Scattering of a Single Plasmon by Three Non-equally Spaced Quantum Dots System Coupled to One-Dimensional Waveguide

Transcription

1 Scaein of a Sinle Plasmon by Thee Non-equally Spaced Quanum Dos Sysem Coupled o One-Dimensional Waveuide Nam-Chol Kim,,* Myon-Chol Ko, Depamen of Physics, Kim Il Sun Univesiy, Pyonyan, D P R Koea Absac: Scaein popeies of a sinle plasmon ineacin wih hee non-equally spaced quanum dos coupled o one-dimensional suface plasmonic waveuide is invesiaed heoeically via he eal-space appoach. I is demonsaed ha he ansmission and eflecion of a sinle plasmon can be swiched on o off by conollin he deunin and chanin he inepaicle disances beween he quanum dos. By conollin he ansiion fequencies and inepaicle disances of QDs, one can consuc a half-ansmiin mio wih hee QDs sysem. We also showed ha conollin he ansiion fequencies and inepaicle disances of QDs esuls in he complee ansmission pea nea he zeo deunin. Keywods: Swichin, Scaein, Plasmon, Waveuide * Eleconic mail: yonnam@yahoo.com

2 . Inoducion The lih - mae ineacion has always been a fundamenal opic in physics, and is mos elemenay level is he ineacion beween a sinle phoon and a sinle emie []. Conollin he scaein popeies of a sinle phoon has aaced paicula ineess fo some fundamenal invesiaions of phoon-aom ineacion and fo is applicaions in quanum infomaion[]. Since phoons ae aaced and eaded as ideal caies of quanum infomaion, phoons ae naually consideed o eplace elecons in fuue infomaion echnoloy[]. Recenly, heoeical idea of a sinle-phoon ansiso has also emeed[]. Such a nonlinea device is essenial o many emein echnoloies, such as opical communicaion, opical quanum compue, and quanum infomaion pocessin. Many heoeical[4-] and expeimenal[-5] wos epoed he phoon scaein in diffeen quanum sysems. In he pevious sudies, he auhos have mainly consideed he scaein popeies of a sinle phoon ineacin wih one emie and seveal emies, and hey mainly focus on he case whee he quanum emies ae all equally spaced each ohe[6-9]. The couplin beween meal nanowies and quanum emies is impoan fo conollin lih-mae ineacions[]. The nanowie exhibis ood confinemen and uidin even when is adius is educed well below he opical wavelenh. In his limi, he effecive Pucell faco P Г pl / Г can exceed in ealisic sysems accodin o he heoeical esuls[], whee Г pl is he sponaneous emission ae ino he suface plasmons(phoons) and Г descibes conibuions fom boh emission ino fee space and non-adiaive emission via ohmic losses in he conduco. Boh expeimenal and heoeical invesiaions show ha he emission popeies of QDs can be sinificanly modified nea he meallic nanosucues[,]. Recen invesiaions have fuhe exended ino he eime of ineacions of quanum emies such as QDs and popaain SPs[4]. The ecen poess of he phoonic cysal waveuide could be a sep owad he oal of minimizin he adiaion loss o he suoundin, as some heoeical sudies have indicaed ha he hih sponaneous emission can be achieved fo sysems wih a quanum do doped inside a phoonic cysal waveuide. Moivaed by hese consideaions, we invesiae he scaein of a sinle plasmon ineacin wih

3 hee non-equally spaced QDs coupled o D suface plasmonic waveuide which is a meal nanowie [Fi. ]. Inciden Plasmons QD- QD- QD- Z= Z= L Z=L FIG.. (Colo online) Schemaic diaam of a sysem consisin of a sinle plasmon and hee non-equally spaced QDs coupled o a meal nanowie. i and i ae he ansmission and eflecion ampliudes a he place z i, especively. In his pape, we invesiae heoeically he scaein popeies of a sinle plasmon by non-equally spaced QDs coupled o a one-dimensional(d) suface plasmaic waveuide, whee he ansiion fequencies of he hee QDs can be diffeen wih each ohe.. Theoeical model and dynamics equaions We invesiae he scaein of a sinle plasmon ineacin wih hee non-equally spaced QDs coupled o D suface plasmonic waveuide which is a meal nanowie as shown in Fi.. Unde he oain wave appoximaion, he Hamilonian of he sysem in eal space is iven by [4,5] H i / i dza z a z a z a z e a z a z a z a z ee l e whee ω e () and ω () ae he eienfequencies of he sae and e of he h QD, especively, ω is he fequency of he popaain plasmon field wih waveveco (ω = ν ). σ e = e ( σ e = e ) is he lowin (aisin) opeaos of he h QD, a + (z ) (a + l (z )) is he bosonic opeao ceain a ih-oin (lef-oin) plasmon a posiion z of he h QD. ν is he oup velociy of he coespondin ω, he non- Hemiian em in H descibes he decay of sae ω () e a a ae Г ino all ohe possible channels. = (πħ/ω ) / Ω D e is he couplin consan of he h QD wih a sinle plasmon, Ω is he esonance eney of he h QD, D is he dipole momen of he h QD, e is he polaizaion uni veco of he suface plasmon[4,5]. l l z l e Z z ()

