exist inboth the systems shown in Fig. 1, and that almost identicalresults are obtained in each case, for the same ratio ofstrong to weak carrier Rabi frequencies. The crucial factorin determining the asymptotic amplitude of the amplitudemodulations is the relative intensity of the carrier fields.

An advantage of the double EIT system over the onediscussed previously is that the frequency of the generatedfield can differ considerably from that of the probe field forDoppler-free two-photon resonance, using copropagatingfields. As before [1], we find that high conversion can beobtained using nonadiabatic resonant amplitude- andphase-modulated pulses. Although highly detuned pulsescan be almost completely frequency converted via FWM,using either of the schemes shown in Fig. 1, the propaga-tion distance at which this occurs is greater than thatrequired for resonant pulses. We showed previously thatnonadiabatic resonant pulses, which contain Fourier com-ponents that are detuned from resonance, are also con-verted at a greater distance than adiabatic pulses [1].Here we show analytically, for adiabatic pulses, that thepropagation distance can be cut in half by choosing oneof the K systems in Fig. 1b to be in resonance while theother is highly detuned, rather than taking both K systemsto be equally detuned. Similarly, the distance required forfrequency conversion using resonant nonadiabatic ampli-tude- and phase-modulated pulses is shown numericallyto be shorter for this system than for the coherently-pre-pared system of Fig. 1a. Another similarity between adia-batic detuned pulses and nonadiabatic resonant pulses isthat the integrated intensities of the probe and convertedpulses oscillate with opposite phases as a function of prop-agation distance, unlike those of the adiabatic resonantpulses which approach pulse matching asymptotically.However, in the case of the double EIT system, unlikethe coherently-prepared system, the nonadiabatic pulsescontinue to be absorbed on propagation, even after pulsematching is achieved.

In order to explain why the propagation length at whichmaximum conversion occurs is shorter in the double EITsystem than in the coherently-prepared system, it shouldbe recalled that it is precisely the deviations from perfectcoherence that allow nonlinear processes to occur [24].

188 A. Eilam et al. / Optics Communications 277 (2007) 186–195

For example, we showed [8] that when the coherently-pre-pared system of Fig. 1a interacts with a strong detunedprobe, the distance at which maximum conversion occursdecreases as the detuning decreases, that is, as the coher-ence is increasingly perturbed. When we compare the twosystems of Fig. 1 for the same pump and probe parameters,we find that the two-photon coherence created by the lowerK system is perturbed more strongly by adding the extrapump in the double EIT system, than by adding the probein the coherently-prepared system. For this reason, maxi-mum conversion occurs at a shorter distance in the doubleEIT system. Another important difference is in the absorp-tion that occurs in the double EIT but not in the coher-ently-prepared system when the resonant probe isnonadiabatic. In the double EIT system, the nonadiabaticprobe and generated FWM introduce Fourier componentsthat are detuned from one-photon resonance so that thesystem is effectively shifted away from two-photon reso-nance allowing the probe and FWM to be absorbed inthe Autler–Townes wings. In the other configuration,coherent population trapping (CPT) and two-photon reso-nance is established by the strong pumps. The nonadiabaticweak probe and generated FWM act as if they are detunedfrom resonance, and are therefore not absorbed. Losses bydecay are avoided in both systems by nonadiabatic pulsesbecause the modulations of the nonadiabatic pulse aremuch faster than the decay rate.

As in our previous work [1], we derive analytical expres-sions for the probe and the generated FWM pulses as afunction of time and propagation distance, for the caseof constant pump fields. From these expressions, whichare valid for either adiabatic or nonadiabatic pulses, wederive the result that starting from a probe pulse, amatched probe-FWM pulse pair with the same shape asthe initial probe pulse is gradually obtained. In addition,we show that, starting with a nonadiabatic matched pulsepair or a pair of matched pulse trains, pulse matching isretained on propagation. However, in contrast to thecoherently-prepared system and to the adiabatic doubleEIT system, the pulses gradually become absorbed onpropagation.

As mentioned above, Cerboneschi and Arimondo [15]discussed the formation and propagation of adiabatons[21–23] in the coherently-prepared double K system, shownin Fig. 1a. For example, for the case where all four Rabifrequencies 2Vij (with i = 3,4 and j = 1,2) are initiallyequal and constant in time (here called carrier Rabi fre-quencies 2V 0

ij), apart from small complementary adiabaticamplitude modulations on V4j, they found that after a cer-tain propagation distance within which absorption of oneof the propagation modes occurs, the Rabi frequencieson the lower K system develop complementary modula-tions which are matched with those of the upper K system.Here, we will show for the double EIT system of Fig. 1b,that when the weak fields Vi2 are constant in time apartfrom similar complementary adiabatic amplitude modula-tions, the strong fields Vi1 also develop complementary

modulations after a certain propagation length. Thus wedemonstrate the existence of adiabatons in the doubleEIT system [14]. Moreover, we show for both systems, thatwhen the complementary adiabatic amplitude modulationsare replaced by nonadiabatic sinusoidal modulations, thesecan also be transferred to the constant strong fields, suchthat pulse matching is achieved. Thus we demonstrate theexistence of diabatons that can propagate for long dis-tances without further absorption, in both systems. We findthat the crucial factor in determining the asymptotic ampli-tude of the diabatons is the relative intensity of the carrierfields.

