Autopsy of a quantum electrical current

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Autopsy of a quantum electrical current

Benjamin Roussel

To cite this version:

Benjamin Roussel. Autopsy of a quantum electrical current. Quantum Physics [quant-ph]. Université de Lyon, 2017. English. �NNT : 2017LYSE1285�. �tel-01730943�

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k9 1G1*h_PL ZlLhlJ PShA*a RXR T?QiQMb T2` KQ/2 Q` 2p2M bT`2/ Qp2` KmHiBTH2 KQ/2bX h?2 /2;`22 Q7 +QMi`QH `2+?2/ BM T?QiQM [mMimK QTiB+b- #Qi? BM i?2 KB+`Qrp2 M/ BM i?2 QTiB+H /QKBM KF2b T?QiQM [mMimK QTiB+b  p2`v T`QKBbBM; THi7Q`K 7Q` 2tTHQ`BM; [mMimK T?2MQK2M (>`Q+?2 M/ _BKQM/- kyye)X SQi2MiBH TTHB+iBQMb `M;2 7`QK [mMimK +QKTmiiBQM (EMBHH 2i HX-kyyR)- [mMimK +QKKmMB+iBQM (_2Bb2`2` M/ _2KT2- kyR8c "`mMbi2BM M/ pM GQQ+F- kyy8c a+`MB 2i HX- kyyN) iQ i?2 [mMimK bBKmHiBQM Q7 KQ`2 +QKTHB+i2/ T?vbB+H bvbi2Kb (G2 >m` 2i HX- kyRe)X

aBM+2- i H2bi BM T`BM+BTH2- 2H2+i`QM [mMimK QTiB+b BKb i `2+?BM;  ?B;? /2;`22 Q7 +QMi`QH QM i?2 [mMimK bii2b Q7 i?2 2H2+i`QM ~mB/ BM  [mMimK +QM/m+iQ`- Bi Bb #b2/ QM i?2 bK2 2tT2`BK2MiH iQQHb M/ i?2Q`2iB+H T`/B;Kb i?M T?QiQM [mMimK QTiB+bX h?2 KBM Q#D2+iBp2 Q7 i?Bb +?Ti2` Bb iQ BMi`Q/m+2 i?2K- bi`iBM; 7`QK 2tT2`BK2MiH iQQHb bm+? b 2H2+i`QMB+ bQm`+2b M/ BMi2`72`QK2i2`b- M/ i?2M KQpBM; 7Q`r`/ iQ i?2 i?2Q`2iB+H +QM+2Tib BMbTB`2/ #v [mMimK QTiB+b MQr mb2/ 7Q` bim/vBM; 2H2+i`QMB+ bii2b M/ [mMimK i`MbTQ`iX q2 rBHH }MHHv +HQb2 i?Bb +?Ti2` #v T`2b2MiBM; p`BQmb T`QiQ+QHb 7Q` K2bm`BM; bBM;H2@ M/ irQ@2H2+i`QM +Q?2`2M+2bX "mi #27Q`2 ;QBM; BMiQ i?Bb- H2i mb 2tTHBM r?v-Hi?Qm;? i?2b2 }2H/b Q7 [mMimK i`MbTQ`i M/ T?QiQM [mMimK QTiB+b `2 iQ bQK2 2ti2Mi mMB}2/ #v i?Bb TT`Q+?- i?2B` `2bT2+iBp2 ;QHb `2 MQi i?2 bK2 b Q7 iQ/vX

S?QiQMB+ bvbi2Kb `2 #v Mim`2 M/ +QMbi`m+iBQM p2`v +H2M #2+mb2 T?QiQMb /Q MQi BMi2`+i rBi? 2+? Qi?2`X +imHHv- i?2v `2 bQ +H2M i?i i?2 KBM /B{+mHiv Bb iQ KF2 T?QiQMb BMi2`+i bi`QM;Hv 2MQm;? i?`Qm;?  +H2p2` mb2 Q7 HB;?iĜKii2` BMi2`+iBQMX PM QM2 ?M/- i?2 +H2MHBM2bb Q7 T?QiQMB+ bvbi2Kb HHQrb iQ i`MbKBi [mMimK bii2b Qp2` ?mM/`2/b Q7 FBHQK2i2`b Ub22 (oHHQM2 2i HX- kyR8c uBM 2i HX- kyRd) 7Q` bT2+i+mH` 2tKTH2bV- KFBM; T?QiQMb  THi7Q`K Q7 +?QB+2 7Q` [mMimK BM7Q`KiBQM M/ +QKKmMB+iBQM T`QiQ+QHbX PM i?2 Qi?2` ?M/- i?2 H2p2H Q7 +QMi`QH QM HB;?iĜKii2` BMi2`+iBQM Bb MQr bQ ;QQ/ i?i- BM bQK2 bvbi2K- Bi Bb TQbbB#H2 iQ ;2M2`i2 BMi2`2biBM; +QHH2+iBp2 T?QiQMB+ bii2b (>Q7?2BMx 2i HX-kyyN)X

"v +QKT`BbQM- 2H2+i`QMB+ bvbi2Kb `2 Km+? KQ`2 K2bbv b +QmH/ #2 2tT2+i2/ 7`QK Qm` T`2b2MiiBQM Q7 [mMimK i`MbTQ`iX h?2`2 `2 irQ KDQ`- 7mM/K2MiH /Bz2`2M+2b #2ir22M 2H2+i`QM M/ T?QiQM [mMimK QTiB+bX 6B`bi- 2H2+i`QMb `2 72`KBQMb M/ MQi #QbQMbX h?Bb /Bz2`2M+2 BM biiBbiB+b Bb- T2` Bib2H7- [mBi2 /`biB+X Ai }`bi BKTHB2b i?i- ;Bp2M  b2i Q7 KQ/2b- i?2`2 `2 rv Km+? H2bb ++2bbB#H2 72`KBQMB+ bii2b i?M #QbQMB+ bii2bX AM T`iB+mH`- i?2`2 Bb MQ /B`2+i +QmMi2`T`i Q7 i?2 +HbbB+H

