The Role of Open Channels in Overlapping Resonances Bull. Korean Chem. Soc. 2010, Vol. 31, No DOI /bkcs

Transcription

1 The Rle f Open Channels n Ovelappng Resnances Bull. Kean Chem. Sc., Vl. 3, N. 3 DOI.5/bkcs Rle f Open Channels n Ovelappng Resnances Studed by Multchannel Quantum Defect They n Systems Invlvng Nndegeneate Clsed and Many Open Channels Chun- Lee Depatment f Chemsty, Aju Unvesty, nchen-dng, Yengtng-Gu, Suwn , Kea E-mal: clee@aju.ac.k Receved August 7,, Aepted Septembe 7, Pevus wk n the phase-shfted vesn f the multchannel quantum-defect they (MQDT f a system nvlvng clsed and many pen channels (Lee, C.-. Bull. Kean Chem. Sc., 3, 669 was extended t btan the fmulae f the spectal shape paametes wth the stuctue f a ple extacted explctly f geneal cases nly lmted by nn-degeneate clsed channels. The they was appled t the naw 6p /,3/np J = autnzng Rydbeg sees n baum petubed by the 6p 3/nf states btaned by de Gaaff et al. Key ds: Phase-shfted MQDT, Ovelappng nance, q evesal, Effectve cntnuum, Autnzng Rydbeg sees Intductn A Rydbeg sees petubed by an ntelpe s an mptant system shwng cmplex nance phenmena due t the velappng nance. Multchannel quantum defect thees (MQ- DT, wth the mplementatn f phase enmalzatn, ae extemely pweful f dealng wth the bseved velappng nances. -5 A system ncludng nly 3 channel systems wth ne pen and clsed chsen as the mdel system was studed extensvely by Gust-Suz and Lefebve-Bn, Cke and Cme, ntgen and Fedch and Ueda. 3,4,6,7 They nvestgated the phenmena n cmplex nances, such as ntensty bwng, q evesals and tuly bund states n a cntnuum. Amng these studes, the efmulatn f the phase-shfted MQDT by Ueda t dsentangle the ntelpe specta fm the petubed autnzng Rydbeg sees pvdes a vey useful tl f analyzng velappng nance specta, and was nstumental n Refs. [8] and [9] f detemnng the mutual nfluences between an ntelpe and the autnzng Rydbeg sees petubed by t n the phtnzatn css sectn σ : σ = σ = σ ( ( + eff + eff + eff + ( + eff + eff ntelpe eff whee ndex dentes the Rydbeg sees f an ntelpe, f the petubed Rydbeg sees, stands f the educed enegy f the Rydbeg sees, s the lne shape ndex and eff and eff ae the effectve educed enegy and lne shape ndex f the petubed Rydbeg sees, pectvely. Eq. ( shws that the effect f an ntelpe t the autnzng specta f a Rydbeg sees appeas n tw ways. One s t mdulate the ente shape f the specta s that ts envelpe s gven by the autnzatn specta ( σ = σ ( + /( + ( Intelpe f the ntelpe and the the s t petub each nance peak f the specta f the Rydbeg sees wth ts effect ncpated nt the nance paametes f each nance peak s that and ae eplaced by the effectve eff and eff, pectvely. The petubatn s lcal s that eff ( s n lnge a cnstant functn f the enegy, whch s n cntast t the cnstant f the unpetubed autnzng Rydbeg sees. Ref. [8] epted that the autnzatn css sectn σ f ( s dmnated by the sngula behav f ( eff + eff / ( eff + f the petubed sees nduced by the ple stuctu f eff ( = + eff + ( + + at = and and q evesals ung at the ze suface stuctu f the lne pfle paamete eff gven by eff (3 ( = ( + =, (4 whee s anthe lne shape paamete petanng t the petubed lne wdth eff f sees by the ntelpe. Such ple and ze suface stuctu wee ncpated nt the fm f a map t fllw the change and cntl the pattens f the autnzatn specta n pevus wk. 8 Ueda's fmalsm 7 was extended futhe t systems nvlvng me than pen channel t detemne the le f the addtnal pen channels n channel cuplng and the specta, whch mght be vewed as a cntvesal feld n MQDT wth uncnfmed

2 3 Bull. Kean Chem. Sc., Vl. 3, N. Chun- Lee assetns emanng t be lved. Fan and Cpe epted that a dscete state can nteact wth nly ne type f cntnuum. Mes develped the they futhe and ppsed the use an velappng matx t epent the extent f velap when a cmplete supempstn f velappng nances s n lnge btaned when me than l pen channel s nvlved n the cnfguatn mxng (CM they. Mes's CM they was mplemented nt MQDT t teat the spectal wdth n the slated ce exctatn (ICE specta by Lecmte. Chen challenged ths pblem f handlng the channel cuplng nduced by addtnal pen channels n the MQDT fmulatn wth the appxmate analytc slutn but was nly patally suessful. The auth's pevus wk appled Ueda's fmalsm f ths system nvlvng me than pen channel, and suessfully dealt wth the Matns and Zmmemann's phtnzatn specta f the Rydbeg sees Cu I 3d 9 4s( D nd G 9/ petubed by the ntelpe, 3d 9 4p 4 F 9/, f whch Chen's 4 channel QDT faled. Hweve, ths wk was unsuessful n btanng explct fmulae f the ze sufaces f the lne pfle ndex eff f geneal systems lmted nly by nn-degeneate clsed channels. Ths pape epts the fmulae wth the stuctue f a ple explctly extacted f such geneal systems. Obtanng explct fmulae s mptant f devng the physcal tems that detemne the spectal shape and btanng the ente spectum f pssble pattens f the spectal shape. The fmulae was appled t the Ba 6p / np 3/ J = autnzng Rydbeg sees wth 6p 3/ nf sees as an ntelpe Ovevew In the pence f me than ne pen channel, the ttal autnzatn css sectn s gven by a sum f the css sectns ve all pen channels. F pactcal pupses, t s bette t use the egenchannels f the physcal scatteng matces S n place f the pen channels. If