Investigation of laser pulse propagation in a three level atomic medium in the presence of eit effect

MINISTRYOFEDUCATIONANDTRAININGVINHUNIVERSITY - - HOANGMINHDONG INVESTIGATIONOFLASERPULSEPROPAGATIONINA THREE-LEVELATOMICMEDIUM INTHEPRESENCEOFEITEFFECT Specialty:Optics Code:62.44.01.09 ASUMMARYOFPHYSICALDOCTORALTHESIS NGHEAN,2017 WorkcompletedatVinhuniversity Supervisors:Prof.Dr.DinhXuanKhoa Reviewer1:Prof.Dr.NguyenQuangBau Reviewer2:Assoc.Prof.DrChuDinhThuy Reviewer3:Assoc.Prof.DrTranHongNhung Thethesiswillbepresentedatschool-levelevaluatingCouncilat: …… ….h…………,date…………month… year2017 Thethesiscanbefoundat:NationallibraryofVietnamorNguyenTh ucHaoinformationandlibrarycenter,Vinhuniversity INTRODUCTION In the past decades, the topic of laser pulses propagation withoutdistortion (soliton) has been attracted much research attention of scientistsbecauseoftheirpotentialapplicationsinopticalinformationanddatapr ocessing.Infact,whenthelightpulsepropagatesint h e r e s o n a n t medium, due to the absorption and dispersion that lead to reduction anddistortion of the signal pulse Therefore, to obtain pulse stability, we oftenuseultrashortpulseswithhighintensity.Moreover,mosto f t h e applicati ons in modern photonic devices often require low-intensity lightwith high sensitivity Therefore, reducing the absorption in the resonancedomain is an excellent solution to reduce the intensity of light pulse andincrease theoperatingefficiencyof photonicdevices Currently,aninterestingsolutionisusedtoreduceabsorptioniselectromag netici n d u c t i o n t r a n s p a r e n c y ( E I T ) e f f e c t T h e b a s i s o f E I T i s theresultofquantuminterferencebetweentheprobabilitya m p l i t u d e s withi ntheatomicsystemwhichisinducedbylaserfields.UsingEITtechnique, some research groupshaveobtainedstablelaserpulses(soliton)in EIT medium Most recently, T Nakajima and coworkers studied thepropagation of two short laser pulse trains in a three-level lambdatypeatomicmediumunderEITconditions.Theyobtainedlaserpulsespropagatin g EIT medium without distortion in picosecond domain.Theinitial studies of pulse propagation inEIT mediumo f t e n ignorethe e f f e c t ofDopplerbroadening.Thiscan onlybesuitableforcoldato micmediumor ultra-short laser pulses Moreover, many applications such as opticalcommunications that require working with long laser pulses in nano ormicro-second domain In addition to quantum interference effects of the shift probabilityamplitude,thereisaquantuminterferenceeffectoccursbetweenspon taneous emission channels by the non-orthogonal orientation of theelectric dipole moment is induced by two laser fields This interference willcreateacoherentofatomsiscalledthecoherenceisgeneratedbyspontaneouse mission(spontaneouslyGeneratedCoherence-SGC).Experimentally, SGC effects were observedforthefirsttimebyXiaandetal 1996 in molecular sodium The presence of the SGC makes the mediumbecome more transparent and narrower the line width; larger and steeperdispersion Furthermore, the results also show that the effect of SGC makesan asymmetric medium so that the response of the medium is very sensitivetothephaseoflaserfields Sofar,theinfluencesofSGCandrelativephaseontheopticalpropertieso f t heEITmediumundersteady-stateregimehavepublished However,theseinfluencesonpulsepropagationeffecthavenotbeeninvestigated With the urgency of the issue of research and the reasons mentionedabove,wechooseresearchtopic"investigationoflaserpulsepropagationi nathree-levelatomicmediuminthepresenceofEITeffect" TheaimofthethesisistostudytheeffectsoflasercouplingparametersandD