Wannier-Stark states of a quantum particle in 2D lattices

"!#!$ %
'&)(*',+-.('&/0
31 2"4658:7 9<;/=?> 2@BA 9<;C=?D 2EF2G@BHI5JHLKM ;N)O PRQ6SIT:UWV 2GX:2G@BHIY<M 9<Z
[C\^]`_a`b'cdb'ef]`_hg_Likjemlonqp:rseutkb'c`jeuvx\kw vGy\oeujzb'c'j`{|\k}~vb'c`rsn?#Lxxxy\keujzb'c`j'{|\o}Ivb'c`rsnb'c`F\orsi
SL A U)> AoY 8:S Y N=:II:x
¡§¦od||F»zF§z£¼ ¡|©¼F¢~£¢xzI ¶¡ ¢~¥<£¥§¦k¦kC¢o|¥<¤¦R¨¥§ dm¦k ¦R| ¥z¥<<¨" m¦R¡s¦o ©|©s¥<£½ª?¢k¦oF ©¡ss¦o||z¾I«¬·#»<#¦o«¬©s¿ ©¦o©s'©ªxd<®°'©¯Ld `z®°¦o`¯L¦k `±?À¦o°±? d<dd¨s¢k¥z©< ¦k¥'Ï© d¥<x¨s<´¸Á©F¡s¤u¢o#²o³£¦Gz©s¦o¢k< d©© |®£I¥<d<<<©s0¥<¦R|G`¦kd¢k¹¨s¤)ªk ªx¡s<²RG°³º <¿¤J£:z¢k´ |µGF<©Ld |©¢o¦kª ¤
«h ¡?¦k©sm©¦R d|z d|¥<®R¯~Å `¦o|±F¹I° `|z¦RÆs <¾Ç£¢x|©d¥< ¨¡sdGd§££:¿´ <¥z |¢k©½¢k¤C ¡sG° `¦R |¥GÂ<£Ãu<¦o |Ä^G ¢ ¡s¥zd¾I° `¦k| ¢xªk`¦os¡¥#¦R·I¢o¤
ÒH½CA 9 H :T 9 YzA A = ÝGA 9<Z HHM'A zZ ABÕfHI5J5mHLÝ ×JT ÛqM N M A Ôh
"! x
I
9 HM 9 HM !$# 9 HsM 9 HsM !%
Ý*Z AoYzA ' HY'& YzAkM H & æH O â^Q 2 Ò"AkÝGÛIÛ H5 Y×mÔ¼×mA.'-×JT Û H 9kA S M'AkM(5 )ÕfHYÝ S~TÀU
zÝ Z Z AñAoYzM AN M 9 ' ZA ÔA 9 × HMÖ 8:M`Ö A T:Ö:5 UÞ×JST YÛ3 S CSI5JA A ,T =ÝGSI+ AoT:A U¬T¸/S 9 Ý+ S~Ó H 8:S~U TsAoÖ 'ÛI8:YzÔAoAxTMqÝGz×JH~AoS ÕB5J50=- ÕfYzÖ AoH A U A H TsÔZz×J×uS 9<× Z5 M =
Ôq1S õ ×J Ôq9 S HI5 YzY'M'AxAoM A Ö H >TÀ× UÛ:M2 xH 2 S ä A ½8 H M Ö'S ×u5u9oM`S H 5 T Ö H H A A Ts'zZ:×JSS~5ñ9 'YzZ A A S~ 9<A Z U H ×J 9 N A
zÝGZ A?H YzM AkzM SI8 T:5 UM ×JT Ö Û Y'Ax5M`S A M'TsAo YA U Ý CS KIAo5JAxHLM Ý 9 Yz9oHSIMzT M ×JCT A Û USL× Y'A Y9§× Û 5 NñZsÇSIÖSITÖ ÛI5 × 5JA AIUÇ2¬ ÒH Z'Z8 AM
M N M A Ô HIÕ 9 HI5 US~ H Ô M ×JT¬Sq H Ö'×u9oS 5)5 S~`'×u9 A2
È§ÈR3
É 2s5Ï 46ÇË¼Ë½8È 7G:Í 9<;>=Î>Ñ ?A@CBD7"ÑÌÍ6FÐ E
MÜÀÖ AoA5ß9§ A6Y H8::ÔY ×M HA HIMG Õ N M'Y'ZHIA ÕA 9oM'Sá Ao5JK5m5 H Ao'9<YZ ZqA 5)Ö:; M'S AA Y NzM"×JÖ9 HI5mH ÕA ×JTsAx×mTqÓ M"zHIZ Õ A M 'Ö Z Y'A AxM` A 5JT:Ao9 K'A#AoZ 5uHIAoM ÕHIY S N M2zÒ SLzZ ×JA9
Uú9 T ×
S 8×JU× zSIT
ð
IKJML N O J öQ PSRUT # W× V J ;XT ÓCHL2 Ý Tçð§=GS YM*P ß M zS~S~T:TÀTU × MÇAkY`ÕfÚHIà Y <S 'Y Z ; Aú55 SIS~U`zU ×JAk9 AYzM Ö AoH~YÕ× H YzU)Ak=M'H RçTÀS~× T:M 9 'AkZ Mk2A SIÔ å T Ú
ÖSIT:5 ×mU¬'8:zU Z AA H~×mÕ TÀ'U Z ARAñaõ M `<SLz×J9 Õf H Y ^9 ]Â ]_Z= ] T 5 S ÀAk 5J\M [ U c× b^]_Ak]^Y'] A Ts× M '5 Z SIAñUU M ×mAo Y<A Mo2×mT:Ò U ZARõ A
5Ý × ÕfA '×JÔ AJ HIÕ0ß*Me àsJ ÚdMg<SLhf AkDM Jd 2JKÒ L N × M 5mU5 AoÜ T A AGU 5 ÕfA NÇzZ AiA6e Y'JAxM`Y H T:S~T:95 A
UzZ A ×uABU9 YzzÜ:A ZY<S MVM` AxMÕfAkS Ý Ý ×m'ß*Zàs×JT:Ú°59 SIYzUA V SU MAk×JYzT M Û S N×JY'T:ÖA6U ×uH~9oAoÕS aõ ÖNI`Z=kN2 'M Z á ×u9oA S 9o5× SI×m8 Ô¼'M'×JÖÔAñHIH~Y Õ zSI'T:Z9× SIMBAÖ2 H ×JTU 5 NN
5JH Û½Ý AFß*àsÚdM AkMjd JML N H A 5JM'H YzH~Ú
