Optimal Tracking Control Design of Quantum Systems via Tensor Formal Power Series Method

Transcription

1 5 he Ope Auoao ad Corol Syse Joural, 8,, 5-64 Ope Access Opal racg Corol Desg of Quau Syses va esor Foral Power Seres Mehod Bor-Se Che, *, We-Hao Che,, Fa Hsu ad Weha Zhag 3 Depare of Elecrcal Egeerg, Naoal sg-hua Uversy, Hschu, 33, awa. Depare of Elecrcal Egeerg, Hsupg Isue of echology,achug, 48, awa. 3 Iforao ad Corol Research Ceer Shehe Graduae School, Harb Isue of echology, HI Capus, Shehe Uversy ow, Xl, Shehe 5855, P.R. Cha Absrac: I hs sudy, order o geerae a sequece of desred quau saes (or quau bs for quau coucao ad copuao, s ore appealg o forulae a quau corol syse as a blear sae referece racg syse. A opal racg corol s proposed o acheve he sae-racg by solvg a Halo-Jacob equao (HJE. I order o avod he dffculy solvg he HJE wh a closed-for soluo, he echque of foral esor power seres s eployed o rea wh he HJE o oba he opal racg corol quau syses fro he approxae desg perspecve. If he quau syse suffers fro sochasc paraeer varaos, could be odeled as sae-depede ose. I he suao, sochasc opal racg corol desg s also developed for quau syses. Fally, several exaples are gve o llusrae he desg procedure ad o cofr he perforace of he proposed racg corol ehod.. INRODUCION Modelg ad corol of quau echacal syses have bee dscussed sce 98s [-3]. Oe eeds o corol he dyacs of reacg aos ad olecules a he croscopc level, whch eeds he owledge of quau echacs for exac udersadg ad descrpo of he dyacs. Opal corol has bee used for quau echacal syse [4-7] by solvg he wo-po boudary proble va agrage ulpler ehod. I he early 98s, chess have red o corol checal reacos by properly arragg elecroagec feld [5, 6, 8] o crease he probably of a favorable checal reaco. A feed-forward corol was developed based o he ehod of verse proble va he seleco of Haloa [9]. he checal experes o he eraco bewee he elecroagec feld ad wo or hree level aoc syses led o oe possble geeralao of corollably of quau echacal syses [-3]. Sce a quau echacal syse ca be regarded as a faly of uary operaors, he corollably s based o he uary represeao of e group. hs echque has resolved he quau feed-forward corol o a uary operaor cosruco proble. I order o rea he flucuaos of phoocurre a quau echacal syse, feedbac corol for quau syses appeared quau opcs [4-6]. Feedbac corol va he rasfer fuco ehod was developed for quau echacal [7, 8]. he sochasc Schrodger equao was roduced so ha oe ca corol quau syses uder ose [9-3]. *Address correspodece o hs auhor a he Depare of Elecrcal Egeerg, Naoal sg-hua Uversy, Hschu, 33, awa; el: ; Fax: ; E-al: bsche@ee.hu.edu.w Recely, progress quau elecrocs has revealed he possbly of quau forao echologes, whch are expeced o elae he boleec of oder coucaos ad copuao [7, 8]. I he coveoal opal corol of quau syses [4, 5, 7] oe eeds o solve wo-po boudary equaos o acheve he opal corol o seer he quau sae o approach a equlbru po. I s o easy for a quau syse o rac a sequece of desred saes. I order o ae quau syses useful coucao ad copuao, s assued ha we ca specfy a sequece of desred quau saes wheever we eed, o aer how he evroe of he quau syse would be. I oher words, s presued ha he quau sae ca be corolled o geerae a sequece of desred quau bs for he use of coucaos ad copuao [7],.e., he wo-level sp quau syse should rac saly a sequece of desred saes (or quau bs whch are eeded for coucaos ad copuao. hs presupo s far fro rval by ag o accou he fac ha he quau syses soees eagle wh he evroeal syses, whch resul a osy forao resource. I he suao, robus racg corol s ecessary o guaraee he quau syse o rac ay desred quau sae we eed he presece of uceraes or oses. herefore, he proposed sochasc opal racg corol s dffere fro he coveoal opal corols for quau syses. I shor, he goal of our proposed approach s o acheve he opal racg corol of quau syses wheher uceraes or oses are preseed or o so ha ca also geerae a sequece of desred quau bs for he use of coucaos ad copuao. I hs sudy, he quau syse s odeled by a blear sae space odel. he desred quau sae o be raced s fro a referece odel. he a coroller s specfed o ae he quau syse rac opally a desred /8 8 Beha Ope