ndex s used f the egenchannels, the ttal autnzatn css sectn s gven by the sum ve as fllws: 4παω σ D (5 3 σ = = whee α dentes the fne stuctue cnstant, ω s the fequency f lght and D s the tanstn dple mment t the egenchannel wavefunctn Ψ. Chen epted that amng the egenchannels, nly ne shws mst f the nance behav and the cntbutns f emanng nes can be teated as backgunds t the autnzatn specta: σ = σ + σn wth σ n =Σ σ. F the nance css sectn, Ueda's type f decuplng can be appled as fllws: 9 σ ( + ( + eff eff = K + eff + whee K dentes (4 παω / 3d A wth the tanstn dple mment d t the nance egenchannel and A dentes the avded cssng tem. Refs. [8] and [9] shwed that the de- (6 cuplng (6 f the ntelpe's spectum fm the petubed autnzng Rydbeg sees can be made me pweful by expltng the ple and ze suface stuctu f the lne asymmety paamete eff, such as (3 and (4. Attempts t fnd smla explct ple stuctu fm (6 f systems nvlvng me than pen channel wee unsuessful n pevus wk. Patal suess was btaned f specal cases f a neglgble dect nteactn between the clsed channels and a neglgble tanstn t the pen channels. Ths pape epts the explct fmula f the geneal cases. Befe devng ths geneal explct fmula, let us descbe the system f whch the pent they was develped. Bef ntductn f MQDT fmulatn. Cnsde the phtnzatn enegy ange satsfyng I3 < L < In < E < I < I s that channels and ae clsed t dect nzatn, and channels fm 3 t n ae pen t nzatn. In ths case, Rydbeg sees and cnvegng t the pectve nzatn lmt, I and I, pduce autnzng phtnzatn specta by channel cuplng wth the pen channels fm 3 t n. In ths system satsfyng I < I, the autnzng lnes f Rydbeg sees act as ntelpes f dense autnzng lnes f Rydbeg sees and pduce cmplex velappng nances. Multchannel quantum defect thees (MQDT used t deal wth these velappng nances utlze a few channel nteactn paametes epented by the eactance matx elements K j f cuplng between channels and j. The phase-shfted vesn f MQDT s nmally emplyed t take advantage f the mnmal set f nn-ze eactance matx elements gven by K K 3 L K n K K c 3 Kn K K L K = = K3 K3 c K K L M M M O M K n K n L f the pent system. Let β dente π ( ν + μ whee the effectve quantum numbe ν s defned by E = I j Ryd/ ν j, wth Rydbeg cnstant Ryd f the nzatn thhld enegy I. Then, the abve-mentned egenchannel wavefunctns Ψ whch ae the egenfunctns f the physcal eactance matx K ae gven n tems f the submatces f K n Eq. (7 as K c (tan K β + K c and ae nmally expanded n tems f the eal standng-wave channel bass functns n the pen channel space P and clsed channel space Q as fllws: (7 Ψ = Ψ Z cs δ + Ψ cs π( ν + μ Z (8 P Q In the expansn, the addtnal facts, cs δ and csπ ( ν + μ, t the expansn ceffcents ae ntduced t make Z ( P untay and essental because the standng-waves Ψ defned utsde the eactn zne as Ψ = Φ j( ω f j( R δ j g j( R K j, ( R > R (9 j

3 The Rle f Open Channels n Ovelappng Resnances Bull. Kean Chem. Sc., Vl. 3, N. 33 wth the phase-shfted egula and egula base pa 3 ( f j, g j f j = f jcsπμj g jsnπμj g = g csπμ + f snπμ j j j j j ( ae nt enegy-nmalzed. The egula and egula base pa ( f j, g j n ( belngs t the j th nzatn thhld enegy I j. As s well knwn, the expansn ceffcents nt the clsed channel bass functns n (8 can be btaned as fllws: ( β Z β K K Z δ, c c c cs = tan + cs ( whee β s π ( ν + μ and δ ( P s the egenphase shft btaned fm the cmpatblty equatn ( K δ det tan = ( f the MQDT equatn = c c K + tan β K Z c K tan δ Z (3 btaned by settng the ceffcents f the expnentally nceasng tems f (8 t ze. Pevus ults elevant t the pent fmulatn. As stated abve, nly ne egenchannel dmnates the nance behav and can be egaded as a nance egenchannel, ndcatng that the paametes f the nance dmnant egenchannel can be used as nance paametes. Hweve, a bette way t btan the nance paametes s t cnsde the sum Σ δ f the egenphase shfts that emve the avded cssng nteac- ( q + eff = sn θ + cs θ + ( + eff = + cs θ eff +, (6 (7 whee s als the genealzatn f K /( ξξ channel QDT nvlvng pen and clsed channels n and the s 3 defned as fllws: 9 K K = = ξ θ. ξ ξξcs (8 ξ dentes the mdulus f the clumn vect ξ, whse k th c cmpnent s defned as K k and stands f the cuplng stength f the clsed channel wth pen channels. Smlaly, ξ ξ dentes the scala pduct f tw vects n pen channel space P defned by Σ k PK kk kand stands f the ndect cuplng between clsed channels and. The educed wdth (6 s the same as the scaled wdth used t extact the mnmal numbe f paametes ppsed by Lecmte when many pen channels ae nvlved n the autnzatn specta. In an analgy wth = ct δ, eff satsfes the equatn eff = ct δeff, whee δ eff s the nance phase shft f the petubed Rydbeg sees and satsfes δ = δeff + δ. See Appendx A f me detals. th (8 and (, the tanstn dple mment D [= ( Ψ T ] f Eq. (5 t egenchannel was tansfmed nt ( ( D = d + q + q eff eff eff A C, (9 tns analytcally: 3 det ( tan β + κ det( S = exp δ = det ( tan β + κ exp( δ =, (4 whee d dentes Σ PDZ, A stands f the avded css nteactn and s clse t unty, whse explct fm can be fund n Ref. [9]. Ths wll nt be cnsdeed futhe as t des nt play a sgnfcant le. The lne pfle paametes, and, nw depend n the egenchannels and dente c c whee κ dentes K K ( + K