Quantum Statistical Properties of Resonant Radiation Scattered on Excited Systems

Transcription

1 J Mod Phys,,, 63-7 doi:436/m34 Publishd Onlin Augus (h://wwwsciog/ounal/m) Quanum Saisical Pois of sonan adiaion Scad on Excid Sysms Absac Bois A Vklnko Join Insiu fo High Tmau of ussian Acadmy of Scinc, Moscow, ussia VklnkoBA@yandxu civd Ail 5, ; visd May, ; accd Jun, Th scaing of sonan adiaion on an xcid aom is considd I is shown ha h scaing coss scion calculad wih h hl of quanum hoy of adiaion is fiv ims lag han h on calculad using smi-classical hoy Th quanum hoy dics, in gnal, h chang in innal quanum saisical ois of ligh du o h scaing ocsss on xcid aoms Kywods: Quanum Thoy, Smi-Classical Thoy, sonan adiaion Inoducion Th quanum xcid sysms ossss mabl ois Thy manifs hmslvs mos ominnly in lass and mass, which w cad in h middl of h las cnuy Th hoy of hs dvics was laboad by W Lamb [] on h bas of a smi classical hoy of adiaion which dals wih classical lcomagnic fild La h quanum hoy was oosd [] I is ossibl o sa omiing h flucuaions ois ha boh h smi classical and h quanum hois sul acically in h sam suls fo quanum mans valus Such a fac suld in ovsimaion of h alicabiliy of h smi-classical hoy In 966 ya, Ch Kos dicd h ffc of ligh nhancmn [3] by slciv flcion of sonan adiaion fom xcid mdia All ffos of quaniaiv xlaining his ffc on h bas of smi-classical hoy of adiaion discussd in monogah [4] w unsuccssful [5,6] I was shown la ha quanum fild hoy should b usd insad [7], bu h mahmaically oblms on his way occud vy difficul [8] Th consquncs of such a hoy manifs hmslvs on a macoscoic lvl Th coc dsciion of simulad adiaion lays an scial ol whn h sonan flcion of ligh fom xcid mdia is considd Nvhlss, h a cn woks [9] which m us h smiclassical hoy and Fsnl s fomula o dscib h flcion of ligh fom nhancd mdia Much anion has bn aid cnly o h ffc of h nhancd ansmission of ligh hough h mallic films [,] Th is no agmn bwn hoy and ximn I is blivd ha h nhancmn of adiaion may b xlaind hough h inacion of ligh wih inducd sanding sufac lasmon wavs Thus w dal wih ffcs of simula adiaion, which mans ha on should us h quanum fild hoy Examls shown abov mad us vis h hoy of sonan adiaion scaing on xcid sysms Th convnional ubaion chniqu is no adqua o dscib h sonan scaing and i is ncssay o sum u (Dyson summaion) h infinily long subss of Fynman s ladd diagams I was V Wisskoff and E Wign who consucd such a hoy fo h fis im by considing h inacion of sonan adiaion wih aomic sysms [] Such a summaion of Fynman s diagam ovd o b usful fo h sha of scum lin of sonan adiaion and ffcs of sonan ligh scaing on non xcid sysms Th difficulis mg in h hoy of combind sonan scaing ocsss whn on of h hoons af simulaion mission of xcid aom undgos of lasic scaing on h gound sa of h sam aom Such combind scaing is non-analyic in chag Th summaion of h Fynman s diagams lik his on is no fomd u o now [8] W oos indic way o sima his sum Psn wok dmonsas insufficincy Wisskoff- Wign s mhod and Dyson s mhod of summaion Fynman s ladd diagams fo h calculaions h coss-scions of ligh scaing on sonan xcid sysms and failu of smi-classical hoy of adiaion L h sonan adiaion scas on som sysm h iniial sa of which in inacion snaion is dscibd by wav funcion Th oal wav funcion of lcomagnic fild and scaing sysm is dnod as Th xansion of such funcion ov a bas of scaing sysm wav funcion is i Coyigh Scis