U8:D 9 ABß T S~TT×m×'AoZhY'Ú zá Z 5JH 9<Z M zS~< SLAx M ß á Ú°M zS~R = AxM T N 9oSITúS ×mTs
÷s
k JKL l nm
d JKL N Aoõ Ö × Y P oKT ]
N
D 8M½z× H ÕfHIT 5J5mHLH~Ý#Õ¬Mñß ÕfYzá H Ô Ú°M zS~zZ AkA MU HIARCÜ AT:N ×zM × H 'T Z A ÷sAk§=Ó 8:'ZS~'A × H 9 T H Tsk '×JJKTL 8 l H 8 "MÇp AkKIHI Ú
5
2 Aoõ Ö < # × I J p hf k JML lqsrutv\xw §= Ý Z AoYzA I J yO J # W× V J 2
ÈoÉÊÈzË¼ÌÍÎBÏ½Ð¼ÑÌÈ<ÎBË
Ò AÓ
M AkM¬HIÕ Y 5mA
AoY H H~Ú
ß A TsS~'TZ×uTS 5× AoÖ Y'Ú58:8à S~zM TsS 'Z Y 8:;H Ô Ô M zHSLÛI <AAkSLTM =Aoß*H 8 àsMÚdÜ:M S¸<AoSL5 U Ö:AkS M ;×mzTT ×J9 HLÝ Ý Z:TÞS~×JTØ T ÕfHLSÙHÝ 5m5JSÖ HLUÝ#S^M ×N M US ×uSYz9ºAGM H ÖzZT AA
H~Õ A zZ A5JH T MkÛ~2½ÚdM Ò<S~T:A¼U×JCT AoÛ Û Ö YzHI:5mÛhA Ô HIMÕ H~Õ0AM ×mTM ÛI5JARÚ Ö:S HIY Õ '×u9 5JA#M Ó 8:Y'S~HTs'5m8:A Ô
ÔUS~ 9<AkZ:M½SITS×u9 9<; H Z zZ A Ö:S~×mTÖ T:Ao×mY½T Ná '5mZH 9<Z zHI8:Õ UkN âs~â= 'Z:ÕfH× 5m5JHLÖ ÝA U Ô N
H9 HzZTsAo YY<×M 8Om §'Q × 2WH T å MTæA H~Õ'Z ãCKIA¬A A T 5 S~AoY H Ah=ä0A S~×HIÛ TÀÕGZsUM''S~A × Ak8M = ß zHZ S~A TÖTYz× AoHIY HI= YF5JAã ÔM S~;qÛIS~AkH TÀY'q5 UqSÔqT S~AoAkTÝMkN 2
×JÒ Ô½Z ÖÀA SI89§T:SIÔ Nh '× Z Û 8 H ×JT8 MHIT:z×M'AoT YzK SLz× H T Ô¼H~×JÕ9 zT:Z UA¼8:ß*9§ àsÚdM Ö 8:A Ö 9R Y 8SLÔç''×u×J9 TS
M'A AkÔ¼ ×J9 HA T:UÀ8:H 9§ HIYqM A 8:Ö AkY'5 SLYz'A"'×uH~9 Õ:A ß*O LàsQ ÚdM A T:U AAkUæM S 5JH T Û HLÝ 'Z AkHIMYzA zH ×J9kS Ú5
ÔUM N H M T SLA 5 ÔNqSI2 S 9k9 á A 8AkÖM ×uA U'UqZ AkM '= Z:T:ß*S~S~'8às'ÚdZ M A <NqSL S AkYzM¼A ÝG'Z AoAÇYzzASLè' éRYzêoA ëL9 =Iì/A S~TsíLTÀ ìCU5 î<N TéM ê§zïd8:íLU×mïdé§× Aê× UÙH~9 Õ ×JTÙzÔ Z SA
HM N zZM AoA5 Y Ô HÓ HI8:Õ ÛS 9M ×uHI 5 ß*UH T àsS~ARÚ° Ú M H UÔ ×JÔ M AxA M ×mT TM × SIY'H AæT T:S HY'5)AxÖMM`'NH ×u9oM S A 5Ô 5 S~MoAk`2M z×J9 A 2 AIO2 ðLQGS~A TÀU M M'H Ô:5mAA
M zSLAD Ak'M Z Ý=Ak×J8 TòY'A Z 'Z MA 'Z5J5 AkzHIS~Yz A '×u9oY'S S H^5 õ S~TÀS 5 N A M T:× S~M¬TÀ H~9 ÕMYz=ñAo5× SL H A UóÔ YÖ <YzSHIM z5JS A Ô AkMM
' HZ T NARÚ° SIT:U)="8'× Û 8ÀZsS ÚNô×JT:S~UÖ:×JTÖ Û =S~×JÔqT:U SL M ×JU Ô¼× 5 S N Y S~zÖS~Ö '× YzH^T:õ S ×mÔqN S~'× <H SLT M 2
áM'A A Ô¼N H×J9 T:H UT:U'8:Z 9§Ah HIH Y T SIARÚ°T: U S~T:H ÖUæz×JS~9kÖS Ö 5Y'5 SLH^õ''×J×uÔ¼9 AxS~M'× ÝGH TAoYz= A#ß*M zà8:Ú°UM × zA S~U¼ Ak×JMT ×JY'TöA 9 zA Z TsA
Ö:H~ÕCSIÖE?AkARYzÕM 2 OO ù^÷LQQS~ M'TÀAoUqA SO øx5JQ M'H NE#8 ARMÕ2×JT O ^Û Q zÕfZ HAY MU 9oA SLz'SI × Ak5uM Y ×JT2Û Ò ÔqZ× SLM SIY ×Öõ Ö SIYzHÖSIÖ 9<YzZFH SS9<9 Z Ú
'S~8ÀTÀS U 5J5 M N 8M'Ö:HIÖ 5J5KI× AkAkMMAR'Zõ AZ:S~H 8 T M ARzÚ × UK×mA Ô ×mTA TÕfHM Y × HÔqTÀS~S '5s× Hß TFSIS TCTH × 8AoY'Údà <S Yß*;àÖ Ú°Y'M HzS~5m A AkÔ Mk2
å A Y'AxMÀ 5JA AoY?ÝGABARõ A A A H H~ÕE?ARÕ2
ÝGTúH A#''Ü ZZ T:A U9kÖ S 'ZM'A?A H~9 TÕH Ô¼ïûÖ ëL'5müÀoýLõ þ ÿqM Ö éoAì9Rê§ þYëL 8ì/Ô T:í UH~5 SL'Õ 'Z '×u 9 Ô Akß*Mk2 'àsZ> Ú°HM YUzS~' Z AxA#M Ü:SIYzT:M U O zùSI×mT:=/Ô SkA ÚQ
