Effects of spatial dispersion on Self--induced transparency in two--level media
Zoran Ivic, Dalibor Čevizović, Željko Pržulj, Nikos Lazaridess, George Tsironis
EEffects of spatial dispersion on Self–induced transparency in two–level media
Zoran Ivi´c,
1, 2
Dalibor ˇCevizovi´c, ˇZeljko Prˇzulj, Nikos Lazaridess,
2, 3 and G.P.Tsironis
2, 3 University of Belgrade, Vinˇca Institute, PO Box 522, 11001 Belgrade, Serbia National University of Science and Technology MISiS, Leninsky prosp. 4, Moscow 119049, Russia Crete Center for Quantum Complexity and Nanotechnology, Department of Physics,University of Crete, P. O. Box 2208, Heraklion 71003, Greece
We study the effects of dispersion in carrier waves on the properties of soliton self–induced transparency(SIT) in two level media. We found substantial impact of dispersion effects on typical SIT soliton features. Forexample, the degree of SIT pulse velocity slowing down (acceleration) is determined by the ratio of the incomingpulse frequency over atomic transition frequency - x = ω/ω . Specifically, an immediate pulse stopping ispredicted for absorbing media when pulse duration time exceeds some critical value. In the sharp line limitstopping may emerge only for frequency ratio above unity, while for the inhomogeneously broadened systemsit appears irrespective of the value of x . Analysis performed on the basis of Mcall& Hahn Area theorem impliesthat pulse stopping is achieved when Ber’s absorption coefficient approaches infinity, that is, pulse energy isfully absorbed in the medium. In the case of amplifying media super-luminal motion is predicted as in the caseof resonance. However, there is a lowest value in the frequency ratio below which the pulse velocity tends to thesub-luminal region. These new features of the SIT phenomenon open novel ways on how it may be exploited forthe control of electromagnetic wave radiation in two-level media. This may be achieved by varying frequencyratio.
I. INTRODUCTION
Propagation of short, intense, electromagnetic (EM) pulses,resonantly interacting with two–level atomic media, lead tothe emergence of a number of remarkable coherent coop-erative quantum phenomena such as Dicke superradiance,self–induced transparency (SIT), electromagnetically inducedtransparency (EIT), photon–echo, coherent population oscil-lation ... [1–13]. During the last two decades these phenom-ena attracted particular attention due to the potential for prac-tical applications, i.e., for realization of quantum memories [14, 15] that are devices fundamental for the future technolo-gies for quantum communication and processing [16–21]. Inthat context the achievement of control over the propagationof EM radiation by matter [22–25], and similarly, the manipu-lation of atoms, natural and the artificial ones (superconduct-ing or quantum dot qubits), by light [26], became a really im-portant issue in physics.Great practical successes were achieved using EIT whichturns out to be very useful allowing substantial slowing downand even stopping of light pulses in media composed of nat-ural atoms (atomic vapors [17, 27–34]). In this way, the in-formation carried by the pulse may be temporarily transferredto the medium. Pulses can then be ’revived’ with their origi-nal information intact. Nevertheless, EIT – based techniqueshave certain limitations in potential practical applications dueto the narrow transparency spectrum [35] and vulnerabilitydue to inevitable coupling with the environment leading torelaxation processes ( homogeneous broadening ) and dephas-ing ( inhomogeneous broadening ). Relaxation effects may besuppressed by exploiting high-intensity pulses. However, thismay cause damage to the medium. For that reason, the de-velopment of analogous techniques but adjusted for the mi-crowave domain may be useful. These novel trends rely onengineered media, quantum metamaterials (QMM), built withartificial ”atoms” made typically of superconducting circuits.This had motivated the investigations of a possible emer- gence of collective coherent quantum