RXR a*A1LhA6A* *PLh1sh L. JPhAohAPL k8 HBKBi i?i H2/b iQ +HbbB+H 2H2+i`QK;M2iBbKX MQi?2` +QMb2[m2M+2 Q7 62`KB biiBbiB+b Bb i?i i?2 ;`QmM/ bii2 BM  K2iH- i?2 62`KB b2- Bb }HH2/ rBi?  ?m;2 MmK#2` Q7 2H2+i`QMb- KFBM; Bi Km+? `B+?2` i?M i?2 T?QiQMB+ p+mmKX h?2 b2+QM/ /Bz2`2M+2 Bb i?i 2H2+i`QMb `2 +?`;2/-M/ BMi2`+i i?`Qm;? HQM;@`M;2/ *QmHQK# BMi2`+iBQMX *QK#BM2/ rBi? i?2 7+i i?i 2p2M BM i?2 ;`QmM/ bii2- M 2H2+i`QM TQbb2bb2b  HQi Q7 T22`b iQ BMi2`+i rBi?- *QmHQK# BMi2`+iBQMb H2/ iQ  +QHH2+iBp2- bi`QM;Hv MQM@ HBM2` /vMKB+bX 1H2+i`QMB+ T`QT;iBQM Bb bi`QM;Hv BM~m2M+2/ #v BM2HbiB+ +QHHBbBQMb r?B+? H2/ iQ bi`QM; 2H2+i`QMB+ /2+Q?2`2M+2X hQ bmKK`Bx2 i?Bb Hbi /Bz2`2M+2 BM  HM;m;2 i?i Kv #2 7KBHB` iQ QTiB+BMb,

Ç AM T?QiQM [mMimK QTiB+b- #HHBbiB+ T`QT;iBQM M/ HBM2` QTiB+b +QKTQM2Mib `2 [mBi2 +QKKQMX M 2tT2`BK2MiH M/fQ` 2M;BM22`@ BM; +?HH2M;2 BM [mMimK bii2 KMBTmHiBQM M/ +QMi`QH Bb iQ BM/m+2 bi`QM; MQM@HBM2`BiB2bX

Ç AM 2H2+i`QM [mMimK QTiB+b- *QmHQK# BMi2`+iBQMb im`M KQbi #HHBb@ iB+ [mMimK +QM/m+iQ`b BMiQ MQM@HBM2` QTiB+b +QKTQM2MibX lM/2`@ biM/BM; M/ +QMi`QHBM; i?2b2 MQM@HBM2`BiB2b Bb #Qi?  +?HH2M;2 7Q` 7m`i?2` TTHB+iBQMb M/ #bB+ +QM/2Mb2/ Kii2` T?vbB+b U2H2+i`QMB+ /2+Q?2`2M+2VX

h?2`27Q`2- BM Kv QTBMBQM- Hi?Qm;? i?2`2 r2`2 T`QTQbHb BM i?Bb /B`2+iBQM ("2`iQMB 2i HX- kyyyc AQMB+BQBm 2i HX- kyyRc AQMB+BQBm- kyyec "2`iQMB- kyydc wB#QH/ M/ oQ;H- kyyd)- i?2 KBM Q#D2+iBp2 Q7 2H2+i`QM [mMimK QTiB+b Bb- BM i?2 T`2b2Mi bii2 Q7 i2+?MQHQ;v- MQi iQ T`QpB/2  `2HB#H2 TQr2`7mH THi7Q`K 7Q` [mMimK +QKTmiiBQM Q` +QKKmMB+iBQM HQM; i?2b2 HBM2b #mi iQ T`Q#2 i?2 +QKTH2t- `B+?- KMv@#Q/v /vMKB+b BM/m+2/ #v *QmHQK# BMi2`+iBQMb- mbBM; 2H2K2Mi`v 2t+BiiBQMb bQ i?i r2 +M ;BM M BMbB;?i BMiQ i?2 2K2`;2M+2 Q7 +QHH2+iBp2 [mMimK +Q``2HiBQMb BM i?2 2H2+i`QMB+ ~mB/X M 2H2+i`B+H 2M;BM22` rQmH/ bv i?i 2H2+i`QM [mMimK QTiB+b Bb  THi7Q`K 7Q` 2tTHQ`BM; ?B;?@7`2[m2M+v 2H2+i`QMB+ i`MbTQ`i BM Bib mHiBKi2 `2;BK2- i?i Bb iQ bv i i?2 bBM;H2@2H2+i`QM H2p2H M/ r?2M [mMimK K2+?MB+H 2z2+ib `2 /QKBMMiX Gbi #mi MQi H2bi- QM2 b?QmH/ MQi +QM+Hm/2 7`QK Kv bii2K2Mi i?i i?2 /B{+mHiB2b K2MiBQM2/ BM i?Bb BMi`Q/m+iBQM /Q MQi BKTHv i?i 2H2+i`QM [mMimK QTiB+b Bb B``2H2pMi 7Q` [mMimK i2+?MQHQ;B2bX A K BM/22/ +QMpBM+2/ Q7 i?2 QTTQbBi2, 2p2M /2+Q?2`2M+2 2z2+ib +QmH/ #2 mb2/ 7Q` b2MbBM; TTHB+iBQMb-Kv#2 [mMimK K2i`QHQ;v M/ 2H2+i`QM [mMimK QTiB+b Kv HbQ #2 p2`v `2H2pMi 7Q` #mBH/BM; M2r T?QiQM bQm`+2b M/ /2i2+iQ`b BM i?2 KB+`Qrp2 `2;BK2X M/ r2 b?QmH/ MQi +HQb2 i?2 /QQ` iQ mM7Q`2b22M /2p2HQTK2Mib Q7

ke 1G1*h_PL ZlLhlJ PShA*a RXkXR 2H2+i`QM [mMimK QTiB+b bvbi2Kb BM r?B+? 2H2+i`QMB+ +Q?2`2M+2 rQmH/ #2 T`2b2`p2/ Qp2`  H`;2` /BbiM+2 i?M BM b:fb:H [mMimK >HH bvbi2KbX

