10theorie.ikp.physik.tu-darmstadt.de/tnp/pub/2005_mhild...$5 //2[ $) 6"( ,L/XS3"( )/a$9"( ! /%&+>_/5...
83
Transcript of 10theorie.ikp.physik.tu-darmstadt.de/tnp/pub/2005_mhild...$5 //2[ $) 6"( ,L/XS3"( )/a$9"( ! /%&+>_/5...
! "#
"$%'&)(*+, "&- ./0 " 12"435
6 798:);&<=>),35
?A@CB0DFEHGJILKNMOQPR@SP
TUIWVYXZ[PA\]@^D`_
ab6cCde!fHfhg
"!#$
%'&)(+*-,./&+()()(10
!#"$% &'"()*+$-,."#/%! 0/21&3!*4/5 0) 6"( 78$9"( :"($%/%"()/;)0&3/%&<! =0/?>@#"(*4/5<(>A0/CB?DD6"()&<*4E&/5 GFIHJ/;! .KL/%)!,."()/M0/C&3N.1 6"(*+$9"( ODP/56"%K?(>QDRTSVUWX)/M,.")/%;"( 0&Y(1$%*+8R 0/5 EMZ0/5*+A*+\[3)/%9SD.N]8R/5^>_*+! ,"( '/2[0"$5+!*4/21/5KL ! `a! /-8P/5D! ,J0/-N3)/5*YFcbW0/)/%8R 0#/ad*4/9"))/%&eDEN4/5K"( !6"(! ,)/5KL/5#"( VD0#/5K"(D /%W&3! ,f0/;/5KL ! TS*e"(! !Ng! Y0/f) ,X !Ng)/58 )!KL/M)/5,X!*4/XF
bO;Q/5 0&hSX/,X!KL/a";0I)$9"( 0iKL/5KE/5' 0/)/9"( !#"( 4(>VUWX)/21jQ! 0)/5! 4$% 0&/5 0#"( g"( 0&! E)E&30$%/D/2k0Nl0/m6/5 &4(>R !#"$% &+,.")/%I! 4831$9"( R :"($%/%iFIHJ/?)/5K/5n0/;UWX)/21B?DD6"()&=*4E&/5 T"( 0&g! .)&30$%/C0/CB;"(*+! \1) g8P/5#"())d>_UWX)/?,.")/%W"( 0&eo(1$%*+8P 0/5 .Z0/5*+P*+\[3)/%9FQpq)/%9KL/59S)/5KL/5#"( rD0#/5K"(D /%7>@;DPsN3)/5*4C"()/f&3)$50)#/%&! sKE/5t(>u0/v)8R/5^k0&)+ps^1G! 0 :"()?86")/C#"( 0! TF
bW0/!*4/21/5KL ! (>Ww.6"( E*xN3)/5*4;M&3)$50))/%&y! y,L/5 0/5#"( "( 0&y0/)/%$) $9"( I&/5#"(! ! )&/5?)4/5KL !KL/fB?DD6"()&N3)/5*47"()/f !! 0/%&hFazE! 0$%/f/8R/5^>_*)!*f :"( 0|(>"()*+$,."#/%?! J"(*+8 !!0&/*4&3 :"()/%&q :"($%/8R)/5 31:"( 9Su/Y&/5!KL/<0/Y"$%$%)&3! ,!*4/21&/58P/5 0&/5 .g86"(#"(*4/5)/5)e(>?0/-B?DD6"()&*4E&/5 GF|Z00/5*4)/XSh/+&3)$50#?0/l*4/%$#6"( *4|(>86"($5 /21G0 /+/2[$5!#"( a$)$%$5! -0/|) ,X !N<)/58 !KL/|)/5,X!*4/XF
HJ/C8R/5^>_*~!*4/21/5KL ! 07>@UWX)/M"( 0&YZ0/5*+O,.")/%Wa!Y4&3\h/5)/5 .E! 0&g(>|8R/5D6"( ! `)&/5=)J! EKL/%!,."()/q0//2[$5!#"( n)8R/%$5*YJ""#"( 6"(Nn8P/5D6"( /"(88 !N !! 0/9"(-8P)/5 .:"( M,X#"&3/5 E)Y(>v&3\h/5)/5 .)/5 ,X09FIbW0/WN))/5*t)09d";$% .! E00/2[$5!#"( 48P/%$5*! 0/W/9"( !N)/58 !KL/l)/5,X!*4/XSha0/5)/9"f"(M) ,-! .)/5#"$5 0f 6"()9)/%) 6"( 0$%/%v"(88R/9"(9FbW0/%)/d)/%) !)I"()/! l,LE&"(,X)/%/5*4/5 .Qa!0/86"($5 /21G0 /a/2[3$5!#"( 8$5)/XF 4"&3N 6"(*+$-8P/5D6"( /Y!*f :"()/0/-DP/56"%Ke(>M"()*+$,.")/%+! "(*+8 !!0&/4*4&3 :"()/%&J8$9"( d :"($%/%9F4HJ/+8)/%)/5 .v)/5KL/5#"( d!*v :"( 0fa!&3\P/5)/5 EN))/5*!9/%iS.m0 ! !! ,l>G"$5))a(>O0/C :"($%/XS0"( 0&=DR 0&0"(N=$% 0&3! 09FbW0/e)/%) 6"( 0$%/-)0$5)/g)/%) !! ,s>@)*v8P/5D6"( cl*4)/<$%*+8 /2[6"( c! ]0/-$9"#/-(>?0/<,X#"&3/5 .9FHJ/-D0#/5KL/Y"( c"&&3! 6"( 7m0 0/210$5)/a$))/%/5*4)+DR/CDP/5NL 0&-0/|)!*+8 /C86"($5 /21G0 /v/2[$5!#"( Y8$5)/XF
V E _Z @FE
\Z!I V _T E_)OQD F )*+$M,."#/%! Y8$9"( T :"($%/%FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF F bW0/|UWX)/21B?DD6"()&YpqE&/5 FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF
F F ?*fDP/5;zE#"()/|Ua" FfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF F F UWX)/21B?DD6"()&BC"(*+! !) :"( FMF|F|FMFfFMF|FMF|F|FMF|FMFfF! F F#" $?D0)/5KX"(D /%'FMF|FMF|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF
F#" bW0/|Z6/5*+\1B7DD6"()&sps&/5 FfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF F#"3F ?*fDP/5;zE#"()/|Ua" FfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF F#"3F Z0/5*+rB7DD6"()&YBC"(*+! !) :"( JFMF|F|FMFfFMF|FMF|F|FMF|FMFfF " F#"3F#" $?D0)/5KX"(D /%'FMF|FMF|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF %
& KN@SGLO('*)HE D Z @FE "3F +C/5 0/5#"( ,?)/%gFMF|F|FMF|FMF|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF "3F UWX)/21B?DD6"()&YpqE&/5 MF|FMF|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF
"3F F jI[$5!#"( s(> )*4! -$?8$9"( /.O"($%/% FMF|F|FMF|FMFfF0"1 "3F F bW! !)/%&2.A"($%/%FMF|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF0" "3F F#" pqE&3 :"()/%&3.A"($%/54r)/5 .:"( FMF|F|FMFfFMF|FMF|F|FMF|FMFfF0"1"
"3F#" pq/50E&yF|FMFfFMF|F|FMF|FMF|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF0"16"3F#"3F 7#"( E187$#0 ) s)$#0/5*4/FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF0"9"3F#"3F : ,L/218;C#"4*4/50E& F|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF0"1
< = E P E P < % F bW! !)/%&3.A"($%/54r)/5 .:"( F|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF % % F pqE&3 :"()/%&2.O"($%/54r)/5 .:"( fFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF % 6
> ? O VRG @FE P @ >A F bW! !)/%&3.A"($%/54r)/5 .:"( F|FMFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF06 AA F pqE&3 :"()/%&2.O"($%/54r)/5 .:"( fFfF|FMF|FMF|F|FMFfFMF|FMF|F|FMF|FMFfF061B
)( - A)! "
6"( .* 0/%NI0/a0/%Nla$)e&/%#$5!DR/%Q 6"()/7 e";*+$5)X)$%8$)$9"( /XF)/%w.!)/%&<)*4/;/2[8P/5!*4/5 .#"( T)E! ! V)f)/5KL/9"( P*e"$5)X)$%8$MwE6"( .* 80/5 0*1/5 6"3SQ !!L/<8R/5)$% 0&30$5!K!N>_ 0&DEN! "$#&%('*),+-##.!/0! !1XFB?/vD0)/5KL/%&q0/fKX"( ! ,-(>u0/l)/%#"( 0$%/l(>d*4/5)$5NYDP/5 9t0/f)/5*+8P/5#" 1)/=(>
4.2 KF $? 0/%e")*4/%&hSQ6"(+0/-/2[38 :"( 6"( '(>?80/5 0*4/5 0
)/%w.!)/%-0/w.6"( E* 0/%NLSCa$) "! "( /9"( !Nn)#"(,L/"(Y6"(Y!*4/XF 00/54*e"$5)X#$%8$Yw.6"( E*/2h/%$5+48P/5^k0&3!NLSa$)`"4>@ 0&DEN;|"(8!)#"3S ! /5 TS3"( 0&<p)/5 0/5! !1"9rbW0/5NgD0)/5KL/%&= !w.&g0/5 !!* 4 B?/k69W1! ,+a!0>@$5 YDR/5 i "$5!$9"( O)/5*+8P/5#"()/XF
UWl/2h/%$5)Q"()/$% 0/%$5)/%&l)C"80/5 0*4/5 0 4$9"( ! /%&lUWX)/21jQ! 0)/5! +$% 0&/5 31#"( TS0a$#";!,X! 6"( ! !N$% 0$%/5!KL/%&D.N213 54678:9-$#;/8<7=#q! ! 1A Fo !:"()/%&D.Nc0/)0&3/%=(>>?@8BA@#DCEGFHI8='KJ3L(/My `0/s)#"($%g(>M80) n,.")/%9SjQ! 0)/5! y8)/%&3$5)/%&y0/l86")/l#"( 0! (>"=&3! !)/v,."M(>W"()*4C! .)-"=8R/21$5:"( A#"()/XSV\>u$%E /%&sDR/5 it"e$5!$9"( A)/5*+8P/5#"()/XF7o ;8P/%$5:"( A#"()/ON=0/UX#/21jQ! 0)/5! $% 0&/5 0 "()/Ny"( ! "()*4"()/+! J0/g#"(*4/4! ,X /21G86"($5 /=#"()/XF4r"($5 /%<(>|6"(e! 0&"()/$9"( ! /%&P4EL(/LI#;/y"( 0&0/5N'DR/5Nc0/sUX#/21jQ! 0)/5! #"($%9F=bW0/4*e"( EN.1GDPE&3NW"%KL/2>@ 0$5 c(>WDPX) 0ffN*+*4/5$4 0&/5l/2[E1$)6"( ,L/C(>Tv86"($5 /%iS3a$)<*4/9"( 00/7W"%KL/2>@ 0$5 Y&E/% 0W$)6"( ,L/XF$? 0/a0/5u6"( 0&hSL86"($5 /%9S.a0X)/aW"%KL/2>@ 0$5 gu"( E\1NE*+*4/5$7 0&/5u86"(\1
QSRUTWVYXT[Z]\!^_^_Ta`SbYVYc(d!egf=hUc(cgTWij`STakWTWVYlTEmonSe(Tqpsr7tITWbvug`SVYwWTxr!yzuge|iVYkWi~VYc WW yr,`e(VYiVc|lTaiBnSVd7\,nSVYr7cgir7c nSegToug`Sr7uIT`nSVTair!y^\6nnSTa`\6nbYrE nSTW^_uIT`<\,nSg`STaiegVYkebYTEm*VYcMnSTa`z\,bYV\|nSrnSe(Tug`Sr|m(kanSVYr!cr,ybYVM(Vmqe(TWbYVY(^ 6 n<\,XTWcy"`Sr!^M!DB76g6 ¡¢6|£¤,¥¡;¦B!£§M6|¨©7¤ª©7¢!«!¬|£M¡«M¡M©7g6®(E¯M¤6|°M¡7±²¦³|6´¢Mµ
¶
·
$5 //2[$)6"( ,L/XS3"()/a$9"( ! /%&+>_/5*+ 09FIbW0/5NDR/5Nl0/Z0/5*+\1 ;!#"$;#"($%9FrbW0/"( E\1NE*+*4/5Ns 0&/5C86"($5 /21/2[3$#6"( ,L/!*+8 !/%9SR6"(?o=&/5 .$9"( r>_/5*+ 0*v0f 0lDR/4! 0/e "(*4/e#"()/X #*+! ,o>@/5*+ 0v)-DP/4! 0/g#"(*4/#"()/XSr0/g! .)/5)$#6"( ,L/-(>0/%)/486"($5 /%$9"( ] 0f /9"&)Y0/g)/%w.!)/%&]!,X 31$#6"( ,L/v(>I0/|W"%KL/2>@ 0$5 TFabW 09a y"0/*4I"( !I/2[3$5 !0 8! 0$5!8 /XFbW0/5)/2>_)/XS";80/5 0*4/5 0 g !!L/a0/$% ! /%$5!KL/?$% 0&/5 0#"( g(>T"( ! 086"($5 /%u! E)0/ "(*4/#"()/WQ 0u8PX)!D />_I>_/5*+ 09FIbW0/N*+*4/5N(>P0/a*e"( EN.1GDPE&3NW"%KL/2>@ 0$5 YW$% 0/%$5)/%&<)v0/C8! <(>O0/;86"($5 /%WK:"f0/ / !D$#²)S/78S@8B/8B#"7/7)8=' 6LEF 4I"($5 /%9S0a0X)/f8! -7"( Y! E)/5,L/57*f !!8 /v(>r0/54Q :"( 0$)-$% 0#"( E~"()/7DRX# 09S.a0/5)/9"86"($5 /%Wa!=6"( \> 1G! E)/5,L/5W*f !!8 /%a(>
~"()/7>@/5*+ 0iF
bW0/qD0)/5KX"( `(>|0/sUX#/21jQ! 0)/5! $% 0&/5 0#"()/%$@UWj7'&48)9KL/%&`)JDP/*4)/|&3)(e$5 !9S6! 0$%/C !w.& 4 B;/C"$%*+8 /2[=! E)/5#"$5! ,gN3)/5*YS3! 0)/9"&-(>u"0/l&3! !)/f,."C"?! sjQ! 0)/5! 0;0/%NLF .T 0& y8)8RX)/%&0/v)8R/5^k0&q#"()/(> 4 B;/v"M"e*e"( \>@/%#"( (>u0/lUWX)/21jQ! 0)/5! $% 0&/5 0#"()/l! !1"1B+*-,F?bW0//5K&/5 0$%/l>@|"=$% 0&/5 0#"()/! y !wE&s0/5 !!* 6"CDP/%/5 /%#"(D !0/%&y! !1 A D.N 0/5) q)$9"()/5! ,0F
bW0/s/2[8P/5!*4/5 .#"( ;,LL"( 7W"4)$5)/9"()/s"y8)/sUX#/21jQ! 0)/5! $% 0&/5 0#"()/>@)*x"=&3! !)/e"()*+$,."|"C0/%)/5$9"( ! !Ny8)/%&3$5)/%&D.NyjQ! 0)/5! TFo q0/DP/21,X! ! , !1B1 |#/5KL/5#"( r,X)80|/%&s)<,L/5 0/5#"()/4"g8)/+$% 0&/5 0#"()/+(>.N3&3)(1,L/5 TSha$#J"f$%E /%&$5 X)/)<0/#"( 0)! 8R! E9STDM0/)/%$%*fD! 6"( )+*4 /%$5 /%7! !D!)/%&Y0/M$% 0&/5 0#"( TF
bW0/C*4X)0$%$%/%)^>@ h)/%$) wE0/M),L/5 0/5#"()/f"v8)/|UWX)/21jQ! 0))/5! $% 0&/5 31#"()/lC0/$%*fD! 6"( ](> :")/5|"( 0&y/5KX"(8R#"(!KL/+$% !! ,-! y*e"(,X 0/5$#"(809F.O"#/5$% !! ,y"! EKL/5 .)/%&D.NczE)/5KL/5 7.TS 7 :"(0&/370/5 31br"( 00&/. GSW"( 0&Hn! ! !:"(*0F 4u! ! !!80f"( 0&y0 0)/%&a!y0/ ?DP/5 I8!9/! y8.N3$%M! !19FbW0/e8)/2>@/5)/%&]"()*4f)-DP/4$% /%&JDEN|)/%$) wE0/<"()/g"( !X"( !9SA)! 0$%/+0/5!8$9"( W#"( 0! 0l*e"()$)0/g"9KL/5 /5 ,X04(>7$5)/5 . :")/5N3)/5*49FYpq)/21iKL/59SE0/%#/;"()*4$9"( gDP/aL/58! g*e"(,X 0/5$9"( R#"(80a"(d im6/5 &=)/5 ,X09S&30/)v0/5!W :"(,L/7*e"(,X 0/5$;*4*4/5 .)iF7 0&/5 0#"( Y$9"( 0DR/C"$)/5KL/%&<DEN4 :")/5$% !! ,Y"( 0/21P0/l "(*+8 /l$9"( qDP/v$% /%&s>_"(;DR/5 it0/f)/5*+8P/5#"()/ 0/%/%&/%&hSDI0/W$% 0&/5 0#"( )/%w.!)/%u"( #7!,X+&/5 0!/%9SL"(Ia$#l0/W#"(*+8 /W!*+8 !N"(D0)D0?0/|80) 0iFbWE09SV! s"4)/%$% 0&s#"(,L/XS00/v#"(*+8 /|7L/587! y"*e"(,1 0/5$e#"(8]"( 0&$% /%&>@0/5fDENJ/5K"(8P#"( TF<bWf)/%$# w.0/gf)!*+! :"(f)$% !! ,|$%(P/%/Xr 0/"( ! 9d0/4320)/%53"()*4Q)C /9"9KL/W0/a/5 0)/5*fD /7a0/5)/5D.N0/"9KL/5#"(,L/W)/5*+8R/5#"()/(>V0/a#"(*+8 /W&/%$5)/9")/%9FIbW0/W !Nv&3#"%aD6"$#9S6"(0/d .*fDP/5A(>6"()*4OT)/%&30$%/%&hS(DODEN;O*4/50E&f0/u$% 0&/5 0#"( (>
2000
6
"!#!!$%&')(*+,!-./0.-.+1.2/345!--164789,:3+;*!--.<7=.+>+4?4@!!@16AB5!-;C>D!<FEE EEEGD1.16HF7I5!-$$ $J,.+>16 K!-2H.%L!--+5!-+;/0- !MH>$N7OH>IP0HH>$4./Q!R)ST!-!-GUV!!%&+4@!-RHAB5!-;M16=H46!-<4-3/WH><16%.-.+1.;PXHHY.*-!G=)SZ [$J,$+16#.+7GC]\O!-M!^+5!-+_C1^`+,!a!bHY.N@7cH>_B5!-d-$!-$7eHf16.-.+>16 PL`$+e/0g]h]h,ijk,kl]l]lmni]gDoYp,q)r9m&s,tuq]v^wuxqFtDyzm|,Dk~sxtDhswkkxD,]oDkqFtD]'mLgDhwu S
87 : DY"()*44$ D&3!* &"7"$#/5KL/%&-! !1 A D.N=jQ$57 0/5 ! O"( 0&37W"( hH`/21*e"( * ,/5+"( GF $&L . &2FAZ0l*4 .0l :"()/5vHJ \> ,."( , ;M/5)/5 /</5l"( GF $@pqob &D0)/5KL/%&0/l$% 0&/5 0#"( (>"g#"(*+8 /v(>
5 × 105 23 C"e"()*4 $@)&3!* &4*#" ,FbW0/%)/a/2[8P/5!*4/5 .)d6"%KL/aDP/%/5 +)/58P/9"()/%&D.N+)/5KL/5#"( ,X)80Qa!e&3\h/5)/5 ."( !"( !"()*49S0"( 0&-! 1 (7 0/5 ! GSHn/5*e"( TS6"( 0&3;M/5)/5 /|)/%$%/5!KL/%&-0/*;DR/5 h8!9/>@M0/5!f&3#$%9KL/5/%9F4bW0/g"(D! !!Ny)-$5)/9"()/+ #?) =#DC L^ I88<7<8R/5 0/%&a0 /M 0/5 #"( ,L/|(>I8RX#!D! !!/%a)+0&3N<w.6"( E* 80/5 0*4/5 6"3F
!11B."(3)$)f/5r"( GFT8)8PX)/%&M0/uD0)/5KX"( v(>0"( 00/5O86"#/Q#"( 0! (>;"UWj7 ! c"q8$9"( :"($%/ * % ,FbW8$9"( W :"($%/<l,L/5 0/5#"()/%&DENo$% .)/5^1G8)86"(,."(! ,f :"#/5)9S(a$)>_*"?#"( 0&3! ,C"9KL/XFQbW0/ :")/5O>@)/%wE0/5 0$5Na>G"(7&/5 0/%&->@)* )/%) 6"( 0$%/%;(>Q0/f"()*49SP"( 0&Y&30/|)e0/M! 0&30$%/%&q&3!8R />@)$%/s0/5N,."(0/5-! n0/q*+! !*e"=*e" [3!*e" $@&/58R/5 0&3! ,' 0/y&/5 ! ,)/5 :"(!KL/v)g0/"()*)/%# 6"( 0$%/5&7(>0/l)#"( 0&3! ,gW"%KL/XF|bW0/v8.N3$%C(>d0/%)/N))/5*47$#6"(#"$5)/5)/%&qD.N-0/f#"(e(>Q0/v 31)!)/|! .)/5#"$5 )/5 ,X $
U&
"( 0&40/;)/5 ,Xg(>T 0/5 ! !! , $J
&IDP/5/%/5 <0/?!)/%9FIbW0/7L/5N>_/9"()/6"(W#"(+a$% .) ! /%&D.N=0/C! .)/5 0!oN<(>r0/C :")/5)9FQZ07"l i #"(
U/J0/
0/5 ! !! ,f&*+! 6"()/%d"( 0&+$% 0&/5 0#"()/aDP/56"%KL/%u !!L/"C8R/5^k0&hF $? +0/a0/5
·
6"( 0&hSL"7!,X#"(U/J
88)/%)#/%A0/*4D! !!Nl(>60/W"()*4I! v0/ :"($%/XSL"( 0&0/l! .)/5#"$5 0f&*+! 6"()/XFfo y"g)/58ya! 0/"()* 8P/5M!)/f0/l8)/2>_/5)/%&#"()/Y! 0/Y) ,X !N)/58 !KL/)/5,X!*4/Yg 0/-a!`/9"$#n"()* 8! 0/%&n"(g"! ,X /M)!)/XFIH`! /C! 0$5)/9"! ,e0/C#"(
U/JS30/;>_*4/58R/5^k0&$% 0&/5 0#"()/
6"a 0&/5,L 0/l"+wE6"( .* 86")/|#"( 0! Y)+0/M)(1$9"( ! /%&sps^1G! 0) :"()9FbW0/v0&3/%;(>BE"()$#q/5;"( GFW6"%KL/vDR/%/5 Y8P/5^>_*4/%&a!! s0/fUWX)/fB?D31
D6"()&<*4&/5 GSa$)0/5Ne8)8PX)/%&Y"a0/C*e"(0/5*e"($9"( T>@#"(*4/5PFuo !:"( ! !NLS0/|B?DD6"()&<*4&/5 T"$% 0$%/5!KL/%&YD.N L B?DD6"()&<! !161")+&/%)$5!DP/C! 31)/5#"$5! ,=/5 /%$5) 07! Y0/f8R/5&3$M :"($%/|(>u) !&iFdo YW>_*e"( !* /2h/%$5) !!L/v0/|*e"(,X 0/5$lDP/56"%KC"( 0&s0/|*4/5#"( \1G! 0 :"()C#"( 0! (>u) !&C"()/0&3/%&hF 00/5"(88 !$9"( Y(>O0/CB?DD6"()&g*4&/5 P0/;&/%#$5!8 Y(>A318P/5^k0& 4 B;/C! QN$%XS0"+8R)0a,X :"#9FubW0/|UWX)/MB?DD6"()&Y*4&/5 O"/5 ! A"0/lB7DD6"()&yBC"(*+! !) :"( J(>"4o(1$%*+8P 0/5 .fZ6/5*+r,.";! y8$9"( I :"($%/%"()/M)/5K/5/%&Y! $)6"(8)/5 F
o 1 XSV0/v,X)8s(>dbW0/%&;B( 0)$) $ .Tp "( 0&p 4 +M"()$)! , &7$% 31m0*4/%&0/ypq^1G! 0 :"()-86")/s#"( 0! ! /2[8P/5!*4/5 . * A ,FZr!,X)/ XF&/58$5)M0/l*e"()/5^1GW"%KL/4! E)/5^>@/5)/5 0$%/!*e"(,L/%f&3! ,=0/#"( 0)! s>@)* 0/8P/5^k0&y)=0/+psC! 0 :"(! ,86")/+"( 0&yD6"$)q! .)<0/+8P/5^k0&y#"()/XF$? 0/|D0#/5KL/%a0/M$%*+8 /5)/M X)a(>I86")/M$%0/5)/5 0$%/|! -0/Mpq^1G86")/Ca$#a)/%)))/%&-D.N<0/C#"( 0! sD6"$)<! -0/|8P/5^k0&-86")/XF
bW0/a"&3KX"( .#"(,L/W(>60/%)/N3)/5*4AA0/8PX)!D! !!Nf);)0&3NM0/$#8.N3$%(>)) ,X !N$%)/5 :"()/%&yN))/5*YSVa$#y7 0;>_/9"!D /f! y$%*+8 /2[sN3)/5*4? !!L/) !& !wE&9FybW0/=)N)/5*4l8)iKE&/=0/=8RX#!D! !!NJ)y$% EKL/5e"Y/9"(E !N! E)/5#"$5! ,,."<! .)`")) ,X !Nn$%)/5 :"()/%& )N)/5* ! a$) 0/y8.N3$%<&*+! 6"()/%&D.N<0/C! .)/5#"$5 TF
o -W=/f0&3Ne0/CDP/56"%K7(>A !#"$% &s"()*+$M,.")/%aD ./%$5)/%&)="( s/2[3)/5 6"( O8P/5D6"( q(>Q0/| :"($%/XSV! s)&/5?)4)/5KL/9"( r! 3>@*e"( y 0/+/2[$5!#"( J8P/%$5* /%8R/%$5:"( ! !Nq! y0/)/5,X (>0/86"#/v#"( 0)! TF !*+8 /s*4/50&')J8)DP/Y0/q/2[$5!#"( n8P/%$5* g)"(88 !N"J !! 0/9"(g8P(1)/5 E:"( ,X#"&3/5 .<"$5)X)+0/< :"($%/ * A S 6/,F 6"( !!#"(!KL/5 !NLSW 6"()9 )/%# 6"( 0$%/%6"9KL/|DP/%/5 YD0)/5KL/5)/%&"(7) ,e)/58 09Sa0/5)/9"7! -0/|8P/5^k0&-)/5,X!*4/"s$% .! E00e8P/%$5* "(88P/9"()9F "( !)/5 6"(!KL/<*4/50&])s/2[$5!)/e0/<N3^1)/5* C(1G80) U#"(,X,-8P/%$5)X)$%8.NsKE:"<"( J"(*+8 !!0&/+*4&3 :"( J(>d0/ :"($%/v8R)/5 E:"( GF?o s/2[38P/5!*4/5 . * 5,I 0/lD0)/5KL/%C"(,."(! y"( #="4D)L"&/5 ! ,<(>
kWr!`VYi \ ug`Sr|m(kanr,y ?r!`ScgVcgdUb\!iir,`SXi |!²B²¦ª7!£,(¤E|§¦Bª7W´µ
0/M/2[3$5!#"( Y8P/%$5*~! -0/C/9"( !N=! .)/5#"$5! ,e)/5,X!*4/|"( 0&<#"(0/57 6"()9)/%) 6"( 0$%/%a>_) ,+)/58 09F
>@)/5O !! ! ,;0/u)/%$# $9"( 3&/5#"(! r(>!*4/21/5KL ! a!! f0/dB?DD6"()&*4E&/5 a"( 0&&3#$50)! ,q0/eD6"$g*4/%$)6"( )*4+(>7/2[3$5!#"( ! $)6"(8)/5 "3SI/8)/%)/5 .M0/ E*4/5$9"( d!*f :"( 0v(>UWX)/,.")/%C! $#6"(8)/5 % FfHJ/+)!*f :"()/0/s/2[$5!#"( (>l"8P)/5 .:"( M,X#"&3/5 E-"( 0&"*4E&3 :"()/%&n :"($%/"(*+8 !!0&/D.N8P/5^>_*+! ,J0/-/2["$5!*4/21/5KL ! n(>?0/<8P/5DR/%&cN3)/5*44"( 0&0//5K"( !6"( y(>)/5KL/5#"( ID0)/5KX"(D /%?! y)&/5;)e*4/9")/v0/v)/%8R 0#/XF7UW/%&/%UX#/<,.")/%4! 'KX"(04)/580e !!L/Y&3\h/5)/5 .e E*fDP/5)g(>|!)/%iSum0 ! !! ,>G"$5))(>u0/f :"($%/%|"( 0&sDR 0&0"(Nq$% 0&3! 09SR/+"( )=!*f :"()/+"e(1$%*+8R 0/5 EZ0/5*+T,."7"( 0&-8)/%#/5 .0/M)/%) !)a! $)6"(8)/5 A F
·
(;- - 3*&
!#K8='*$/ "'& !?8<7 8=' J L(/|4|4EEC L|C*7 ]$/EF 6C ' #&4,7 /78S@8<4EI)/!$/'MLH /7$#&% ³) "6L ! LI#?7#²8_4EL(/ML# /7A(/78<7$/HC/"/,/M6C I#DC/|(7E UL4!/M7g*4 ³!/$#|SFvL^ 8=' /5!;7C 8SL LI883$#/& I8BL !D'&(/M 8BE#;/7=8BL#;/ 66|SF EC ' O4|4EEC$ 8SL## ML 8& L "6LI !;L#?7#²8 77]/!A(/8<7 / 7 = U(/B8=/# 46O/78SI8<4E/!$/H/ $#D8BWL|C ",6C #DC8=' $# L*4!/|7g*47 5 := $ 46L 8B "$#?6C
d! #"%$ & (' #"I*)+)i #"
bW0/|0/%)/5$9"( r> #"(*4/5Y)=&/%)$5!DR/l"()*4;! 8P/5E&3$v8R)/5 E:"( ;7,X!KL/5 D.NM0/dB?DD6"()&v*4E&/5 GFrbWO*4&/5 .W"O! !:"( ! !Nl"(88 !/%&f! l) !&f#"()/d8.N3$%)?&/%)$5!DP/*e"(,X 0/5*YFI !11B,&(|/"7'M8 -r/5*+8 9NL/%&4"4EL(/LI#/.gECvKL/5) +(>W*4E&/5 h)+&/%)$5!DP/CDPX) $|"()*4a! Y8$9"( h :"($%/%?"(a9/5))/5*+8P/5#"()/* % ,F
bW0/J L(/|7)B0;44EEC H= "8SL#²B# $9"( DP/v&/5!KL/%&q>@)* 0/B;"(*+! !) J831/5#"()a>@aDPX) $|"()*47! s"( Y/2[)/5 6"( O8R)/5 E:"( G
H =
∫
dx ψ† (x)
(
− ~2
2m∇2 + Vlat (x) + VT (x)
)
ψ (x)
+1
2g
∫
dx ψ† (x) ψ† (x) ψ (x) ψ (x)
$ F5&
! sa$#ψ(x)
"( 0&ψ†(x)
"()/l0/lDPX) $|m6/5 &y8P/5#"());>@|"e,X!KL/5 J"()*+$#"()/XS
Vlat(x)?0/8$9"( r :"($%/l8P)/5 .:"( u"( 0&
VT (x)M"( y 9a !NK"(N! ,
/2[)/5 6"( P8P)/5 .:"( $@/XF ,0FQ86"(#"(DP !$;#"(88! ,+8R)/5 E:"( &2Fdo gW*4&/5 h$% .#"$5
1
· 5
! E)/5#"$5 +DR/5o/%/5 +0/W"()*4AI")*4/%&hFIbW0/86"(#"(*4/5)/5Q(>6A)0^1G#"( ,L/80)/50&8P)/5 .:"(
g = 4πa ~2
m
f0/+(1GDR&3N! .)/5#"$5 )/5 ,XTSra0/5)/a &/5 0)/%r0/ /21GW"%KL/a)$9"()/5! ,; /5 ,XTFIbW0/um0))A)/5*t(>0A/%wE6"( r" 0 31
! E)/5#"$5! ,<BC"(*+! !) s8P/5#"();"( 0&0/f)/%$% 0& 0/f&/%)$5!DP/%0/l"()*1o"()*! E)/5#"$5 TFbW0/|8$9"( O :"($%/M8R)/5 E:"(
Vlat(x)! 0/M&3!*4/5 0) )/9"&
Vlat(x) = V0 sin2
(2π
λx
) F $ F &
a!Y0/CW"%KL/5 /5 ,Xλ(>I0/M :")/59F )*+! ,+0/f"()*4)+DR/C! -0/C i/%
K!D#"( 6"( O#"()/;0/;m6/5 &18P/5#"())ψ(x)
† "( 0&ψ(x)
$9"( -DP/C/2[86"( 0&/%&Y! =0/H"( /5?D6"9
ψ(x) =
I∑
i=1
w0(x− ξi)ai
"( 0&ψ†(x) =
I∑
i=1
w∗0(x− ξi)a
†i
S $ F#"&
a!I0/W E*fDP/5d(>V :"($%/a!)/%u"( 0&
ai
"( 0&a†i0/"( !! :"( g"( 0&+$5)/9"(
8P/5#"())9Sa$#sDP/5Ng0/M$%*+*v#"()?)/5 :"( 0
[ai , aj
]
−=[a†i , a
†j
]
−= 0
"( 0& [ai , a
†j
]
−= δij .