phenomena in QMMsand their implementation in design of quantum technologicaldevices [36–43]. The use QMMs had shown that, in paral-lel with EIT–based techniques, the use of SIT could be veryuseful. Specifically, employing SIT could make possible tophase out out inhomogeneous broadening , the main obstacleto quantum coherence, and actually turn it into an advantage.This is due to the fact inhomogeneous broadening is requiredfor the emergence of SIT. We note that the whole concept ofSIT relies on inhomogeneous broadening since the Area the-orem that is fundamental relies on it. Additionally, relaxationeffects (homogeneous broadening) may be avoided using (ul-tra)short light pulses with the duration far less than both trans-verse and longitudinal relaxation times.To recall, SIT is a lossless propagation of an optical pulsethrough an otherwise opaque optical medium composed ofa large number of inhomogeneously broadened two-levelatoms. According to the McCall – Hahn area theorem , whichis the main theoretical result of the whole concept, pulsestravel through the media with no gain or loss when their area θ ( x ) ∼ (cid:82) dtE ( x, t ) = nπ , where E ( x, t ) is the magnitude ofelectric field. For even (odd) n pulses are stable (unstable).When the pulse area is below π , the pulse gradually weakenswith traveled distance and finally disappears being absorbedby the medium; on the other side, pulses whose input areasare slightly above π increase their area up to a π after whichpoint continue stable propagation as π –soliton. Finally, allpulses with the areas nπ for n > split in two, three, etcsolitons.While the area theorem is a purely theoretical result, ob-tained under very restrictive assumptions, it is neverthelesssurprising how accurately it describes most of the main fea-tures of SIT as confirmed by numerous experimental results[3]. For that reason, at least concerning the original problem –SIT in a system of atomic vapors , it’s further theoretical con-sideration could be superfluous. However, recent proposals[39–43] of the practical applications of SIT effect in media a r X i v : . [ qu a n t - ph ] S e p built of artificial atoms, quantum metamaterials (QMM) forexample, reopens the importance of theoretical studies espe-cially those effects which have not been considered. One suchissue is the study of the dispersion effects, i.e. dependence ofthe carrier wave frequency on its wave vector – ω = ω ( k ) . Inthe original context [4–7] it was assumed that the dispersionlaw satisfies the simple relation ω ( k ) = ck , where c is thespeed of light in the medium. However, as shown recently inthe context of SIT in QMM [41–43] and exciton SIT [44, 45]this issue cannot be neglected easily and, under some circum-stances, turns out to be of particular interest resulting in somevery peculiar features.In this article, we study the effects of dispersion on SITwithin the original model introduced by McCall and Hahn[4, 5]. We base our study on a system of reduced Maxwell–Bloch equations (RMBE) obtained from the original one byeliminating fast oscillating terms in accordance with slowlyvarying envelope and phase approximation (SVEA) [3] lead-ing, in final instance, to well known solutions [3–7]. A briefand very instructive pedagogical overview may be found in[7]. II. EVALUATION OF THE DISPERSION LAW
The starting point of our analysis is RMBEs found throughconsistent application of SVEA. ˙ S x = − (∆ + ˙ φ ) S y , a) ˙ S y = (∆ + ˙ φ ) S x + κ E S z , b) ˙ S z = − κ E S y . c) (1)Here, S i ( x, t ) , E ( x, t ) and φ ( x, t ) are new slow dynamicalvariables corresponding, respectively, to transformed atomicfunctions, envelope and phase of the EM pulse. More pre-cisely, S x and S y correspond to ’dispersive’ and ’absorp-tive’ components of induced polarization of the medium: P ( x, t ) = N d (cid:104) s x ( z, t ) (cid:105) , where s x – quantum mechanical ex-pectation value of a x component of Pauli spin matrix ( σ x ) ina state being the superposition of ground and the excited ones.In terms of ’slow’ variables it reads s x = S x cos Ψ( x, t ) + S y sin Ψ( x, t ); Ψ( x, t ) = kx − ωt + φ ( x, t ) ; N stays forthe concentration of atoms, d is a dipole transition matrix ele-ment, ∆ = ω − ω ( k ) with ω and k are the frequency and thewave vector of a carrier wave and ω corresponding to atomtransition frequency. The angular brackets in the last systemrefer to the fact that in practice we deal with a system of the in-homogeneously broadened atoms. That is, ω corresponds toan atomic transition frequency that is different for each atom .In the case of a system composed of a large number of atoms,all of these frequencies may be taken continuously distributedaround some mean value. Since the large wavelength ( λ (cid:29) d )EM pulse interacts simultaneously with large number of IHbroadened ”atoms” the collective back–action on the propa-gating pulse must be described in terms of the average ”polar-ization” as follows– (cid:104) .... (cid:105) = (cid:82) ∆0 ( .... ) G (∆ (cid:48) ) d ∆ (cid:48) where G (∆) is normalized to unity ( (cid:82) ∞ d ∆ G (∆) = 1 ) line–shape function.Finally, c = c /n is the speed of light in TL medium with in-dex of refraction n . In addition to the above system, there aretwo more equations arising as a result of the transformation ofa single second order equation to a two first order ones for theamplitude and phase. c ω (cid:16) k − ω c (cid:17) E + (cid:16) ˙ φ + kc ω φ (cid:48) (cid:17) E = γω ω (cid:104) S x (cid:105) , a) ˙ E + kc ω E (cid:48) = γω ω (cid:104) S y (cid:105) , γ = 4 π N d. b)(2)The necessity for the consideration of dispersion effectsmay be viewed on the basis of equations (2) and (3). Wefirst recall that the simplest analytic solutions of these equa-tions exist in resonance ω = ω and for stationary phase ˙ φ ≡ φ (cid:48) = 0 . In that case the system of equations (2) and(3) greatly simplifies and may be solved using trigonometricparameterization in terms of Bloch angle ( S y = S sin θ and S z = S cos θ ) [3–7], which, satisfies sine–Gordon equation.In addition, employing the usual initial conditions that popu-lation inversion S = S z ( −∞ ) ≡ ± , from system (2), weobtain that S x = const ≡ . Its immediate consequence isthe dispersion law ω = ± kc (see eq. 3.a) that holds onlyat resonance. Out of resonance, this relation is only approx-imate and a correct treatment requires determination it’s trueform. In this case parametrization in terms of Bloch anglesis again possible but demands an ad hock assumption knownas factorization ansatz introducing the spectral response func-tion : S y (∆) = F (∆) S y (∆ = 0) . This approach leads onceagain to the SG equation for the Bloch angle. Knowing that ) S y (∆ = 0) = S sin θ and that ˙ θ = − κ E SG equation forBloch–angle became ¨ θ + kc ω ˙ θ (cid:48) = γω κ ω (cid:104) F (∆ (cid:105) sin θ. (3)Solutions of this equation are well known, they are π soli-tons. For the spectral function we use known relation [3–5]connecting it with the detuning ∆ and pulse duration time τ p However, eq. (4) and thus its solutions still contain a singleyet undetermined parameter k that appears in the evaluationof the soliton delay ratio ( v/c ), absorption coefficient ( α ) andthe area theorem. In other words, all these functions are fuc-ntions of k whose explicit knowledge is required to examinethe potential usability of SIT in practical applications. Thus,the relation (3) is useless in that respect. in order find the dis-persion law, we need to go back to the system and assumethat the pulse propagates undistorted in a soliton form. Thisenables us to take that all system variables depend on spatialcoordinates and the time only through the variable τ = t − xv (i.e. passing to moving frame). Accordingly, systems (1) and(2) become S x,τ = − (∆ + φ τ ) S y , a) S y,τ = (∆ + φ τ ) S x + κ E S z , b) S z,τ = − κ E S y . c) (4) (cid:16) φ τ + G ω Γ (cid:17) E = γω ω Γ (cid:104) S x (cid:105) , a) E τ = γω ω Γ (cid:104) S y (cid:105) , b) Γ = 1 − c kωv , G = ω − c k ω . c) (5)The last two equations, combined with the third and