Chapter 3: Optical Linear Response

Transcription

1 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy Chap : Opcal La sps. Elccal/Opcal Suscpbly a h Dlcc Csa I Chap, was shw ha h sps a w-lvl sysm a classcal lcmagc l s gal la. I suas wh chc s as, whch s h cas, a much smpl pcu h ac bw a w-lvl sysm a aa ca b ba. I hs pcu, h ac s scb by la sps hy a pcal pcsss ca b scb ms smula mss a smula absp as. Opcal pcsss ca als b scb ms smula mss a smula absp as wh h upp a lw gy lvls a sgl sc lvls bu css a cllc clsly spac gy lvls. I hs la cas, Fm s Gl ul ca b us cmpu h as as. W als scuss cs u uaum mchacal c pcal la sps, such as ak sas a lcmagcally uc aspacy... Elccal/Opcal Suscpbly h pcal la sps a mum s scb by h suscpbly csa ha a la by, a h lcc I h ucy ma, h suscpbly las h uc macscpc plaza sy vc h lcc l, P E I h m ma h abv ua bcms, P ' E ' ' Sc al lcc ls h uc plaza sy mus als b al, h suscpbly * mus sasy h c,. h magu h vc P uals h umb pl mms p u vlum a ps h c h pls. P P.. Suscpbly a Mum w-lvl Sysms wh Fas Dchc as W wll calcula h suscpbly a h lcc csa a mum ma up w-lvl sysms wh sysms p u vlum. F smplcy, w wll assum ha all h w lvl sysms a h sam way, as shw h Fgu blw. A mum w-lvl sysms Supps hs mum s xc by a mchmac lcmagc l h m,

2 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy E E E E cs h plaza sy P s la h lcc l by h pcal suscpbly as llws, E E P * F a w-lvl sysm acg wh a mchmac lcmagc l h Hamla s, H wh, E E E... W hav ma h ag wav appxma wg h abv Hamla. h as makg h ag wav appxma s xpla blw. h uas h lms h sy max ca b v m h ua, H, a h sul s, W assum ha h chc a s much as ha h laxa a as wll as h ab ucy.g., gup III-V smcucs s.-. ps a s - ps. Sc chc ccus vy as m scals, w ca ga u h uas h -agal cmps h sy max. I s bs ac u h as m pc xplcly. L,. W assum ha h l was u a a sm m h m pas s ha a m all h al ass hav w a h, ' ' ' ' ' W hav xplcly slv sc *. Sc,, w ca pull ' ' us h gal,

3 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy ' ' ' Sc, w g, Smlaly, h -agal cmps h sy max ca b us calcula h plaza sy as llws. call ha -agal cmps h sy max cay ma abu h supps bw h upp a lw gy gsas. A uaum sa whch s a supps upp a lw gy gsas csps a scllag lc chag sy, h ws, a scllag chag pl. h pl mm p ca by h scllag lc chag sy a sgl w-lvl sysm s gv by, p ac h plaza sy P umb pl mms p u vlum} s, } { P ac I spc ma, h plaza sy has -z cmps ly h c h appl l. W h kp ly hs cmp smplcy a w, P. } { ac Sc,... wh. s h pcal pl max lm, w g,.... } { P ac

4 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy 4... c c E Cmpag h abv ua wh, E E P * A assumg ha h ccupacy c, h ppula c, vas s slwly m ha ca b ak as a csa, w g,. ha, as u. I w ha ma h ag wav appxma a plac h ac m h Hamla, wh, h w wul hav ba h sc -sa m s h backs h xpss abv. Wh h ucy s cls h lvl spacg h h s sa m s much lag ha h sc m a h ag wav appxma s a g appxma. Bu wh s a mv m, xampl wh s s calculag h acv x a ma ucs much small ha, h h -sa m ca b g. has al a magay pas, " '. A skch h ucy pc h al a magay pas s shw h Fgu blw. h lcc csa a mum s,