2 Opal racg Corol Desg of Quau Syses he Ope Auoao ad Corol Syse Joural, 8, Volue 5 referece sae. I he coveoal opal corol quau syse, oe eeds o solve a wo-po boudary proble by he agrage ulpler ehod [4-7]. I he proposed opal racg corol of quau syse, Bella dyac prograg equao s used hs sudy. Based o Bella dyac prograg [4], he proposed opal racg corol proble of quau echacal syses eeds o solve a HJE. Because he HJE s a olear paral dffereal equao, ca o be easly solved aalycally o oba a closed-for opal racg corol desg of quau echacal syses. For he coveece of desg, a esor foral power seres approach s eployed o solve he HJE approxaely. Afer solvg he esor foral power seres for he HJE [5], we ca approxae he opal racg corol of quau echacal syses. If he paraeer varao ose ad approxao error ca be odeled by a sochasc process, he he corolled quau echacal syse ca be cosdered as a sochasc sae referece racg syse. I he suao, a opal sochasc racg corol s also developed o acheve a desred referece sae racg. Because he paraeer varao ose ad odelg error always exs praccal case, he proposed sochasc racg corol desg for quau echacal syse s uch poeal for praccal applcao. Because he quau sae wll collapse o soe egesae whe we observe, we fd he opal corol pu by sulag he feedbac syse odel wh copuer wh whch here s a odel o represe he quau echacal syse, ad eforce acually he corol wh ope loop o he acual quau echacal syse [4, 5]. he syseac bloc dagra s show Fg. (. racg corol sgal s geeraed by he coroller usg he forao of racg error o ae he sae x ( of a quau syse rac a desred sae xd ( opally. I suary, he a corbuos of hs paper are as follows. Frs, our corol approach ca rac ay desred referece sae, whereas he coveoal opal quau corol [4-7, 6-8] oly ca regulae he quau syse o a fxed quau syse sae bu ca o rac ay desred referece sae. Secod, a esor foral power seres approach s eployed o solve he HJE approxaely, sce ca o be easy o solve a HJE aalycally. hrd, our corol approach ca be suable for a opal sochasc racg corol whle he paraeer varao ose ad odelg error exs. referece sae r( Corol law u( Quau syse Model Fg. (. Sysec bloc dagra of racg corol of quau syses. he opcs laer secos are suared as follows. he corolled quau syse s descrbed he sae space seco II. he opal corol racg corol sraegy s proposed seco III, whereas he opal sochasc racg corol s furher dscussed seco IV. he echques of esor foral power seres wll be eployed o solve he olear paral dffereal HJE ad he he opal racg corol s expressed wh a esor foral power seres seco V. I seco VI, several sulao exaples are gve o llusrae he desg procedure ad o cofr he racg perforace of he proposed corol desg ehod. Fally, a cocluso s ade seco VII.. SAE SPACE DESCRIPION OF QUANUM MECHANICA SYSEMS Cosder a quau echacal syse o be corolled as follows [3-5]: h & ( Hˆ ( ( where ( s he sae of he quau syse, defed a (fe or fe-desoal coplex Hlber space H. h s he Plac s cosa dvded by. Ĥ s a Haloa operaor represeg he eergy of he syse. he eergy cludes he ec eergy ad poeal eergy. By coveo, we orale ( (, (Merbacher 998 [8]. he resul cofors o he probably cocep,.e., ( dsplays he dsrbuo of he egesaes of he Haloa operaor. he egesaes sasfy [9], H ˆ E,,,3,... ad are orhogoal o each oher, where s he -h egesae of he quau syse, ad E eas he egeeergy of he -h egesae. Cosder a quau syse wh oraled egesaes,, 3, wh whch he sae ( ca be decoposed as [9]: ( c( + c( + c3( ( where c ( s he probably aplude of he -h egesae ad c ( expresses s probably. Because ( (, we ca fd ha c (. he corol of a quau syse s geeraed by he exeral poeal felds, ad he felds ca fluece he Haloa operaor. he corolled quau syse s descrbed as follows [6, 3-34] h & ( ( Hˆ + u Hˆ ( (3 where H ˆ are he exeral Haloa operaors ha are flueced by he exeral elecroagec feld, ad u ( are relaed o he sregh of he elecroagec feld. We