2. The model

2.1. Bloch equations for double K system

Let us first consider the double K system. Each jji ! jiitransition (with j = 1,2 and i = 3,4) interacts with an elec-tromagnetic field

~Eijð~r; tÞ ¼ ð1=2ÞxijEijðrÞ exp½�iðxijt � kijzþ uijÞ� þ c:c:

ð2Þwith unit polarization vector xij, frequency xij, wave-vectorkij and initial phase uij, whose detuning from the transitionfrequency is Dij and whose Rabi frequency is 2Vij(z) =lijEij(z)/�h. The Bloch equations for the double K system[8,25] are given by

_q11 ¼ iðV 13q031 þ V 14q

041 � V 31q

013 � V 41q

014Þ � c12q11 þ c21q22

þ c31q33 þ c41q44; ð3Þ_q22 ¼ iðV 23q

032 þ V 24q

042 � V 32q

023 � V 42q

024Þ þ c12q11 � c21q22

þ c32q33 þ c42q44; ð4Þ_q33 ¼ iðV 31q

013 þ V 32q

023 � V 13q

031 � V 23q

032Þ � c3q33 þ c43q44;

ð5Þ_q44 ¼ iðV 41q

014 þ V 42q

024 � V 14q

041 � V 24q

042Þ � c4q44; ð6Þ

_q021 ¼ iðV 23q031 þ aV 24q

041 � V 31q

023 � aV 41q

024Þ

� ðC21 þ iD21Þq021; ð7Þ_q031 ¼ iðV 31q11 þ V 32q

021 � V 31q33 � V 41q

034Þ

� ðC31 þ iD31Þq031; ð8Þ_q032 ¼ iðV 32q22 þ V 31q

012 � V 32q33 � a�V 42q

034Þ

� ðC32 þ iD32Þq032; ð9Þ_q041 ¼ iðV 41q11 þ a�V 42q

021 � V 31q

043 � V 41q44Þ

� ðC41 þ iD41Þq041; ð10Þ_q042 ¼ iðV 42q22 þ aV 41q

012 � aV 32q

043 � V 42q44Þ

� ðC42 þ iD42Þq042; ð11Þ_q043 ¼ iðV 41q

013 þ a�V 42q

023 � V 13q

041 � a�V 23q

042Þ

� ðC43 þ iD43Þq043; ð12Þ

where a = exp(iU) and U = u31 � u32 + u42 � u41 is theinitial relative phase, ckl is the longitudinal decay rate fromstate jki ! jli, ci is the total decay rate from state jii, andCkl ¼ 0:5ðck þ clÞ þ C�kl is the transverse decay rate of the


off-diagonal density-matrix element q 0kl, where C�kl is therate of phase-changing collisions. The rapidly oscillatingterms have been eliminated by the substitutions:

q0ij ¼ qij exp½�iðDijt þ kijz� uijÞ� ð13Þ

and

q021 ¼ q21 expf�i½ðD31 � D32Þt þ ðk31 � k32Þz� ðu31 � u32Þ�g;ð14Þ

q043 ¼ q43 expf�i½ðD41 � D31Þt þ ðk41 � k31Þz� ðu41 � u31Þ�g:ð15Þ

It is only possible to write the Bloch equations in this formwhen the multiphoton resonance condition, x31 � x32 +x42 � x41 = 0, is satisfied. This condition can be rewrittenin terms of the one-photon detunings as D31 � D32 = D41 �D42 = D21, where D21 is the two-photon or Raman detun-ing. In all our calculations, D21 will be taken to be zeroor very small.

2.2. The double K system interacting with two strong pumps

Let us consider the special case where strong cw laserfields interact with the j1i ! j3,4i transitions. Resonantor detuned laser pulses then interact with the j2i ! j3,4itransitions. We are interested in studying the propagationof laser pulses with initial Rabi frequencies V32(t,z = 0)and V42(t,z = 0). Assuming q11, q33, q44, and q 043 to be neg-ligible, q22 ’ 1, and U = 0, we find from Eqs. (7), (9), and(11) that