K c c = K K K. Σ δ = fm K = n the phase-shfted MQDT yelds the 3 d equalty n Eq. (4. Fm (4, the nant phase shft δ s equvalent t the phase f det(tan β + κ and the squae f the mdulus f det(tan β + κ was btaned as fllws: 9 ( ( C = + + eff eff, (5 whee dentes tan β / ( =, and dentes ξ, and stands f the educed enegy and spectal wdth f the Rydbeg sees, pectvely. The effectve educed spectal wdth and enegy eff and eff f petubed Rydbeg sees wee btaned as fllws: ξ Z csα ξ ξ Z ( cs β ξ T D D = = T ( e d ξξ d T D D = = T ( e d ξξ d whee csα and cs β ae the dectn csnes f f the unt length nt the axes ξ and ξ Z, pectvely. The at f the dectn csnes cs β / csα gves the at f cntbutns f Rydbeg sees and n egenchannel, and wll be dented as. Its the equatn s = ( ξ / ξ/( ξ / ξ. The emanng paamete eff n (9 s the effectve lne shape paamete and was btaned as fllws:

4 34 Bull. Kean Chem. Sc., Vl. 3, N. Chun- Lee ( = ( cs θ ( cs q / q θ + eff +. eff cs θ eff + ( can be btaned as cs θ =. ( The the paametes becme th the substtutn f (5, (9 gves the autnzatn css sectn fmula, whch takes the fm f (6 when =. Nte that the decmpstn f the autnzatn css sectn f the systems nvlvng clsed and me than pen channel eles n the dmnance f the ttal css sectns by the nant egenchannel. Such decmpstn wuld be exact f the dynamcs at the excted states alne ae cnsdeed, ndcatng that the decmpstn s a me fundamental natue f the velappng nances. Ths fact deves fm the exact slatn f the nance phase shft δ nt δ eff + δ, even when me than pen channel s nvlved. See Appendx A f pf. Devatn f the geneal explct fmulae f q eff. Addng addtnal pen channels t the fmulatn s tublesme n the MQDT fmulatn. One f the pblems s caused by the cnnectn pblem f the egenchannels at dffeent eneges. T classfy the egenchannels nt nance and nn-nance nes, the cntnuty f the fst devatve f the dynamc quanttes, such as the egenphase shfts δ needs t be sacfced. Ths means that the dynamc quanttes unde ths cnnectn mght nt have cntnuus fst devatves because the cnnectn s nt based n achevng smth vaatns n the physcal quanttes as a functn f enegy but smply stng nt nance and nn-nance pats. Knks caused by ths dscntnuty ae nt nduced by eal physcs but by cnnectn pblems. The emval wll geatly smplfy the nvestgatn wthut lsng any valdty f the analyss asscated wth q evesals. One way f emvng the knk s t use the aveage value. The the way s t use slutns at the mst emte pnts fm the knks. = tan β = cpnds t such a pnt that was aleady utlzed n Ref. [9]. The analytcal slutn f the egenchannel pblem f K Zk = tan δz k at ths pnt can be gven by Z k = K k / ξ ξk / ξ whee Z k dentes the k th cmpnent f the nance egenchannel (see Appendx B f pf when me than pen channels ae nvlved. Nte that ths slutn cpnds t the effectve cntnuum 4,4 Fan s a state 5 f autnzng Rydbeg sees actng as an ntelpe. Ths suggests that the pnts chsen t emve the knks egad the ntelpe as a stuctued backgund f the petubed autnzng sees. The lne pfle ndex eff has n me cnstant. Its enegy vaatn fllws the stuctued backgund pvded by the ntelpe. Cnsde btanng the nant lne pfle ndex eff ( = f the petubed dense Rydbeg autnzng sees. Let us stat by btanng the slutn-specfc 3 paametes =, and at. Fst cnsde btanng defned by ( ξ / ξ/( ξ / ξ. Because ξ = ξ Z = ξ ( ξ / ξ = ξ ξ Z ξ θ, = ξ Z = cs D csθ = d ξ D = d ξ (3 whee d s gven by D ( ξ / ξ. If θ = and d s egaded as D 3 n the 3 channel QDT, and n (3 have the same fms as thse f and n the 3 channel QDT, pectvely. Cnsde efff =. Nte that nly the secnd tem f the ght sde f ( s slutn-specfc. Dente the backet pat f the secnd tem as q and ntduce and t defned by D ξ /( D ξ and cs θ /, pectvely. Substtutn f / = /( cs θ and cs θ [ + q / ( q ] = + t fm ( and (3 tansfms q nt = cs θ + ( + t +. (4 Fm the substtutn f (4 and (6 nt ( and afte sme manpulatns, we btan q = θ θ eff + + ( cs + ( + cs + sn θ( + + cs θ( + ( ( q (5 whee eff s dented as eff f cmpact vsblty. Placng θ = nt (5 educes eff t eff n the 3 channel QDT : q + ( + ( qeff θ = = (c eff. ( ( (6 It was aleady shwn that and have the same fm as and n the 3 channel QDT, pectvely, when θ = and d s egaded as D 3. Ths tgethe wth (6 means that the gss featu f the autnzatn specta depend nly n the scala pduct f ξ ξ. If addtnal pen channels cannt alte the value f the scala pduct, the pence cannt be felt n the specta. A pevus study 9 cnsdeed the case f the null tanstn D = t the pen channels fequently encunteed n the specta. In ths case, ( =, but ts pduct wth d s fnte. By takng the lmt f ( =,, multplyng

