ABSORPTION-FREE SUPERLUMINAL LIGHT PROPAGATION IN A V-TYPE SYSTEM

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1 Amenin Jounl of Physis 0 vol. 4 issue ABSORPTION-FREE SUPERLUMINAL LIGHT PROPAGATION IN A V-TYPE SYSTEM S. W. Rbiei* Kh. Sidi B. Ruzbhni M. Mhmoudi 3 Islmi Azd Univesity - Snndj Bnh Snndj In Physis Detment Univesity of Kudistn Snndj In 3 Physis Detment Znjn Univesity Znjn In *E-mil: w.biei@uok..i; E-mil: ksidi@uok..i; 3 E-mil: mhmoudi@isbs..i Reeived 5 Ail 0 Abstt Disesion nd bsotion oeties of wek obe field in thee-level V-tye tomi system is studied. By lition of indiet inoheent um fields the effet of oulting ue levels on otil oeties of the tomi medium in esene of stong oheent um field is investigted. It is shown tht the sloe of disesion swithes fom ositive to negtive just by hnging the intensity of the oheent o indiet inoheent um fields. It is demonstted tht the bsotion-fee sueluminl nd subluminl light ogtion e in this system.. Intodution In the st few dedes thee hve been temendous inteests in the study of subluminl nd sueluminl light ogtion [ 5]. The gou veloity of light ulse n be edued in Bose- Einstein ondenste of sodium tom gs [] nd in hot gses [3] nd even hlted in vo of Rb toms [4]. Also it n exeed the vuum light seed nd n even beome negtive [5]. These exeiments e bsed on the ft tht eletomgnetilly indued tnseny (EIT) nd eletomgnetilly indued bsotion (EIA) [6] led to disesion ofile with sh ositive o negtive deivtive [7 8]. Anomlous disesion ws fist studied in mehnil osilltos [9] nd ws lte lied by Sommefeld nd Billouin [0] to light ogtion in bsotive oque mteils. They showed theoetilly tht inside n bsotion line the disesion n be nomlous esulting in gou veloity fste thn the vuum seed of light. Suh n nomlous veloity es due to the wve ntue of light [ ]. Tlukde et l. hve shown femtoseond lse ulse ogtion hs swithed fom sueluminl to subluminl veloities in n bsobing dye by hnging the dye onenttion [3]. Shimizu et l. wee ble to ontol the light ulse seed with only few old toms in high-finesse miovity by detuning the lse fequeny fom vity esonne fequeny-loked to the tomi tnsition [4]. In seies of es [5 ] Chio nd owokes showed theoetilly tht nomlous disesion n ou inside tnsent mteil. It ws edited tht by using gin doublet [8] it is ossible to obtin tnsent nomlous disesion egion whee the gou veloity of light ulse exeeds with lmost no ulse distotion. Wng et l. used gin-ssisted line nomlous disesion in tomi esium gs nd the gou veloity of lse ulse in thei exeiment exeeded nd ould even beome negtive while the she of the ulse ws eseved [5]. The inoheent uming fields n lso ly n imotnt ole in the ontolling of the gou veloity of light in disesive medi