RXk "mBH/BM; #HQ+Fb Q7 2H2+i`QM [mMimK QTiB+b

RXkXR h`MbKBbbBQM +?MM2Hb

ZmMimK QTiB+b 2tT2`BK2Mib `2Hv QM i?2 7+i i?i- /m`BM; T?QiQM T`QT@ ;iBQM- BMi2`+iBQM rBi? Kii2` Q++m`b BM  p2`v +QMi`QHH2/ rv- i?`Qm;? H2Mb2b M/ KB``Q`b r?B+? ?p2 #22M bT2+B}+HHv KMm7+im`2/ M/ rBi?BM bKTH2b r?2`2 HB;?ifKii2` BMi2`+iBQM Bb T`Q#2/X AM  [mMimK i`MbTQ`i HM;m;2- [mMimK QTiB+b /2Hb rBi? #HHBbiB+ T?QiQM T`QT;iBQM KQbi Q7 i?2 iBK2- 2t+2Ti BM bT2+B}+HHv 2M;BM22`2/ /2pB+2bX h?Bb 2tTHBMb r?v i?2 +QM+2Tib M/ T`/B;Kb Q7 [mMimK QTiB+b }M/ i?2B` #2bi BM+`MiBQM rBi?BM #HHBbiB+ [mMimK +QM/m+iQ`bX PMHv i?2v T`QpB/2  K2/BmK 7Q` 2H2+i`QMB+ T`QT;iBQM KBMBKBxBM; mMrMi2/ b+ii2`BM; 2p2MibX

 bvbi2K Q7 +?QB+2 7Q` i?Bb `2 k. 2H2+i`QM ;b2b- BM r?B+? 2H2+i`QMb `2 +QM}M2/ BM irQ /BK2MbBQMb r?2`2 i?2v +M T`QT;i2 HKQbi 7`22HvX am+? ;b2b `2 +QKKQMHv 7QmM/ BM ?B;?@7`2[m2M+v i`MbBbiQ`b- 7QmM/ BM KB+`Qrp2 /2pB+2b bm+? b +2HH T?QM2bX h?2v TQbb2bb  ?B;? 2H2+i`QMB+ KQ#BHBiv- BKTHvBM;  HQM; 2HbiB+ K2M 7`22 Ti? 7Q` 2H2+i`QMbX AM i`M@ bBbiQ`b- b r2HH b BM 2tT2`BK2Mi BMpQHpBM; [mHBiv k. 2H2+i`QM ;b- i?2 ;b Bb 7Q`K2/ i  k. b2KB+QM/m+iQ` ?2i2`QDmM+iBQM #2ir22M b: M/ b:HX i i?2 BMi2`7+2- +QM/m+iBQM M/ pH2M+2 #M/b #2M/BM; +`2i2  i`BM;mH` r2HH Q7 TQi2MiBH r?2`2- T`QpB/2/ i?2 i2KT2`im`2 Bb HQr 2MQm;?- 2H2+i`QMb `2 F2Ti BM i?2 HQr2bi 2M2`;v H2p2H Q7 i?2 r2HH- F22TBM; i?2K KtBKHHv +QM}M2/ BM i?2 /B`2+iBQM T2`T2M/B+mH` iQ i?2 BMi2`7+2X 1H2+i`QMb +QK2 7`QK /QMQ` iQKb i?i `2 HQ+i2/ Ryy MK rv 7`QK i?2 2H2+i`QM ;b M/ i?2`27Q`2- i?2 `2bmHiBM; /BbQ`/2`2/ TQi2MiBH Bb bKQQi?2` i?M BM  MQ`KH K2iHX h?Bb H2/b iQ ?B;? KQ#BHBiB2b M/- i +`vQ;2MB+ i2KT2`im`2b- 2H2+i`QMb +M i`p2H KQ`2 i?M ky ƒK rBi?Qmi 2tT2`B2M+BM; Mv 2HbiB+ +QHHBbBQMbkX Ai im`Mb Qmi i?i b+ii2`BM; Bb KQbiHv BM2HbiB+ M/ +QK2b i p2`v HQr i2KT2`im`2b 7`QK 2H2+i`QMf2H2+i`QM BMi2`+iBQMbX

h?2 M2ti `2[mB`2K2Mi 7Q` 2H2+i`QM [mMimK QTiB+b Bb i?2 T`+iB+H kX h?2 2HbiB+ K2M 7`22 Ti?le Bb `2Hi2/ iQ i?2 KQ#BHBiv Q7 i?2 k.1: M/ iQ i?2

2H2+i`QMB+ /2MbBiv, i?2 +QM/m+iBpBiv Bb ;Bp2M #vσ = nee2τe/m∗= neeμ r?2`2 m∗Bb

i?2 2z2+iBp2 Kbb Q7 +QM/m+iBQM 2H2+i`QMb M/ τe i?2 2HbiB+ K2M 7`22 iBK2X h?Bb

H2/b iQle= vFτe= m∗μvF/eX h?2 62`KB p2HQ+Biv HbQ /2T2M/b QM i?2 /2MbBivX lbBM;