5 N A × M?ÛIA T AoY S 5)M Y 8:9§z8 Y'A2
> å`4½2 I H A Tz×JS 5/A T AkY'Û N I ÕfHY SGSIT:U
AÒ × AoZÛIY A8 TH M Õ= 8T:ß9R'á × HÚ°MT zhfS~H~ Õ AkM 'Z 9oA S~T2 AoKHIC 5 A 8H 'SA× H 5 T AkY H TÀÖ SLAo'Y× KISLA½Ak HI5 A N Y¼ÛHLAU KIARAkÜK Y T 5 A zUZ AkA MS H~á M Õ 5JH zZ9<Z AA
ÖAoKHI×5 8U'× H T H Ö AoY SL R,HIY6O ~ÕfHIQ Y ÔçU Ao ÛA zT Z:AkSLY S~6 'A½Z S~TÀ× U M T I JS 8 o: I zJ Z 2
H~HDÕ U:Ao'UYZ ×zA¼× HHIM T:YN SM M 5J5A NIÔúH =s=C5 HÇzZ A AÇTCMA A 8 × ÛYzAA T 'KHZ:S S~5 8 Añk 5 MJMAxLÓ l A 8:SLz × HS TY'AÕfY'HIAxAæY M`H'M Z T:A½S~A T:Ao9 KÜ A#HI5M 8<'SLH× H AkT MÚ
Z'ZÀÖAkSLY Ô¼qSL×m' 'Z ×uS~A¬T Y'AxCZ Ó H 8:88 × T:YzU AU:U S Y ÀNH 9SI8H 9kT:T:9 UUS Ô½×m'Y ×NæÖ H T × 9 2?H Z TÀå UU ×Ý z× SH N MT M ÖzZ YzS HLYzKIA A ThÖ ×mÔ¼×J9RT Ö × HE?93M'AARUÕT 2 O¿T kN Q
'Ô Z H A Ô AY Ts8zT:89oÔ S~'× YzH A T Ö YzH~AkÕ M'A 'TsZ <A¬SL'Ak× KIH HIT 5 2 8z× H T H Ö AkY S~ HIY ÔqSL Y × õ ×JTæzZ A
ßÞA Y'H AkA Ý A ÝHIÚ A M H 5 M'AI2 M A Ú
'S × M H T 'Z A UºA Ö T S H CT HL9 Ú KIZ A AkU = Y ßÔ¼×m×má ''×uZºÚdS~M T z'SLZ A × AkÛIM A T ×JT3M zSÞSLU ×mAkÔ M? HIT 5Õ SL× '`ZzT:×JA9 S A AkKI9o9oHSSI5 T 8zC× H A DT U H ARÖÔÜ AoT Y A T S U Ú
ÝGHYH HLUK× Aob Y#AoYzH A T TsA "á á 5J5JH H 9<9<Z¬Z¼ÖÖ AoAkYY ×× HH UU M2S å T¬M'M'HzZ9R×uA SL A U ÝÖ Yz×zHIZq:5m'A Z Ô A zÝZ AoH YzA9 H S Ô Y'A Ú
Öß H áT A Ú°MTz S~M AxHIM Õ z9oZ S~AT M ÀzAS~'×J×uTs9 Ü:YzH AoU5 U 8:2¬9 A Ò U Z H AkTY'Ao5 NÕfHIYz×mTæA z'Z Z A A T 9oH S 'M'× AúH T H~Õ HIÕ9 H zÔ Z A Ú
ÔH~Õ A 'T Z M AB8 Y Ü:SLAo 5 A U Ö Ao Y × H U S MYz=A × 9 2 AIH 2 Ö =Y ×mT×JÔ 'A Z A×JTs9o SAkÛIMÀAoY<H~M Õ § + Y S~'× H T:S 05 - U× Y'A 9§'× H T
R
R
øI
v R
v ]
R
YzHLK ×JU A U¸9 H T:U×m'× H T øI¼× MM SLz× M`Ü:A U)= ÝA U ARÜ T A ß á Ú
M AzSL AkAoM KSHIM5 'Z H A T H H T AoÚ Y Z AoY HIÔ¼Y½×HLz×JKISIAoT Y A × ÛA A T HÕ 8TÀ9§'H × H T Mh5JH~H Õ zZ AAoY M N H M`Ú
Ô 8 '× T v Ö hf SLR 2 M ×mT Û ''Z Z A½9 @BÔ¼Y SIÔ Ô AkTÙYzM`Ú á V A T:9<ZöT AoÖCAoYz×ÛIAkU Y
'ZY S~AúT MMzÕfHS~Y'Ôq×u9qSL zAk× HY Ô T= Ý ×JTsZ ×uH 9<ZÇ'Z × MAú KI8 A M 9§ 'HZ Y A?Ö Û H S~ 8 A ÛITsAG'×uÝ S 5 Z=×J9<'Z½Z Aú Y AoSIKT HIM5 Õf8H'Y × Ô H T M
H Ö AoY SL HIY 9oSIT CA Ö YzAkM'A Ts A U ×JTú'Z ABÕfHY Ô
p'
& p ) ( ù
¸A q A q Ao õ Ö # ×
hf "% !$# U
+ , * # R p + * # R p & p ,
! ~
Ý ZAF×u9<ARZ õ YzH AoKAA S 5JM M ×Û M Y AS~T H M'5 SLAk'M × H TÀS 5AF×mHIT Y K S AoY Y ×uS~T:Û 9 A 2 '5Z A AoY Z:S~ KHLAoK5 AoY
'ÝGZ A 9oS~T Ö YzH T <SLTs A z× Z A Th9 U HHT Y U ×JT:S~ 'Ak×JÔMM'H 'ZÀU SL×JzT Z A U× D Y'A 9§z× H T:T SLzHI× Õ zZ NIA =
Ü:Ak5 Uh9 H ×mTÀ9R×uU AkM"Ý ×m'Zú'Z A - Ú S õ × M
!
