Chapter 2: Semi-Classical Light- Matter Interaction

Transcription

1 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) Chapr : Sm-Classal Lgh- Mar Inraon. A Two-lvl Ssm Inrang wh Classal Elromagn Fld n h Absn of Dohrn.. Hamlonan for Inraon bwn Lgh and a Two-lvl Ssm Consdr a wo-lvl ssm, sa an lron n a ponal wll or n an aom wh wo nrg lvls, nrang wh lromagn radaon of frqun. Th lr fld a h loaon of h wolvl ssm s, E ne o os E o os( + ) In h absn of h lr fld h Hamlonan of h lron s, H o Th nrg dffrn bwn h uppr and h lowr sa s. In h prsn of h lr fld, h ponal nrg of h lron of harg q s, q r. E qr. ne o os So h Hamlonan boms, H H o qr. n Eo os o ha h Hamlonan s m-dpndn. In h wo dmnsonal Hlbr spa onssng of onl sas and, and assumng r r 0, h abov Hamlonan an b wrn as, H os os whr h frqun s rlad o h dpol momn of h sas, qeo r. n qeo r. n In h so alld roang wav appromaon onl h mporan rsonan rm n ah osn rm s rand (w wll dsuss hs n dal lar n h ours) and on obans, H p p p p Th Hamlonan abov s usd o dsrb h nraon of a lassal lromagn fld wh a wolvl ssm. An of h mhods usd for h m-ndpndn wo-lvl ssm problm an b usd o solv h m-dpndn problm as wll.

2 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs).. Soluon Usng h Shrodngr Pur W assum a m-dpndn soluon of h form, and plug no h Shrodngr quaon, H d d o g h followng quaons for h offns, d d d d Th dunng s dfnd as. Th abov quaons an b solvd usng appropra boundar ondons. I s onvnn o dfn a frqun as. Suppos, 0, hn h soluon s, sn os sn In h absn of dunng (.. 0 ), h probabls of fndng h lron n h uppr and lowr lvls oslla wh a frqun qual o. Ths osllaons ar alld ab osllaons, and h frqun s alld h ab frqun. Th mamum valu of h populaon dffrn,, s +. In h prsn of dunng, h populaons oslla wh a frqun qual o and h mamum valu of h populaon dffrn,, s. Thr s no smpl wa o nopora dohrn and/or populaon da from h uppr lvl no h lowr lvl n h Shrodngr quaon. To nlud hs w hav o us h dns opraor formalsm...3 Soluon b Transformaon o a Tm-Indpndn Hamlonan Consdr h m-dpndn Hamlonan dsrbng h nraon of a wo-lvl ssm wh lgh, p p p p H W dfn a unar opraor B as follows, B p To undrsand h ff of h unar opraor B on h sa suppos ha s wrn as follows,

3 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) 3 hn, B Th opraor B booss h nrg of h lowr lvl b. I s as o prov h followng wo dns, p p B H B H B whr, H Th Shrodngr quaon for h sa s, H Suppos w dfn a nw sa as follows, B Thn dffrnang boh sds wh rsp o m, and usng h abov dns, gvs, H Th abov quaon shows ha h orgnal problm wh a m-dpndn Hamlonan Ĥ s quvaln o a problm wh a m-ndpndn Hamlonan Ĥ. On an solv h m-ndpndn problm asl usng an of h mhods of h las Chapr. In parular, som nsgh s oband b usng h gnsas of h Hamlonan Ĥ. Suppos, for smpl, ha h dunng s ro. Thn h wo gnsas, and h orrspondng gnnrgs, of h Hamlonan Ĥ ar, v v Th slng of h gnnrgs s. Suppos h nal sa s, 0 0 v v. Thn, ) ( ) ( sn os ) ( ) ( 0) ( ) ( v v H

4 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) Fnall, h dsrd quanum sa an b oband as follows, B os sn Th abov rsul agrs wh h on oband n.. whn dunng s ro. Th mos nrsng asp of h abov hnqu s ha shows ha h m-dpndn problm an b mappd ono a mndpndn problm. W know ha n an m-ndpndn problm h gnsas pla a spal rol. Egnsas ar saonar sas and whnvr an arbrar sa s wrn as a suprposon of saonar sas, h probabl for h sa o b found n an on of h saonar sas upon 0 0 v masurmn rmans m-ndpndn. Suppos, ( ) ( v B B ) ( ) v. Thn, And hrfor, ( ) ( 0) ( ) ( 0) Th probabls of fndng h lron n h uppr and lowr lvls rman m-ndpndn whn h nal sa orrsponds o an gnsa of h m-ndpndn Hamlonan Ĥ. Ths onluson rmans vald, of ours, vn whn dunng s non-ro.. Opal Bloh Equaons.. Soluon Usng h Dns Opraor and Opal Bloh Equaons Hr w wll us h dns opraor approah n h Shrodngr pur. Sarng from h dns opraor quaon, d H, H H d on an drv h dffrnal quaons for,,, and b akng mar lmns of h dns opraor quaon wh rsp o h sas and. Th rsul s, d p p d d p p d d d p d d p 4

