We investigate the propagation of a soliton-pair in 3-level atomic system out of resonance. We present an explicit analytical expression of the soliton-pair shape. We show that the speed of the soliton-pair can be controlled. Furthermore, we derive a condition for completely stopping the light

Boutabba, N.

Eleuch, H.

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Appl. Math. Inf. Sci.7, No. 4, 1505-1509 (2013) 1505Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.12785/amis/070433SLOWING LIGHT CONTROL FOR A SOLITON-PAIRN. Boutabba1,2,∗ and H. Eleuch1,31Universit́e de Montŕeal, C.P. 6128, Succ. Centre-Ville, Montreal (QC), H3C 3J7 Canada2Physics Department, University of Carthage, Tunis 1040, Tunisia3Ecole Polytechnique, C.P. 6079, Succ. Centre–Ville, Montreal (QC), H3C 3A7 CanadaReceived: 28 Nov. 2012, Revised: 17 Feb. 2013, Accepted: 5 Mar.2013Published online: 1 Jul. 2013Abstract: We investigate the propagation of a soliton-pair in 3-level atomic system out of res nance. We present an explicit analyticalexpression of the soliton-pair shape. We show that the speed of the soliton-pair can be controlled. Furthermore, we derive a conditionfor completely stopping the light.Keywords: Soliton-pair, slowing light1. IntroductionConsiderable attention was paid in the three last decadesto the non linear optical media [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19], mainly the Solitons formationin electromagnetically induced transparency medium(EIT)[20,21,22,23], as they have important physicalfeatures[24,25]. In particular, spatiotemporal solitonsknown as STSs [26]. They are pulses, which maintaintheir shape by propagating in the medium due to thebalance between the group-velocity dispersion,diffraction, and nonlinear self-phase modulation.Moreover, EIT is an important tool for realization ofcontrollable atom-light coupling, such as themanipulation of optical pulse propagation through atomicand atom-like media via slow [27,28,29] and stored light[30,31,32,22]. Among the interesting potentialapplications of EIT and slow light is the practicalrealization of a quantum memory. Electromagneticallyinduced transparency (EIT) has mainly been studied forthree-level systems. In particular, a considerable intereshas been dedicated to the Lambda-configuration where apair of optical pulses propagates without absorption. Thismedium can be made experimentally [33] if, a weaksignal pulse propagates through the medium pumped by astrong control field. The group velocity of the weak signaldepends on the control field intensity, and can be reducedto zero. In this case the photonic information of the pulseis completely transferred to the atoms. This coherenttransfer is reversible, which means that the pulse may beretrieved[34]. In another context, Park and Shin [35]showed a systematic method for constructing families ofmulti-component pulses in EIT media. H. Eleuch and hisco-workers [36,37], derived analytical solutions ofsolitons and pair of solitons in dissipative resonant media.In this paper we investigate the soliton-pairpropagation in the three level dissipative media out ofresonance, we elaborate an explicit analytical expressionof the pair soliton shape and we derive a condition forcompletely stopping the light.2. ModelThe medium considered here is a three level atom in theΛ−configuration interacting with two non-resonantelectromagnetic fields. The two atomic transitions areexcited by two variable laser fields.The three-level atom is described by quantum systemwith three energy levels|0〉 , |1〉 and|2〉 . The restriction totwo lower energy level is valid if the frequency of theinteracting waves are distant enough to all otherfrequencies. In this model we take into account the ratesγ1,2 of radiative decay from the higher level|0〉 to thelevels |1〉 and|2〉 and neglecting the other dissipationeffects.This three-level system is irradiated by a light beamcontaining two monochromatic fieldsE1 and E2 whichpropagate with polarization adequate to couple the opticaltransitions (|0〉 ↔ |1〉 and|0〉 ↔ |2〉). We suppose that the∗ Corresponding author e-mail:n boutabba@yahoo.frc© 2013 NSPNatural Sciences Publishing Cor.1506 N. Boutabba and H. Eleuch: Slowing light control for a Soliton-pairlevels|1〉 and |2〉 are decoupled (transition|1〉 ↔ |2〉negligible). In fact a free atom has at least two states atsame parity between which an electrical dipole transitionis not allowed. Furthermore, we assume that the twooptical fields have slowly varying envelopes [34,35,36]:1ω1∣∣∣∣∣∂E1∂ t∣∣∣∣∣<<∣∣∣E1∣∣∣;1ω2∣∣∣∣∣∂E2∂ t∣∣∣∣∣<<∣∣∣E2∣∣∣(1)In the rotating wave approximation the Hamiltonian of thesystem isH = H0+HiH0 = ∑i=1,2εia+i aiH = ∑i=1,2gi(a+0 aiEi +a+i a0E∗i)(2)Where H0 is the field free Hamiltonian andHi is theinteraction of the atom with the electromagnetic fields.ai,a+i are respectively the annihilation and creation fermionsoperators of the atomic leveli, verifying theanti-commutation relation[ai,a+j]+= δi j. The twodipole transition matrix elements which are assumed to bereal are denoted byg1 andg2. The dynamical evolution ofsystem is governed by the master equationddtρ =1ih̄[H,ρ ]+£ρ (3)The irreversible decay part in the system is denoted by£ρand corresponds to the incoherent processes. It is given by:£ρ =γ12[a+0 a1ρ ,a+1 a0]+γ12[a+0 a1ρ ,a+1 a0]+γ22[a+0 a2ρ ,a+2 a0]+γ22[a+0 a2ρ ,a+2 a0](4)The time-varying density matrix elements verify thefollowing evolution equationsddtρ10 = iω10ρ10− id1E∗1 (ρ00−ρ11)+ id2ρ∗21E∗2 −(γ1+ γ22)ρ10ddtρ20 = iω20ρ20− id2E∗2 (ρ00−ρ22)+ id1ρ21E∗1 −(γ1+ γ22)ρ20ddtρ21 = id1E1ρ20+ id2ρ∗10E∗2 + i(ω20−ω10)ρ00ddtρ j j = id j(E jρ j0−E∗j ρ∗j0)+ γ jρ00 f or j = 1,2ddtρi j =ddt(ρ ji)∗ f or i, j = 0,1,2ddtρ00 = −ddt(ρ11+ρ22) (5)The last equation is derived fromtr (ρ) = 1. di are thecoupling constants.ω10 andω20 represent the two atomictransition frequencies and we have:ω10= ε0−ε1h̄ andω20=ε0−ε2h̄ .ρ10 and ρ20 terms oscillate at the respective drivingfield frequency and theρ21 oscillate with frequencydifferences of the two light fields. So, we can define theslowly varying amplitudes of the off-diagonal densitymatrix elementsρ10, ρ20 andρ21 through the relations:ρ j0 = ρ j0exp(iω j0t) f or j = 1,2ρ21 = ρ21exp(i(ω20−ω10) t) (6)The off-diagonal elements, which describe the atomiccoherences, can be decomposed into imaginary part andreal parts :ρ j0 = χ j0+ iψ j0ρ21 = χ21+ iψ21 (7)The Hermitian propriety of the density matrix ensuresthat the diagonal elementsρ11,ρ22 andρ00 must be real.These terms are the level populations and determine theinternal energy of the atom.δ1 = ω10 − ω1andδ2 = ω20 − ω2 are the detunings between the laserfrequencies and the atomic transitions frequencies. Wehave developed in this section, the evolution of the atomicparameters. In the next section, we explore thepropagation of the fieldsE1 andE2 through the mediumand its spatial and temporal dynamical behavior.3. Analysis of the fields propagationThis section deals with the analysis of soliton propagationin the medium described above. The signal fieldE j forj = 1,2 are described by the Maxwell equations for aslowly varying approximation (SVA) [38,39]:∂E j∂ t + c∂E j∂x = ig′ −ρ j0. Moreover, the condition forsoliton-pair propagation is expressed asE j (x, t) = E j (x− vgt)The propagation constants of the fieldsg′i, which areconsidered to be real, are given byg′i =2πε0Ngi(ω2+ δ2).