3 5 he Ope Auoao ad Corol Syse Joural, 8, Volue Che e al. ca corol he quau sae o ay desred sae by desgg u ( as corol pus. I order o apply he corol law he quau syse, we eed frs o reodel he Schrödger equao he arx represeao [9] o le he sae ad egesaes of he quau syse be he for of vecors, ad he he Haloa operaor be he for of arces. he reodeled dyac syse he arx represeao s + h & ( ( H u ( H ( (4 where he sae vecor ( c ( c (, ad H,,,..., H are Hera arces. If he quau syse o be corolled s fedesoal or hgh-desoal, for he coveece of corol desg, we oly cosder o corol a sall uber of egesaes whose egevalues are close o each oher ad far away fro hose of he oher res egesaes. I he suao, we use a weaer corol o acheve a ear desred sae, o ecessarly o arrve a he far egesaes wh larger egevalues, whch are rucaed he desg proble. herefore we ca jus dscuss he syse sae ha chages o he space whch s coposed of paral egesaes, ad he deso of he quau syse (4 s reduced sgfcaly. Geerally, he corol objecve quau syses s o corol he probables c (,,,... for whch he quau syse s respecvely a he egesaes,,,.... We wll develop a racg corol law by usg he opal corol ehod. Obvously, he desred sae corol desg quau syses ca be forulaed as a opal racg proble. 3. OPIMA RACKING CONRO IN QUANUM SYSEMS Cosder a quau syse of deso. he dyac odel of he quau syse s of he followg for + h & ( ( H u ( H ( (5 where ( C ; H C ; H C ; u R,,,,, ad H, H are Hera arces. hs quau syse sae s defed coplex space. I order o decopose he quau echacal equao o real ad agary pars, he followg real syse sae vecor ad arces are eployed o deoe boh real par ad agary par of quau syse (5: Re[ ( ] I[ H] Re[ H] x (, I[ ( ] G Re[ H] I[ H], h I[ H] Re[ H] G Re[ H] I[ H],,,..., (6 h I hs suao, he coplex quau echacal dyac equao (5 ca be equvalely represeed by he followg real quau echacal syse: + x& G x u G x (7 where x R ; G R ; G R, u R,,,,. Sce H, H are Hera, G, G are sewsyerc arces wh all her egevalues he agary axs. Furherore, fs wh x, where x x + x x. Assue he desred sae o be seered s xd R wh x. Before he opal racg corol s developed d for he quau syse (7 o acheve he desred sae, he racg cos fuco s defed as follows: J ( x( r( F ( x( r( + f f o f f f ( ( xr Qo x r + u d (8 where F ad Q are syerc posve defe weghg arces, ad are posve ubers. he referece odel s defed as r &, r( xd.e., he referece sae could be ay desred quau sae x d, or r ( xd for. he opal referece racg corol proble s o specfy u, u,..., u such ha he cos fuco J (8 s ed. I he coveoal opal quau corol [6], oly he oal corol eergy f ud s ed so ha s o suable for racg desg of quau syses. However, hs sudy, he corol sraegy s o e he eral racg error, he racg pah error ad he corol effor sulaeously. he radeoffs are depede o he specfcao of weghgs F, Q ad by desgers. hs opal racg corol s suable o oly for he wo-level quau syse bu also for oher ul-level quau syses. I s dffcul o fd he opal soluo of a olear racg corol syse drecly. I could be rasfored o he followg opal sae regulao proble. x e R r 4, %x : x r I I (9 where s he augeed sae ad x% s he racg error. he he opal racg syse (7 ad (8 ca be augeed as he followg opal sae regulao proble:

4 Opal racg Corol Desg of Quau Syses he Ope Auoao ad Corol Syse Joural, 8, Volue 53 & A u B,,..., ( + f J f Ff + Q + u d ( here G G A, B,,,..., I I F F[ I I], Q Q[ I I]. I I he opal regulao proble of quau echacal syse ( ad ( s o specfy corol varables u (,,,..., o ae J as sall as possble. hs s a blear opal regulao proble, he Bella dyac prograg echques [4] olear opal regulao proble wll be eployed o rea hs opal corol proble of quau syses. he we ca have he followg resul. heore : he opal racg corol of he quau echacal syse (7 s gve as o V u B,,,..., ( where V(, > s solved fro he followg olear paral dffereal HJE V V V V Q+ A B B 4 V(, F f f f f (3 Proof: See Appedx. 4. SOCHASIC RACKING CONRO UNDER SOCHASIC PARAMEER PERURBAION I he corolled echacal syse (4, he oal Haloa operaor H s defed by he ec eergy ad poeal eergy he vara poeal feld. Soees, here s sysec varao or uceray, whch s due o eperaure varao, odelg error, chage of agec oe of ucleus, perurbao of equpe, ec.. he ucera varao H could be decoposed as H (, where he sochasc par s absorbed by he ero ea whe ose ( wh u varace ad he deersc par s absorbed by H. I he suao, he dyac quau syse (4 could be odfed as he followg Iô dffereal syse h d( ( H + u ( H ( d + H ( dw (4 where dw ( d ad W s he sadard Browa oo process, ad HdW eas he sochasc perurbao Haloa operaor. By he slar rasforao procedure, he quau echacal syse wh sochasc paraeer perurbao could be odfed fro ( as follows d ( A+ u ( B d+ CdW (5 G where C wh I[ H] Re[ H] G Re[ H] I[ H]. h Suppose he perurbave quau syse s corolled o rac he desred referece sae r ( xd by g he followg regulao cos fuco for sochasc opal racg corol f J Ef Ff + Q + u d (6 where E deoes he operao of expecao. he we have he followg resul for sochasc quau echacal syses. heore : he opal racg corol for quau echacal syse wh sochasc perurbaos (4 s gve by o V u B,,,..., (7 whch V(, > s obaed by solvg he followg HJE V V V V Q+ A B B 4 V(, + C C V(, F f f f f (8 Proof: See Appedx. Fro he aalyss above, we eed o solve he HJE (8 before we desg he opal racg corol (7 for perurbave quau syse (5. herefore, he os pora wor o rea he opal racg corol proble he quau echacal syse s o solve he HJE (3 or (8 a frs. However, s dffcul o fd a closed-for soluo for (3 or (8. I hs sudy, for he coveece of desg, he echques of esor foral power seres wll be used o solve HJE (3 or (8 fro he approxae perspecve. Rear : he evroeal perurbao quau syses ples a dsspave behavor called decoherece. he opal feedbac corol of such d of quau syses has bee suded [7]. Sce hs d of odel s ore coplcaed ha he odel (4 [35], we leave for furher research. he odel (4 ca be used o deal wh he corol of oe parcle whose sae ca be represeed by a sae vecor ad s uecessarly characered by a desy arx.