5 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) Th dunng bwn h fld frqun and h nrg lvl sparaon s, Th soluon s mos asl oband, and nsghful as wll, f on uss a formalsm orgnall dvlopd o ra prsson of nular spns. W dfn hr nw ral quans,, and as follows, Th vor s, o ha h - and -omponns of h vor ar rlad o h off-dagonal lmns of h dns opraor, and h -omponn of h vor s rlad o h oupaon probabl dffrn bwn h uppr and h lowr nrg lvls (also alld h populaon dffrn). On an now rwr h quaons for h dns mar lmns n rms of h quans,, and, d d d d d d If on dfns a vor frqun as, hn h quaons abov an b wrn n vor form vr ompal as, d X d o ha all fas m dpndns (on h sal of or ) prsn n h quaons for h dns mar lmns ar absn n h quaon for h vor and hs s h man advanag of workng wh. Th quaon for rsmbls h quaon of h magn momn M of a lassal spn n a onsan magn fld B, d M B X M d whr s h gromagn rao (rao bwn h spn magn momn and h spn angular momnum). Th magn momn quaon was drvd b Fl Bloh n 946. Thrfor, h quaon for h vor s somms alld h opal Bloh quaon. Th followng fas ar no hard o prov and follow drl from h vor quaon for ) Th magnud of h vor dos no hang wh m., 5

6 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) ) Th vor us a prod moon and h angular frqun s qual o h magnud of h vor. ls durng roaon. Th vor s alwas normal o h plan of roaon. ) Th plan of roaon s h plan n whh h p of h vor Plan of roaon (h fron fa of h on) Crl of roaon (sold ln) Con.. Th Bloh Sphr Th vor ompll harars h quanum sa of an lron nrang wh an lromagn wav. -as Bloh Sphr ( ) -as -as Th -omponn of h vor quals and gvs h populaon dffrn bwn h wo-lvls and an hav valus bwn + and -. Th ohr omponns of h vor apur h ohrn n h ssm and ar rlad o h off-dagonal omponns of h dns opraor. Sn h magnud of h vor dos no hang wh m, and f a m 0 h quanum sa of h ssm s hn 0 0, h magnud of h vor s 6

7 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) un. Thrfor, roas wh m bu s p alwas ls on a sphr of radus un whh s alld h Bloh sphr, as shown n h Fgur. Cas of Zro Dunng: If h nal quanum sa of h lron s, hn 0 0, and h vor p a m 0 s a h loaon ndad b h blak do on h sphr n h Fgur. If lromagn fld wh ro dunng (.. 0) s urnd on a 0, hn sn, h p of h vor roas n h - plan (wh -as as h as of roaon) as shown n h Fgur. Th - plan s h plan of roaon n hs as. Th p of h vor follows a rl formd whr h plan of roaon nrss wh h Bloh sphr. Ths rl s alld h rl of roaon. A an m, h projon of h vor ono h -as (.. h - omponn) gvs h populaon dffrn. Th projon ono -as and -as gvs h ohrns (.. h off-dagonal omponns of h dns mar). A m whn, h p rahs h souh pol, and h populaon dffrn gos o ro bu h ohrn s mamum. A hs sag, h lron wavfunon (oband b dr ompuaon usng h Shrodngr quaon) s, Whn, h p rahs h posv -as, h lron s n h uppr lvl, and h populaon dffrn s mamum (+) bu h ohrn s mnmum. Th lron wavfunon a hs sag (oband b dr ompuaon usng h Shrodngr quaon) s, Whn 3, h populaon dffrn gos o ro agan. And whn, h populaon dffrn s a s mnmum valu of -, h lron s n h lowr sa, and h p of h vor s a h sarng loaon shown b h blak do n h Fgur. Thus, n on ab prod (gvn b ) h populaon dffrn gos hrough on ompl l. Ths phnomnon s alld ab floppng or ab osllaon. Cas of on-zro Dunng: Usng h Bloh sphr pur, on an vsuall solv man problms. Suppos hr s som dunng so ha h as of roaon, gvn b h dron of, s slghl ld awa from h ngav -as, as shown blow n h Fgur. Th vor sars from h loaon of h blak do, bu now h plan of roaon s no h - plan bu a plan ha onans h blak do (sn hs s h nal sarng pon) and s also prpndular o h dron gvn b. Th bs wa o s wha hs mans s o look a h sphr from h op (.. from abov h orh pol) as shown n h Fgur blow. As h dunng s nrasd, h plan of roaon of h vor ls. Th p of h vor, of ours, alwas rmans on h Bloh sphr and n h plan of roaon. As h dunng nrass, h frqun of roaon (gvn b ) nrass and h p of h vor 7