!#. - ! - ]

v
v ÒY SI T MÕfHY Ôq S~'×v H T Y'×mTsA Y'H UAk8:M 9 AkM AqSM T AoAÇÝ HIÕ 5 S~`'A¼×u9 AHIY Ö Û AoY × H 5.U / Y 5mÚ
5J5Y H P 8×JT A H TAoYA1H 0 M S~ T: U AkÝ U 8:5J'H9 ×JÔ Ax'MoZ 2 D 2A c×M'43 M'>H 9R ×uSL zA Z U Ý ×m '× Zq ×mT:zvSZ A hf á T Ao× Ý
ÝSLZM`z×u×J9<9 ZqÛ × Ö* 2M - 0 × z×mÔUF# ×Akhf M6S#M Z T 6 HIY - Aoá Y 'Z:9<SIZB* T¼- '×J'Ô Z A## HY h7f × Û6 ×J T:6 S !85 - á 5JH 9<A Z½ ' ×JÔ Añ5A AoKR¼?H~Ú 2=
5 8H'H×J× TYH T ,H Ö AoAkY M SL H× Y0M HLKAoABY Õf'HZ Y SIA T:T U AoÝ , á 5JH 9<ZF× '×JÔ 9A 3 =×m'TñZ 'Z 'AG×JÔ Y'H zS~ A U
9 U×JT:S~ Z:S 'Z Ô
p'
& - p ( â
- 3 WA q : <; v 3 AR õ Ö # ×
hf % !+= U

ks
+, * - # R p , * - - "! - ]
Ò Z A T=Ö Y'AxMÀ Tz×mT Û zZ AFÝ S KAÕ 8:T:9§z× H ThS M
D
k - W?A >A@ ;CB; mD D ;)G -IH JK-ML G -IH JK-NL / A : ;O B; v 3 ;FE
I^
ÝGABÛIA GzZ A ÔqSL Y × õ AkÓ 8:S~'× P H T
D P
qS v\xw D
x
m P -CQ ; @ ;CR ; ; ¸A
; ! = E
;
E
D P ;CR
ÝHIÕ Z AoYzAFTA H -AAoQ ; @Y ; HIU Y A T H AxM 'Z A o - Ú U A Ö A T:U A TsÇÔqSL Y × õAk5mA Ô A Ts M
zZ Ö DSL P âs§
kðs
-AQ ; @ ;R ; G J
-IH A q @ ; O B ; - 3 A @ ; O B ; H UV-WL ]
à ×mÔ¼Y × 5 S Y H Y õ'Z A 9oS M'A =I'Z A Y 8T:9kSL'× H T HIÕ 'Z A ×mT Ü T× A 8T:× Ú
<S NÔqSL × xðR=
H X - H H Y - H &[Z]\_^ H X - H H Y - H &a`b\c^ o÷
Ý Z:×J9<Zæ5J× 5M Ö YzAkM 8HÔ M'AkA M Uö×mATö'H Z A Ú TAk8Y Ô AoY ×J9kS 5 À9oH S 5 9o8 5 S~Y '× H T H M =S~8 H H~Ú
ÔqS 5mHS~T'×uÛ 9oS 'Z NA ×mÔ¼- Ú UÖ × Y'A 9§'× zH Z T 2 T ÒT Y 8:Z T:9oS~Ô¼'××mH'×uT S~TH~Õ zZ 8A T:ÔqUS~S YN× õB9 HLT:KIUAkY×z'× Z T A
×J9 T:H U T:AoUõ ×z×X H - Tº= S Y 5mH - T ÛU H'AkZ M A T !dH - ½Ú U9<× YzZ:A SI9RT 'ÛI× H A T z2 Z A Ò Z Z AoA Y TºÔ½'×m'Z ×uAqS~T A × ÀÛAH T 8K T:S U5 S8 Y AkN M
I HI zS~×JT A U NT8Ô AoY ×u9oS 5 U×uS ÛH T:S 5 × kS~'× H T HIÕ zZ A Y 8T:9oS~ A U
ÔqS~å TqY × zõ Z A 9 HI YzY Y'SIAxT M MÖ ÕfHHTÀY Ô UqA U½H 9 zHZ HIAY ÓU8:×mTÀS~SLTs 'Ax8:M Ô =I'Z YzAkA M'8H T:T×mSIGT:9 9 AoAx5JMo5À2 Ý ×m'Z S Y'A S
/ Q P0 0 9 H TszSI×mT 5
AzYzZAo5 Ô A M 8 ÖÖ HIYY M × M½KIA HLMÝ Y 0T Uß M'× 5b á AoYzÚdHAM zTsSL M AkHIM 8MkÕ 2Þ:5 S~Ò A `Z '×uAú9 AxY'M MH8 =/S~5 S~TÀA Ù z×JA95 SIAkM9<Z S YzHIA A Õ
SÕfIAkT:YUúSL 9 NöHIU YzÔÇYzAkNñ8 M 5Ö Ö zH×mÖ ×JT:Ô½5mUAx×mMq×J'T × ÛIH~Õ5 N¼/ 'R Z SIAFeT 0A 2 TS~'Ao>:× Yz8Û T ×Y AkzMGZ AoHIY Õ zÔ Z'Z HIAYz8A× Y#T ="ß S M½zá SLÕ Ú°8M TÀzU S~9§'SLAk× H `M zT U×J9 × HIÕ ÚÕ =
zg Z AÇ Ó 8: S M ^×J]^Ô ]_H] Ô× M A 'TsZ zA 8Ôh+¿á =:5mH z9<Z Z AÇ A TSIT:AoYzU Û - × Ak×JT:M U I Aoõ SIIT:f TUQ > R h o - o _- ]^ ]_]H Z 0 AkY'× AM
zÕfYzZ H AñÔ M 8 'Z 5A#SLÕ'SI'×u9§9 A zZ:×JT:SLU GAoSFõ #9<Z:U S~H T TÛA"H H~Õ U oA Ö- A ×JT:T U XH ÓCTa2 o k- ð2#Ò9kS~ZT × M#CÕfA HI5J9 5mHHLÔÝ#MÚ
Ö!8-A Ú T U M AoS~ÛI YzA AoU AñH~NÕÕfM YzZAoA× Õ U '×JH T Ô Û zzZZ AA 'á ×JÔ 5JH A½9<Z HIY '× Z Û Ao×JTHIYz×mA T Ô,X9kÓCS~2 T âC A 2 S~> ÖHIÖ Y 5 × 'A Z U)A =
SÛIT:A T UFAkzY Z S Ao5m5 YzNqARÕfT HH Y'AMT I Akf Q Y'> HR oS~Ô¼- Ö o 5- ×m'08:× MU jASiÖ AoI Y f × H 2UÒ ×u9 ZÕ 8 8:M T:=:9§S zM'× HM 8T Ô¼H~Õ×mT o Û - S Ý Y×zS Z Ú
z×JTs× H T:AkY'S K 5 S U5 × Y'/ A R¼9§'=G× H 'TZ A H~Õ M <'SLZ 'A×u9 Ü:Ü:Ak5 AkU)5 U¸=/×JT ×JT:AUS8:9<9 Z AkM1Õ 80 T:U SIÔ A Ts zS 5 A T ×JUAoYzA Û T N Ú