5 54 he Ope Auoao ad Corol Syse Joural, 8, Volue Che e al. 5. OPIMA RACKING CONRO VIA ENSOR FORMA POWER SERIES Due o he dffculy of obag he closed-for soluo for he HJE (3, he echques of esor foral power seres wll be eployed o solve he olear paral dffereal HJE fro he approxao po of vew. A he begg, he forulas abou esor ad foral power seres [5, 36] wll be roduced. 5.. esor Foral Power Seres Approach he -h order esor he -desoal space s a aheacal objec ha has dces ad copoes ad obeys cera rasforao rules. Here he esor spaces are defed by he esor produc [36] of vecors. Because he basc properes of esors are slar o he vecors, we ca wre he esors as he vecor fors. So we wll use roecer produc o dcae ay esor produc below. For exaple, cosder wo vecors -desoal x y space x, y x y, he he esor produc of he s xy xy x y. Afer ha, we eed o fd he for of he xy xy lear operaors o esor space. Rear : e { e,... } deoe he bass of vecors R, ad { e e... e,,.., ; } j j,,.., deoe he bass of esors R. Sce R s a Hlber space, we ca cosder lear operaors defed o R. e ( R be he space of lear operaors defed o R. We ca defe a lear operaor P whch has he relao [36] P l, l,..., l P,,..., l l l j j,,..., ( e e... e ( e e... e (9 P l, l,..., l here s he correspodg elee of he operaor,,..., P. Now he er produc of wo esors ( x... x ad ( y... y s defed as follows ( x... x, ( y... y xj, yj xj yj j j Ad he we defe he adjo * P of P wh he relao * P ( v... v, ( u... u ( v... v, P( u... u P Fro equao (, P s self-adjo f P l, l,..., l,,...,,,..., l, l,..., l ( ( If we deal wh he esor he vecor for, ad use he roecer produc o dcae he esor produc, lear operaors o he esor space ca be represeed as arces, ad self-adjo operaors becoe syerc arces. If we se v,...,, v w,..., w R, he wo esors ( v... v ad ( w... w could be refored o wo correspodg vecors wh elees, ad he operag arx o he s a ( ( arx,.e., he arx P has he elee l, l,..., l P he,,..., ( + ( + ( ( -h colu ad ( + + ( ( -h row. he ( l l l For exaple, cosder wo vecors x y x, y x y, he he equao ( y y P( x x ca be rewre vecor for as follows,,,, yy P, P, P, P, xx,,,, yy P P, P, P, xx,,,, yy P, P, P, P, xx,,,, yy P, P, P, P, xx ea [3]: We oe x s a ( vecor wh he - es roecer produc of x, ad he operag arces P,Q R ( (. he ( ( ( x P( x x ( x P( x x ( x P( x ( x ( x P( x x ( + jx ( P Q ( + jx ( ( (3 here P s syerc, ad ( P Q eas a roecer produc. ea : o geerale equao (, becoes e e... e, e e... e l l l f h lh h oherwse ( x P( x Cx ( x P ( C( x x ( (4

6 Opal racg Corol Desg of Quau Syses he Ope Auoao ad Corol Syse Joural, 8, Volue 55 here P ( R ad P ca be represeed as a ( ( syerc arx, C s a arx, ad ( s a C ( ( arx relaed o he arx C. he elee l, l,.., l ( ( C of he arx ( sasfes,,.., C l, l,.., l ( ( C,,.., C, + C, C, f,,..., l l Cl, f,..., j l l j j j lj+ j+,..., l, bu ( lj j ohers (5 he dealed aalyses abou ea ad ( C are gve Appedx 3. ea 3: Cosder he followg operao equao ( x P( x xc Cx ( x R( x (6 x he he arx R could be represeed as follows R ( C P ( C + P ( C P ( C (7 C where he arx ( s he sae as (4, ad P s syerc. he dealed dervao of ea 3 s gve Appedx 4. he calculao echques of esor power seres approxao ha we eed o solve (3 ad (8 are ready as show above. Now we use hese echques o hadle he quau racg corol desg proble. We se he soluo V(, of he HJE (3 or (8 o be of he esor foral power seres for where V(, ( P( ( (8 4 R ad P ( are couous, e-varyg (4 (4 syerc arces. Before usg he esor power seres o solve he HJE (3 ad (8, soe relaed dervaos should be gve a frs. Subsug (8 o (3 wh he help of (3 ad (4, we oba V ( P & ( ( (9a V A ( P ( ( A ( (9b V V B B 4 ( r Pr( r( B( r 4 r ( s Ps( s( B( s s ( ( r+ s ( Pr( r( B r s ( P( ( B ( (9c s s r+ s herefore, fro (9, (3 becoes ( P & ( ( Q+ ( P( ( A( ( ( r+ s ( Pr( r( B r s wh he eral codo ( P( ( B ( (3 s s r+ s f f f f f f f V(, F ( P( ( (3 Now, equag le powers (3 leads o he followg Rcca-le equaos P & ( + PA ( + AP ( + Q for (3a P & ( + P ( ( A + ( AP ( ( Pr( r( B ( Ps( s( B r+ s rs, for,3,... (3b wh eral codos P (, ( f F P f,,3,... (3c where he calculaos of ( A, ( B,, 3,... are gve (5. Afer solvg P ( fro he Rcca-le dffereal equao (3, we ge he opal racg corol fro ( as follows o u ( P( B(,,,..., (33 By duco, we ca rewre (3b as P & ( + P ( ( A + ( AP ( ( r( B Pr( ( s( B ( Ps( ( { ( ( r+ s rs, ( P B ( P B } + ( ( ( ( (34 r r s s ad so P ( s syerc.

