Proeedings of the 6th WSEAS Interntionl Conferene on Miroeletronis, Nnoeletronis, Optoeletronis, Istnul, Turkey, My 7-9, 7 1 Swithing properties of n ritrrily exited nonliner eletron-wve diretionl oupler ANDREAS F TERZIS Deprtment of Physis University of Ptrs Ptrs, Rion, GR-654 GREECE terzis@physisuptrsgr http://wwwphysisuptrsgr/ EMMANUEL PASPALAKIS Mterils Siene Deprtment University of Ptrs Ptrs, Rion, GR-654 GREECE psplk@uptrsgr http://wwwmtersiuptrsgr/ Astrt: We theoretilly investigte the swithing properties of nonliner eletron-wve diretionl ouplers y pplying the oupled mode theory Anlytil results for the proility of finding the eletron wve in eh wveguide, for ritrry initil ouption of eh wveguide, re presented Our findings revel tht severl tehnologilly importnt ses n e hieved, suh s signifint or omplete eletron trnsfer etween the wveguides or trpping of the eletron wve in the initil stte, depending on the struturl prmeters of the devie In ddition, for speifi vlues of the initil stte of the system the importnt se of symmetry reking is relized Key-Words: - Eletron-wve diretionl oupler, swithing, ritrry initil onditions, eletroni osilltions, self-trpping, symmetry reking 1 Introdution Severl studies hve proposed devies tht ould relize oherent eletron trnsport An importnt devie in this re of quntum eletronis is the eletron-wve diretionl oupler [1-1] This devie hs lso een reently proposed for pplitions in quntum omputtion [11-15] In prtiulr study, Tsukd et l [7] proposed nd nlyzed numerilly with the use of oupled mode theory nonliner eletron-wve diretionl oupler This devie onsists of two losely sped, prllel eletron wveguides with extremely smll pitnes They showed tht this devie exhiits either omplete eletron trnsfer from one wveguide to the other or eletron self-trpping to the initilly exited wveguide for speifi vlues of the system prmeters A symmetry reking effet is lso found in the se tht oth wveguides re initilly exited in oherent superposition stte In lter work [16] we revisited the nonliner eletron-wve diretionl oupler of Tsukd et l [7] nd presented nlytil results for the proility of finding the eletron wve in eh wveguide for the se tht only one wveguide is initilly exited We lso lulted the ritil length of the devie In ddition, we proposed new type of devie, ent nonliner eletron-wve diretionl oupler nd studied its swithing hrteristis In this rtile we study further the swithing hrteristi of the nonliner eletron-wve diretionll oupler of Tsukd et l [7] nd otin nlytil results for the proility of finding the eletron wve in eh wveguide for the se tht the system is initilly in n ritrry superposition stte These results show the generl ehvior of the system With the nlytil solution we lso mnge to explin the effet of symmetry reking tht n our in this system [7] nd lso show the ruil dependene of this effet to the form of the initil stte of the system This rtile is orgnized s follows: in the next hpter we present the si equtions sed on oupled mode theory In hpter 3 we present nlytil results for the se of ritrry initil onditions Then, in the following three hpters the ehvior of the system for severl ses of initil onditions is presented Finlly, we summrize our findings in hpter 7