zà ×J×m9kÔÇS 58 5 M <8 S~T Ú°Ak H S~8 TÀM'U5 NIM == zM`Z A AñÖ:S M Yc× SLA A H~U Õ zZ N A 'áZ AqY × 5JA 5mTH Ao8Yz×JTÛ N¬ H ×JTsT A AoYz× KMS Y'5 A U/ 8:R 9 eA 0U 2
Õf HNÙY SSúHIYÕ SI 9§ 5 HISLY ''Õf0×uH9 2óYA YSÒ b ZA × 9RM A Y'5 UAxM 8 Nú5 A SñYzAkzM'×mÔA Ô Ao:Ú Ö 5mAxAoÛ M Y × H 'A ZUA¬×u9BH Ü:Ö T AoAoA¬5Y 'HI8 YzzS~SL×Jz2ÇT × HA å UT
zCOmkZ A LAk Q ÝGM'A AoA 9oTWS 'Z:MÀx'S~ZMl A k á Ö 5mY'H SúHL9<K Z ×uU ÖA U AoS~Y '×`H Z'×uU*A 9 9 S~H ×JT:TT:U¸US×zzZ Ôq× H A TS Ö HIT AoÕ Y ×'9 H ×uH 9 U Ô H~A Õ T UæM '8 Z OmY AIS R UQ × Y 5 ×× KTN Ú
×JT T A Ûº'×u9 ÕfHG Y 98 A¬õ HI'Y Z Y''H Z 8 A Û Z39 H ShT:U8×mT'× ×mH ÇT 9 AoH~5JÕ 5 = + YY'S~AxM'Ö× H A T:9RS '×5 KI×mAkN 5 N-=0ÕfHI× MBY Õ '8 Z 5mAÜ:5m5JÔqA U S Û~k Ú
zM Z Y A 8:9R Ó'8 8:YzS AIM 2 × A T AoYzÛ N M Ö A 9R Y 8Ô HIÕ zZ A M N M A Ô Z:S M S M ×JÔ¼× 5 S Y
ßA H 5 A MM'A H Ý Y'A Y H AoY Û A
T8Ô AkY 9 ×u9oS TÀ5 9 Ö 8:Y'U H 9 A'Z:U×8 YzAI2 9R'× D T5 zZ H ×m8'ZqÛ ZóS 'Z ÔqA3S YzA ;U8:9 9 TÀA Uó9 á TY ×JT× 5J5JH '8Z ×JT
& - p
ð
>M 8 å`4½ 2 SIT: U A HT M Ao×zYz× Û H N T ×JH~TsÕ 'AkZ Y'KA"S ÛI5 YzYH PS8:RT:U ß á Ú° S~ T:q U v Y' A Ö S A M SLS A Õ U 8T: NB9R'z× HZ T A
H~Õ 'Z AB] ÜÀAo5 UU× YziA 9§z× H T Ao ÛAoS Y<Y M 9§< S~T 2 Ö:S Y SIÔ A AkYzMjhf =
R
= =×JTs
&¸x
H T A SIÖÖ YzH SI9<Z¸U AxM 9 Y × ÀA UWS CHLKIA × M zZ A Ô HM 9 H T M × M A Tsk=
ÝGHA¬ÕfH H 8TÀAkYUW×úHYÔ Ý HIYzA H 9 H T KYzA AoT5 × A Tsq YH UYz×uH S ÛH T:H S 5 × H~AÕ zZ A A AkHsKIHHIY 5 8 ÚÚ
'T:× S~ T AI2Ö å T H S~z Z AoYGÝ×zHZ Y U 8M G=×JÖ T HIY ×mÔ¼U Ak×JY T:S HñN Ü TÀU<SLz'Z × AT ß á zÚdZM Ö A 9 9R Y 8UÔh× =
ÝGAM'Y HI5Jõ KIA H zZ AM AY × ÛA T A K S H5 8 A#AkÓ AÇ8:S~ '× MH T MH~ xÕ ÝAÇ×mH 'ZqAkzY Z A HI Y Y 8: T:9o2S~ A U M
ÔqS SLYzAkM ×8 5 9 HIT Õ ' Z 8:A 9RU ×uS U ÛH T¬T:S z5 Z× kS~'× SH T× = H T A 'Z H zSIÖ ×mT M½S~A × ÛA T ùK S 5 8 D AkM
I f o o Ý ×zZ Ó 8:S M ×JÔ H Ô A Ts'8Ô o o U ARÜ T A U
×JS TæYzA zZ 8A¬T× HIÕfHIY Y× ÛÔ ×JT:SS 55JH áT ÛY × 5Jz5mHZ 8A ×mT U× Y' AH 9§T zAI× H2 T á HIA Õ 9oS~zZ8 AhMÀ Ü:'AoZ 5 U)A¬= ß I f á Ú°o SI T:o U~M
× M S3S~Ö TÀAoUöY × H U ×u9 YzÕAk8M TÀÖ 9§A '9§× zH × KT Ao5 H~N Õ2ÇCÒ H Z 'ZnA½A o T AoYzSIÛ T:× AkU M6oH zÝSI×m×T zAZ U Ö ×JAkTÞY × 'H ZU × MM
Ý S^Nö9oSITæzZ AQ > T R C- A 8 - M'A UÙ H 9 H T M Y 8À9§q'Z A 9 H Ô½Ö 5mA A ß á Ú
ÝGM Ö AoA AY<9§M Ö YzH Y Ak8M'Ô A YzTsAo5 I fY'AxH M 8 o 5 I Mo HI Õ §= S h T 8Ô Õf HAo^YY ]_×J9k]^S]H5A 0I9o2 S Ao5 å 9oY T38 H 5 'S~Z '×A H T T H ARHIõ AÕ zM`Z A A 9§5 'U× H× M`T Ú
ÖS~TÀU × Ô T H U AoY SLSLz × A T K S 5 f 8 Ako Mú HIo Õ ' Z AºM zzSLZ z×J9 Ö Ü:Ak× 5 nU U×J9ñÖ TsR z×JS R IR =
H H R X 0p2
E
È§È§ÈRÉ Ë¼FÐ E 7GÍÈ§>Ñ 6=öÍ 7@ÀFÐ =Ì @
å TÞ å 'ZM × × M M ×mT9oAkSMM M`YA 8:=9R ' × KIA ß Hß á CÚdM Ao<Ú°Û SLA ×J T AxAkM Ý Y'Û S×m'YzAxZ¼AÇM 'Û Z × YzKAA A M'T A Ö: M S N¬Y S zZ 5mAhA A MÖ9kSYzHM'A U8:HI9§Õ H~ Õ 2
A T ÛBAoYzH Û z× SLAk5 Mk2å T S~A?T:ÝÛIUöYzZ:H SL ÕfHI5J 5mHLá Ý#MGT 2 ÝGAY<Yz× MAkM ÝGYS ×JA 9R H H 8 8 M Y<ñMÀkzAo5mZ YKAkM A"8H ÔY'A SIT:5 S 5 N Y Ú
×JH~T Õ 'Z AT MNFÖ 'A Z 9R Y 8Ô 8O T:U% *SIE?T:U A I >% × 2 9 T ×uU zZ S Ö:S