μ = Ryy Kk_o−R_b−R_M/v

RXkXR "lAG.AL: "GP*Ea P6 1G1*h_PL ZlLhlJ PShA*a kd #BHBiv iQ +QM/m+i 2H2+i`QMb 7`QK QM2 TQBMi iQ MQi?2` BMbB/2 i?2 ;b HQM; +QMi`QHH2/ QTiB+H Ti?bX ZmMimK rB`2b +QmH/ #2  bQHmiBQM #mi i?2b2 `2 MQiQ`BQmbHv /B{+mHi iQ KMm7+im`2X  KQ`2 T`+iB+H bQHmiBQM +QK2b 7`QK i?2 mb2 Q7 2/;2 +?MM2Hb BM [mMimK >HH bvbi2Kb, i +`vQ;2MB+ i2KT2`im`2b- BM i?2 T`2b2M+2 Q7  bi`QM; 2ti2`MH T2`T2M/B+mH` K;M2iB+ }2H/ UivTB+HHv 72r i2bHbV- i?2 k.1: 2Mi2`b i?2 `2;BK2 Q7 BMi2;2` [mMimK >HH 2z2+i (pQM EHBixBM; 2i HX- RN3y)X AM i?Bb `2;BK2-i?2 #mHF Bb BMbmHiBM; M/ +QM/m+iBQM Bb QMHv TQbbB#H2 HQM; `2;BK2-i?2 2/;2b Q7 i?2 bKTH2X 6m`i?2`KQ`2- +QM/m+iBpBiv Bb [mMiBx2/ HQM; i?2b2 2/;2b-#v BMi2;2` KmHiBTH2b Q7 i?2 BMp2`b2 Q7 pQM EHBixBM; `2bBbiM+2

RK =

h

e2 ≈ k8.3 FǕ. URXRV

h?2 BMi2;2` KmHiBTH2 +Q``2bTQM/b iQ [mMiBx2/ 2H2+i`QMB+ +QM/m+iBQM +?MM2Hb i i?2 2/;2 Q7 i?2 bKTH2b- ν #2BM; i?2 MmK#2` Q7 +?MM2HbX h?2 Q`B;BM Q7 +QM/m+iBQM +?MM2Hb +M #2 mM/2`biQQ/ i?`Qm;? };X RXRX .m2 iQ bi`QM; K;M2iB+ }2H/ "- GM/m H2p2Hb TT2` M/ `2 }HH2/ mT iQ i?2 62`KB H2p2H r?B+?- BM i?2 [mMimK >HH `2;BK2- 7HHb #2ir22M irQ bm+? H2p2HbX h?2v `2 b2T`i2/ #v bvM+?`Qi`QM 2M2`;v e" /m∗X aTBM /2;2M2`+v Bb HB7i2/ #2+mb2 Q7 i?2 K;M2iB+ }2H/ #v w22KM bTHBiiBM;X AM i?2 #mHF Q7 i?2 bKTH2- Bi `2[mB`2b  HQi Q7 2M2`;v iQ ;Q 7`QK QM2 }HH2/ GM/m H2p2H iQ M 2KTiv QM2- KFBM; i?2 #mHF M BMbmHiQ`X i i?2 2/;2- ?Qr2p2`- i?2 +QM}M2K2Mi TQi2MiBH #2M/b i?2 2M2`;v H2p2Hb r?B+? rBHH i?2M +`Qbb 62`KB H2p2H M/ 2t+BiiBQMb +M #2 +`2i2/ i  pMBb?BM; 2M2`;v +Qbi BM i?2 i?2`KQ/vMKB+ HBKBi- i?mb KFBM; i?2 2/;2 +QM/m+iBM;X h?2 MmK#2` Q7 +?MM2Hb Bb 2[mH iQ i?2 MmK#2` Q7 H2p2Hb i?i +`Qbb 62`KB 2M2`;v r?B+? Bb T`2+Bb2Hv i?2 MmK#2` Q7 }HH2/ GM/m H2p2HbX h?Bb MmK#2` Bb HQr2`2/ r?2M i?2 K;M2iB+ }2H/- M/ i?mb i?2 bvM+?`Qi`QM 2M2`;v- Bb `Bb2/X .BbQ`/2` THvb M 2bb2MiBH `QH2 BM i?2 2tBbi2M+2 Q7 bi#H2 THi2mb BM i?2 K;M2iB+ }2H/ (M/Q 2i HX- RN3kc >HT2`BM- RN3k)X i 2p2M ?B;?2` K;M2iB+ }2H/b- QMHv i?2 }`bi GM/m H2p2H Bb T`iHv }HH2/ #mi /m2 iQ *QmHQK# BMi2`+iBQMb- i?2 2H2+i`QM ~mB/ Kv 2Mi2` MQi?2` #mHF@BMbmHiBM; T?b2, i?Bb Bb i?2 7`+iBQMH [mMimK >HH `2;BK2 /Bb+Qp2`2/ #v hbmB 2i HX (RN3k) Ub22 (aiQ`K2`- RNNN) 7Q`  MB+2 `2pB2rV-  p2`v BMi2`2biBM; bi`QM;Hv +Q``2Hi2/ T?b2 BMpQHpBM; +QHH2+iBp2 2t+BiiBQMb rBi? 7`+iBQMH +?`;2 M/ bTBM (Gm;?HBM- RN3j)X .2bTBi2 i?2 BMi2`2bi Q7 i?Bb 2tQiB+ T?b2- r2 b?HH QMHv 7Q+mb QM i?2 BMi2;2` [mMimK >HH `2;BK2 BM i?Bb i?2bBbX

b /pQ+i2/ #v "ɃiiBF2` (RN33)- [mMimK >HH 2/;2 +?MM2Hb THv i?2 `QH2 Q7 2H2+i`QMB+ rp2;mB/2bX h?2v ?p2 b2p2`H BMi2`2biBM; T`QT2`iB2bX

k3 1G1*h_PL ZlLhlJ PShA*a RXkXR y ω −/2 /2 μ ωw ω+ " = B 2z y x 

6B;m`2 RXR, P`B;BM Q7 +QM/m+iBQM +?MM2HbX PM i?2 H27i- r2 +M b22 i?2 GM/m H2p2HbX 6Q` 2+? bTBM- i?2v `2 b2T`i2/ #v i?2 +v+HQi`QM TmHbiBQM ωcX h?2 bTBM /2;2M2`+v Bb HB7i2/ #2+mb2 Q7 i?2 K;M2iB+