Proeedings of the 6th WSEAS Interntionl Conferene on Miroeletronis, Nnoeletronis, Optoeletronis, Istnul, Turkey, My 7-9, 7 Theoretil Model For the theoretil nlysis of the nonliner eletronwve diretionl oupler we use oupled mode theory [1-3,7] Aording to this pproh the wvefuntion ψ ( z) of the eletron propgting long the z diretion n e written s liner omintion of the eigenfuntions of the individul wveguides ψ, ψ, ( z) ( z) ( z) ψ = ψ + ψ (1) If this wvefution is sustituted into the Shrödinger eqution we otin the following set of nonliner oupled differentil equtions (we use ħ = 1 in this pper) d( z) i = Ω( 1 ( z) ) ( z) + k( z), () d( z) i =Ω( 1 ( z) ) ( z) + k( z), (3) where k is the wveguide oupling onstnt per unit length In the derivtion of Eqs () nd (3) symmetri wveguide struture is ssumed The oeffiient k rises due to eletron tunnelling etween the wveguides The first terms in the right hnd side of Eqs () nd (3) re due to the Coulom hrging effet tht exists due to the extremely smll pitne of the struture Also, Ω= Q /( C) where Q is the totl hrge on the wveguides per unit length nd C is the effetive pitne of the wveguide [7] We note tht k, Ω re tken positive in our work The quntity of interest is the proility of finding the eletron wve in eh wveguide t speifi distne, given y P ( z) ( z), P ( z) ( z) = = 3 Anlytil Results for Aritrry Initil Conditions A onvenient quntity for otining nlytil results is the differene of these proilities p( z) = P ( z) P ( z) As P ( z) + P ( z) = 1, we find P ( z) = 1 + p( z) /, P ( z) = 1 p( z) / Our gol is to otin nlytil results for ritrry initil onditions t the input of the wveguide oupler ( ) =, ( ) = with, eing omplex numers with 1 + = We omine Eqs ()-(3) nd otin du = Ωp( z) v( z) k p( z), (4) dv p( z) u( z) = Ω, (5) dp ku( z) =, (6) ( ) * ( ( ) ( )) * where v( z) Re ( z) ( z) u( z) = Im z z Here, = nd p + u + v = 1 We re interested for nlytil results in the se of = u = u ritrry initil onditions p( ) p, ( ) v = v with the onstrint p + u + v = nd ( ) 1 By omining Eqs (5) nd (6), we otin dv( z) Ω dp( z) = p( z), whih y integrtion gives k the following eqution Ω v( z) = v + p ( z) p k (7) Then, from Eq (4) we otin Ω Ω 3 uɺ ( z) = p Ωv k p( z) p ( z) (8) k k If now we differentite Eq (6) nd use Eq (8) we otin seond order differentil eqution for p( z ) d p = p 4 kv 4 k p( z) Ω Ω (9) 3 Ω p ( z) For ritrry initil onditions, the solution of Eq (9) is [17,18] DΩ p( z) = D n ( z z) k, (1) k where 1 p ζ k = 1 +, (11) ζ + 4ξ u ( ) D= sign p p ζ + ζ + 4ξ u (11) k ξ = Ω, (11) ζ = ξ ( ξ+ v ), (11d) ros( p / D) k dx z = D Ω, if u =, 1 k sin x nd
Proeedings of the 6th WSEAS Interntionl Conferene on Miroeletronis, Nnoeletronis, Optoeletronis, Istnul, Turkey, My 7-9, 7 3 ros( p / D) z k dx = sign ( u ) D Ω 1 k sin x if u, (11e) Oviously, for the se of p = 1, ie the se tht only the wveguide is initilly exited, then u = v = nd Eq (1) redues to Ω p( z) = n kz k =, (1) 4k tht is identil to Eq (1) of Ref 16 The ehviour of Eq (1) is governed y the vlue of prmeter k nd in turn prmeter ξ In the se k =, p( z) Dse h[ D ( z z )] tht 1 = Ω nd the proilities in either wveguides reh stedy stte s the eletron-wve propgtes in the wveguide In the ses tht k< 1 or k > 1 then the evolution of the system is periodi 4 Results for the u = se We strt with the se tht u = This n our, for exmple, in the se tht oth, re rel or imginry As p + u + v =, v is not n 1 independent prmeter s v =± 1 p We n esily find the ritil vlue of the nonlinerity prmeterξ, ξ The vlue ξ speifies the vlue of ξ t whih k = 1 Therefore, ξ is given y ξ = ( v ± ) 1 / In our study we onsider only positive vlues for the nonlinerity