DYNAMICS OF A SUPER RADIANT DISSIPATIVE SYSTEM OF ELECTRONS TUNNELING IN A MICRO-CAVITY

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1 QUANTUM MECHANICS DYNAMICS OF A SUPER RADIANT DISSIPATIVE SYSTEM OF ELECTRONS TUNNELING IN A MICRO-CAVITY ELIADE ªTEFÃNESCU, AUREL SÃNDULESCU Cntr of Advancd Studis in Physics of th Romanian Acadmy Cala 13 Sptmbri nr. 13, Sct. 5, Bucharst, Romania Rcivd May 5, 005 W discuss th supr-radiant dissipativ tunnling of lctrons in a prfctly tund micro-cavity of a p-i-n smiconductor structur with quantum dots. Our dscription is basd on a quantum mastr quation with microscopic cofficints, dpnding on two-body potntials, dnsitis of th dissipativ nvironmnt stats, and tmpratur, that is in accordanc with th dtaild balanc principl. For this systm, w obtain Maxwll-Bloch quations with xplicit microscopic dissipativ cofficints, whil taking into account a currnt that could b injctd in th dvic, and th fild dissipation and propagation, that ssntially dtrmin th suprradiation procss. W ar spcially intrstd in th absolut valus of th suprradiant puls amplitud and in th spac and tim-distributions as functions of physical charactristics and opration conditions of th systm. Du to th planar distribution of th quantum dots, at a low dnsity of ths dots th supr-radiant xponnt is 3, not as it is for a volum distribution. Du to Rabi oscillations, in an undr-dampd systm th supr-radiant xponnt dcrass with th quantum dot dnsity, tnding to 1.5. Powr dnsitis of th ordr of thos that could b absorbd from Sun at th lvl of our plant ar asily obtaind for ralistic valus of th systm paramtrs and opration conditions. PACS numbrs: Ca, Fk Dspit a long history, th atom-fild intraction is still an activ fild of invstigation spcially du to th dissipativ procsss [1], standing at th basis of important tchnical applications an xampl is a nw amplifying dvic basd on th dissipativ supr-radiant tunnling, rcntly proposd by M. Asada []. Th supr radianc of a systm of lctrons prdictd by Dik [3] has bn intnsivly studid taking into account various physical ffcts as: (1 th statistical distribution of th lctron stats [4], ( lvl dgnracy ffcts [5, 6], (3 Langvin forcs acting on th atomic systm [7], (4 transvrs ffcts [8], (5 compting of thr-photon and on-photon transitions [9], (6 th supr radianc spctrum [10], (7 xistnc of photon gaps [11 14], (8 spontanously gnratd cohrnc ffcts [15], (9 supr radianc supprssion by scattring stfan@pcnt.ro Rom. Journ. Phys., Vol. 50, Nos. 7 8, P , Bucharst, 005

2 630 Eliad ªtfãnscu, Aurl Sãndulscu [16]. Howvr, a dtaild study of th supr-radiant powr as a function of th main physical charactristics of a spcific systm do not yt xist in litratur, and th accordanc of th dissipativ supr-radiant dynamics with th dtaild balanc principl [17 19] has not yt bn discussd. W invstigat th supr-radianc of a p-i-n smiconductor structur with quantum dots and a prfctly tund micro-cavity (Fig. 1. Hr th lctrons dcaying btwn wll-dtrmind nrgy lvls on on sid of this cavity buildup a supr-radiant fild transmittd out on th othr sid. Th insulating rgion i nabls a thorough control of th transition dipol r 01 0 r 1 that dtrmins th mattr-fild coupling. W dscrib this systm by a Hamiltonian S F H = H + H + V, (1 (a Supr-radiant smiconductor structur. (b Supr-radiant lctromagntic puls. Fig. 1. Supr-radiant dcay in a quantum dot p-i-n structur micro cavity.