7 56 he Ope Auoao ad Corol Syse Joural, 8, Volue Che e al. 5.. Sochasc racg Corol Desg Now we furher vesgae he opal racg corol of a sochasc quau syse (4. Afer subsug (8 o (8 wh he help of (3, (4, (6, ad (7, we could also solve hs HJE. he addoal er (8 should be cosdered as V ( C ( ( (C P ( (C+ P ( (C P ( (C ( (35 ad hus he HJE (8 becoes ( P & ( ( Q+ ( P( ( A( ( r+s ( P r ( r (B r s ( P s ( s (B ( r+s + ( ( (C P ( (C+ P ( (C P ( (C ( wh he eral codo f f f f f f f V(, F ( P( ( (36 Now, equag le powers (36 leads o he followg Rcca-le equaos P & ( + PA ( + AP ( + Q+ CPC ( for (37a P & ( + P ( ( A + ( AP ( ( Pr( r( B ( Ps( s( B r+ s rs, ( C P C P C P C + ( ( ( + ( ( ( (, for wh he eral codos,3,... (37b P (, ( f F P f for,3,... (37c Afer solvg he Rcca-le equaos (3 o oba P (, P (,..., P (,..., he opal racg corols (7 for perurbave quau syse (4 becoe o u ( P( B(,,,..., (38 I geeral, he esor foral seres have fe ers o be solved. However, s dffcul o plee hgh-order ers P ( due o her coplcaed copuao. I hs sudy, oly a few pora ers of P ( are cosdered o approxae he opal racg desg. A wo-level sp syse ad a hree-level quau syse are boh dscussed below o llusrae he desg procedure of he proposed racg corol Coparsos wh Oher Approaches Now we provde a dealed coparave aalyss of he proposed schee wh he exsg schees [4-7,6-8]. Our proposed schee has he followg advaages: Frs, our corol approach ca rac ay desred referece sae, whereas he coveoal opal quau corol [4-7, 6,8] oly ca regulae he quau syse o a fxed quau syse sae bu ca o rac ay desred referece sae. Secod, he proposed corol sraegy ca e he eral racg error, he racg pah error ad he corol effor sulaeously. By coras, oly he oal corol eergy s ed [6], oly boh he oal corol eergy ad he eral sae error are ed [4,5,7, 8] ad he oal corol eergy, he eral sae error ad he sae eergy are ed [6, 7]. herefore, hose schees [4-7, 6-8] are o suable for racg desg of quau syses, sce her corol sraeges are o o e he racg pah error. hrd, a esor foral power seres approach s eployed o solve he HJE approxaely sce s sll o easy o solve a HJE aalycally, whereas here s o ehod cocerg solvg a HJE [7] alhough s opal corol s also relaed o a HJE. Fourh, our corol approach ca be suable for a opal sochasc racg corol whle he paraeer varao ose ad odelg error exs, whereas [4, 6, 7, 6, 8] ca o deal wh he sochasc opal corol of quau syses. Bu, our proposed schee has oe dsadvaage ha s soewha uch copuaoal coplexy due o a esor foral power seres. However, our uercal copuao e s uch reduced sce here s o erave procedure o search he opal corol. By coras, here s uch copuaoal coplexy [5-7, 7-8] due o a exrcable erave procedure o solve wo coupled dffereal equaos,.e. he sae ad cosae equaos, ad he updae he opal corol a each erao sep. Sce [4-7, 6-8] are o suable for racg desg of quau syses, he corollers are que dffere fro ours so ha coparsos of he copuaoal coplexy of her schees o ours are o easly dscussed. Moreover, here s o ehod cocerg solvg he HJE [7], he coparso of he copuaoal coplexy of her schee o ours s o easy o dscuss eher. 6. SIMUAION EXAMPES I hs seco, several sulao exaples of wo-level ad hree-level quau syses are gve o llusrae he desg procedure of he opal racg corol by he esor foral power seres ehod. wo-level Quau syses are he os basc quau syses wh pora applcaos, especally coucaos ad copuao [6]. A wo-level quau syse eas a syse whch has wo eergy levels ( oher words, he syse has wo ege-