_q032 ¼ iðV 32 þ V 31q012Þ � ðC32 þ iD32Þq032; ð16Þ

_q042 ¼ iðV 42 þ V 41q012Þ � ðC42 þ iD42Þq042; ð17Þ

where

_q012 ¼ iðV 13q032 þ V 14q

042Þ � ðC21 � iD21Þq012: ð18Þ

In order to study pulse propagation, we solve the Maxwell–Bloch equations, in the paraxial approximation, which maybe written in the form [26]

d

dzþ 1

cd

dt

� �eV ij ¼ ia0ijq0ij; ð19Þ

where j = 1,2, i = 3,4, eV ij ¼ V ij=C31 is the dimensionlessRabi frequency, and a0

ij ¼ pxijNl2ij=c�hC31 is four times

the unsaturated line-center absorption coefficient for thejji ! jii transition [27]. In our numerical work, we solveeither the full Maxwell–Bloch equations (Eqs. (3)–(12)and (19)) for all four fields, or the restricted set of Max-well–Bloch equations (Eqs. (16), (17), and (19)) for onlythe weak pulsed fields. The restricted set of Maxwell–Blochequations have been solved analytically for the adiabaticcase by Cerboneschi and Arimondo [13,14], Raczynskiand Zaremba [16], Raczynski et al. [17], and Li et al. [18],and the equivalent Maxwell–Schrodinger equations havebeen solved by Wu and Yang [19]. The full set has beensolved using dressed states by Cerboneschi and Arimondo[15].

2.3. Analytical solution

In order to obtain an analytical solution for the casewhere the pulses propagate either adiabatically or nonadia-

batically, we take the Fourier transforms of Eqs. (16), (17),and (19), and obtain

d

dzbeV 32 ¼ �a32

beV 32 þ j32beV 42; ð20Þ

d

dzbeV 42 ¼ �a42

beV 42 þ j42beV 32; ð21Þ

where ~x ¼ x=C31, x indicates the Fourier transform of x,and

a32 ¼ �ia032

ðjeV 41j2 � m�21m42ÞA

� ixc; ð22Þ

a42 ¼ �ia042

ðjeV 31j2 � m�21m32ÞA

� ixc; ð23Þ

j32 ¼ �ia032eV 14eV 31=A; ð24Þ

j42 ¼ �ia042eV 13eV 41=A; ð25Þ

where x is the Fourier variable

mij ¼ eDij � ~x� ieCij ð26Þ

and

A ¼ �m�21m32m42 þ jeV 31j2m42 þ jeV 41j2m32: ð27ÞThe solution to these ordinary differential equations is gi-ven by [27]beV 32ðzÞ ¼

1

gþ � g�f½j32

beV 42ð0Þ � ðg� þ a32Þ beV 32ð0Þ� expðgþzÞ

� ½j32beV 42ð0Þ � ðgþ þ a32Þ beV 32ð0Þ� expðg�zÞg;

ð28ÞbeV 42ðzÞ ¼1

gþ � g�f½j42

beV 32ð0Þ � ðg� þ a42Þ beV 42ð0Þ� expðgþzÞ

� ½j42beV 32ð0Þ � ðgþ þ a42Þ beV 42ð0Þ� expðg�zÞg;

ð29Þ

where

g� ¼ �1

2ða32 þ a42Þ �

1

2½ða32 � a42Þ2 þ 4j32j42�1=2

: ð30Þ

When all the fields are on resonance, and assuming thateC32 ¼ eC42 ¼ eC31 � eC, eC21 � eC, and a032 ¼ a0

42 � a0, wefind that

gþ ¼�ia0cþ i~xCð~xþ ieCÞ

cð~xþ ieCÞ ; ð31Þ

g� ¼i~xC½~xð~xþ ieCÞ � a0c=C� ðjeV 31j2 þ jeV 41j2Þ�

c½~xð~xþ ieCÞ � ðjeV 31j2 þ jeV 41j2Þ�: ð32Þ

In order to determine the characteristic length and thegroup velocity [15,17], we expand g± in a Taylor series in ~x


gþ ¼ �a0eC � ia0=eC � ieCC=ceC ~x; ð33Þ

g� ¼ �a0cþ CðjeV 31j2 þ jeV 41j2Þ

icðjeV 31j2 þ jeV 41j2Þ~x: ð34Þ

We see that the terms proportional to expðgþzÞ in Eqs. (28)and (29) decay within a characteristic length of a0z ¼ eC,whereas those proportional to expðg�zÞ propagate with a‘‘slow’’ group velocity [17,19]

vg

c¼ CðjeV 31j2 þ jeV 41j2Þ

a0cþ CðjeV 31j2 þ jeV 41j2Þ: ð35Þ

It should be noted that the same group velocity is obtained,even when the detuning of the upper K system is non-zero.