$ F % &
bW0/w0(x − ξi)
"()/0/,X) 0&#"()/4H"( /5M> 0$5 0 $@&/5 0)/%&D.Nq0/l! 31&/2[
0&f $9"( !)/%&]"(+!)/
iFybW0/=6"(8R/=(>70/=H"( /5l> 0$5 0iSu/XF ,0Fq0/5!
a&3TSP;&/58R/5 0&3! ,- q0/f :"($%/l8P)/5 .:"( 1h;;$50$5:"( Ia0/5 y&3#$50)! ,0/&/58P/5 0&/5 0$5N4(>P0/a*4E&3 :"()/%&e :"($%/?"( 0&+0/86"(#"(*4/5)/5)(>h0/B?DD6"()&BC"(*+! !) :"( ! Y)/%$5 "3F F#"3F
: /5a!! ,40/|BC"(*+! !) 8P/5#"() $ F5&0! ,+0/|/2[386"( 0 0;"(DRiKL/M,X!KL/%
H =
I∑
i=1
(a†i ai+1 + a†i+1ai )
∫
dx w∗0(x)
(
− ~2
2m∇2 + Vlat (x)
)
w0(x− δ)
+
I∑
i=1
a†i ai
∫
dx VT (x) |w0(x− ξi)|2
+
I∑
i=1
a†i a†i ai ai
g
2
∫
dx |w40(x)|
F
$ F A &
Z0+)/5a!! ,0/=m0)+)/5* 0/9"()/%+ 0/5!,X.DP4088! ,JW"e")*4/%& 1δ
0/ :"($%/4)86"$5! ,0Sha$)J|"#*4/%&y)=DR/$% 0#"( Ef9KL/5|0/+)N)/5*YFvbW0/)/%$% 0&)/5* |# :"()/%&y> )*x0/m0)|! E)/5,X#"( GS )*+! ,6"(f0/4/2[3)/5 6"( 8P)/5 .:"(
VT (x)f ia !NK"(N! ,s 0/e)$9"( /g(>a0/4 :"($%/e)86"$5! ,
δSA0/
· 5
(V1&3:"(,L 6"( u$% E!D 0f$9"( yDR/l 0/5,X /%$5)/%&hFl q0/ :";)/5* !Ns 0/*9KL/5?"( ! T :"($%/f!)/%a)/5*e"(! 09S0)! 0$%/M 31!)/M! .)/5#"$5 q7"))*4/%&- !NLF
gd4/C"()/7&/9"( !! ,a!=)/5,X :"(W"| :"($%/78P)/5 .:"( GS3E00/am0)! .)/5,X#"( d! ]/%w.6"( $ F A &;|! 0&/58P/5 0&/5 .(>a0/4!)/+! 0&/2[
iF=bW0/4! E)/5,X#"(
$9"( eDP/7$9"(/%&=d>_"M,X!KL/5 <N3)/5*"( 0&g&/2m0 0/70/?86"(#"(*4/5)/5)W(>T0/7UWX)/21B7DD6"()&YBC"(*+! !) :"( T
J =
∫
dx w0(x)
(
− ~2
2m∇2 + Vlat (x)
)
w0(x− δ)$ F#6&
εi =
∫
dx VT (x) |w0(x− δ)|2 $ F 2&
U = g
∫
dx |w0(x− δ)|4 F $ F#B&
bW0/f86"(#"(*4/5)/5J)/58)/%)/5 .)C0/ 8 &##?7 = =#% /8BE#%I8=' 0 S
εiM"g!)/21&/58R/5 0&/5 E
/2[)/5 6"( V8R)/5 E:"( R"( 0&Uu0/$#²8<W"|8BBL# /8BE7#&%I8='EFQbW0/! E)/5,X#"( /9"&3! ,f)
J"( 0&
U$@/%wE6"( 04$ F#6&"( 0& $ F#B&& !Ne&/58P/5 0&< g0/; :"($%/;8P)/5 .:"( $_"
&3)$50))/%&hS.0/6"(8P/7(>P0/H"( /5> 0$5 g"( )|&/58P/5 0&3! ,v g0/a :"($%/8R)/5 E:"( &2F 7 0#/%w.0/5 E !NLS 0/d8.N3$%O(>0/uB?DD6"()&|*4&/5 E"()/&3!KL/5 fD.NC0/#"(X
U/J"( 0&
εi/JF?H`!q0/%)/v86"(#"(*4/5)/5)C0/vUWX)/lB?DD6"()&qBC"(*+! !) :"(
m0 6"( ! !Ng)/9"&
H = −J∑
i
(
a†i ai+1 + a†i+1ai
)
+∑
i
εia†i ai +
U
2
∑
i
a†i a†i ai ai
F $ F#&
/ M )
< ! ]&#" |:%$2'&)(*& l&3)$50))/%&]! ]0/=8)/5KE086"(#"(,X#"(8c0/=UWX)/21B7DD6"()&*4&/5 ")*4/%6"(M0/v :"($%/48R)/5 E:"( uM (e$5/5 . !N&/%/58)=0)/ $9"( !)/%&y! ,X /86"($5 /"%KL/?>@ 0$5 09FQbWW"( ! 9)v&/%#$5!DR/;#"()/%a!! g0/;B7DD6"()&g*4&/5 VDEN0/+E$%$586"( J E*fDP/5f(>0/4 :"($%/+!)/%9F ! ,X /+)#"()/! "<N))/5* (>
I :"($%/|!)/%7"( 0&N86"($5 /%$#6"(#"$5)/5)/%&YDEN<0/
I1G8 /
n1, n2, · · · , nIQ*+sbYir kE\,bYbTWm,7r7iTWuge(ir!c-iBn`STWc(d,nSe!nSTa`S^_TEm \,y"nST`DnSegT.tg`SV nSVYie-u(eMiVYkWVYiBn.-D`SV\!c0/ \6l|Vm1,!r7iTWugegfir!c 32 E45 µ *nSe(T mgVYikWr6lT`STa`:r!y²nSe(T1,7r7iTau(e(ir!c T76@TWkan 98 n mgTWik`SVtITaiUnSe(T]nS(cgc(TWbYbYVYc(d r!yDu\!V `STEmTabTakan`Sr7cgi&nSeg`Sr!(d7e \,c]nSegVYc VYcgi(b\,nSVYc(d t\6``SVYTa`;tITan$.TWTWc]n=?ri(uITa`SkWr!cm(kanSr,`Si ,7r7iTau(e(ir!c:=(cgkafnSVYr!c µ<; => ? egT \,``<\!cgd7TW^_Tac|nsr!yD\_uIr!nSTac|nSV\,b&t\6``SVYTa` tITan$.TWTac n=.r_?TWbYbisr!ynSe(T]r!ugnSVYkE\!b&b\6nnSVYkWTtITae\ElTWi iVY^_VYb\,`
@
· 5
a! ∑In=1 ni = N
SP! 0$%/f0/l E*fDP/5|(>"()*4;! s0/l :"($%/lM")*4/%&q)DP/f$% 0#"( E9FUN< ! ,g),X"( ! A$%*+8PX! 0;(>
NDPX) 07
I :"($%/
!)/%?"l*4E&/5 O)86"$%/M$% 00$5)/%&-a$#s$% 0))(>r0/| .*fDP/57#"()/%
∣∣n1, · · · , nI
⟩=∣∣n1
⟩⊗∣∣n2
⟩⊗ · · · ⊗
∣∣nI
⟩ F $ F! &
jd"$#vM81GKL/%$5)q(>l 8<7#;/LI !?WL|C "M8Y)/58)/%)/5 .)q"c! ,X /!)/ia! !)
$%$586"( E*fDP/5niFdbW0/C .*fDP/5?)#"()/%7DP/5N
⟨n1, · · · , nIα
∣∣n1, · · · , nIβ
⟩= δαβ
0 0*e"( !!N $ F15&∑
α,β
∣∣n1, · · · , nIα
⟩⟨n1, · · · , nIβ
∣∣ = 1
$%*+8 /5)/5 0/%) S $ F &
! |a$#|0/u*4A"()/u/5K"( !6"()/%&|D.N; ! ,7),Xv"( ! .$%*+8PX! 0 n1, · · · , nI 0&/5A0/u$% 0#"(! . ∑Ii=1 ni = N
FQzE! 0$%/QDRX# 0A"()/d"( ! i/%&v)aDR/u! v"( v/%wE6"( #"()/;"(u0/7 "(*4/!)/a0/?&3!*4/5 0 g(>hu*4&/5 V)86"$%/,X)9)/5*4/5 0&0 !Na!c0/= .*vDR/5g(>C"()*44"( 0&!)/%9FZ0
NDRX# 0+
I :"($%/-!)/%l0/
&3!*4/5 0 s
D =(N + I − 1)!
N !(I − 1)!
$ F!"&
Z0Q/2[0"(*+8 /XS(>@I"7)N)/5*t(>6DPX) 0I
6!)/%
D = 462S>_r0/$%*+8PX!
(>8DPX) 0
8!)/%
D = 6435S0"( 0&+>_
12DPX) 0
12)!)/%
D = 1352078F
$'&&1 U?G (7 1(E UW/2>_)/v! E)E&30$5! ,0/lBC"(*+! !) J8P/5#"()M/D/2k0Ns&3#$50);09 8P/5#"()) ]0/gZ0$)86"$%/XF 6"( ,X0/g)0/g .*vDR/54#"()/ $ F! &2SQ8P/5#"())9Sa$#" P/%$5l"g)8R/%$5\m6$+!)/4"()/e&/2m0 0/%&hF+Z0|/2["(*+8 /XSr"( J8R/5#"()
nia$#
"$5) s!)/i&/2m0 0/%&-DEN
ni = 1 ⊗ 1 ⊗ · · · ⊗ n ⊗ · · · ⊗ 1$ F % &
! =a$)n0/
i1G-8P/5#"()! =8)E&30$59FIHJ/C"( )"(DD)/5KE:"()/ $ F % &dD.N
niS3! Ya$#Y0/M! 0&/2[
i&/5 0)/%0/|)!)/Ca$)
n8P/5#"()/%7 TF
bW0/lBC"(*+! !) q8P/5#"();(>u0/lUWX)/21B?DD6"()&*4&/5 A! q)/%$% 0&qw.6"( E#" 1 ->_aDPX) 0aa,X!KL/5 YDEN=/%wE6"( F#3
H = −J∑
i
(
a†i ai+1 + a†i+1ai
)
+∑
i
εia†i ai +
U
2
∑
i
a†i a†i ai ai
S $ F A &
? egT kar7^_u(bYTanSTac(TWii | µ VYi`STWiBn`SVYkanSTEm nSr-nSe(T]^_r|mTWbiu(\!kWT]iu(\!c(cgTEm tM nSe(T(`SiBn TWcgTa`Sd!t\,cm ¶
· 5
a!e0/? 0/5 ! !! ,l)/5 ,XJ"( 0&g0/! .)/5#"$5 -)/5 ,X
UFIbW0/?86"(#"(*4/21
)/5εiQ"?)!)/21&/58R/5 0&/5 Eu/2[)/5 6"( 38P)/5 .:"( GSL/XF ,0FI"?#"(88! ,;8P)/5 .:"( GFQbW0/
a†i"( 0&ai
"()/f$5)/9"( J"( 0&s"( !! :"( 8P/5#"())7a$#yDP/5N<0/v$%*+*v#"())/5 :"( 0
[ai , aj
]
−=[a†i , a
†j
]
−= 0
"( 0& [ai , a
†j
]
−= δij
F $ F!6&bW0/%)/a8R/5#"())$5)/9"()/4$
a†i&rQ)/5*4iKL/ $
ai
&I";86"($5 /7"(u)!)/i! +0/W>@ ! ia! ,
>_" T
a†i∣∣n1, · · · , ni, · · · , nI
⟩=
√ni + 1
∣∣n1, · · · , (ni + 1), · · · , nI
⟩
ai
∣∣n1, · · · , ni, · · · , nI
⟩=
√ni
∣∣n1, · · · , (ni − 1), · · · , nI
⟩$ F 2&
bWu"( ! 9u)M&/2m0 0/0/W .*fDP/5d8P/5#"()ni = a†j ai
SLa$)e$% E)u0/"()*4"(?!)/
iSR"( 0&qDEN-0! ,g0/l$%*+*f#"()M)/5 :"( 0 $ F!6&7 0/l$9"( s)/5a!)/l0/
! .)/5#"$5 86"(?(>r0/|BC"(*+! !) 8P/5#"()9
H = −J∑
i
(
a†i ai+1 + a†i+1ai
)
+∑
i
εini +U
2
∑
i
ni (ni − 1)F $ F!B&
Z0g0/>_ ! 9a! ,c&3)$50) n/"))*4/0/qB;"(*+! !) 8P/5#"()g 0<)DR//2[8 !$5!-!*4/21&/58P/5 0&/5 .9F pq)/%9KL/59SC/$% 0&/5s"#"( 0 :"( 6"( |! EK"(:"( E :"($%/e"( 0&J /9"9KL/e|0/4/2[3)/5 6"( d8P)/5 .:"( 86"(f(>a0/4BC"(*+! !) :"( $ F!B&>@$% EKL/5 /5 0$%/XF
M&3#$50))/%&J/9"( !/59SO0/4/2[)/5 6"( d86"(#"(*4/5)/5fa$)]$% .) f0/+8.N3$%(>T0/?*4&/5 Rd0/;#"(l(>
U/JFQZ0 :"(,L/;K"( !0/%W(>
U/J0/ $ )/58 !KL/5&d! .)/5^1
"$5 J&*+! 6"()/%|9KL/5M0/l! 0/5$+/5 0/5,XNq"( 0&y.0C088! ,DR/5o/%/5 :"($%/!)/%)88)/%))/%&hFuo g0/; E*fDP/5D6"W&3)$50))/%&-"(DP9KL/XS30/;,X) 0&#"()/C! 0/M) ,X !N<)/58 !KL/M)/5,X!*4/Ma)/58)/%#/5 .)/%&YDEN<0/|! ,X /C .*vDR/5;#"()/
∣∣ψ0
⟩=∣∣n, n, n, · · · , n
⟩ S
a0/5)/n = 1, 2, 3, ...
Fao Y?)#"()/f/2[0"$5 !Nn"()*4;"()/v8! 0/%&s"(;/9"$)y!)/XF
UN]! ! ,(>|$)6"(,L/%&"()*44!geD.K04aENc4g$9"( ! /%&`"( =#;/& ³I8B)$#%g86")/X! 0$%/M088! ,gDR/5o/%/5 y!)/%7$%*+8 /5)/5 !NY88)/%))/%&0/5)/M 0$)6"(,L/l#"( 08P9F?bW0/v! 0 :"(! ,<8)8R/5oN;&3!KL/5 yD.NY0/v86"($5 /21G86"($5 /! .)/5#"$5 1R! 0 :"());(>I?! 0&"()/f$9"( ! /%& LI88 $#;/ ³I8SL!/2S6! $% .#")?)4E#DC) S 7! /!)O_13#DC*7!/LI#)P! 0 :"())9SE! ga$#g0/?#"( 08Pd88)/%))/%&&30/g)y86"($5 /21G :"($%/! .)/5#"$5 0 $@)/%/ *# ,a>@+&/5#"(! &2F]UN&/%$5)/9"! ,J0/#"(
U/J0/v088! ,-)/5* (> $ F!B&;$9"( 0MDR/l 0/5,X /%$5)/%&J"( .N*4)/XFlB?/5 0$%/XS
0/Y,X) 0&#"()/q$% 0)g(>C*4)/s6"( n"! ,X /Y E*fDP/5=)#"()/YDP/%$9"(0)/Y0/
¶;¶
· 5
! E)/5^1!)/+088! ,-/9"(L/5 0|0/+) ,< E$9"( !#"( J(>0/l>@*4/5MpqM! 031 :"()7)#"()/XFubW&/5 $9"( !#"( $%)/%8P 0&)4",X) 0&)#"()/|$% 0! ,e(>u" "$#.E",L 47$#?I8BBLI#<(>h E*fDP/5#"()/%iSL! ea$#g E*fDP/5)#"()/%ua!e0/?"()*4D 0$#0/%&="(W"M! ,X /!)/76"%KL/;"C*f0$#g i/5/5!,X.6"( g)#"()/%ua!="( ! R"()*4&3!D)/%&s9KL/50/|a0 /M :"($%/XF
Hn! /+&/%$5)/9"! ,-0/l#"(U/J
0/! .)/5#"$5 )/5* (>$ F!B&?,L/5)C*4)/"( 0&<*4)/M!)/5 /5KX"( .9S6 .! T! -0/C !!*+!
U/J −→ 0"( ! h .*vDR/5;#"()/%"()/|$%$531
8/%&hFubW0/fN))/5*~$9"( ! /%&-)DP/C! s"8P/5^k0&-86")/XF+M/5 0/5#"( ! !NLS00/M,X) 0&#"()/f(>Q"+UWX)/21B?DD6"()&YN3)/5*~a,X!KL/5 D.N
∣∣ψ0
⟩=
D∑
j=1
c(0)j
∣∣n1, n2, · · · , nIj
⟩ S $ F!&
! ]a$)Df0/=&3!*4/5 0 (>0/gZ0$)J)86"$%/e,X!KL/5 ]D.N $ F!"&l"( 0&
c(0)j
0/$%/(e$5/5 .|(>d0/
j1G E*fDP/5|#"()/XFMbW0/%)/$%E/(e$5/5 E)f"()/D#"(! 0/%&yD.N
) !K! ,40/C!*4/21G! 0&/58P/5 0&/5 .|z$)&3! ,L/5;/%wE6"( T
H∣∣ψν
⟩= Eν
∣∣ψν
⟩ S
! 4a$# ∣∣ψν
⟩ Q0/ν1G4/5 0/5,XN+/5!,L/5 0#"()/?"( 0&
Eν0/$%)/%8P 0&3! ,f/5 0/5,XN
/5!,L/5 EK"( !0/XFbW/5!,L/5 .KX"( !0/M8)D /5* # !KL/%&< .*4/5$9"( ! !N<D.N</2[8)/%#! ,+0/BC"(*+! !) :"( %$ F!B&! 0/C .*fDP/57D6")aa!Y0/M*e"(\[Y/5 /5*4/5 .)
⟨n1, n2, · · · , nI
∣∣ H∣∣n′1, n
′2, · · · , n′I
⟩ F $ F &
o c)&/5+)$% 00$5+*e"(\[ 0/<6")q 'iKL/5e"( ! $%*fD! 6"( 0e(>
n1, · · · , nI"( 0& n′1, · · · , n′I
0&/5M0/+$% 0)#"(! .) ∑Ii=1 ni =
∑Ii=1 n
′i =
NSVa$#*e"(L/%
D ×D*e"(\[y/5 /5*4/5 .)9FlzE! 0$%/v 0/9"()/%); 0/5!,XEDRM088! ,
7"#*4/%&hS0/C .*vDR/5;(>r 0 31G9/5)e*e"(\[Y/5 /5*4/5 .)7 !N ∼ 5DS30,XTF
&&² E &
"!$#bW0/y,X) 0&#"()/J"( 0&0/s>@/5 /2[$5!)/%&#"()/%7 0/MD#"(! 0D.N</2[0"$5&3:"(,L 6"( !#"( s(>I0/MB;"(*+! !) *e"(\[-"( ! i )/5KX"( !6"()/e)/5KL/5#"( !*+8 /eD0)/5KX"(D /%9F JD0#/5K"(D /e/g"( !)/9"&3Ny0)/%&J! J0/&3)$50# '"(DRiKL/<v0/-"9KL/5#"(,L/YE$%$586"( c E*fDP/5e"(4"s)8R/%$5\m6$<!)/
iFJo
|D#"(! 0/%&D.Ny$9"( $5 :"(! ,0/4/2[38P/%$5#"( JK"( !0/e(>0/4$%$586"( J E*fDP/58P/5#"()7(>I0/
i1G!)/X
¶6
· 5
ni =⟨ψ0
∣∣ ni
∣∣ψ0
⟩ $ F 5&H`!-0/C,X) 0&#"()/ $ F!&W 0/M$9"( -)/5a!)/M0/M/2[38R/%$5#"( YKX"( !0/|"( 0&<,L/5)
ni =
D∑
j=1
D∑
k=1
(
c(0)j
)∗
c(0)k
⟨n1, n2, · · · , nIj
∣∣ ni
∣∣ n1, n2, · · · , nIk
⟩ F
zE! 0$%/-0/Z0$)D6"ee"( `/5!,L/5 ED6"=(>;0/sE$%$586"( ` .*fDP/5g8P/5#"()9S0/+/2[8P/%$5#"( K"( !0/! y0/+* $9"( yDP/+/9"! !Ns/5KX"( !6"()/%&J"( 0&yD.Nq0! ,-0/0,L 6"( !!NY(>I0/|Z0$)=D6" 0/f"(!KL/%;"(
ni =
D∑
j=1
D∑
k=1
(
c(0)j
)∗
c(0)k n
(k)i δjk
=
D∑
j=1
‖c(0)j ‖2n(j)i
$ F 1 &
a0/5)/g0/=! E)/5,L/5n
(k)i
"()/g0/= E*fDP/54(>;"()*4+ 0/i1Gc!)/=(>70/
k1
.*vDR/5|)#"()/XF7Z0;#"( 0 :"( 6"( I! .KX"(:"( .M :"($%/%;;D0)/5K"(D /la! ! rDR/!)/C! 0&/58R/5 0&/5 E;"( 0&-)/58)/%#/5 .)0/Mm0 ! !! ,+>_"$5)
N/IF
"! ! # 00/5WD0)/5KX"(D /0/ E*fDP/5KX"(:"( 0$%/C"("|)!)/ia$)JW"f"( )<0)/%&q! 0/+&3)$50)) "(DRiKL/XF+bW0/ .*fDP/5|KX"(:"( 0$%/+)/5KL/9"(
! 3>@*e"( + f0/$%*+8PX! +(>00/d,X) 0&#"()/! l)/5*4I(>0 .*fDP/5Q#"()/%9F y0/+) ,X !Ns)/58 !KL/+)/5,X!*4/XShGF /XFv>_M"= :"(,L/#"(
U/JSh0/,X) 0&#"()/
$% 0)O"(88)9[!*e"()/5 !Nf(>6"a)! ,X /Q .*vDR/5I#"()/d"( 0&ME0O0/u .*fDP/5rK"(:"( 0$%/a! ! QDR/*e"( ! GFl $% E#"9SA"( ! Q E*fDP/5f#"()/%|$% E!D)/+)<0/,X) 0&#"()/! c0/Y8P/5^k0&c)/5,X!*4/XSa$))/%) !)4! "y :"(,L/54 E*fDP/5eKX"(:"( 0$%/XFbW0/ .*fDP/57KX"(:"( 0$%/f"(a0/
i1Gs!)/Ca,X!KL/5 YDEN=0/M/2[38P/%$5#"( YK"( !0/
σi =(⟨ψ0
∣∣ n2
i
∣∣ψ0
⟩−⟨ψ0
∣∣ ni
∣∣ψ0
⟩2)1/2 S $ F "&
a0/5)/g0/<)/%$% 0&])/5* ! 0/<)w.6"()/g)ll0/<)wE6"()/g(>?0/=$%$586"( .*fDP/54"(v0/
i1Gc!)/XFqbWv*4/9"( 0iSQ 0/g6"v)y$9"( $5 :"()/=0/g$%$586"(
.*fDP/5+"( 0&J0/e/2[38P/%$5#"( KX"( !0/e(>a0/e8P/5#"()n2
i
! )&/5v)$%*+8)/0/y .*fDP/5KX"(:"( 0$%/
σi(>0/
i1G !)/XF !0,X 0/J .*fDP/5KX"(:"( 0$%/
8)9K&/%e! 3>_*e"( )&/5 E\> Nc0/86")/Y(>M0/)N)/5*YS!e4 0g!)/%&"="( n)&/5g86"(#"(*4/5)/5<)J&/5 E\> N0/s8P/5^k0&`)pqg! 0 :"()=86")/#"( 0! *! ,d"?09a q Y0/M /2>@6"( 0&Y&/f(>rm0,X)/ FXS0! Y0/M)/5,X q(>
¶u
· 5
E FE EU/J
E
E
E
E
E
σ
E FE EU/J
E
E
E
E
E
c2 max
-D ]B+ '!'+] C>$ (-+16σ+7!Bu 16] 1.$+!
C2max
(D. !+,!.^16!+ |!-$+!-
U/ \D!$*/
6C>,+'+
6!$ PL4@7 4+$^S9+7
8CY+
+8!$ PX7$7 4+$^SZ 4!!-1. HY!-.+,!@4 !--+4@!-+4+](-@+,!>!] !
+] C>$B()@+>16σ@B5!- +7.HY.+7D$+,!$
0/l86")/l#"( 0! (U/J)cr = 4.65
Sha$#"|&/5)/5*+! 0/%&yDENspq .)/21 7W"( $9"( $5 :"( 0 *1LS ,S0/<k00$56"( 0=$)6"( ,L/ !! 0/9"( !Nca!'0/Y! E)/5#"$5 )/5 ,Xq"( 0&Y&+ 0a! 0&3$9"()/v"l#"( 0)! Y8R! E9F
"!#! "$# zE!*+! :"(d$#6"(#"$5)/5$%(>R0/W,X) 0&#"()/;"()/W)/21KL/9"( /%&-DEN<0/M*e" [!*v* $%/(e$5/5 .
c2max = max(|c(0)i |2
) $ F % &
(>70/=/2[386"( 0 $ F!&2F KX"( !0/c2max ≈ 1
! 0&3$9"()/%6"(l0/e,X) 0&#"()/∣∣ψ0
⟩ $% 0))(>A"v! ,X /; .*vDR/5a#"()/21! <$% .#"9S0"v*e"( ! PKX"( !0/c2max
*4/9"( 06"(0/f#"()/C7"8P/58RX)! (>r*e"( EN< .*fDP/5?)#"()/%9F
$"!&%(')%# 00/5=)!*+8 /sD0)/5KX"(D /se0/q/5 0/5,XN',."(8TSaa$#n<">_/9"()/g(>?0/gpqv! 0) :"()v86")/XFyofl&/2m0 0/%&c"f0/</5 0/5,XN&3\h/5)/5 0$%/DP/5/%/5 0/M,X) 0&#"()/v/5 0/5,XN
E0"( 0&Y0/Mm0)/2[3$5!)/%&#"()/
E1S0"( 0&)+!
$9"( YDP/M/5KX"( !6"()/%&Y&3!)/%$5 !N=> )* 0/;m0))4/5 0/5,XN-/5!,L/5 .KX"( !0/%9
Egap = E1 −E0$ F 1A &
Zr!,X)/ F $ /2> &I&/58$5)0//5 0/5,XN+(>h0/Wm0) A |/5 0/5,XN4/5!,L/5 0#"()/%d(>O"MN3^1)/5* (>
6DPX) 0u
6)!)/%d"(d&3\h/5)/5 .d)/58 )!KL/! .)/5#"$5 <)/5 ,X0
UF $? 0/
¶
· 5
E $E E E ,E E$.+!-!
ν
E
E
E
E
E
Eν
[J]
E $E EU/J
E
$E
Egap
D D ^H + !- 4./Q!B z!^! E .+.-\.$+]()4$9/ \D!$/
6C>,+ +
65!-$ /07Du.-.+,!+,!$-1Z!-+ !-.+!
U 4+$=*!
+,!$-1Z!-+ |!-$+!- PXC>!!-G!!H>SU = J, 5J, 20J, 50J
+ 1$+ . ! @+.+$\ HVC>.!|9$.+V!-+7 +>7 !^|! 16!$7 |!^! P0+ ! (.-\ 46/Q! / !-H4!^S+ ! UV!! 1$$ P
U/J > 4.65J 0SZ B,! +7V7D '!-$+.-\
HEgap
(D. ! +,!-.^1Z!-+ |!-$+!-U/0 \D|!-. /
6CY+> +
6!F PL4@7
4+FS+78CY+>+
85!-$ PL7F7 4+FSZ
$5 /9"( !Ne)/%/%6"(a)/5KL/5#"( h/5 0/5,XN4,."(80a8R/5 =8Y"(!,X0/5W! E)/5#"$5 )/5 ,X0$ m0,X)/ F &2FtbW0/s,X#"(8 `0/s!,X.-6"( 0&n&/q09=0/y/5 0/5,XN',."(8(>DPX) $CN3)/5*4a(>A$%*+*4/5 0#"()/Mm0 ! !! ,>_"$5))4$
6"()*4a
6!)/%"( 0&
8"()*4v
8!)/%&?KL/5)0|0/+! E)/5#"$5 )/5 ,X
UF+bW0/4DP/56"%Kf(>"
,X)9a! ,J/5 0/5,XN,."(8]a!! 0$5)/9"! ,#"(U/J
$9"( DR/=/2[8 :"(! 0/%&]DEN$% 0&1/5! ,"s)N)/5* a!c$%*+*4/5 0#"()/<m0 ! !! ,y>G"$5)9Sd/XF ,0F .*vDR/5g(>786"($5 /%/%w.6"( a E*fDP/5;(>Q!)/%
N = IF &3)$50))/%&/9"( !/5! Y7#/%$5 TS3>@?*e"( !