2 Amenin Jounl of Physis 0 vol. 4 issue [3 4]. The double-λ setu is nothe sheme whih ovides vey ih setum of henomen bsed on tomi oheene [5]. Reently the disesion nd the bsotion oeties of wek obe field in simle multi-level quntum systems hve been onsideed [6 8]. They showed tht by using oheent nd n indiet inoheent um field the gou veloity of light n be ontolled. Thee hve been only few exeimentl nd theoetil studies in whih both fee bsotion sueluminl nd subluminl light ogtion in single system hve been elized. In this tile the disesion nd the bsotion oeties of wek obe field in thee-level V- tye tomi system is investigted. By lition of indiet inoheent um fields the effet of oulting ue levels on otil oeties of the tomi medium in esene of stong oheent um field is investigted. It is shown tht the sloe of disesion hnges fom ositive to negtive just by djusting the intensity of the oheent o indiet inoheent um fields. It is demonstted tht the bsotion fee sue-luminl nd subluminl light ogtion e in this system.. The Model nd Disussions The model onsists of losed thee-level V-tye tomi system with gound level nd two ue levels nd 3 s shown in Fig.. A stong oheent ouling field with Rbi fequeny Ω =E h dives tnsition while wek tunble obe field with Rbi fequeny Ω =E 3 h is lied to tnsition 3. Hee j ( j = 3) e the tomi diole moments nd E ( E ) is the mlitude of the ouling (obe) field. By indiet inoheent ums with tes nd the oultion of the gound level n be umed to the exited levels nd 3 vi some unseified uxiliy sttes [9 30]. The sontneous dey tes fom sttes nd 3 to the gound level e denoted by γ nd γ 3 esetively. The density mtix equtions of motion unde the otting wve oximtion nd in otting fme e: whee ( ) ρ = iω ρ + iω ρ iω ρ iω ρ +γ ρ +γ ρ + ρ () * * ρ = Ω ρ Ω ρ γ ρ + ρ (b) * i i ρ = iω ρ iω ρ γ ρ + ρ () * * * * ( i ) i i i ρ = Δ Γ ρ +Ωρ Ωρ Ωρ (d) 3 * * * ( i ) i i i i( ) i i ρ = Δ Γ ρ + Ω ρ Ω ρ Ω ρ (e) ( ) * ρ = Δ Δ Γ ρ + Ω ρ Ω ρ (f) Γ ij e the oheene dey tes given by

3 Δ =ω ω nd Δ 3 ( )( ) ( )( ) ( )( ) Γ = γ + + Γ 3 = γ Γ = γ +γ 3 3 Amenin Jounl of Physis 0 vol. 4 issue =ω ω e the detuning of the ouling field nd the obe field esetively. Note tht the intefeene due to the diffeent sontneous emission hnnels hs been ignoed. The esonse of the tomi system to the lied fields is detemined by the susetibility χ whih is defined s [3] * ( N )( ) χ= ƒ ε ρ Ω (3) whee N is the tom numbe density in the medium. The el nd imginy ts of χ oesond to the disesion nd the bsotion esetively. The so-lled gou index n = v is lso intodued whee is seed of light in the vuum nd v g = + πχ ω + πω ω χ ω ( ) ( ) ( ) is the gou veloity of the obe field [5 3]. Hee χ is the el t of χ. Eq. (4) shows tht when χ is negligible the sloe of disesion hs the mjo ole in detemintion of the gou veloity nd the gou index. In ou nottion the ositive (negtive) sloe of disesion oesonds to the inese (edution) of the gou index nd the ositive (negtive) vlue of g g () (4) χ = Im χ shows the ttenution (mlifition) of the obe field. It is ent tht negtive gou index mens negtive gou veloity. The stedy stte solutions of Eqs. () fo the wek obe field oximtion i.e. Ω γij se of tuned ouling field ( Δ = 0 ) e Ω +Γ ρ = γ Γ γ + Ω (5) Ω +Γ ρ = ρ Γ γ + Ω ( 33 3) * ( i )( ) (5b) ρ = γ ρ (5) ρ = Ω Γ ρ ρ (5d) * iω Ω ( ) ( ) ( ) ( ) ( ) ρ 3 = ρ ρ33 ρ ρ. Γ3 iδ + Ω Γ3 iδ Γ Γ3 iδ (5e) Fo simliity fom this oint on it is ssumed tht the Rbi fequenies ( Ω nd Ω ) e el numbes the sontneous dey tes e the sme fo ll levels i.e. γ =γ 3 =γ nd othe metes e nomlized to the sontneous dey te γ. In the following using the deived 40