8 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) roas n rls of smallr and smallr rad (h rl of roaon formd b h nrson of h plan of roaon and h Bloh sphr). Th mamum populaon dffrn, gvn b h -omponn of h vor, drass from +, and a larg dunngs s no longr vn posv a an m durng h dnams. -as Plan of roaon whn =0 (h - plan) Plan of roaon whn s small -as Plan of roaon whn s larg Bloh sphr sn from abov h orh pol.3 A Two-lvl Ssm Inrang wh Classal Elromagn Fld n h Prsn of Dohrn and Populaon Da.3. Opal Bloh Equaons wh Dohrn and Populaon Da W know ha lrons n marals or n aoms n hghr lng nrg lvls usuall om down o lowr nrg lvls b gvng off hr nrg o phonons or o an ohr non-radav hannl. In a wo-lvl ssm, on an nlud rlaaon of h populaon from h uppr lvl o h lowr lvl va suh nonradav mhansms. W assum ha h populaon rlaaon m s T. On an also nlud h ffs of dohrn b assumng ha h off-dagonal omponns of h dns mar da wh a m onsan T. In phsal ssms, h sam sarng mhansm s ofn, bu no alwas, h sour of boh populaon rlaaon (.. T ) and dohrn (.. T ). Whn hs s ru, T and T ar rlad; T T. Ohrws, T T. In h prsn of populaon rlaaon and dohrn, h quaons for h lmns of h dns mar bom, d p p d T d p p d T 8

9 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) d d T p d d T p Th quaons for h omponns, ar now as follows, d d d T T,, and d d d T On an no longr wr h abov s of quaons n h ompa form, d X d Also no ha h magnud of h vor s no onsrvd anmor. Th abov quaons hav a wll dfnd sad sa soluon, T T TT T T TT T T T TT Th abov prssons for h sad sa show ha lromagn radaon, no mar how srong, anno ra populaon nvrson n sad sa (.. mak h -omponn posv n sad sa). Opal Conrol b Shor Pulss: Shor pulss of lgh an b usd o onrol and/or prpar dsrd quanum sas of a wo-lvl ssm. For ampl, onsdr a suaon whr dunng s ro and populaon rlaaon and dohrn ms ar suffnl long. Suppos, 0. adaon s swhd on a m 0 and swhd off a m gvn b. Suh a radaon puls s alld a puls. A h nd of h puls, h valu of s appromal + and populaon nvrson s ahvd. Th quanum sa of h wo-lvl ssm a h nd of h puls s, 9

10 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) 0 Th populaon wll, of ours, rla no h lowr sa afr h puls baus of T. Smlarl, as shown arlr, a puls (.. a puls whos duraon n m s ) an b usd o ak an lron from an nal sa no a lnar suprposon of and gvn b,.4 Phoon Eho Eprmns Phoon ho s a usful prmnal hnqu o harar dohrn ms n marals. Consdr an solad wo-lvl ssm (n h absn of an radaon) n whh h man poson of h lron n boh sas, and, s ro, and, r r d ow onsdr h suprposon sa, 0 Th man poson of h lron n suh a suprposon sa s, d r os Th harg dns assoad wh h lron wavfunon osllas n spa wh a frqun rlad o h nrg lvl dffrn. Elron n suh a suprposon sa an hrfor rada lromagn nrg, jus lk a lassal dpol annna. adaon from on suh lron s oo dfful o d. adaon from man suh lrons n a ollon of wo-lvl ssms would also b dfful o d unlss all h lrons wr osllang n-phas. Th rk s o g man lrons o oslla n-phas. Elron sas n man marals, suh as smonduors, an b modld as a ollon of wo-lvl ssms. In mos ass, lron nrgs of h uppr and/or lvls would hav a rang of valus. In ohr words, h nrg dffrn, or h dunng masurd wh rsp o som fd, would no b h sam for all h wo-lvl ssms. For som ssms, would b ro, for som would b posv, and for som would b ngav. Thrfor, harg osllaons n dffrn wolvl ssms would no sa n-phas for long. o ha h suprposon sa, 0 orrsponds o a vor wh h followng omponns, 0 sn os Th argumn of h osn and h sn s also h m-dpndn phas of h harg osllaon rlav o. If for wo dffrn wo-lvl ssms h vors wr o bom dnal a an pon n m, hn a ha momn h lron harg n hs wo wo-lvl ssms wll b osllang n-phas.