YÝU× 'å A 9§ÝzA × HS AoTMYzMñMÛ Z HIHLÕ MÝ 'TZ 5 A×JT¬Ü:AkzÛ Z5 UA 'Ö Z YzeAúAo0K 2 ÛI× HYzH 8 M 8MTÀMÙ A 9§Aoßzõ × H á T¬Ú°M '8 Z:5JA S~ SIÕfT:HÛ:YU 2 MÇY SLY'MzA × Ö H HLAT:Ý#SLS M 5
''ZZ ×mAñA'Z SIYzAoT 5 TÛISL5J'A × KIAN Ö Ö HsS M Y ×m×m`9§'z<× ×mHS~T TT M / H~ R Õ zZ ÕfHIAkiYM'D Añhf M 8S~T SIS~T:TÀS~UUÔ¼M R Ö S M =S > Õ× ] 8 T: 9§ z× H Z 2T ßH~A Õ
YzA 9oS 5m5 'Z:S~×mT¼zZ A 9 H T M ×uU AoYzA U¼9oS MÀ?HIÕ S MÀ Ö:S Y S 5JA Ö H A Tz×JS 5
>U× å`M 4½Ö Ak2 Yzð:M × H T E?AY'AkS 55 S~' ×5JHARÕT §qI fS~T: o UÙ ×JÔqo S Û Õf×JHIT:Y S Y 'N Z A½ Y Û× ÛY'HZs8<ÇT:U Ö:S ß Y áM¼Ú°H~M ÕzS~' Z AkAM
SI T:] U¸= U] øc× b=AkS~Y'T:A UæT K S ÕfYz5 8 H AkÔ MqHIH Õ ÖÞzZ H¬A CÖ HH ' A H TsÔq'×uS 25 Ö:Ò S Z Y ASIÔ M N A M AkA Y ÔçaÖ: S Y SÚ =
Ô A AoY<M S YzAFhf =MR R ] :=S~T:U/o 2
zZ AFÒ Z ASIT:U ÔqM S~Z:×JTS KA U×c b AoAkYzH½Y'A ÝT:9 ×uUA zZ CAÕfH ÝGY AoS~A TT N MÀ Ö:S Y S P : 5mA 2 S~T:U T H T Ú
M'A ÖÀÜ S Y S Aú5JÝA Ö H A Ts'×uS 5uM A× M5 'Z:AkS~Y q'Z M'A AI2ºM 8 Ò S~M T:U M M*5mO 5 % Q M > R Y 0½A Z:S KA
S>× Û:T2 ð×m S Ý ×JUZ'×uZö9<Z ×JT3M Z zHLZ Ý#M 'SLZ 'A U× M 9kÖ S AkYzM × H T YzZAo× 5 SLz× × H ×T 8O % o SL oU N
s ] Õf=HY ] zøZ =A SIÖ T:H U A T2öz×JS Ò 5 Z A I UÝ × Y'×mA '9§Z z× H ÕfT YzH ÔH~Õ 'H Z ÖA Ü:HÇAk5CUæH '× M H Ô P ÷:==
2 AI2 2WÒ A 5 A¬HIÕ A M Ü:Ak5
z××JT Z A½>K × SÛ:5 2 8 Añ2 H~Õ å 'hZ ×A¼MhM M'9kAoS ZA 5JTA U S~×mTó Ô¼5 ÖSI>T:× ×mÛÀ9<'8:;h2 U ð 9 H SsT ¼M <'S~zZÀZTsSL6úS 'Y'<Z ASLAºz'×JZ9 ß A¼á M SIÚUÙÔ S~A S~T:TÀUS U MM
Û S~A×JT S M Ü T:AoY<×M AÇH Ý ×uY'UAk'5 ZS H M O × M ×m TÀ9 Yz A S M'A U 2ßA ÕfHS Y 5JM'H 9oAoS Yz5 A9o8 5 SL A U Ú
zÛZ 5mAxM U× Ö S Y 9R× zS~T T S~ '§× = Ý T ×m'Z % Yo &Ùo ù 2å Ý S MÕfH 8UTÀ× b U½'Z:TsS~'S~Z T A
Ý M i$O iFO Y Y'A 5J5 M 5m5JAoY
zÒ Z:S~ZS~T:8 T U M ='Z Õf×uAHIUY zÔ Z Ö A Y SIST 9§zA ×J%T 9kSAoYz5 Û ÖN 8 M'Y A Ö% Ö:HS M'Y A S~= '0×H SH T T A C9oNA SI ÖTÙÝG×J9kAoTS A AoT ÛINBz5JAÔÇZ 9RAB8:M9<'8Z Z :A Ô¼S~ S T:S~U TÀMoU 2
Ý ×uAUzZ M`ÕfA HIY z Z AYz A S 5 Ö:Ý S Y AkHIY'AÕ zZ AA?MÝ Ö A 9§ Y 8:HIÔ Õ 2 AGD#ß T Aoõ Ú° 9 A Öz× MH T¼× MÚ
zÖ Z Y'H^õ 9o×JS Ô¼S~ Ak5 Nú9 H P×JT:9o ×JU AxZM?Ý ×m''Z¬Z zZ Añ×uUÝ z×uZ U'Z 'H~Z Õ 'Z A á á 5JH S~9<TÀZ U S~S~TÀÖ U
×JHIT¬Õ 'zZ Z AA S SI:T:M'AU T:M 9 'AñZ AH~NÕ zÝGZ AoA¼YzAM <ÕfSLH '8×u9T:UqÜÀAo 5HñU Õf2 HY Ô AkSÛI5JAM 9RY '8À×JT 9§Û'8 zY'Z ABAÇM ×JÝ Ô¼×uU× 5 zS Z Y
H ß'Z:A S~ Ö YzM H Z 9 HLAoÝ A TúU ×JÝ Th×z>Z¬× Û:'2Z A 2 S~T:S 5 N M × M?HIÕ 'Z A U A 9kS^N Y SL AñH~Õ 'Z A
ßHIÕ á ÚdM A <SL H AkM = Ý 5mAoZ:õ×J9<A ZæAo× YzM Û U A AoY Ô¼ ×mT A# U å N I'Z A 2Þ×mÔqå S Û ×JT:A S Y NÞM`ÖÀAS HIY Õ
SHM'K 'A ×Z ÖÀH 8S YM`9 S5 N Ô½5JÛ A Ö × KIÖ A H T A TsT Nq'×uzS Z 5 AFN'= AkZ VÓ A 8:% S~U 'A × Ö H A T T: U A Ô T:9 A % V % T3'V Z % R 9oS q× M
xø
V % R V %- Rº9 HsM V %- R M ×mT Ý HIZ Y AoYzA AV %- R Y - M zA S~AoTÀY<U M MÕfHM'AY 'Z Ahf Ý ×uU'Z HIÕ ß* àsÚYz] AkM'H TÀS~T:9 AkMkA 2
>U A Ö A 'T:Z U A T:Ö:9 S A S~ Ô ^øI# × MBM 8Z HLÝ U T ×JTÞ>× ÛÀ¬2 S~÷ T: U N¬R S M'HI5 ×J U 5 ×JT AII2FFÒ 'ZZ A
ÔqS õ ×JÔñ8:Ô S YzH 8TÀU P HY × Û ×JT:S~ AkMÇÕfYzH Ô SÖ A SI; Ú°5 ×m; A