prmeter, so we only dopt the positive root nd onlude tht 1 v ξ = (13) The ritil vlue of nonlinerity prmeterξ orresponds to the vlue ove whih the system undergoes extended periodi eletron trnsfer with the vlue of p( z ) hnging from p to p This is prtilly the ehvior of the n( x k ) ellipti funtion for k < 1 Atully, for vnishing k- prmeter the n funtion eomes osine funtion For vlues of ξ elow ξ suppression of eletroni trnsfer is found nd self-trpping to the initil stte of the system is otined This gin silly depits the ehvior of the n( x k ) ellipti funtion for k > 1 Finlly, in the se tht k = 1, s we hve noted ove the n funtion eomes the seh funtion nd no osilltions re found The ehvior nlyzed ove n e seen in Fig 1, where we plot the sptil evolution of the proilities for positive nd negtive vlues of v As n e seen from Figs 1(e) nd 1(f) the ehvior of the self-trpping region n e different For ertin vlues of Ω one otins self-trpped osilltions etween the initil vlues [s in Fig 1(e)] nd for other vlues of Ω there re osilltions ove nd elow the initil vlues [s in Fig 1(f)] Fig1 We present ( ) P z (solid urve) nd P ( z ) (dshed urve) s funtion of the normlized propgtion distne for initil onditions = 8, = This leds to p = 6, u =, v = 8 The prmeters of this figure re () Ω= 1k, () Ω= k, () Ω= 11k, (d) Ω= 1 k / 9, (e) Ω= 1k, nd (f) Ω= 15k We note tht for ξ < v, the n ellipti funtion eomes dn ellipti funtion Also, in the se tht ξ > v the osilltions re etween the initil vlues [s in Fig 1(e) where ξ = 833 ] while for ξ < v the osilltions re ove nd elow the initil vlues [s in Fig 1(f) where ξ = 667 ] Of speil interest is the se where ξ = v, then k eomes infinity nd we hve stle sttionry (time-independent) solution We now py some more ttention to the se of lmost equl initil popultion in oth wveguides, ie the se tht p We first note tht in the
Proeedings of the 6th WSEAS Interntionl Conferene on Miroeletronis, Nnoeletronis, Optoeletronis, Istnul, Turkey, My 7-9, 7 4 se tht p = then v =± 1 nd, for 1 v =+ ξ = ( 1 v) / = (14) 1, for v = 1 As we n see from Eq (1) in this se p stys zero t ll distnes However, in the se where the system is in superposition stte suh tht p is very lose to zero ut not extly zero with v = 1 p 1 we find tht for vlues of the non-linerity prmeter elow its ritil vlue, ξ, symmetry reking sitution is hieved This n e seen in Figs nd 3 Fig4 The sme s in Fig for initil onditions = 49, = 51 This leds to p =, u =, v = 9998 The prmeters of this figure re () Ω= 1k, () Ω= k Fig The sme s in Fig 1 for initil onditions = 49, = 51 This leds to p =, u =, v = 9998 The prmeters of this figure re () Ω= 1k, () Ω= 9k, () Ω= 15k, (d) Ω= k Fig3 The sme s in Fig for initil onditions = 51, = 49 This leds to p =, u =, v = 9998 By symmetry reking we men tht for ξ > ξ we otin smll mplitude osilltions with the proility hnging etween the initil proilities of the two wveguides, while for ξ < ξ osilltions with lrge mplitude ours nd even omplete trnsfer to one of the wveguides is hieved In the ltter se the wveguide tht will e minly exited during propgtion depends on the sign of p This is explined s follows: for negtive v we otin negtive ζ, whih ours for [ v ] < ξ < ξ = (1 ) / ~ 1, nd the oeffiient in front of the ellipti Joi funtion eomes D= sign( p ) p ζ sign( p ) ζ Also, 1 p k = whih is lmost unity, ut lwys lrger ζ thn unity As ζ ξ( ξ 1) the prmeter D otins mximum vlue pproximtely t D sign( p) for ξ 5 So, in this se for ξ = 5, depending on the sign of the smll initil vlue of p, we otin omplete trnsfer to wveguide (for positive p ) or wveguide (for negtive p ) This is shown espeilly in Figs (d) nd 3(d) The spontneous reking effet is lso dependent on the sign of v This n e seen in Fig 4 For positive
