aa r X i v : . [ phy s i c s . op ti c s ] N ov Driving light pulses with light in two-level media
R. Khomeriki , , J. Leon ( ) Laboratoire de Physique Th´eorique et AstroparticulesCNRS-IN2P3-UMR5207, Universit´e Montpellier 2, 34095 Montpellier (France)( ) Physics Department, Tbilisi State University, 0128 Tbilisi (Georgia) A two-level medium, described by the Maxwell-Bloch (MB) system, is engraved by establishing astanding cavity wave with a linearly polarized electromagnetic field that drives the medium on bothends. A light pulse, polarized along the other direction, then scatters the medium and couples tothe cavity standing wave by means of the population inversion density variations. We demonstratethat control of the applied amplitudes of the grating field allows to stop the light pulse and tomake it move backward (eventually to drive it freely). A simplified limit model of the MB systemwith variable boundary driving is obtained as a discrete nonlinear Schr¨odinger equation with tunableexternal potential . It reproduces qualitatively the dynamics of the driven light pulse.
PACS numbers: 42.50.Gy, 42.65.Re
Introduction.
Manipulation of light with light has be-come one of the hottest research spots in quantum op-tics this last decade. A widely studied field of researchmakes use of electromagnetically induced transparencyin three-level systems, which allows to slow down, andeventually stop, a light pulse [1, 2, 3]. Another interstingresearch option uses resonantly absorbing Bragg reflectors (RABR) which consist in a periodic array of dielectricfilms separated by layers of a two-level medium [4, 5, 6],allowing a light pulse not only to be stopped and trapped[7], but also to be released by scattering with anothercontrol pulse, thus creating a “gap soliton memory” [8].The fundamental process underlying such novel lightpulse dynamics is the cooperative action of nonlinearityand periodicity. Still, a serious drawback when makinguse of RABR is the built-in periodic structure that re-stricts both the freedom of pulse parameter and of pulsedynamics.We propose to prepare a two-level system (TLS) byestablishing a standing electromagnetic wave in a givenpolarization direction, and then to scatter a light pulse,orthogonally polarized. The incident pulse then feels the electromagnetic induced grating through the coupling me-diated by the population density, as described by the gov-erning Maxwell-Bloch (MB) system [9, 10]. The freedomin the choice of the standing wave parameters (in partic-ular the boundary amplitudes) allows us to demonstrateby numerical simulations as in Fig.1 that the incidentlight pulse not only can be stopped but also can be re-leased back to the incoming end. Engraving a mediumwith a cavity standing wave is a method previously usedto create two dimensional waveguide arrays in stronglyanisotropic photonic crystals [11].In a TLS of transition frequency Ω, the MB systemis considered in the isotropic case for a plane polarizedelectromagnetic field propagating in direction z . Thetime is scaled to the inverse transition frequency Ω − , thespace z to the length Ω c/η ( η is the optical index of themedium), the population inversion to the density of ac- tive dipoles N , the energy to the average W = N ~ Ω / p W /ǫ and the polarization to √ ǫW .The resulting dimensionless MB system then reads E tt − E zz + P tt = − γ E t , P tt + P + αN E = − γ P t , (1) N t − E · P t = − γ (1 + N ) . where E and P , denote vectors in the tranverse plane,e.g. E = ( E x ( z, t ) , E y ( z, t )). The coupling eventuallyresults in a unique dimensionless fundamental constant α = 2 µ c / (3 η ) | µ | N / ( ~ Ω)( η + 2) / µ is averaged over the orientations [10])and by the normalization of the population inversion den-sity: N = − N = 1 in the excited state. The dimension-less dissipation coefficients are γ = ( c/η )( A / Ω) resultingfrom the electric field attenuation A , then γ = 1 / (Ω T )and γ = 2 / (Ω T ) resulting respectively from the popu-lation inversion dephasing time T and the polarizationdephasing time T .In the strong coupling case α ∼ i ( ωt − kz )] on a medium at rest,namely ω ( ω − ω ) = k ( ω − , ω = 1 + α, (2)The upper edge of the stop gap at frequency ω corre-sponds to k = 0 and dω/dk = 0. Numerical simulations.