4 Assumin ha a sinle plasmon is incomin fom he lef wih eney E = ħω, hen he eiensae of he sysem, defined by H ψ = E ψ, can be consuced in he fom za z za z, e dz e, (),, l l, whee, denoes he vacuum sae wih zeo plasmon and hee QDs bein unexcied,, e denoes he vacuum field and only he h QD in he excied sae and e () is he pobabiliy ampliude of he h QD in he excied sae. Φ +,(z) (Φ +,l(z)) is he wavefuncion of a ih-oin (a lef-oin) plasmon a posiion z. Fo a sinle plasmon inciden fom he lef, he mode funcions Φ +, (z) and Φ +,l (z) iz i zl can ae he foms as followins, z e, z L e,, i zl L L z L L e,, i zl L iz, z L L, z e,, e, l i zl i zl L z L e, L z L L e, l z L L, l and, l, especively, whee L is he spacin beween he fis and he second QDs and L beween he second and he hid QDs. Hee and ae he ansmission and eflecion ampliudes a he place z, especively. By subsiuin Eq. () ino H ψ = E ψ, we obain a se of equaions as: e i il e, e i il e il i e, e e,, i il e e, e i L L, e 4 i e, e i i L L e,, J /, e, whee,, i /,,, il il il il condiions of he mode funcions, i. e.,,, e i L L 4 e e. By ain he bounday, whee =, 4 =, ino accoun, we obain he ansmission and he eflecion ampliudes, especively. By evaluain he ansmission coefficien T e e e 4

5 and eflecion coefficien R, we can find he scaein popeies of a sinle plasmon in he lon ime limi.. Theoeical analysis and numeical esuls Fis of all, we pay aenion o he dependency of ansmission speca on he spacin beween he QDs. Fiue shows ypical ansmission speca vesus L fo he cases ha all he hee ansiion fequencies of he QDs ae equal o each ohe, whee we se he phases L as (a) L =.π, (b) L =.π and (c) L =.7π, especively. Now, we suppose ha J i = J = cons ( =,, ). As shown in Fi. (a), he ansmission coefficien vesus phase L exhibis peiodic popeies, he peiod of which is L =π. In ohe wods, he peiod is a half of he wavelenh of inciden plasmon calculaed in ems of spacin L, whee we oo he elaion = / λ ino accoun. We also found ha his oscillaoy paen of ansmission speca dose no depend on he spacin beween he fis QD and he second QD. Fiue (b) shows ha sein he spacin beween he fis QD and he second QD as L =.π esuls in he swichin beween he complee ansmiin and half ansmiin by conollin he spacin beween he second QD and he hid QD. Wha is moe ineesin is ha, whaeve he spacin beween he second QD and he hid QD, L, is, sein he spacin L as L =.7π esuls in he consan ansmission coefficien (.84) [Fi. (c)]. The esul implies ha hee is a possibiliy o build a body wih such a consan ansmission coefficien as needed, which can povide a new idea of desinin quanum opical devices such as a file o a swich in is sinle-plasmon level. Fi.. (Colo online) Tansmission speca of a popaain plasmon ineacin wih hee i non-equally spaced QDs ( =.ω, i =,, ) vesus L, whee he fequency is in unis of ω πν / L and J = -4 ω : (a) L =.8π, (b) L =.7π, (c) L =.7π. 5