Proeedings of the 6th WSEAS Interntionl Conferene on Miroeletronis, Nnoeletronis, Optoeletronis, Istnul, Turkey, My 7-9, 7 5 v, with v 1, we otin smll mplitude eletroni osilltions etween the two wveguides, s in this se ξ nd we re in the regime tht ξ > ξ even for very smll vlues of ξ The only thing tht hnges with the hnge of ξ is the period of osilltions 5 Results for the v = se We ontinue with the se tht v = An exmple of this se is the sitution tht either or is imginry nd the other is rel In this se we get muh simpler expression for the k prmeter s now ζ does not depend of the initil onditions ( ζ = ξ ) From this expression we n find the ritil vlue of the nonlinerity prmeter s funtion of p (this tht orresponds to k = 1) Esily we otin tht the ritil vlue is ξ = 5p (15) Therefore, we hve the sme ritil vlue oth positive ( u ( u = 1 p ) of u ξ for =+ 1 p ) or negtive vlue Fig5 The sme s in Fig 1 for initil onditions = 8, = i This leds to p = 6, u = 8, v = The prmeters of this figure re () Ω= 1k, () Ω= 5k, () Ω= 5 k / 9, (d) Ω= 6k In Figs 5 nd 6 we oserve tht for vlues of the nonlinerity prmeter ove the ritil vlue p z hnging etween the osilltions ours with ( ) mximum vlues D nd D We stress tht, in this se, D n e lrger thn p, so the osilltions n our etween vlues of the proility tht re lrger thn the initil proilities in the wveguides Also, for ξ < ξ suppression of osilltions nd trnsfer is found nd self-trpping of the system lose to its initil stte ours Results for this re shown in Figs 5 nd 6 As it is expeted, y ompring the results for negtive nd positive initil u, we otin similr results tht only differ y phse ftor The symmetry reking sitution long the lines disussed in the se of setion 4 does not our here, s in this se for very smll vlues of p the ritil vlue ξ eomes lso very smll nd we re lwys prtilly in the regime tht ξ > ξ Fig6 The sme s in Fig 6 for initil onditions = 8, = i This leds to p = 6, u = 8, v = 6 Results for the u, v se This is the se tht oth, re omplex in generl From the generl expression for the prmeter k [Eq (11)] we otin the ritil vlue of the nonlinerity prmeter s funtion of p After some lger we find tht the ritil vlue is given y p ( v± 1) ξ = (16) ( v 1) We onentrte on the positive ritil vlue nd get p 1 u v ξ = = (17) (1 + v) (1 + v ) Even in this most generl se tht ll vlues of u, v, p re non-vnishing the ehvior of the system is rther similr to tht disussed in the two previous ses The system exeutes eletroni p z osilltions etween the two wveguides with ( ) hnging etween the mximum vlues D nd D one the nonlinerity prmeter is ove the ritil vlue ξ In this se, too, D n e lrger thn