2 64 B A VEKLENKO f f f f i i i Th m conaining is win saaly Th scala oduc du o ohogonaliy of sca s wav funcions f f is qual o zo Assum ha h incidn ligh is in quanum cohn sa [3] and is quanum man lcical sngh is no qual o zo E (,) in all sac oins a abiay insan of im W a insing in quanum man valu of oao E of h flcd ligh E (,) E (,) f E (,) f f E (,) f ( c) ( n) E E W sa ha h fis m of h igh hand sid of Equaion () dscibs h so-calld cohn scaing channl wih mdium uning o h iniial quanum sa af scaing (g lasic scaing) Th scond m of h igh hand sid of Equaion () dscibs h non-cohn scaing ocsss wih h mdium changing iniial quanum sa (Comon scaing, aman scaing and inducd adiaion of ligh) Th la is vy imoan W sss onc again ha h cohn Hisnb-Kams scaing and inducd adiaion of ligh a dscibd by diffn scaing channls I mans ha if h scaing mdia consisd only of h non-xcid aoms h fis m of Equaion () would dscib h cohn Hisnbg-Kams scaing whil h scond on would dscib h diffusion scaing If h xcid aoms a sn in h mdium hn du o h inducd adiaion ocsss i is imossibl o avoid h snc of h non-cohn channl vn if only h slciv scaing is und ou invsigaion Th oal masud lcical sngh E (,), ha is h lf hand a of Equaion (), may b valuad saaly using h smi-classical hoy of adiaion if on nglcs h flucuaion oical ocsss and hi influnc on E (,) Th gion of validiy of h smi-classical hoy of adiaion is vy lag bu i dos no man ha E (,) dscibs h bilina fild chaacisics L us consid h ngy chaacisics of lcomagnic fild dscibd by nomal oao oduc N E Such valu should b simad fom blow () using h following ocdu On s ino accoun ha ck ik ick i ick (,) i k E k k k, k V wh k and k a h annihilaion and caion hoon oaos in sas dscibing by wav vco k and olaizaion indx Ths oaos oby h convnional commuaion laions ; k k kk Consid lcomagnic fild as a ansvs on (,), k dnos h uni lina olaizaion vcos, V is h quanizaion volum Sinc h oaos k and k a muual conuga han i( kk) ic( kk) k k k k k k k k k k Now kk kk i( kk) ic( kk) k k k k kk kk i ic k k ( kk) ( ) k k k k If h lcomagnic fild osssss h chaacisic fquncy and chaacisic wav lngh and w a insing in im and sac valus much lag hn / and h following inqualiy occus kk kk kk kk i( kk) ic( kk) k k k k kk kk i ic k k ( kk) ( ) k k k k kk i( kk) ic( kk) k k k k kk i ic k k ( kk) ( ) k k k k Now i is non difficul o s ha N E (,) Thus kk k c kk i ick V ( kk) ( ) k k k k c kk V E (,) i( kk) ick( ) k k k k () E ooss h oouniy o sima Coyigh Scis

3 B A VEKLENKO 65 N E fom blow Th validiy of obaind inqualiy dos no dnd on aicula quanum sa on which h avaging is fomd and dos nohing o do wih ubaion hoy Bu if such inqualiy is alid o ach m of igh hand si of N( E ) f N( E ) f (3) f N( E ) f W find ha N( E ) f f f f E E Th las fomula can b win in as N( E ) f f f f E E E Tha ssss h imoanc of h cohn scaing channl whn h scad ligh is no classical and N( ) E E Inqualiy (4) allows o sima N( E ) (4) in h smi-classical aoximaion Th valu E E can b calculad using h convnional smi-classical hoy oaing wih non quanum lcomagnic fild ( ) Th calculaion c f E f E can b fomd using only h cohn scaing channl Evn in xnsiv mdia such ocdu may b fomd wih h hl of wav funcions [4] Thus on can avoid of maix dnsiy fomalism scific fo non cohn scaing channl Pincial Equaions L h lcomagnic fild scas on an aom siuad a a oin wih adius-vco and fo h s of simliciy osssss only on obial lcon wih coodina L h aom osssss only wo ngy lvls Zman`s sublvls wih diffn magnic numbs a ossibl L h fquncy of incidn adiaion is in a quasi sonanc wih h aom ansiion fquncy L Schoding quaion fo aom and adiaion is as follows i HaHhH, wh H () ( ) a U () d m () H A () () d mc a h Hamilonian of h non-inacing aoms and an inacion Hamilonian in Schoding snaion Than () ( ) b, () ( ) b, i Th following communiaion laions a assumd ; b b fo h lcon caion oao b and annihilaion oao b in h sa dscibd by wav funcion Th aicula fom of communicaion laions in ou cas of on lcon in h aom dos no lay any ol By U ( ) w dno h onial ngy of aom lcon Th Einsin summaion ul is assumd ov all aing indics houghou h a Th Hamilonian of f lcomagnic fild and vco-onial oao a as follows H h ck kk k c ik i A () k k k k k kv In od o aliz h calculaion oc mniond in inoducion w swich o h inacion snaion wih h hl of uniay oao () x ( U Ha H h) i In his icu () S,,, x S T H( ) d i (5) ( ) ( ) H x A ( x) ( x) d mc, x, wh, i ( x) ( ) b i ( x) ( ) b, S is h scaing oao, is h aomic ngy in sa, T is h im-oding oao and Coyigh Scis