Previously [1], we showed analytically and numericallythat pulse matching occurs in the coherently-prepared Ksystem regardless of whether the laser pulses are adiabaticor nonadiabatic. Pulse matching has also been shown tooccur in the double EIT system in the adiabatic limit[13,15]. We now consider whether it also occurs when thepulses are nonadiabatic. Analytically, from Eqs. (28) and(29), and numerically, we find that if a pair of pulses areinitially matched according to the expression

V 32ðt0; 0Þ=V 42ðt0; 0Þ ¼ V 31=V 41; ð36Þthey will propagate so that the ratio V32(t 0,z)/V42(t 0,z) re-mains constant. However, in contrast to the previous case[1], and the adiabatic limit, both the probe and FWM fieldsare gradually absorbed on propagation. In addition, it canbe shown both analytically and numerically that if only oneof the pulses is initially present, the probe and generatedpulses gradually become matched on propagation, accord-ing to

V 42ðt0; zÞ=V 32ðt0; zÞ ! V 41=V 31: ð37Þ

However, the Rabi frequencies themselves tend to zeroasymptotically [17]. In our analytical calculations, we cal-culated V32(t 0,z)/V42(t 0,z) using the full expressions for ai2

and ji2 (Eqs. (22)–(25)) and found that Eqs. (36) and (37)hold provided a0z exceeds the characteristic length eC.The calculation is rather complicated but can be simplifiedby assuming that jm�21mi2j � jeV i1j2 which is valid providedthe pumps are sufficiently intense. Thus pulse matching inthis double EIT system can be obtained for nonadiabaticpulses as well as for adiabatic pulses.

2.4. Frequency conversion in the adiabatic approximation

We now write simple expressions for frequency conver-sion from eV 32 to eV 42 in the adiabatic approximation, forthe case where eC32 ¼ eC42 ¼ eC31 � eC, a0

32 ¼ a042 � a0. We

assume that the initial pulse is either very long or verydetuned so that ~x can be neglected in Eq. (26). In addition,we assume that eD21 � eC and eC21 � eC, and neglect termsproportional to m�21 in Eqs. (22), (23), and (27). Then insert-ing the approximate expressions for a32, a42, and g�, in Eqs.

(28) and (29), and inverting the Fourier transformations,we obtain

eV 32ðt0; zÞ ¼eV 32ðt0; 0Þ

jeV 31j2 þ jeV 41j2

jeV 31j2 þ jeV 41j2 expia0zðjeV 31j2 þ jeV 41j2ÞjeV 31j2m42 þ jeV 41j2m32

" #;

ð38Þ

eV 42ðt0; zÞ ¼eV 32ðt0; 0ÞeV 13

eV 41

jeV 31j2 þ jeV 41j21� exp

ia0zðjeV 31j2 þ jeV 41j2ÞjeV 31j2m42 þ jeV 41j2m32

" #:

ð39Þ

From these equations, it is easy to confirm that Eqs. (36)and (37) hold in the adiabatic approximation.

Let us consider three cases of frequency conversionfrom eV 32 to eV 42: in the first, eD42 � 1 and eD32 ¼ 0, inthe second, eD32 ¼ eD42 � eD � 1, and in the third,eD32 ¼ eD42 ¼ 0. We will then compare the results obtainedwith those from our previous work on the coherently-pre-pared K system [1], shown in Fig. 1a. When eD42 � 1 andeD32 ¼ 0, the exponential in Eqs. (38) and (39) can beexpressed as

expia0zðjeV 31j2 þ jeV 41j2ÞjeV 31j2m42 þ jeV 41j2m32

’ expia0zðjeV 31j2 þ jeV 41j2Þ

jeV 31j2eD42

exp�a0zeCðjeV 31j2 þ jeV 41j2Þ2

jeV 31j4eD242

;

ð40Þ

so that for a0z� eD242jeV 31j4=ðjeV 31j2 þ jeV 41j2Þ2eC

eV 32ðt0; zÞ ¼eV 32ðt0; 0Þ

jeV 31j2 þ jeV 41j2

jeV 31j2 þ jeV 41j2 expia0zðjeV 31j2 þ jeV 41j2Þ

jeV 31j2eD42

" #;

ð41Þ

eV 42ðt0; zÞ ¼eV 32ðt0; 0ÞeV 13

eV 41

jeV 31j2 þ jeV 41j21� exp

ia0zðjeV 31j2 þ jeV 41j2ÞjeV 31j2eD42

" #:

ð42Þ

From these expressions, we obtain

eV 32ðt0; zÞeV 32ðt0; 0Þ

�� ¼ 1� 4

jeV 31j2jeV 41j2

ðjeV 31j2 þ jeV 41j2Þ2

"

sin2 a0zðjeV 31j2 þ jeV 41j2Þ2jeV 31j2eD42

!#1=2

; ð43Þ


�� ¼ 2jeV 31jjeV 41jjeV 31j2 þ jeV 41j2

sina0zðjeV 31j2 þ jeV 41j2Þ

2jeV 31j2eD42

!��:ð44Þ


It can be seen from Eqs. (43) and (44) that the Rabi fre-quencies oscillate [1,8,9,25], and that the maximumconversion