8 Opal racg Corol Desg of Quau Syses he Ope Auoao ad Corol Syse Joural, 8, Volue 57 values ad wo egesaes. A wo-level syse s used o geerae a sequece of desred quau bs for quau copug ad daa coucaos [6], whch could be forulaed as a racg corol proble of quau syse. Soe of he syses have he propery ha hey have wo close levels far away fro ohers, ad ca be approxaed by he wo-level syses. he sp / syses are wo-level syses ad have bee suded exesvely. Here we cosder a sae racg corol proble of a sp / aoc ucleus [6]. A wo-level-sp parcle (here eas a aoc ucleus s corolled by a exeral agec feld slar o he oe he uclear agec resoace (NMR experes [6]. Frs we fx agec feld B he - dreco, ad he fxed feld deeres he dreco of sp ad decdes he egesaes of he sp. he agec feld he x y dreco s vared o chage or corol he dreco of sp. he vared feld ca be corolled by wo orhogoal copoes ux ad u y, or oe of he. he syse s descrbed Fg. (. Here he corol purpose s o seer he sp fro oe egesae o a desred sae o for a quau logc gae quau copug [6, 3, 34], ad he al sae could be prepared by Ser-Gerlach apparaus [37]. Fg. (. wo-level sp corol syse. Cosderg he corol of a sp / ucleus, he dyac equao of he sp syse s derved soe ex boos of quau echacs [9]. he quau dyac equao s gve as follows h & ( x u x h h h B ux u y ( B sp + sp - (39 where s he gyroagec rao assocaed wh he sp syse, B s he fxed copoe of he agec feld ad u x, u y are he copoes of he agec felds vared wh he racg corol laws. he sp / syse has egesaes :, :. u y y B B By rescalg (, ( becoes (, B ux ( becoes ux (, ad uy ( becoes B uy (. Equao (39 ca be rewre as u x uy & ( + + ( B B (4 u u x y Ad furher rescalg ux (, uy (, equao B B (4 ca be rewre as & ( + ux + uy ( (4 For coo NMR devces, we se he fxed agec feld B as esla (for exaple: he NMR devce, Bruer AC, aes he hydroge ucleus wh he resoace frequecy MH. By corollg he hydroge ucleus wh he gyroagec rao rad s esla 4.58 MH / esla, he e of he sulaed resul (4 ca be scaled 8 wh, ad he corol agec feld wll be B scaled wh I hs desg exaple, hree racg corol cases are cosdered wh oe corol pu, wo corol pus, ad oe corol pu wh sochasc paraeer perurbao, respecvely. Exaple : Sp syse racg wh oe corol pu Frs we cosder he sp syse corolled by a agec feld he y -dreco. he dyac equao of he corolled sp syse s gve by & ( + u ( (, ( (4 By (6, he correspodg sae space of (4 ca be rewre as x& Gx+ ugx, x( x where G, G ad x (43 correspods o he sp-up sae :. Ad he referece odel for he desred sae s gve by r& r r x (44, ( d

9 58 he Ope Auoao ad Corol Syse Joural, 8, Volue Che e al. where xd correspods o he sp-dow sae :. herefore, we seer he sp fro he sp-up sae x o he sp-dow sae x d. For he racg cos (8, we specfy, 4, F Q f 8 7 where f * s s afer recoverg he scale of real quau syse. Afer regulag, we have he augeed quau syse ( ad he correspodg opal corol cos fuco (. he arces ( ad ( are show as follows. A, B, F Q (45 For he splcy of copuao, oly a s cosdered o approxae he corol law. Afer recoverg he scale, he sulao resul s show Fg. (3. So he probably,.e. c ( x ( + x (, of he sp-up sae ad 3 he probably,.e. c( x( + x4(, of he spdow sae are sulaed Fg. (3 o cofr he racg perforace of he proposed ehod. I s see ha he sae of he wo-level quau syse ca rac he desred sae que well wh oly oe corol pu. Because he Rcca-le equaos (3 could be effcely solved by he oolbox Malab, he proposed opal racg corol (33 by esor foral power seres could be easly calculaed, especally for a case hs exaple. probables agec feld(esla.5.5 c ( + c ( c ( x +x 4 c ( x +x corol pu u( x e (sec x 7 Fg. (3. he probables of egesaes (upper half sde, ad he correspodg corol sgal (lower half sde for he sae racg corol he wo-level quau syse by a agec feld he y -dreco exaple. Exaple : Sp syse racg corol wh wo pus Suppose wo corol pus wh he agec felds he x-dreco ad y-dreco are eployed o corol hs sp quau syse. he dyac equao of he sp syse wh wo corol pus s gve by: & ( + u( + u( (, ( (46 he al value of x (7 s se as x( x [ ]. Suppose we corol he quau sae wh a agec feld o rac he followg desred quau sae as (, f x d ( ] as ( f, f,.e., he sp syse s o be corolled fro he sp-up sae a e o he sp-dow sae wh f [, ] ad he bac o he sp-up sae wh f. [, f ] Slar o he above exaple, he weghgs he racg cos (8 are gve by