÷
CH~Õ A Z:S K × YzHIAkY?M'H H~Õ V %- R - ?SIAkT:5 UhÛ × M#ARõ 2 Ö 5 S~×JT A U N zZ A ÖZ A T H Ô A T H T
>H~Õ z ZHIYAÇÜ:SAo5T UhTÀH TS~'ZÚdTsMÀ A Ö:zU 8S A TY 9oSTS^N 5JA×JY T S~Ö AH O¿xU A Q ATsÖ '×uA S T:5 U S~M6T:H U T¬Y 'SLZ zA¼× H Ó T:8:S S5 M U×JÔ × Y'HA 9§Ô '×A H T T Ú
'U 8:A ÔÖ×J29R > A HUY ×JT¬'Z >A × Ö:ÛÀS2 Y ð ' ×u 9R8 25 S ßY 9kAFS ÝM'A H 8 5 U 5 P×m; A ÷½ 'H ZT × M H U A A Ö zZ A T:A U 9 A H T:Ô¼9 AÖ 5 ×× MÚ
9oU S~A 9oA SÛ N CY ASLZ: A S K × M× HIS~YT HIÕ H V Ö % A T0Ö 2Y'HÒ Z 5JA A6Ô HM Õf9RH× Y5m5 SLzzZ ×mAT Û Ö 9<YzZ:AkM'S A Y TsSIF9R UAoS^YN H~2 Õ ázZ ARA Ú
M'A A A Y A A A M6H A½Ó M H A
Ô'¼9oZSI× A8 Û ZsS K Ao'ÀZY A S Û9 UAH T × 9kM KIS^<A N S~T; × A A SLTsT HL KIUH Ao×JY Ö T' Z T:Y'HAGU UYz8ÀA U9 ThA8:9 'zAZ Z UÇA á T HY8:×zS5J× 5mH H ×JT 8Ô ×JH~T Õ Ô V f H % T Ts= AIzÝ 82ÔZ Ò AkZ Y'×mAA
f
YzUM'AAoH A9§ zM × H ×mTúT >HIÕ× ÛÀÕf'HI2 Z Y÷ AM ÜÀZ M HLAo5Ý UÙz5mSI5Z T:AB U3K S ÝG5 8 H ]AkMUH~× bÕ AqAoV YzY% A TsÕfH HYGK M'Si H 5 Ô8 AkAM¼ Y f HIS~Õ '× xH MF2öT:MS å 5 U××5JMÚ 5
SYz~A TÀÖ UöTÞYzH z'UZ ZÀ8:A SL9 AxHM z'SIZ:×mSS~T A U¸ÔqH~Õ SU 'A Z Ö AA T:MÀU Ö:A T:S Y9ÇS A 'Z V5Jf A % 9oS SLM`zA× V f2 % V YVHLÝG% AoVKAxAo% M'Y M'×=)A Ts'ZzÔq×J× S M 5mS 5 × N M
TÒ H Z 8 MK =S ×J5 T¼×JU 'ZÕfHIA YF9oFS M` A#HIIÕ= M Ý Y'Z H AoT YzÛ A 9 'H Z 8A Ö 5U ×mTA 9kÛñS^ÀN A Y ÝSL AkAqA T¼K S Y ÝG× AkH MFU Ý AoÛ× 5 Y'U Ak5 AkN M 2 å`4½2 A Y AH~Õ AÛY'H ß ÚdM AxM M Õ H
H~Y ÕÕfA Y'AkAM U H ÔôM 'Z A A U AkM 2 9 Y ×mÖ'× H T H~Õß*àÚ°M zS~ AB N S½Ô A S~ThU A 9kS^N HI> Õ 'Z AF÷:ÜÀAo5 U¬9kUS^N× YzA 9RSL' × H T zZ ×JTh'Z A 8T:9kS U M'AFH~á ÕM'A <Ö:SLS Y S S 5mA S Ö H8: T:A 9§Tsz'× ×uS T 5
M`H5 Y'KA 2Ò A M A
M A Ú H A 5 AkM
SL × ×mT 8 9o× T
S9kSY'A5 9RS~8 TÇ5 SL= ×m TsA U Ak×uY UÕfÖ HIHIY?9R85 M`SLH zÔ× H ATY SLH zZ S × H Y'T: U:S×mS 5 Y ZUS Y× Y'N UA 9§SI'T:HI× H UqÕ T 'UMZ S A HIÕ ÔZ zA Z U SIA6TÇU Ü:U Ao'A 5 U9oSÛ N U Y×JH T SL M A
È É ÑBÎBËÑ>=Ð1@ÀÈ§ÎBË
ÕfHYUU ] ¼K S~5 TÀU AxM¼ H~ Õ I= A YzAkM A Ö A 9§z× KY Ao5 N Aú2¼ÕfHÒ Y Z A ÔqAkM'S A õ ×mÔÇÛ85mÔçAxM S~TÀYzU A
ßM ÞAGMzMM z8:Û U× A U A ß H S~TT:× 5JAkARY0õYzA AkM'H AoT:YzÛ SIT:9 M AxM A ×mTÇY S HIM Õ N M AA6Ôúß =^ÔqS~×JT Ao5Y'N Ú Ô¼×JT:×JUT×u×J9oÔñS~ 8A Ô U NqS z8 Z A + AoYzY''HZ _Y - U S Yz9oMkS^2 N SL zZ S~T S
Uá × 5mH 9o9<8 Z M ×JzT S~ AkzMkZ 2 V 9 HLÔ¼ÝGAoÖ KAoY = ÀT A 9oSI8 N MÀ Ö zZ 9§Añ 5 8SLÔ' AkY S 'YzZ AFYzAo5 SLS~TA TUú× H
''ZZ AAHIß <S~S~T×JTT × A AoU Y'ÚYzà AkzM S 8 Y 5; M M 9k<S~SLT AkMCA NWA S MS× 5 N> H Yz8 ARYÕfHI× AkY Y Ôñ 8 Y 5 SIS~T AMÕfUúHY ×JThÔqS~ Ak'Y× H Ô TM = O LQ X#Z2 N XMo22sE?1¬AoA K/TÀ2 U ä A A `=" 2> 2 D =Û L8÷5J~5JùH~ÚdE k8 â A U:S =XSI2T:XU 2sX:1¬2A 13T:U 2 A V H S~T TÀÛ U =
H~K Õ 5 'Z A¼2 ß S~T T:Ak×ÛIAk5JYÀ Úà zS YÛ ; YzAkA M'H TÀM S~T:9 AkMkH 2Ò Z A TÞ5 'Z ß*A¼àÕfHÚ°M 5m5JHLÝ Ak×JMT Û Y'× AM O ð~Q 134½2 2á"E S S~M × z S A Y TU)= K= 2ZsN àMkHI25JH Ò0Ô H UH S^TN = ×JSI=T ð~÷ ×J8xâI= âIðsZ 2 N M ×u9 MñÒH US^NI=
5ÕfJHHIS 5J9k5J×JS HLU 5 Ý#×c MA UÕf×uYz H Õ 8T:9§z× AH 9RT '×JM AoT ÛS A 5JH zT ZAkÛY 'S Z A NH~Ô¼UÕ× ÖßYzA 9§z'×Ú°×uH9 T <S~HI× Õ M = 'Z A5mH Ü:AkÛ 5zU S~2 A# Ò Ü:SAkZ5 × M O ÷IQ X138 2s5 N 46x58:7 âI9<â;/==D Ö 22ðIE 22s@BH5mHLKM ;sN=~V 2sX:2s@BHIY<M 9<Z¼S~T:Uq> 2ã ×mÔ Ú