K"( !0/%+(>U/J
0/-N3)/5* ! ]0/-)8R/5^k0&86")/Y"( 0&'"( ! W E*fDP/5e#"()/%$% .!D)/Y)J0/-,X) 0&#"()/XF`H`!'! 0$5)/9")! ,#"(
U/JSd0/-! E)/5#"$5
DR/5o/%/5 q0/f"()*47DP/%$%*4/%*4)/l"( 0&Y*4)/|&*+! 6"( E|"( 0&-.0;Z0$)-#"()/%a!g*4)/?6"( = 0/;"()* 8R/5!)/?"()/;88)/%))/%&hSEDP/%$9"(0)/70/5Ne"()/;/5 0/5,L/5\1$9"( ! !N| 3>_"9KL#"(D /XFQ l0/ !!*+!I(>P"( l! 3m0 !)/! E)/5#"$5 4)/5 ,X
USX !N|0/
Z0E$#-#"()/|a!q 0/f"()* 8P/57!)/f)/5*e"(! 09S0a$#s6";",X) 0&#"()/l/5 0/5,XN
E0 = 0F ;0/|)="&3*+\[)/%C(>dZ0$)Y)#"()/%7a!qg?*4)/l"()*4?8R/5;!)/
! ]0/gm0)+/2[$5!)/%&c#"()/-"( 0&0/<) ,! .)/5#"$5 ))/5 ,XU"( /5 0/5,XN
,."(8s"(88R/9"()iFbW0/M8 ; Y0/|!,XE?(>rm0,X)/ F 099S06"(70/|/5 0/5,XN-,."(8
Egap|! 0$5)/9")! ,s ia !Na!0/4! E)/5#"$5 ]))/5 ,X
U! 0/e8P/5^k0&
86")/XS0D?! 0/M)/5,X y(>Q0/|86")/f#"( 0! q"(U/J ≈ 5
S00/v 8R/f#"()
¶
· 5
)M! 0$5)/9"#/a84)1! 40/apqu! 0 :"()u)/5,X!*4/XFQbWd! 0&3$9"()/%u6"(u0/am0)
/2[$5!)/%&)#"()/e! ]0/gpql! 0 :"())/5,X!*4/el&*+! 6"()/%&D.NJ E*fDP/5+#"()/%a!J/2[0"$5 !Ns 0/&D !Ns$%$58/%&!)/XFvzE! 0$%/0//5 0/5,XNy(>d0/+Z0$)q#"()/%a! 0/l&D !NE$%$58/%&y!)/vC"(88)%[3!*e"()/5 !N
USVKX"(:"( y(>
UDENs"g$%/5^1
#"(! "(*4 E$)6"( ,L/%a0/C/5 0/5,XN=,."(8Egap
DENe0/C#"(*4/M"(*4 .9F <$% .#")9S>_fN3)/5*4M! J0/e8R/5^k0&)/5,X!*4/+! Ja$#"( ! dZ6E$#y)#"()/%f$% .!D)/g)0/v,X) 0&#"()/l0/v/5 0/5,XNq8P/%$5* $ /2>@76"( 0&y&/v(>um0,X)/ F &7?#"(0/5*4TF
$"!&%(' ! !"!$# bW0/g/5 0/5,XNy#"( 0^>@/5∆E
l"=K"( !6"(D /gD0)/5K"(D /e! $9")/(>d!*4/21&/58P/5 0&/5 .vB;"(*+! !) :"( 0
H(t)SRa$)"()/l! !)/%&q! $)6"(8)/5) %
"( 0& A F-bW0/g/5 0/5,XN#"( 0^>_/5∆E
l&/2m0 0/%&]"f0/e/2[38R/%$5#"( ]K"( !0/g(>a0/! !:"( rB;"(*+! !) q8R/5#"()
H0 = H(t = 0)"( 0&Y0/|$5)/5 E;#"()/ ∣
∣ψ, t⟩ *+! E0
0/</5 0/5,XN(>70/-#"(! ,#"()/XFo !:"( aB;"(*+! !) '8P/5#"()+*4/9"( 0! 6"($9")/XS06"(0/M#"(! ,e#"()/ ∣
∣ψ, 0⟩ 7"( Y/5!,L/5 0#"()/f(>
H0
∆E =⟨ψ, t
∣∣ H0
∣∣ψ, t
⟩−⟨ψ, 0
∣∣ H0
∣∣ψ, 0
⟩ $ F 6&
bW<D0)/5KX"(D /q8)9K&/%g0/y/5 0/5,XN'a$#g#"(L/5 nD.N0/q)N)/5* 0&/58P/5D6"( TSV"( 0&-E0! 0&3$9"()/%?/2[3$5!)/5*4/5 E?(>I0/|N3)/5*YF
" !! # UWX)/|ju! 0)/5! s$% 0&/5 0#"( s#"(L/%8 :"$%/fDR/5 i "8P/%$5\m6$;#"( 0! <)/5*+8R/5#"()/XS0\>O0/; 9/%)! ,X /;86"($5 /M/5 0/5,XNe /5KL/5 a"()/$%$58/%&hFdzE! 0$%/?)N)/5*4&/%)$5!DP/%&4D.N+0/?B7DD6"()&e*4E&/5 P"()/?"))*4/%&4)|DP/"(M9/5)-)/5*+8P/5#"()/XSA0/+$% 0&3! (>DP/5! ,- 9/5f6"( J0/#"( 0)! J)/5*18P/5#"()/"( !"%N3W#"(^m6/%&hFIo e)&/5d)v&/2m0 0/?" "6LI#DC*7#;/MI8<'7W"M8BBLI#-/)/2>_/5 q0/|>_*f :"( J(> 4r/5 )X)/+"( 0& $? 0#"(,L/5 *!"/,S6a$##"%N3;6"(C"g$% 0&/5 31#"()/Q8)/%)/5 .\>P 0/7(>R0/O#?I8 E .LM4=8S /_"EL(/"6L !D"6 = A4$%$58/%&hFIbW0/ 6"(#"( ID!#"( M"()/|0/f/5!,L/5 EKL/%$5))M(>u0/L#.7)W4ELCA C*7#;/7=8BAI8B!q&/2m0 0/%&DEN=0/C*e"(\[Y/5 /5*4/5 E)
ρ(1)ij =
⟨ψ0
∣∣ a†j ai
∣∣ψ0
⟩ $ F 2&
! qa$) ∣∣ψ0
⟩ 70/f,X!KL/5 s,X) 0&)#"()/XF?bW0/lE$%$586"( y E*fDP/5C(>"4 6"(31#"( dD!#"( dM0/+$%)/%8P 0&3! ,s/5!,L/5 .KX"( !0/XF4bW0/+*e"$5)X#$%8$e$%$586"( J#"(m6/%&\>V0/W#"(|(>R0/aE$%$586"( 4 E*fDP/5"( 0&0/W)#"( V"()* E*fDP/5u)/21*e"(! 0m0 !)/M! Y0/C0/5*4&3NE 6"(*+$9"( r !!*+! $
N, I −→ ∞ a! /N/I = const.
&2FbWE09ST/+&/2m0 0/+0/+$% 0&/5 0#"()/>@#"$5
fc = NC/NFlZ)* 0/#"$%/+(>0/
¶
· 5
0/21GDR&3Nl&/5 0)!Nl*e"(\[lTrρ
(1)ij = N
>_ ! 9u&3!)/%$5 !NLSE6"(Q! m0 !)/N3)/5*40/M :"(,L/%7/5!,L/5 EK"( !0/v /%)77/%w.6"(
N/IFbW7!*+8 !/%6"(?0/|$% 0&/5 0#"()/
> #"$5 Y?"( !"%N3 :"(,L/576"( 1/I"( 0&Y)/5*e"(! 0am0 !)/XF
""! ! !! # py"$5)X)$%8$9"( ! !NLSr8P/5^k0&3!NyC0/+8)8P/5N(>"N))/5*~)lk69 a!07> $5 TFubW0/M)8R/5^k0&<>@#"$5 (>u"l !w.&$9"( YDR/&/2m0 0/%&ya!! J0/(1_k0&J*4&/5 GCHJ/+$% 0&/5v"gk0&y! "=DP/a$)J*49K! ,ga!"4KL/5 $5!N
vF7o 0/v)/%?> #"(*4/l(>u0/fDP/f !N0/v8P/5^k0&
> #"$5 <(>T0/k0&g*49K! ,0FQ g> #"(*4/?0/7! 0/5$;/5 0/5,XNe(>T0/?)8R/5^1k0&<> #"$5 s7/5KX"( !6"()/%&YDEN-0/|&3\P/5)/5 0$%/v(>Q0/f/5 0/5,X/%?! s"+N3)/5* a!"M*49K! ,DR/;"( 0&g 0/?a!-"MDP/;"()/%9FQZ)*0/7! 0/5$?/5 0/5,XN="( 0&e0/KL/5 E$5!oN
v0/|8P/5^k0&<> #"$5 s$9"( -DR/MD#"(! 0/%&hF
6"( ,X0/|)+a*e"$5)X)$%8$f8$5)/| 0/|$9"( &/2m0 0/M0/f8R/5^k0&-> #"$21
fs(>00/UWX)/21B?DD6"()&N3)/5*YFTHJ/$% 0&/5Q"a :"($%/a!l$5N3$5 !$DR 0&1
"(NY$% 0&3! 09SP;*4/9"( 0?6"(;0/"()*4;$9"( y08qDR/5o/%/5 y0/l!)/%1"( 0&
I&3!)/%$5 !NLFuHJ/;>@)$%/;0/M"()*4W! <0/;)N)/5*)l*49KL/CDEN="vKL/5 E$5!oN
vs! <0/
:"($%/MD.N-" +M"( !! /5TDPX9F
∣∣ψboost
⟩=
N∏
i=1
exp
(i
~vsmxi
)∣∣ψ⟩ S $ F B&
! sa$#m;0/f*e")C(>"e86"($5 /XS
N0/l .*vDR/5M(>d86"($5 /%|"( 0&
xi0/
8RX! ]8R/5#"()(>0/i1G86"($5 /XFs 8RX! ]86"$%/g0/e8P/5#"()
xi
!*+8 !N0/<KX"(:"(D /xi"( 0&.0+0/<DPEX)8P/5#"()+)/%&30$%/%&c)"q86")/
exp (iϑ(xi))
ψboost(x1, · · · , xN ) =N∏
i=1
eiϑ(xi)ψ(x1, · · · , xN )S $ F &
a!0/v*e"( EN.1GDPX) W"%KL/2>@ 0$5 ψ(x1, · · · , xN ) =
⟨x1, · · · , xN
∣∣ψ⟩ ! J$%(1
)&3! 6"()/)/58)/%)/5 E#"( TF bW0/86")/ϑ(xi)
g$% 0/%$5)/%&`)J0/sKL/5 E$5!oNvsKE:"
vs =~
m∇ϑ(xi)
F $ F#"1 &bW*4/9"( 09S66"(70/MKL/5 $5!N<m6/5 &
vs)/% !)> )* "+86"(:"( OKX"(:"( s(>Q0/
86")/ϑ(xi)
F<HJ/g"))*4/e"-)#"( 86")/4oal(>θ9KL/5l0/4 /5 ,X
L(>a0/
:"($%/M! -0/C>_ ! 9a! ,4>G" T
ψ(x1, · · · , xk + L, · · · , xN ) = eiθψ(x1, · · · , xk, · · · , xN ) ∀k S $ F#"5&
¶ 1
· 5
a$#J|/%wE!K"( /5 EM)Y"gk69 a!0/KL/5 $5!Nvs = ~θ
mL
FlbWCk6i C 0 31&3))!86"(!KL/e"( 0&!)#"( 6"( da$#"()/+8)8P/5/%f(>0/48P/5^k0&3!NLF 6" 1 ,X0/l)g0/v*e"$5)X#$%8$l&/2m0 ! ,X!KL/5 J"(DP9KL/XSP0/v8R (>d0/lN3)/5*YSa$#s>_ ! 9;0/fk69 ?&/2m0 0/%&"?0/v)8R/5^k0&> #"$5 TF?bW?> #"$5 q$9"( $5 :"()/%&> )*&3\P/5)/5 0$%/gDP/5/%/5 ]0/4,X) 0&)#"()/g/5 0/5,X/%l(>0/4oa)/%&"( 0&- 0 31Goa)/%&)N)/5*4iF
o ')&/5e)D#"(! 0/-,X) 0&#"()/(>C0/-oa)/%&'N3)/5* /Y*e"(8'0/ !#"(N<#"( 0^>_*e"( s E)+0/|BC"(*+! !) :"( TFuHJ/M$% 0&/5
Bθ =N∏
i=1
Uθ,i =N∏
i=1
exp
(iθ
Lxi
) $ F#" &
"0/?DRXW8R/5#"()>@a"vN))/5*(>N86"($5 /%9FdbW0/
Uθ,i"()/? $9"( V !#"(N
DPEX)v8P/5#"())4"$5! ,q !)/iFsbW0/=8R/5#"()
Bθv 0l !#"(NLSQDP/%$9"(0)/
!f&/%M 0|)/%)8R/%$5|0/e$5N$5 !$+DP 0&0"(N$% 0&3! 0v(>W0/4BC"(*+! !) :"( Tf/W"( .0/?"()*4)|,."(! =0/86")/
θ/δD.N+088! ,v> )*!)/)f!)/XS.a!
δ0/
:"($%/g86"$5! , 1Q|! 0$5 !0&/%"( )088! ,DP/5/%/5 ]0/e)!)/%1"( 0&
IF=bW0/
$5N3$5 !$MDP 0&0"(N<$% 0&3! 07 &)/%w.!)/|"86"#/C,."(! (>θ>_aa08TF
bW0/fB;"(*+! !) 8P/5#"()a>_?"N3)/5*~a!Y0/C86")/|aθ)/9"&70/5
Hθ = B†θHBθ
$ F#"1"&
a!H0/4BC"(*+! !) :"( (>a0/+ 0 31Goa)/%&N3)/5*YFg E)E&30$5! ,s0/4UWX)/21
B?DD6"()&YBC"(*+! !) :"( %$ F!B&W 0/M,L/5)
Hθ = B†θ
(
−J∑
i
(
a†i ai+1 + a†i+1ai
)
+∑
i
εini +U
2
∑
i
ni (ni − 1)
)
BθF
zE! 0$%/a0/7E$%$586"( g .*fDP/58P/5#"()d&/%Q 0d$#6"( ,L/0/a8PX! e(>T";86"(^1$5 /C0/;8PX! -8R/5#"()
xj(>A0/
j1G-86"($5 /|"( 0&<0/C8P/5#"()
niSa$#
$% E)C0/l"()*4M"(C!)/iSP$%*+*v)/XF;bW0/vDPEXC8P/5#"();$9"( qDR/l/2[386"( 0&/%&
! "br"%N 7)/5/%9Sa$)s$% 0!)(>A8P9/5)7(>xjS0"( 0&<E0a!$%*+*f)/%7a!
0/a E*fDP/5d8P/5#"()9SL)FrbWu*4/9"( 0Q0/a! .)/5#"$5 <"( 0&4/2[3)/5 6"( 8R)/5 E:"( )/5*x(>0/+B;"(*+! !) :"( ]"()/ 0f" P/%$5)/%&DEN
Bθ"( 0&yD.Nq0! ,-0/8)8P/5N
U†θ,i = U−1
θ,i
(>I !#"(N-8R/5#"())? 0/C,L/5)9
Hθ = −J∑
i
(
B†θ a
†i ai+1Bθ + B†
θa†i+1ai Bθ
)
+∑
i
εini +U
2
∑
i
ni (ni − 1)F
bAm0,X)/=l09 0/eDPXv8P/5#"() $ F#" &v"$5) 0/g088! ,y)/5*/)/5a!)/e!M! "=8)E&30$5v9KL/5| :"($%/g!)/%M! 0)/9"&J(>W86"($5 /%9F4bW0/486"($5 /%
¶
· 5
"()/v! 0&3)! ,X6"(D /g"( 0&! 0$%/ $ F#" &M" h/%$5)|"( ! Q86"($5 /%M! 0/l :"($%/4 0/$9"( Y/2[8)/%)0/
Uθ,i! $ F#" &WDEN
Uθ,i → U′θ,i = exp
(iθ
Lξini
) S $ F#" % &
a!ξi0/|$%E)&3! 6"()/v(>Q0/
i1G :"($%/v!)/|"( 0&
ni0/vE$%$586"( s .*fDP/5
8R/5#"()?"()!)/iFQbW0/5)/2>@)/XS60/CDRX8P/5#"())/9"&
Bθ =
I∏
i=1
exp
(iθ
Lξini
) F $ F#" A &
"( /2[0"(*+8 /M/|/5K"( !6"()/|0/|/2[8)/%#
B†θa
†i ai+1Bθ =
I∏
j=1
exp
(
− iθ
Lξjnj
)
a†i ai+1
I∏
k=1
exp
(iθ
Lξknk
) F $ F#"16&
HJ/M$% 0&/5;"Z0$)-#"()/f -0/M!,X.9F am0)a0/|8P/5#"()Uθ
"$5)7 -#"()/|"( 0&-8)&30$%/%?"l8)&30$57(>I/2[38P 0/5 .:"( O)/5*4
exp
(iθ
Lξ1n1
)
· · · exp
(iθ
Lξjnj
)
exp
(iθ
Lξj+1nj+1
)
· · · exp
(iθ
LξInI
) S$ F#"92&
a!07" P/%$5! ,40/C#"()/C!)#/5 \>^FQbW0/5 -0/M#"()/;$#6"( ,L/%&-D.N<"l08Y(>r 0/86"($5 /l> )* !)/
j + 1)<!)/
j,."(! ! ,-0/l>_"$5) √
nj + 1√nj+1
FlzED0)/21w.0/5 E !N0/lDRXC8R/5#"()| q0/ /2>@;(> $ F#"16&?"$5)M y#?F#"()/+"( 0&$5)/9"()/%?"( )4"l8)&30$57(>Q/2[8P 0/5 .:"( A)/5*49
exp
(
− iθ
Lξ1n1
)
· · · exp
(
− iθ
Lξj(nj + 1)
)
×
exp
(
− iθ
Lξj+1(nj+1 − 1)
)
· · · exp
(
− iθ
LξInI
) S
! 'a$)nj
"()/Y0/E$%$586"( ` .*fDP/5)<(>;0/! !:"( ?Z0$)c#"()/XFnbW0/%)/)/5*4"()/;"( !*4X$9"( 0$%/5 ! /%&ga!e0/78)/5KE0 0/%9S !Ne"C86")/)/5*e"(! 0W&30/)0/M088! ,0
exp
(iθ
L(ξj+1 − ξj)
)
= exp
(iθ
Iδδ
)
= exp
(iθ
I
) S
! ]a$#δ0/< :"($%/-86"$5! ,0FZ0+0/-)/%$% 0&hS0/5*+!:"( `"&/.^! E4)/5*
0/4,L/5)|0/=$% .^,."()/%&/2[38)/%) (> $ F#"1B&2F-Zr! 6"( ! !NLSr0/eBC"(*+! !) :"( >@a)/%&-DP 0&0"(N-$% 0&3! 0)/9"&
Hθ = −J∑
i
(
eiθ/I a†i ai+1 + e−iθ/I a†i+1ai
)
+∑
i
εini +U
2
∑
i
ni (ni − 1)S
$ F#"1B&
¶ @
· 5
- -H ! HY.%N7 /0^16!+
fs
/ \D!$/10CY+>
+105!-$ (D$ !-B+!-.^16!+=|!-$+!-
U
+=!- -.+ /Y! H>$N7 !- UV!!z+%4@! H!--+5!-+
U/J ≈ 5 − 8+
1$+$'! -H=7D$1.4+B/>! HY.N@7/0^16!+
$E E+,!$-1Z!-+ !-.+!
U [J ]
E
E
E
E
E
f s
! Ya$#e±iθ/I "()/|0/|)(1$9"( ! /%& 77 /|86")/C>G"$5))9F
bW0/,X) 0&)#"()/e/5 0/5,X/%|(>0/a)/%&"( 0&y 0 31Goa)/%&E
(0)θ
"( 0&E(0)
N3)/5* "()/<D#"(! 0/%&D.N/2[0"$5+&3:"(,L 6"( !#"( '(>70/<BC"(*+! !) :"( 0Hθ
"( 0&
HFQbW0/;&3\h/5)/5 0$%/;DR/5o/%/5 -0/%)/;/5 0/5,X/%W0/7! 0/5$C/5 0/5,XNg(>O0/7k69a! ,
8P/5^k0&h
E(0)θ −E(0) =
1
2Msv
2s
$ F#"1&
Hn!0/eKL/5 E$5!oNvs = ~θ
mL
"( 0&JD.N! .)&30$5! ,q0/g8P/5^k0&J>@#"$5 K:"0/M*e"#
Ms = fsNm 0/|D#"(! 0
fs =2mL2
~2N
E(0)θ −E(0)
θ2
$ F % &
a!m0/=*e")+(>? 0/Y"()*YS
N0/= .*fDP/5e(>M"()*44"( 0&
L0/< /5 ,X
(>d0/l :"($%/XF|bWM/%wE6"( y;K"( !&s>_;*e"( ! Ioa|"( ,X /%θF;UNY)/58 :"$5! ,
0/M8)/2>G"$5)?(>I0/M! 0/5$|/5 0/5,XN~2
2m
DEN<0/M 0/5 ! !! ,g)/5 ,XJ"( 0&Y0/
/5 ,XL(>0/e :"($%/4D.N0/e .*vDR/5+(>a :"($%/g)!)/%
I 0/+m0 6"( ! !N,L/5)"(
/2[38)/%) ->_a0/M8P/5^k0&<>@#"$5 Y! Y0/C :"($%/X
fs =I2
JN
E(0)θ −E(0)
θ2
$ F % 5&
Zr!,X)/ F#"J0ie0/s8R/5^k0&'> #"$5 fs(>f"J)N)/5*(>
10DRX) 0g
10!)/%CKL/5)0C0/l! E)/5#"$5 )/5 ,X
UFvbW0/+6"(8&/%$5 !! 0/! q0/)/5,X
U/J ≈ 5 − 8! 0&3$9"()/%;0/f86")/|#"( 0! sDR/5o/%/5 q0/f8P/5^k0&y"( 0&pq
! 0 :"()786")/XF
6
· 5
h6! M )
]+)/%$5 0/-B7DD6"()&]*4&/5 >@4"s(1$%*+8R 0/5 EgZ0/5*+a,."&3^1$50))/%&hFJbW0/< E*fDP/5e#"()/=D6"4"( 0&]0/-B;"(*+! !) '8P/5#"()+! .)&30$%/%&"( 0&<0/M*e"(! D0)/5KX"(D /%?"()/C)/5K/5/%&hF
9< ! ]&#" |:%$2'&)(*& 6"( ,X0/g)Y0/gUX#/21B7DD6"()&]*4&/5 $@)/%$5 F &M0/e :"($%/e8P)/5 .:"( W")*4/%&<)DP/M (e$5/5 . !N-&/%/58s"( 0&"( ! O"()*4;"()/C! -0/5!,X) 0&#"()/XS6.00/5N|$9"( fDP/u&/%)$5!DP/%&fDENM E$9"( !)/%&fH"( /5O>@ 0$5 09F $%*+8RX)! l(>0/%)/! ,X /21G86"($5 /f#"()/%a>@ao486"($5 /f8R/%$5/%a)/58)/%)/5 .)/%&D.N<
I1G8 /%
n(c)1 , · · · n(c)
I "( 0& n(d)1 , · · · n(d)
I $ F % &
! la$#n
(c)i
I0/ E*fDP/5u(>V86"($5 /%d"(Q!)/i(>V8P/%$5/%
c"( 0&
I0/W E*fDP/5
(> :"($%/+!)/%9F+zE! 0$%//4"()/+&/%#$5!D! ,"<N))/5* (>>@/5*+ $4"()*4|a$)DR/5N0/4I"( !W/2[$5 !0 8! 0$5!8 /XS"(l*4X 0/-"()*(>;"! 0&+"( ! 9/%&"(="!)/XF bWE09S0/qE$%$586"( n .*vDR/5)Y"()/s)/%$5)/%&n)
n(c)i = 0, 1 FbWv!*+8 !/%l6"(0/g .*fDP/5+(>;"()*4(>;"s$%/5#"(! )8R/%$5/%l$9"( 0+/2[3$%/%/%&
0/s E*fDP/5Y(>f :"($%/!)/%9FbW0/q*4&/5 M86"$%/qg0/5 $% 00$5)/%&DENn"( ! $%*+8RX! 0 $ F % & 0&/50/|$% 0)#"(! .)
I∑
i=1
n(c)i = Nc ≤ I
"( 0& I∑
i=1
n(d)i = Nd ≤ I
$ F % "&
! -a$)Nc
0/|86"($5 /M .*vDR/5;(>Q)8R/%$5/%cFubW0/|/5 /5*4/5 E)?(>I*4&/5
86"$%/|"()/M0/M E*fDP/5?#"()/%
∣∣n(c)
1 , · · · n(c)I α
⟩⊗∣∣n(d)
1 , · · · n(d)I β
⟩ F $ F %1% &
bWd*4/9"( 06"(d0/?*4&/5 R86"$%/?$% 0)(>A"( ! R$%*vD! 6"( 0a(>T0/78PX)!D / .*fDP/5D6"l#"()/%(>0/eoy8P/%$5/%9FsbW0/=&3!*4/5 0 c(>?"s! ,X /g)8R/%$5/%Z0E$#=)86"$%/M
Dc =
(
I
Nc
)
=I!
Nc!(Nc − I)!
S $ F % A &
"( 0&gE0W0/C&3!*4/5 0 -(>O0/CZ0E$#g86"$%/C(>r"f(1$%*+8R 0/5 E?Z0/5*+T)N)/5*DFF = Dc × Dd
FZ0+/2["(*+8 /XSu>@4"qN3)/5* (>?)/%/=>_/5*+ 0Nc = 3"( 0&
I = 6 :"($%/g!)/%|0/4&3!*4/5 0)
Dc = 20F Z0/5*+\1Z0/5*+W*+\[3)/
6²¶
· 5
a!Nc = Nd = 3
6!)/%e6"
DFF = 400F o `$%*+86"() a!0/
UWX)/21B?DD6"()&c*4&/5 0/-&3!*4/5 0 '4/2[3)/5*+ !N])/%&30$%/%&'&30/-)y0/ 4I"( !8! 0$5!8 /XFbW0/M .*vDR/5;#"()/CD6"a7"( #+0 0*e"( (⟨n(c)
1 , · · · , n(c)I α
∣∣⊗⟨n(d)
1 , · · · , n(d)I β
∣∣
)(∣∣n(c)
1 , · · · , n(c)I γ
⟩⊗∣∣n(d)
1 , · · · , n(d)I δ
⟩)
= δαγδβδ$ F % 6&"( 0&Y$%*+8 /5)/∑
αβγδ
∣∣n(c)
1 , · · · , n(c)I α
⟩⊗∣∣n(d)
1 , · · · , n(d)I γ
⟩⟨n(c)
1 , · · · , n(c)I α
∣∣⊗⟨n(d)
1 , · · · , n(d)I δ
∣∣
= 1$ F % 2&
"( !)/5 6"(!KL/v 0#"( >_;"e$%/5#"(! qZ6E$#Y#"()/f(>"eZ0/5*+\1Z0/5*+I*+\[)/v! "l :"($%/fa)/9"( !)/%&-DEN<"))!,X ! ,40/f"()9 ↑ "( 0& ↓ )0/|8P/%$5/% c "( 0& d
∣∣↑↓, ↑, · · · , 0, ↓
⟩ $ F % B&! Ya$# ↑ $ ↓ &&/5 0)/%7/2[0"$5 !N< 0/M>@/5*+ q(>I8P/%$5/% c $
d&a"(0/f"$%$%)&3! ,
!)/XFdbW#"()/C/%wE!K"( /5 E7)∣∣1, 1, · · · 0, 0
⟩⊗∣∣1, 0, · · · , 0, 1
⟩ F $ F % &o e$% .#"))l"MDPX) $N3)/5* 0/76"u)f0)/$5)/9"( $
c†&u"( 0&="( !! :"(
$c&8P/5#"())a$#sDP/5Ng0/#²8BG1$%*+*v#"()7)/5 :"( 0
[ci , cj
]
+=[c†i , c
†j
]
+= 0
"( 0& [ci , c
†j
]
+= δij
$ F A &"( 0&+E0Q! .)&30$%/0/ 4I"( !V/2[$5 !0 48! 0$5!8 /XFQbW0/%#/a8R/5#"())$5)/9"()/ $
c†i&
)/5*4iKL/ $ci
&W"v>_/5*+ (>Q8P/%$5/%c"(?!)/
iF
c†i∣∣n
(c)1 , · · · , n(c)
i , · · · , n(c)I
⟩⊗∣∣n
(d)1 , · · · , n(d)
I
⟩=
=∣∣n
(c)1 , · · · , n(c)
i +1, · · · , n(c)I
⟩⊗∣∣n
(d)1 , · · · , n(d)
I
⟩ $ F A 5&"( 0&
di
∣∣n
(c)1 , · · · , n(c)
I
⟩⊗∣∣n
(d)1 , · · · , n(d)
i , · · · , n(d)I
⟩=
=∣∣n
(c)1 , · · · , n(c)
I
⟩⊗∣∣n
(d)1 , · · · , n(d)
i −1, · · · n(d)I
⟩ $ F A1 &o 0/$9"#/(>h"( .Nl(>V0/a$%$586"( + .*fDP/5)QQ&3\h/5)/5 .I>@)* 0, 1 0/a#"()/a! EK"( !&s"( 0&-0/M)/5* KX"( 0/%9F
6 6
· 5
9 ( 1 U? G (7 1(E zE! 0$%/?/M&/%)$5!DP/?o+&3\h/5)/5 .8P/%$5/%"v088! ,+)/5*>@a/9"$)- 0/%$%/%)#/5N 10/C088! ,e86"(7(>I0/MZ0/5*+\1Z0/5*+IB;"(*+! !) :"( s)/9"&a0/5
T = −J∑
i
(
c†i ci+1 + c†i+1ci
)
− J∑
i
(
d†i di+1 + d†
i+1di
)
,$ F A "&
! la$#c†i
$ci
&OI0/a$5)/9"( $_"( !! :"( &u8R/5#"()u(>R0/m0)d"( 0&d†
i
$di
&>@Q0/#/%$% 0&+8R/%$5/%iFrUW486"(!)d(>P8P/5#"())dDP/5Nv0/?"( .\1$%*+*f#"()W)/21 :"( $ F A &2F +M/5 0/5#"( ! !NLS00/M 0/5 !! ,<)/5 ,X0
J(>IDRs8R/%$5/%?"()/|! 0&/21
8R/5 0&/5 E9SDa>_88RX#/%7"! ,X /J7")*4/%&hF
zE! 0$%/?"(u*4X)d 0/86"($5 /?(>T"|8R/%$5/%u"( ! 9/%&g <"M$%/5#"(! g)!)/a0/5)/ 04(1GDR&3NY! E)/5#"$5 qa!! q0/f)8R/%$5/%?!))/5 \>F $? !NY0/|! E)/5^18R/%$5/%;! 31)/5#"$5 g$% .!D)/%W)|0/7BC"(*+! !) :"( TFubW0/! E)/5#"$5 g86"((>P0/?Z0/5*+\1Z0/5*+OB?DD6"()&YB;"(*+! !) :"( sa0/5
V = Ucd
I∑
i=0
n(c)i n
(d)i
$ F A% &
a0/5)/Ucd
A0/! .)/5#"$5 4)/5 ,XDP/5/%/5 o?>@/5*+ 0I(>V)8R/%$5/%c"( 0&
dS
"( 0&n
(c)i = c†i ci
v0/e E*fDP/58P/5#"()+(>78P/%$5/%c"(!)/
iFZr! 6"( ! !NLSI0/
Z0/5*+OB?DD6"()&YB;"(*+! !) :"( s)/9"&
HFF = −J∑
i
(
c†i ci+1 + c†i+1ci
)
− J∑
i
(
d†i di+1 + d†
i+1di
)
+ Ucd
I∑
i=0
n(c)i n
(d)i
F$ F A1A &
6"( ,X0/M)40/|UWX)/|B?DD6"()&YN3)/5*~! )/%$5 F 0/|/5 0/5,XNY/5!,L/5 0#"()/%"()/=D#"(! 0/%&DEN) !K! ,y0/=!*4/21G! 0&/58R/5 0&/5 E<z$)&3! ,L/54/%wE6"( ca$)*4/9"( 0-&3:"(,L 6"( !#"( (>f0/yBC"(*+! !) *e"(\[PF bW0/yBC"(*+! !) *e"(\[`! .*fDP/5?)#"()/C)/58)/%)/5 .#"( y$% 0!)(>I0/M*e"(\[-/5 /5*4/5 .)(⟨n(c)
1 , · · · , n(c)I α
∣∣⊗⟨n(d)
1 , · · · , n(d)I β
∣∣
)
HFF
(∣∣n(c)
1 , · · · , n(c)I γ
⟩⊗∣∣n(d)
1 , · · · , n(d)I δ
⟩)
$ F A 6&bW0/M&3:"(,L 6"( !#"( s8)iKE&/%70/|/5 0/5,XN</5!,L/5 0#"()/%
∣∣ψν
⟩=
Dc∑
α=1
Dd∑
β=1
c(ν)αβ
∣∣n(c)
1 , · · · n(c)I α
⟩⊗∣∣n(d)
1 , · · · n(d)I β
⟩ $ F A 2&
a!c(ν)αβ
0/|$%/(e$5/5 .)?(>r0/i1G/5 0/5,XN-/5!,L/5 0#"()/XF
6
· 5
&&² E &+C/5 0/5#"( ! !NLSA0/eD0#/5K"(D /%l"()/4&/2m0 0/%&J! 0/g#"(*4/*e"( 0/5l"f! J0/4UWX)/21B?DD6"()&l*4E&/5 GFrB?/5)/XS0/5Nl"()/dD/2k0N|)/5K/5/%&! l)/5*4I(>V"o(1$%*+8P 0/5 .Z0/5*+rN3)/5*YF
"!$#bW0/*4/9"( J$%$586"( .*vDR/5v"(|)!)/i(>
8P/%$5/%cg&/2m0 0/%&"g0/s/2[38P/%$5#"( `KX"( !0/s(>v0/s"$%$%)&3! ,'$%$586"(
E*fDP/5?8P/5#"()9
n(c)i =
⟨ψ0
∣∣ n
(c)i
∣∣ψ0
⟩ F $ F A B&∣∣ψ0
⟩ "( Y"(D!#"(N<#"()/C(>A0/CN3)/5*YS3/XF ,0Fd"( -/5!,L/5 0#"()/ $ F A 6&2S6"( 0&n
(c)i
0/vE$%$586"( q .*vDR/5C8R/5#"()C(>u8P/%$5/%
c$% .! ,<0/f .*fDP/5C(>u"()*4
"(W!)/iFro <$% .#")f0/?8)/?DRX) <N3)/5* 0/?*4/9"( <E$%$586"( < .*vDR/5
(>;0/=)N)/5* l! 0/<! .)/5KX"( n
(c)i = (0 − 1)
&30/=)s0/ 4I"( !a8! 0$5!8 /XFo q$9"#/f(>"( q#"( 0 :"( 6"( I! .KX"(:"( .M :"($%/v0/f*4/9"( $%$586"( y .*fDP/5C!)/21G! 0&/58P/5 0&/5 .