We first proceed with numer-ical simulations of the MB equations (1) submitted to thefollowing boundary-value problem E (0 , t ) = (cid:18) E (0)1 cos( ω t ) E ( t ) cos( ω t ) (cid:19) , (3) E ( L, t ) = (cid:18) E ( L )1 cos( ω t )0 (cid:19) , (4)where E ( t ) is the slowly-varying low-amplitude pulse en-velope E ( t ) = E max cosh[ µ ( t − t )] . (5)The carrier frequencies are chosen close to the gap edge ω , inside the passing band for the grating field, in thegap for the incident pulse, namely ω > ω , ω < ω , | ω j − ω | ∼ O ( ǫ ) (6)for j = 1 ,
2, where ǫ is our small control parameter. Weset from now on α = 1. FIG. 1: (Color in line) A typical result of a numerical simu-lation of (1) under boundary values (3-4) with parameters E (0)1 = E ( L )1 = 0 . ω = 1 . E max = 0 . ω = 1 . t = 1000, µ = 1 / y , trapped bythe grating induced by the stationary wave. The pulse is even-tually released back at time 4500 by increasing the amplitudeon the incident side E (0)1 by 25%. We have chosen damping coefficients such as to reach afast stabilization of the lattice created by the stationarydriving: in the x -direction we have set γ = γ = 0 . γ = 0 . y -direction they have beenset to zero in order to see clearly the process. Undersuch boundary data, with the parameter values indicatedin the caption, we obtain the result displayed in Fig.1.More precisely, the applied grating field (polarized along x ) is settled smoothly and, at time t = 1000 the incidentpulse enters the medium. It is naturally trapped and seen to oscillate about the center z = 30. At time t = 4500the left-hand-side amplitude E (0)1 is increased from 0 . .
5, which produces the reverse motion of the storedpulse. Actualy one may paly with the driving amplitudes E (0)1 and E ( L )1 to drive the pulse back and forth. Interpretation.
The process described above is nowunderstood, within a multiscale analysis of MB equa-tions, in terms of a discrete nonlinear Schr¨odiger modelwith variable coefficients related to the variation of theboundary grating field amplitudes. Thanks to (6), a so-lution of (1) under boundary values (3-4) can be soughtunder the form E ( z, t ) = (cid:18) ǫE ( ζ, τ ) ǫ E ( ζ, τ , τ ) (cid:19) e iω t + c.c. (7)with the slow variables ζ = ǫz and τ n = ǫ n t . The secondslow time τ is meant to capture the nonlinear dynamicsof the low-amplitude ( ǫ ) long duration ( ǫ − ) incidentpulse. Note that form assumption (6), the frequency shiftfrom ω is contained in the slow time variations with τ = ǫ t .Inserting then (7) in the MB system (1) we eventuallyobtain (with vanishing damping) i ∂E ∂τ − ω − ω ∂ E ∂ζ = ω + 34 ω | E | E , (8) i ∂E ∂τ − ω − ω ∂ E ∂ζ − ω + 12 ω | E | E + ω − ω E E ∗ = ǫ (cid:18) − i ∂E ∂τ + ω + 34 ω | E | E (cid:19) , (9)where star means complex conjugation. The boundaryvalues (3-4) for the x -component E imply from equa-tion (8) that E ( ζ, τ ) is a periodic stationary solutionof the nonlinear Schr¨odinger equation with frequency ν = ( ω − ω ) ǫ − . It acts then as an external periodicpotential in the evolution (9).In order to take into account variations | E | ( ζ ) result-ing from the variations of the boundary driving (3) weset (remember ν τ = ( ω − ω ) t ) E = | E | ( ζ ) e iν τ , | E | ( ζ ) = V ( ζ ) + ǫ V ( ζ ) (10)where V ( ζ ) is purely periodic while V ( ζ ) describesthe aperiodic inhomegenities of | E | ( ζ ) induced by theboundary values.We then seek a solution of (9) on a suitable orthonor-mal basis of Wannier functions ϕ j ( ζ ) which are localizedwith respect to the site j [14], within the one-band ap-proximation [15], where j actually indexes the minima ofthe periodic potential V ( ζ ). A solution of (9) is soughtas E = X j e iν τ F j ( τ , τ , ) ϕ j ( ζ ) , (11)and the equation for the coefficients F j is worked out byinserting (11) in (9) and by projecting on a chosen ϕ j . Inthe tight-binding approximation [16] we eventually obtain i ∂ F j ∂τ − (Ω + ν ) F j + Λ F ∗ j = ǫ (cid:2) − i ∂ F j ∂τ + Ω j F j − Λ j F ∗ j + Q ( F j − + F j +1 ) + U |F j | F j (cid:3) , (12)where the coefficients are given from the Wannier basisby (integrals run on ζ ∈ R )Ω = ω − ω Z ϕ ′′ j ϕ j + ω + 12 ω Z V ϕ j , Λ = ω − ω Z V ϕ j , Ω j = ω + 12 ω Z V ϕ j , Λ j = ω − ω