4 Amenin Jounl of Physis 0 vol. 4 issue exession fo susetibility χ in tems of given system metes the esonse of the tomi system to the lied oheent nd inoheent fields is studied. It hs to sy tht the min inteest is in disesion nd bsotion oeties of the obe field ound zeo detuning Δ = 0. Fig.. The oosed level sheme. A thee-level V-tye tomi system diven by stong oheent ouling nd wek tunble obe filed. The indiet inoheent ums e denoted by the dshed uve line. Fig.. () Rel nd (b) imginy ts of susetibility vesus obe field detuning fo metes γ = 0 = = Ω = 0.5 (solid) 0.7 (dshed).7 (dot-dshed) nd 4 (dotted). Fist the fmili effet of the ouling field in bsene of the indiet inoheent ums ( = = 0 ) is mentioned. In this se due to lition of the ouling field the bsotion line fo obe field slits into two bsotion lines nd onsequently the sloe of disesion hnges fom negtive to ositive. In Fig. the el () nd imginy (b) ts of the susetibility e lotted vesus the obe field detuning. It shows tht inesing the vlue of the Rbi fequeny Ω leds to the ene of tnsent egion between the bsotion lines ound zeo 4

5 Amenin Jounl of Physis 0 vol. 4 issue detuning Δ = 0 whih is n EIT sitution ootionl to lossless subluminl light ogtion. Fo the next ste while the stength of the seond indiet inoheent um is zeo ( = 0 ) the effets of diving tnsition by the ouling field nd oulting level 3 is investigted. To get ough view of this ondition it will be useful to dw some ontou lots fo ng nd χ vi the ouling Rbi fequeny Ω nd the fist indiet inoheent um stength while nd Δ e zeo. Using Eqs.(5) nd djusting sle bitily by letting Nƒ ε = nd 3 0 πω = γ the exliit exessions fo n nd χ in tems of Ω nd t the zeo 0 obe field detuning ( Δ = 0 ) e nd g { } ( ) ( )( ) ( ( ) ) ( ) n = γ A B Ω + Ω (6) g esetively whee ( ) ( )( ) χ = 4 + Ω AB + + AB (7) A= Ω ( ) ( ) B= Ω. Fig. 3. Contou lots fo ng = 0 (ed solid lines) nd 3 (blk-dshed lines) () χ = 0 (edsolid line) nd (blk-dshed lines) (b) vi Ω nd. The ommon metes e γ = = 0 Δ = 0 nd Δ = 0. 4

6 Amenin Jounl of Physis 0 vol. 4 issue Fo = 0 equtions 6 nd 7 e used to dw the lot ontous of Fig. 3 nd Fig. 3b esetively. In Fig. 3 the sueluminl egions e oloed nd the subluminl egion is left white. In ddition the seting ontous ( ng = 0 ) e lotted with ed-solid lines. In this figue the ontou lines fo ng = nd 3 e lotted in blk-dshed style. Fig. 3b illusttes the bsotion oety of the medium χ. In this figue the bsotive nd mlifying egions e seted by the ontou line of χ = 0 (ed-solid line) nd the dkness of the oloed egions is mesue of thei bsotion o mlifition mgnitude fo the obe field. Theefoe lighte olo is used to illustte the moe tnsent egion. Finlly the ontou lines of χ = nd e lotted in blk-dshed style. As Fig. 3 illusttes thee e two egions of nomlous disesion nd one egion of noml disesion. As detemined by Fig. 3b the fist egion of negtive disesion is due to the existene of n bsotion line ound zeo obe field detuning nd ous whee Ω <. The solid lines in Fig. n be egded s n exmle of this egion. The othe egion of negtive disesion hens whee Ω > while oding to Fig. 3b it omnies by gin. Fo the egion of noml disesion s Fig. 3b demonsttes the bsotion oety n tke eithe smll ositive o lge negtive vlue ootionl to negligible ttenution o stong mlifition of the obe field esetively. Fig. 4 nd Fig. 5 e esented to lify how ositive nd negtive disesions n ou in the gin egion of Ω nd lne. By the indiet inoheent um of stength >γ (see Eq.5) the invesion of oultion between levels nd 3 is estblished. In this sitution the system shows mlifition fo the obe field. Then thee exists gin dee with ositive sloe of disesion ound zeo detuning (see the ed-solid lines of Fig. 4). By lying the ouling field level is slit into two symmeti dessed levels d + nd d with fequeny diffeenes of Ω fom the oiginl level. When the ouling field is wek thee still seems to be one gin dee nd the dessed levels e indistinguishble. By stengthening the ouling field the gin dee is tnsfomed to two gin dees (see Fig. 4). This is ootionl to hnging the noml sloe of disesion to nomlous (negtive) disesion whih leds to exeeding the gou veloity fom (seed of light in vuum) o even my use its beoming negtive. When the dessed levels e tly distinguishble system my still exhibits onsideble gin ound zeo detuning whih oesonds to existene of noise in the medium fo obe ulse with middle fequeny of ω. 3 By hoosing lge Rbi fequeny fo the ouling field the gin beomes lesse nd the nomlous egion beomes wide. Then to hieve tnsent medium with nomlous 43