6 Fiue shows he ansmission speca vesus L when he wo QDs have he same deunins.ω and he anohe has a deunin.ω. As shown in Fi. (a), he ansmission speca exhibi an oscillaoy paen wih half of he wavelenh of he popaain plasmon as a peiod in ems of spacin beween L. Fom Fis. (b) and (c), we also found ha he ode of he QDs can influence on he scaein of he inciden plasmon, which is quie diffeen wih he case of wo QDs sysem [7]. Especially, we can consuc a half-ansmiin mio wih hee QDs sysem by conollin he ansiion fequencies of QDs and sein he spacin L as.7π, as shown in Fi. (c). Fom he esul shown in Fi., a half-ansmiin mio fo a sinle inciden plasmon can be buil wih hee QDs by conollin he deunins and he spacins beween QDs, which is valuable fo he pacical applicaions. Fi.. (Colo online) Tansmission speca of a popaain plasmon ineacin wih hee non-equally spaced QDs vesus L, whee ω πν / L and J = -4 ω : (a) L =.8π, = =.ω, =.ω, (b) L =.7π, = =. ω, =.ω, (c) L =.7π, = =.ω, =.ω. Now, we can also conside he ansmission speca vesus he fequency of inciden plasmon ω. When all he hee QDs have he same ansiion fequencies (Ω = Ω = Ω =.), he ansmission speca ae shown in Fis. 4(a) and 4(b). Thee appeas a shap complee ansmission pea nea he zeo deunin when he spacins ae equal o each ohe, L = L =(.+.5n) λ, n =,,, [Fi. 4(a)]. As is well nown [7], a sinle plasmon swich has wo saes (on and off) fo conollin he ansmission of a sinle plasmon, in which he swich is off fo T =, and on fo T =. As shown in Fi. 4(a), he swich is off when he popaain plasmon is esonan wih QDs and on when he popaain plasmon is deuned wih QDs a lile, which could be applied in desiin swiches of sinle plasmons. When he spacins ae diffeen wih each ohe, L L, hee doesn 6

7 appea a complee ansmission pea bu a semi-pemeable pea of ansmission nea he zeo deunin [Fi. 4(b)], which is quie diffeen wih he case of equal spacins as shown in Fi. 4(a). Anyway, he esuls shown in Fi. 4 show ha he spacin beween QDs influence he scaein popeies of an inciden sinle plasmon ealy. Fi. 4. (Colo online) Tansmission speca of a sinle plasmon ineacin wih hee QDs vesus inciden fequency of a sinle plasmon: (a) L =L =.π (b) L =.π, L =.4π, whee Ω =Ω =Ω =.ω, ω πν / L and J = -4 ω. Fiue 5 shows he ansmission popeies of a sinle plasmon ineacin wih nonequally spaced QDs havin diffeen ansiion fequencies. The solid line in Fi. 5 shows he ansmission coefficien when he spacin L is lae han he spacin L (L =.6π, L =.π). The dash-do line in Fi. 5 shows he ansmission coefficien when L < L (L =.π, L =.6π). As we can see easily fom Fi. 5, when L = (.+.5n) λ, L = (.5+.5n) λ (n =,,, ), whee we oo he elaion = / λ ino accoun, hee appeas a vey shap complee ansmission pea nea he fequency Ω =Ω =.ω, which is quie ineesin. In conas, hee is no shap complee ansmission pea nea he zeo deunin and hee appeas a complee ansmission pea a (Ω +Ω ) /, when L < L (L =.π, L =.6π). Fom Fi. 5, we found ha he inepaicle disances influence he scaein popeies of sinle inciden plasmon and dopin QDs havin he same ansiion fequencies esul in he complee ansmission pea nea he zeo deunin, which could find pacical applicaions. 7