Proeedings of the 6th WSEAS Interntionl Conferene on Miroeletronis, Nnoeletronis, Optoeletronis, Istnul, Turkey, My 7-9, 7 6 p, so the eletroni osilltions n our etween vlues of proility tht re lrger thn the initil proility vlues In the se tht ξ is smller thn ξ the system is trpped in its initil stte nd performs osilltions of smll mplitude etween the two wveguides In ddition, s in the se of Setion 5 the symmetry reking sitution does not our in this generl se Only in the se tht u tkes very smll vlues (suh tht it n e ssumed tht it prtilly goes to zero) we n reover the symmetry reking result of Setion 4 7 Summry In this work we hve studied the swithing hrteristi of nonliner eletron-wve diretionl oupler in the se tht the devie t the entrne is prepred in generl superposition stte We first otined nlytil results for the proility of finding the eletron wve in eh wveguide Then, we nlyzed the swithing ehvior of the system for speifi initil onditions For different prmeters of the system extended eletroni osilltions etween the two wveguides, self-trpping in the initil stte nd symmetry reking n e otined Aknowledgements We thnk the Europen Soil Fund (ESF), Opertionl Progrm for Edutionl nd Votionl Trining II (EPEAEK II), nd prtiulrly the Progrm PYTHAGORAS II, for funding the ove work Referenes: [1] J A del Almo nd CC Eugster, Quntum fieldeffet diretionl oupler, Applied Physis Letters, Vol 56, No 1, 199, pp 78-8 [] N Tsukd, AD Wiek nd K Ploog, Proposl of novel eletron wve oupled devies, Applied Physis Letters, Vol 56, No 5, 199, pp 57-59 [3] R Q Yng nd JM Xu, Anlysis of guided eletron wves in oupled quntum wells, Physil Review B, Vol 43, No, 1991, pp 1699-176 [4] L Thylen nd O Shlen, Swith energy nlysis of quntum interferene diretionl-ouplers, IEEE Journl of Quntum Eletronis Vol 8, No 1, 199, pp 31-33 [5] DW Wilson, EN Glytsis nd TK Gylord, Supermode nlysis of eletron wve diretionl oupling using multilyer wve-guide pproh, Journl of Applied Physis, Vol 73, No 7, 1993, pp 335-3366 [6] W-P Yuen, Improved oupled-mode theory for prllel eletron wve-guides, Journl of Applied Physis, Vol 74, No 5, 1993, pp 3341-3345 [7] N Tsukd, M Gotod, M Nunoshit nd T Nishino, Nonliner eletron-wve diretionl oupler, Physil Review B, Vol 53, No 1, 1996, pp R763- R766 [8] RQ Yng, Eletron trnsfer effiieny in quntum well wveguide ouplers, Journl of Applied Physis, Vol 8, No 3, 1996, pp 1541-1546 [9] OE Rihev nd P Vsilopoulos, Effets of Coulom intertion nd tunneling on eletron trnsport in oupled one-dimensionl systems: From the llisti to the diffusive regime, Physil Review Letters, Vol 83, No 18, 1999, pp 3697-37 [1] P Pingue, V Pizz, F Beltrm, I Frrer, DA Rithie nd M Pepper, Coulom lokde diretionl oupler, Applied Physis Letters, Vol 86, No 5, 5, rt no 51 [11] A Bertoni, P Bordone, R Brunetti, C Jooni nd S Reggini, Quntum logi gtes sed on oherent eletron trnsport in quntum wires, Physil Review Letters, Vol 84, No 5,, pp 591-5915 [1] R Ioniioiu, G Amrtung nd F Udre, Quntum omputtion with llisti eletrons, Interntionl Journl of Modern Physis B, Vol 15, No, 1, pp 15-133 [13] R Ioniioiu, P Znrdi nd F Rossi, Testing Bell s inequlity with llisti eletrons in semiondutors, Physil Review A, Vol 63, No 5, 1, rt no 511 [14] MJ Gilert, R Akis nd DK Ferry, Dul omputtionl sis quit in semiondutor heterostrutures, Applied Physis Letters, Vol 81, No,, pp 484-486; Vol 83, No 7, 3, pp 1453-1455 [15] GB Akgu, LE Reihl, A Shji nd MG Snyder, Bell sttes in resonnt quntum wveguide network, Physil Review A, Vol 69, No 4, 4, rt no 433 [16] E Psplkis nd AF Terzis, Swithing properties of nonliner eletron-wve diretionl ouplers, Journl of Applied Physis, Vol 99, No 4, 6, rt no 4374 [17] GP Tsironis nd VM Kenkre, Initil ondition effets in the evolution of nonliner dimer, Physis Letters A, Vol 17, No 4, 1988, pp 9-1 [18] S Rghvn, VM Kenkre nd AR Bishop, Phse-nonlinerity interply in smll quntum systems, Physis Letters A, Vol 33, No 1-, 1997, pp 73-78