In this paper we analyze and discuss the solitons propagation in a left-handed four-level atomic system coupled to three laser fields. The system is analyzed through the density matrix formalism. We show that by varying the external probe field amplitude in time, the soliton propagation is obtained in the left-handed medium . Our scheme allows to engineer under certain conditions, the soliton propagation in the four- level atomic system while maintaining the negative refractive index

Boutabba, N.

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Appl. Math. Inf. Sci. 14, No. 4, 729-732 (2020) 729Applied Mathematics & Information SciencesAn International Journalhttp://dx.doi.org/10.12785/amis/140421Soliton Propagation in a Left-handed Four-Level AtomicSystemN. BoutabbaFatima College of Health Sciences, Institute of Applied Technology, Al Mafraq, P O BOX 3798, UAEReceived: 7 Sept. 2019, Revised: 1 Nov. 2019, Accepted: 5 Nov. 2019Published online: 1 July. 2020Abstract: In this paper we analyze and discuss the solitons propagation in a left-handed four-level atomic system coupled to threelaser fields. The system is analyzed through the density matrix formalism. We show that by varying the external probe field amplitudein time, the soliton propagation is obtained in the left-handed medium . Our scheme allows to engineer under certain conditions, thesoliton propagation in the four- level atomic system while maintaining the negative refractive index.Keywords: Solitons, density matrix, negative refractive index, four level system1 IntroductionConsiderable attention was paid to the formation oflocalized electromagnetic waves propagating withoutspreading shape, called solitons in nonlinear opticalmedia [1, 2]. Ultra slow optical solitons were extensivelystudied in electromagnetically induced transparency (EIT)media [3–6] where the wave propagation of a weak probefield displays large suppression of optical absorption anda significant reduction of group velocity. This leads toimportant physical features [7, 8]. In particular, thespatio-temporal solitons known as STSs [9] aremultidimensional pulses which maintain their shape in thelongitudinal (temporal) and transverse (spatial) directions.This is due to the balance between the group-velocitydispersion (GVD), diffraction, and nonlinear self-phasemodulation. On the other hand, non linear optical media,known as left-handed materials have shown interestingnonlinear optical phenomena such as the harmonicgeneration, parametric amplification, sub-wavelengthimaging, and solitary wave propagation [10, 11]. Thesematerials, known as Metamaterials (MMs), are artificiallyengineered materials that have unique unnaturalproperties. In fact such media are able to focus light into aspot much smaller than the wavelength, which lead torealize a perfect lens [12]. Moreover, the electromagneticproperties of such materials can be controlled bymodulating the design of the cell and bringing severalpromising applications [13, 14]. These laters are of greatsignificance for developing applications in meta-materialsdue to their properties [15, 16]. In this paper it is assumedthat the material is a four level system which exhibitsLeft-handed properties under certain conditions and in arange of frequency. Our aim is to define how themeta-material properties can be associated with solitonpropagation and to investigate how this behavior can becontrolled. This paper is organized as follows: In the firstsection we present the model of a four level atomicsystem interacting with three electromagnetic fields. Inthe second section, we analyze and describe our system interms of the density matrix formalism. Then we presentthe results followed by a discussion and concludingremarks.2 The modelIn this model, the different levels denoted by (k, l) where(k, l = 1,4) are coupled to the corresponding (k, l) fieldsas follows: the level |3〉 is coupled to the level |4〉through the field Ec while the level |2〉 is coupled by theweak probe field Ep to the level |3〉. The electric andmagnetic fields of Ep are coupled to the level pairs |3〉 to|2〉 and |2〉 to |1〉, respectively, while the weak signalfield Es drives the transition |1〉 to |3〉. The Rabifrequencies of the corresponding fields are Ωc, Ωp, Ωs,also denoted by, the Ωkl , corresponding to the (k, l)levels. The frequency detuning δkl corresponds to ∆c, ∆p,∗ Corresponding author e-mail: Nadia.Boutabba@fchs.ac.aec© 2020 NSPNatural Sciences Publishing Cor.730 N. Boutabba: Soliton propagation in a left-handed...Fig. 1: A schematic diagram of a four level atomic systeminteracting with three electromagnetic fields.