}2H/-M/ BMBiBHHv /2;2M2`i2 G}2H/-M/m H2p2Hb `2 b2T`i2/ BM 2M2`;v #v i?2 w22KM 2M2`;v ωZX h?2 H2p2Hb `2 }HH2/ mT iQ +?2KB+H TQi2MiBH μX

i i?2 2/;2 Q7 i?2 bKTH2b- i?2 +QM}M2K2Mi TQi2MiBH #2M/b i?2 H2p2Hb mTr`/bX q?2`2b BM i?2 #mHF- i?2 +QM/m+iBQM Bb T`2p2Mi2/ #2+mb2 Q7 i?2 2M2`;v ;T- QM i?2 2/;2- r2 ?p2  K2iHHB+ #2?pBQ`X PM i?2 `B;?i-r2 +M b22 i?2 2/;2 +?MM2Hb BMbB/2 i?2 bKTH2b +Q``2bTQM/BM; iQ i?2 bBimiBQM /2TB+i2/ QM i?2 H27iX

6B`bi- i?2v `2 +?B`H r?B+? K2Mb i?i 2H2+i`QMB+ 2t+BiiBQMb 2tT2`B2M+2 MQ #+Fb+ii2`BM;X a2+QM/- i?2 MmK#2` Q7 +?MM2Hb +M #2 imM2/ #v +?M;BM; i?2 2ti2`MH K;M2iB+ }2H/- KFBM; Bi  THi7Q`K Q7 +?QB+2 7Q` 2tTHQ`BM; /Bz2`2Mi bBimiBQMbX h?B`/- 7Q` 2t+BiiBQMb M2`#v 62`KB 2M2`;v i?2v TQbb2bb HBM2` /BbT2`bBQM `2HiBQM- r?B+? 2KT?bBx2b i?2 MHQ;v rBi? T?QiQM T`QT;iBQMX AM i?Bb i?2bBb- r2 rBHH 7Q+mb QM HQr@2M2`;v 2t+BiiBQMb- M/ i?mb Hrvb +QMbB/2`  HBM2` /BbT2`bBQM `2HiBQMX 6BMHHv-BMi2`+?MM2H imMM2HHBM; Q7 2H2+i`QMb ?TT2Mb QMHv Qp2` H`;2 T`QT;iBQM H2M;i?b UivTB+HHv b2p2`H ?mM/`2/b Q7 KB+`QK2i2`bV bQ i?i 2/;2 +?MM2Hb +M- BM  }`bi TT`QtBKiBQM- #2 pB2r2/ b BM/2T2M/2Mi +?MM2Hb 7Q` 2H2+i`QMB+ rp2bX hQ #2 KQ`2 bT2+B}+- bBM+2 i ν = 2- i?2 irQ +?MM2Hb `2 bTBM TQH`Bx2/- BMi2`@+?MM2H imMM2HBM; `2[mB`2b  bTBM ~BT r?B+? Bb  ``2 2p2Mi #v bKTH2 /2bB;MX i ν = 3- irQ +?MM2Hb rBHH TQbb2bb i?2 bK2 TQH`BxiBQM #mi i?2 62`KB p2HQ+Biv Bb MQi i?2 bK2 BM i?2b2 irQ +?MM2Hb (EmK/ 2i HX- kyRR)- T`2p2MiBM; imMM2HBM; 2p2Mib 7`QK ?TT2MBM; /m2 iQ 2M2`;v M/ KQK2MimK +QMb2`piBQMX

h?2 MmK#2` Q7 2/;2 +?MM2Hb KB;?i b22K  /2iBH `B;?i MQrX "mi r2 rBHH b22 BM +?Ti2` j i?i Bi THvb  +`m+BH `QH2 BM mM/2`biM/BM; *QmHQK#

RXkXk "lAG.AL: "GP*Ea P6 1G1*h_PL ZlLhlJ PShA*a kN

6B;m`2 RXk, ZmMimK TQBMi +QMi+iX h?2 [mMimK TQBMi +QMi+i Bb +QMbiBimi2/ Q7 irQ iQT ;i2b QM iQT Q7 i?2 ;bX "v TQH`BbBM; i?2 ;i2b M2;iBp2Hv- Bi Bb TQbbB#H2 iQ ;Q 7`QK  bBimiBQM BM r?B+? #Qi? +?MM2Hb ;Q2b +`Qbb i?2 ZS* UQM i?2 H27iV iQ  bBimiBQM BM r?B+? #Qi? +?MM2Hb `2 `2~2+i2/ UQM i?2 `B;?iVX "v imMBM; i?2 pQHi;2- Bi Bb TQbbB#H2 iQ }M/  `2;BK2 BM r?B+? i?2 Qmi2` +?MM2H Bb T`iBHHv `2~2+i2/ M/ T`iBHHv i`MbKBii2/- `2HBbBM; M 2H2+i`QMB+ #2KbTHBii2`X