Cmax ¼ jeV 42ðt0; zÞ=eV 32ðt0; 0Þjmax ’2jeV 31jjeV 41jjeV 31j2 þ jeV 41j2

ð45Þ

occurs on the first oscillation at a distance

a0zmax ¼ pjeV 31j2eD42=ðjeV 41j2 þ jeV 31j2Þ: ð46ÞLet us now consider the case where eD32 ¼ eD42 � eD � 1.The exponential in Eqs. (38) and (39) now reduces to

expia0zðjeV 31j2 þ jeV 41j2ÞjeV 31j2m42 þ jeV 41j2m32

’ expðia0z=eDÞ expð�a0zeC=eD2Þ;

ð47Þ

so that for a0z� eD2eCeV 32ðt0; zÞeV 32ðt0; 0Þ

�� ¼ 1� 4

jeV 31j2jeV 41j2

ðjeV 31j2 þ jeV 41j2Þ2sin2 a0z

2eD� �" #1=2

; ð48Þ


�� ¼ 2jeV 31jjeV 41jjeV 31j2 þ jeV 41j2

sina0z

2eD� �� : ð49Þ

0 40 80 1200

0.2

0.4

0.6

0.8

1

α0z

norm

aliz

ed in

tens

ity

I42I41

a

0 40

0.2

0.4

0.6

0.8

1

norm

aliz

ed in

tens

ity

c

Fig. 2. (a) Time-integrated relative intensities I41,42 of the generated and convera0z, for the incident pulse of Eq. (55), with n = 8, (C31s)2 = 10, eV 31 ¼ eV 32 ¼ 1intensities I42,32 of the generated and converted pulses in the scheme of Fig. 1b(55), with n = 8, (C31s)2 = 10, eV 31 ¼ eV 41 ¼ 10, and initial Rabi frequency eV 32

Thus the maximum conversion for this case is the same asin Eq. (45), but now occurs at a distance

a0zmax ¼ peD: ð50ÞThis distance is the same as was obtained for the coher-ently-prepared K system [1]. However, we see from Eqs.(45) and (46) that almost complete conversion can be ob-tained at half this distance by choosing jeV 31j ¼ jeV 41j,eD42 � 1, and eD32 ¼ 0.

We now consider the case of zero detuning. For this caseeV 32ðt0; zÞeV 32ðt0; 0Þ

�� ¼ 1

jeV 31j2 þ jeV 41j2jeV 31j2 þ jeV 41j2 expð�a0z=eCÞh i

;

ð51ÞeV 42ðt0; zÞeV 32ðt0; 0Þ

�� ¼ jeV 13jjeV 41jjeV 31j2 þ jeV 41j2

1� expð�a0z=eCÞh i; ð52Þ

so that

Cmax ¼ jeV 42ðt0; zÞ=eV 32ðt0; 0Þjmax ’jeV 13jjeV 41jjeV 31j2 þ jeV 41j2

ð53Þ

and a0zmax is of the order of eC. This is the same distance aswas obtained for the case of the coherently-prepared K sys-

0 40 80 1200

0.2

0.4

0.6

0.8

1

α0z

norm

aliz

ed in

tens

ity

I42I32

b

8 12α0z

I42I32

ted pulses in the scheme of Fig. 1a, with all fields resonant, as a function of0, and initial Rabi frequency eV 41ð0; 0Þ ¼ 0:01. (b) Time-integrated relative, with all fields resonant, as a function of a0z, for the incident pulse of Eq.ð0; 0Þ ¼ 0:01. (c) As in (b) but using adiabatic pulses.

0 4 8 12 16 20–1

–0.6

–0.2

0.2

0.6

1

Γ31

t’

norm

aliz

ed R

abi f

requ

ancy

initial V41

min V41

max V42

α0z=24.43

a

0 4 8 12 16 20–1

–0.6

–0.2

0.2

0.6

1

Γ31

t’

norm

aliz

ed R

abi f

requ

ancy

initial V32

max V42

min V32

α0z=16.22

b

Fig. 3. (a) Comparison of the initial Gaussian pulse amplitude eV 41 withthe pulse amplitudes eV 41 and eV 42, at the value of a0z where the integratedintensity of eV 42 reaches its maximum value and eV 41 its minimum value.Parameters as in Fig. 2a. (b) Comparison of the initial Gaussian pulseamplitude eV 32 with the pulse amplitudes eV 32 and eV 42, at the value of a0z

where the integrated intensity of eV 42 reaches its maximum value and eV 32

its minimum value. Parameters as in Fig. 2b.


tem [1]. As in that case, the maximum conversion can onlyreach 50%, as opposed to almost 100% in the case of largedetuning.

As we will show in Section 3, higher conversion can beachieved at resonance using nonadiabatic pulses. The con-version can be improved even more by using nonadiabaticpulse trains [28]. This occurs because for nonadiabaticpulses, the converted pulse overlaps the original pulse, thatis

V 42ðt0; zÞ=V 32ðt0; 0Þ ! V 41=V 31 ð54Þat much shorter distances than those required for the pulsematching of Eq. (37). This point is illustrated in Fig. 3,below.