∆s as ∆c = ωc −ω34, ∆p = ωp −ω23 and ∆s = ωs −ω13,and the decay rates γkl are γ4,2,3 from the higher level |4〉to |3〉, |3〉 to |2〉 and |3〉 to |1〉. This is illustrated byFigure 1. This four-level system is irradiated by a lightbeam propagating along an arbitrary direction, withpolarization adequate to couple the optical transitions,and containing three monochromatic fields.E p and Ec andEs are assumed to be slowly varying functions in thefollowing sense [11–14]:1ωi∣∣∣∣∂Ei∂ t∣∣∣∣<<∣∣∣E i∣∣∣The Hamiltonian describing the interaction of the fourlevel atom with the fields has the expression:H = H0 +HiH0 = ∑iεia+i ai〈k|Hi|l〉 = −12h̄Ωkle−i(ωk−ωl)t .Where H0 is the field free Hamiltonian and Hi is theinteraction of the atom with the electromagnetic fields. aiand a+i are the annihilation and creation fermionsoperators of the atomic level i. ai and a+i verify theanticommutation relation[ai,a+j]+=δi j. The levels |2〉and |3〉 have opposite parity.Thus we can denote by d23 the electric dipole element〈2|d23|3〉 6= 0. In the same way, µ23 denotes the magneticdipole element 〈1|µ23|2〉 6= 0 between levels |1〉 and |2〉 .Moreover, we have:D = εωEB = µωH (1)We recall the Fourier transform of the curl equation forelectric field in Maxwell’s equations by:B =k∧Eω(2)To study the evolution of interaction between theatom and fields we use the density matrix formalismunder rotating-wave approximation. The density matrixoperator is defined byρ = ∑k,lρklei(ωk−ωl)t〈k|〉. (3)The time-varying off-diagonal density matrix elementsverify the evolution equations:ddtρkl = (−iδkl − γkl)− i ∑m=1,4(Ωmlρkm −Ωkmρml). (4)We neglect the irreversible decay part in the system whichcorresponds to the incoherent processes. In term of thebasis set of the bare atom {|1〉 , |2〉, |3〉, |4〉} we have:a+k al = |k〉 〈l| and ρ =4∑k,l=0|k〉ρkl 〈l| (5)The coupling constants are denoted by gi =−idiℏas wellas the mi j =−i fi jℏ. The off-diagonal elements describe theatomic coherences. The ρ34, ρ23 as well as ρ13 termsoscillate at the respective driving field frequency and theall others oscillate with frequency differences of the twolight fields. Hence, we can define the slowly varyingamplitudes of the off-diagonal density matrix elementsthrough the rotating wave frame [12]. The signal fieldE j f or j = 1 is described by the Maxwell equations for aslowly varying approximation (SVA) by [6, 14, 16, 17]∂E j∂ t+ c∂E j∂ z= ig′−ρ j.In general, the propagation constants of the fields g′i aregiven by g′i = 2Πε0Ngi(ωk + δk). ε0 is the vacuumelectric constant, N is the atomic dipole density and c isthe velocity of light. vg represents the group velocity ofthe soliton. One can vary more than one field signal toobtain a soliton pair propagation [6]. Here, we limit ouranalysis to one variable field amplitude in time, since wewant to maintain the negative refractive index property inthe medium. In addition, the condition for solitonpropagation is expressed as:E j(x, t) = E j(x− vgt) (6)We introduce a moving coordinate z = x − vg(t) whichpropagates with the pulse velocity. In this new movingcoordinate we have ∂∂ t = −vg∂∂ z Thus, the motionequations of the density matrix off diagonal elements inc© 2020 NSPNatural Sciences Publishing Cor.Appl. Math. Inf. Sci. 14, No. 4, 729-732 (2020) / www.naturalspublishing.com/Journals.asp 731the dipole and RWA for this system can be written asfollows:ddzρ11 =−1vg(2g3ρ33 +2g1ρ22 + iΩs(ρ13 −ρ31))ddzρ22 =−1vg(2g2ρ33 −2g1ρ22 + iΩp(ρ23 −ρ32))ddzρ33 =−1vg(−2(g2 +g3)ρ33 +2g4ρ44 − iΩs(ρ13 −ρ31)−iΩp(ρ23 −ρ32)+ iΩc(ρ34 −ρ43))ddzρ12 =−1vg(−2(g1 − i(∆s −∆p))ρ12 + iΩpρ13 − iΩsρ32)ddzρ13 =−1vg(−2(g2 +g3 − i∆s)ρ13 + iΩpρ12 + iΩcρ14+iΩs(ρ11 −ρ33))ddzρ14 =−1vg(−(g4 − i(∆s +∆c))ρ14 + iΩcρ13 − iΩsρ34)ddzρ23 =−1vg(−(g2 +g3 − i∆p)ρ23 + iΩsρ21 + iΩcρ24+iΩp(ρ22 −ρ33))ddzρ24 =−1vg(−(g1 +g4 − i(∆p +∆c))ρ24 + iΩcρ23 − iΩpρ34)ddzρ34 =−1vg(−(g2 +g3 +g4 − i∆c)ρ34 − iΩsρ14 − iΩpρ24+iΩc(ρ33 −ρ44))ddzΩp =−d′2g2vg(c−vg)ρ32To analyze the system we use the Mathematica Softwareto numerically solve the density matrix equations. Thefollowing experimental parameters were taken intoconsideration:Ωs = 0.15Γ , Γ1 = 0.05Γ , Γ2 = 0.03Γ ,Γ3 = 0.01Γ , Γ4 = 0.1Γ ,Ωc = 2Γ ;∆c= -1.5; ∆s= 1.5;Fig. 2: The refractive index in the considered fourlevel atomicmedia.Fig. 3: The soliton in the negative refractive index four-levelatomic media3 PerspectiveIn this paper we investigated the soliton propagation in aleft-handed atomic