5 The Rle f Open Channels n Ovelappng Resnances Bull. Kean Chem. Sc., Vl. 3, N. 35 and dvdng bth the numeat and denmnat f (5 by d and d, pectvely, we btan ( csθ + + t + t eff = sn θ ( + + cs θ ( + (7 whch s the mpved vesn f Eq. (4 f Ref. [9]. Nte that thee s n smple ple at = n ths case, whch s n cntast t (5 and (6. Eq. (7 s als educed t eff f the systems nvlvng clsed and pen channel when θ =. Systems ae smetmes encunteed whee the null tanstn dple mments D t the pen space and thee ae null ndect cuplngs between clsed channels s that θ s π/. In ths case, t s ze but t s nt ze and becmes /, whee = K /( ξξ. Smlaly, cs θ ( + s nt ze but becmes. Takng these nt aunt, (7 can be educed futhe t ( + + eff = + + q. (8 Nte that the ze suface at + = n (7 dsappeaed. The quadatc fm f the numeat f the ght sde f (8 enjys seveal dstnctve elatnshps. The unty f the cnstant tem means that the tw ts ae multplcatve nveses f each the and lmt the values f duble ts t ±. The sum f the ts gven by s equal t K D /( D. Ths suggests that f appaches ze, ne f the ts appaches nfnty wheeas the the appaches ze, whch s equvalent t the statement that ne f the ts appaches the nance pnt f the ntelpe, wheeas the the t becmes ff-nant. Ths means that q evesal us at the nance peak f an ntelpe and anthe q evesal takes place aund the mnmum f the css sectns. Ths tpc was dscussed extensvely fm the vewpnt f Feshbach type s they f nance n Ref. [5]. The sgn f the duble t was detemned by the sgn f K D / D, suggestng that the elatve sgn f D / D and the sgn f K ae nvlved n the ntefeence patten that detemnes the spectal shape. Theefe, ths study shws the paametes nvlved n the spectal shape and hw they affect the shape. Anthe fequently bseved q evesal patten, whee the autnzng satellte peaks f the petubed sees lean twad the cental peak due t the ntelpe, s elated t the sgn f f (8. The fequently bseved q evesal patten f the satellte peaks leanng twad the cental peak cpnds t the negatve sgn f. If the sgn s pstve, the q evesal patten wuld be such that the satellte peaks f petubed sees lean utwad fm the cental peak. Ths elatnshp f the sgn f the ceffcent f the hghest de tem f the numeat f the fmulae f eff t the q evesal patten s geneal and apples t all cases. Table lsts the ceffcent f the hghest de tems f the numeat f eff that detemne the q evesal pattens n vaus cases. The shape f phtnzatn css sectn (6 s detemned by eff, whch n tun s detemned nly by,, and θ ( I and μ ae als needed f. All the paametes D, D, d, ξ, ξ, θ and K must have been cnsdeed nstead f mee,, and θ f the analytcal fm f eff s nt knwn. Althugh the numbe f paametes ae educed cnsd eably, the functnal behav f ze sufaces f eff ( = f q evesal analyss s stll detemned by as many as 5 paametes,,,, θ and, and cannt be vsualzed n 3 dmensnal space. On the the hand, f the tanstn dple mments t the pen channels ae ze, nly 3 paametes ae needed f the ze sufaces f eff ( and can be vsualzed, as was dne n Ref. [9]. Bth (5 and (7 shw that when θ =, ( +, whch s the secnd de ple at the denmnat n (7 cancels + ut n the numeat and becmes a smple ple. In the wds, ne f the ze sufaces, + =, s tansfmed t a smple ple. Ths change leaves the q evesal pattens unalteed but huge ntensty bwng appeas aund the smple ple as a new phenmenn f systems nvlvng nly pen channel. In cntast, a huge peak due t ntensty bwng dsappeas when addtnal pen channels ae ncluded and the autnzatn specta vay me smthly. The le f pen channels n nance phenmena and ts lmtng cases. If me than pen channel s nvlved n autnzatn specta, nly ne egenchannel acts as a nance egenchannel and the thes nly cntbute t the backgund specta. Ths means that the numbe f effectve cntnuum emans unalteed when the numbe f pen channels nceases. An ncease n the numbe f pen channels nceases the dmensnalty f the space spanned by the pen channels and s ampaned by an ncease n the ndect cuplng paths that ae equal t the numbe f pen channels, makng t easy t ntefee destuctvely. In ths space, let us epent the egen channel cpndng t the nance pat by the vect Z, cuplng f the clsed channels wth pen nes by vects ξ and ξ and the tanstn dple mments t pen channels by D Z d P space Table. The tems whse sgns detemne the q evesal pattens. β α Case Geneal D = R csθ D =, θ = π / θ = π / Tems ξ θ ξ Fgue. Vaus vects and dectn csnes n pen channel space P.

6 36 Bull. Kean Chem. Sc., Vl. 3, N. Chun- Lee the vect D as shwn n Fg.. The degee f destuctve ntefeence n ndect cuplng s epented n ths space by the angle θ that tw cuplng vects ξ and ξ make. The magntude f the lne pfle ndces and ae pptnal t the dectn csnes f Z wth unt vects f ξ and ξ, pectvely. The pjectn f the tanstn dple mment vect D t Z detemnes ts actual cntbutn t the css sectns. As the ndect cuplng ( ξ ( ξ = K K va the pen channel cmpses ne cmpnent f the scala pduct f tw cuplng vects ξ and ξ, destuctve ntefeence can be vsualzed as the scala pduct f tw cuplng vects ξ and ξ n pen channel space P. Cmplete destuctve ntefeence cpndng t the null ndect cuplng us when tw vects ae pependcula t each the, when θ = π /. The maxmum cntbutn f ndect cuplng us when tw vects ae paallel t each the s that θ = π. In ths case, all the behavs f the autnzatn specta f the nant egenchannels ae the same as thse f the 3 channel QDT, except at the dscntnuty pnts, as aleady shwn n (6. F the ze ndect cuplng case cpndng t ξ ξ =, defned by (8 appaches nfnty whle csθ emans fnte and equal t K / ξξ. Let us dente ths fnte pduct as. Numeus specta belngng t ths null ndect cuplng case have been bseved, patculaly by Hgevst s gup. 6-9 One f the specta wll be cnsdeed n the next sectn. They appled extensvely the cncept f assgnng ne effectve cntnuum t ne clsed channel ppsed by Cke and Cme. 4 They als ntduced the smplfcatn t c the ppsal and lmted the nnze