n(c)i = Nc/I
F
! ! # ;0/;)v0/ 4I"( !h8! 0$5!8 /C0/? E*fDP/5aK"(:"( 0$%/C&/% 0C8)9K&/*f0$#! 3>@*e"( J Z0/5*+QN3)/5*49FCoC;&/2m0 0/%&s>_M8R/%$5/%
c"(7)!)/iD.N
(σ(c)i )2 =
⟨ψ0
∣∣
(
n(c)i
)2 ∣∣ψ0
⟩−⟨ψ0
∣∣ n
(c)i
∣∣ψ0
⟩2 S $ F A &
bW0/Q E*fDP/5A8P/5#"()n
(c)i
"$5)T f)#"()/%ha!f$%$586"( | .*vDR/5)n
(c)i = 0, 1 !Nqa$)J!*+8 !/%|6"(
(n(c)i )2 = n
(c)i
FbW.09SO0/+!)/l! 0&/58R/5 0&/5 Ef E*fDP/5KX"(:"( 0$%/M! s"l#"( 0 :"( 6"( A! .KX"(:"( .? :"($%/M
(σ(c)i )2 =
Nc
I−(Nc
I
)2 F $ F#61 &
o J$9")/4(>"< 0 31G#"( 0 :"( 6"( ! .KX"(:"( .lN))/5* 0/+ .*vDR/5vK"(:"( 0$%/e|!)/&/58P/5 0&/5 .?D! .KX"(:"( .? 0&/5K"(:"( s(>r0/M#"(
U/JF
" !! # o c0/Y$9")/-(>|"Z0/5*+\1Z0/5*+?*+\[)/s$% 0&/5 31#"( J(>"<! ,X /+8P/%$5/%Ma! ! Q 0|$%$5f! 0$%/0/4"()*4|DP/5NyZ0/5*+\1 ;!#"$21#"()$%9FOB;9/5KL/59S 0/d$% 0&/5 0#"( v(>3a>_/5*+ 09Si0/Q#(1$9"( ! /%& "6L ! L/!B8<4EL/L#;/5SR?! ! r8PX)!D /XF;ju8P/%$5:"( ! !NY0/f! EKL/%!,."( J(>u0/l86")/v#"( 0! >@)* "$% 0&/5 0#"()/-(>7>@/5*+ 86"(!)l)" J3EC*67#):L|L !;!)<> "7'*737 $@U 7az &
6
· 5
8R/5^k0&7$5)/5 E !NY(>I,X)/9"(?! .)/5)/% * % S A ,FWbW78PX)!D /MDENY$% .) \1 !! ,?0/W"()*1o"()*! .)/5#"$5 0IK:"?"7*e"(,X 0/5$21_m6/5 &+Z0/%ED6"$)+)/%) 6"( 0$%/ *!6 ,FbW0/%)/M)/%) 6"( 0$%/%?E$%$5?"(?$%/5#"(! s)/5 ,X0?(>I0/|*e"(,X 0/5$Mm6/5 &hFUN<*e"( ! K"(:"( 0l(>a0/m6/5 &"() 0&0/%)/+KX"( !0/%f 0/g$9"( $)6"( ,L/g0/+! E)/5#"$5 0> )* "(#"$5!KL/f))/58 !KL/v"( 0&<K$%/MKL/5)#"3F
! "%$#bW0/s"( 6"( ,])J0/s8P/5^k0&c>@#"$5 ! 0/s$9")/(>f"DRX) $gN3)/5* |0/H;C*? 7³%('8_Sra$)$)6"(#"$5)/5#/%"Y#"( 08Pv8)831/5N-! s>@/5*+ yN3)/5*49FbW0/ ?0&/v/5!,XEM?"g&3!)/%$5?8)DP/f>@?0/|*4/5#"( \1! 0 :"()86"#/#"( 0!) =! ="fN))/5* (>O/5 /%$5$9"( P$)6"(,L/%&g>@/5*+ 0W"( 0&g.08)8R 6"( T)40/|$% 0&30$5!KE!oNLF
bW0/4;0&/M/5!,XE>_7"l*+\[)/f(>ro+>_/5*+ $|8P/%$5/%a&/2m0 0/%&YD.N
f(c)D =
I2
JNc
E(0)θ −E
(0)
θ2
S $ F#65&
a!f
(c)D
0/ ;0&/s/5!,X.Y(>f)8R/%$5/%c"( 0&
Nc0/s .*fDP/5-(>|86"($5 /%9F
E(0)θ
"( 0&E(0) "()/+0/e,X) 0&#"()/e/5 0/5,X/%l(>a0/4a))/%&"( 0& 0 31Goa)/%&
B;"(*+! !) :"( TFQbW0/5Ne"()/D#"(! 0/%&gD.N+) !K! ,f0/;z$)&3! ,L/5/%wE6"( 0(>h0/ 0 31Ga)/%&sN))/5* $ F A1A &a"( 0&
HFF,θ = −J∑
i
(
eiθ/I c†i ci+1 + e−iθ/I c†i+1ci
)
− J∑
i
(
eiθ/I d†i di+1 + e−iθ/I d†
i+1di
)
+ Ucd
I∑
i=0
n(c)i n
(d)i
F
a$)r0/Z0/5*+B?DD6"()&B;"(*+! !) :"( l>_I"?N3)/5* a!4"( /2[)/5 6"( 386")/a
θF
6
· 5
6
6 3 "& L)! "
!#8='*/ "7'& !.8<7 8=' 8B= gL 8BBL# L^ #D8 q/7Ag/8<7/ !/ !;" = A 0;44EEC/!A(/78<7/ $/ CI/"*/!/MEC $!/78 # L(7|SF L(728B$ 7)EgL &8BL# L^ I#²8 /!A(/78<7/ =# %*#?7E ;/ % 7# ' -8<"7'##", ²C*8S= /-L^gL |=#%30;44EECH/7A(/78<7/(/ $ (/-8=' 346(/!" "'&#$/!/L^ "B8SI8BL# 63C/"/,/M6C ]=#D = "AC 3767#²8# 77"6 8='&LC(/ :$ = s46L 8B =#?EC
hO*)
w.6"( E* *4/%$)6"( $%-0/s)/5*+8R#"( |/5KL ! (>l"#"()/qg,X!KL/5 nDEN0/!*4/21&/58R/5 0&/5 ECz$)&3! ,L/5C/%w.6"(
i~∂
∂t
∣∣ψ, t
⟩= H
∣∣ψ, t
⟩ S $ "3F5&
a0/5)/ ∣∣ψ, t
⟩ r0/!*4/21&/58R/5 0&/5 Ed#"()/W(>V0/WN3)/5*"( 0&HA0/WBC"(*+! !)
8R/5#"()9FQo 40/am0))u86"(#"(,X#"(8=(>h)/%$5 e0/7!*4/21/5KL ! <(>TN3)/5*4&/%)$5!DR/%&4D.N 8B=)<=#DC* !;#DC*7#²8dB;"(*+! !) =8R/5#"())a! ! 6DP/&3)$50)#/%&e"( 0&40/,L/5 0/5#"( a>@* (>M"q!*4/21/5KL ! n8P/5#"()4a! ! aDP/<,X!KL/5 TFHJ/-$% 0&/540/! !:"( T#"()/C(>A0/CN3)/5* ∣
∣ψ, t0⟩ "(a!*4/
t = t0F $? 0/;8PX)!D /;"9Ng)&/5!KL/
"( y/2[38 !$5!M/2[8)/%# q>@C0/!*4//5KL ! 8P/5#"();C)=8 !,-0/+"( 0#"(>@a0/M#"()/ ∣
∣ψ, t⟩ "(7!*4/
t
∣∣ψ, t
⟩= U(t, t0)
∣∣ψ, t0
⟩ S $ "3F &
a!U(t, t0)
!#"(Nl!*4/21/5KL ! =8R/5#"()9SE! .)|0/?z$)&3! ,L/5/%wE6"( $ "3F5&2F=bWf /9"&f)s"Y&3\h/5)/5 .:"( W/%w.6"( >@|0/e!*4/21/5KL ! ]8P/5#"()
6 1
· 2
U(t, t0)
∂
∂tU(t, t0) = − i
~HU(t, t0)
S $ "3F#"&
a!Y0/f) !
U(t, t0) = exp
(
− i
~H
∫ t
t0
dt′) F
Z070/|#"()/ ∣∣ψ, t
⟩ >_ ! 9a0/5 ∣∣ψ, t
⟩= U(t, t0)
∣∣ψ, t0
⟩ $ "3F % &
= exp
(
− i
~H (t− t0)
)∣∣ψ, t0
⟩ F
Z0|"e,X!KL/5 N))/5* a!J"( yB;"(*+! !) :"( H 0/v6";)=&/5!KL/l0/v!*4//5KL(1
! '8P/5#"()9FZ6+"q!*4/21G! 0&/58P/5 0&/5 .gN))/5*0/<*4X)DEKE04"%N]))/9"( !)/|aa)+#"(L/f"&3KX"( .#"(,L/f(>r0/|/5!,L/5 ED6"7(>I0/|BC"(*+! !) :"(
H
H∣∣n⟩
= En
∣∣n⟩ $ "3F A &
$ !*4/21G! 0&/58P/5 0&/5 .lz$#E&3! ,L/5M/%wE6"( &7a!s0/ljQ!,L/5 0#"()/% ∣∣n⟩ "( 0&s0/
/5 0/5,XN]/5!,L/5 .KX"( !0/%EnFqbW0/g! !:"( #"()/ ∣
∣ψ, t0⟩ /2[386"( 0&/%&! ]0/</5!,L/5 31
D6" ∣∣n⟩ DENy0! ,Y0/+D6"v&/%$%*+8RX)! (>W0/+&/5 .!oN8R/5#"() 1 =∑
n
∣∣n⟩⟨n∣∣
∣∣ψ, t
⟩=∑
n
∣∣n⟩⟨n∣∣ψ, t
⟩=∑
n
cn∣∣n⟩.
$ "3F#6&
a!|0/d$%*+8 /2[M$%/(e$5/5 .)cn :=
⟨n∣∣ψ, t
⟩ F;9J0/u!*4/u/5KL ! v8P/5#"()$9"( JDP/4"(88 !/%&/9")! !NRM! a$#0/+!*4/21G! 0&/58P/5 0&/5 0$5NJ(>
HW"|0)/%&hF4UN
! E)E&30$5! ,g0/|/2[86"( 0) (>r0/|! !:"( O#"()/ $ "3F#6&W 0/M,L/5)
∣∣ψ, t
⟩= exp
(
− i
~H (t− t0)
)∑
n
cn∣∣n⟩ S
a0/5)/M0/lz$)&3! ,L/5?/%wE6"( $ "3F A &W$9"( YDP/|"(88 !/%&h
∣∣ψ, t
⟩=∑
n
cn exp
(
− i
~En (t− t0)
)∣∣n⟩.
$ "3F 2&
bWl*4/50&)/%wE!)/%l0/g# ! (>?0/g$%*+8 /5)/-/5!,L/5 8)D /5*YSda$)!*+8 !Nl 0I>_/9"!D />@I*e"( .N+N3)/5*49ObO|) !KL/0/a/5!,L/5 8)D /5* .*4/5$9"( ! !N 0/l6";)g$#0EX#/"eD6";)=,L/5?0/l*e"(\[q)/58)/%)/5 .#"( J(>d0/BC"(*+! !) 8P/5#"()9FdbWBC"(*+! !) *e"(\[<6")DP/M&3:"(,L 6"( !)/%&D.N-#"( 0&0"()&- E*4/5^1$9"( )/%$) wE0/%9F- *e"( .N$9")/%iSO 0/e$9"( "(fDP/%f$%*+8)/<"=>@#"$5 (>a0//5!,L/5 08P/%$5*YS6&30/|0/C!,X&3!*4/5 0 s(>I0/C8)D /5*YF
6
· 5
=)&/5W)f! E)/5,X#"()/C0/Cz$#E&3! ,L/5/%wE6"( =>@N))/5*4da$#"()/7 0/2["$5 !Nf# !K"(D /! l)/5*4Q(>60/a" >@)/5*4/5 E 0/%&+*4/50E&iSX 0/$9"( Nf);) !KL/0/I8)D /5* "(88)%[3!*e"()/5 !NLFIbW0/5)/u/2[3O"W :"(,L/Q E*fDP/5A(>3 .*4/5$9"( .*4/50&a$)Y! E)/5,X#"()/f&3\h/5)/5 .:"( I/%w.6"( 0aa!!,Xs"$%$5#"$5NY"( 0&-8R/5^>_*e"( 0$%/XF
bW0/a! .)/5,X#"( <(>:8B$ 7)WC* !;7#?C*7#²8u)N)/5*4Q$9"( eDR/a8P/5^>_*4/%&+D.N8 !!! ,80/v!*4/f! E)/5K"( I! s#/5KL/5#"( r*e"( ! I)/580;(>
∆tSV! a$#y0/vBC"(*+! !) :"(
")*4/%&-)DP/M$% 0#"( .9F $? 0/C6"a)4/5KX"( !6"()/C0/M!*4/21/5KL ! s8P/5#"()
U(t, t0)>@W/9"$)Y)D31G! .)/5KX"( R! -)&/5W)lD#"(! =0/;#" ./%$5)Ng(>O0/CN3)/5*YF
HJ/;"(88 !N4*4/50&e! <$%*fD! 6"( -a!-"v)/%$% 0&1)&/57"(88)%[3!*e"( Y(>0/!*4/21/5KL ! 8P/5#"()C>@C0/+!*v :"( 0f&3)$50)#/%&y! q0/+$#6"(8)/5) %"( 0& A F
M )
4Q)D! ,e0/M8)8P/5/%;(>r !#"$% &y"()*+$|,.")/%! 8$9"( O :"($%/%?D.N<0! ,/2[3$5!#"( 06"DR/%/5 <D ./%$5(>O)/%$%/5 E/2[38R/5!*4/5 E#"( * A S S 5,h"( 0&g0/%)/5$9"( *!B ,?PF bW0/%)/)/%$) wE0/%=8)9K&/*v0$)! 3>@*e"( 0/s8.N3$%=(>) ,X !Nq$%)/5 :"()/%&N3)/5*4f"( 0&J/%8P/%$5:"( ! !Nq0/ /5!;7*&C)+ L@88=#;/& @8SL!²'&(/|8BE#;/7=8BL# $GzZO1pq&2F
;\P/5)/5 EC"(88)L"$#0/%C6"%KL/vDR/%/5 q/5*+8 9NL/%&q)48)DP/|0/%)/v) ,X !NY$%)/21 :"()/%&gN3)/5*4D.N=/2[3$5!#"( TFQZ0W! 0#"( 0$%/*+;)/5! 0/5W/5a"( GF'* A ,P/2[0"(*+! 0/%&<"fUWX)/,."W! -0/Mpsa)/5,X!*4/MDEN<"(88 !N! ,e"l8P)/5 .:"( T,X#"&3/5 E9Fdo =0/| :"( ,X6"(,L/|(>0/UWX)/B7DD6"()&l*4&/5 $@/%wE6"( F!B&hI$%)/%8P 0&I)C" !! 0/9"(I! 0$5)/9")/(>V0/W/5 0/5,X/%
εi"$5)X)I0/W :"($%/XFIUNf)/58P/9"(! ,MQ/2[8P/5!*4/5 .da!4&3\P/5^1
/5 . !! 0/9"(?,X#"&3/5 .) $ ! !) &W 0/|D0)/5KL/%?/2[3$5!#"( 0;"(7$%/5#"(! KX"( !0/%9F -0/8$5)/+(> ! 8B#"M ³7)<'&L 5 "B8SI8BL#;/0/%)//2[3$5!#"( 0|$%$5Ca0/5 0//5 0/5,XN(>?"()*4"(l 0/5!,X.DP! ,!)/%9KL/5)$%*4/%0/g)/58 !KL/g! .)/5#"$5 TFbW0/%)/6"(8 6!/L#?# ",!/C"()/?"C&3!)/%$5d! 0&3$9"()(>h0/ 7#?7%AO%@ !-a$)g! 0&3$9"()/%d0/8)/%)/5 0$%/v(>d"( yps?! 0 :"(! ,<#"()/XFbW! !! ,=";/2["(*+! 0/%&q0/%)/5$9"( ! !NY! *!B ,A"( 0&-a86"(7(>IaPF
00/5<)/%$# w.0/y=0/y/2[3$5!#"( DEN!*4/21&/58P/5 0&/5 .Y8P/5D6"( 09FzE(>@/5 /J/5"( GF * 5,f/2["(*+! 0/%& 0/zZO1pqg86")/#"( 0! DENU#"(,X,`8R/%$21)X)$%8.N%*!/,FMps&3 :"( J(>d0/l#"(88! ,-8P)/5 .:"( Qa!J"e> )/%wE0/5 0$5N
ωmod"&&MY&/5D6"( 0& ±ωmod)/5 :"(!KL/+)<0/+ :")/5C>@)/%wE0/5 0$5NLF+bW|&/2m0 0/%v"(
/5 0/5,XN&3\P/5)/5 0$%/(>∆E = ~ωmod
a$#q! qK/5 (>d86"($5 /21G0 /+/2[$5!#"( 0#"( 0^>@/5)v/5 0/5,XN)-0/g"()*+$e,."9F<bWf*4/50&Jf /%)|)/5 0)!!KL/4)U $)
6.@
· 2
X)$5! ! :"( 0v"( 0&hSP !!L/! q :"($%/%Ca!y !! 0/9"(M,X#"&3/5 E)9Sh0/9"(! ,Y(>0/N3^1)/5* a! ! Q 0ME$%$59Ff EKL/%!,."( J(>&3N 6"(*+$9"( Q/2[3$5!#"( 0|(>6"(ME! 0&y"()/86"(7(>Ia-"( 0&Y/5)/f"( )4! .KL/%!,."()/%&! *!B/,F
o -0/C>_ ! 9a! ,e)D0)/%$5 00/C0/%)/5$9"( O> #"(*4/5g>_a0/M E*4/5$9"( !*v :"( (>v0/yoc)/%$# w.0/%iS?! !! ,"( 0&*4&3 :"(! ,'0/q :"($%/XSM"()/&3)$50#)/%&hF
9 (7 ( K' & ( (; : (; & # bW=)/%$5 a! ! ;D/2k0N&3)$50)g0/*4/%$#6"( *4-(>f/2[$5!#"( 0! y0/+pq;)/5,X!*4/XFvUa"$9"( ! !NLSP/4"))*4/v6"(M8)/%&*+! 6"( E !Ny86"($5 /21G0 //2[$5!#"( 0;#"(L/f8 :"$%/N<)! 0$%/f0/l"()*4C"()/l)/586"(#"()/%&sD.NY!,XqD6"(/5)M"( 0&0/f8R/5D6"( 0M"()/v"#*4/%&Y)eDR/|) (e$5/5 . !NY i N<$% ! /%$5!KL/l/2[$5!#"( 0"()/= 0/5,X /%$5)/%&hF]bW0/<,X) 0&#"()/=>@4"sN3)/5* (>
I :"($%/-!)/%4"( 0&
N nIDPX) $e"()*4 $n = 1, 2, 3, . . .
1I$%*+*4/5 0#"()/+m0 ! !! ,>_"$5) &C! J0/+ .*vDR/5D6")/58)/%)/5 .#"( q7"(88)%[3!*e"()/5 !N-,X!KL/5 YDEN<"+)! ,X /MZ0E$#-#"()/X
∣∣ground
⟩=∣∣n, n, · · · , n
⟩ FjI[$5!! ,e#"()/-! '0/-86"($5 /21G0 /8$5)/ N] 0/s"()* 080+>@)* 0/
(i+ 1)1GY)40/
i1G!)/3Ne,X!KL/%70/C 0*e"( !)/%&)#"()/
∣∣excited
⟩=
1√n+ 1
√n
a†i ai+1
∣∣ground
⟩
=∣∣n, n, · · · , (n+ 1), (n − 1), · · · , n
⟩ FB;/5)/XS
a†i"( 0&
ai
"()/W0/aDPX) $$5)/9"( g"( 0&e"( !! :"( =8R/5#"())u>@u)!)/iS
"(,."(! TF C/%/58J! y0/4pq;)/5,X!*4/ $U J
&; !Ns0/! E)/5#"$5 )/5* (>0/UWX)/MB?DD6"()&YBC"(*+! !) :"( )/5*e"(! 09
H ≈ U
2
I∑
i=1
ni (ni − 1)
bWE09S0/|/5 0/5,XN- 0/C6")8)iKE&/|>@7"l08
∆E =⟨excited
∣∣ H∣∣ excited
⟩−⟨ground
∣∣ H∣∣ ground
⟩
=U
2
(
(I − 2)n(n− 1) + (n+ 1)n+ (n− 1)(n− 2)
)
− U
2In(n− 1)
= U
bW0/5)/2>_)/XS?"N))/5* ! 0/qpqg)/5,X!*4/sa$)ne8P/5DP/%&DEN`"( `/2[3)/5^1 6"( A>@)$%/l$9"( y)/%8P 0&a!"486"($5 /21G0 /+/2[$5!#"( y\>u0/l"9K"(! :"(D //5 0/5,XN$%)/%8P 0&)0/|! .)/5#"$5 y)/5 ,XTS
∆E = UF
· 5
! "! ! #Z0W"MN3)/5* (>P>_/5*+ 0d 0/6"d)|)/%8P/%$5u0/4r"( !08! 0$5!8 /7a$)=&E/%Q 0d"( ! io|&/5 E$9"( 0>_/5*+ 0Q)MDP/W! +0/#"(*4/#"()/XFWHJ/v$% 0)&/5C"eN))/5* (>
I :"($%/l!)/%7m0 ! /%&Ya!