firstone of system 4, may be easily integrated to give S z and phase.It is trivial in the sharp line limit, while for the finite broad-ening we have to employ the factorization ansatz which en-ables one to expres (cid:104) S y (cid:105) through S y . For that purpose wetook S y in factorized form and its average as: (cid:104) S y (∆) (cid:105) = (cid:104) F (∆) S y (∆ = 0) (cid:105) ≡ (cid:104) F (∆) (cid:105) S y (∆ = 0) with F (∆) = τ p . Now we multiply the last expression with F (∆) F (∆) .This simple manipulation yields (cid:104) S y (cid:105) = (cid:104) F (∆) (cid:105) F (∆) S y .Employing this approach in the last equation we may com-bine it with the third one in 4 which finally yields: S z = S − ωκ Γ2 ω γ F (∆) (cid:104) F (∆) (cid:105) E . (6)We note that S is the initial population of TLS where S = − means that all TLS’s is in their ground state, while S =+1 means that we have all TLS’s in the excited state.We may now focus on the equation for phase–the first onein 5. We first differentiate it with respect to τ , then we use firstequation in 4 to eliminate (cid:104) ˙ S x (cid:105) . Also, in a final step we usesame trik as above to express (cid:104) S y (cid:105) through S y . This finallyyields : φ ττ E + 2 φ τ E τ + (cid:16) ˜∆ − G Γ (cid:17) E τ = 0 , ˜∆ = (cid:104) ∆ F (∆) (cid:105)(cid:104) F (∆) (cid:105) Its integration yields: φ τ = 12 (cid:16) ˜∆ − G Γ (cid:17) Here, we used the initial condition lim τ →−∞ E = 0 . At thisplace, it is necessary to recall that the phase φ was introducedthrough overall phase ψ ( x, t ) = kx − ωt + φ ( x, t ) in which lin-ear terms in x and t are already accounted for independentlyof φ , which, therefore, cannot contain terms linear in ( x and t ). That is, we must have take φ τ = 0 . In such a way, lastrelation implies: ˜∆ − G Γ = 0 (7)This is quadratic equation for the pulse wave vector k . Itmay be solved for k as function of ˜∆ , ratio ω/ω and pulsevelocity as a parameter. However, in a view of the analysisof experimental data, it is more convenient to examine k independence of the pulse duration instead of its velocity.For that purpose we use known solutions: E = E sech( τ /τ p ) E = (cid:114) γS κω Γ (cid:104) F (∆) (cid:105) , E τ p = 4 κ , (8) to obtain Γ = γκS ω x (cid:104) F (∆) (cid:105) ω τ p and eliminate it from (7).Our final results, expressed in terms of a dimensionless vari-ables: K = kcω – the carrier wave quasimomentum, the pulsevelocity V = vc and frequency ratio x = ωω , read: K ± = (cid:115) x − νS ω (cid:104) F (∆) (cid:105) ω τ p ,V ± = Kx − νS (cid:104) F (∆) (cid:105) ω τ p . (9)In the last expressions sign + ( − ) corresponds to initial pop-ulation inversion S = − ( S = 1 ). Here ν = γκ ω , here-after called material parameter, characterizes the strength offield–atom interaction. Apparently, the pulse propagation isdetermined both on its characteristics (duration time) and bythe properties of the material.At this stage, before detailed discussion of the pulse prop-agation, we perform some preliminary calculations, in orderto see how the initial conditions are reflected on the nature ofthe solution. To this end we derive an alternate form of thedispersion law. K ± = − ˜∆ ω V ± (cid:115) ( x + ˜∆ ω ) + ˜∆ ω ( 1 V − . (10)From the expression for the pulse amplitude we immediatelyfind that the existence of solutions requires Γ S > . In thecase of simple dispersion ω = ck it may be addressed to sub–luminal or super–luminal propagation for S = − or S = 1 .The same conclusion holds here and may be easily provedon the basis of (10). Note that condition Γ S > may berewritten as (cid:16) − KxV (cid:17) S > . (11)That is, S = − requires K > xV , which, together (10), af-ter some straightforward calculation, yields x (1 − V ) > implying sub–luminal propagation. The same reasoning re-sults with x (1 − V ) < for S = 1 leading to super–luminalmotion. III. DISCUSSIONA. Sharp line limit