7 Amenin Jounl of Physis 0 vol. 4 issue disesion i.e. sueluminl light ogtion stong enough ouling field should be lied while the oultion is inveted fo the obe field tnsition [6 7]. If = 0 s Fig. 3 shows Ω > is the suffiient ondition to swith the obe ulse ogtion fom subluminl to sueluminl by lition of the indiet inoheent um of stength. Fo Ω = is the theshold vlue t whih the sloe of disesion hnges fom ositive to negtive. On the othe hnd the steeest nomlous sloe of disesion ous when < Ω < 3 nd stong enough indiet inoheent um oultes level 3. Fig. 5 is esented to see how n inese in uses the gin dees to beome moe distinguishble nd onsequently the nomlous sloe of disesion beome steee. In Fig. 5 fo =. (solid line) the sloe of disesion ound zeo obe field detuning is ositive. With inesing the indiet inoheent um te to =.5 (dshed lines) the sloe of disesion beomes zeo nd it is negtive fo = 3 (dsh dotted lines). In Fig. 5b the obe field bsotion uves e lotted. It shows tht the subluminl nd sueluminl light ogtion ound zeo obe detuning is omnied by onsideble gin. It is time to investigte the otil esonse of the medium to oulting level while the tnsition 3 is diven by the oheent field nd the oultion between levels nd 3 is inveted due to the indiet inoheent um with te >γ. In Fig. 6 the el () nd imginy (b) ts of χ e lotted vesus obe field detuning. The ommon metes e Ω =.69 nd =.5 γ. The fist lots (solid lines) oesonding to = 0 e esented fo omison only. They show tht the sloe of disesion is lmost zeo ound zeo obe field detuning nd thee is onsideble mount of gin due to the tly ombined gin dees. By oulting level with te =γ (dshed lines) the sloe of disesion beomes negtive ound Δ = 0. The bsotion uve vies oding to the obe field detuning so tht the mximum edution in gin ous ound zeo obe field detuning while beyond Δ =±Ω γ the gin ineses slightly. By inesing the vlue of to.43γ (dshed-dotted lines) the egion of nomlous disesion beomes wide nd its negtive sloe ineses in bsolute vlue. The bsotion uve undegoes vitions simil to those of the evious se ( = γ ). The vlue of the gin fo the ltte se ( =.43γ) vnishes t Δ = 0. The oe vlue of esulting in zeo gin ound Δ = 0 n be obtined by + 4Ω Ω. ( Ω ) = + Ω + (8) 44

8 Amenin Jounl of Physis 0 vol. 4 issue Fig. 4. () Rel nd (b) imginy ts of susetibility vesus obe field detuning fo the metes γ = =.5 = 0 Ω = 0.7 (solid).7 (dshed) 4.0 (dot-dshed) nd 6.0 (dotted). Fig. 5. () Rel nd (b) imginy ts of susetibility vesus obe field detuning fo the metes γ = Ω =.69 = 0 =. (solid).5 (dshed) nd 3.0 (dot-dshed). 45