BMi2`+iBQM 2z2+ib QM 2H2+i`QMbX

RXkXk 1H2+i`QMB+ #2KbTHBii2`

lbBM; [mMimK >HH 2/;2 +?MM2Hb- 2H2+i`QMb +M #2 ;mB/2/ rBi?BM i?2 bKTH2 HQM; QTiB+H Ti?b r?B+? `2 /2}M2/ i?`Qm;? i?2 2M;BM22`BM; Q7 i?2 k.1: 2/;2X AMi2`2biBM;Hv i?2`2 `2 irQ rvb Q7 /QBM; i?BbX h?2 }`bi QM2 Bb #v 2i+?BM; i?2 bKTH2- ;BpBM;  /2}MBiBp2- biiB+ 7Q`K iQ i?2 2/;2 #v +iBM; QM i?2 +QM}M2K2Mi TQi2MiBHX "mi Bi Bb HbQ TQbbB#H2 iQ /2TH2i2 i?2 ;b rBi?Qmi iQm+?BM; i?2 bKTH2- #v //BM;  M2;iBp2Hv +?`;2/ iQT ;i2 r?B+? rBHH `2T2H 2H2+i`QMb HQ+HHvX Aib 2z2+i +M #2 imM2/ #v +?M;BM; i?2 M2;iBp2 /X+X pQHi;2- HHQrBM; iQ b?T2 i?2 k.1: i rBHHX h?Bb 2H2+i`QbiiB+ `2TmHbBQM 2z2+i Bb mb2/ iQ +`2i2  imM#H2 2H2+i`QMB+ #2KbTHBii2`X h?2 T`BM+BTH2- /2TB+i2/ QM };X RXk- Bb iQ T`iBiBQM i?2 ;b rBi?  M2;iBp2Hv +?`;2/ iQT ;i2 i?i rBHH THv i?2 `QH2 Q7  dzb2KB i`MbT`2Mi KB``Q`ǴX am+?  /2pB+2 Bb +HH2/  [mMimK TQBMi +QMi+i UZS*V (pM q22b 2i RN33c pM >Qmi2M 2i RNNkc pM q22b 2i HX-RNNR)X Ai Bb TQbbB#H2 iQ +?M;2 i?2 TQi2MiBH Q7 i?2 ;i2 7`QK  bBimiBQM BM r?B+? 2/;2 +?MM2Hb 7QHHQr i?2 2/;2 Q7 i?2 bKTH2- iQ  bBimiBQM BM r?B+? i?2v `2 +QKTH2i2Hv `2~2+i2/X AM #2ir22M- Bi Bb TQbbB#H2 iQ imM2 i?2 ;i2 pQHi;2 bQ i?i 2H2+i`QMb BM i?2 Qmi2` +?MM2H +M 2Bi?2` #2 iQiHHv `2~2+i2/ Q` i`MbKBii2/ #v imMM2HHBM;X AM i?Bb `2;BK2- Bi Bb TQbbB#H2 iQ imM2 i?2 i`MbKBbbBQM Q7 i?2 ZS* #v +?M;BM; Bib TQi2MiBHX

h?2 ZS* Bb i?mb  +`m+BH iQQH BM 2H2+i`QM [mMimK QTiB+bX Ai THvb i?2 `QH2 Q7 M 2H2+i`QMB+ #2KbTHBii2` M/- b bm+?- rBHH #2 T`2b2Mi BM 2p2`v 2H2+i`QM [mMimK QTiB+b 2tT2`BK2MiX Ai Bb M 2bb2MiBH 2H2K2Mi Q7

jy 1G1*h_PL ZlLhlJ PShA*a RXkXj BMi2`72`QK2i2`b #mi HbQ THvb  F2v `QH2 BM i?2 bQm`+2 mb2/ #v :X 6ĕp2 i G#Q`iQB`2 SB2``2 B;`BMX

RXkXj aQm`+2b

hQ T2`7Q`K 2H2+i`QM [mMimK QTiB+b 2tT2`BK2Mib- r2 M22/ bQm`+2b i?i +M 2KBi bBKTH2 M/ r2HH@+QMi`QHH2/ 2H2+i`QMB+ 2t+BiiBQMbX HH bQm`+2b mb2/ BM 2tT2`BK2Mib `2 T2`BQ/B+ bQm`+2bX h?2 M2+2bbBiv iQ #2 #Qp2 i?2`KH MQBb2 BKTQb2b iQ ;Q iQ ;B;?2`ix 7`2[m2M+B2bX  T`iB+mH`Hv BMi2`2biBM; ivT2 Q7 bQm`+2 `2 bBM;H2@2H2+i`QM bQm`+2b i?i 2KBi 2t+iHv QM2 2H2+i`QM- Q` 2t+iHv QM2 2H2+i`QM M/ QM2 ?QH2 2t+BiiBQM T2` T2`BQ/X *HbbB+H /`BpBM;

h?2 bBKTH2bi 2H2+i`QMB+ bQm`+2 Bb- Q#pBQmbHv- +HbbB+H /`BpBM; BM r?B+? i?2 2H2+i`QMB+ ~mB/ Bb 2t+Bi2/ #v K2Mb Q7  +HbbB+H pQHi;2 ;2M2`iQ`X am+? bQm`+2b `2[mB`2  +QMM2+iBQM #2ir22M i?2 2H2+i`QM ;b M/ i?2 +QtBH +#H2 TQr2`2/ #v  +HbbB+H pQHi;2 ;2M2`iQ`X

 }`bi TQbbB#BHBiv Bb iQ #mBH/ M P?KB+ +QMi+i /B`2+iHv BMiQ i?2 bKTH2X h?2 P?KB+ +QMi+i Bb  #mHF Q7 K2iH BM +QMi+i rBi? i?2 2/;2 +?MM2HX h?2 T`QtBKBiv #2ir22M i?2 2/;2 +?MM2H M/ i?2 +QMi+i KF2b i?2K bi`QM;Hv +QmTH2/- #Qi? #v +T+BiBp2 M/ imMM2HHBM; 2z2+ibX M P?KB+ +QMi+i rBHH i?mb BKTQb2 Bib TQi2MiBH iQ Qmi;QBM; 2H2+i`QMb M/ 2p2M //  /X+X T`iX AM i?2 #b2M+2 Q7 Mv X+X 2t+BiiBQM TTHB2/ iQ Bi-i?2 P?KB+ +QMi+i ;2M2`i2b M 2[mBHB#`BmK /Bbi`B#miBQM 7mM+iBQM i Bib +?2KB+H TQi2MiBH M/ i2KT2`im`2X