4 66 B A VEKLENKO ( ) ( ) A ( x) A ( x) A ( x) c ikikc c (6) ikikc kk kk kv kv k k 3 Cohn Scaing Channl Suos ha h iniial sa of h fild was dscibd ( k, ) and was in quanum cohn sa [3] by n ( ) f k n n! Th amliud of iniial adiaion was A ( x) f A ( x) f a a ikikc ikikc k k k a k c kv W a insd in h adiaion amliud af scaing in scond od of ubaion chniqu Th oblm of Fynman s diagams summaion will b discussd blow In Equaion (5) i is sufficin o consid h sum () () (3) S S S S, wh S H ( ) d i () T S H ( ) d ( i), (3) T S H 3 3 ( ) d 3!( i) If h hoon scas in h cohn channl hn h aom ss in iniial sa So in scond od of ubaion chniqu w a insd in h consucion ( c) () A ( x) A ( x) S cc (7) () W chang h im-oding oduc of h aom oaos by h nomal oding on Fo h scaing () oao Ŝ w g () S T A ( x ) A ( x ) G ( x, x) A ( x ) ( x) dxdx mc i, wh TA is h im-oding oao acing only on h lcomagnic fild oaos and T N x x i G x x ( ) ( ) (, ), G ( x, x ) E ( ) ( ) ( ) G E de E G (8) E i If h aom undgos h acion of xnal andom filds h fini widh of is ngy lvls can b n ino accoun by lacing h m i by i / wih h sam sign bcaus i is govnd by h causaliy incil Th sam sul follows fom summing u (Dyson summaion) h ladd Fynman`s diagams fo xcid aoms du o hi inacion wih lcomagnic vacuum Fo h sam ason fomula (8) can b win as E G E i wihou scifying h valu W ino accoun ha, TA A ( x) A ( x) (9) id ( x, x ) NA ( x ) A ( x ) wh D ( x, x) is no h oao funcion Th fis m in (9) dos no lay any ol in lcomagnic fild scaing ocss Finily () S ( x ) G ( x, x) i mc A ( x ) A ( x ) A ( x ) A ( x ) ( x ) dxdx ( ) ( ) ( ) ( ) () Th igh hand sid ms of his qualiy a sonsibl on scaing ocsss of lcomagnic fild by boh h non xcid aom and xcid on 3 Scaing on Non-Excid Aom Subsiuing (6) and (8) ino Equaion () and n h limi, w find () c kc kc ik ( k) S f b b k f k k ivk mc k kc i Coyigh Scis

5 B A VEKLENKO 67 Though on dno h h quanum numb of iniial sa of aom In diol aoximaion ( ) ( ) ρ d ρ ρ ρ Th limi is no ncssay bu i ms h calculaions siml Accoding o (7) w nd o calcula h consucion A ( c) ( ) () ( x) f A ( x) S f cc L us us h following qualiis conncing any smooh funcion f ( k ) and limis V, k ck c f( k ) ik ( ) k k k V ( ) 3 ik ( ) kckc f( k) dk sin k V k f ( k ) n n wh n H w ino accoun only h m dscibing divg wav Th nglcd m uns ino zo by infinily small inval of ingaion ov k ha is suosd Finily nn ( c) ik ik ikc A ( x) cc k () 4 mc ck i 3 Scaing on Excid Aom Th scond m in Equaion () af h sam y of ansfomaion shown in a 3 yilds nn ( c) ik ik ikc A ( x) cc k () 4 mc ck i If on s ino accoun h widh of aom s ngy lvl in sa dscibd by han i is ncssay o lac in Equaion () Th validiy of Equaions () and () a sicd by domain / wh is h adiaion widh of xcid sa of aom 4 Non Cohn Scaing Channl Th scond od ubaion chniqu gs ( n) () () ( x) f S A ( x) S f A O by n ino accoun only h scaing ocsss w hav in xlici fom A ( n) ( ) ( ) ( ) ( x) f A dx A A dx f cc ( i) mc Following h ocdus dscibd in a 3 w hav nn A ( n) i ik ik ikc ( x) k ck cc mc (3) If w ino accoun h fini widh of aom ngy lvl han in fomula (3) i is ncssay o chang kc i kc i kc i Coyigh Scis