5 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy 5. h lcc csa ps h ccupacy h gy lvls hugh h m. Smula Emss a Smula Absp.. a Euas h Dagal Cmps h Dsy Max W w lk a hw lgh acs h agal cmps h sy max, a, h cas as chc a cs abv Sc... h uas h agal cmps h sy max a. Subsug h xpsss h -agal cmps h sy max m Sc.. h abv ua, a gg ms ha a aply scllag whch g as, bas, Smlaly, h sc m s h backs abv wul hav b abs h ag wav appxma. hs m s -sa, a s usually vy small cmpas h sa m wh h ucy s cls h lvl spacg. h abv uas shw ha wh chc s as h agal lms h sy max by smpl a uas. h s m h gh ha s h a uas abv cag pss -aav laxa m h upp lvl h lw lvl. h sc m h gh ha s scbs ass bw h lvls caus by h lcmagc l. h ass a w yps. h as a m h upp lvl h lw lvl wgh by h ccupacy h upp lvl s,

6 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy s m hs s h smula mss a. h l causs h lc, ps h upp lvl, mak a as h lw lvl a gv s gy h l. h as a m h lw lvl h upp lvl wgh by h ccupacy h lw lvl s, s ab hs s h smula absp a. h lc, ps h lw lvl, absbs gy m l a maks a as h upp lvl. h c bw h smula mss a smula absp as appas h a uas abv. h smula mss a absp as p h sua sgh h lcc l a h lca h w-lvl sysm hugh h sua ab ucy. I gs h -sa ms, h h smula vs p sc p u vlum s.. smula mss mus smula absp vs p sc p u vlum ca b la h magay pa h suscpbly, E s s m s ab Im W wll s la h cus ha h uay h sua backs abv s h ph sy a h lca h w-lvl sysm Smula Absp Smula Emss W s ha wh chc s as, gag u h -agal cmps h sy max suls smpl a uas h agal cmps. hs a uas scb pcal ass bw h upp a lw lvls ms smula mss a absp vs... A Sm-Classcal Dscp aa Emss a Absp h scp aa mss a absp vlp mgh sm vy m wha s kw m classcal physcs. Fm classcal physcs w kw ha h a wk a by by a c s gv by F v. wh F s h c a v s h vlcy h by. I h c a/ vlcy a ucs m,.g. susal, h h m avag a wk h by s F. v. makably, hs smpl xpss ca xpla smula mss a absp as wll. Cs a w-lvl sysm acg wh aa. Supps h l s gv by, E E cs h Hamla h ag wav appxma s, 6

7 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy H W assum z ug,. I h lc was ally h upp sa, h h a gy mss s m h sy max uaw Sc.., s m A h lc was ally h lw sa, h h a gy absp s, s m w w v h abv suls usg h classcal xpss, F. v. I h al sa h lc s h, assumg chc, h lc wavuc ms s Sc.., cs s I h al sa h lc s h, assumg chc, h lc wavuc ms s Sc.., s cs h ma ps h lc ms ca b u m, h ma vlcy h lc s, v F h w al sas, h ma vlcy s, s cs v s cs h c acg h lc s c wh a u--phas by 8-gs wh h c wh. h, h m avag a gy absp by h lc h w cass s, E. s E. v E. s W s ha h gy absp a cass magu wh m. hs s bcaus h supps bw h lw a h upp sas aks m bul up m z. S a w hav clu h cs chc. Dchc sys supps bw h upp a lw sas a sups lc m. h pbably P ha chc has sup lc m ll m s, E. Isgly, h vlcy h lc s -phas wh h 7