opplerbroadeningontheprocessoflaserpulsepropagationindifferentpulsedomains.Studyingon influences of the non-orthogonal orientation of electric dipole moments and the relative phase onthelaserpulsepropagationinthepresence ofincoherentpump Chapter1 PULSEPROPAGATIONINRESONANTMEDIA 1.1 Interactionbetweentwo-levelatomsandlight Weassumeasingleopticalfieldpropagatinginthezdirectionhave form:   → Ez,tz,t e ikzt c.c, (1.1) where z,t ist h e e n v e l o p e f u n c t i o n , ωis i s t h e f r e q u e n c y o f l i g h t , a n d k = ωis/ccisthewavenumber 1.1.1 DensityMatrixFormalism Realm e d i a o f t e n c a n n o t b e s u f f i c i e n t l y d e s c r i b e d b y a s i n g l e w a v e functionand,inthatcase,thedensitymatrixformulasisnecessary, 12  cc12*  11  c        c * (1.2)   c c2  21 22 , 2  For both mixed and pure states, the density matrix must have unittrace (1122 1) so that probability is conserved in our closed system.Thedensitymatrixoperator’sevolutionaccordingtotheLiouvilleequation:  i  (1.3)   H, t  In matrixform,theatom-fieldinteractionHamiltonianis:  → →  H  →1→  d.E  d*E     (1.4) 1.1.2 Atomicevolutioninrotatingwaveapproximation In the rotating wave approximation (RWA), to simplify theHamiltonian,weintroducetheunitary transformmatrix:  1 i t (1.5) U e 1  0 e ikzt  Hamiltonianintherotatingwavebasisas:   HRW UHU†i U U†  t  In termsofthedetuning,  HRW   (1.6)  21,theHamiltonianinthisbasisis: → →   d12.E e ikzt   (1.7) → → ikzt  d 21.E   e     Inmakingtherotatingwaveapproximation,wenegle ctthefast→ oscillating 2itt e r m s , a s s u m i n g t h a t t h e e n v e l o p e f u n c t i o n   z,t  v aries slowlyc o m p a r e d t o t h e c a r r i e r w a v e U s i n g t h i s a p p r o x i m a t i o n , w e f i n d totalHamiltonianinthe RWA:   → →       d12.  (1.8) d * RWA     *  H   → → 21        Intheabove,wehaveidentifiedtheRabifrequency, d21.z,t  z,t   (1.9) 1.1.3 Rabifloppingandpulsearea Theatomicpopulationisperiodicallytransferredbetweenthegroundan dexcitedstates, (1.10) 22 tsin2t /2 , ate x a c t r e s o n a n c e ( = ) T h i s p r o c e s s , k n o w n a s R a b i flopping,is illustratedi n F i g u r e T h e  0 t0ti s s o calledpulsearea,allow determinestheexcitedstatepopulation, 22, atanytimet Fig1.1.Rabifloppingofthepopulationintheexcitedstate Usingapulseareaisconvenient fordescribingthepulsetimedependen tenvelopefunction,defined:  (1.11) z  z,d,  Fig1.2.The Gaussianwithpulsewidth0=1andp u l s e area0t=2 1.1.4 TheMaxwell’sandwaveequations FromMaxwell’sequations,weleadtothewaveequation, → → 1 2→ E 2  2 c  t E c 0t P (1.12) 1.1.5 Theslowly-varyingenvelopeapproximationwaveequation Waveequationintheone-dimensionalform: 2→ 2→ 1 2→   0 , z2E c2 t2E t P (1.13) –→ whereP ist h e p o l a r i z a t i o n F o r a d i e l e c t r i c m e d i u m o f t w o - levelatoms withNisparticledensity,thepolarizationisexpressed: –→ PNTr → d, → ˆ   (1.14a)   N d12 21d 2112 , → →   RWeikzt  RWeikzt N d1221 d 2112  (1.14b) (1.14c)  Takingthederivatives(1.14c)andcollectingsame-frequencytermsinMaxwell's waveequation(1.13),we obtained: 2 2  →  →  → 1 2→ →  → k   2ik  2  2   2i  2  z  z z  z c  →  2RW  RW 2 RW  0Nd12  12   12 2i 12 t2 t    (1.15) Intheslowly-varyingenvelopeapproximation(SVEA):  k ; z  2 k ; z z2 (1.16a)   ; t  2  t t2 (1.16b) Similarly,therotating-wavevariablesareslowly-varyingbydefinition: RW RW 12  12 (1.17) t UsingtheSVEAandRWAandnotingthatk=ωis/c c1/c  0 ,wefind c,slowly-varyingwaveequation: i → RW    → 1 (1.18) z ct  2 c Nd1212 or,intermsoftheRabifrequency,  (1.19)  2iRW, z