 b2+QM/ TQbbB#BHBiv Bb iQ +QMM2+i i?2 +HbbB+H ;2M2`iQ` iQ  iQT ;i2- i?mb +`2iBM;  +T+BiBp2 +QmTHBM; #2ir22M i?2 ;2M2`iQ` M/ i?2 k.1: (JBbBQ`Mv 2i HX- kyRd)X M BKTQ`iMi /Bz2`2M+2 rBi? i?2 P?KB+ +QMi+i Bb i?i Bi Bb QMHv TQbbB#H2 iQ i`Mb72` X+X 2t+BiiBQMb, MQ 2H2+i`QM imMM2HBM; Bb HHQr2/ #2ir22M i?2 +QtBH +#H2 M/ i?2 k.1:X JQ`2Qp2`-/2T2M/BM; QM i?2 ;2QK2i`v- i?2 7`2[m2M+v /2T2M/2M+2 Q7 i?2 `2bTQMb2 Q7 i?2 ~mB/ +M #2 [mBi2 +QKTHB+i2/ (:`2MB2`- kyRR)X LQM2i?2H2bb- B7 i?2 ;2M2`iQ` b2M/b  bBM;H2@iQM2 pQHi;2- i?2 P?KB+ +QMi+i M/ i?2 iQT ;i2 rBHH 2t+Bi2 i?2 ~mB/ BM i?2 bK2 rv- mT iQ  `2MQ`KHBxiBQM Q7 i?2 pQHi;2 KTHBim/2X

G2pBiQM bQm`+2

AM ;2M2`H-  +HbbB+H /`Bp2 ;2M2`i2b  +QMiBMmQmb bi`2K Q7 2t+BiiBQMb r?B+? `2 2H2+i`QMf?QH2 TB`b M/ i?2B` MmK#2` ?b ~m+imiBQMbX :2M2`B@

RXkXj "lAG.AL: "GP*Ea P6 1G1*h_PL ZlLhlJ PShA*a jR +HHv- Bi +MMQi #2  ;QQ/ bBM;H2@2H2+i`QM bQm`+2X

>Qr2p2`- bm+?  bQm`+2 Bb p2`v ~2tB#H2, T`QpB/2/ i?i QM2 TQbb2bb2b  rp27Q`K ;2M2`iQ`- Bi Bb TQbbB#H2 iQ ;2M2`i2  HQi Q7 /Bz2`2Mi M/ BMi2`2biBM; 2H2+i`QMB+ bii2b (:#2HHB M/ _2mH2i- kyRk)X _2K`F#Hv-KQM; HH Q7 i?2K- QM2 H2/b iQ  bi`2K Q7 bBM;H2@2H2+i`QM 2t+BiiBQMb QM iQT Q7 i?2 62`KB b2 b T`2/B+i2/ #v G2pBiQp 2i HX (RNNe)X h?Bb Bb Q#iBM2/ #v /`BpBM; rBi?  GQ`2MixBM +m``2Mi TmHb2 Q7 /m`iBQM τ0 M/ iQiH +?`;2 q = −e bm+? b i?2 p2`;2 +m``2Mi V/(t)/RK Bb ;Bp2M #v

i(t) = _πτq 0

1

1 + (t/τ0)2. URXkV

q?2M TTHvBM;  /`Bp2 rBi? q = −ne- QM2 Q#iBMb M 2t+BiiBQM +HH2/  G2pBiQp [mbBT`iB+H2 Q7 +?`;2−ne- Q` n@G2pBiQMX Ai Bb HbQ TQbbB#H2 iQ ?p2  TQbBiBp2 +?`;2 q = ne- M/ BM i?Bb +b2- i?2 `2bmHiBM; 2t+BiiBQM Bb M 2t+BiiBQM Q7 +?`;2 ne- i?i r2 rBHH +HH ?2`2  n@MiB@G2pBiQMX _2K`F#Hv- i?2 n@G2pBiQM Bb  +QHH2+iBp2 2t+BiiBQM Q#iBM2/ #v //BM; n > 0 2H2+i`QMb BM  aHi2` /2i2`KBMMi QM iQT Q7 i?2 62`KB b2 (E22HBM; 2i HX- kyyec :`2MB2` 2i HX- kyRjc "iiBbi 2i HX- kyR9)X h?2 n@MiB@G2pBiQM +Q``2bTQM/b iQ i?2 `2KQpH Q7  aHi2` /2i2`KBMMi Q7 n 2H2+i`QMb #2HQr i?2 62`KB H2p2HX h?2b2 2t+BiiBQMb `2 iBK2 HQ+HBx2/ `QmM/ i?2 +2Mi2` Q7 i?2 GQ`2MixBM TmHb2X h?2B` 2M2`;v +QMi2Mi Bb  /2+vBM; 2tTQM2MiBH, i?2v `2 HQ+HBx2/ BM 2M2`;v bT+2 +HQb2 iQ i?2 62`KB H2p2HX

1p2M KQ`2 BMi2`2biBM;Hv-  bmK Q7 GQ`2MixBM pQHi;2 TmHb2b Q7 +?`;2 eQ`−e rBHH Hrvb H2/ iQ  bii2 bQH2Hv +QKTQb2/ Q7 2H2+i`QMb M/ ?QH2bX h?Bb HHQrb i?2 +`2iBQM Q7 T2`BQ/B+ bQm`+2b #b2/ QM GQ`2MixBM +m``2Mi TmHb2b r?B+? ?p2 #22M /2KQMbi`i2/ #v i?2 ;`QmT Q7 .X*X :HiiHB BM  k.1: i x2`Q K;M2iB+ }2H/ (.m#QBb 2i HX- kyRj#)X