6 68 B A VEKLENKO L us find now h oal amliud of lcomagnic fild scad by xcid aom nn ( c) ( n) ik ik ikc ( x) ( x) ( x) cc k 4 mc ck i A A A (4) 5 Smi-Classical Thoy of adiaion Th s quaions fo fild oaos ( x) in Hisnbg snaion is h following and A ( x) ( x) i, U( ) A ( x) ( x) m mc A( x) ( x), ( x) c m This s quaions is quivaln o h on mniond a a 3 Now A ( x) A ( x) (, x x) ( x) ( x ) dx mc (5) H A ( x) is givn by h fomula (6) and ( x, x) A ( x); A ( x) ( ) i c nn c ( ) 4, (6) W a insd in h scond od ubaion xansion This man ha h ( x) oao has b valuad in h fis od of ubaion chniqu ( x) ( x) G (, x x) A ( x ) ( x) dx mc (7) Subsiuing (6) and (7) ino (5) w find A ( x) A ( x) ( x, x ) ( x ) G ( x, x ) A ( x ) ( x ) dxdx H c mc (8) Fo h man valus, h sam sul can b obaind ih by avaging (8) wih subsqun bing u h colaos, o using h smi-classical hoy Af alizing in (8) h subsiuion fo h scad fild w hav ( ) A ( ) ( ) A x x A x a a ikikc ikikc k k k nn A ( x) a cc (9) 4 mc ck i ik ik ikc k k Th u sign dscibs h scaing of lcomagnic fild on h non-xcid aom whil h low on dscibs h scaing on xcid aom On should ino accoun h widh of aomic ngy lvl by lacing in dominao i wih i / i( )/ By comaing Equaion (9) wih Equaions () and (4) w find ha in h aoximaion w usd boh h quanum hoy and h smi-classical hoy sul in h sam xssions fo h scad amliud A ( x) Namly h ncssay coinciding in such suls lads o h qualiy of consans in Fomulas (9), () and (4) 6 Bilina Fild Chaasisics In his a w a insing in h following consucion shown in inoducion ( ) NE x E ( x) In od o calcula his valu in foh od of ubaion xansion i should us h Fomula (3) Bu i is no woh o do i Th sai calculaion shows ha fo sonan fild scaing ( ) h consucion S NA ( x) A ( x) S, () (3) Coyigh Scis

7 B A VEKLENKO 69 which aas in such aoximaion a non-cohn channl suls in ngaiv valu This fac vidnly conadics wih h osiiv dfiniion of xssion f N( E ) f Such conadicion was found bfo in fnc [5] wh diffn modl has bn considd In od o consuc h osiiv dfiniion of h non-cohn channl using ubaion s i is ncssay o avag h oduc ( ) E x E ( x) ov h wav funcion () () (3) S S S Bu doing his w find h ms ooional o h sixh od of chag I mans ha such consucion may b achivd only by using high od ms of ubaion chniqu Thus on can no sic onslf h by h ms of lov od of ubaion chniqu So h convnional ubaion hoy fo ( ) NE x E ( x) is oblmaic Fo hs asons w sima h conibuion of non-cohn ocsss using inqualiy () f N EE f f E f () Thn w us h sam mhod o sima h conibuion of cohn channl Thus accoding o h quanum hoy using Equaion (), Equaion (3) and Equaion () on gs fo h scaing by xcid aom in wo lvl aoximaion h following fomula; nn NEE k qu 4 mc ck ck 4 4 Whil accoding o h smi classical hoy on gs nn NEE 4 k scl mc ck 4 Th aio of suls of hs wo calculaion mhods fo h sonan scaing fquncy is qual o NEE qu 5 N EE scl Th sam valu chaacizs h aio of scaing coss scions qu / scl This sul dos no dnd on W no ha fo h scaing of lcomagnic fild on non-xcid aoms his aio is qual o on Th dndnc of aio qu / scl fo scaing on xcid aom as a funcion of scaing fquncy by is shown in h Figu 7 Conclusions Th valuaions h scad fild amliud of sonanc scaing lcomagnic fild on an xcid aom can b fomd qually wll using boh h Hisnbg snaion and Schoding on In ou aoximaion h boh calculaions lad o h sam suls Th sam suls follow also fom h smi-classical hoy of adiaion, which dals wih classical lcomagnic fild In gnal, h ubaion chniqu is no sufficin o dscib h sonanc scaing ocss and w nd o sum u h ladd Fynman diagams Such ocdu is no difficul o b fomd using any of hois mniond abov In h oh cas w dal wih calculaion of h quanum man valus of bilina oducs of h fild oaos EE H i is mo convnin o dal wih Sch- / qu scl ck / Figu Th yical dndnc of aio qu / scl fo scaing of lcomagnic fild on xcid aom as a funcion of scaing fquncy ck / Coyigh Scis