8 Fi. 5. (Colo online) Tansmission speca of a sinle plasmon ineacin wih hee QDs vesus inciden fequency of a sinle plasmon: We se Ω =Ω =.ω, Ω =.ω, ω πν / L and J = -4 ω. The solid line(blue) displays he ansmission speca when L =.6π, L =.π, and he dash-do line(ed) displays he ansmission speca when L =.π, L =.6π. 4. Conclusions In summay, we invesiaed heoeically he swichin popeies of a sinle plasmon ineacin wih hee non-equally spaced QDs which ae coupled one dimensional plasmaic waveuide. We showed ha he ansmission and eflecion of a sinle plasmon can be swiched on o off by dynamically unin and chanin he spacin beween he QDs. Sein he inepaicle disances popely esuls in he swichin beween he complee ansmission and half ansmission. Especially, we can consuc a half-ansmiin mio wih hee QDs sysem by conollin he ansiion fequencies and inepaicle disances of QDs, which is valuable fo he pacical applicaions. We also showed ha conollin he ansiion fequencies and inepaicle disances esuls in he complee ansmission pea nea he zeo deunin. Ou esuls may find a vaiey of applicaions in he desin of he quanum opical devices, such as nanomios and quanum swiches, and in quanum infomaion pocessin. Acnowledmens. This wo was suppoed by he Naional Poam on Key Science Reseach and Key Poec fo Fonie Reseach on Quanum Infomaion and Quanum Opics of Minisy of Educaion of D. P. R of Koea. Refeences [] J. M. Raimond, M. Bune, and S. Haoche, Rev. Mod. Phys. 7 () 565. [] A. Wallaff, D. I. Schuse, A. Blais, L. Funzio, R.-S. Huan, J. Mae, S. Kuma, S. M. Givin, and R. J. Schoelopf, Naue 4 (4) 6; P. Ko, W. J. Muno, K. Nemoo, T. C. Ralph, J. P. Dowlin, and G. J. Milbun, Rev. Mod. Phys. 79 (7) 5. 8

9 [] D. E. Chan, A. S. Søensen, E. A. Demle, and M. D. Luin, Na. Phys. (7) 87. [4] J. -T. Shen, S. Fan, Op. Le. (5) ; J. -T. Shen, S. Fan, Phys. Rev. Le. 95 (5). [5] J. -T. Shen, S. Fan, Phys. Rev. A 79 (9) 87; M. Badfod, J. -T. Shen, Phys. Rev. A 85 () 484. [6] Z. R. Gon, H. Ian, Lan Zhou, and C. P. Sun, Phys. Rev. A 78 (8) 586. [7] J. -T. Shen, S. Fan, Phys. Rev. A 79 (9) 88. [8] G. Rosami, M. Shahabadi, A. Kusha, and A. Rosami, Applied Opics, 5 () 59. [9] M. -T. Chen, Y. -Q. Luo, P. -Z. Wan, and G. -X. Zhao, Appl. Phys. Le.97 () 99. [] Y. C. Shen and J. -T Shen, Phys. Rev. A 85 () 8. [] M. -T. Chen, X.-S. Ma, M. -T. Din, Y. -Q. Luo, and G. -X. Zhao, Phys. Rev. A 85 () 584. [] Luas Neumeie, Main Leib, and Michael J. Hamann, Phys. Rev. Le. () 66. [] B. Dayan, A. S. Pains, T. Aoi, E. P. Osby, K. J. Vahala, H. J. Kimble, Science 9 (8) 6. [4] T. Aoi, B. Dayan, E. Wilcu, W. P. Bowen, A. S. Pains, T. J. Kippenbe, K. J. Vahala, H. J. Kimble, Naue 44 (6) 67. [5] K. M. Binbaum, A. Boca, R. Mille, A. D. Booze, T. E. Nohup, H. J. Kimble, Naue 46 (5) 87. [6] L. Zhou, Z. R. Gon, Y. X. Liu, C. P. Sun, F. Noi, Phys. Rev. Le. (8) 5; T. S. Tsoi, C. K. Law, Phys. Rev. A 8 (9) 8. [7] Nam-Chol Kim, Jian-Bo Li, Zhon-Jian Yan, Zhon-Hua Hao, and Qu-Qian Wan, Appl. Phys. Le. 97 () 6. [8] W. Chen, G. -Y. Chen, and Y. -N. Chen, Op. Expess 8 () 6. [9] Nam-Chol Kim, Myon-Chol Ko, Qu-Quan Wan, axiv [physics. opics] (). [] H. Wei and H. -X. Xu, Nanophoonics, ()

10 [] D. E. Chan, A. S. Søensen, P. R. Hemme and M. D. Luin, Phys. Rev. Le. 97 (6) 5. [] H. M. Gon, L. Zhou, X. R. Su, S. Xiao, S. D. Liu, and Q. Q. Wan, Adv. Func. Mae. 9 (9) 98. [] Zhon-Jian Yan, Nam-Chol Kim, Jian-Bo Li, Mu-Tian Chen, Shao-Din Liu, Zhon-Hua Hao, and Qu-Quan Wan, Opics Expess 8 () 46. [4] H. Wei, D. Rachfod, X. Li, H. X. Xu, and C. K. Shih, Nano. Le. 9 (9) 468.