I/2>_/5*+ 0?(>d 0/
8R/%$5/%"( 0&I/2>_/5*+ 0(>O"f)/%$% 0&e 0/XS&/5 0)/%&eDEN ↑ E"( 0& ↓ !SE)/%8P/%$5!KL/5 !NLFZ0/5*4)/XST/4")*4/) ,<)/58 DP/5/%/5 0/voY)8R/%$5/%9F C&3^1
$50))/%&C! v$#6"(8)/5 "Z0/5*+\1Z0/5*+3N))/5* T&/%)$5!DP/%&|DENM"B;"(*+! !) f8P/5#"()$ F A1A &?a!"g088! ,1G)/5* >_M/9"$)J8P/%$5/%|"( 0&J"( y! .)/5)8P/%$5/%^1G! E)/5#"$5 )/5*YF ,."(! =/;$% 0&/50/7 !!*+!W(>O) ,l)/58 !KL/7! E)/5#"$5 0a 0/5,X /%$5! ,0/C088! ,e)/5*YFubW0/|BC"(*+! !) 8P/5#"()0/5 Y)/9"&
HFF ≈ UFF
I∑
i=0
n↑i n
↓i
Jv8$5)/g0/4,X) 0&#"()/=|8)/%#/5 .)/%&D.N"< !! 0/9"($%*fD! 6"( c(>?"( ! Z0E$#=)#"()/%Wa!/9"$))!)/;$%$58/%&-DEN=/2[0"$5 !N= 0/|"()* (>u"( .N<8P/%$5/%9FQZ0! 0#"( 0$%/XS/f/2[3$5!)/M0/M)#"()/
∣∣α⟩
=∣∣· · · , ↑
︷︸︸︷ i
, ↓, · · ·⟩ S
! -a$)s"( "()i $ ↑ ↓ &*4/9"( 0/2["$5 !N< 0/;>_/5*+ Y(>r0/|"$%$%)&3! ,4oNE8P/"(46"(e!)/XF`bWeZ0$)c#"()/XSa$)`g"( /5 /5*4/5 Eg(>M0/-,X) 0&#"()/XSWE$%$58/%&qa!J"( y"()* ↑ "(M!)/ i "( 0&qa!"( J"()* ↓ "(;0/v 0/5!,XEDR! ,!)/
i+ 1F ! T0/5)!)/%7"()/|$%$58/%&<a!/2["$5 !N< 0/f"()*~\>u"( .N-8P/%$5/%9F
U/%$9"(0#/(>M0/24I"( !M/2[3$5 !0 `8! 0$5!8 /s !N'Z6/5*+ 0=(>|&3\h/5)/5 .<#"()/%$9"( q6"()/f0/v "(*4/v!)/XF?bW0/5)/2>@)/XSP !NY088! ,<)="e!)/fa!J"( q"()* (>&3\P/5)/5 E78! $9"( Y$%$59
∣∣excited
⟩= a†↓,ia↓,(i+1)
∣∣α⟩
= a†↓,ia↓,(i+1)
∣∣· · · , ↑, ↓, · · ·
⟩
=∣∣· · · , ↑↓, 0, · · ·
⟩ S
a!a†↓,i
"( 0&a↓,i
0/C>_/5*+ $f$5)/9"( q"( 0&q"( !! :"( q8P/5#"())?"(?!)/i(>I8P/%$5/% ↓ FubW0/|/5 0/5,XN< 0/%/%&/%&<>_7"08-)e"l 0/5!,XEDR! ,<!)/Ca0/5
∆E =⟨excited
∣∣ HFF
∣∣ excited
⟩−⟨α∣∣ HFF
∣∣α⟩
= UFF − 0 = UFFF
$ "3F#B&
¶
· 2
∆E = U
∆E = U
"()*4
:"($%/!)/%
∆E = U
∆E = U
"()*4 :"($%/|!)/%
- u$1Z! / V ,!-.-+49H>!$+!-4z-7D$+,!$ |!-.$H>$^7$+! $+C4FHH+ ! +$,CY+=!D! 4'.'!45! $+D/0^16FHH+ (.'!|' 5!-$$
6"( ,L0)+0/CDPX) 0a! -0/|8)/5KE0786"(#"(,X#"(8q"l(1$%*+8R 0/5 EMZ0/5*+,."d! g"( g8$9"( V :"($%/8P)/5 .:"( 6a! ! R09`)/%) 6"( .DR/56"9KEa0/5 g/2[$5!)/%&a!s"( s/5 0/5,XN
∆E = UFFF
9 ( ! :( (; &
4Q)D! ,e"()*+$;,.")/%W! <8$9"( h :"($%/%WD.Ng! !! ,0/?8P)/5 .:"( h)/5KL/9"( WKX"( !31"(D /a! 3>_*e"( e 4) ,X !N$%)/5 :"()/%&e)N)/5*4I! 4/2[38P/5!*4/5 . * A ,V"( 0&0/%N*!B/,FAbW0/d! !! ,;T)/9"( !#/%&fDENf"a8P)/5 .:"( ,X#"&3/5 .u"$5)X)A0/u :"($%/XFI f)/5*4(>u0/lB7DD6"()&s*4&/5 GSV0/l 31!)/l/5 0/5,XN
εi! 0$5)/9"#/%? !! 0/9"( !Na!s0/l!)/
! 0&/2[i
εi = iεtiltS $ "3F#&
! =a$)i0/;!)/7! 0&/2[g"( 0&g0/?86"(#"(*4/5)/5
εtilt$% E) W0/7/5 0/5,XNe&3\h/5^1
/5 0$%/vDR/5o/%/5 0/5!,XEDR! ,Y!)/%iF7>^S6! sK/5 (>u0/v86"($5 /21G0 /+/2[$5!#"( 0&3)$50#)/%&q! y#/%$5 "3F FXSh0//5 0/5,XNq&3\h/5)/5 0$%/DR/5o/%/5 J 0/5!,XEDR! ,s!)/%$%)/%8P 0&;)40/l/5 0/5,XNY 0/%/%&/%&>@C"408q 0/fa! ! ID0)/5KL/"+)/%) 6"( E;DP/216"9KE9F
UW/%&/%?0/v!KE:"( I$9")/f(>u088! ,g>@)* !)/v)e!)/XSV 0/2[37 0/9"()/%; 0/5!,X31DPv088! ,q"( )Y8PX)!D /="(v i/5! !! ,0SID9SQ! 0$%/40/=BC"(*+! !) :"( 0$ F!B&d"( 0&+$ F A1A &I&M 0d&3!)/%$5 !N+$% 0/%$50/%)/#"()/%9S.!)d"M)/%$% 0&e)&/5u8)(1$%/%)ua!g"; 9/5d8)D6"(D! !!NLFbW0/%)/W)/%) 6"( 0$%/7a! ! 0E$%$5"(d! !! ,
εtilt = U/2"Q0ia +! lm0,X)/ "3FXFroIQ"( );8RX))!D /)C08+iKL/5Q*4)/W6"( +M)!)/%9SD! 0$%/|0/%)/|"()/M8)$%/%))/M(>I!,X0/57)&/50/5N-E$%$5aa! 9/5?8)D6"(D! !!/%9F
6
· 5
E: : (; : (E bW0/g! EKL/%!,."( `(>M"()*+$<,.")/%! '8$9"( a :"($%/%+DENo(1G80) `U#"(,X,8R/%$5)X#$%8.N+"D ./%$5(>T)/%$%/5 E/2[38R/5!*4/5 E) * S /,FIo g$% .#"))l"(88 !N.1! ,l"C,X#"&3/5 Ed*4/50E&ed 00)$%/58!D /7)f/2h/%$5)u !!L/7U $)=X)$5! ! :"( 0Q0/9"(! ,|/2P/%$5)9FIpq)/%9KL/59S.0/a/2[3$5!#"( e/5 0/5,XNlQ 0/9"(D /aKL/5Nv8)/%$5)/5 !NLF </2[8P/5!*4/5 .aWa& 0/;DENg*4&3 :"(! ,+0/M"(*+8 !!0&/|(>O0/C :"($%/; :")/5)a!"v> )/%wE0/5 0$5N
ωmodFQbWa,L/5 0/5#"()/%o4&/5D6"( 0& ±ωmod
)/5 :"(!KL/M)0/ :")/5W> )/%wE0/5 0$5N=a$)s&/2m0 0/v"( /5 0/5,XN
∆E = ~ωmodF
)&/5M)<8P/5^>@* $%*+8)/5f)!*f :"( 0f(>a"()*+$,.")/%C! L|C& @8<EC8$9"( A :"($%/%;! q0/|>@#"(*4/5q(>u0/vB?DD6"()&*4&/5 I 0/f6";)<"$%$% .>@0/=!*4/21&/58R/5 0&/5 0$5Nc(>;0/ $ *4&3 :"()/%&&l :"($%/XF]bW0/< 0/21&3!*4/5 0 6"( #"($C :"($%/|8P)/5 .:"( T)/9"&
V (x) = V0 sin2(kx)
F
Z0W"(*+8 !!0&/?*4&3 :"()/%&g :"($%/%#"($?8R)/5 E:"( Rd*4E&3\m6/%&=DENe"|!*4/21&/58R/5 0&/5 Ef>G"$5)va$#8)iKE&/%l0/=X)$5! ! :"( (>0/g"(*+8 !!0&/=D.N"-> )/21w.0/5 0$5N
ωmod"( 0&Y"+&3!*4/5 0 /%)7*4E&3 :"( s"(*+8 !!0&/
F
V (x, t) = V0 sin2(kx) (1 + F sin(ω t))F $ "3F! &
Y)&/5)4#"( 0 :"()/M8P)/5 .:"( O! .)40/| :"( ,X6"(,L/f(>I0/|B7DD6"()&Y*4E&/5 0/W6"u);m0,X)/u0i'0/a!*4/21&/58P/5 0&/5 0$5N4(>
V " h/%$5)Q0/a 0/5 ! !! ,
)/5 ,XJ"( 0&0/a! .)/5#"$5 =)/5 ,X
UFIo 4"( 6"( ,XNl)C0/7&/2m0 ! e,X!KL/5
! Y)/%$5 FM0/C!*4/21&/58P/5 0&/5 .; 0/5 ! !! ,g)/5 ,Xa,X!KL/5 YD.N
Jij(t) =
∫
dx w0 (x− ξi, t)
(
− ~2
2m
∂2
∂x2+ Vmod(x, t)
)
w0 (x− ξj, t)S
! =a$#w0 (x− ξi, t)
"()/C0/;H"( /5^1_>@ 0$5 0 $9"( !)/%&Y"(W0/i1G< :"($%/
!)/>@Q0/W i/%uD6"( 0& $@&/5 0)/%&eD.Nv0/a! 0&/2[0&2FIbWI)/5* ,X!KL/%Q0/a/5 0/5,XN
>@ '&L !!D=#%l>@)* )!)/i)v!)/
j13 !N+ 0/9"()/%^1G 0/5!,X.DP^1G088! ,4W")*4/%&hS
0/5)/2>@)/XS0 0/v"(!KL/%?"(a0/|!*+8 /5?/2[8)/%))
J(t) =
∫
dx w0 (x, t)
(
− ~2
2m
∂2
∂x2+ Vmod(x, t)
)
w0 (x− δ, t)S $ "3F15&
! la$#δ = ξi+1−ξi
Q0/W :"($%/a86"$5! ,0FIbW0/aH"( /5I>@ 0$5 0"()/W!*4/21&/58R/5 0&/5 E;"/5 ! N=0/5!?6"(8P/|&/58P/5 0&7 0/| :"($%/|8P)/5 .:"(
Vmod(x, t)
Z0)6"( ! 9 :"($%/;8R)/5 E:"( W!W/9"/5>_W"( -"()* )'&L !Y)+"( =0/5a!)/XS
· 2
)!)fW"%KL/2>@ 0$5 a! ! DP/+D)L"&/5 0/%&hFq $% E#"9Sr>_v&/%/58P/5v :"($%/%v0/"()* a! ! ODR/ !D=##?6C<)4"+!)/3Ne!)aW"%KL/2>@ 0$5 s#"(0/5 6"()9|F
bW0/|! .)/5#"$5 y)/5 ,X
U(t) =4π~
2a m
∫
dx |w0(x, t)|4S $ "3F &
$ a!a 0/K/51G"9KL/J)$9"()/5! ,c /5 ,X &="( )&/58P/5 0&- n0/y6"(8P/q(>f0/
W"%KL/2>@ 0$5 "( 0&E0C!;&/58P/5 0&M q!*4/vDP/%$9"(0)/l(>d0/X)$5! ! :"(! ,< :"($%/8P)/5 .:"( GF
o v)&/5A)?/9"#/Q0/# ! v(>00/u! E)/5,X#"( '$ "3F15&O"( 0& $ "3F &P0/H"( /5>@ 0$5 0v$9"( DR/e"(88)9[!*e"()/%&D.NJ"2+|"(0):"( JW"%KL/2>@ 0$5 ! "-6"(*4 $8P)/5 .:"( GSI"(88)%[3!*e"(! ,q"<! ,X /4/5 ! d(>W0/ :"($%/+8P)/5 .:"( $ "3F! & $W8B³%('8B)47=#DC=#%<"(88)%[3!*e"( &2
w(x) =1
√√πσ
exp
(
− x2
2σ2
) $ "3F!"&
a!σ0/4a&3(>a0/+|"(0):"( TSAa$)&/58P/5 0&f 0/+ :"($%/48P)/5 .:"( GF
bW0/f"#*4/%&-6"(*4 $M8P)/5 .:"( Aa0/5
Vh(x, t) =m
2
(V0(1 + F sin(ωmodt))k
2
m
)
x2 S $ "3F % &
"(,."(! TSa!0/M*4E&3 :"( > )/%wE0/5 0$5Nωmod
"( 0&-0/|"(*+8 !!0&/FF
+&3)$50)#/%&DR/2>_)/g>@0/<H"( /5+>@ 0$5 0e"q#"(0/546"( ! i :"($%/a! ! M)/% !-! "]D)L"&/5-+|"(0):"( N0/y86"(#"(*4/5)/5
σa! ! MDP/%$%*4/y :"(,L/59F
bW0/f*4/50E&s)m0,X)/l77&/58P/5 0&/5 0$5NLSh"?"(88 !/%&s! * ,Q"( 0& *!B ,S0?)*+! !*+)/|0/(+;)X)^184u!#"/5K!T/5 0/5,XN=> 0$5 6"( G
E[w] =
∫
dx
[
~2
2m
∣∣∣∣
dw(x)
dx
∣∣∣∣
2
+ Vh(x, t) |w(x)|2 +1
2
4π~2a
m|w(x)|4
]
$ "3F A &
o .)/5,X#"(! ,+a/%wE6"( <0! ,0/ +|"(0):"( w
$@/%w.6"( 3"3F!"&da!<0/;86" 1#"(*4/5)/5
σ /9"&u)f"( 4/2[8)/%# +>_Q0/a/5 0/5,XN+""?> 0$5 e(>P0/ +M"(0#:"(
a&3
E[w] =⇒ E (σ) =~
2m
√2
σ2+
m√2Ω(t)2σ2 +
4π~2a √
2πmσ
S $ "3F!6&
! ga$)<0/?"(DD)/5K:"( Ω(t) = V0(1+F sin(ωt))
W"d0)/%&e>_d$5 :"(!NLFubW0/#"( 6"(N8R! E)?(>Q0/l/5 0/5,XN<> 0$5 6"(
E(σ)"()/vD#"(! 0/%&DENY)/5! ,e0/
&/5!KX"(!KL/|a!-)/%8P/%$5)σ)9/5)
√2mΩ2σ4 − ~
2
m
√2 − 4π~
2a √2πmσ2
= 0$ "3F 2&
· 5
z !KE! ,47/%w.6"( Y /9"&7)4"( Y!*+8 !$5!7/2[38)/%) <>_σ(t)
σ4 =~
2
m2Ω2
(1 + 2
√πa σ
) $ "3F!B&
bW/%wE6"( *v0DP/) !KL/%& ! )&/5q)',L/50/a&3σ"(y"!*4/
tF
0ia c! *!B ,a 0/-$9"( 0/5,X /%$5+0/<! E)/5#"$5 '86"(4(>?0/</5 0/5,XN> 0$21 6"( $ "3F A &a"( 0&-,L/5)a0/f!*+8 /5?"(88)%[3!*e"(
E[w] =
∫
dx
[
~2
2m
∣∣∣∣
dw(x)
dx
∣∣∣∣
2
+ Vh(x, t) |w(x)|2]
S $ "3F!&
a$)Y8)iKE&/%;"( /2[8 !$5!?/2[8)/%)) <>@a0/Ma&3σ(t)
σnon−int(t) =(mV0k
2(1 + F sin(ωmodt)))−1/4 $ "3F &
jQK"( !6"(! ,l0/! E)/5,X#"( $ "3F15&d"( 0& $ "3F &I0! , $ "3F &d"a&3g(>T0/ +M"(0^1:"( 0
φ(t) /9"&)40/M/%w.6"( 0
J(t) =~
8mk2σ2exp
(4k4σ4 − π2
4k2σ2
)
×
×(ek
2σ2
(π2 + 2k2σ2(V0(1 + F sin(ωt))σ2 − 1) − $ "3F 5&
− 2V0(1 + F sin(ωt))k2σ2)) S
a!σ = σnon−int(t)
"( 0&
U(t) =2√
2πa ~σ(t)
= 2√
2πa ~(mV0k
2(1 + F sin(ωt)))1/4 F $ "3F 1 &
>@)/5a&/5 .\>@NE! ,e0/|$% 0#"( .>_"$5)U0
$ D.NY"(*+8 !!0&/F −→ 0
&> )* /%wE6" 1 $ "3F 1 &a 0/;m0 6"( ! !NY"(!KL/%?"(
U(t) = U0 (1 + F sin(ωt))1/4 F $ "3F "&
J$9")/e(>0/+!*4/21&/58P/5 0&/5 ./2[38)/%) >_J(t)
/+0)/+0/>_ ! 9a! ,y"(8318)%[3!*e"( *4/5 . 0/%&! *!B ,O! 0)/9"&-(>I!*+8 /5*4/5 .! ,</%wE6"( %$ "3F 5&
J(t) = J0 exp(−F sin(ωt))F $ "3F % &
bWvD6"#/%&c ]0/Y"#*+8 c(>C"s !! 0/9"(4&/58R/5 0&/5 0$5N(>?0/<iKL/5 :"(8(>0/H"( /5M> 0$5 0f! J/%w.6"( $ "3F15&;a!J0/4"(*+8 !!0&/e(>W0/ :"($%/8R)/5 E:"( Ta$)saK"( !&=>_7*e"( ! h*4&3 :"( q"(*+8 !!0&/%
FF
· 2
o 'e#/%$5 '/s&3)$50)40/Y*4/50E&e0)/%&c)J/5KL !KL/q"N3)/5* (>|"()*4! `8R/5&3$Y8P)/5 .:"( g! !*4/XF +C/5 0/5#"( ! !NLSW0/!*4//5KL ! (>|w.6"( E*N3)/5*4a,X!KL/5 YD.N=0/M!*4/21&/58P/5 0&/5 .|z$)&3! ,L/5?/%wE6"(
∂
∂t
∣∣ψ, t
⟩= − i
~H∣∣ψ, t
⟩ $ "3F 1A &
a!q0/v)#"()/ ∣∣ψ, t
⟩ "( 0&s0/lB;"(*+! !) y8R/5#"()HF ?*4/5$9"( r$9"( $5 :"( 0
)/%wE!)/)-/2[8)/%#C0/+B;"(*+! !) :"( J! J"=D6"C! )&/5|)=,L/5M0/+BC"(*+! !) *e"(\[hF`bW0/B;"(*+! !) '*e"(\['! .*fDP/5eD6")4)/58)/%)/5 E#"( `e"J86"())/*e"(\[c(>C"s.,L/&3!*4/5 0 $@)/%/-)/%$5 F5&2F +&3#$50))/%&]! c)/%$5 "3F) !K! ,l0/;z$)&3! ,L/5/%wE6"( $ "3F 1A &uD.Ne# !KE! ,l0/7/5!,L/5 EK"( !0/;8)D /5* 0I/(e$5/5 .9FIju8P/%$5:"( ! !N|!*4/21&/58P/5 0&/5 .uN3)/5*4O &l)/%w.!)/);&3:"(,L 6"( !)/0/|BC"(*+! !) Y*e"(\["(/9"$)s!*4/%)/58s"( 0&-E07$% 0*4/| )(> 7 4 !*4/XF
pq)/q/(e$5/5 .-"()/ E*4/5$9"( C# !KL/5)4>_g)&3! 6"(N&3\h/5)/5 .:"( C/%wE6"( 0$ $Cju &2qHJ/q/5*+8 9N'*e"(! !Nc0/s)(1$9"( ! /%& 7#"( .18?$)0 ) )$#0/5*4/ $ 7 &2Sa$#qC"+*+\[3)/l(>u0/vjQ /5C/2[38 !$5!C"( 0&s!*+8 !$5!C*4/50E&y"( 0&Y 09a y)DP/= 0$% 0&3! 6"( ! !N]#"(D /XF 00/5+)/%$# w.0/<l0/</2[8 !$5! : ,L/218;C#"*4/50& $ : ; &2FObW0/%)/+*4/50&|a! ! uDP/+&3)$50))/%&y! ]"(88 !$9"( )YB?DD6"()&N3)/5*47"( 0&<0/v"&3K"( E#"(,L/%;"( 0&-&3#"&3KX"( .#"(,L/%?a! ! TDP/M !! 0/%&hF
# " # bW0/yjQ /5^1G*4/50&"()/qm0)-)&/5 0*4/50&>@0/?! .)/5,X#"( (>A)&3! 6"(N=&3\P/5)/5 E:"( T/%w.6"( 0iFQ =0/C$9")/?(>0/vz$#E&3! ,L/5;/%w.6"( %$ "3F 1A &0/5NY"()/C,X!KL/5 YD.N
∣∣ψn+1
⟩=
∣∣ψn
⟩+ ∂
∂t
∣∣ψn
⟩∆t
jQ /5;/2[8 !$5!9S∣∣ψn+1
⟩=
∣∣ψn
⟩+ ∂
∂t
∣∣ψn+1
⟩∆t
︸ ︷︷ ︸
ju /57!*+8 !$5!9S
! ya$) ∣∣ψn
⟩ &/5 0)/%M0/+)#"()/"(C0/n1Gy!*4/%)/58"( 0&
∆tC0/!9/(>
!*4/%)/58TFH(tn)
C0/+B;"(*+! !) J8P/5#"()v"(C!*4/tn = n∆t
FfZ0M! 0#"( 0$%/XS0/</%wE6"( >_f0/</2[38 !$5!*4/50E&*4/9"( 06"(0/e 0/2[3l)#"()/ ∣
∣ψn+1
⟩ 0/l$5)/5 .|#"()/ ∣
∣ψn
⟩ 8 !0;0/)(1$9"( ! /%& 8B$ !/8< !D)&# "M8BL# ∂∂t
∣∣ψn
⟩∆tF?bW0/
&/5!KX"(!KL/M! 0&/C0/M!*4/%))/58-> 0$5 -,X!KL/5 YD.N=0/fz$)&3! ,L/5?/%wE6"(
Q + ir!bYlTa` VYi r,y;r!`<mTa`n
V y nSegT:Ta``Sr!` VYir!yr,`<mgTa`O
`
∆tn+1
´
·
$ "3F 1A &W)0/|/5KL ! /%wE6"( 0am0 6"( ! !N<)/9"&
∣∣ψn+1
⟩=
∣∣ψn
⟩− i∆t
~H(tn)
∣∣ψn
⟩ S $ "3F 6&
∣∣ψn+1
⟩=
∣∣ψn
⟩− i∆t
~H(tn+1)
∣∣ψn+1
⟩ F $ "3F 2&
bW0/+&3\P/5)/5 0$%/DP/5/%/5 J0/%)/-*4/50&MC6"(M0/!*4/%)/58y>@ 0$5 (>0/g/2[38 !$5!+*4/50E&$9"( ]DR/</5KX"( !6"()/%&&3!)/%$5 !N0! ,y0/=8)/%)/5 .+#"()/-"( 0&B;"(*+! !) ]*e"(\[ 1d! ]$% E#"9SQ0/g!*+8 !$5!*4/50E&! .KL !KL/%0/
(n + 1)1G
#"()/W! +!)I!*4/%)/58+>@ 0$5 TSLa$#4Q 0QNL/5Q 09a TFQUNf)/5a!! ,f/%wE6"( $ "3F 2&W 0/M,L/5)
(
1 +i∆t
~H(tn+1)
)∣∣ψn+1
⟩=∣∣ψn
⟩ S
a$)Q"?#/5r(>6 !! 0/9"(Q/%wE6"( 0r>@r0/#"()/ ∣∣ψn+1
⟩ FAbWE09S! +)&/5r);) !KL/"( q)&3! 6"(Nq !! 0/9"(M/%w.6"( y !!L/l0/4z$)&3! ,L/5|/%w.6"( 0! ,-"( $ !D ""=8*4/50E& 0/=6")s# !KL/<"q)/5(>7 !! 0/9"(4/%wE6"( 0! ]/9"$#c!*4/%)/58TFJbW0/DR/5 0/2m0C(>d;/2PC?0/fDP/5)/5;)#"(D! !!Ns(>0/%)/l"( !,L!*4M$%*+86"()/%&q)/2[8 !$5!*4/50&9F
9< ( *& !& (
! # ?*4/5 . 0/%&J/9"( !/5C0/ 7#"( .18?$)0 # J)$)0/5*4/M"e*+\[E1)/f(>Q0/|o=jQ /57*4/50&78)/%#/5 .)/%&s! 0/M8)/5K0786"(#"(,X#"(8TF 7 !!L/0/MjQ /5a*4/50E&Y!aa)/%$% 0&-)&/5! <!*4/XFubW0/M!*+8 !$5!a!)/5#"(!KL/|/%wE6"( )/9"&
∣∣ψn+1
⟩=∣∣ψn
⟩− 1
2
(i∆t
~H(tn)
∣∣ψn
⟩+
i∆t
~H(tn+1)
∣∣ψn+1
⟩) S $ "3F B&
a$)e$9"( 4DP/W)/5a!)/5 ="( 6"( ,L0) !Nl)|0/a!*+8 !$5!djQ /5/%w.6"( 4! E)M !! 0/9"(/%w.6"( 09
(
1 +i∆t
2~H(tn+1)
)∣∣ψn+1
⟩=
(
1 − i∆t
2~H(tn)
)∣∣ψn
⟩ F $ "3F &
bW0/!,X.M6"( 0&&/|"g*e"(\[1GKL/%$5)|8)E&30$5|(>E 0ia JKX"( !0/%9FfjuwE6"( $ "3F &7$9"( DP/v)/5a!)/5 J! .)=0/+)(1$9"( ! /%& : 57A->_* (>0/+"(88)%[3!*e"( (>r0/C!*4/21/5KL ! y8R/5#"()9
U(ti+1, ti) =1 − i∆t
2~H(tn)
1 + i∆t2~
H(tn+1)
$ "3F#"1 &
1
· 2
bWd0/ UE#;)F "7'&L /L# !DWL! |%@I8SLI|a$)-/5KL !KL/%W"f#"()/7D.N∆t
! g!*4/XF )*+! ,q6"(0/gBC"(*+! !) c8R/5#"()6"()&3 !N$#6"( ,L/%! ]0/g!*4/=! .)/5KX"(
[t, t + ∆t] 0/-*e"9N'/5K"( !6"()/sDR`B;"(*+! !) :"( 0=! /%wE6"( $ "3F#"1 &g"(40/
#"(*4/|!*4/f"( 0&<,L/5)
U(ti+1, ti) → U′(ti+1, ti) =1 − i∆t
2~H(tn+1/2
)
1 + i∆t2~
H(tn+1/2)
S $ "3F#"5&
a$#=W" #B8S7Al>@*(>O/%w.6"( $ "3F#"1 &"( 0&eE0W"#)/%d0/;$% 0)/5KX"( (>0/e 0*(>0/e)#"()/e9KL/5l!*4/XFYbW0/e!*4/
tn+1/2= ∆t(n + 1/2)
*4/9"( 06"(0/fB;"(*+! !) :"( Y7/5K"( !6"()/%&! -0/M*+&&3 /|(>I0/M!*4/M! .)/5KX"( GF
"! " # o <)&/5)v8P/5^>@* 0/;!*4/21/5KL ! Y!^1)/5 \>T)*4/8)/586"(#"()N++6"u)|DR/a& 0/X $? 0/6"d)M,L/5 0/5#"()/0/7088! ,*e"(\[ $@(V1&3:"(,L 6"( d/5 /5*4/5 E)|(>d0/B?DD6"()&qBC"(*+! !) *e"(\[&! y E*fDP/5D6"C)/58)/%#/5 .#"( ]"( 0&q0/D6"C!)#/5 \>^F|o |$9")/l0/%)/+"()/+D#"(! 0/%&D.N"l! ,X /C)(>@W"()/C86"$)X"(,L/Ca$)-)/%wE!)/%W0/? E*fDP/5(>A :"($%/C!)/%9S E*fDP/5(>a"()* N8R/%|"( 0&y0/ E*fDP/5f(>W"()*4M(>W/9"$)JoNE8P/XFfbW0/! !:"( u#"()/l>_0/+!*4/21/5KL ! M0/+,X) 0&#"()/e(>0/4BC"(*+! !) J*e"(\[$% 0! ,(>0/?088! ,*e"(\[="( 0&="f8P)/5 .:"( P*e"(\[4>_"v$%/5#"(! g! E)/5#"$5 Y)/5 ,X
U$@#/%/|)/%$5 F5&2FbW0/f!*4/v/5KL ! q!))/5 \>u#"()?D.N- L"&3! ,=0/|! !:"( r#"()/XS00/v088! ,
*e"(\["( 0&c0/< E*fDP/5eD6"9FcbW0/<088! ,*e"(\[c+))/%&c! c0/ "6LI)!D6!/!/MEC ",L &#VLI!I8Wa$#s7)/%w.!)/%&D.N-0/| !! 0/9"(;/%w.6"( q) !KL/59FWbW0/ E*fDP/5D6"WW))/%&<! <0/ 4E(/7/51D ./%$59FubWaD .^/%$57"( ! ia)l8R/5^>_* "( ! 8P/5#"( 0W 0/%/%&/%&hS/XF ,0FQ$9"( $5 :"(! ,0/; 31)!)/78P)/5 .:"( O"( 0&=/2[)/5 6"( P8P)/5 31:"( P86"(W(>r"f$%/5#"(! =>_E$#e#"()/C"/5 ! T"a"(88 !NE! ,+$5)/9"( Y"( 0&-"( !! :"( 8P/5#"()))+#"()/%iF
bW0/7/5K"( !6"( e(>PKX"(0uD0)/5KX"(D /%uu"( )|/5 0$9"(80) :"()/%&+! g"C$5 :")9AbW0/D ./%$5)7(>I$5 :")a*v0?"( )DP/C! !:"( !)/%&-a!0/|N3)/5*~86"(#"(*4/5)/5)9F
jdw.6"( "3F ,X!KL/%v0/g !! 0/9"(/%w.6"( 0l6"(l6"v)sDP/e) !KL/%&"(l/9"$#!*4/%)/58TF`HJ/-)/5a!)/-!e! '0/- !#"(N>_* "e&3)$50)#/%&c! c0/-8)/5K086"(#"(,X#"(8T
(
1 +i∆t
2~H(tn+1/2
)
)∣∣ψn+1
⟩=
(
1 − i∆t
2~H(tn+1/2
)
)∣∣ψn
⟩ F $ "3F#" &
bW0/f*e"($%/%? YDP&/%?(>u/%w.6"( $ "3F#" &&3\P/5; !N<D.N<0/f!,X 10! ^1&/5)|#"%KL/7*4/5*4N+/;$% 00$5d0/7 /2> d6"( 0&e&/a*e"(\[+>@)* 0/7088! ,
·
*e"(\[+"( 0&0/W8P)/5 .:"( GFQbW0/W!,XEu6"( 0&4&/W(>h/%w.6"( +$ "3F#" &rQ/5KX"( !6"()/%&KE:"-"<*e"(\[1GKL/%$5)f8)&30$5f(>W /2> M6"( 0&&/*e"(\["( 0&0/4$5)/5 Ev#"()/)/%8R/%$5! ,=0/$)6"( ,L/(>!,X q! q;8P/5#"( TFMbW0/l)/5;(> !! 0/9"(C/%wE6"( 0e) !KL/%&c0)! ,0/;psZ 4 7 ;x86"$#"(,L/%$ KL/5) % F#"&+D.N M"%K+* 1 ,FbW0/*e"($%/%W! </%wE6"( $ "3F#" &6"%KL/;0/M#"(*4/C0$5)/M"W0/CBC"(*+! !) -*e"(\[! ]0/<(R1&3:"(,L 6"( \1G86"( $ 9/5)q*e"(\[/5 /5*4/5 .)<"()/=$% 0#/5KL/%&&2Sd0/<&3:"(,L 6"( 86"(;C 0 31G9/5)Fv $% E#"9ST&30/l)<Z0E$#s#"()/%Ca!J 0/C9/5)=86"($5 /%8R/5!)/XSL0/;B;"(*+! !) =*e"(\[e*+!,XE6"9KL/79/5)X! e0/78R)/5 E:"( P86"(9F ?/5KL/5^10/5 /%)9SV0/v86"())!N(>u0/v*e"(\[C$% 0)/5KL/%&hF;bW0/f 0/5t#"()/v786")#/%&)0/MD0)/5K"(D /%?D .^/%$59S3a$#s/5KX"( !6"()/%7"( 0&Y))/%0/|&0"(#"3FubWa8)$%/%)a)/58R/9"()/%&- E! T0/|$%*+8 /5)/f!*4/21G! .)/5KX"( O6"aDP/%/5 $%9KL/5)/%&hF
z !KE! ,0/;!*4/21&/58P/5 0&/5 .;z$#E&3! ,L/5/%wE6"( <0! ,0/;$%*vD! 6"( (>0/7#"( E18?$)0 ) l)$)0/5*4/"( 0&M0/ ;psZ 4 7 ;'86"())/Q !! 0/9"(A) !KL/5T3KL/5N/5 ! >@v"( ! v8)D /5*4iFYbW0/4*4/50&fKL/5NJ#"(D /ga!)/%8P/%$5f)$)6"( ,L/%M! y0/)/580!9/XSPE0C!M"( ! if"=$%L"())/+"( 6"( !N)Ca!"g#"(0/5M :"(,L/!*4/%)/58Y),L/5?"wE6"( !!#"(!KL/|8$5)/XF
&3#"9aD6"$)4(>Pd"( 0&8)D6"(D !N4"( ! !*+8 !$5!d) !K! ,M)! 0/%uI0/W !! 0/9"(/%w.6"( J8)D /5*YSAa$)]$% 0*4/%v )|(> 7 4 !*4/XFeZ0f/2[0"(*+8 /XSr/5KL !K.1! ,"yN))/5* (> 6qDPX) 04 6q :"($%/Y!)/% $ .*fDP/5eD6")4&3!*4/5 0 % 6 &#"(L/%|"(DR*"1 =)/%$% 0&;>_ ! 1 1 =!*4/%))/580 1O"=N3)/5*(> BgDRX) 0C B=!)/%$@&3!*4/5 0 -6 % " A &#"(L/% "1 )/%$% 0&8P/5)/58 $ DP s"#"( 0&0"()&34 7'&2F
4Q)*+! ,! c6"(4$9")/=+$#6"( ,X! ,J>@)*0/ ;pqZ/4 7 ; ) !KL/5)y0/4 : Cz $'86"$#"(,L/WDEN+z$#0/5 l/59F "( GF * " ,rbWQ)(>@"()/W86"$#"(,L/a$%*vD! 0/%d&3\1)/%$5Q"( 0&l!)/5#"(!KL/a) !KL/5)Q"( 0&l!)r8R/5^>_*e"( 0$%/Wr /%)I&/58R/5 0&/5 Eu 0/)N)/5*!9/Q6"( ?pqZ 4 7 ;+F &&3! 6"( ! !NLS0/u86"$#"(,L/W"( ! 9r86"(#"( ! /5 3$%*+8! ,; zp-4*e"$#! 0/% $ *f !\1G8)$%/%))f*e"$)! 0/% &;a!]"( "(0)8$50M8R/5^>_*e"( 0$%/,."(! TF !0,X+r86"$)X"(,L/WA 0INL/5r>@ ! !N|!*+8 /5*4/5 .)/%&+! +Q$%&///5)/"(D /v)=/%!*e"()/l!)C8P/5^>@*e"( 0$%/XFMo;)0 &sDP/v8RX))!D /f)</2[0"(*+! 0/vDPX) $N))/5*4W8Y)
10!)/%7"( 0&
10"()*4? Y#"( 0&0"()&26 % D!*e"$#! 0/%9F
9 ]( .! <&
)&/5s)',L)cD!,X,L/5q)N)/5* !9/%Y/J!*+8 /5*4/5 .)/%& 0/ : ,L/218;C#"*4/50E&c"0)/%&DENU#"( 31p %! ,L/5 *!B/,F +"( '/2[8 !$5!+*4/50&]!4&E/% 0)/%w.!)/f)4) !KL/M !! 0/9"(;/%w.6"( 0iFbW0/v$5 :")$9"( : ,L/218;C#"+?">@
@
· 2
)&/5*4/50&<a$)aD6")/%&- -0/|/2[38 !$5!jQ /5*4/50&<8)/%)/5 .)/%&! -)/%$5 TFol,."(! 0"$%$5#"$5ND.N#"(E! ,y*e"( ! /5:"( a)/580l! 0)/9"&(>?0/e>@ ! !*4/%)/58"( 0&)*+*+! ,0/%)/8TFZ0e! 0#"( 0$%/XS0/$5 :")$9"( 7>@n)&/5: ,L/218;C#"e#"(L/% % :"( I)/580M"( 0&0/*+*+! ,<8y$9"( 0$%/5 ;C0/v 9/5)&/5;/5))/5*4 $@)/%/ * % ,&2F +C/5 0/5#"( ! !NLS60/|!,X0/57)&/5;(>I/2[38 !$5!7*4/50&; 0M (e$5/5 .ML/%/58q0//5)M*e"( ! I! q$%*+86"() )g0/ 7#"( .18?$)0 # *4/50&hFAbWE0r//5*+8 9N|0/ 7 46EC@C*6C 78=' LEC*7 &#%*7) 88SC*4/50&|>@)*F0 77#", " !;!/ * % ,S.a$#g0#/%"( ="&0"(8!KL/;)/580!9/?$% .) R)|)/%&30$%/?!)/5)9F ? !!L/a0/a!,X! 6"( 3>@* (>PI*4/50&+D.N !'* 5467% * 1A ,SL!QQ/%w.!88P/%&a!8!*+#/%& :(/,')B ! * 6 ,T86"(#"(*4/5)/5)9F
B;9/5KL/59S0/|"( !,L!* &3&< 0WNL/5W8R/5^>_* "W/2[8P/%$5)/%&h m0)a/2[38 !$5!KL/5) g(>R0/Wm>@e)&/5 : ,L/218;C#"f"( !,L!* "dwE!)/>G"QD"( )MKL/5N 0#"(D /XFbW0/<8P/5^>@*e"( 0$%/<,."(! cW"+ X+&30/<)q0/-!,X \m6$9"( .e)/%&30$5 (>Q0/f))/58s!9/XFWbW0/f/5*vDR/%&&/%&m>@)&/5 : ,L/218;C#"+*4/50&Ya!y)/5831!9/l$% .) IC !!,XE !N->_"))/5?6"( q0/ 7#"( E18?$)0 ) *4/50&DC!; :"$)3"$%$5#"$5N<! &3!)/%$57$%*+86"(# TF
A, "
!# 8='*$/ "7' !?8<7 8=';/7$ & I8BL#;/ L^ 4EL(/ML#" @8SL/=#KL !?8B"6 : ³I88B",!/6 !DE)/M7#²8<6C I#DCKCI 3767#²8 "6LI# s%&W@8BBL#/2L^ $#D8<7W"M8BBLI# /8BE#%I8='@/ $ "$#&%VM"M8SL!/#DC /!A(/78</7/.g!/ 6 C$/-"/,/|EC !# 8=' ,/8 ! I78= C$/-" */!/8=' 6!/& 8=/ ML/78SI8BL#D7A !;78 4EI8BL#;/L^8=' 7/M/!A(/78< / 47A228B= "8<EC ³I88B#"! !;LI8<7#²8B - !# 8=' I$# ! 8U8=' /7$ & @8BBL#/3L^3CA#D" "B8SI8BBLI#;/ | LC ³I8<EC I88B", !;L)8<7#²8B 6 !DE7/M7#²8<6C L "6L# (7#S7# ", $ #D8B=8BS!/8='*ELI%('&L 8_8='*$/4"'& !?8<7E %(7# =#2#?I8 E &#B8=/]8='&I8 E#;/H8='&@8
c = ~ = 1
) #" 9*)
4)/%$5 4/8)/%)/5 E0/a)/%) !)d(>O"MN3)/5* (>hDPX) 0u! g"M! !)/%&4 :"($%/XF &3#$50))/%&-! #/%$5 -"3F F ! !! ,40/f :"($%/|8P)/5 .:"( A7"+!*+8 /f*4/50E&)e8)DP/f0//2[$5!#"( 8R/%$5* (>"g) ,X !N$%)/5 :"()/%&N3)/5*YF; q)/5*4(>r0/MUWX)/|B?DD6"()&-*4&/5 GS6"l! !)/%&- :"($%/f8R)/5 E:"( T*4/9"( 09S6"(?"+ !! 0/9"( !N! 0$5)/9"! ,e 31!)/|/5 0/5,XN-/5 E)/5)0/M! E)4B;"(*+! !) :"( T
H = −JI∑
i=1
(
a†i ai+1 + a†i+1ai
)
+
I∑
i=1
εini +
I∑
i=1
ni(ni − 1)S $ % F5&
! a$)εis"! ,X /21G86"($5 / 31!)//5 0/5,XN$% E!D )0/9KL/5#"( !