11 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) Phoon ho hnqus ar abl o ahv n-phas harg osllaons n a ollon of wo-lvl ssms wh vn dffrn valus of. Consdr suh a ollon of wo-lvl ssms nrang wh radaon of frqun. Suppos a shor and srong ( ) lromagn puls s usd o h wo-lvl ssms assumd o b all nall n h ground sa. Th dnams of ah wo-lvl ssm ar govrnd b h quaons, d d T d T d d d T Assum ha dohrn and populaon rlaaon ms ar vr long. gh afr h puls, h vor quals ŷ for all h wo-lvl ssms, as shown n Fg.(a) blow. Th subsqun voluon of h wo-lvl ssms s aordng o h quaons, d d d 0 d d d Th abov quaons show ha h vors of wo-lvl ssms wh dffrn dunngs mov apar n h - plan as shown blow n Fg.(b). Afr som m, sa T d, anohr shor and srong ( ) lromagn puls s usd. Th puls roas h vors of all wo lvl ssms b 80-dgrs around h -as and h fnal posons of h vors ar as shown n Fg.() blow. gh afr h puls, h fr voluon of h wo-lvl ssms s agan aordng o h quaons, d d d 0 d d d Th puls hangd h sgn of h omponn of vr wo-lvl ssm, bu dd no hang h sgn of h omponn. Consqunl, h vors of wo-lvl ssms wh dffrn dunngs, whh wr movng apar n h - plan bfor h puls, now sar movng losr wh m. A m qual o al T d afr h puls, h vors of al wo-lvl ssms om oghr, as shown n Fg.(d). A hs pon n m, lrons n all h wo-lvl ssms ar osllang n-phas. Consqunl, h radaon md b hm an asl b dd. Baus of dffrn dunngs, harg osllaons n dffrn wo-lvl ssms wll soon go ou of phas wh ah ohr. In aual prmns, a wak bu dabl radaon puls s dd a h momn whn h lrons n all h wo-lvl ssms ar osllang n-phas. Ths puls s alld h phoon ho puls. ow suppos dohrn s prsn. Th ohrns wll g rdud h longr h m dla T d s ompard o h dohrn m T and, hrfor, h srngh of h phoon ho puls wll also g rdud as T d s mad longr. Thus, f h srngh of h phoon ho puls s masurd as a funon of h m dla T d hn hs nformaon an b usd o ra h dohrn m T (assumng T T ).

12 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) (a) (b) () (d).5 ams Frngs and Aom Cloks Consdr h problm of drmnng h frqun of a radaon sour wh hgh aura (fraonal aura br han on par n 0 4 ). On wa o do hs would b o mak an aom, or a wo-lvl ssm, nra wh h radaon from h sour. Suppos h nrg lvl sparaon of h wo-lvl ssm s known o b wh a vr hgh aura. Th frqun of h sour s assumd o b los o bu h dunng s unknown and nds o b drmnd wh hgh prson. Suppos, on onsdrs h followng shm: h wo-lvl ssm s mad o nra wh h radaon from h sour for a duraon T n. Immdal afrwards, h uppr sa oupaon (.. ) s masurd. Th quson s whhr hs absorpon prmn an b usd o drmn. W wll assum ha dohrn and populaon rlaaon ms ar vr long. Suppos h duraon Tn s hosn suh ha h radaon appars o h wo-lvl ssm as a puls ( T n and ). Bu