3. Numerical results for nonadiabatic pulses

3.1. Frequency conversion

We now present numerical results for the conversion ofeV 32ðt0; zÞ to eV 42ðt0; zÞ, via FWM, for the case where the inci-dent pulse is resonant and nonadiabatic. In our previous

work on conversion of resonant nonadiabatic pulses inthe coherently-prepared K system [1] of Fig. 1a, we dis-cussed the efficient frequency conversion of an incidentsinusoidal wave enclosed in a Gaussian envelope. As thenumber of oscillations increased, the conversion becamemore efficient but occurred at a longer propagationdistance. Here, we compare the frequency conversionobtained in the coherently-prepared K system with thatobtained in the double EIT system of Fig. 1b. In each case,the initial probe is of the formeV ijðt0; 0Þ ¼ eV ijð0; 0Þ exp½�ðt0 � t00Þ

2=s2� cos½nC31ðt0 � t00Þ�;

ð55Þwhere nC31 is the frequency of the cosine wave, t00 is thetime at which the Gaussian envelope reaches its maximumvalue (chosen to be some arbitrary time after the two-pho-ton coherence is established by the strong cw fields in thecase of Fig. 1a), and C31s is the width of the Gaussian enve-lope. In Fig. 2a, we plot the relative integrated intensities

I4j ¼R1�1 jeV 4jðt0; zÞj2dt0R1�1 jeV 41ðt0; 0Þj2dt0

ð56Þ

of the pulses with frequencies x4j, as a function of the prop-agation distance a0z, for the case where eV 31 ¼ eV 32, n = 8,and (C31s)2 = 10. In addition, we assume that eC41 ¼eC42 ¼ eC, and a0

41 ¼ a042 ¼ a0. It should be noted that max-

imum frequency conversion occurs at a shorter distancethan in Fig. 4 of Ref. [1]. This is due to the fact that theGaussian envelope used here is broader than in our previ-ous work, so that although there are more oscillations, theoverall effect is more adiabatic. This is reflected in the de-creased frequency conversion compared to the previouscase.

In Fig. 2b, we plot the relative integrated intensities

I i2 ¼R1�1 jeV i2ðt0; zÞj2dt0R1�1 jeV 32ðt0; 0Þj2dt0

ð57Þ

of the pulses with frequencies xi2, as a function of the prop-agation distance a0z, for the case where eV 31 ¼ eV 41, n = 8,and (C31s)2 = 10, for the double EIT system of Fig. 1b,assuming that eC32 ¼ eC42 ¼ eC, and a0

32 ¼ a042 ¼ a0 (assumed

in all the figures). We first note that, for both systems, thepulse intensities oscillate with opposite phases on propaga-tion, although all the fields are resonant. This is quite dif-ferent from the adiabatic, resonant case (see, for example,Eqs. (51) and (52)) where the pulse intensities vary expo-nentially until pulse matching is achieved asymptotically,but resembles the detuned, adiabatic case, as can be seenfrom Eqs. (43), (44), (48), and (49). The reason for this sim-ilarity derives from the nonresonant Fourier componentsintroduced by the sinusoidal time dependence of the probefield, given in Eq. (55).

It can be seen from Fig. 2a and b that the maximumintensity conversion, achieved on the first oscillation, isgreater for both cases than the 25% achievable in the adia-

0 5 10 15 209.9

10

10.1

Rab

i fre

quen

cy

0 5 10 15 209.95

10

10.05

0 5 10 15 209.95

10

10.05

Γ31t’

0 5 10 15 209

10

11

0 5 10 15 209.95

10

10.05

0 5 10 15 209.95

10

10.05

Γ31t’

a b

c d

e f

Fig. 5. Propagation of resonant diabatons with equal carrier Rabifrequencies eV 0

42 ¼ 10, and complementary amplitude modulations [n = 3,(C31s)2 = 10, eV 32ð0; 0Þ ¼ �eV 42ð0; 0Þ ¼ 0:1] initially on upper K system: (a)initial values of eV 41 (grey) and eV 42 (black), (b) initial values of eV 31 (grey)and eV 32 (black), (c) eV 41 (grey) and eV 42 (black) at a0z = 110, (d) eV 31 (grey)and eV 32 (black) at a0z = 110, (e) eV 41 (grey) and eV 42 (black) at a0z = 150,(f) eV (grey) and eV (black) at a0z = 150.


batic case (see Fig. 2c). In addition, the maximum intensityconversion is slightly greater for the coherently-preparedsystem than for the double EIT system, but is achieved ata greater distance. In Section 2.4, we found a similar resultfor the case of adiabatic pulses where one of the K systemsinteracts with resonant fields whereas the other interactswith detuned fields. The weak probe and FWM fields expe-rience far greater absorption in the double EIT case than inthe coherently-prepared case. This can be seen by compar-ing the asymptotic behavior of the integrated intensities. InFig. 2a, and Figs. 2–4 of Ref. [1], we see that the gapbetween the oscillating intensities decreases on propaga-tion, so that pulse matching in the sense of Eq. (37) isachieved. In addition, the intensities of both fieldsapproach an asymptotic value of 0.25, as in the adiabaticcase (see Eqs. (51) and (52)). The gap between the intensi-ties in Fig. 2b also decreases on propagation, in agreementwith Eq. (37). However, in contrast to the coherently-pre-pared system or the adiabatic double EIT system, the probeand converted pulses experience continuous absorptionuntil they become completely absorbed at long propaga-tion distances.