10 Opal racg Corol Desg of Quau Syses he Ope Auoao ad Corol Syse Joural, 8, Volue 59 F Q, 5, ( here f f 7 s real syse. Afer regulao rasforao, he arces A, F, ad Q are he sae as (45, ad B, B are gve as follows B, B For he splcy of copuao, a s cosdered. Afer recoverg he scale, he sulao resul s gve Fg. (4. I s see ha he racg perforace wh wo corol pus s uch beer ha ha wh oly oe corol pu, especally he rase respose. hs s because wo corol pu has ore degrees of freedo o apulae he corol effors o rac he desred referece sequece ha oe corol pu case. probables agec feld(esla.5 c ( + c ( c ( x +x 4.5 c ( x +x 3 c ( x +x 3 c ( x +x corol pus u( x 7 4 u ( u ( u ( u ( e (sec x 7 Fg. (4. he probables of egesaes (upper half sde, ad he correspodg corol sgals (lower half sde for he syse racg corol he wo-level quau syse by wo agec felds he x -dreco ad he y -dreco, respecvely, exaple. Exaple 3: Sp racg syse wh sochasc perurbaos Cosder he followg sp syse wh sochasc perurbaos d( ( d + u ( ( d.. + ( dw.. (47 ( he al value of x (7 s se as x( x [ ] ad he referece odel for he desred sae s gve by r &, r x d Afer regulao rasforao, we have he augeed quau syse ad he correspodg opal corol cos fuco show (5 ad (6. he arces A, B, F, ad Q are he sae as (45, ad he oher weghgs (5 ad (6 are show as follows C, 4, f ( here f 7 s real syse. For he splcy of copuao, a s cosdered. Afer recoverg he scale, he sulao resul s show Fg. (5. probables agec feld(esla.5.5 c ( + c ( c ( x +x corol pu u( x c ( x +x e (sec x 7 Fg. (5. he probables of egesaes (upper half sde, ad he correspodg corol sgal (lower half sde for he sochasc syse racg corol by a agec feld he y -dreco exaple 3.

11 6 he Ope Auoao ad Corol Syse Joural, 8, Volue Che e al. Because of sochasc perurbao, he racg perforace hs exaple s worse ha ha Exaple. However, he racg perforace s sll very sasfacory. Obvously, he proposed racg corol ehod s robus o he sochasc perurbao quau syse ad has uch poeal applcao o praccal corol desgs. Exaple 4: Sae racg corol for a hree-level syse I order o cofr he proposed racg corol ehod o ore coplex quau syse, le us cosder he followg hree-level quau syse [8] & ( H( + u( Hra(, (. (48. where H. deoes he self Haloa 3. of he hree-level quau syse wh each dagoal elee represeg he eergy for each syse level, H ra deoes he elecroc dpole raso arx ad u ( deoes he elecrc feld whch duces rasos bewee hree levels of hs quau syse. Here we oe ha all calculaos are carred ou aoc us (a.u.. Suppose we corol he quau sae o rac he followg desred quau sae as (, f d ( ] as ( f, f,.e., he quau syse s o be corolled fro he egesae a e o he egesae wh [, f ] ad he f o he egesae 3 wh [, f ], where f 3 a.u.. he weghg arces F ad Q are he sae as he oes Exaple, ad 5. he sulao resul s show Fg. (6. Dscusso: By he esor power seres o approxae he soluo of he olear HJE (3, he opal racg corol could be easly desged by solvg he Rccale equaos (3, whch could be easly solved wh he help of oolbox Malab. herefore he proposed ehod could splfy he desg procedure eve hough he approxao procedure sees coplcaed. Furher, hese exaples we have red oher ehods o solve he HJE of opal racg corol desg of quau syses, e.g. fuy approxao echque, bu he echque of esor foral power seres s spler desg procedure ad beer racg perforace. Furher, he coveoal quau corol desg oly corol he syse o a equlbru sae. Bu he proposed opal racg corol desg could ae he quau syse rac a sequece of desred saes accordg o he referece odel. herefore, he proposed racg ehod s ore poeal o geerae a desred sequece of quau bs for quau coucao ad copuao. probables elecrc feld.5.5 c ( c ( c ( + c ( + c 3 ( c 3 ( c 3 ( c ( c ( corol pus u( e (a.u. Fg. (6. he probables of egesaes (upper half sde, ad he correspodg corol sgal (lower half sde for he sae racg corol by a elecrc feld exaple CONCUSIONS I hs sudy, he sae space odel for quau syses s cosruced ad he he opal racg corol of quau syse s proposed o acheve ay desred sae based o Bella dyac prograg ehod. I order o avod he dffculy solvg he paral dffereal HJE equao drecly, he echques based o esor foral power seres s eployed o oba he approxae opal racg corol law of quau syses. he sochasc opal racg corol desg for quau syse wh ucera perurbao s also developed based o Iô-ype dffereal equao o ee he praccal osy evroe. Ule he coveoal opal corol s oly o sable he quau syse o a equlbru sae, he proposed racg ehod ca rac ay desred sae geeraed by a refereced odel. Several sulao exaples of wo-level or hree-level quau syses are gve o llusrae he desg procedure of he opal racg corol by he esor foral power seres ehod. Eve wh secod order esor power seres approxao, he racg perforace s very sasfacory. I s also foud ha he racg perforace wh wodreco corol pus s wh shorer rase respose ha wh oe-dreco corol pu. Eve he desg procedure s coplcaed because of he requree of opal racg perforace ad approxag soluo of HJE by esor power seres, he corol law s very sple he resul. herefore, he proposed opal racg corol desg s suable o geerae a sequece of desred saes (or bs wh uch poeal applcao o quau copuao ad coucaos. APPENDIX : Proof of heore :