ε0 is the vacuum electric constant,N is the atomic dipoledensity andc is the velocity of light.vg represents thegroup velocity of the soliton-pair. We introduce a movingcoordinatez = x− vgt which propagates with the pulses’svelocities. In this new moving coordinate we have∂∂ t =−vg∂∂ z and∂∂x =∂∂ z . We assume in this work that thetwo spontaneous emission rates are approximately equal:γ1 = γ2 = γ.We suppose also, that−E2 is real which givesχ20 = 0. The complete set of the evolution equations formedium-fields interaction (Maxwell-Bloch equations) arec© 2013 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci.7, No. 4, 1505-1509 (2013) /www.naturalspublishing.com/Journals.asp 1507obtained from Maxwell equations and the master equationddzχ10 =δ1vgψ10−α2ψ21+Γ χ10ddzψ10 =δ1vgχ10−2α1χ11−α2χ21+Γ ψ100 =δ2vgψ20+α1ψ21ddzψ20 = α2(−1−ρ11)−α1χ21+Γ ψ20ddzχ21 = α1ψ20+α2ψ10ddzψ21 = α2χ10ddzρ11 = 2α1ψ10−Γ (1−ρ11−ρ22)ddzρ22 = 2α2ψ20−Γ (1−ρ11−ρ22) (8)ddzα2 = −d2g2vg(c− vg)ψ20= −k2ψ20ddzα1 = −d1g1vg(c− vg)ψ10= −k1ψ10Whereα1andα2 are both variables and related to thefield amplitudes by the following expressions:α1 =d1E1vg; α2 =d2E2vg(9)We deal here with the case whereα2 = Aα1 = Aαwhich means that the two pulses constituting theamplitudes of the pair have the same shape but withdifferent amplitudes. The new constantΓ is defined byΓ = γvg .The optical fields 1 and 2 have slowly varyingamplitudes, in this case, we can neglect the variation ofthe curvature and we can assume that the third and theforth order of the differentiations are negligible. Afteralgebraic manipulations and differentiation of theMaxwell-Bloch equations, we obtain a non-lineardifferential equation:0= Bp2+Cp+Dp(d pdα)+E(d pdα)+F (10)where we introduce a new variablep describing thefield evolution of the fieldα byp(α) =dαdz(11)The expressions of the constants are:B = −3C = −Γ αD = 4αE = 4α2 (12)F = (−A2−δ1k2δ2k1)α44. Shapes of Solitons and stopping lightconditionIn this section we derive an analytical explicit shape ofthe soliton, so we must find an explicit relation betweenαandz. From the definition of thep function we obtain thefollowing relation between the field amplitudeα and thelocal coordinatez, for small field amplitudeα,we get :z =∫dαp=∫dαS1α +S2α2=−2S1tanh−1[2S2α −S1S1](13)whereS1 = 4−ΓS2 = ±X= ±√15(δ1k2δ2k1+A2) (14)Herewith we obtain two possible expressions of thesoliton pulse shapes (The plots in Fig.1 represent the twopossible shapes of the solitons, the used parameters arefrom realistic values [40,41,42,43,44,45]) :α(1) (z) =−S12X(1− tanh(S1z2))α(2) (z) =S12X(1− tanh(S1z2)) (15)In order to calculate the velocity of the soliton, wedetermine, first the maximum amplitude of the soliton. Itis given by the following relation:αmax=S1X=4−ΓS2=4− γvgS2(16)The expression of the group velocity for the solition pair isthen:vg =γ4−Xαmax. From the fact thatvg ≤ c, we deducea condition for the soliton propagationαmax≤4− γcX, (17)c© 2013 NSPNatural Sciences Publishing Cor.1508 N. Boutabba and H. Eleuch: Slowing light control for a Soliton-pair-5 0 5-0.025-0.020-0.015-0.010-0.0050.000Α(a)-5 0 50.0000.0050.0100.0150.0200.025Α(b)Figure 1: Soliton shapesα(1)(a) andα(2) (b) as function of z forthe following parameters (in the SI units):ω2 = 1.5∗108;ω1 = 5∗108;d2= 3∗10−32;d1= 1.5∗10−32;γ = 5∗108;A = 1.5;g2 =0.15∗108;g1 = 0.17∗108;δ1 = 0.03∗108;δ2 = 3δ1.wherec is the velocity of light in vacuum.Furthermore, for δ1δ2 < 0 another propagationcondition for the soliton-pair should be verified:|A|>√−δ1d2g2ω20δ2d1g1ω10(18)The two cases whereA = ±√− δ1d2g2ω20δ2d1g1ω10are veryinteresting as the soliton’s velocity vanishes which meansthat the soliton-pair can be stopped. This can be used togreatly reduce noise, allowing information to betransmitted more efficiently in the media[46]. Moreover,storing the energy information at a desired time, is animportant challenge for the quantum informationprocessing. 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SLOWING LIGHT CONTROL FOR A SOLITON-PAIR

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