In Fig. 3, we compare the temporal shapes of the pulsegenerated at the FWM frequency, and the depleted probepulse, with the original probe pulse, at the propagation dis-tance where maximum intensity conversion occurs. Fig. 3acorresponds to the case considered in Figs. 2a and 3b tothat considered in Fig. 2b. In both cases, the time depen-dence of the generated pulse is almost identical to that ofthe initial pulse. However, in the double EIT case the gen-erated pulse is displaced to slightly longer retarded timesdue to a reduction in the group velocity (see Eq. (35)).

20 25 30 35 400.9

1

1.1

Rab

i fre

quan

cy

20 25 30 35 400.998

1

1.002

20 25 30 35 400.998

1

1.002

20 25 30 35 409

10

11

20 25 30 35 409.98

10

10.02

20 25 30 35 409.98

10

10.02

a b

c d

e f

Fig. 4. Propagation of resonant adiabatons with carrier Rabi frequencieseV 032 ¼ eV 0

42 ¼ 1, and eV 031 ¼ eV 0

41 ¼ 10, and complementary Gaussian ampli-tude modulations [n = 0, (C31s)2 = 10, eV 32ð0; 0Þ ¼ �eV 42ð0; 0Þ ¼ 0:1] ini-tially on probe fields: (a) initial values of eV 32 (grey) and eV 42 (black), (b)initial values of eV 31 (grey) and eV 41 (black), (c) eV 32 (grey) and eV 42 (black)at a0z = 110, (d) eV 31 (grey) and eV 41 (black) at a0z = 110, (e) eV 32 (grey) andeV 42 (black) at a0z = 150, (f) eV 31 (grey) and eV 41 (black) at a0z = 150.

3.2. Adiabatons and diabatons

Cerboneschi and Arimondo [15] discussed the formationand propagation of adiabatons in the double K system,shown in Fig. 1a. In Fig. 4, we show that adiabatons canalso be produced in the all-resonant double EIT systemof Fig. 1b [14]. We begin with constant strong carrier fieldson the j1i ! j3,4i transitions and weaker carrier fields on

31 32

0 5 10 15 200.9

1

1.1

Rab

i fre

quan

cy

0 5 10 15 200.999

1

1.001

0 5 10 15 200.999

1

1.001

Γ31t’

0 5 10 15 209

10

11

0 5 10 15 209.99

10

10.01

0 5 10 15 209.99

10

10.01

Γ31t’

a b

c d

e f

Fig. 6. Propagation of resonant diabatons with carrier Rabi frequencieseV 041 ¼ eV 0

42 ¼ 1, and eV 031 ¼ eV 0

32 ¼ 10, and complementary amplitudemodulation [n = 3, (C31s)2 = 10, eV 32ð0; 0Þ ¼ �eV 42ð0; 0Þ ¼ 0:1] initiallyon upper K system: (a) initial values of eV 41 (grey) and eV 42 (black), (b)initial values of eV 31 (grey) and eV 32 (black), (c) eV 41 (grey) and eV 42 (black)at a0z = 110, (d) eV 31 (grey) and eV 32 (black) at a0z = 110, (e) eV 41 (grey) andeV 42 (black) at a0z = 150, (f) eV 31 (grey) and eV 32 (black) at a0z = 150.


the j2i ! j3,4i transitions. Complementary Gaussianamplitude modulations (n = 0 in Eq. (55)) of the samewidth as in Fig. 2 are added to the probe carrier fields.We see that complementary Gaussian modulations appearon the pump fields after a certain propagation distance, insuch a way that the pulse matching of Eq. (37) is achieved.These then propagate unchanged.

In Fig. 5, we show the formation of diabatons in thedouble K system for the case where all the carrier Rabi fre-quencies are equal and all the fields are resonant. The com-plementary amplitude modulations are initially on theupper K system. On propagation, they are transferred tothe lower K system, so that pulse matching is achieved.Thereafter the pulses continue to propagate unchanged.In Figs. 6 and 7, we show the formation of diabatons onpropagation for the configurations shown in Fig. 1a andb, respectively. The pump Rabi carrier frequencies in eachcase are taken to be 10 times those of the probe. We seethat the amplitudes of the modulations are the same inboth cases, once pulse matching is attained. However, thisamplitude is much smaller than in the case where all thecarrier Rabi frequencies are equal (see Fig. 5). Thus thecrucial factor in determining the asymptotic amplitudesof the modulations seems to be the ratio of the probe topump carrier Rabi frequencies. As expected from pulsematching, the ratio of the amplitude modulations on thepump and probe is equal to the ratio of the pump andprobe carrier Rabi frequencies.