values f K m t the nes wth ndces satsfyng m= nc + ( n c : the numbe f clsed channels. Ths lmtatn yelds null ξ ξ j f dffeent clsed channels and j. Reduced wdths and enegy f ths null ndect cuplng case ae smplfed as fllws: q eff = + + =. eff + (9 Snce thee s nly ne path f dect cuplng K, n destuctve ntefeence takes place and eff s always. Destuctve ntefeence between dect cuplngs mght have been pssble f thee wee me than clsed channels. Eq. (9 shws that eff and eff becme and, pectvely, f K s ze, meanng that the nances ae slated as a matte f cuse because n ths case, the ndect cuplng f clsed channels and the dect cuplng ae ze. If the dect cuplng K s tuned n between clsed channels and n the absence f an ndect ne, the petubatns f the ntelpe begn t affect the spectal wdth, educe the enegy f Rydbeg sees and cntbute t addtnal tems pptnal t K. Stctly, t s nt K but that s equal t K /( ξ ξ. Ths hghlghts the mptance f the vect lengths f the cuplng vects ξ and ξ besdes the angle between them. Althugh f the null ndect cuplng case, the scala pduct ξ ξ epentng the ndect cuplng n space P s ze, and thee s nly dect cuplng between the clsed channels whch has n epentatn n space P, the addtnal pen channels may stll affect the autnzatn dynamcs. The abve ult shws that ths s ndeed the case, and the pen channels affect the dynamcs thugh the vect lengths f the cuplng vects. The effect f dect cuplng n the autnzatn dynamcs s dmnshed as the vect lengths f cuplng vects ncease. Ths suggests that dect cuplng s nt fully cmpatble wth the autnzatn dynamcs, and ths ncmpatblty n cnjunctn wth the uncetanty pncple f quantum mechancs causes deplazng quantum aveagng. th ths ntepetatn, the meanng f can be gven as the deplazed stength f the dect cuplng. Let us examne the behav f eff f (9, whch s the measue f the autnzatn ate f a petubed Rydbeg sees and whse enegy vaatn s caused by cuplng fm the ntelpe. The maxmum f eff us at the nance pnt max = f an ntelpe sees and s gven by eff = + K /. Ths ndcates that the maxmum autnzatn ate f the petubed sees nceases wth nceasng dect cuplng stength K wth an ntelpe. The ncease n the maxmum ate f sees wth nceasng K may be due t ethe the leakage f sees nt sees the leakage f sees nt sees. Intetngly, the maxmum ate f sees deceases as the ate f sees gauged by nceases, ndcatng that the autnzatn f sees blcks the autnzatn f sees. Ths als suggests that an ncease n the maxmum ate f sees wth nceasng K s caused manly by the leakage f sees nt sees. Nte that the maxmum ate appaches nfnty as the autnzng ate f sees measued by appaches ze. Hweve, cautn shuld be taken when makng an ntepetatn. The ange affected by the ncease n the maxmum ate s gven by. Theefe, althugh the maxmum ate f sees appaches nfnty, ts affected ange als appaches ze, meanng that the nfnte ate s seldm ealzed n the specta. Cnsde the lne shape paamete eff when θ = π /. Its unpetubed vesn s ze fm (3. Its value cmes puely fm the petubatn by dect cuplng and s pptnal t : / eff = q eff + +, (3 whee dentes q cs θ. F nly dect cuplng, / eff s meanng that the leakage due t the dect cuplng educes the magntude f eff and the veall shapes f the autnzatn specta becme clse t the ntelpe ne. Eq. (3 becmes (8 when D = ze, cnfmng the valdty f the devatn. Thus fa, the behav f the dynamc paametes f the lmtng cases was examned: ethe θ =, π θ = π /. F the cases lyng n between, the scaled spectal wdth (6 s gven by the sum f tw ncheent tems wth the weghts gven by

7 The Rle f Open Channels n Ovelappng Resnances Bull. Kean Chem. Sc., Vl. 3, N. 37 cs θ and sn θ. One tem cpnds t the lmtng case f θ = π and aunts f the change n wdth due t ntefeence between dect and ndect cuplngs wth an ntelpe epented by the at, = K / ξ ξ. The the tem s the unpetubed case and stands f the unpetubed cntbutn fm sees. Nte that the θ = π / case cntans cntbutns fm bth tems, as shwn n (9. Snce the ple and ze suface stuctu f eff geatly affect the autnzatn specta, let us cnsde the effect f addtn pen channels n the ple and ze suface stuctu f eff. Thee s ne me ze suface f eff when me than pen channel s nvlved. Ths ze suface becmes a smple ple f θ = s that tw nances ae cmpletely supempsable: eff = ( + q + ( cs θ + ( + cs θ ( + sn θ( + + cs θ( + [ + ( q + q ( q+ q ] θ = ( + ( + (3 Althugh thee s n dffeence n the fact that a lne pfle ndex changes sgn at, the natue f the pnt f = s cmpletely dffeent. Applcatn t Ba 6p / np3/ J = autnzng Rydbeg sees wth 6p 3/ nf sees as an ntelpe. Let us apply the they t the naw 6p /,3/np J = autnzng Rydbeg sees n baum petubed by the 6p 3/ nf states btaned n Ref. [9]. They used 5d 3/ 6p 3 3/ P as an ntemedate level t excte the 6pnl J = autnzng Rydbeg sees. F J =, thee ae fu 6pnp sees, tw cnvegng t the 6p / lmt and tw t the 6p 3/ lmt. In addtn, ne apdly autnzng 6p 3/ nf 5/ sees us. Tw sees cnvegng t 6p /np / and 6p /np 3/ ae nt cupled stngly and sees 6p / np / may be mtted wthut a ntceable change n the maj natue f the specta belngng t the 6p /np 3/ sees f the test f the cuent they. Usng ths smplfcatn, the 7 channel phase-shfted MQDT calculatn, whch satsfactly epduces the bseved specta, can be educed t the ne nvlvng 4 channels: 