8 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy 8 P P I w avag s h xpss. v E wh spc P, a als assum a as chc a, h al sul s,. v E h abv xpsss shw ha h a wk h lc s gav, h lc gvs gy h l, wh h al lc sa s. A h a wk h lc s psv, h lc absbs gy m h l, wh h al lc sa s. h c h w cass ass m h phas h lc m. h as ba usg h classcal pcu ag wh hs ba usg h sy max appach. h absp mss aa by a lc s hus a w-sp pcss; h aa l s causs h lc mv, a pg whh h sulg lc vlcy s -phas u--phas wh h c acg, gy s absb m by h lc... Applca Fm s Gl ul Fm s Gl ul gvs wll as as wh h pssbl al sa s a sgl sa bu a cllc clsly spac gy sas. Supps ha h sgl upp gy lvl s plac by a cllc clsly spac gy lvls. h Hamla s, H Supps. Assumg ly lg chc ms.., w ca calcula h as a m h lw gy lvl h upp gy lvls u smula absp usg Fm s Gl ul, 4 ab s I s sucv cmpa h sul ba abv m ha ba usg h sy max appach usg mhs scuss Sc.., ab s I h lm, h Lza ucs h backs abv ca b appxma by la ucs wh ga wghs ual. I hs lm, h sc m h backs s csv gy a s cbu, a h al sul s h cal h ba abv usg h Fm s Gl ul.

9 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy 9. Spaus Emss Cs h a uas h agal cmps h sy max h absc ay aav ppula laxg mchasm, ha,.. E E h m-avag l gy sy s E. h, bh h smula mss a absp as a ppal h gy sy h l a h lca h w-lvl sysm. I h lcmagc l was c a cavy vlum V, a ach ph has a gy ual, h h ph sy wul ual E. I s h al umb phs h cavy h, V Ω. h, bh h smula mss a absp as a ppal h al umb phs h cavy. I 96, Alb Es p u ha a mpa g, spaus mss, was mssg h abv a uas. H ma hs scvy usg agums ha llw m h csscy sascal hmyamcs a hs agums w ma wll b h ull uaum hy aa was vlp. I h cx u scuss w-lvl sysms, hs agum ca b sa as llws. Supps assums a w-lvl sysm hmal ulbum wh h lcmagc l a cavy. h cavy supps jus a sgl aa m ucy. Sascal mchacs aa ms, as alay vlp by Max Plak hs 9 pap, shw ha hmal ulbum h umb phs a lcmagc m ucy s, K B Supps als, smplcy, ha h gy lvl spaa xacly uals. Sascal mchacs als cas ha hmal ulbum h ccupa h upp gy lvl a h lw gy lvl mus b la as, K K B B I hmal ulbum w mus als hav a say sa,

10 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy Bu h abv say sa c, wh us h ppula a uas abv, gvs h sul, whch claly cacs Eua abv. Es cclu ha h a uas h agal cmps h sy max, as w abv, a cc cmpl a bs. x h pblm, Es assum ha a lc h upp gy lvl mus b abl spausly m a ph v h absc ay smula by phs alay h cavy a mak a as h lw gy lvl. cpa spaus ph mss a h a uas, Es a a xa m as shw blw,. C V. C V h csa C ca b m by aga vkg h hmal ulbum cs, K K B B K B h abv h las ca gv h valu C a h sul s,. V C Cmpag h spaus mss a wh h smula mss a, s s ha h spaus mss a uals h smula mss a assums ly ph h m. O ca w w h cmpl ppula a uas as llws clug bh -aav laxa a spaus mss,

11 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy.. V V.. V V. Opcal Lss a Amplca Smula absp a mss suls gy h bg absb m h aa l by h mum bg as m h mum h aa l. h, smula absp a mss mus als cas cas h sgh h l. A sm-classcal ml hs ac h mum h l ca b vlp usg h amla la sps ucs. Supps h lcc l h ac Hamla s u a lcmagc wav avllg h z-c. I h m-hamc a, h l s, k E E., h wavvc k sass h sps la, c k H, s h acv x h mum. call ha, Spaus Emss