3 3 Dynamics of lctrons tunnling in a micro-cavity 631 with thr trms: H S = ε i ici + ci, i = 0, 1 for th systm of lctrons, HF = ω 1 ( a a + + for th supr-radiant fild mod of frquncy ω=ω 10, and th intraction potntial V = i ijrij c+ ω i cj A. ( dpnding on th transition frquncis th potntial vctor ij ikr ω, ij on th dipol momnts r ij, and on + ikr A = K a a +, whr K = α λ 0, whil α 0 =, λ is th wavlngth, and V is th V 4πεc quantization volum. W also tak into account th xistnc of a complx dissipativ coupling of th systm dscribd by th Hamiltonian H with (1 othr lctrons in th profound clustrs of th n-rgion, ( th crystallin lattic, (3 th fr lctromagntic fild coupld with th systm of lctrons, and (4 othr dissipativ lmnts of th smiconductor structur coupld with th suprradiant mod, that ssntially dtrmins th radiation procss. Th dscription of this procss dpnds on th systm-nvironmnt intraction modl and on th approximations usd to rduc a quantum dynamical quation to a mastr quation [0 3]. In this papr, w considr an xplicit quantum mastr quation, with twobody potntials btwn th systm and nvironmnt particls as dscribing th most probabl dissipativ procsss, of singl-particl transitions of th systm c+ c corrlatd with singl-particl transitions of th nvironmnt [19, 4]: i j (3 N d ρ ( t = i [ H,ρ ( t ] + λij ([ c+ i cjρ ( t, c+ jci ] + [ c+ i cj,ρ ( t c+ jci ]. dt ij, = 1 (4 This quation for an N-lvl systm has N 1 xplicit dissipativ cofficints F B ij ij ij ij λ =λ +λ +γ, (5 with trms dscribing dcay/xcitation procsss that for transition nrgis much highr than tmpratur, ε ji T, j > i, hav th approximat xprssions: λ ij F = π αiv F βj 1 fα F ( εji gα F ( εji (6a

4 63 Eliad ªtfãnscu, Aurl Sãndulscu 4 λ F ji = π αiv F βj fα F ( εji gα F ( ε ji, for a dissipativ nvironmnt of Frmions, λ ij B = π αiv F β j 1 + fα B ( εji gα B ( εji λ B ji = π αiv F βj fα B ( εji gα B ( εji for a dissipativ nvironmnt of Bosons, and (6b (7a (7b α 0 γ ij = 3 r ij ε ji + c ji ε / (8 T 1 for th fr lctromagntic fild. In ths xprssions, V F, V B ar dissipativ two-body potntials, gf α ( ε ji, gb α ( ε ji ar dnsitis of th nvironmnt stats, and f F α ( ε ji, f B α ( ε ji ar th Frmi-Dirac and rspctivly Bos-Einstin distributions. W rmark that ths cofficints satisfy th dtaild balanc principl for a dissipativ nvironmnt of Frmions [19]: F F λji f α ( εji ji ε / T = =, λf 1 f F( ε ij and of Bosons, or, particularly, of th fr lctromagntic fild α ji (9 B B λji f α ( εji ji ε / T = =. λ B 1 + f B( ε ij α ji (10 That mans that in fact dissipation is a ncssary part of th quantum dynamics, othrwis this principl making no sns. Considring a two-lvl systm, i, j = 0, 1 with ngligibl dimnsions in comparison with th fild wav lngth, and a prfctly tund cavity, from (4 with th Hamiltonian (1 on obtains th Maxwll-Bloch quations d ut ( = γ ut ( gg (0, twt ( (11a dt d wt ( = γ [ wt ( w0 ] + Φ+ ( TgG (0, tut ( dt Gzt (, z= 0 Tc Gzt (, z= 0 = γ EMGzt (, z= 0 g ω ut ( t z 4εV Gzt (, + c Gzt (, = 0, z> 0 t z (11b (11c (11d

5 5 Dynamics of lctrons tunnling in a micro-cavity 633 with th polarization ut, ( th population wt, ( th lctromagntic fild amplitud Gzt (,, a particl flux Φ that could b injctd in th dvic, th mattr-fild coupling cofficint g= r01, th dcay rat of th supr-radiant mod γ EM, and th dissipativ cofficints γ =λ +λ +λ +λ (1a γ = ( λ 01 +λ10 w ω / T 0 =, ω 0/ T (1b (1c whr T is tmpratur. Th scond cofficint of th last trm of Eq. (11b taks into account th coupling with th two countr propagating wavs in th cavity, whil T dscribs th dcras of th mattr-fild coupling du to th fild radiation through th output mirror. From Eq. (11c w notic that th polarization ut ( in th quantization volum V = 1/ N 3 /, N bing th numbr of quantum dots on th ara unit, is a sourc for two fild variations: (1 a timvariation of th fild in th cavity Gzt (, z = 0, and ( a fild flow through th mirror with th transmission cofficint T, whr th transmittd fild GT ( z, t = TG ( z, t propagats according to Eq. (11d. For T = 0 Eqs. (11 dscrib a closd cavity, whil for T = 1 thy corrspond to an opn supr radiant structur. Whn th dissipativ cofficints ar nglctd, ths quations satisfy consrvation rlations: ( TV d (0 d 11( 0 (nrgy dt W, t + ω dt ρ t = (13a ( Tu ( t + w( t = 1 (Bloch vctor, (13b whr W(0, t =ε G(0, t is th nrgy dnsity of th lctromagntic fild, whil ρ 1 11( t = [1 + w(] t is th population of th uppr lvl. For T = 0, ths rlations dscrib a closd cavity, with a factor for th two countr propagating wavs, whil for T = 1 thy corrspond to an opn structur, with a singl radiation mod. In comparison with othr Maxwll-Bloch quations usd in th supr-radianc domain as Eqs. (47, (48, (49 in [7], our quations (11 hav xplicit xprssions of th dissipativ cofficints (1a, (1b with (5, (6, (7, (8, that satisfy th dtaild balanc conditions (9, (10. Mor than that, in (11c w considr a currnt injction I = NΦ, and a dissipation of th supr-radiant fild, that, as it will b shown in th following, is ncssary for th solution convrgnc.