6p /np 3/, 6p 3/nf 5/ and tw pen channels. In the specta, the 6p 3/ nf 5/ sees acts as an ntelpe t the autnzng Rydbeg sees 6p /np 3/. The magntudes f channel cuplng wth the clsed channels ae the paametes t be nputted f pen channels. In addtn t thse, the the natue f the pen channels s nt equed f the MQDT calculatn. The pent they clafes ths pnt and shws that pen channels ente the fmalsm nly n the fm f ξ and θ. F cnceteness, the expemental dentfcatns wth 6sl f the decay f the 6pnp sees and 5d l f the decay f the 6p 3/nf sees, whch ae cmpatble wth the selectn ule, Δ l =±, may be specfed. The numbe f pen channels des nt affect the ft pvded geate than ne because the cntbutns mght be egaded n an aveage sense as a cnstant functn f enegy, as n the cnfguatn mxng they f Fan. They nly affect the ft thugh the angle θ between the Table. MQDT paametes f the 6p /np 3/ and 6p 3/nf 5/ J = sees f Ba n 4 channel analyss. = = = 3 6p /np 3/ 6p 3/nf 5/ 6sl = 4 5d l I (cm μ D.5 5 Nnze K matx elements 4 K =., K 3 =.8, K 4 =.58 Ba 6p /np 3/ petubed by 6p 3/nf 5/ t t t Enegy (cm Fgue. (a Spectum ultng fm 4 channel QDT wth the paametes n Table. The css sectn σ s dawn wth a sld lne, the envelpe wth a dtted lne and σ max wth a dashed lne. Effectve quantum numbes ae shwn n the σ spectum. The uppe ntege values ae n * f the petubed Rydbeg sees and the lwe ne s n * f the ntelpe. The nablty f σ max t fllw the peak maxma aund n * = 8 s due t nstumental smthenng. (b The spectum cpndng t the nant egenchannel cmpnent. (c The lne pfle specta f the petubed sees dawn wth Eqs. ( and (8. cuplng vects ξ and ξ. The elevant paametes n Table wee cped fm Ref. [9] f self cntanment. Nte that n ths system, the nteactn f clsed channel wth pen channels s nn-ze nly wth channel 3,.e., ξ = [.8 ] and clsed channel nly wth channel 4,.e., ξ = [.58] s that ξ ξ =, tw cuplng vects ae pependcula t each the s that θ = π /. Each clsed channel cuples wth ts wn effectve cntnuum, ultng n null ndect cuplng. Althugh clsed channels d nt nteact ndectly va pen channels, they can nteact dectly wth each the but wth the stength K educed by ξ ξ. Fg. (a shws the spectum calculated usng the 4 channel QDT wth the paametes n Table. Fg. (b shws the spectum f the nance pat and cnfms the assetn that t cmpses mst f the ttal css sectns. Fg. (c shws the spectum f the lne pfle ndex eff f the petubed autnzng Rydbeg sees. The staddlng ntevenng e- (a (b (c

8 38 Bull. Kean Chem. Sc., Vl. 3, N. Chun- Lee Ba 6p /np 3/ petubed by 6p 3/nf 5/ (3 channels 4 (a 3 4 Rts f ( = eff tan β = Enegy (cm t Enegy (cm (b tan β = Fgue 3. Phtnzatn css sectn and eff specta calculated usng the 3 channel QDT wth the same paametes as thse f Fg. except that K 4 =. The lne pfle spectum s dawn wth (C. snance peak cpndng t 6p 3/ 7f 5/ aund whch the petubed autnzng Rydbeg sees change sgn n q can be dentfed n Fg. (a. The clseness f the q value t ze n a wde ange f eneges aund the staddlng peak makes the specta fllw the specta f the ntelpe me smthly. If K 4 s ze, the system s educed t a 3 channel ne nvlvng pen channel. Fg. 3 shws the spectum calculated usng the 3 channel QDT,.e., wth the same paamete as thse f Fg. except that K 4 =. Nte the cmpletely dffeent appeaance and natue f the tw specta. It s dffcult t dentfy the sepaate nance peaks f the ntelpe at n = 7 n Fg. (a. Instead, t acts as a stuctued backgund t the petubed sees. On the the hand, t s clealy dscenable as a sepaate peak n Fg. 3(a. Hee, cautn shuld be taken when ntepetng the naw peaks f sees n Fg. (a as the wndw-lke nances due t the velappng nawng bseved by Mes. 4 In that case, the peaks shuld le n the mddle f the gnal peaks f sees, as clealy shwn by Gust-Suz and Fan 3 n cntast t the spectum n Fg. (a. Fg. 3(a shws huge ntensty bwng aund the ples, whch s the newly emeged featue by the change fm 4 t 3 channels. thn the tctn f n tanstn t the pen channels and null ndect cuplng, the specta can stll shw a dffeent type as K 4 vaes. Adng t Fg. 4, the ts f eff ( = dsappea afte megng nt ne wth value unty at K 4 gven by K D /( D, whse value s.74. (If the sgn f K D / D s negatve, K 4 wuld be gven by K D /( D wth the duble t cpndng t the value f. The staddlng ntevenng q evesal aund the cental peak due t the nte lpe wuld then dsappea, as shwn n Fg. 5, whse spectum cpnds t K 4 =.9. Ths applcatn shws that the ex plct fmula f eff btaned n the pevus sectn can be used as a pweful tl f detemnng the ente ange f pssble spectal pattens q t K 4 t (.74,. Fgue 4. Behav f the ts f eff ( = and elevant paametes as a functns f K 4. Ba 6p /np 3/ petubed by 6p 3/nf 5/ ( K 4 =.9 3 (a Enegy (cm Fgue 5. The spectum ultng fm 4 channel QDT wth the same paametes as thse f Fg. except that K 4 =.9. Result and Dscussn A Rydbeg sees petubed by an ntelpe s ne f the mptant systems shwng cmplex nance phenmena due t the velappng nance. Ueda s fmulatn 7 a smla ne by Cke and Cme 4 pvdes the decmpstn f the autnzatn specta nt thse f the ntelpe and the petubed autnzng Rydbeg sees by t. Althugh the decmpstn pvdes an analyss tl f examnng the mutual effects between the ntelpe and petubed autnzng sees, each decmpsed pat was fund t be pne t the cntamnatn wth ples. The ples ae nt pent n the undecmpsed fm. They nly emege n the decmpsed fm and ae cnsdeed ntnsc n the decmpsed fmulae and cannt (b (c