four-level system coupled with threefields of laser. The stokes strong field was considered as aconstant while the probe field was variable. The systemwas described by means of the density matrix formalismunder the rotating wave approximation. Varying theexternal probe field amplitude in time, the propagation ofsoliton was simulated in the four-level atomic systemwhile maintaining a negative refractive medium. This wasachieved in a large detuning interval under suitableexperimental conditions. In the future, we aim toinvestigate an analytical analysis of the system toexamine the effect of fields amplitudes, the detunings, andthe probe field on the existence of the solitons. This canbe achieved by deriving the full analytical solutions of thedensity matrix.AcknowledgementThe authors are grateful to the anonymous referee for acareful checking of the details and for helpful commentsthat improved this paper.References[1] Kivshar, Y.S. and Luther-Davies, B. Dark optical solitons:physics and applications. Physics reports, 298(2-3)., 81-197,1998.[2] Haus, H.A. and Wong, W.S. Solitons in opticalcommunications. Reviews of modern physics, 68(2).,423, 1996.[3] She, Y., Wang, D., Zhang, W., He, Z. and Ding, J. Formationand interaction characteristics of two-component spatialweak-light soliton in a four-level double Λ type system.JOSA B, 27(2)., 208-214, 2010.[4] Yang, W.X. and Lee, R.K. Slow optical solitons viaintersubband transitions in a semiconductor quantum well.EPL (Europhysics Letters), 83(1)., 14002, 2008.c© 2020 NSPNatural Sciences Publishing Cor.732 N. Boutabba: Soliton propagation in a left-handed...[5] Boutabba, N. and Eleuch, H. Slowing Light Control for aSoliton-Pair. Applied Mathematics & Information Sciences,7(4)., 1505, 2013.[6] Boutabba, N. and Eleuch, H. Soliton-pair propagation underthermal bath effect. Mathematical Modelling of NaturalPhenomena, 7(2)., 32-37, 2012[7] Du, Y., Zhang, Y., Zuo, C., Li, C., Nie, Z., Zheng, H., Shi,M., Wang, R., Song, J., Lu, K. and Xiao, M. Controllingfour-wave mixing and six-wave mixing in a multi-Zeeman-sublevel atomic system with electromagnetically inducedtransparency. Physical Review A, 79(6)., 063839, 2009.[8] Wu, Y. and Yang, X. Electromagnetically inducedtransparency in V, Λ , and cascade-type schemes beyondsteady-state analysis. Physical Review A, 71(5)., 053806,200.[9] Malomed, B.A., Mihalache, D., Wise, F. and Torner,L. Spatiotemporal optical solitons. Journal of Optics B:Quantum and Semiclassical Optics, 7(5)., R53, 2005.[10] Powell, D.A., Shadrivov, I.V. and Kivshar, Y.S.Asymmetric parametric amplification in nonlinear left-handed transmission lines. Applied Physics Letters, 94(8).,084105, 2009[11] Shahvarpour, A., Gupta, S. and Caloz, C. Schrödingersolitons in left-handed SiO 2–Ag–SiO 2 and Ag–SiO2–Ag plasmonic waveguides calculated with a nonlineartransmission line approach. Journal of Applied Physics,104(12)., 124510, 2008.[12] Cowan, R.D. The theory of atomic structure and spectra, 3,Univ of California Press, 1981.[13] Shahvarpour, A., Gupta, S. and Caloz, C. Schrödingersolitons in left-handed SiO 2–Ag–SiO 2 and Ag–SiO2–Ag plasmonic waveguides calculated with a nonlineartransmission line approach. Journal of Applied Physics,104(12)., 124510, 2008.[14] Vendik, I.B. and Vendik, O.G. Metamaterials and theirapplication in microwaves: A review. Technical physics,58(1)., 1-24, 2013.[15] Shadrivov, I.V. and Kivshar, Y.S. Spatial solitons innonlinear left-handed metamaterials. Journal of Optics A:Pure and Applied Optics, 7(2)., S68, 2005.[16] Cheng, X., Zhuang, B., Dai, X., Su, W. and Wen, S. Darksoliton solutions to the nonlinear Schrödinger equation forultrashort pulse propagation in metamaterials. Journal ofNonlinear Optical Physics & Materials, 18(02)., 271-284,2009.[17] Eberly, J.H. Transmission of dressed fields in three-levelmedia. Quantum and Semiclassical Optics: Journal of theEuropean Optical Society Part B, 7(3)., 373, 1995.N. Boutabbaareceived the PhD degreein Quantum Physicsat the university of Carthage,Tunisia. Her researchinterests are in the areasof Laser-Matter Interactions,Non-linear Optics, Solitons,Lef-thanded media includingthe mathematical methods forcomplex systems. She has published research articles inreputed international journals of Quantum Optics. Inaddition, she is a referee in many journals of optics.c© 2020 NSPNatural Sciences Publishing Cor.

Soliton Propagation in a Left-handed Four-Level Atomic System

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