9 Amenin Jounl of Physis 0 vol. 4 issue Fig. 6. () Rel nd (b) imginy ts of susetibility vesus obe field detuning fo the metes γ = =.5 Ω =.69 = 0 (solid).0 (dshed) nd.43 (dot-dshed). b Fig. 7. () The gou index ( ng ) vesus fo the metes = 0.0 Ω =.0 (solid).7 (dshed) 4.0 (dot-dshed) nd 6.0 (dotted); (b) The gou index vesus Ω fo the mete = 0.0 =.5 (solid).0 (dshed) 3.0 (dot-dshed) nd 5 (dotted); ()The gou index vesus while = ( Ω ) defined in eqution (8) fo the metes Ω = 0.5 (solid) 0.75 (dshed).0 (dot-dshed) nd.7 (dotted). Othe ommon metes e γ = Δ = Δ = 0. 46

10 Amenin Jounl of Physis 0 vol. 4 issue Fig. 7 dislys the effet of inoheent nd oheent um fields on the gou index. Fig. 7 shows the gou index vesus indiet inoheent um te fo Ω =.0 (solid).7 (dshed) 4.0 (dot-dshed) 6.0 (dotted) while = 0.0. Fig. 7b dislys gou index vesus oheent um field Ω fo =.5 (solid).0 (dshed) 3.0 (dsh-dotted) nd 5.0 (dotted) while = 0. In Fig. 7 the gou index is lotted s funtion of fo Ω = 0.5 (solid) 0.7 (dshed).0 (dot-dshed) nd.7 (dotted) while the seond indiet inoheent um hs te ( ) = Ω defined in Eq.(8). It shows tht even fo Ω < the obe ulse n ogte sueluminlly while the medium is tnsent if oe vlues fo nd e hosen. 3. CONCLUSION In onlusion we hve ontolled the disesion nd the bsotion of wek obe field in thee-level V-tye tomi system. By lition of indiet inoheent um fields the effet of oulting ue levels on otil oeties of the tomi medium in esene of stong oheent um field hs been investigted nd it hs been found tht line ositive o negtive tnsent disesion ould ou between the doublet bsotion o gin lines esetively. Then the bsotion fee sueluminl light ogtion hs been estblished in this system. REFERENCES. L.V.Hu S.E.His Z.Dutton C.H.Behoozi Ntue (London) (999).. M.O.Sully Ntue (London) (003); M.O.Sully M.S.Zubiy Siene 30 8 (003) 3. M.S.Bigelow N.N.Leeshkin R.W.Boyd Siene (003); M.S.Bigelow NNLeeshkin R.W.Boyd Phys. Rev. Lett (003). 4. D.F.Phillis A.Fleishhue A.Mi R.L.Wlswoth M.D.Lukin Phys. Rev. Lett (00); M.Bjsy A.S.Zibov M.D.Lukin Ntue (London) (003). 5. L.J.Wng A.Kuzmih A.Dogiu Ntue (London) (000); A.Dogiu A.Kuzmih L.J.Wng Phys. Rev. A (00). 6. A.Lezm S.Beio A.M.Akulshin Phys. Rev. A (999); A.V.Tihenhev A.M.Tumikin V.I.Yudin Phys. Rev. A (R) (999). 7. S.E.His Phys. Tody (997) 8. M.Xio Y.Q.Li S.Z.Jin J.Ge-Bnlohe Phys. Rev. Lett (995). 9. Lod Ryleigh Phil. Mg. XLVIII 5 (899). 0. L.Billouim Wve ogtion nd Gou Veloity Ademi Pess New Yok M.Bon E.Wolf Piniles of Otis Cmbidge Uni. Pess Cmbidge L.D.Lndu E.Lifshitz Eletodynmis of Continuous Medi Pegmon Oxfod Md.A.I.Tlukde Y.Amgishi M.Tomit Phys.Rev. Lett (00). 4. Y.Shimizu N.Shiokw N.Ymmoto M.Kozum T.Kug L.Deng E.W.Hgley Phys. Rev. Lett (00). 5. R.Y.Chio in Amzing Light R.Y.Chio Ed. Singe New Yok

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