9 The Rle f Open Channels n Ovelappng Resnances Bull. Kean Chem. Sc., Vl. 3, N. 39 be emved. Indeed, the bseved specta wee nfluenced geatly by the ple stuctu, whse effect was tned dwn by the cntbutn fm cncuent ncheent pcesses. th the explct factng f the ple stuctue smetmes wth the help f the decmpstn nt the patal factns, the pevus study tuned the pblematc ples nt pweful tls f examnng autnzatn specta. 8 hen addtnal pen channels ae nvlved, futhe cntamnatn wth knks and ples us because f the cnnectn natue f egenchannels nt the nant and nn-nant nes ppsed by Chen. Hs ppsal t cnstuct nant dmnant and nn-nant dmnant egenchannels was the ctcal step t the extensn f the Ueda type f decmpstn t the systems nvlvng me than pen channel, whch allwed us t apply Ueda s fmulatn t the nant egenchannel and teat the cntbutns f emanng nn-nant pat as a backgund. Hweve, such attempts wee hampeed by the dffculty t btan fmulae wth the stuctu f ple explctly expsed except f sme patcula cases, and nly patal suess was epted. 9 Ths study examned ths pblem agan. The emval f knks was acheved usng the slutns at the mst emte pnts fm the knks. The slutns cpnded t the effectve cntnuum 4,4 Fan s a state 5 f the autnzng Rydbeg sees. In physcal tems, such emval f knks helps eplace the petubed dynamc cnstants petanng t the pen channels wth n stuctue n the autnzatn pcesses f Rydbeg sees wth a stuctued backgund that s pvded by the envelpe f the ntelpe. F example, the lne pfle ndex eff s n lnge cnstant. Its enegy vaatn fllws the stuctued backgund pvded by the ntelpe. The explct fmulae f the petubed autnzng Rydbeg sees thus btaned f the geneal cases shw the stuctu f tublesme ples and extemely valuable ze sufaces f the spectal shape paamete q. thut knwng such stuctu, the behav f the dynamc paametes wuld nt be cntlled easly because f the pecula behav due t the sngulaty. The explct fmulae helped dentfy the paametes pnsble f the bseved phenmena, such as f the fequently bseved q evesal patten, whee the autnzng satellte peaks f the petubed sees lean twad the cental peak due t the ntelpe. The table was gven t shw the tems that ae pnsble f the bseved phenmena, n patcula f the q evesal, whch s the mst mptant sngle paamete that detemnes the spectal lne shapes f autnzatn. In de t emphasze the effect f the addtnal pen channels n the autnzatn specta, the case f exteme devatn fm the pen channel systems was studed n depth. Such exteme devatn cpnds t the null ndect cuplng case, whch us when the ndect cuplng paths between the clsed channels va the pen channels n space P (cnssted f pen channels ntefee destuctvely and thee emans nly dect cuplng between the clsed channels, whch has n epentatn n space P. Althugh n cuplng f the clsed channels us n space P, the pen channels may stll affect the autnzatn dynamcs nt by ndect cuplng epented by the scala pduct between the cuplng vects n space P but by the vect lengths f the cuplng vects n space P. The effect f dect cuplng n the autnzatn dynamcs deceased wth nceasng the vect lengths f the cuplng vects n space P. Ths suggests that the dect cuplng s nt fully cmpatble wth the autnzatn dynamcs, and ths ncmpatblty n cnjunctn wth the uncetanty pncple f quantum mechancs causes deplazng quantum aveagng. th ths ntepetatn, the meanng f was gven as the deplazed stength f the dect cuplng. The pent they was appled t the naw 6p /,3/ np J = autnzng Rydbeg sees n baum petubed by the 6p 3/nf states btaned by de Gaaff et al., 9 whch s ne f the sees f expements that apples the cncept f assgnng ne effectve cntnuum t ne clsed channel ppsed by Cke and Cme. 4 The expement cpnds t the case f null ndect cuplng between the clsed channels n u they. T emphasze the effect f addtnal pen channels t autnzatn specta, the tansfmatn fm the bseved autnzatn spectum nt the ne f the 3 channel system was pefmed by tendng ne f the cuplng paametes t ze. A cmpletely dffeent spectum was btaned: the nance peaks n the 4 channel system act as a stuctued backgund t the petubed autnzng sees, wheeas t s clealy dscenable as a sepaate peak n the 3 channel system and huge ntensty bwng aund the ntelpe nance peak us. In addtn t ths tansfmatn fm a 4 t 3 channel system, the they was used t dentfy the pssble dffeent types f specta by vayng the same cuplng paamete. One f the lmts f ths applcatn was that the numbe f clsed channels needs t be educed fm 5 t t apply the they. Theefe, futue wk wll need t extend the they t systems wth me than clsed channels, n patcula t systems that stll keep the scheme f a petubed autnzng sees by an ntelpe but allw degeneacy ethe bth f the petubed sees and ntelpe. Acknwledgments. Ths study was suppted by the Aju unvesty each fellwshp f 9 unde cntact N. 954CORS. Appendx Appendx A: Decmpstn f the Resnance Phase Shfts nt the Incheent Sum. Let us btan the explct fm f the nance phase shfts δ whch can be btaned fm the phase f tan β + κ. Fm tan β + κ = Cexp( δ and afte sme manpulatns, the fllwng can be btaned: ( β ( ( ξ β ξ ξ ξ K Ce δ = tan tan + +. (A Nte that tan β / ξ = tan β / =. It s als equal t ct δ. Eq. (A can then be wtten as fllws: ( ( ( δ Ce = + cs θ (A Nte that Eq. (A s symmetc. The next cucal step s t beak ths symmety n the fmula t ncpate the asymmety