12 Opal racg Corol Desg of Quau Syses he Ope Auoao ad Corol Syse Joural, 8, Volue 6 he opal corol ( ad ( s obaed by eas of he soluo V(, > of he followg Bella dyac prograg equao [4] V(, V(, Q+ ( A+ ub + u u (,.., (49 wh he eral codo V( f, f f Ff. Usg he echque of copleg of square, (49 becoes V V Q+ A+ u (,.., V V ( u + B ( u + B V V B B, V( f, f f Ff 4 (5 Fro he rgh had sde of (5, ca be foud ha he u s acheved f he opal corols are specfed as o V u B,,,..., (5 o Uder he specfcao of he opal corol u (5, he dyac prograg equao (5 becoes he followg Halo-Jacob equao (HJE V V V V Q+ A B B (5 4 V( f, f f Ff herefore heore s proved. APPENDIX : Proof of heore : By Bella dyac prograg prcple for olear sochasc opal corol syses, we ge V(, V V H (,, u,, (53 u ( V( f, f f Ff where V(, > ad he geeraled Haloa fuco H s defed as V V H (,, u,, : Q u + + (, (, ( + + V V V A u B C C Q+ A V V + ( u + B ( u + B V V V(, B B + C C (54 4 Here he er V(, C C appears due o he Iô dffereao sochasc process [38-4]. Obvously, fro (54, follows ha (,,, V, V (,, o, V, V H u H u u ( where (55 o V u B,,,..., (56 ad he dyac prograg equao (53 becoes he followg HJE V V Q+ A V V V(, B B + C C (57 4 V(, F f f f f herefore heore s proved. APPENDIX 3: Before provg ea, he followg fac should be frs proved. Fac ( x P( x Cx ( x Q( x x ( (58 where P s syerc ad he copoes of Q are of he followg for [3] l,..., l l,..., l Q,..., P,..., ',..., C ', ' (59 Proof of fac : he rgh had sde of (58 ca be expressed as follows : l ( ( x Q( x x x Q x x (6 l l l l l Now we defe he esor, h f h. f h he he lef had sde of (58 ca be expressed as follows: ( x P( x x Cx

13 6 he Ope Auoao ad Corol Syse Joural, 8, Volue Che e al. l l ( xl x l P x x C, hxh x j j ll h jhl, l l l x x x x P x x l l l, j l+ l ll xl x l P x x, jx x C, hxh + h + (sce l, jcj, h Cl, h j l l xl x l x hxl x l P, Cl hx x + h ll l l xl x l P, C hx x x hx x + h ll + exchagg l (by h l l hl+ l xl x l P, Ch l x x h ll l l xl x l P, h Ch x x + h ll + (by exchagg l j j j he frs er ad usg P syerc l l xl x l P, h Ch x x + h ll (by replacg h wh l l xl x l P C x, x + ll ll x l x l ll P C x, x + (6 Coparg (6 ad (6, we have proved (59. herefore he Fac s proved. Proof of (5: Afer provg he equaos (58 ad (59, for he purpose of applcao, we eed o facore he arx Q o be P ( C as follows ( x P( x x ( Cx ( x Q( x ( x P C x ( ( ( (6 So he copoes of he arx Q sasfy l,..., l l,..., l q,..., q,..., P q,..., q C,..., q q q Q... ( ( (63 Coparg (63 wh (59, we ca fd he elee l,..., l ( ( C,..., sasfes l,..., l ( ( C,..., C, + C, C, f,,..., l l l (64 Cl, f,...,,..., bu j l l j j j lj+ j+ l lj j ohers herefore ea s proved. For exaple, for a -desoal sae, we have C, C, ( C, C, C, C, C, C, C, C, + C, C, ( C C, C, + C, C, C, C, C, 3C, C, C, C, C, + C, C, C, C, + C, C, C, C, C, + C, 3 ( C C, C, C, C, C, C, C, C C, + C, C, C, C, C, C C C C + C C C C 3C Appedx 4: Proof of ea 3: ( x P( x x C Cx x ( x P( x x C Cx x x ( x P( x x C C x ( x P( C( x Cx x, +,,,,,,,,, (by usg (4

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15 64 he Ope Auoao ad Corol Syse Joural, 8, Volue Che e al. [39] B. S. Che ad W. Zhag, Sochasc H / H corol wh sae depede ose, IEEE rasacos o Auoac Corol, vol. 49, o., pp , 4. [4] W. Zhag ad B. S. Che, H corol for olear sochasc syses, SIAM Joural o Corol ad Opao, vol. 44, o. 6, pp , 6. Receved: Aprl, 8 Revsed: Aprl, 8 Acceped: Jue 6, 8 Che e al.; cesee Beha Ope. hs s a ope access arcle lcesed uder he ers of he Creave Coos Arbuo No-Coercal cese (hp://creavecoos.org/lceses/by-c/3./ whch pers uresrced, o-coercal use, dsrbuo ad reproduco ay edu, provded he wor s properly ced.