Here 1  c t   2N dc12 , 12 (1.20) istheatom-fieldcouplingparameter 1.1.6 Inhomogeneousbroadening Tocalculateforthisinhomogeneousbroadening,weneedtoaverage ρ12t h r o u g h thepolarization in Maxwell'sequation: 12 (1.21)  12g   d  Thea p p r o p r i a t e d i s t r i b u t i o n f u n c t i o n , g ( ),c a n b e d e r i v e d f r o m t h e Maxwell-Boltzmanndistributionforgases.Intermsofthedetunings,itis:   0 2 p p , g    exp    kv m    kvm   (1.22) where k2/c ist h e w a v e n u m b e vm ist h e m o s t p r o b a b l e 2kBT / m r, velocitythatcorrespondstoaDopplerwidthDgiven D 2l n 2vmp/ c0 (1.23) 1.2 Interactionbetweenthree-levelatomsandlight In this section, we consider the cascade three-level atomic mediumexcited by laser field and coupling laser field as in Figure 1.3 The weaklaserdrivetotransitions|1| 2andthestrongcouplinglaserfieldexcited transition|2|3.Wedenote 21 and 32 arethepopulationdecayrateof thestates 2a n d 3, respectively.Wewritethesumofthetwofieldsinthe carrier-envelopeformas: → kz t →   → z,t ei  c.c i  k z  t (1.24)  z,t e Ez ,t p p p c c c Fig.1.3.Thecascadethree-levelatomschemeexcitedwithlaserandcouplinglaserfield In the framework of semiclassical theory, the evolution of densitymatrixoperatorρofofthesystemcanberepresentedbythefollowingLiouvil le equation:  i  (1.25)   H,  t Here,HisthetotalHamiltonian:  1  →→ H d21E The  identified,  d12E→  → → d32E  → → d23E    nmLnm  (1.26)  3  n,m , (1.27) isthepopulationdecayratefromlevel here, n m isthecoherencedecayrate,  L    nmnm here,  nm n  2 nm   mn m tolevel n , and nm ,   nm nmm n (1.28) nmm n isthedensity matrixoperatorsifn =mandtheelectric m dipoleoperatorsifnm 1.2.1 Hamiltonianinteractionintherotatingwaveapproximation We make the rotating wave approximation (RWA) as before For thethree-level cascade-system, the unitary transformation that takes us to therotating frame is:  i k pz  t e t U ei2    p TheHamiltoniantransformsas HRW UHU†i 0 0 e ikczct     (1.29)  U † U, t (1.30) andis, therefore: H RW p → → i k pz  t  d E21 e    d→ E→ e i k pz  t 12   → →0 p p      d →  d.23E e c c  E e   21p d* →→ 12p d→ * → 32c  c  c c   p   → →   d p  23c   c      *     → p→ RWA H  d  ,(1.31) ikz t  we find:     →  ikz t    p  *  c2   c ,(1.32) 2   c   →→ Where, z,t  p 2d12p z,t   z,t  dcc z,t ,  c is the Rabi frequenciesoftheprobeandcouplingpulses,respectively 1.2.2 Thelaserpulsepropagationequationsintheslowlyvaryingenvelope approximation Intherotationalwavereferencesystem,wederivethewaveequationforeach field:    2i RW, 1 p12 z ct p  RW    2i c2  z c t c (1.33a) (1.33b) Here,t h e a t o m f i e l d c o u p l i n g coNn ds t a2 n t s ( a l s o rNe df e r2 r e d t o a s p r o p a g a t i o n p12 constants)are  p  2 0c c23 and c 2 0c ,respectively 1.2.3 Coherent populationtrapping 1.2.4 Electromagneticallyinducedtransparency The EIT is a quantum interfering effect that leads to the propagationoflightthroughthemediumwithoutbeingabsorbedintheresonantfr equencyoftheatom.EITcanbeexplainedbasedonthequantuminterferenceofthe probabilityamplitudesoccursbetweenthedifferentexcitedp a t h w a y s i n a n a t omicsystemasshowninFigure1.4.Oneterm, whichisduetoexcitationbytheresonantfield  p only,i.e.,adirectpath fromstate tostate ;Anadditionalterm,whichisduetothepresence oft h e s e c o n d f i el c, i e , a n i n d i r e c t p a t h f r o m s t a tostate to d te state andb a c k t o t h e s t a t H e n c e , t h e t o t a l t r a n s i t i o n a m p l e itude vanishest h a t l e a d i n g t o t h e t r a n s p a r e n c y f o r t h e p r o b e l a s e r b e a