Another point to notice is that there seems to be someslowing down of the nonadiabatic amplitude modulationsin Figs. 5 and 6 which relate to the case of the coher-

20 25 30 35 400.9

1

1.1

Rab

i fre

quan

cy

20 25 30 35 400.999

1

1.001

20 25 30 35 400.999

1

1.001

Γ31t’

20 25 30 35 409

10

11

20 25 30 35 409.99

10

10.01

20 25 30 35 409.99

10

10.01

Γ31t’

a b

c d

e f

Fig. 7. Propagation of resonant diabatons with carrier Rabi frequencieseV 032 ¼ eV 0

42 ¼ 1, and eV 031 ¼ eV 0

41 ¼ 10, and complementary amplitudemodulations [n = 3, (C31s)2 = 10, eV 32ð0; 0Þ ¼ �eV 42ð0; 0Þ ¼ 0:1] initiallyon probe fields: (a) initial values of eV 32 (grey) and eV 42 (black), (b) initialvalues of eV 31 (grey) and eV 41 (black), (c) eV 32 (grey) and eV 42 (black) ata0z = 110, (d) eV 31 (grey) and eV 41 (black) at a0z = 110, (e) eV 32 (grey) andeV 42 (black) at a0z = 150, (f) eV 31 (grey) and eV 41 (black) at a0z = 150.

ently-prepared K system. This is in agreement with theresults of Cerboneschi and Arimondo [15] who found sim-ilar slowing down both analytically, using dressed states,and numerically for adiabatons in the case of Fig. 1a. Thisresult cannot be predicted from the analytical treatmentpresented in our previous paper [1] where it was assumedthat the fields interacting with the lower K system do notchange on propagation. However, if we allow q21 to changeon propagation, it can be shown [29] for the case whereV4j� V3j, that vg/c� 1 where vg is the group velocityexperienced by V4j.

4. Conclusions

We have shown that nonadiabatic, resonant amplitude-and phase-modulated pulses can be converted with greaterefficiency than adiabatic resonant pulses in a double K sys-tem in the double EIT configuration of Fig. 1b (see Figs. 2and 3).

We have compared the double EIT configuration ofFig. 1b with the coherently-prepared system of Fig. 1a,which we discussed previously [1]. In both cases, the inte-grated intensities of the nonadiabatic resonant probe andconverted pulses oscillate with opposite phases as a func-tion of propagation distance, unlike those of the adiabaticresonant pulses which approach pulse matching asymptot-ically. Two main conclusions arise from comparing the sys-tems. First, maximum conversion can be obtained at ashorter propagation distance in the double EIT configura-tion, and second, greater absorption occurs on propagationin the double EIT system. However, at short propagationdistances, these effects counteract one another so that themaximum conversion efficiencies are approximately equalin both configurations. In addition, the two systems differin their asymptotic behavior. In the coherently-preparedsystem, the nonadiabatic pulses stop absorbing once pulsematching is achieved, whereas the double EIT system con-tinues to absorb both the probe and converted pulses, evenafter pulse matching is achieved.

As before, by solving the Maxwell–Bloch equationsusing Fourier transforms, we have derived analyticalexpressions for the probe and the generated FWM pulsesas a function of time and propagation distance, for thedouble EIT configuration. From these generally validexpressions we derived the result that starting from a non-adiabatic probe pulse, a gradually matched probe-FWMpulse pair with the same shape as the initial probe pulseis obtained. In addition, we showed that, starting with anonadiabatic matched pulse pair or a pair of matched pulsetrains, we obtain matched propagation of these pulses.However, the propagating nonadiabatic pulses are contin-uously absorbed on propagation. This is in contrast toadiabatic pulses which propagate without loss, and in con-trast to the coherently-prepared configuration whereabsorption of the nonadiabatic pulses occurs until theirintensities reach the values obtained from the adiabaticapproximation.


Finally, we demonstrated the formation of adiabatonsin the double EIT system, and of diabatons in both thecoherently-prepared K system and the double EIT system.Both the diabatons and adiabatons satisfy pulse-matchingconditions. In addition, the asymptotic amplitude of thecomplementary amplitude modulations is proportional tothe ratio of the pump to probe carrier frequencies, and isthe same in each of the configurations.

References

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011802(R).[21] R. Grobe, F.T. Hioe, J.H. Eberly, Phys. Rev. Lett. 73 (1994) 3183.[22] M. Fleischhauer, A.S. Manka, Phys. Rev. A 54 (1996) 794.[23] A. Kasapi, M. Jain, G.Y. Yin, S. Harris, Phys. Rev. Lett. 74 (1995) 2447.[24] A. Rosenhouse-Dantsker, A.D. Wilson-Gordon, H. Friedmann, Opt.

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Enhanced frequency conversion