6 634 Eliad ªtfãnscu, Aurl Sãndulscu 6 In th following, w tak a rsonanc nrgy ω 10 = 01. V, and calculat th powr dnsity (Poynting vctor amplitud Szt (, = TS0( zt, with S0( z, t = = cwzt (, = cg ε ( zt, from Eqs. (11 with cofficints containing th main physical paramtrs of th systm. From (1b with (5 8, th dcay rat gts a tmpratur-indpndnt trm and a tmpratur-dpndnt on, coming from th coupling of th systm to th Frmion and rspctivly to th Boson part of th nvironmnt: 0 F B ω/ T γ + 1 =γ +γ, (14 ω/ 0 T 1 that, ssntially, mans a tmpratur dpndnc givn by th dtaild balanc rlations (9, (10. For simplicity, w tak into account only th Boson componnt of th dcay rat that dpnds on tmpratur, and considr γ =γ / =γ EM. In Fig. w rprsnt th tim volution of an opn structur (T = 1 for two cass: (a undr-dampd, and (b ovr-dampd, whil in Fig. 1 (b th powr dnsity is rprsntd in spac, for th undr-dampd cas with th sam paramtrs but diffrnt valus of th transmission cofficint T. In ths rprsntations, w considrd th initial condition of a thrmal stat w (0 = 0. 4, T = K, Φ = 0, and an initial polarization u(0 = w 0 w (0 in accordanc with th quilibrium condition: w(0 w0 u(0 0. Fig. 3 illustrats othr two ffcts dcrasing th supr-radiation fild amplitud: (1 th propagation of this fild in an opn structur, that diminishs th mattr-fild coupling, and ( th fild mod dissipation. From Fig. 3 (a, w notic that for th tunnling of a packt of N lctrons, th fild propagation and dissipation hav somhow similar ffcts, diminishing th puls amplitud without changing its shap (grn and magnta curvs in comparison with th blu curv. Howvr, from Fig. 3 (b w notic that whn a currnt I is injctd in th structur, th two trms of th fild quation, of propagation and of dissipation, hav qualitativly diffrnt ffcts. Only through th fild dissipation trm (th magnta curv T = 0, or th rd curv T = 1, Eqs. (11 hav a finit asymptotic solution of th lctromagntic fild dnsity of nrgy 1 ( (0 I 1 3 γ γ W G N / w0 N / ε, = ω T + γ ε, 4γ EM g whil th population is w0 + I Nγ w( =. g 1 + ( T G(0, γ γ (15 (16

7 7 Dynamics of lctrons tunnling in a micro-cavity 635 γ F =. B s, 0 γ = (a γ F = s, 0 B γ = (b Fig.. Th population/polarization dcay gnrating a supr-radiant puls in an opn structur (T = 1, for an initial population lctrons/m 6, a dipol momnt r 01 = 10 nm and two valus w (0 = 0. 4, a numbr of lctrons (dnsity of quantum dots B γ whil T = K. of th dcay rat N = 10 16

8 636 Eliad ªtfãnscu, Aurl Sãndulscu 8 (a Quantum tunnling without injctd currnt (Φ = 0. (b Quantum tunnling with an injctd currnt I = N Φ = 0 ma/mm. Fig. 3. Supr radiant puls gnration in quantum tunnling with and 16 without injctd currnt, for w (0 = 0. 4, N = 10 quantum dots/ m, r B 7 1 =, and γ = 10 s, γ = nm F 0 Othrwis, th continuous polarization cratd by th injctd currnt I dtrmins a continuous incras of th fild (th blu curv T = 0, and th grn curv T = 1, that is a non-physical solution this incras is limitd by fild dissipation.without a fild dissipation ( γ EM = 0, for an injctd currnt abov a thrshold, th dnsity of nrgy (15 bcoms, whil th population diffrnc (16 bcoms 0, that mans a violation of th dtaild balanc