8 7 B A VEKLENKO öding snaion o wih inacion snaion, which giv addiional oouniis o sum u h Fynman diagams Th l snaions allow us o sn h scaing ocss wih h hl of wo comonns: cohn (lasic) and non-cohn Such comonns could b valuad indndnly Th analysis of non-cohn channl shows ha h Dyson s summaion of ladd Fynman s diagams by scaing of sonan lcomagnic fild on xcid aoms is no sufficin Oh summaion mhods a vy unwildy [8] In sn wok w oos h siml mhod of simaion fom blow h suls of h non-cohn scaing channl As a sul w find ha h smi-classical hoy of adiaion ssnially undsimas h coss scion of sonanc scaing Th quanum hoy in is un shows h violaion of qualiy N EE E E in scad adiaion vn if such qualiy ook lac in h incidn lcomagnic fild So h quanum hoy suls in a chang of quanum saisical sucu of lcomagnic fild du o scaing This can no b obaind wih h hl of smi-classical hoy of adiaion This chang of innal quanum fild sucu by is scaing on xcid aom manifss islf on macoscoic lvl Namly such ffc ms imossibl using h h smi-classical hoy of adiaion 8 fncs [] W E Lamb, Th Thoy of Oical Mass, In: C DWi, A Blandin and C Cohn-Tannoudi, Eds, Quanum Oics and Elconics, Univsiy of Gnobl, Houchs, Nw Yok-London-Pais, 965 [] M O Scully and W E Lamb, Quanum Thoy of a Oical Mass, Physical viw Ls, Vol 59, No, 967, 8-6 [3] C J Kos, 9A4-Las Acion by Enhancd Toal Innal flcion, IEEE Jounal of Quanum Elconics, Vol QE-, No 9, 966, [4] B B Boyko and N S Pov, flcion of Ligh fom Enhancd and Non Lina Mdia, Nauka and Tchnika, Minsk, 988 [5] F Cybulski and C K Caniglia, Innal flcion fom fn Exonnial Amlifying gion, Jounal of h Oical Sociy of Amica, Vol 67, No, 977, 6-67 [6] T C Biba, N S Pov and I Z Dilavday, Th flcion of h Ligh fom Non-Homognous Mdium wih Limid gion of Enhancmn, Jounal of Alid Scoscoy, Vol 3, No, 98, 6-7 [7] B A Vklnko, Acion of h Nois Pocsss on Infnc Pois in sonanc Skoscoy, Izvsia Vouzov Physika, No 9, 983, 7-75 [8] B A Vklnko, BGusaov and Y B Shkunov, Slciv flcion of sonanc adiaion fom Excid Mdia, Jounal of Eximnal and Thoical Physics, Vol 86, No, 998, [9] J Skaa, Fsnl Funcion and h faciv Indx of Aciv Mdia, Physical viw E, Vol 73, No, 6-7 [] I I Smolyaninov and Y-J Hung, Enhancd Tansmiion of Ligh hough a Gold Film du o Exciaion of Sanding Sufac-Plasmon Bloch Wavs, Physical viw B, Vol 75, No 3, 7, -4 [] T W Ebbsn, H G Lzc, H F Ghami, T Thio and P A Wolff Exaodinay Oical Tansmission hough Sub-Wavlngh Hol Aays, Nau (London), Vol 39, No 6668, 998, [] V Wisskof and E Wign, Bchnung d Naulichn Lininbi und Gund d Diacschn Lichhoy, Zischif fu Physik, Vol 63, No /, 93, [3] Glaub, Oical Cohnc and Phoon Saisics, In: C DWi Ed, Univsiy of Gnobl, Houchs, Nw Yok-London-Pais, 965 [4] B A Vklnko, ms on Kams Hisnbg Fomula and som Pois of inducd adiaion, Izvsia Vuzov Physika, No 6, 987, 3-4 Coyigh Scis