8R)/5 E:"( GFdbW0/εi"()/M&/2m0 0/%&D.N
εi = iεtiltS
¶
·
a0/5)/εtilt
40/Y!)/21G! 0&/58P/5 0&/5 .</5 0/5,XNc&3\P/5)/5 0$%/YDR/5o/%/5 n 0/5!,X.DP! ,!)/%iF
Z6C"eN))/5* ! 0/lps?! 0 :"();)/5,X!*4/U J
/")*4/|6"(C/2[$5\1#"( 0C"()/v"&/%wE6"()/5 !N&/%)$5!DP/%&Y! Y0/f86"($5 /21G0 /l8$5)/ $@)/%$5 "3F F5&2FZ0/5*4)/XSu0/4,X) 0&#"()/=(>?"-N3)/5*(>
NDPX) 0v"( 0&
I = N :"($%/
!)/%a,X!KL/5 YD.N∣∣ψ⟩
=∣∣1, 1, 1, 1, · · · , 1, 1
⟩ F
Y&3)$50)#/%&! )/%$5 "3F F SC\>0/J :"($%/J! !! ,`/%wE6"( -0/J! E)/5#"$5 )/5 ,X
εtilt = USdGF /XF!S0/-/2[)/5 6"( 78R)/5 E:"( 8)9K&/%e/5 0,X/5 0/5,XN]>_
"( s"()* )08KC@L]:#q0/M! !)/%&- :"($%/| E)e"( 00/5;"()*47!)/XFHJ/;!*v :"()/;"vN3)/5* (>
6DPX) 0a!
6 :"($%/?)!)/%d0! ,$5N3$5 !$DP 0&1
"(N]$% 0&3! 09Sd6"(+*4/9"( 0iS"()*44$9"( c08cDP/5/%/5 cm0)4"( 0& :"+!)/<&3\1)/%$5 !NLFQpq)/%9KL/59S"|$%/5#"(! g#"(
U/Jd$#0X)/5 TS3"(da$)=0/7)!*f :"( <#"(L/%
8 :"$%/XF ;! !:"( u#"()/XST0/v,X) 0&)#"()/4(>d0/ 0 31G! !)/%&JBC"(*+! !) :"( $ % F5&;$%*+8)/%&hFo )&/5?)g/2[8 )/f0/f/2[$5!#"( q)8R/%$5*(>u?N3)/5* /8P/5^>@*4/%&v0/!*4//5KL ! l>_O :"($%/! !)A! v0/d#"( ,L/d>@)*
εtilt = 0)
2UF
bW0/=/5KL ! c#"()la!0/g 0 31G! !)/%&cB;"(*+! !) :"( '"( 0&!)f,X) 0&)#"()/XF > )/5|"g!*4/
t = 0.1 J−1 0/ :"($%/! !;C! 0)#"( . !Nq"(88 !/%&"( 0&sL/58M E! 0/+!*v :"( /5 0&f"(M!*4/
t = 20 J−1 F |/9"$)!*4/%)/580/+D0)/5KX"(D /%/5 0/5,XN<#"( 0^>_/57"( 0&- .*vDR/5?K"(:"( 0$%/v"()/|/5K"( !6"()/%&hF
Zr!,X)/% % FXS % F SR"( 0& % F#"e! ! !0#"()/|0/f)/% !);(>I0/v!*f :"( 0?>_?N3^1)/5*4Ma!
U/J = 50, 20,"( 0&
10Fvo J"( ! Am0,X)/%|0/+/5 0/5,XN#"( 0^>_/5MC! ! !0^1
#"()/%&c c0/= /2>@l6"( 0&c)!)/XS"( 0&0/= .*vDR/54KX"(:"( 0$%/Y"(0/<!)&c!)/=&/58$5)/%&g +0/a!,X.d(>h/9"$)em0,X)/XFIbW0/&/5 0!N8 )$ 9/58 ) &r0i'0/D0)/5KX"(D /%|"7> 0$5 (>d!*4/+"( 0&s :"($%/v! !! ,
εtiltSV! ya$)y0/l*e"(,X \1
0&/;(>T0/7D0)/5KX"(D /?u! 0&3$9"()/%&=D.N+0/;$% #$)0/5*4/XFQbW0/;$% )/9"$#0/%>@)*x&0"(yD !0/>_M 9/5lKX"( !0/%MKE:"-,X)/%/5 J8J)-)/%&y>_| :"(,L/5|*e"(,X !0&/%iFbW0/l8 )M"(DRiKL/(>d0/v&/5 0)!N8 );! ! !0#"()/l0/"%KL/5#"(,L/%&yKX"( !0/%C9KL/5C0/>@ ! O!*4/
t = 0)tmax) = 20 J−1 Fa sDRqN))/5*49S)/%) 6"( 0$%/%C$%$57\>Q0/
:"($%/=! !! ,/%w.6"( l0/g! .)/5#"$5 ')/5 ,Xεtilt = U
"l8)8PX)/%&"(DRiKL/XFpq)/%9KL/59S3)/%) 6"( 0$%/%7E$%$5a"(> #"$5 6"( PKX"( !0/%(>
US3"&3#$50))/%&=! =#/%$5
"3F FX")/%) 6"( 0$%/"(+0/- :"($%/-! !! ,εtilt = U/2
! 0&3$9"()/%="q08')y0/ 0/2[3v 0/9"()/%)v 0/5!,XEDR! ,J!)/XF &&3! 6"( ! !NLSu! ]0/=$9")/g(>
U/J = 50"( 0&
U/J = 20)/%) 6"( 0$%/%4"(
εtilt = U/4"()/<$5 /9"( !NK!D /XSQa$#c! 0&3$9"()/<080
iKL/5>_!)/%9Fo $% 0$5 !0 TS|"(Y*e"( ! /5-#"(X
U/J0//2[$5!#"( 8P/%$5* -D)L"&
6
·
E
E
E
E
E
E[J
]
E
E
E
E
E
E
σ
E E E E E $EEεtilt [J ]
E
$E
t[J−
1]
E E E E E $EEεtilt [J ]
E
$E
t[J−
1]
-44|!--!F \D|!-. /6CY+ +
6!F /u !-4!F7 4!!@16 !
! ^!-U0/J0 = 50
+ !- 46/Q! >+7 7D ! $+.-\ !^+/0.B@ 7.H@1Z!-$7 Y+ !,! !=+]CY. (@+1.@ B+ =49$ $ "!F=C$(C4F /0+1Z!-+/u!=P0([email protected] D@-S +7=]!-.-+4HY!-.+,!@4
εtiltP0-$+,!-4 D^S L !-
H>!.+,!-47DY$$+16 C>.!|9$.+ +.]C>-+ !$$ $ + !-H / ! 7D.+!|\H4!- !'!-=6% )($-F7 7!^ 9-$74++!$+.-\ !^+>|/0$ H4!- 7D.+!F! !-4!+
εtilt = U
"( 0&$% .! E009F $? 0/0/5=6"( 0&hS6"(8`8R/9"(3g"(88P/9"(=! 0/)) ,X !N)/58 !KL/C)/5,X!*4/XS3>_*+! ,4#"(0/576"(8-)/%) 6"( 0$%/f0$5)/XSa$)W! -KL/5N,LE&s"(,X)/%/5*4/5 E;a!y0/v8$5)/v(>u86"($5 /21G0 /+/2[$5!#"( 09F;bW0/v#/%w.0/5 0$%/(>!*f :"( 0fa!J&/%$5)/9"! ,
U/J$ m0,X)/% % F21 % F#"&;0if"( )=0/l)/(>W"
D6"$),X) 0&'! '0/- E*fDP/5eKX"(:"( 0$%/Y&30/-)0/Y! 0$5)/9"! ,*4D! !!oN'(>;0/"()*49F
·
E
E
,E
E
E
E[J
]
E
E
E
E
E
E
σ
E $E E E Eεtilt [J ]
E
FE
t[J−
1]
E FE E E Eεtilt [J ]
E
$E
t[J−
1]
- Dz - 44!^!F \]!$/6CY+>B+
6!FB/ !-4!F7 4!!@16
!9!- ^!-U/J = 20
+ !- 4./Q!>+7 @7D ! .+$\ !^+/0.'7D$H@1Z!F7 D+ !-,! !-+]CY. (-+16 @ B+ 4'. F !F C.-(C4$ /0+>1Z!+ /Y! P0([email protected]> D^S +>7=]!$+4HY!$+,!@4
εtiltP0 .+!^4D^S L !
HY!-.+,!@4 7DY$$+16 CY6!|'.$+ +$,CY+ 5!-$$ $ +V!HV/ !- 7D$+5!|\H4!^ !'!=6% )($-$7 7!- -$7 4+9+ !.+.-\ !^+/0. H4!^ 7.+!$!- !45!-+
εtilt = U
·
E
E
E
E
E
E[J
]
E
E
E
E
E
E
σ
E $E Eεtilt [J ]
E
$E
t[J−
1]
E FE Eεtilt [J ]
E
$E
t[J−
1]
> 44!^!-$B \D|!-.#/6C>,+B+
65!-$ / !-4!F7 4!!-1.
!'! -!U/J = 10
+ ! 46/Q!B+>7 7D ! $+.-\ !^+>|/[email protected]@1Z!-$7 ]+ !,! !=+]CY. (@+1.@ B+ =49$ $ "!F=C$(C4F /0+1Z!-+/u!=P0([email protected] D@-S +7=]!-.-+4HY!-.+,!@4
εtiltP0-$+,!-4 D^S L !-
H>!.+,!-47DY$$+16 C>.!|9$.+ +.]C>-+ !$$ $ + !-H / ! 7D.+!|\H4!- !'!-=6% )($-F7 7!^ 9-$74++!$+.-\ !^+>|/0$ H4!- 7D.+!F! !-4!+
εtilt = U
·
< )9 #" 9*)
o Y7#/%$5 TS60/|)!*f :"( 0C(>QK"(0?$% 3m0,X#"( 0;(>QDPX) $|,."#/%! "(*+8 !!0&/-*4&3 :"()/%&c8$9"( a :"($%/%9F 0ia c! #/%$5 "3F F#"3Su0/-"(*18 !!0&/+*4&3 :"( J(>0/ :"($%/8P)/5 .:"( d8)9K&/%f"( /5 0/5,XN
∆E = ωmoda!ωmod
0/4*4&3 :"( J>@)/%wE0/5 0$5NLF=UNyKX"(:"( (>a|*4&3 :"( >@)/21wE0/5 0$5Ns 0/+$9"( 8)DP/l0/*e"( .NE1GDR&3Ns! y0/ :"($%/XShGF /XF!SO/2[3$5!)/0/N3)/5*"&3#$50))/%&<! )/%$5 "3F FXFubW0/C*4E&3 :"( q(>r0/| :"($%/M6"a)DP/C#"( 0^1 :"()/%&q)=0/B7DD6"()&q*4&/5 GSPa$) /9"&9Sh"M0ia q! )/%$5 "3F F#"3SP)g0/86"(#"(*4/5)/5)
Jmod(t) = J0 exp(−F sin(ωmodt))
Umod(t) = U0(1 + F sin(ωmodt))1/4F
bW0/%)/a"()/W0/!*4/21&/58P/5 0&/5 .d86"(#"(*4/5)/5)d(>60/aUWX)/B?DD6"()&+BC"(*+! !) :"( TSa!<0/C$% 0#"( .)
J0$ 0/5 ! !! ,4)/5 ,X &W"( 0&
U0$ ! E)/5#"$5 )/5 ,X &"
/5 ! A"a0/M*4&3 :"( q"(*+8 !!0&/FF
o y)&/5;)g8P/5^>@* "4!*4/21/5KL ! /l6"%KL/l)g$#0EX#/l"( y! !:"( I#"()/XFbWQ#"()/ID#"(! 0/%&+DENv/2[0"$5I&3:"(,L 6"( ! "( 4(>V0/! !:"( 0B;"(*+! !) *e"(\["(l0/=!*4/
t = 0>@+"Y,X!KL/5 #"(
U0/J0
FybWv*4/9"( 0iSQ0/e!*4/21/5KL ! "( !W"%N3g#"()ea!0/Y/5 0/5,XN'/5!,L/5 0)#"()/XSW,L/5 0/5#"( ! !Nc0/-,X) 0&#"()/XFbW0/#"(
U0/J0
"( )+&/2m0 0/%70/C)/5,X!*4/|! -a$)s0/|!*v :"( #"(L/%a8 :"$%/XFbW0/l)/% !);8)/%)/5 E)/%&y! y;#/%$5 "()/l,L/5 0/5#"()/%&qD.Ns8R/5^>_*+! ,<!*4/21
/5KL ! 0<>_<)/5KL/5#"( C*4E&3 :"( n>@)/%w.0/5 0$5/%ωmod
FtbW0/q)/%8P 0)/q(>f0/N3)/5* &3! ,0/=!*4/21/5KL ! 'l*4/9")/%&KE:"0/</5 0/5,XN#"( 0^>_/54"( 0&0/M E*fDP/57KX"(:"( 0$%/XF
HJ/M)0&3N<"lN3)/5*~(>I)\[eDRX# 0a Y\[=!)/%a(>u"( Y"(*+8 !!0&/|*4&3 :"()/%&8$9"( I :"($%/XFCbW0/f :"($%/l6"C$5N3$5 !$fDP 0&0"(N$% 0&3! 09SPGF /XF!SR86"($5 /%|"(0/4m0)v!)/e*e"%N08)Y0/g :"f!)/="( 0&JK$%/4KL/5)#"3F-UN0/%)/g$% 0&3! 0 0/|"9KL&aDR 0&0"(N=/2P/%$5)iFQZA!))9S3/C g"(7"lN))/5* a!-0/C! E)/5#"$5 )/5 ,X
U0 = 20 J0T/$%*+8)/W!*4/a/5KL ! 0Q>_I*4E&3 :"( 4> )/%wE0/5 0$5/%
! g0/7#"( ,L/ωmod = (0− 40)J0
! g)/580(>∆ωmod = 0.1J0
"( 0&g/5K"( !6"()/%&g0/D0)/5KX"(D /%"(a/9"$)-!*4/%)/58TFQZr!,X)/ % F % 0iW0/;/5 0/5,XNg#"( 0^>_/59SEa$#-,X!KL/5 <D.Ne0/;/5 0/5,XNg/2[8P/%$5#"( <K"( !0/;/5K"( !6"()/%&=a!<0/;!*4/21/5KL !KL/%&-#"()/∣∣ψ, t
⟩ K:"
E =⟨ψ, t
∣∣ H0
∣∣ψ, t
⟩−E0
a!E0
0/s/5 0/5,XN'(>C0/! !:"( ;)#"()/"( 0&'0/! !:"( ;BC"(*+! !) `8P/5#"()
H0 = H(t = 0)FfzE09a ;0/f> ! I&0"(#"<"7>@ 0$5 (>d!*4/+"( 0&*4&3 :"(
·
E
$E
E
E
E[J
0]
E
E
E
E
E
σ
E $E E E Eω [J0]
E
$E
t[J−
10
]
E FE E E ,Eω [J0]
E
$E
t[J−
10
]
9 4'. 7.+5!|\ H4! + !- 46/Q! >+7 7D !- .+.-\ !--+/0. /0+>1Z!+W/!=
tP0($!-1$4zD^S +>7 D7D4!+ /0-$J,.+>16\
ω !- H4! CY(
! .+.-\ !^+/0.
)($-F7 (.z!
(D$ =D7D4@!-+ /0-$J,$+16\ω
([email protected] -$7 4+ 7D$+!-$! .+$\U0 = 20 J0
! B1^ + '4@7 ]HY$16!!-=+V-$+>+1. + !-HU !! 1. =F + !--,! 3!-+] C>$()@+>16
> )/%wE0/5 0$5Ns"( 0&s0/v!*4/+"9KL/5#"(,L/%&/5 0/5,XNY#"( 0^>_/59F $? 0/D0)/5KL/5)|"e!8 /(>)/%) 6"( 0$%/% 1R;!8 /v)/%/5*4?)gDR/l"e!,X 6"()/l(>dN3)/5*4?a!
6DRX) 0;
6!)/%9F )/%) 6"( 0$%/= 0/9"(D.N0/=*4E&3 :"( ]>@)/%w.0/5 0$5N
ωmod = U0$9"( DR/
/2[8 :"(! 0/%&D.N"86"($5 /21G0 /q/2[$5!#"( n(>M0/Y,X) 0&)#"()/XSaa$)$% 0)*e"(! !NJ(>0/gZ0$)J)#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ ! 0/gpqf)/5,X!*4/XF v&3)$50))/%&
! n)/%$5 "3F Fy 0/q6"<)8)iKE&/y0/y/5 0/5,XNU0
)8- 0/"()* .)"( 00/5O! frZ0$)|#"()/XFrbWO/5 0/5,XNfA"%KX"(! :"(D /W"(r"W*4&3 :"( f>@)/%wE0/5 0$5N
ωmod = U0"( 0&YE0; /9"&7)g0/v"4)/%) 6"( 0$%/XF7bW0/l$%/5 .)&y(>u?!8 /f
!!,X. !N-\>@)/%&-(q0/∆E = ωmod
1G !! 0/M)9W"()&7!,X0/5a>@)/%wE0/5 0$5/%9F
1
·
E
FE
E
E
E[J
0]
E
E
E
E
E
σ
E E ,E E Eω [J0]
E
FE
t[J−
10
]
E E E E Eω [J0]
E
$E
t[J−
10
]
- D +$\V!^+/0. +7 !+] C>$ ()@+>16=/V\D!.I/6CY+> +
64@!!@16 !$ ! ^!-
U0/J0 = 50 K 4'. 7.+5!|\ H4!^ !- $+.-\
!--+|/0$ +7 +]CY.B(@+1. '/0+16!+ / != +7 =]74!+ /0FJ,.+1.\ωmod
([email protected]$7 4+ 7.+!$'!- /0-$J,.+>16\
ωmod = U0/J0
bW0/|!*e"(,L/%7 -0/M!,XE?(>QZA!,X)/ % F % 09 0/| .*fDP/57KX"(:"( 0$%/
σi = 〈n2i 〉 − 〈ni〉2
S
! ]a$)iv0/e)!)/4! 0&/2[hFszE! 0$%/e0/g :"($%/g|#"( 0 :"( 6"( W! .KX"(:"( .
"( 0&E0M0/+ E*fDP/5fKX"(:"( 0$%/|!)/! 0&/58P/5 0&/5 .9Sr !Nq0/lk00$56"( 0"(0/em0)v)!)/g"()/e)09a TFbW0/g&/5 0!N8 v ]0/4!,X.(>am0,X)/ % F % 0i0/; .*vDR/57K"(:"( 0$%/M"> 0$5 Y(>A!*4/C"( 0&<*4&3 :"( <>@)/%w.0/5 0$5NLS0/;8 "(DP9KL/q09=0/s!*4/21o"%KL/5#"(,L/%& E*fDP/5=KX"(:"( 0$%/XF +; D6"( ! !NLSa0/q0$5)/+!*+! :"(+)y0/-/5 0/5,XN]#"( 0^>_/59F $? 0/-D0#/5KL/%4"q)/%) 6"( 0$%/<!8 /"( 0&'"( "&&3! 6"( GSEX)$5! ! :"()Nv0$5)/W! 0/W&/5 0!oNf8 9FI $% .#")Q)?0/W/5 0/5,XN#"( 0^>_/59Su0/=!*4/-"%KL/5#"(,L/%&'8 (>;0/= .*fDP/54KX"(:"( 0$%/<09e"s$% 0#"( E
·
D6"$),X) 0&hFbWvD6"$#E,X) 0&&3!)/%$5 !N]$% 0/%$5)/%&)q0/g#"(U0/J0
+DEN! 0$5)/9"! ,0/e! .)/5#"$5 c)/5 ,X
U0$@f i/5! ,y0/+ 0/5 ! !! ,y)/5 ,X
J0&088! ,=a! ! ADP/f88)/%#)/%&q"( 0&s0/v .*fDP/5;K"(:"( 0$%/la! ! A&3 "(88R/9"(9F;
0/d !!*+!I(>V"( v! 3m0 !)/! E)/5#"$5 4)/5 ,XTS0/)N)/5* $9"( vDP/W")*4/%&"O0//!=#% 57Z6E$#4#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ SLE09S.0/5)/7u 0fK"(:"( 0$%/7! e0/7$%$586"(
.*fDP/59FQbW0/7X#$5! ! :"()N40$5)/! 40/7 .*vDR/5K"(:"( 0$%/78 )u&30/)M0/X)$5! ! :"(! ,|)/5 ,X0
J(t)"( 0&
U(t)a$#4&/58R/5 0&+
sin(ωmodt)F/4Q ! ,|0/
K"( !0/%W(>Aa)! 0/7>@ 0$5 -! Y"( Y"(#"%Nωmod
KL/5)0t /9"&W)la0$5)/XF
Zr!,X)/ % F A &/58$5v0/g)/% !)f>@"Y*e"( EN.1GDPE&3NJ)N)/5*(>7\[DRX# 0v \[c :"($%/!)/%ga!n"( `! .)/5#"$5 )/5 ,X
U0 = 50 J0Fo $% .#")g)
0/-8)/5KE0=$9")/XS=N))/5* e&/%/58P/5e! '0/spsg)/5,X!*4/XF ,."(! TSa 0/$9"( s)/%/M0/f!8 /21G)/%) 6"( 0$%/l0$5)/f! 0/f/5 0/5,XN-#"( 0^>_/5 $ !*e"(,L/%; 0/ /2> m0,X)/ % F A &+"+/5 ! C"+! '0/- E*fDP/5eKX"(:"( 0$%/ $ !*e"(,L/%g '0/-!,X.=(>m0,X)/ % F A &2Fs 0/4!*4/21o"%KL/5#"(,L/%&8 (>0/g .*fDP/5K"(:"( 0$%/= 0/=$9"( )/%/"( y"&&3! 6"( I)/%) 6"( 0$%/l 0/9"(M"4*4&3 :"( q> )/%wE0/5 0$5N
ωmod ≈ U0/2 = 25 J0S
a$)J"( )=09C0/v!8 /l8R/9"()!,X 6"()/XF;bWE09SP! q0/f>@0/5C&3)$50) /$9"( ! ?0$)"D 0$)n(>M8R/9"(3g"y)/%# 6"( 0$%/XF '0/$9")/(>|N3)/5* "(DP9KL/6"(? &DP/
2)/%) 6"( 0$%/%M"(
ωmod = 50"( 0&
ωmod = 25SR/9"$#q$% 0)!! ,<(>
38R/9"(39F .O"()/5;! ;$)6"(8)/5C/va! ! A&3#$50)7N3)/5*4?(>d&3\P/5)/5 E;!9/%C"( 0&
DR 0&0"(N<$% 0&3! 09S0a$#q"( )+)09 "&3\h/5)/5 .;0$5)/|(>I)/%) 6"( 0$%/%9F W*4/5 E 0/%&Y! Y)/%$5 2"3F F#"l0/f/2[3$5!#"( "(
ωmod = U0W! 0&30$%/%&-DEN
0/<"(D0#8 (>;"q80) '(>?l>@)/%wE0/5 0$5NLF]zE!*+! :"( !NLS4#"()/<$9"( cDR//2[3$5!)/%&=D.Ng80) 0a(>O6"( \>r(>O0/C/5 0/5,XNLS3a$#-)/% !)! -"v)/%) 6"( 0$%/M"(0/C*4E&3 :"( ->@)/%wE0/5 0$5N
ωmod = U0/2F $%$%)&3! ,X !NLS6! 0$%/C0/|"(D0)D q(>
480) 0a /%)8)86"(D /|0/C)/%) 6"( 0$%/fa/9"(L/59FZr!,X)/% % F#6Y"( 0& % F Y09 0/+K"(:"( ](>0/e*4E&3 :"( "(*+8 !!0&/4!
&3\P/5)/5 El)/5,X!*4/%(>;"sN3)/5*(>7)\[DRX# 0v \[J :"($%/<!)/%9FsZA!,X)/ % F#6&/58$5)v0/=/5 0/5,XNJ#"( 0^>_/5+"( 0&Jm0,X)/ % F 09l0/g .*fDP/5K"(:"( 0$%/=KL/5^104!*4/y"( 0&'>@)/%wE0/5 0$5NLF bW0/*4&3 :"( "(*+8 !!0&/
Fg$#6"( ,L/%&n! 0/
$% !*+ 0> )*F = 0.01
8<)F = 0.1
FdbW0/C)9a0i &3\P/5)/5 Ea#"(XU0/J0
S! 0$5)/9"! ,>@)* )8')DP)*YFUN$%*+86"(! ,0/-)ie(>;m0,X)/ % F y 0/$9"( ])/%/e0/eD6"$#E,X) 0&(>0/g .*vDR/5KX"(:"( 0$%/=&/%$5)/9")! ,s)i"()&!,X0/5#"(X
U0/J0
FvbW0/v88P/5M)9|! ! !0#"()/0/+N3)/5* ! 0/8P/5^k0&y)/5,X!*4/"(
U0/J0 = 3S6a$)y097KL/5N-/9"(Y/2[$5!#"( s! $%*+86"(# s)e0/M#"(0/5
ps^1G !!L/N3)/5*4O! v0/#/%$% 0&l"( 0&v!)&f)i|FIZ0A0/*e"( ! *4E&3 :"( 4"(*18 !!0&/
F = 0.010/d8P/5D6"( +$9"( v !N|DP/d#/%/5 f! l0/d E*fDP/5AKX"(:"( 0$%/a"
@
·
F = 0.01 F = 0.05 F = 0.1
U0
=3J
0
E ω [J0]
E
$E
E
E
t[J−
10
]
E ω [J0]
E
ω [J0]
U0
=10J
0
E FE Eω [J0]
E
$E
E
E
t[J−
10
]
E $E Eω [J0]
E FE E
ω [J0]
U0
=20J
0
E $E E E Eω [J0]
E
$E
E
E
t[J−
10
]
E FE E E ,Eω [J0]
E $E E E E
ω [J0]
E $E E EE [J0]
- > +$\ !^+/0. /> \]!$"/6C>,+ +
6!F ! 7D u.-.+,! +,!-.^1Z!-+
!-.+!U0
+7=]74!+=H4!>7D$F = Vmod,0/V0
91^H4! !-9$+.-\!--+|/0$z+! != +,!.-(4
t = (0−30) J−10
(D. ! =D7D4@!+=/0-$J,$+16\ωmod =
(0 − 2) U0
- ! +,!.^16!+ |!-$+!-U0 = 3J0, 10J0, 20J0
P0!H !-CY!! S D! 1.4=+B !- =]74!+ /0FJ,.+1.F
F = 0.01, 0.05, 0.1
·
F = 0.01 F = 0.05 F = 0.1
U0
=3J
0
E ω [J0]
E
$E
E
E
t[J−
10
]
E ω [J0]
E
ω [J0]
U0
=10J
0
E $E Eω [J0]
E
$E
E
E
t[J−
10
]
E $E Eω [J0]
E FE E
ω [J0]
U0
=20J
0
E FE E E Eω [J0]
E
$E
E
E
t[J−
10
]
E $E E E Eω [J0]
E FE E E ,E
ω [J0]
E E E E E Eσ
> D C>$ ()@+>16 /9 \D|!-. /6C>,+ +
65!-$ ! 7DY$$+! +,!-.%
1Z!-+ |!-$+!-U0
+7 =D7D4@!-+ =H45!-7D$F = Vmod,0/V0
z1^ H4! !=+]CY. N>1Z!>!+ + !- ! =+,!.-(4
t = (0 − 30) J−10
(]$ !-==]74!+/0FJ.+1.\
ωmod = (0 − 2) U0
G - M! +!-.^16!+ !-.+!U0 =
3J0, 10J0, 20J0
PQ!HK!3C>!! S !- 1.4=+ ! =]74!+K/0-$J,$+D%16$
F = 0.01, 0.05, 0.1
;¶
·
"f$)6"( ,L/;! e0/;X)$5! ! :"()Ne0$5)/C"(ωmod ≈ 4J0
F )) ,L/5d*4&3 :"( "(*+8 !!0&/% 0/CD0)/5KL/5)?"( )l"v/9"(g)/%) 6"( 0$%/|"(
ωmod ≈ 4J0"(88P/9"(! ,+!
0/v/5 0/5,XN=#"( 0>@/59FWbW0/|)/%$% 0&)9 09?0/|N3)/5* "(U0/J0 = 10
S6a$)? !!,X. !NY"(DP9KL/|0/f8R/5^k0&)4pq! 0 :"()786"#/M#"( 0! TF $? 0/|$9"( )/%/v0/l!8 /l8R/9"()!,X 6"()/4"(,."(! TSh$5 /9"( !NK!D /f! y0//5 0/5,XNY#"( 0^>_/5|"/5 ! W"f! 0/+ E*fDP/5lKX"(:"( 0$%/+>_|*4&3 :"( "(*+8 !!0&/%
F ≥ 0.05F-bW0/
!)&q)9t$%)/%8P 0&;)="e#"(U0/J0 = 20
SRa$#y)09C$5 /9"(;)!,X 0?(>d)/%1 6"( 0$%/%C! s0/v E*fDP/5CK"(:"( 0$%/"( !)/9"&3Nq"(C*e"( ! r*4E&3 :"( J"(*+8 !!0&/%C! 0/M E*fDP/57KX"(:"( 0$%/XF
UN4$%*+86"(! ,0/7$%/5 E#"( P8R/9"(3"(&3\P/5)/5 E#"(XU0/J0
"(DP9KL/?0/?8P/5^1k0&)+ps! 0) :"()786")/M#"( 0)! %$ m0,X)/% % F#6e"( 0& % F 2&2S0 0/Cm0 0&76"(0/;$%/5 .#"( h8P/9"( /! " !D/; .)l0/?>@)/%w.0/5 0$5N
ωmod = U0a0/5 =,L! ,&/%/58P/5! E)
0/upqT)/5,X!*4/XFr M0/u !!*+!O(>"aKL/5N; :"(,L/u#"(U0/J0
0/u,X) 0&#"()/d(>3N3)/5* |"(88)%[3!*e"()/5 !NJ&/%)$5!DP/%&yD.Ns0/+Z0E$#q)#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ SPa$#
)/%wE!)/%7/2[0"$5 !Ng0/f/5 0/5,XNE = U0
)+DR/M/2[$5!)/%&Y! -)/5*4?(>Q"+86"($5 /21G0 //2[$5!#"( $@)/%$5 "3F F5&2FeUNq&/%$5)/9")! ,0/+#"(
U0/J0
)9W"()&f0/486")/#"( 0! |/5 0/5,XNyM! 0$5)/9")/%&J&30/)"&3*+\[3)/%f(>a"&&3! 6"( d E*fDP/5#"()/%iF
o e)&/5u)v"( 6"( !N)/W0/7/2[3$5!#"( g*4/%$)6"( *4! +0/$9")/(>T"C*4&3 :"()/%& :"($%/M/M6"%KL/l"l*4)/|&/5#"(! /%&Y ="(70/M)/%) 6"( 0$%/%9Fuo -0/Mpq)/5,X!*4/XS0/a&*+! 6"( EdZ0$)v)#"()/iS.")*+! ,|";N3)/5*t! la$#+0/W .*vDR/5d(>P"()*4/%wE6"( a0/M E*fDP/5?(>I :"($%/|!)/%9S
∣∣ψ0
⟩≈∣∣1, 1, 1, · · · , 1
⟩ S
a! ∣∣ψ0
⟩ &/5 0! ,e0/C,X) 0&#"()/XF $?D.K0 !NLS6 0/C6")4/2[3$5!)/C?#"()/! )&/5?)+#"( 0^>_/5?/5 0/5,XN<! .)e0/fN3)/5*YFWzE! 0$%/f"+#"()/ ∣
∣ψ, t⟩ "(!*4/
t
$#6"(#"$5)/5)/%&DENg0/f$%E/(e$5/5 E)cν(t)
∣∣ψ, t
⟩=
D∑
ν=1
cν(t)∣∣n1, · · · , nIν
⟩ S
a!I0/| .*fDP/57(>I :"($%/M!)/%7"( 0&
D0/M&3!*4/5 0 (>r0/M .*vDR/5D6")
$@)/%/;)/%$5 F &2S 0/;$9"( -"( 6"( !N3)/?0/C/2[3$5!#"( 0WDEN4#"$)! ,+0/C$%E/(e$5/5 E)&3! ,40/|!*4/|/5KL ! TF
bW0/5)/2>_)/XSW/0&3N'"N3)/5* (>6!)/%+a!