GM/m [mbB@T`iB+H2 bQm`+2

MQi?2` bQm`+2- i?i /2T`ib 7`QK +HbbB+H +m``2Mi- Bb i?2 QM2 mb2/ #v :X 6ĕp2 i G#Q`iQB`2 SB2``2 B;`BM (6ĕp2 2i HX- kyyd)X Aib BMi2`2bi Bb i?i Bi Bb #H2 iQ 2KBi 2t+BiiBQMb i?i `2 bB;MB}+MiHv rv 7`QK i?2 62`KB bm`7+2X AM i?Bb `2bT2+i- i?Qb2 2t+BiiBQMb `2 +HH2/ GM/m [mbB@T`iB+H2b BM 2H2+i`QM [mMimK QTiB+bX

hQ +?B2p2 i?Bb- r2 M22/  /2pB+2 i?i +M `Bb2 i?2 2M2`;v Q7 2H2+i`QMb #27Q`2 b2M/BM; i?2K BMiQ i?2 +?MM2HX h?Bb /2pB+2 Bb /2TB+i2/ BM };X RXjX Aib KBM +QKTQM2Mi Bb  [mMimK /Qi- r?B+? Bb MQi?BM; KQ`2 i?M M 2H2+i`QMB+ +pBivX AMbB/2 i?Bb /Qi- 2M2`;v H2p2Hb `2 /Bb+`2i2- b2T`i2/ #v M 2M2`;v ;T ΔX PM iQT Q7 i?Bb /Qi i?2`2 Bb  ;i2 r?B+? Bb

mb2/-jk 1G1*h_PL ZlLhlJ PShA*a RXkXj #v +T+BiBp2 +QmTHBM;- iQ b?B7i i?2 2M2`;v Q7 i?2 H2p2Hb BMbB/2 i?2 /QiX P7 +Qm`b2- Bi Bb HbQ M2+2bb`v i?i i?2 /Qi Bb bQK2?Qr +QMM2+i2/ iQ i?2 QmiTmi +?MM2HX h?Bb +QMM2+iBQM Bb K/2 i?`Qm;?  [mMimK TQBMi +QMi+i rBi? imM#H2 i`MbKBbbBQM DX

D

Vg(t)

6B;m`2 RXj, GS bQm`+2X h?Bb bQm`+2 Bb +QKTQb2/ #v M BbHM/ U[mMimK /QiV +QmTH2/ iQ i?2 k.1: i?`Qm;?  ZS*X h?2 /`BpBM; Bb /QM2 i?`Qm;?  ;i2 QM iQT Q7 i?2 /QiX

h?2 QT2`iBM; +v+H2 Bb b?QrM QM };X RX9X h?2 iQT ;i2- /`Bp2M #v  b[m`2 pQHi;2- Bb mb2/ iQ HQr2` M/ `Bb2 i?2 2M2`;v Q7 2H2+i`QMb #`mTiHvX h?Bb b?B7ib  H2p2H BMBiBHHv #2HQr i?2 62`KB bm`7+2 M/ i?mb TQTmHi2/ rBi? M 2H2+i`QM M/ #`BM;b Bi #Qp2 i?2 62`KB H2p2HX h?2M- i?Bb 2H2+i`QM imMM2Hb BMiQ i?2 QmiTmi +?MM2H- HH Qi?2` 2H2+i`QMb #2BM; #HQ+F2/ #v i?2 SmHB T`BM+BTH2X PM+2 i?2 2H2+i`QM ?b 2b+T2/- r2 +M HQr2` i?2 /Qi 2M2`;v H2p2HbX h?2M- M 2H2+i`QM 7`QK i?2 2/;2 +?MM2H +M }HH i?Bb 2KTiv H2p2H- H2/BM; iQ M Qmi;QBM; ?QH2X q?2M /`Bp2M T2`BQ/B+HHv- i?Bb /2pB+2 +`2i2b M 2H2K2Mi`v X+X +m``2Mi r?2`2 QM2 2H2+i`QM Bb 2KBii2/ /m`BM; ?H7  T2`BQ/ M/ QM2 ?QH2 Bb 2KBii2/ 7Q` i?2 M2ti ?H7 T2`BQ/X P7 +Qm`b2- i?Bb B/2HBx2/ pBbBQM rQ`Fb r?2M i?2 2b+T2 iBK2 1/γe Q7 i?2

2H2+i`QM Bb b?Q`i2` i?M i?2 ?H7 T2`BQ/ Q7 i?2 /`BpBM;X h?Bb K2Mb i?i i?2 /Qi Kmbi #2 QT2M 2MQm;?- bQ i?i i?2 2H2+i`QM ?b iBK2 iQ ~v BMiQ i?2 +?MM2HX PM i?2 Qi?2` ?M/- i?2 KQ`2 r2 QT2M i?2 /Qi- i?2 rB/2` i?2 2M2`;v H2p2Hb #2+QK2X aBM+2 Bi Bb BKTQ`iMi iQ ?p2 QMHv QM2 2M2`;v H2p2H i?i ;Q2b 7mHHv #Qp2 M/ #2HQr i?2 62`KB bm`7+2- i?2`2 Bb bQK2 QTiBKH i`MbKBbbBQM +Q2{+B2Mi M/ pQHi;2 KTHBim/2 i r?B+? r2 +M /Q #Qi? 7Q`  ;Bp2M pQHi;2X hQ bmKK`Bx2- r2 2tT2+i iQ `2+Qp2` i?Bb QTiBKH `2;BK2 r?2M r2 ?p2 i?2 7QHHQrBM; ?B2``+?v 7Q` 2M2`;B2b,

hf  γe Δ/2  eV. URXjV

AM i?2 +b2 Q7  b[m`2 /`BpBM;- i?2 +m``2Mi bbQ+Bi2/ iQ  bBM;H2 2H2+i`QM /2+vb 2tTQM2MiBHHv 7i2` i?2 #2;BMMBM; Q7 i?2 2KBbbBQM- #2+mb2

RXkX9 "lAG.AL: "GP*Ea P6 1G1*h_PL ZlLhlJ PShA*a jj URV URV UkV UkV UjV UjV U9V U9V −e Vg(t) t Δ Vg Δ Vg Vg Δ Vg

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