9 9 Dynamics of lctrons tunnling in a micro-cavity 637 principl, that rquirs a population invrsion for cancling th diffrnc btwn th nvironmnt-assistd dcays and xcitations. Th basic problm discussd from th bginning of th supr-radianc domain [3], and that continus to b of intrst [14], is th xponntial dpndnc of th supr-radiant puls amplitud on th numbr of tunnling lctrons (no injctd currnt I = 0. In Fig. 4 w rprsnt this dpndnc for (a Absolut valus. (b Supr-radiant xponnt. Fig. 4. Th N -dpndnc of th supr-radiant puls amplitud S M 6 for diffrnt valus of dcay rat for a dipol momnt r01 = 10 nm, 16 N = 10 quantum dots/ m, and (0 0 4 w =..

10 638 Eliad ªtfãnscu, Aurl Sãndulscu 10 diffrnt dcay rats. For a strongly dissipativ cas, th supr-radiant xponnt is approximatly 3, whil for a wkly dissipativ cas, this xponnt dcrass du to th Rabi oscillation that tnds to limit th amplitud of th supr-radiant puls whn th frquncy of this oscillation incrass with th quantum dot dnsity. As a conclusion, from a quantum mastr quation ntirly satisfying th quantum-mchanical and dtaild balanc principls w drivd Maxwll-Bloch quations for a supr-radiant structur with quantum dots and a prfctly tund cavity. In comparison with othr Maxwll-Bloch quations prviously usd in th supr-radianc domain, ths quations hav xplicit microscopic cofficints, with a tmpratur dpndnc according to th dtaild balanc conditions, and includ an injctd currnt and a dissipation of th fild that is ncssary for thir intrnal consistncy. W invstigatd th dissipativ suprradiant tunnling having in viw principal charactristics that dtrmin th supr-radiant powr. For ralistic valus of th systm paramtrs w obtaind supr-radiant powr dnsitis comparabl to th powr dnsity providd by Sun at th lvl of our plant, that is approximatly kw/m, thus suggsting th application to an fficint transformation of th solar powr to micro-wavs, whil this fficincy is monitord in th framwork of a microscopic modl. REFERENCES 1. D. J. Atkins, H. M. Wisman, and P. Warszawski, Phys. Rv. A 67 ( Masahiro Asada, J. Appl. Phys. 94 ( R. H. Dick, Phys. Rv. 93 ( Fritz Haak and Roy J. Glaubr, Phys. Rv. A 5 ( A. Crubllir, Phys. Rv. A 15 ( A. Crubllir and M. G. Schwighofr, Phys. Rv. A 18 ( D. Poldr, M. F. H. Schruumans, and Q. H. F. Vrhn, Phys. Rv. A 19 ( F. P. Mattar, H. M. Gibbs and S. L. McCall, and M. S. Fld, Phys. Rv. Ltt 46 ( I. V. Jyotsna and G. S. Agarwal, Phys. Rv. A 50 ( G. S. Agarwal and R. R. Puri, Phys. Rv. A 43 ( K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rv. Ltt. 65 ( E. Yablonovich and T. J. Gmittr, K. M. Lung, Phys. Rv. Ltt. 67 ( Sajv John and Tran Quang, Phys. Rv. Ltt. 74 ( Nipun Vats and Sajv John, Phys. Rv. A 58 ( Wi-Hua Xu, Jin-Hui Wu, and Jin-Yu Gao, Phys. Rv. A 66 ( M. Hirasawa, T. Ogawa, and T. Ishihara, Phys. Rv. B 67 ( A. K. Rajagopal, Phys. Ltt. A 46 ( G. W. Ford, R. F. O Connll, Phys. Rv. Ltt. 8 ( E. Stfanscu, Physica A, in prss. 0. R. Xu and Y. Yan, X.-Q. Li, Phys. Rv. A 65 ( C. Anastopoulos and B. L. Hu, Phys. Rv. A 6 ( Lornza Viola and S. Lloyd, Phys. Rv. A 58 ( P. J. Dodd and J. J. Halliwll, quant-ph/ v1 0 Jan E. Stfanscu and A. Sandulscu, Int. J. Mod. Phys. E 11 (