6DRX) 0g"(
U0/J0 = 20F
o '$% .#"g)0/-8)/5K0e$9"( $5 :"( 09SW/Y$% 0&/5= !N]0/Y)/%/-*e"(! 8P/9"( $@)/%/<m0,X)/ % F#B3S /2>@46"( 0&'&/5&2Sa"( 0&c8P/5^>_* !*4/21/5KL ! 0-"(40/
6
·
E
$E
E
E
E[J
0]
E E E Et[J−1
0
]
E E
E
E
E
E
> ./Q!9+7=@7DF H4!9 !- )(.^'.+.-\ !--+/0.F H4!z+ !-,! 7.H@1Z!^ !-'!='$(4D!-+ /Y16+,!-CD!-+/!|!^!- ∣
∣ψ, t⟩ !!-=]74!+
/0FJ.+1.\ωmod = 20.8J0
7!!-$7 4+V44!^!-$ !-V.(4D!+ / ! D1^`|!^! ∣
∣1, 1, 1, 1, 1, 1⟩ >!- 4@7V4+ !=16+,!-C!+V/zD1^` !-!F B! +
7DC4\ D1$16H$7=5!- ,+7! 7$7 4+BB!-!-$ B!=!|' 7DC4\ D1$16H$7=5!-$$ + 1.+ $ 164$4\ !- 1^44.+C>.!|9$.+ !- !-! ∣
∣1, 1, 1, 1, 1, 1⟩ +7 !- D1^`
|!^!FB5!- + 7DC4\ D1.16HF7 ! B@1^ @ 7D-(.+ C,\ H!@1646%&4 16!-!+$
*4E&3 :"( f> )/%wE0/5 0$5/%ωmod = 15.3 J0, 20.8 J0, 25.5 J0
FA 0)/9"&v(>0&3!)/%$5 !N/5K"( !6"(! ,-D0)/5KX"(D /%C&3! ,=0/l$9"( $5 :"( 09SP/f)/%$%)&q0/l$%E/(e$5/5 E)|(>"( !
462D6"7#"()/%a>@7/9"$#!*4/%)/58TFbW0/%)/|$%/(e$5/5 .)C"( ! 9 0a)e"( 6"( !N3)/
0/-$%*+8RX! '(>;0/-#"()/ ∣∣ψ, tn
⟩ "(4/9"$)c!*4/%))/58tn(>?0/-/5KL ! TS
"( 0&)|D#"(! 4! 3>_*e"( g +0/a*4/%$#6"( *4ua$#e /9"&+)M/2[$5!#"( 4! 40/$9")/M(>Q"+)/%) 6"( 0$%/XFdo Y)&/5)+)/%&30$%/M0/| :"(,L/f"(*4 .;(>r! 3>_*e"( 0/$%E/(e$5/5 E)O8)iKE&/XS(/$5 :")\m6/%&f0/dZ0$)M#"()/%r"$%$%)&3! ,?)a0/&3!D (>A0/M$%$586"( Y .*vDR/5)9FZ07/2[0"(*+8 /XS0"( ! T#"()/%aa!/2[0"$5 !N< 0/M&D !NE$%$58/%&n!)/s)/58)/%)/5 ."$5 :")9S7"( ! C#"()/%=a!n&D !NnE$%$58/%&!)/%$% 0!)/M"( 00/59S0"( 0&=#v TFQo <)/5*4(>Aa$5 :")\m6$9"( -)$)0/5*4/M 0/C$9"( * 8g0/?)wE6"()/7(>T0/?*e"(,X !0&/;(>T0/?$%/(e$5/5 .) |ci|2 $@! 0$%/ |ci|2 d0/8)86"(D! !!Ns(>u0/
i1GyZ0$)#"()/5&7$%)/%8P 0&3! ,<)="g$%/5#"(! q$5 :"#7>_?/9"$#
!*4/%)/58TFZr!,X)/% % F#B-"( 0& % F#-0i0/)/% !)|(>W0/%)/+$9"( $5 :"( 09F+bW0/+8 |
0/C /2> a6"( 0&-&/M(>Am0,X)/ % F#B&/58$5)a0/M!*4/21o"9KL/5#"(,L/%&/5 0/5,XN=#"( 0^>_/5W>@$%*+86"() TFbW0/Y8 g `0/-!,X.=6"( 0&)&/Y(>Cm0,X)/ % F#B! ! !0)#"()/%e0/$% .!D 0l)Y0/g$%/5 .#"( 8R/9"(Pf0/g&)/%&J !! 0/g09v0/e)wE6"()/e(>a0/*e"(,X !0&/=(>?0/=$%/(e$5/5 .4(>?0/g#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ SIa$)+")*4/%&
u
·
E E ,E Et[J−1
0
]
E E
E
E
E
E
E E ,E E
t[J−1
0
]
E E
E
E
E
E
- > 6/Q! +7 @7D$ H4! !=16=H>,!+V/ !=\D!$ !-! / !HY$` !
ωmod = 15.3 J0
Y+7 !-H4! + !- -! !ωmod = 25.5 J0
7D!!F74+?@ !-?1.] 16.+,! / !?D1^`T!-!- ∣
∣1, 1, 1, 1, 1, 1⟩ !-?4@7T4+ !
1.+,!-CD!+ /44u|!^!FzB! 16!4\=+ 7C4\=D1.16HF7 5!- D+>7! 7$74+ !-!-$B! !|'=7C4\ D1.1.H$7 5!-$$
)yDP/-&*+! 6"( . !N'$%$58/%&! '0/-pq+)/5,X!*4/XFcbW0/Y) !& !! 0/-0i+0/$% E!D y(>uZ0$)Y)#"()/%a!y/2["$5 !N- 0/f&D !NY$%$58/%&)!)/XF $? 0/v$9"( )/%/XS36"(7"l&/%$5)/9")/M(>r0/M$%E/(e$5/5 Ea>_0/MZ6E$#=#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ /9"&
)"( ! 0$5)/9")/-(>70/-$%/(e$5/5 .)4(>?0/-#"()/%la!'&D !N$%$58/%&c!)/%9S"( 0&K$%/KL/5)#"3F4pq)/%9KL/59SrZ0$)#"()/%Ma!s&D !NE$%$58/%&J)!)/%f"( )$% E!D)/7)M0/7#"()/XFIbW0/%)/Z0$)+#"()/%d"()/ 0d&3!)/%$5 !N+8R8 :"()/%&+> )* (>0/#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ S! 0$%/ur/2[$5!#"( l &f)/%wE!)/0/d/5 0/5,XN
∆E ≈2U0
F7bW0/%)/v#"()/%M"()/|*4)/v !!L/5 !NY8R8 :"()/%&sK:"486"($5 /21G0 //2[$5!#"( 0;(>#"()/%;a!y 0/&D !NE$%$58/%&y!)/XSVa$)y &J"( )e/2[38 :"(! y0/v&/5 :"9NL/%&! 0$5)/9")/f(>r0/|$%/(e$5/5 .)7KE)!D /M! <m0,X)/ % F#B3F
Zr!,X)/ % F#! ! !0#"()/%0/y/5KL ! (>l0/)*+*4/%&$%E/(e$5/5 E)s"(<0/*4&3 :"( Y>@)/%w.0/5 0$5/%
ωmod = 15.3 J0$ /2>@6"( 0&Y&/5&"( 0&
ωmod = 25.5 J0$ 8 -0/C!,XE &2Fdo =DPY$9")/%a 0/M$9"( #/%/?0/M "(*4/CDR/56"9KE?">_W0/$%/5 E#"( 68P/9"(RA0/Z0$)+#"()/%Qa!g 0/&D !N+E$%$58/%&g!)/"()/a8P8 :"()/%&g"(0//2[8P/5 0)/(>00/dZ0$)f)#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ F $? 0/$9"( "( #7)/%/d0/d8P8 :"(
(>OZ0$)4)#"()/%da!=v&D !Ne$%$58/%&e)!)/%9S.D0/5!$%/(e$5/5 .W*e"( ! /5"! =0/?$9")/?(>O0/;$%/5 .#"( P)/%) 6"( 0$%/;8R/9"(PFI <$% .#"))v0/;$%/5 .#"( P8R/9"(PS 0/|D0#/5KL/5)?"( YX)$5! ! :"( q(>r0/M$%/(e$5/5 .)?a!s"l :"(,L/5a>@)/%w.0/5 0$5NLF
o )&/5g)],L/5-"( `!*+8)/%)) n(>f0/q/2[$5!#"( 0<*4/%$)6"( *4</s6"9KL/"g$5 X)/5; E"(;0/vZ6E$##"()/%;a!q 0/l&D !N$%$58/%&y!)/XF7bW0/%)/lZ0$)
·
E E E Et[J−1
0
]
E E
E
E
E
- > FE ^H 7D$H@1Z!- !?16% 16.+,!- /'D1^` !-!F B5!- 1Z!-4\V+7C4\[D1.1.H$7I! G47[4+ ! 16+,!-CD!-+V/ 7DF1Z! H>!@1646%4 16!$7 !-!-$ / ∣
∣1, 1, 1, 1, 1, 1⟩ !-
!$ !|'W-.HF$+! |!^!F 1.++$1Z!-$7 !∣∣1, 1, 1, 1, 1, 1
⟩ (] + PL7$7 4+)S +7!|'VPX7D!!-$7 4+)S77D!+>4H.
#"()/%I$9"( +"(,."(! +DP/DR&3!K&/%&l! .)M&3\h/5)/5 .u$5 :")#/%9S! 0$%/0/5Nf! 0$5 !0&/W#"()/%a$)="()/7&3!)/%$5 !N$% 0/%$5)/%&e)M0/a8)/pqd#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ K:"M"C08
)4"l 0/5!,X.DP! ,=!)/ $ 86"($5 /21G0 /l/2[$5!#"( &2S0/XF ,0F
a†1a2
∣∣1, 1, 1, 1, 1, 1
⟩=
√2∣∣2, 0, 1, 1, 1, 1
⟩ F
?!)/%$5 !Nq$% 0/%$5)/%&y*4/9"( 09Sh6"(C0/l*e"(\[y/5 /5*4/5 E|(>0/BC"(*+! !) :"( DR/21/%/5 -0/%#/?o+#"()/% 0 31GKX"( ! ,0FubW0/5)/2>_)/XS30/?86"($5 /21G0 /f/2[3$5!)/%&#"()/%|"()/$5 :")\m6/%&qD.Nq0/&3#"( 0$%/DR/5o/%/5 0/+&D !NqE$%$58/%&J"( 0&y 0 31E$%$58/%&n!)/XF ! CZ0E$#'#"()/%=a!`0/q&3#"( 0$%/
d = 0$ 0/q&D !N`"( 0&
0 31E$%$58/%&!)/"()/l&3!)/%$5M 0/5!,X.DP) &;"()/l&3!)/%$5 !Nq$% 0/%$5)/%&y)e0/l8)/psC#"()/ ∣
∣1, 1, 1, 1, 1, 1⟩ Sd = 1
1#"()/%M"()/v$% 0/%$5)/%&q)d = 0
#"()/%;KE:"g"08TS6"( 0&Y)4 TF -0/M> 0/5;&3)$50) TS/M /9"%KL/v0/
= 1!,X q"( 0&-0)/
dn1#"()/C"W"l"(DD)/5K:"( <>@W"( ! P)#"()/%9S.a$#"()/?$% 0/%$5)/%&<DEN
n080)"
d = 01#"()/X
∣∣2,
x
0, 1, 1, 1, 1⟩
︸ ︷︷ ︸
d0
−→∣∣2, 1,
x
0, 1, 1, 1⟩
︸ ︷︷ ︸
d1
−→∣∣2, 1, 1, 0, 1, 1
⟩
︸ ︷︷ ︸
d2
F
Zr!,X)/ % F! ! ! !0#"()/%-0//5KL ! (>v0/J$%/(e$5/5 .)s(>lZ0$)#"()/%&3!)/%$5 !Ns$% 0/%$5)/%&q) ∣
∣1, 1, 1, 1, 1, 1⟩ $@) !&s !! 0/5&2ST"( 0&qZ6E$##"()/%;a$)"()/
$% 0/%$5)/%&f)7A#"()/dKE:"?"( f86"($5 /21G0 /W/2[$5!#"( "( 0&v 0/$@&)/%&v !! 0/5&h $@&0"0/%&v !! 0/5&T0809FAbW0/
d01#"()/%Q"()/*e"(! !Nf$%$58/%&hSX! 0$%/u0/5N|DP/5 0/2m0
> )* 0/g&3!)/%$5v$% 0/%$5 )-0/gZ0E$#J#"()/ ∣∣1, 1, 1, 1, 1, 1
⟩ F< E)/5)/%! ,X !NLS
d11#"()/%9SIa$#c"()/=$% 0/%$5)/%&]) ∣
∣1, 1, 1, 1, 1, 1⟩ D.N"( c"&&3! 6"( W08"()/
"( !*4Xa 07$%$58/%&-! $%*+86"() Y)d21#"()/%9F
Zr!,X)/ % F1M09W0/M#"(*4/M)!6"( -! =0/M$9"#/M(>A0/;8R/9"(3"(ωmod =
15.3 J0$ /2> C6"( 0&&/5&C"( 0&
ωmod = 25.5 J0$ !,XE|6"( 0&&/5&2Fl $%*+86"(^1
·
E E ,E Et[J−1
0
]
E E
E
E
E
E
E E ,E E
t[J−1
0
]
E E
E
E
E
E
- ^H 7D$H@1Z!- ! 1.] 16.+,!- / D1^` !-!FB5!-T16!4\ +7DC4\?D1$16H$7W! + ! H>F`D!
ωmod = 15.3 J0
P046/Q!ZS +7ωmod = 25.5 J0PX,!^S6 B47 4+ ! 16+,!-CD!-+ / 7D-$16! H!-1.4.%n4 16!$7V!-!F
/ ∣∣1, 1, 1, 1, 1, 1
⟩ ! !-.!|9?$HF$+,! |!^!-$ 16++$16!$7 ! ∣∣1, 1, 1, 1, 1, 1
⟩ (]@+PX7F7 4+FS+7 !|9VPX7!!-$7 4+)S775!-+4H$
) c)y0/-$%/5 E#"( )/%) 6"( 0$%/-8P/9"(0/-Z6E$#]#"()/%4a$)`"()/< 04&3!)/%$5 !N$% 0/%$5)/%&=)v0/78)/;ps#"()/C"()/*f0$#Y) ,L/59I g0/?$9"#/?(>T0/?)/%)(1 6"( 0$%/;8P/9"(RS"(
ωmod = 15.3 J00/
d11)#"()/%"()/?8P8 :"()/%&<&*+! 6"( . !NLFbW0/
86"($5 /21G0 /M/2[$5!)/%&d01)#"()/%W0/5*4)/5 \>@/%0i"v8P/58RX)/%&gX)$5! ! :"( =a!
"l#"(0/5?!,X-> )/%wE0/5 0$5N=! -DRY8P/9"(39Sωmod = 15.3 J0
"( 0&ωmod = 25.5 J0
F
·
E
$E
E
E[J
0]
E
$E
E
E[J
0]
E $E E E Eω [J0]
E
$E
t[J−
10
]
E $E E E Eω [J0]
E
$E
t[J−
10
]
+.-\ !^+/0. / \D|!-.I/4CY+ +
44@!!@16 5!-$ PX4./Q!^S +7
5C>,+ +
54@!!-1. 5!-$]C>! !' ^!-
U0/J0 = 20 ]B+ @z! .+$\!--+|/0$
/0+1Z!-+?/ !=+7 D7D4!+ /0-$J,$+16\ωmod
P04'. $^S +7V!-!-=.%)(.^$7=(4F P0HH>$ $^SZ + C>!V1.F+ 1$+ $
2%nHY$` +>!
HJ/e")*4/40/40$5)/e(>a0/+)/%# 6"( 0$%/%f)-DP/4"( /2h/%$5v(>W0/+m0 !)/!9/(>R0/ :"($%/%9FAbW0/5)/2>_)/XSL/K"(/%&l0/ .*vDR/5u(>h"()*4u"( 0&l)!)/%A! 0/>@ ! ia! ,=!*v :"( 09FZr!,X)/ % F ! ! !0#"()/%;0/f/5 0/5,XN-#"( 0^>_/5?(>uN3)/5*4(>
4$ /2> g6"( 0&&/5&e"( 0&
5$ !,XEg6"( 0&`&/5&+)!)/%9SDR`a!n"( n"9KL/5#"(,L/
m0 ! !! ,M>_"$5)(>1DRX) e8R/5u!)/XFIUWg)!*f :"( 0W"()/a8R/5^>_*4/%&g"(d0/#"(
U0/J0 = 20FvbW0/+&/5 0)!Nq8 )|090/+/5 0/5,XNq#"( 0^>_/5|";> 0$5 (>!*4/
"( 0&<*4E&3 :"( ->@)/%wE0/5 0$5Nωmod
S0/M8 )"(DP9KL/f&/58$50/M/5 0/5,XNg#"( 0>@/59S!*4/21o"%KL/5#"(,L/%&9KL/5C0/va0 /v)!*f :"( q> )*
t = 0)
20 J−10
F?o s$% E#")-0/
6DPX) N3)/5*4|&3)$50#)/%&DP/2>@)/XSO0/%)/40i"<o(1G8P/9"()/%) 6"( 0$%/
0$5)/XF ,."(! TS0/)/%) 6"( 0$%/> )/%wE0/5 0$5Nωmod = U0
S()/5,."()&3! ,M0/W$%/5 .)&(>r0/C)/%) 6"( 0$%/XS6a! "(,X)/%/5*4/5 .;a!-0/M86"($5 /21G0 /l/2[3$5!#"( 8$5)/XF
Zr!,X)/ % F!"?! ! !0#"()/%I0//5 0/5,XNf#"( 0>@/5Q(>7DRX) 0I
7!)/%'$ /2> I6"( 0&
1
·
E
$E
E
E
E[J
0]
E
$E
E
E
E[J
0]
E $E E E Eω [J0]
E
t[J−
10
]
E $E E Eω [J0]
E
$E
E
E
t[J−
10
]
- z+.-\V!--+/0. /BV\]!$/7CY+ +
74@!!-1. !F P046/Q!ZS +>7
8C>,+ +
84!!-1. 5!-$CY!- ! ^!
U0/J0 = 20 ]B+@ !- $+.-\ !^+/0.
/0+1Z!-+ /9!= +7 =D7D4@!-+ /0-$J,.+>16\ωmod
P049$ F-S +7 ! !-=6%)(.^F7 (4$VPXHHY. $^SZ T
7C>,+ \D|!-.f
3%&H>F` +!--
=4 ! !-6C>,+D%
65!-$ \D|!-. !
8CY+D%
8!F \D!$I
4HY$`
-$++16 !-1Z!-
&/5&7"( 0&8DPX) 0?
8!)/% $ !,X.;6"( 0&)&/5&"(0/f#"(
U0/J0 = 20FabW0/
71GDPX) N))/5* 0iC"g!*+! :"(C)/%) 6"( 0$%/0$5)/+"?0/
61GDPX) yN3)/5* 1
"(,."(! s0/f$%/5 .)&q(>Q0/M!8 /|8P/9"(<7$5 X)/|)40/C>@)/%w.0/5 0$5Nωmod = U0
"8)8PX)/%&D.Nl0/W86"($5 /21G0 /7/2[$5!#"( +8$5)/XFQbW0/a#"(*4/W0 &Q0/>_I0/
8DPX) )N)/5*YSV"( !0,X0/|)/%# 6"( 0$%/f$% 0)?(>
48P/9"(9S3a0X#/f$%/5 .)&
7"( #+$5 X)/M)0/C>@)/%w.0/5 0$5Nωmod = U0
FZr!,X)/ % F % )09"] :"($%/J(>
5!)/%<a!
10DPX) 0"(
U0/J0 = 20S
a$#)094"$)/2[3$5!#"( c8P/%$5*Y4bW0/<) ,L/%l8R/9"(]"(88R/9"(! ,y! 0/4/5 0/5,XNy#"( 0^>_/5v"(
ωmod = U0$ /2> M6"( 0&&/+(>m0,X)/ % F % &C)/58)/%)/5 E)
0/d86"($5 /21G0 //2[$5!#"( 0r(>00/u8)/dpq^1#"()/d)d01#"()/%iF,.T/2>@A(>0!A 0/
$9"( J)/%/+"g)/%) 6"( 0$%/e"(U0/2
SPa$#J! 0&3$9"()/%M0/4"(D0)D J(>Y80) 0
·
E
$E
E
E[J
0]
E
E
E
E
E
E
σ
E E ,E E Eω [J0]
E
$E
t[J−
10
]
E E ,E E Eω [J0]
E
$E
t[J−
10
]
> +$\ !^+>|/0$PX4./Q!z+7 @7DFS +7+] C>$ (-+16 P0-,! +7 7D)S/
10C>,+ +
54@!!@16 !FzB! ^!
U0/J0 = 20 7D.+!|\ H4!-'7D.H@1Z!'!
C.-(C4$ /0+16!+ / !-=+>7 =D7D4@!-+ /0-$J,.+>16\ωmod
!=H4!^ + !-H44>|!--!'!F(4F )(.^$7 (. !=
t = (0−20) J0
F74+B+ !-B.+$\!^+/0. H4!^ 7D.+! !=/0FJ.+1.\
ωmod = U0/J0
@ \D!$I F++1.$+$C]\=! /0FJ.+1.F
ωmod = U0, 2U0, ∼ 3U0
(>I0/l/5 0/5,XNE = U0/2
S0a$#"( #4 /9"&?)40/f/2[$5!#"( y(>u0/f8)/vps#"()/XFIbW0/7! EN+8R/9"(e"(
U0/3! 0&3$9"()/%a"M)/%/?80) <"(D0)D TS3/9"$#=(>T0/
/5 0/5,XNE = U0/3
F &&3! 6"( ! !NLS)/%# 6"( 0$%/%W"(88P/9"(a"(! E)/5,L/5*v !!8 /%a(>U0
0/*v !\1G8R/9"(+0$5)/"(ωmod = 2U0
I,L/5 0/5#"()/%&D.NlM86"(#"( ! /5 86"($5 /210 /a/2[3$5!#"( 0u(>P0/W8)/apqu#"()/Wa!e";80) e(>V0/7/5 0/5,XN
E = 2U0F
bW0/+)/%) 6"( 0$%/g"( ∼ 3U0! 0&3$9"()/%
386"(#"( ! /5 d86"($5 /21G0 /=/2[3$5!#"( 09F=bW0/
8)86"(D! !!N(>u0/%)/v/2[3$5!#"( 0;?)/%&30$%/%&! q$%*+86"( y)40/v)/%) 6"( 0$%/%"(ωmod = U0
"( 0&ωmod = U0/2
S6! 0$%/M0/fBC"(*+! !) :"( s&/% 0;$% 0/%$570/8)/|ps)#"()/ ∣
∣2, 2, 2, 2, 2⟩ a!s")#"()/ ∣
∣3, 3, 1, 1, 2⟩ F
Zr!,X)/ % F A 090/-!*v :"( (>C"y)N)/5* (>7DPX) 0+
6 :"($%/
!)/%+"(l0/=#"(U0/J0 = 20
F $% .#"+)q0/<86"(#"(*4/5)/54)/5)&3)$50#)/%&
@
·
E
$E
E
E
E[J
0]
E
E
E
E
E
σ
E $E E E Eω [J0]
E
$E
E
E
t[J−
10
]
E $E E E ,Eω [J0]
E
$E
E
E
t[J−
10
]
- D +.-\ !^+/0. PX4./Q! +7=7FS +7 +] C>$ ()@+>16 P0-! +7=@7D)S/
7CY+ +
64@!!@16 5!-$ B! + -!
U0/J0 = 20 7.+5!|\?H4!-7D.H@1Z!
!-C.-(C4$ /0+>1Z!+ /z!=+7?=]74!+?/0FJ,.+1.\ωmod
!-H4!^ +!-HV44!^! !F ()4$ )(.^$7 (.B!-=
t = (0 − 40) J0
F7 4++V!$+.-\!--+/0.H4!-B7.+! ! /0FJ.+1.\
ωmod = U0/J0
/9"( !/5+! c+$#6"(8)/59Sd+N3)/5*6"4"q 0 31$%*+*4/5 0#"()/<m0 ! !! ,y>G"$5)9FbWl)/% !)l! c"Y) ,L/5D6"$),X) 0&]! 0/g .*vDR/5+KX"(:"( 0$%/ $ !,X.6"( 0&&/;(>Tm0,X)/ % F A &d/5KL/5 -"( :"(,L/
U0/J0
FdzE\[g(>T0/C)/5KL/5 =DPX) 0d &=>@*~"pqa! 0 :"(! ,e#"()/v"a&3)$50))/%&-/9"( !/5! $)6"(8)/59SD70/C)/5*e"(! ! ,DPX) <a&/5 E$9"( !#/%&=iKL/5a0/Ca0 /; :"($%/|"( 0&<,L/5 0/5#"()/%Wk00$56"( 0&30/C)!)Q*4D! !!oNLFQbWd"(#"( ,L/5*4/5 .8)E&30$%/%"7)/%# 6"( 0$%/0$5)/&3\h/5)/5 .Q>@)*6"(d(>P0/N3)/5*4Ia!e$%*+*4/5 0)#"()/m0 ! !! ,M>G"$5)9SLa$#4!*+8 !/%9S.6"(u0/ E*fDP/5f(>d8P/9"(M"(88P/9"(! ,Y&/58R/5 0&M y0/!9/(>d0/l :"($%/4";/5 ! d"M 0/Cm0 ! !! ,4>_"$5)9F
Zr! 6"( ! !NLSI/g!*v :"()/%&]"6DPX)
6!)/e)N)/5*a!DP%[1GDR 0&0"(N$% 31
&3! 09FbW0/4m0,X)/% % F!6 $U0/J0 = 20
&v"( 0& % F $U0/J0 = 50
&|! ! !0#"()/e0//5 0/5,XN`#"( 0^>@/5-"( 0&n .*vDR/5-KX"(:"( 0$%/y(>v0$)N3)/5*49F 7*+86"()/%&)0/
·
E
$E
E
E
E[J
0]
E
E
E
E
E
E
σ
E $E E E Eω [J0]
E
$E
t[J−
10
]
E FE E E ,Eω [J0]
E
$E
t[J−
10
]
> 9 7D$+!|\ H4! + !-46/Q! ! .+$\ !--+|/0$ /z \D!.M/
6CY++
6!F / 4!!@16'B!CY C>+7\ 1.+7D!+>!!-9^!-
U0/J0 = 20
4'. =B !B$+.-\ !--+/0. H4!!-$7 ($ D7D4!+ /0FJ.+1.\ωmod+7 != -)\ 4+ +=!B!-=.%&)(.^F7 H4!97.+!$ ! 1..+,!-@7 /Y! H>F`D
+ !$+ωmod ≈ (15 − 25) J0
H4!- +?!--! 44!^!- ! +] C>$N1Z!-!-+
σ! !- >-! 5!- Y!49$ H4! !N16!!+ (.B!-==+7
D7D4!+ /0FJ,.+1.\ωmod ≈ (15−25) J0
F!- ^H C>( Y!- !-=6% )($-F7N1Z!-!-+$
$5N$5 !$+DP 0&0"(N$% 0&3! 0fo*4)/+8R/9"(3f"(88P/9"(|! J0/+)/%# 6"( 0$%/e0$21)/XS6D?0/|)/%) 6"( 0$%/f> )/%wE0/5 0$5N-C"(,."(! ! ,L&s"(,X)/%/5*4/5 .Ca!s0/f8)/21&3$5 (>r0/|86"($5 /21G0 /v/2[$5!#"( Y8$5)/XFdo Y$%*+86"() sa!Y0/|#"(*4/N))/5* a!+$5N$5 !$DP 0&0"(Nf$% 0&3! 09SX0/a&3+(>60/)/%) 6"( 0$%/W0$5)/#"%N370/l#"(*4/XF? sDRy$9")/%9SV0/v&3)#"( 0$%/fDP/5/%/5 0/fo<)/5?8P/9"(37
∆ωmod ≈ 10 J0FQUW- 0 31$5N$5 !$M)N)/5*4a)09 "( )4"v/9"(=)/%) 6"( 0$%/M0$21
)/f"(a0/C>@)/%w.0/5 0$5Nωmod = U0/2
FbW0/?)!*f :"( <! ! !0#"()/%&=! em0,X)/ % F!Bf"(,."(! Y"fN))/5*(>
6DRX) 0
6 :"($%/-!)/%la!DR9[DP 0&0"(N$% 0&3! 04"( 0&c"( c8R/58PX)/%&6"(*4 $
&¶
·
E
$E
E
E
E[J
0]
E
E
E
E
E
E
σ
E E E E Eω [J0]
E
$E
t[J−
10
]
E E E E Eω [J0]
E
$E
t[J−
10
]
- D +.-\ !^+/0. PX4./Q! +7T7D)S +7 +]CY. ()@+>16 P0-,! +77FS /
6C>,+ +
64!!-1. 5!-$ B! -!
U0/J0 = 50+7=CY CY+7-\=16+>7D5%
!-+. 7D$+!|\=H4!-7D$H@1Z!'! C.-(C4$9z/0+16!+ /!= +7=D7D4@!-+/0-$J,$+16\
ωmod
'! H4!- + !H 44!^!-V!$V()4$ )(.^F7 (.=!V!=t = (0 − 20) J0
-$7 4+ + !- $+.-\?!^+>|/0$H4!^ 7D$+!- !- /0-$J,$+16\
ωmod = U0/J0
#"(88! ,J8P)/5 .:"( GSa$#')/%)/5*fD /%40/Y"$56"( 7/2[8P/5!*4/5 .#"( ?)/5809FbW/2[3)/5 6"( a8R)/5 E:"( ! .)&30$%/%&c),Xc0/- 31!)/</5 0/5,XN
εi(>?0/-UWX)/
B?DD6"()&BC"(*+! !) :"( TF- J$9"#/e(>"<6"(*4 $e#"(88! ,s8R)/5 E:"( 0/%#/+86" 1#"(*4/5)/5)7)/9"&
εi = 4Vtrap,0
(i
I− 1
2
)2 $ % F &
a!I0/ E*fDP/5f(> :"($%/+!)/%|"( 0&
Vtrap,00/l#"(88! ,Y8R)/5 E:"( "(C0/
)/5-!)/%9F bW=6"(*4 $y#"(8gN*+*4/5$qa!)/%8R/%$5=)0/s$%/5 E)/(>0/y :"($%/XS;a0/5)/0/y8R)/5 E:"( |KX"( 0/%9F bW0/8P/%$5* -!*+! :"(Y)0/|)N)/5* a!0;"l6"(*4 $v#"(88! ,e8P)/5 .:"( GS0!?09;")/%) 6"( 0$%/M>_0/*4&3 :"( y> )/%wE0/5 0$5N
ωmod = U0"( 0&J"( "&&3! 6"( Q/9"(q)/%) 6"( 0$%/e"(
6
·
ωmod = 3U0F
$% !0 TSA>@v0/4*4&3 :"()/%& :"($%/g 0/e$9"( "%N6"(f0/e)/%) 6"( 0$%/> )/%wE0/5 0$5/%-*4!KX"()/%&D.Nn0/y86"($5 /21G0 //2[$5!#"( 8$5)/J"()/! ,LE&"(,X)/%/5*4/5 .a!<0/;)/%) 6"( 0$%/%D#"(! 0/%&<! <0/C!*f :"( TF?/5KL/50/5 /%#9S3/$9"( 0C/2[38 :"(! y0/vm0 0/v0$5)/+(>u0/%#/v)/%# 6"( 0$%/%9SPa$)&/58P/5 0&q y0/!9/d(>00/d :"($%/W"A/5 ! "r f0/ .*fDP/5Q(>V"()*4Q"( 0&|DP 0&0"(Nf$% 0&3! 09F
>
·
E
$E
E
E
E[J
0]
E
E
E
E
E
E
σ
E E E Eω [J0]
E
$E
t[J−
10
]
E E ,E Eω [J0]
E
$E
t
[J−
1]
- > ] CY+1 !z+ 5!-$'/+ =H4!7D =D7D4@!-$7 4@!!@16 B5!-++D% 16\D1641 CY+7-\ 16+>7D5!-+ +7 BH-C>4@1 !^HH+BH>!.+,!-4! + +!-.^16!+!-.+!