13 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) s unkown. To ovrom hs problm, on an dplo a suffnl srong puls suh ha. Afr h puls, h valu of s, If, h abov prsson shows ha s los o un rrspv of h valu of. Thrfor, hs smpl absorpon prmn wll no b abl o drmn wh hgh prson. Aom ohrns an b plod o oban muh br auras, as w wll s now..5. ams Frngs Consdr a wo lvl ssm prpard n h ground sa,.. 0. A m 0, a srong puls ( ) of duraon T n and dunng s h ssm. Th dnams of h wo-lvl ssm ar govrnd b h quaons, d d T d T d d d T Assum ha dohrn and populaon rlaaon ms ar vr long. Afr h puls, h sa vor quals ŷ (Fg.(a) blow). Th subsqun fr voluon of h wo-lvl ssm s aordng o h quaons (Fg.(b) blow), d d d 0 d d d Th vor roas n h - plan wh a frqun qual o for a duraon T (Fg.(b) blow). Afr hs duraon T, a sond puls of duraon T n nras wh h wo-lvl ssm. If h T m, whr m s an posv odd ngr, and h sa vor m duraon T s suh ha quals ŷ (Fg,() blow), hn h sond puls wll mak h sa vor qual o ẑ and h wo-lvl ssm wll b n h lowr sa afr h sond puls. On h ohr hand, f h m duraon T s suh ha T m, whr m s an posv vn ngr, and h sa vor quals ŷ (Fg,(d) blow), hn h sond puls wll mak h sa vor qual o ẑ and h wo-lvl ssm wll b n h uppr sa afr h sond puls. I s no dfful o show ha undr h assumpons and vr long dohrn and rlaaon ms, h valu of a h nd of h sond puls s gvn b os T, and h oupan of h uppr sa s gvn b 0.5 os T. Thrfor, f on maks a masurmn on h wo-lvl ssm afr h sond puls and drmns whhr h ssm s n h uppr sa or h lowr sa, hn, knowng T, h dunng of h radaon from h nrg lvl sparaon of h wo-lvl ssm an b drmnd 3

14 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) o a vr hgh aura. o ha frqun aura n hs as s no drmnd b h m of nraon T n of h wo-lvl ssm wh h radaon bu b h duraon T bwn h wo pulss whh an b mad vr long subj o h onsrans mposd b dohrn and rlaaon ms. (a) (b) () (d).5. Csum Aom Cloks Th prnpl dsussd abov s usd n aom loks whh ar vr sabl and hgh-prson frqun sours. In a Csum aom lok, radaon from a unabl F osllaor s oupld o Csum aoms whh hav an nrg lvl sparaon los o ~9.9 GH. Th frqun of h F osllaor an drf ovr m and nds o b lokd o a sabl rfrn wh hgh prson. Th prnpl of opraon of h lok s as follows. Csum aoms ar prpard n h ground sa and mad o nra w wh pulss from h F sour. Th duraon bwn hs pulss s T. A h nd of h sond puls, h uppr sa oupan of h Csum aoms s drmnd. Ths produr s rpad man ms whl h frqun of h F sour s vard. A pal plo of h masurd uppr sa oupan afr h sond puls vs h dunng of h F sour s shown n h Fgur blow. 4

15 Quanum Ops for Phoons and Opolrons (Farhan ana, Cornll Unvrs) Uppr sa oupan Th obsrvd osllaons ar alld ams frngs (afr orman ams). o ha h wdh of h nr frng (n H) s T (s h ns). A mor dald analss shows ha f on dos no mak h assumpon and also dos no assum ha dohrn and rlaaon ms ar nfnl long, hn h obsrvd nvlop of h osllaons an also b rprodud. Usng fdbak from hs masurmns, h frqun of h F osllaor s adjusd and lokd o h nrg lvl sparaon of h Csum aoms. In an aual Csum founan lok (s Fgur abov), Csum aoms ar prpard n h ground sa and hurld upwards agans grav (lk war n a founan). Th pass hrough a mrowav a av w; on hr wa up and hn agan on hr wa down. Whl passng hrough h av, h aoms nra wh h radaon from h F sour. Fnall, h sa of h Csum aoms s dd and h l s rpad. nl, ulra-old aoms wh mpraurs lss han K hav bn usd o rdu dohrn and rlaaon ras rsulng n vr sabl aom loks. Th roo-mansquar fraonal frqun sabl oband n pal Csum aom loks s gvn b h prsson, T ~ T a Hr, T s h l m (m ndd o ompl on masurmn), s h ngraon m usd n smang h frqun, and a s h numbr of aoms usd n on l. Tpal valus of T and 5 a ar sond and 6 0 aoms, rspvl. Ths gvs, ~ s n sonds r sabl F sours, wh fraonal sabls smallr han 0-6, hav bn rald usng hs prnpls. 5