UO M`× AkY'AjA 9§Xz×ÓCH 2 T 2 øI Ô °Q ß*'×JZ ×uhàsÚdU> M <HISLY AxTS~M TS N SY'A9RY N SLá '5J× H H 9<TÀZ S 5 áÝ US ×KIY'S~AxA T:M 9§U '×mT¼× H SzT Z A H~T Õ Y 'SI'Z ZT AM'KIÜ:AoAkY<5M'UU A O øLQ Ô13Ao2Y 46 58:8À7 T9<Ö;C8 =D 5 ×2M EFZ A 2U:@B HI5JHLKM ;NI=sV 2XÀ2@BHYzM 9<Z=ZN Mo2E?AoK/2
ä IA IÌ 2s X:= 2 âÖ 2 xá6âIââ 8:S~TsZ'8:N MkÔ 2E#àAAoÔ¼K/2 ×J9 D 5 S M'MkC2 =IÖù: o2 ÷s=) ù E ^
UM Ö × Y'H A T:9§Uz× ×JH T T Û2 A T ×mAo×J×YzÛ N > H Y S~T:S¬U T M H Z:T S ÚdKIMÀA Ö:S S Y Ü S T×m5J A AÇÖ Ý H ×J UA zTsZ z2 ×JS 5 × K '#Z A > HI9 YHY'zYzZ ARA Ú
è'é<í CÖ:SAFY ÝG AoHI5JÕ 5 'Z AoA6Û5mM A Ö A 9RA Y 8ÕfHIÔhY =YzZ A6 S~T:2 U Ý ×JUzZ M S YzAM ÔqS 5m5 S~T:U O ù~Q 13 II2I46s5 8À7 2 9<;C=D 2EF2@BHI5JHLKM ;NI=sV 2XÀ2@BHYzM 9<Z= ZN Mo2E?AoK/2
9oSITßÞA S 5uM`H"ÕfH T 8T:US?9§ T H U Ts Y × K ×uS 5 U A Ö A T:U A T:9 AH~Õ zZ AYzAkM'H T:SIT:9 A
A 2C` 462 57 =)xøIð~2À÷ EF 2Àkâ@BâIHIâs5JHLKKM ZN Mo2E?2ÀXÀAkK/2:M2 @BX HIY<M =:LX:÷2 xâIâ2 â 2
ä
3
1
;NI=V
9<Z0= Ö á6
Ý KIAkYzM'AF5 ÕfA AñH~Õß*àsÚdM AxM H A YzA H H~Õ O ^Q 8:9<;/=ÀD
'Ü Z T ×uA¼UAx'M Ü:Z 'AoZ 5 U A ×mT 2 U Aá 9oA S^9oN SIHI8 × Õ MÀ z'Z ×J'Ô A Z AqÖ YzK HIS 5 S8 A½× 5 HI×mÕ NI<zSL=IZ SBA¼9 Y'H AxÔ¼M`TúH Ö T:5 '×uS~9oZ T:S~9 AÇAUU × Ý C×u9RUA 'Z:z× ZºS TK U × HIARYÚ O ~Q åT 8:S~'TsZ zA 8Ô z× Û àZsA Ô½Ú° ×u×m9 TÀ5 US Mz×mT Mk2Û S~ÖÖ Ö 2 Y'H^=:õ ù×JÔ¼â~÷ S~' × ~H T Is'Z 2 A AkKIH5 8z× H T
H Ö 2 Ao Y SL HIY SIE T:Uº× MA × ÛA T Õ 8 TÀ9§'× H T 2 MÝGAoYzA 9 H T M ×uU AoYzA U N
H~H~ÕÕ 'Z A#AMÜ:8 AoYz5 K × K MGS 5K Ö YY'HA S 2 Ò × 5 ×A Nñ× MA Aoõ Ö 5JA A 9§ M A U Ý Z AHIT¼Õ zZ A A UYz× Y'HIA 9§' × H 5T Ú
×mN'ZUNT:SIÔ½U×u9 × M × M"S Yz×AkM'U AoYzKIA UZ ÕfHIU Y"zÕ 8SIz× 8 YzU A Öz8:8 UN5 ×J9kSLz×'H Z T 2 Ö S × O â~Q Ò Z YzH ×J8 8Û C= Zö zZ á A Ö:S~Ö =AoðY ùÝø A k8 â M'A âU ×JÔ A T M × H T 5mAxM'M¼K S Y ×uS 5JAkM
Ý 5Z AoYzA zZ AH S~M Ô¼Ö 5 ×z8:UYzAAGH~Õ AzZ A"M <A SL'A ×u9 Ü:A Ao5 UñSIT:UFY zZ AGA M Ao9oY<S M#5mA HIU Õ
'×m TsZ AS~Y'T:H M 9<UN ;¬8:M 9 9 A A Ô T 'Z 2 zA DSITT 5 FH Ak'SY × T:H TS~'zH~× ZKIÕ Ak'5 ZN×mTÀ=AFU H M 9oT Ö S A 5JAT:9kUúU S~T S~TsÔ¼M'AÖ Ö: 5 S×m'hf 8:S~U Ô A % S~ TÀU
Í 7 7GÍ 7Ë>Ñ 7D@
O¿PxQ@BD 5JY M'S H M T S~H 6Nsä S Y<2 M Ç;/= 2/E @ 8 × YzM'AM T×uS M 2 ;N å T M '×m'8 A½H~Õ ZN M ×J9 M =CùIùIIðù Omk~Q ÕfHI2Y Úd4½zZ 2A ã Ö Z:H S H A = TEFz×JS 205 X S~Z: T2 ; A =0S~T:U 2 ×m8= ZN Mk2 ä A ' 2 D
OmRQ>àH 2 9 á 2 5mäH 9<H ZTÀ= U ãH T2 = àZsAkN YkMk2 2 D = k)âs=~øIâI0ð øIøIxø âIð~4½÷2~ã/ä A T2 Ak Y 2Y= äS~YzHT:9 U2xSIEF82= OmIoQ 132 =:S~âs2 jD xCâIAoâs5 øI= ã 2 Z 2 X ; Mo2GÒ0AkHIYx2 >c× 2 =?âIâ xâIù~÷
O àE HLKCá 2 ZN=:MkI2Iã ðIXâ Ò k âLùs =2 ùIð~÷ xâIùI÷s°Q 2IEF2 V H~ÕM zSU AkY =
ZN Mk2ã2àHLK/2 C=÷sù kâIðsI 4½2 V 2ß S~TT× AoY i
=
ZN Mk2
E#2AoE?K/Ao2 K/2 )=÷sð kâI ù D 2 2E S ×mT HLK ×mz9<ZæS~T:U XÀ2ã S~;/=
k
M
ZN
á =:~ðseø xâs^
"!
"
$!
% '&
(
)
'*
+,
./
"
#
!
!
-

Download Report

Wannier-Stark states of a quantum particle in 2D lattices

Paperzz.com

Your Paperzz