U0 = 20+7 !-++$44+ !-.+!
J0 = 1 $H16!F7 @ !-$+.-\ !^+/0.
PX4./Q! >+7 @7DFS +7V! +]CY. N16!!+=P0-,! +7 7FSZ =7D$+!|\ H4!- !-BC>$(C4F /0+1Z!-+=/Y! +7 D7D4!+ /0FJ,+1.\ )!-BH4!- C>(R! (4$ )($-F7 (.B!-!-= / ! 4@!+ !^HH+ HY!-.+,!@4@Vtrap = U0
! ! C>+7$ PX!$1+7
6S +7
Vtrap = 0+?! 1..+,!-/ !
4@!!@16 P0CY6!|'.$+V!F3+7
4SZ
3 (N ! "#
!#G8='*/ "7'& !?8<8='; /!= ³I8BL#;/L^ 8& L)"6L ! L#?#²8 |77L# %@/ =# # L !?8B=)"6 : ³I88B#"!6!?E!/|7#²8<EC 13#D ³L6%@L*/7 A 8SL 8=' ", )" ³I8BL#;/ L^ 8=' J3L(/|%@/M!/!;78 ;4E@8BBL#/47A 8B= "8<6C I#DCL|C ³I8<EC ³I88B",!/ !D "B8 C*7/ EC$/-" */!/MEC
) #" 9*)
YC)/%$5 s0/v/2[$5!#"( y8R/%$5*(>"+o(1$%*+8P 0/5 .M>_/5*+ q)N)/5*a!
6 :"($%/g!)/%vv8)DR/%&D.NJ :"($%/g! !! ,0Fqjd"$)c(>0/gs>_/5*+ $
8R/%$5/%?$% .!D)/%3"()*49F 6"( ,L0?)g0/f&3#$50) s>@C"+DPX) $fN3^1
)/5*+! e)/%$5 "3F F SX0/7Z0/5*+6B?DD6"()&4B;"(*+! !) :"( $ F A1A &I$9"( 4DP//2[3)/5 0&/%&! -0/|#"(*4/C"9NRdD.N=! .)&30$5! ,="( s"&&3! 6"( r 31!)/|/5 0/5,XN<)/5*
HFF = −JI∑
i=1
(
c†i ci+1 + c†i+1ci
)
− JI∑
i=1
(
d†i di+1 + d†
i+1di
)
+ Ucd
I∑
i=0n
(c)i n
(d)i +
I∑
i=0εi
(
n(c)i + n
(d)i
) F
+C/5 0/5#"( ! !NLS/9"$)-8P/%$5/% &<6"%KL/C!)Wia Y 31!)/C/5 0/5,XNe)/5*YFQo =a$9")/XS/2[3)/5 6"( V8P)/5 .:"( RW")*4/%&g)|DP/0/7 "(*4/>_dDPe! 0&(>h>_/5*+ 09FbW0/C86"(#"(*4/5)/5
εia&/2m0 0/%&D.N
εi = iεtiltS
a!εtilt
0/!)/21G! 0&/58P/5 0&/5 .f/5 0/5,XNq&3\P/5)/5 0$%/lDP/5/%/5 J 0/5!,X.DP! ,!)/%"( 0&
i0/v!)/|! 0&/2[hFabW0/l/2[3$5!#"( y8P/%$5* 8)DP/%&sD.N-KX"(:"( s(>u0/
·
E
E
E
E
E[J
0]
E
E
E
E
E[J
0]
E FE Eεtilt [J0]
E
$E
E
t[J−
10
]
E FE Eεtilt [J0]
E
$E
E
t[J−
10
]
- D +.-\ !^+>|/0$ / \]!$#/ !|9=/0$=+@1 H>F16$ +64!!-1.!F.
z1^WH>F16$1.+,!-CD!F3!. 6/Q! +7W@7D !- \D!.O! ! -!
Ucd/J = 5 ,! >+73@7D !
Ucd/J = 10 $H@1Z!F7 @ ! .+.-\ !^+/0.
/0+>1Z!+ / !-= +7W4!!-1. !-4!+εtilt
PL7D.+!|\ H4!- +W! 49$- S +>7 !!-=6% )($-F7.+$\!^+>|/0$B(.'!- B4 4!+ F7 4+ 7D.+!F!!-4!+
εtiltB@1^V16--$H>+7z!=! +,!$-1Z!-+ !-.+!
Ucd
:"($%/<! !! ,εtilt
FJo 0/<) ,X !N)/58 !KL/=)/5,X!*4/XSu0/=,X) 0&#"()/-&/% 0;$% .#"(! yZ0E$#-#"()/%a!q&D !NYE$%$58/%&s!)/%9FG>I0/f/5 0/5,XNY&3\h/5)/5 0$%/DP/5/%/5 ]!)/%l9KL/5)$%*4/%l0/e! .)/5#"$5 c/5 0/5,XNJ&30/4)0/4! !9SA0/eN3)/5*)/%8P 0&a!-"|86"($5 /21G0 /C/2[3$5!#"( TF$? 0/?6")fL/%/58=! g*+! 0&hS6"(9S&30/)M0/ 4I"( !V/2[$5 !0 48! 0$5!8 /XS.! .)/5#"$5 <E$%$5)d !NvDP/5/%/5 eM>_/5*+ (>Q&3\P/5)/5 E?8P/%$5/%9F
Zr!,X)/% A FC"( 0& A F ! ! !0#"()/?0/?!*v :"( -(>r"Mo(1$%*+8P 0/5 .a>_/5*+ N3)/5* "(&3\h/5)/5 .a#"(X
Ucd/JS! 0$5)/9"! ,+>@)* "l/9"( !Ng! E)/5#"$5! ,eN3)/5*
a!Ucd = 5 J
8J)Ucd = 50 J
F= ]"( ! d$9")/%|0/e!*f :"( #"()|a!0/f,X) 0&#"()/v$%)/%)8R 0&3! ,=)40/f#"(
Ucd/J"(?0/|!*4/
t = 0a!0/
0 31G! !)/%&]B;"(*+! !) :"( TF v!*4/t = 0.1 J
0/4! !v|!*+*4/%&3:"()/5 !N"(88 !/%&
·
E
E
E
E
E[J
0]
E
E
E
E[J
0]
E FE E E Eεtilt [J0]
E
$E
E
t[J−
10
]
E E E E Eεtilt [J0]
E
$E
E
t[J−
10
]
+$\ !--+|/0$ / \D|!-.M/ !|'=/0.- +1 HY$16$ +64@!!@165!-$$
z1^ H>F16$ 16+!-CD!-$3!-. 6/Q!=+7 @7D ! \D!$O! ! ^!-
Ucd/J = 20'-,! +>7T@7D !
Ucd/J = 50 .H@1Z!-$7 @!- $+.-\ !--+/0.
/0+1Z!-+W/ != +7 4@!!-1. !-4!+εtilt
PL7D.+>!|\ H4!- + !- 4'.- S +7 !-! .%&)(.^$7$+.-\!--+/0.B($'! B4 4@!-+ -$7 4+7D$+!-$'!-!45!-+
εtiltB1^ 16-$H>+79! !- +,!-.^1Z!-+ !-.+!
Ucd
a!"=$% 0#"( E|86"(#"(*4/5)/5εtilt
.! Q0/+!*v :"( /5 0&|"(t = 63 J
FvbW0/m0,X)/%a090/M/5 0/5,XN=#"( 0^>_/5a">@ 0$5 Y(>A!*4/|"( 0&= :"($%/M! !
εtiltS0"( 0&
!*4/21o"%KL/5#"(,L/%&g"u"?>@ 0$5 e(>R0/W! !9FrbW0/W!*4/21o"9KL/5#"(,X! ,vI8P/5^>_*4/%&+9KL/50/Ca0 /M!*4/|86"( TF
bW0/M)/%wE0/5 0$%/M(>I!*v :"( 0! ! !0#"()/|0/M$)6"( ,L/|(>r0/|/2[3$5!#"( s8R/%$21* a!c0/<K"(:"( '(>;0/<#"(
Ucd/JF +/9"(]! .)/5#"$5 n)/5 ,X0
$ m0,X)/ A F5&; 0/+D0)/5KL/%|"=D)L"&J"( 0&$% .! E00f8P/%$5*YSha0/5)/9"M! 0/) ,X !Ne)/58 !KL/|)/5,X!*4/ $ m0,X)/% A F &W"&3)$5)/5)/M8P/%$5*~a!Y6"(8-8P/9"("(88R/9"()9F
bW0/?)) ,L/%d)/%) 6"( 0$%/M"(εtilt = Ucd
! 0&3$9"()/%0/7!K:"( 6)/%) 6"( 0$%/C&30/)g088! ,)<"= 0/5!,XEDR! ,s!)/XSO"M&3)$50)#/%&y"(DP9KL/XFlZ0M0/+!*v :"( 0
1
·
a!#"(XUcd/J ≥ 10
8P/9"(M"(;>@#"$5 6"( rKX"( !0/%M(>Ucd
"(88R/9"(9FfZ0M! !! ,
εtilt = Ucd/20/l/5 0/5,XNs&3\h/5)/5 0$%/lDR/5o/%/5 0/2[3; 0/5!,XEDR! ,Y)!)/%?
∆E =
UcdFAo v0/W#"(*4/W)$#0/5*4/XSX)/%) 6"( 0$%/%u"(
εtilt = Ucd/3"( 0&
εtilt = Ucd/4$ 088! ,
"$5)X)4!)/% &$%$59F
< )9 #" 9*)
o +u)/%$5 40/a)!*f :"( 0d(>PM>_/5*+ $8P/%$5/%I! g"( e"(*+8 !!0&/*4&1 :"()/%& :"($%/8P)/5 .:"( 6"()/8)/%)/5 .)/%&hF I&3)$50))/%&v! +)/%$5 F#"3F 0/WZ0/5*+B?DD6"()&gBC"(*+! !) :"( <$% 0!)(>Tol! 0/5$)/5*4d>@/9"$)<8R/%$5/%iFrZ00/5^1*4)/XSr! 0$%/l>_/5*+ 0|6"%KL/e)-DR/5Nq0/ 4I"( !/2[$5 !0 J8! 0$5!8 /="( 0&0^1#"( ,L/a 31!)/! E)/5#"$5 0d"()/a")*4/%&hSL !Nv! .)/5^18P/%$5/%I! .)/5#"$5 g"()/8PX^1!D /XFQbWE09S30/;86"(#"(*4/5)/5)(>O0/CZ0/5*+TB?DD6"()&<B;"(*+! !) :"( "()/;0/; 31 0/5 ! !! ,4)/5 ,X0
J(>r/9"$)-8P/%$5/%"( 0&=0/;! .)/5^18P/%$5/%W! .)/5#"$5 ))/5 ,X
UcdFQ <aRS3/C)/%$5))/5 \>_/%W)4"! ,X /; 0/5 ! !! ,e)/5 ,X
JSGF /XF
0/86"(#"(*4/5)/5)J(>VDRlE! 0/5$)/5*4u"()//%w.6"( GFIbW0/a"(*+8 !!0&/*4E&3 :"(
(>I0/C :"($%/M8P)/5 .:"( O!*+8 /5*4/5 .)/%&s"!a"7& 0/M>@a0/MUWX)/MB?DD6"()&*4&/5 O! q)/%$5 "3F F#"3F7bW0/|!*4/21&/58P/5 0&/5 .C86"(#"(*4/5)/5)M(>Q0/fBC"(*+! !) :"( $ F A1A &a"()/
J(t) = J0 exp(−F sin(ωmodt))
Ucd(t) = Ucd (1 + Fsin(ωmodt))1/4 S
a!0/+$% 0#"( E|86"(#"(*4/5)/5)J0"( 0&
UcdSP0/*4&3 :"( y>@)/%w.0/5 0$5N
ωmodS
"( 0&0/e*4&3 :"( "(*+8 !!0&/FF-bW0/4#"(
Ucd/J0
$% .) l0/e8.N3$%v(>MN3)/5*YSPa$#*4/9"( 0M6"(C>@C :"(,L/l8PX!!KL/#"(X
Ucd/J0
0/+N3)/5*! =0/C) ,X !N4)/58 !KL/?)/5,X!*4/XFzEN3)/5*4a!=$%*+*4/5 0#"()/;m0 ! !! ,l>G"$5))9SGF /XF70/v* (>u0/"()*4;(>uDPs)8R/%$5/%;/%w.6"( ;0/v .*vDR/5M(>u :"($%/l!)/%88)/%#)/%e0/s#"( 08P=(>|86"($5 /%<! n6"(<$9")/s"( 0&>@* "( n! 0 :"(! ,86")/XF $? -0/M0/5a6"( 0&hS>@7"&*+! 6"( EE! 0/5$C)/5*
J0 > Ucd0/|"()*4
$9"( ]*49KL/-"( !*4Xv> )/%/5 !NLFs ]$%*+86"() ])0/gDRX# 09SI/5KL/5 >_l"( ]! 3m0 !)/) ,e 0/5 ! !! ,e)/5* 0/C*4D! !!oN-a)/%$5)/%&Y&30/M)40/ 4r"( !A8! 0$5!8 /XF
o -)&/5a)8R/5^>_* "v!*4/M/5KL ! s"( -! !:"( T#"()/;aD#"(! 0/%&-D.N=$%*18! ,v0/W,X) 0&#"()/7$%)/%8P 0&3! ,f)C0/#"(
Ucd/J0
FIbW0/a8)86"(,."( g0/5 g8R/5^>_*4/%&+0! ,|0/7B;"(*+! !) =8R/5#"() $ F A1A &Ia!40/a!*4/21&/58P/5 0&/5 .86"(#"(*4/5)/5)a,X!KL/5 Y"(DP9KL/XF ;! ,!*4/C/5KL ! -0/;/5 0/5,XNg#"( 0^>_/5! E)0/N3)/5*M/5K"( !6"()/%&hF C&3)$50))/%&q! )/%$5 F#"3F 0/l .*vDR/5MKX"(:"( 0$%/
·
E
$E
E
E
E[J
0]
E
$E
E
E
E[J
0]
E $E Eω [J0]
E
$E
E
t[J−
10
]
E $E Eω [J0]
E
$E
E
t[J−
10
]
+$\ !--+|/0$ / \D|!-.M/ !|'=/0.- +1 HY$16$ +64@!!@165!-$$
z1^ H>F16$ 16+!-CD!-$3!-. 6/Q!=+7 @7D ! \D!$O! ! ^!-
Ucd/J0 = 5-!+7 @7D !
Ucd/J0 = 10 $H@1Z!F7 ! $+.-\ !^+/0.
/0+1Z!-+?/z!-=+7 =]74!+?/0FJ,.+1.\ωmod
PX7D$+!|\ H4!^ +?!=49$ S+7 !V!=6% )(.^$7W.+$\ !^+/0. (.=!- B4 4@!-+ 3 F7 4+7D.+!F'! -$++16 /0-$J,.+>16\
ωmod = Ucd
σ&/%? 0C)/5KL/9"( I*f0$#! 3>@*e"( y>_?>@/5*+ $+N3)/5*4;&30/l)g0/ 4I"( !
8! 0$5!8 / $@&3)$50))/%&Y! Y)/%$5 F#"3F#"&2FHJ/M6"%KL/f$%*+8)/%&Y0/M!*4/|/5KL ! q(>Q"4N))/5* (>
6 :"($%/v!)/%aa!
6>@/5*+ 09S
3"()*4l(>/9"$)]8R/%$5/%iF=bW0/+m0,X)/% A F#"s"( 0& A F % ! ! !0)#"()/40/
/5 0/5,XNv#"( 0^>_/5d&3! ,|0/W!*4/7/5KL ! g(>V0/%#/7"(#"( ,L/5*4/5 .)W"(Q0/a#"(XUcd/J0 = 5, 10, 20,
"( 0&50F
bW0/->_/5*+ 0<09 "J!*+! :"(gDR/56"9KE-"g0/DPX) 04! n0/Y8)/5K0$)6"(8)/59FbW0/y8$5)/(>v86"($5 /21G0 /J/2[$5!#"( 0-8)/%&3$5)")/%) 6"( 0$%/J"(0/|*4&3 :"( s>@)/%wE0/5 0$5N
ωmod = UcdS6a$#y70/l/5 0/5,XNY 0/%/%&/%&s)e8 :"$%/
0/l"()* (>/9"$#J8R/%$5/%; y0/v "(*4/l)!)/XFMzE!*+! :"(M)g0/lDRX# $vN3)/5*49S0/+*f !\1G8P/9"(J0$5)/e(>a0/e)/%) 6"( 0$%/%f)/%/5*4f 0f)-DP/4/2[38 :"(! 6"(D /4DEN
@
·
E
$E
E
E
E[J
0]
E
$E
E
E
E[J
0]
E $E E E ,Eω [J0]
E
$E
E
t[J−
10
]
E E ,E E Eω [J0]
E
$E
E
t[J−
10
]
- D +.-\ !^+>|/0$ / \]!$#/ !|9=/0$=+@1 H>F16$ +64!!-1.!F.
z1^WH>F16$1.+,!-CD!F3!. 6/Q! +7W@7D !- \D!.O! ! -!
Ucd/J0 = 20 -!=+7W7 !
Ucd/J0 = 50 $H16!F7W@ ! .+.-\ !^+/0.=
/0+>1Z!+ /9!==+7? D7D4!+ /0-$J,$+16\ωmod
PL7D.+!|\ H4!- +?!=49$ - S+7 ! ! .%&)(.^$7 $+.-\W!^+/0. (. ! B4 4@!+ 3 -$7 4+7D$+!-$'!- F++>16 /0FJ.+1.\
ωmod = Ucd
86"($5 /21G0 /v/2[3$5!#"( 09FbW0/aZ6/5*+0,."Q!*f :"()/%&4a!+0/W#"(X
Ucd/J$ m0,X)/ A F % &r09u"( )CKL/5N
/9"(4)/%) 6"( 0$%/%d>@0/7*4&3 :"( e>@)/%wE0/5 0$5Nωmod = Ucd/2
SEa$)=! 0&3$9"()/%0/f"(D0#D (>roe80) 07(>I0/|$%)/%8P 0&3! ,e/5 0/5,XNLF
1
) $CT '
*,vZWF M"( \>@iKLSRzE9F +;,X! GS .uF 4rF4u!#"/5K!GSRzRFRzE! ,."( 1 ' 6L!A L^ J3L(/M)9 =#;/78<7=# :LI#DC*7#;/MI8BL#K=# ;E !!;6C(/M!/1 : /5KPFTpqE&hF4u.N39F%$ !11&% 61"3F
* ,vpFdBvF 0&/5)) TS 0F : FQjQ 00/59Sap]F : FQpy"(0/59S8 - 1 + 4!/|7gI8BL#L^3J3L(/M7)9 =#;/78<7=# :L#DC*#;/I8BL# $# O= &8< 1 8SL"; !;L1Pz$5/5 0$%/$ !1 A & !1B3F
*#" ,(;+F0UCFM"%K9SVpF $lF6ps/5/%9SVp]F : F 0&3)/5iSU8- - 1.J L(/|7)B9-$#;/8<7=# :L#)C*7#;/MI8BL# =#2 (/OL^3>?L|C 1 8SLI /14QEN9F : /5KRF .T/59F $ !1 A & "11613F
* % , F."(3)$)TS 7;FU0&/59SY6FE2F 7!#"$XS8] -1 :LI CHJ3L(/LI##"31 8SL/ $# + !.8B#"6I I88B",!/1 4u.N39F : /5KPF .T/59F $ !11B& "! 1B3F
* A ,vpF+;)/5! 0/59S $lFVpy"( 0&/5 GSPbMFVjd) !! ,L/59S]8 - 1 I#²8 !²' (/M8BE#;/7=8BL#!ELH/5!;7*&C 8SL 0 LI88 =#;/& @8SL-$# 3%@/LZ& 8BW",L CI8SLI /1 ;"()/ $ 1 & "13F
*#6 ,(;+FdU#"( 31ps %! ,L/59S 0F F ; ! ,X6"(*YS ;+FU 0/5 1 9 "=8S@8BBL#/2L^J L/M7)9 $#/78<7=# "6L#DC*#;/I8<7/ $# L !?8B"6 @88B#",7/21 4u.N39F : /5KRF $ 1 % & A "16!"3F
* 5,vbW! fzE(>@/5 /XSEB?/5 ! ,vps!LS7:"( gz$)0GS 8 /1(WI#;/!B8BBLI# 7WLI >D8BEL#% "A !#²8<7E"M8B=#% > !; CK8SLG+ L@88 !#;/& @8SL1 4QEN9F : /5KRF.T/59F $ 1 % & !"1 % 1"3F
*#B ,(?/5! hHtF 0$5)(> 1 s!/78=L7 !;7 !²'*A(/!$ 1$? &/5 .DP, $ 1 5&2F
*# ,vZWF +C/5D6"()& 1 ' LI88 8S ") !#;/ ³I8SL&;E#;/7=8BL# 1szE8! ,L/5]bO#"$5)$ !192&2F
*! , : F : TS1;+F(U 0/5 1;> 5!;7*&C=8BA#DC !#D8<7L|767# ", _@88<7!# L^ 8BE"6L ³CJ L/L#;/3$# + !?8B"6 I88B",!/1 4QENiF : /5KRF +$ 1 1"& 1"!61 $ : &1E/2184Q! E9 "!$#&%('*),+*- - - TF
*1, +F +FhUa"() GS : FPbMFOz$9"( /5#"( 1.L7 CI)< "$#? ;#²8 & L#D8< :7 ³L ")%@L7=8='fML L#.7)aC= 7#;/7BL#? J3L(/M2L|C* 1 4QEN9F : /5KRFU $ !1 &1 A XF
1@¶
* , 6F;+F0Z)/%/5$#iSVBlF0pq /5 1 >D8BEL#%)"6L5!D "$#%2 ! #/!L#VLI8=' ! &EI#DCCI/MLWC7E6C J L/M7)4|4EEC L|C* 1 4QENiF : /5KPFU $ !116& 61XF
*!" , $lF 4A/5 )X)/XS .uF $? 0 "(,L/5 13J L/M7)9 $#/78<7=# :LI#DC*7#;/MI8BL# #?C &BC 7) "+1 4u.N39F : /5KPF - $ ! A 6& A 63F
* % ,vp]F+;)/5! 0/59S 7;F F : /5,."( GS lFhzRF X! 1 WL47$#&% 8=' "B8SI8BBLI#G/ !"M8BL^+ ! %@(/ =# 8=';J s>²)BJ 9 "WL/,/L(7 6%= 1 4QEN9F : /5KPF .T/59F $ 1 A & 9 % 1"3F
* A ,vp]F HtFVa/5 /5! TS 0F : F DP(1 zE6"/%/59S Fuz$)!)9/5PSOM8 - 1 ;L8B#",7/ I#DC/ !7 &BCB8BA$# /8BWLI#% "AK=#²8<7E"M8B=#% s7!3%*(/1*;"()/ $ 1 A & 1" A B3F
*!6 ,vj7FQbW/%! ,."3SUCFB0F u/56"X"(9SBlFQbMF 7;FdzE)E(>1 '*E7/,'&L ³C I#DC 6!/LI#D# ",!D' 7#DL 7#?$# & 8BW",L C %EL#?C)S/8SI8< ",L $ "$/!L#;/1 4u.N39F : /5KRF +$ !11"&% 1 % F
* 5,vps$)6"/5 ; GShB?/5 ! ,-pq!LSPbW! zE(>@/5 /XS]8 I /1 77BL#²#"@8SL/=# L !?8B"6 ³I88B#"! +47/M7|=#% s7!=)S/n",!/ CA#DI"7/I#DC =#²8<77)"M8BBLI#;/1 4Q.N39F : /5KRF.T/59F $ 1 A & 1B1 % 1"3F
*!B ,(;|"()/5 U#"( 31p %! ,L/5 1 OA#D"7/ L^:J L/M7)9 $#/78<7=# "6L#?C*7#;/MI8<!/-$#L !D)8B"6 I88B",!/1 4QTF F0/% 1 ? !KL/5)!N-(> $7[E>_)& $ 1 A &2F
*! , 6FOzE)/5 ,L/59SrzRFP 0NL/XS F4rF,7!EX"(9S 83 /1sJW%|% >2!;"M8BEL(/"6L !DA L^J3L(/|7)B9 =#;/8<7$# :L#?C*7#;/MI8<1 4u.N39F : /5KPF .T/59F $ !11& % A 613F
* ,vp]F; GSLBlFipq!LSXbMFLzE(>@/5 /XS²8s /1&> 5!;7*&C8SL0 LI88.=#;/& @8SL 8BE#)/7=8BL# =# LI#?_8& L -#?C8='E6 C=#;/!L#;/1 . 6"( I(> .TibO/5*+8P/5#"()/4u.N3$% $ 1 A & 61" A F
* , +M)& JUW"9NE*YS7;F 0F4r/5$) 1 -EL&#DC)S>?8SI8< -WL !778BS!/ L^ |%#.8B#", $ "AE !!;ECJ3L(/|7) :L#?C*7#;/|EC ;47C (/1 4Q.N39F : /5KRF.T/59F* $ !116& 63F
* 1 ,vbW!* M"9KE 1 .1 # &#;/!A 8B!") ! I88<77# 8B !EL#²8S 8=' L|C 1 7pxbO#"( 0#"$5 0 -py"(0/5*e"($9"( rz(> o"()/ - $ 1 % & !163F
* " , : . "!#%$#&"!'( &%$*)"!"+,*"-.+#"+$-+#./.-!0-1#"!02$+#0F* % ,fHn! ! !:"(*BlF 4Q)/%#9SIz3"( FAbO/5L NLSrH`! ! !:"(*bMF u/5)/5 !! ,0SQU:"( 4rF
Zr :"( 0/5N 1F&!"6 "M !;7/ =# 17W"(*fD&3,L/? !KL/5)!N 4u)/%) 1)/%$% 0&/%&3 TF $ !1 A &2F
* 1A ,vj7F7Z0/5 !DP/5, 1 L]:)WLEC*7%"M (/!/!"6 &#%*7) &88S ML! & / :B8=' /78< !²/7/.g",L#²8BEL #DC 8=' 7= !!D "#"6@8BBL#8SL/ML ' E@88BWI#;/L|7 !DEL*47 57/21 z bA/%$) $9"( : /58P $ !161& B3F
* 6 , 6F : F7W"TS F BlF;|"(8 1 7p bO#"( 0#"$5 0M ypq"(0/5*e"($9"( z(>@W"()/* $ !11 & XF
1 6
r $< $
v/5))/5IzE)/5 ! /d* E$#.)/$) 4Q)(>^F ;9F : DR/5 : M>A&3/p ,X !$#L/5!A% 3>_/5!,X ,q&3/%#/5 DR/5!&0"( L/5 TSQa/<"(0$)]>v)/5! c"( 6"( !)/5 0&/%+ E)/5)/%))/ 0&g)/5! 0/;*4!K/5)/5 0&/ ()/5 0&<*4/5! 0/5W,L/%#"(*v)/5 <D0/5!,L/5 R/5!W! =&/5+;88R/XF
M"( J,L/5D+KL4"( ! /5**4/5! 0/5* ;?*+*+! !!) 0/5 Z0/5 !\['z$)*+!9Sd&/5!*1*4/5f>|>G"$) !$#0/< 0&]"(0)#/5^1_>_"$# !$)0/2+C/%8$#0/</5! ](h/5 0/% $?v6"(9SI 0&/%)/5 E !$)q%* u/5)( 0&3 7 0#/5)/% DP/5!),L/5D/5)/%7DR/5!,L/5 E#"(,L/5 6"(9F
C/5Q,L/%#"(*f)/5 ebW0/%/21 D)/5! ! ,v&/%d 0!)Q> ;M/5 8.N3!* $).)/$)>W>@)/5 0&3 !$)0/v 0&"( ,L/5 0/5*4/ *4X8 ()/f&0"( L/5 TF M"( =,L/5E7/5DP/%) 31&/5)l"( ]&3/ 36&#DC*N M"%K& $? 0&3)/5X"3SQbW0*e" : TSIpq"(:"vH"(,X 0/5 0& u/5)/5 6"+HJ/5TF
7 /5*4/5 05;C08P/5 &0"( L/4$)&0" >9Sr&0")|/5f$#]"( (7$#.^184QEN!L/5&3/%)/ DP/5!a% ;7)/5,L/5 /%)/5 6"(9S 0&"( Z)/5 0&!*+*4/5KL/5^>,XD6"(9F
o$#e&0"( L/7bW! X"fzE8P/5u 0&+*4/5! 0/5p!DP/5 0/5! X !:"MBC"( e&0" >9S.&0"/C!*+*4/5?"( Z)/5 0&/v&0"! 0&hF
M"( =,L/5EC"( q&3/fps!,X !/%&/5?&/5 ;M/5!sU 0/5;bW0/%N-+C)8%$ $7[>@)&&)9a/4;9F ;|"()/5 U#"( 31ps %! ,L/57>a!)/C>G"$) !$#0/ ? .)/5)% ,0F
UW/%) 0&/5)/5 M"( ?,L/5Er"( M*4/5! 0/ups)/5.h/%#/5 )/dB?! &C 0&M*4/5! 0/5 fU0&/5p$#6"/5 GS9>T&3/d"( 6"( !)/5 0&/XS(DR/%&3! ,X ,L) X)/Q 0&M 0/5*0&3 !$)0/ 7 E)/5)% ,! .^/5,X !$#0/5?B?! 0$)E ()/5 0&*4/5! 0/%;zE0&3!*49F
1D
1
!#"$ %!&('!)!*,+-.0/2135476+98
B7/5*+!sKL/5)$)0/5)/$)TS|&3/ ;!8 *e"(DR/5!y 0/B?! \>@/ ?!)/5q .s*+!s&/5 "( ,L/5,L/5DR/5 0/5 0/5 ! /5 c 0&cB?! \>_*+!)/5 ! "( ,L/2>@/5!,X+%c6"(DP/5 TF ! /YzE)/5 ! /5 TS&3/W"(0Q&3/%#/5 0/5 ! /5 4/5 E 0*+*4/5 ea)&/5 TSL)! 0&"( Q# $)0/WL/5 . !$#4,L/5*e"$)E)&/5 TF?/%#/ DR/5!?6"(?! q,X /5$)0/5M&/5( !$#0/5MZ0* 0E$#sL/5! 0/5 4Q31> ,LDP/5")&/MKL,L/5 /5,L/5 TF
C"(*4)#"&39S0&/5 !B3FBX ! 1 A
pq"(0B?! &
![[XLS] · Web view86453 28 35 7 0 0 87324 4 5 1 0 0 88881 4 5 1 0 0 87326 5 5 0 0 0 87327 3 5 2 0 0 88895 4 5 1 0 0 87329 4 5 1 0 0 87332 1 5 4 0 0 87083 14 30 16 0 0 89274 2 5 3 0](https://static.fdocuments.in/doc/165x107/5af0fdb97f8b9ac2468eca92/xls-view86453-28-35-7-0-0-87324-4-5-1-0-0-88881-4-5-1-0-0-87326-5-5-0-0-0-87327.jpg)