10 3 Bull. Kean Chem. Sc., Vl. 3, N. Chun- Lee mpsed by the physcal stuatn f the apdly vayng sees petubed by a slwly vayng ne, the ntelpe, f a gven enegy nteval. Let us call the apdly vayng sees sees and the slwly vayng ne sees. Snce sees vaes slwly, t wll act as an envelpe t sees. Ths can be ncpated nt (A by factng - ut: ( q ( δ Ce = ( ( + cs θ. (A3 (A3 can be tansfmed by sepaatng the tems n the backet n the ght sde f (A3 nt eal and magnay pats, as fllws: ( ( Ce δ = (A4 ff e eff whee eff and eff ae the ntnsc spectal wdth and educed enegy f the petubed sees n ( defned n (6 and (7, pectvely. Substtutng (5 nt (A4 yelds ( eff ( ( + ( + / δ e = eff. / Snce exp( δ j = [( j /( j+], (A5 ndcates that eff (A5 δ = δ + δ, (A6 wth j cpndng t eff and, pectvely. Eq. (A6 means that nance scatteng s decmpsed nt tw ncheent tems: ne n the ntelpe and the the ne n petubed autnzng Rydbeg sees. The ncheent decmpstn (A6 f the nance phase shft s cmpatble wth Ueda s decmpstn f css sectns ( and (6 but hlds f wde systems. F systems nvlvng clsed and pen channels, bth (A6 and ( ae exact. Hweve, f systems wth addtnal pen channels, (A6 s stll exact whle ( hlds f each egenchannel wth an addtnal avded cssng tem wth the egenchannels. Appendx B: Obtanng Z at. F the sht-ange eactance matx (7, n whch K s a null matx, the physcal eactance matx K s gven by c c K (tan β + K K. F systems nvlvng clsed channels, we may cnsde tw clumn vects, ξ and ξ, defned n the pen channel space by { K P} and { K P}, c pectvely. Then K can be wtten as [ ξ ξ ] and ( K = ξξ tan β + ξξ tan β K ξξ + ξξ D T T T T whee D dentes det(tan β + K. In the lmt tan β, K T becmes ξ ξ /tanβ and ts nant slutn s gven by ξ, as can be cnfmed by the fllwng egenvalue equatn: T ( ξ ξ ξξ ξ ξ Kξ ξ ξ ξ. T = = = = tan δ tan β tan β tan β Appendx C: The K 3 = case f the 3 channel systems nvlvng pen channel In the case f K 3 =,.e., ξ =,, and ae sngula, wheeas emans fnte. cannt be defned because t s always nfnte. The dynamc quanttes that cntan them eque caeful teatment. Tw examples ae as fllws: ( + q ξ +. eff ξ Let us emve the sngulates n eff and eff. F eff, let us shw the fnal ult: K eff = tan β. Next cnsde emvng the sngulates fm eff. It s bette t begn fm ts geneal equatn: eff + ( + =. ( + q ( + q Althugh all dvege, + and + ae dmnated by at the lmt ξ because the ple f s f the secnd de wheeas the thes ae f the fst de. The lmt s btaned as qq K D + = +. eff ξd 3tan β Eq. (C yelds the t f eff as tan K D β = D. (C Fm the ults btaned thus fa, the envelpe becmes σ σ ntelpe ξ and the petubed spectum f sees becmes ( + eff + eff eff + ξ β + ξ = ( D tan K D K D 3 D 3 tan β K + tan β Pevusly, eff was fund t have tw smple ples at

11 The Rle f Open Channels n Ovelappng Resnances Bull. Kean Chem. Sc., Vl. 3, N. 3 and. Nte that the values ae nfnte n the pent case and ae educed t ne smple ple at. The pent analyss shws that the decease n the numbe f smple ples fm t causes a lage change n the functnal behav f eff. Expementally, whethe K 3 s abslutely ze almst ze may nt be clealy detemned. Snce physcs shuld nt depend n small uncetantes n the measuement, a lage change n the behav f eff shuld nt affect the bsevables, such as autnzatn css sectns. Ths mght suggest that the css sectns ae small at ths tublesme egn. The abve fmulae cannt be used f D 3 = ze, whch s the case f Fg. 3. In ths case, the de f the ples f and s wth pect t ξ, wheeas that f s. Ths means that the numeat, +, f the envelpe s dmnated by, and ( + /( + tends t as ξ. In addtn, the ze f D 3 shuld be taken nt aunt. In ths case, the envelpe σ ntelpe cannt be defned as t s ze because D 3 =. D 3 can be extacted fm σ ntelpe and placed t nt the numeat ( eff + eff f the petubed dense specta and then edefne eff as Dq 3 eff. Nte that D 3eff= whle Dq 3 effs nt. Let us ntduce the ntatn q eff f Dq 3 eff. Then fm (C, D D K eff = + ξ ξ tan β (C The easn f ntducng the new ntatn s t keep the fm f Ueda pattnng at least n appeaance. The newly pattned envelpe was btaned by 4 σ = ntelpe π αω 3 Refeences. Seatn, M. J. Rep. Pg. Phys. 983, 46, 67.. Fan, U.; Rau, A. R. P. Atmc Cllsns and Specta; Academc: Oland, U.S.A., Gust-Suz, A.; Fan, U. J. Phys. B 984, 7, Cke,. E.; Cme, C. L. Phys. Rev. A 985, 3, Cnneade, J. P.; Lane, A. M. Rep. Pg. Phys. 988, 5, ntgen, D.; Fedch, H. Phys. Rev. A 987, 35, Ueda, K. Phys. Rev. A 987, 35, Ch, B.; Lee, C.-. Bull. Kean Chem. Sc., 3, Lee, C.-. Bull. Kean Chem. Sc., 3, Lecmte, J. M. J. Phys. B 987,, Chen, S. Eu. Phys. J. D 998, 4, 3.. Matns, M.; Zmmemann, P. Z. Phys. D 993, 7, Lee, C.-. Bull. Kean Chem. Sc. 9, 3, Mes, F. H. Phys. Rev. 968, 75, Fan, U.; Cpe, J.. Phys. Rev. 965, 37, A De Gaaff, R. J.; Ubachs,.; Hgevst,. Phys. Rev. A 99, 45, Abutaleb, M.; De Gaaff, R. J.; Ubachs,.; Hgevst,. Phys. Rev. A 99, 44, Bente, E. A. J. M.; Hgevst,. J. Phys. B 989,, De Gaaff, R. J.; Ubachs,.; Hgevst,.; Abutaleb, M. Phys. Rev. A 99, 4, Fan, U. Phys. Rev. 96, 4, 866.