12 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy a,. h, k Im c c W assum ha h wavvc h wav s cmplx, k k' k" k k ' k " k' k" h wav wll cay amply wh sac pg h sg k ", Ez E kz k z E " csk ' z Sc, k ' k " Im c h sg k " s h sam as h sg h magay pa. Supps. h Im a h wav amplu cays xpally wh sac. I hs cas, smula absp xcs smula mss. call ha, E s m s ab Im I, a c ha s call ppula vs, h Im, a smula mss xcs smula absp. As a sul, h wav amplu gws xpally wh sac. h amplca aa a ppula v mum s h pcpl us lass. Phas Pps h aa Em va Smula Emss: I s sg ha h aa m va smula mss as chly.. -phas wh h smulag aa l sulg a xpal gwh h l wh sac. hs s supsg kps m h sm-classcal pcu aa mss vlp Sc... I h lc s h upp gy lvl, h aa l causs h lc sclla. hs scllas a ch a h phas s m by h phas h smulag aa l. h scllag lc aas lk a pl aa. I a mum ma up such w-lvl sysms, h aa m hs scllag pls as up -phas h c ppaga h smulag aa l sulg a xpal gwh h aa l wh sac. La h cus w wll s ha h sua s vy aa m spausly. Spausly m aa has a am phas..4 La sps Fucs a Kams-Kg las I hs Sc w gv sm gal chaacscs la sps ucs ma. Suscpbly s a xampl a la sps uc. I las h uc maal plaza h appl lcc l. La sps ucs a us chaacz h la sps

13 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy maals small pubas. Oh xampls clu h lccal cucvy, h magc suscpbly, h hmal cucvy, c. Cs a la sps uc ha las h appl smulus S h sulg maal sps, S wll gal b cmplx wh al a magay pas, ' " I m ma w hav, ' S ' ' s h vs Fu asm. Sc h ucs wh mus sasy h c, * ' " " a S a al, Ay physcal sysm mus b causal. hs mas ha h sysm ca sp b h smulus s appl. A cssay a suc c causaly s ha mus b z. h, ca w a causal physcal sysm, ' S ' ' I ucy ma, causaly cas h lca h pls h cmplx uc. h sul s ha h al a magay pas h uc a p bu la h llwg way, ' ' ' ' 4 " ' " ' ' " " 4 ' ' ' ' hs a h Kams-Kg las. Ms physcal sysms ca sp ucy smul. h, ms cass, ' " Kams-Kg las mply ha kws h magay pa a sps uc all ucs, h h al pa ca b u jus by ga a vc vsa. I s ga uly xpms. F xampl, lgh lc spcscpy whch h lc cc a maal s masu v a w ag ucs s us u h cmpl absp spcum h maal whu acually masug absp cly. O cus, h xpss h suscpbly ha w calcula hs chap als bys h Kams-Kg las..5 Ic Ecs Opcal sps: Dak Sas a EI Quaum mchacal c ca la sg pcal cs. H, w scuss w cs ha a la; ch ppula appg ak sas a lcmagc uc aspacy.

14 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy 4.5. Ch appg a Dak Sas Cs h h-lvl sysm shw h gu blw. W assum ha h lw w lvls a ga. h pcal max lms bw h upp a h w lw sas a, Supps h l a h lca h h-lvl sysm s, E E cs W, E E.. h Hamla bcms, H W w cs a al sa ha s a supps h lw w sas, b a F pcal absp happ, h lva Hamla max lm bw hs al sa a h al sa,, mus b -z, b a Supps h csas a a b a such ha, h h Hamla max lm bw h abv al sa a h al sa,, s z. hs mas ha h aa wll cupl hs al sa h upp sa a h pcal absp wll ak. h sa a m wll smply b, O way usa h absc absp s ha h amplus ass bw h w lw sas a h upp sa sucvly sulg a cmpl caclla absp. h pblm wh hs schm s ha s cul alz xpmally. h w lw sas wll lkly b xacly ga a s h lav phas wll chag wh m as,