In this case line shape function tends to delta function sothat: ˜∆ = ∆ and (cid:104) F (∆) (cid:105) = F (∆) which significantly sim-plifies further calculations and both, dispersion law and ve-locity as function of frequency ratio x = ω/ω attain simpleanalytic forms: K ± = (cid:115) x − νS (1 − x ) τ − x ) τ ,V ± = Kx − νS τ − x ) τ (12) k c / ω ω/ω τ =0.25τ =0.25 τ =1τ =1.94τ =4 S =-1 k c / ω ω/ω F1F2F3 τ =3τ =0.25τ =1 FIG. 1:
Sharp line limit : illustration of the soliton dispersion law k ( ω ) for a few different values of scaled pulse width: τ = ω τ p .Upper pane – absorbing media S = − .Lower pane – amplifyingmedia media S = 1 . We have graphically presented our results on figures (1) and(2) for absorbing ( S = − ) and amplifying ( S = 1 ) initialconditions.In both cases, around resonance ω/ω ∼ and for shortpulses τ < we observe similar results as those obtainedwithin the linear approximation ω = ck . That is, for ab-sorbing media, only subluminal motion is possible v < c .While dispersion law attains simple linear functional depen-dence and velocity gradually decrease as a function of dura-tion time. When pulse duration, for a given ν , exceeds somecritical value corresponding a minimum of the function τ = x ν (1 − x ) − (1 − x ) x , (13)sudden vanishing of k is observed when ω/ω ≥ . Thisindicates immediate pulse stopping. For example, τ crit ∼ . for ν = 1 . Above the critical pulse width two branchesin dispersion law appear. First branch lies in the interval
Sharp line limit : Pulse velocity delay v/c versus dimension-less pulse width ω τ p for a few values of the frequency ratio. Upperpane – absorbing media S = − . Lower pane – amplifying media S = 1 . B. Influence of the inhomogeneous broadening
For simplicity we took line shape function in the Lorentzianform: G (∆) = 2 τ ∗ π
11 + ∆ τ ∗ (14)where τ ∗ stays for the inhomogeneous broadening relaxationtime. This choice enables us easy analytic evaluation of theaverage values in the expressions for dispersion law and ve-locity delay: (cid:104) F (∆) (cid:105) = 11 + τ p τ ∗ , (cid:104) ∆ F (∆) (cid:105) = 2 πτ ∗ ln τ ∗ τ p − τ p τ ∗ , (15) ˜∆ = 2 πτ ∗ ln τ ∗ τ p − τ p τ ∗ . Accordingly, dispersion law and pulse delay became: K ± = (cid:115) x + 4 S νω τ ∗ π y ln yy − , y = τ p τ ∗ ,V ± = K ± x − νS ( ω τ ∗ ) y y (16)Our results are visualized in Fig.(3). In upper pane we haveplotted, in dimensionless units, dispersion law as function offrequency ratio ( x = ω/ω ) for a few different values of ratioof the pulse width over the inhomogeneous broadening relax-ation time – y = τ p /τ ∗ . In lower pane we have presentedpulse velocity as function of y = τ p /τ ∗ . Dispersion law ex-hibits substantially different behavior with respect to that ob-served within the sharp line limit. This particularly concernsthe near resonance case ( ω/ω ∼ ) where our results do nottend to the known ones obtained in the resonance. This is theconsequence of the appearance of the constant shifts, deter-mined by the ratio y , in the expressions for dispersion law andpulse delay (16). In both cases these shifts tend to zero when τ ∗ (cid:29) τ p and our results approach those obtained within thestrict resonance.Dispersion relation for finite values of y exhibits very spe-cific behavior for absorbing and amplifying media. That is, forabsorbing media, for each particular value of y there is a min-imal value of frequency ratio ( x = ω/ω ) below which thereis no solutions for K , that is pulse does not exist. When x exceeds this minimal value K ( x ) monotonically increases ap-proaching linear dependence for large x . For large y this limitis approached for the un–physically large values of x (cid:29) when presented theory of SIT does not hold.For the amplifying media, starting from some minimalvalue, K monotonically increases approaching, again, lineardependence for large x . In contrast to absorbing media pulseexists for all x .In the absorbing media, for each x , pulse velocity exhibitssimilar behavior as well as in the sharp line limit: as a function of y it gradually decay towards zero approaching it for somecritical value specific for each x .In the amplifying media super–luminal pulse motion is pre-dicted: for each the particular value of the frequency ratio, v ( y ) exhibits qualitatively the same behavior as well as in thecase of resonance with no accounted for effects of dispersion(represented with curve constructed of (cid:5) ). That is, as a func-tion of y , v/c monotonically increases from unity to infinity,while each curve may be recovered from the some particularone, say x = 1 , by simply rescaling frequency ratio. k c / ω ω/ω y=1.25 y=0.999y=0.5 S =1S =-1y=0.5 y=0.999 y=1.25 v / c τ p /τ * ω/ω =1ω/ω =1.25ω/ω =0.5S =1 S =-1 FIG. 3:
Illustration of the impact of the inhomogeneous broad-ening on SIT pulse properties : Upper pane – carrier wave quasi–momentum versus frequency ratio. Dispersion relation for absorbingmedia ( S = − ) is visualized by full lines, while to amplifying me-dia correspond dotted lines. Blu full line corresponds to resonancecase ω = kc . Lower pane – pulse velocity delay. Full lines cor-respond to absorbing media S = − . Dotted lines correspond toamplifying media. Curves indicated by (cid:5) stand for resonance. IV. CONCLUDING REMARKS
Our study reveals some new features of the SIT phe-nomenon stemming from the spatial dispersion of the carrierwave ω ( k ) . We found that the properties of the SIT pulse sub-stantially depends on the frequency ratio ( x = ω/ω ). Thisparticularly concerns the delay of the pulse velocity whichmay be controlled by means of varying of x . In that sense themost interesting consequence is the possibility of full stoppingof SIT pulse. In particular, in the sharp line limit, for eachvalue of x > EM pulse of is fully stopped (absorbed) by themedium provided that its width (duration time) exceeds somecritical value. In the case of inhomogeneously broadened me-dia each EM pulse gets stopped irrespectively on the value of x , that is there is no any limitation on the value of x whichmay be arbitrarily low provided that pulse is wide enough.In order to relate dispersion with absorption coefficient wederive the Area theorem from the system (2) ∂θ∂x = β θ, β = γω G (0) κ c k ≡ γω τ ∗ κ c k . (17)Apparently, absorption coefficient due to /k dependencetends to infinity, indicating its full absorption for k (cid:55)→ . In conclusion, our results point to possible new means ofthe control of propagation of EM waves. It relies on the pre-diction of possible dramatic influence of the frequency ra-tio x = ω/ω on carrier wave of SIT pulse and its veloc-ity whose vanishing may be expected for a convenient choice of x and pulse duration time. In systems built of naturalatoms, this may not be easily realized due to small valuesof material constant. Nevertheless, tunability of the param-eters of artificial atoms may enhance the predicted effect andmake it possible. Also, by applying an additional driving fieldlike in EIT, mixed induced transparency (SIT+EIT), may beachieved where the best features of both effects were exploitedas shown in [46]. Acknowledgments
This work was partially supported by the Ministry of Edu-cation, Science and Technological Development of RepublicSerbia, Grants No. III - 45010 and OI - 171009, the Ministryof Science and Higher Education of the Russian Federation inthe framework of Increase Competitiveness Program of NUST”MISiS” (No. K2-2019-010), implemented by a governmen-tal decree dated 16th of March 2013, N 211. NL also acknowl-edges support by General Secretariat for Research and Tech-nology (GSRT) and the Hellenic Foundation for Research andInnovation (HFRI) (Grant no.: 203). [1] M. Gross and S. Haroshe, Superradiance: An essay on the the-ory of collective spontaneous emission, Phys. Reports 93, 301–396, (1982)[2] R. Dicke, Coherence in Spontaneous Radiation Processes Phys.Rev. 93 (1954)[3] L. Allen, J. Eberly, Optical resonance and two-level atoms,Courier Dover Publications, 1987.[4] S. L. McCall and E. L. Hahn, Phys. Rev. , 457 (1969).[5] G. L. Lamb, Rev. Mod. Phys. , 99 (1971).[6] J. C. Eilbeck, J. D. Gibbon, P. J. Caudrey,[7] N. Theodrakopulos, Nonlinear Physics (Solitons, Chaos, Dis-crete Breathers)