Z V ϕ j ,U = ω + 34 ω Z ϕ j , ǫ Q = ω − ω Z ϕ ′′ j ± ϕ j . Note that translational invariance guarantees that theabove coefficients are j -independent, except of course forΩ j and Λ j that bear the aperiodic inhomogeneity of theexternal potential | E | ( ζ ).Equation (12) can now be solved first for the τ -dependence of F j as a linear system, which provides thena discrete nonlinear Schr¨odinger coupled system for the τ -dependent amplitudes. This is done by seeking a so-lution under the form F = G j e − i ∆ τ + G − j e i ∆ τ (13)for which the leading order of (12) furnishes the two cou-pled linear equations − h Ω + ν − ∆ i G j + Λ (cid:0) G − j (cid:1) ∗ = 0 , − h Ω + ν + ∆ i(cid:0) G − j (cid:1) ∗ + Λ G j = 0 , The dispersion relation and the relation between G j and (cid:0) G − j (cid:1) ∗ automatically follows as∆ = q(cid:2) Ω + ν (cid:3) − Λ , (cid:0) G − j (cid:1) ∗ = Λ Ω + ν + ∆ G j . At next order we readily get for G j the discrete non-linear Schr¨odinger equation, the parameters of whichare greatly simplified if we note that in our numericalsimulations (for sufficiently deep lattice) Ω ≫ Λ andΩ j ≫ Λ j . In such a case the equation reads i ∂ G j ∂τ − Ω j G j = Q [ G j +1 + G j − ] + U |G j | G j . (14)The electric field envelope in the y -direction of polariza-tion reads E = X j e − i Ω τ G j ( τ ) ϕ j ( ζ ) , (15) in terms of the solution of the chosen Wannier basis.The dynamics of the pulse is thus interpreted out of thediscrete nonlinear Schr¨odinger model (14) as the actionof the potential Ω j that translates the applied variationsof the boundary driving in the x -direction of polarization. Application.
In order to illustrate the above inter-pretation, we proceed now with numerical simulationsof (14) where τ = t to read quantities in physical di-mensions. The potential Ω j models the variations of | E | away from a purely periodic function as soon as E (0)1 = E ( L )1 . We setΩ j = (cid:26) , t < , − . j , t >
650 (16)and obtain the Fig.2 which shows the same qualitativebehavior as Fig.1
10 20 30 1000 2000 300000.51
Sites Time| Ψ | FIG. 2: (Color in line) Plot of the solution of (14) with theparameters Q = 0 . U = 0 . G ( t ) =0 . e − . it / cosh[( t − / j in (16). Conclusion and comments.
We have shown that theMaxwell-Bloch system, the celebrated fundamental andgeneral semi-classical model of interaction of radiationwith matter, may serve as a tool to store and drive lightpulses in an arbitrary way by conveniently using laserlight to engrave the medium with a controllable stand-ing wave pattern. The extreme genericity of MB modeltogether with the freedom in the boundary values of theengraving field, constitute a decisive advantage over othertechniques to store and manipulate light pulses.The process is understood by deriving a discrete non-linear Schr¨odinger model where the external tunable po-tential actually translates the effects of boundary drivingvariations. Our purpose was simply to provide a qualita-tive interpretation.A more detailed study, reported to future work, wouldrequire first to use exact solutions of (8) for the gratingfield (e.g. in terms of Jacobi elliptic functions), secondto construct the corresponding most adequate Wannierbasis, third to evalute precisely the effect of boundarydriving variations on the factor Ω j , and last to study thefull system (12) that couples F j to F ∗ j . Acknowledgements.
Work done under contract CNRSGDR-PhoNoMi2 (
Photonique Nonlin´eaire et Milieux Mi-crostructur´es ). R.K. aknowledges invitation and supportof the
Laboratoire de Physique Th´eorique et Astropartic-ules and USA CRDF award [1] C. Liu, Z. Dutton, C. Behroozi, L.V. Hau, Nature 409(2001) 490[2] A. Andre, M.D. Ludkin, Phys Rev Lett 89 (2002) 143602[3] M. Bajcsy, A.S. Zibrov, M.D. Ludkin, Nature 426 (2003)638[4] A.E. Kozhekin, G. Kurizki, Phys Rev Lett 74 (1995) 5020[5] A.E. Kozhekin, G. Kurizki, B. Malomed, Phys Rev Lett 81 (1998) 3647[6] N. Ak¨ozbek, S. John, Phys Rev E 58 (1998) 3876[7] W. Xiao, J. Zhou, J. Prineas, Optics Express 11 (2003)3277[8] I.V. Mel’nikov, J.S. Aitchison, Appl Phys Lett 87 (2005)201111[9] L.C. Allen, J.H. Eberly,
Optical Resonance and Two-Level Atoms , Dover (NY 1987)[10] R.H. Pantell, H.E. Puthoff,