15 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy ul h caclla u c s mga a h absp wll ccu. A b schm s shw blw. W w assum w spaa l c h h-lvl sysm, E E cs E E cs h ug ach l s, W h cuplg csas as, E. E. h Hamla bcms, H h ass way usa h yamcs sa m h Schg ua h m-p Hamla, H h a w sa as llws, B wh, B xp h, jus lk Sc.., ca b shw ha sass a Schg ua wh a mp Hamla, H H 5

16 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy s a gsa h Hamla Ĥ a s ca h sa, h wll als ca a wll h acual sa. S w all h gsas a ggs h Hamla Ĥ a s h s a gsa wh h u pps. h s such a gsa pv h llwg c s sas, By ug h ucs h c aas, h abv c ca always b m. h s gsa s, h cspg ggy s. h,, h, a, B I s w asy s ha h al sa hs sa has wgh h upp sa a h h h lvl-sysm s ppa h ca al sa h h wll b pcal absp. hs s a xampl lcmagcally uc ak sa. h us ha ass w s hw s ppa a h-lvl sysm such a al sa. W asw hs us x..5. Elcmagcally Iuc aspacy W w cs h sam h-lvl schm h pvus Sc, bu assum, W assum ha a sg pump bam cupls h lvls a. A wak pb bam s us masu h absp bw h lvls a. I h ws, w sk h la sps h h-lvl sysm ms a suscpbly h pb bam. h slu s smpl s b m ha w ly sk a slu la. Hwv, sc h pump bam s sg, w wll sk a sps all s. Sc w sk h la sps, w ca assum hughu h 6

17 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy calcula ha h agal cmps h sy max chag a hav h llwg valus, W cs a mum ma up cal h-lvl sysms wh sy. I say sa, h h suscpbly s by h pb bam h ag wav appxma ca b w as,. E S h -agal cmp h sy max. h ua s, h sc l shws ha s cupl. S w as wll. h ua s, h las m h gh ha s ca b g sc wll gv a hgh cbu. h as m pc a ca b ac u by assumg, h sulg uas ca b slv assumg say sa, g, W w assum, smplcy, ha.. h pump bam has z ug. Als, sc,, w g, Fally,. 7

18 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy * ha abv xpss h suscpbly s sasy h c bcaus w ma h ag wav appxma. hs s a pblm sc w a s h slu a ucs a away m h sac. W assum ha h chc bw h lw w lvls has a lg cay m,... Opcal absp s ppal h magay pa h suscpbly. W pl h magay pa as a uc h ug valus h a h Fgu blw. I ca b s ha h pcal absp gs z wh h ug gs z. h pak h absp ccus bh ss z ug wh, hs phm s call lcmagcally uc aspacy EI. h mum has b ma aspa by h pump bam cuplg lvls a. ha h absp a z ug wll b small bu xacly z. u h g hs z absp, w lk a h -agal cmp h sy max wh h chc s small a h ugs a bh z, Cmpa h abv xpss wh h ak sa u Sc.5., Isgly, h sa cspg h valu ba abv s h ak sa. h ac h pb a pump bams pu h lc, ally h gu sa, h ak sa, a c h ak sa h lc s app h. hs als xplas why h absp s 8

19 Quaum Opcs Phcs a Oplccs Faha aa, Cll Uvsy z wh h chc a. h gy h ak sa ps h chc bw lvls a. EI s h xmly ssv h chc as. M gally, h cs EI a h sam as hs u al h xsc a ak sa, h al pa h suscpbly EI s als sg. h shap au h z absp sac suls vy shap aus h al pa h suscpbly, as shw h Fgu blw. hs shap aus mak h acv x h mum hghly spsv a h h gup vlcy lgh avllg hugh h mum ca b ma vy small. EI has b us applcas wh slw lgh s s, sampl, pcal lay ls. 9