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46DE22@=>=>$C+2@7$]032KE203J3KEK<6 -3K<(+CK<6)DED<2 'C$'=>657",<6*$t,?65=1-%6)K<$',?J3K<$H032J37 273DE265=>6032 Q%(+DE(D%732203657,?2&527(+7127,<6)K?$'+6)7,?2 BCDE2 -3J" 27,?J2K<6&/A3676)**@2=2@,<6032,<6)= - 65K?$',<J3K?$ 7"J3**$BC0$I,?(&/A36F*@6 -$CK ,?2@&)6)*@*6 - (CD<D<(C73( (&5&)J-3$CK<6 *@(1DE,<6)DED<(DE,?$',?(3B*$x&5(+7CJ3K?$C852(+76H&/A36F=2732@=>2@8(%8Q$*7" 6)76)K<C2$ "J365**$&/A6 'C6)036,?J,E,?6g*6 -$CK ,?2@&)6)*@*6 (&)&5J-$CKE6:*@(DE,/$I,?( (+730$C=6)7,?$C*6'LN2 " &/A36 *7" 27,?J32@8)2@(+736n73(+7 -3J" 032KE6 uD<6u,/$C*@6u&5(+= - (+K ,/$C=6)7,<( DE2$ - 65&)J3*@2$CKE6u06)**@(8)65K<(x$CD<DE(+*J,<( 03( '+6 6)7365K<+2$*@2?Q%6)K?$F650 6)76)K<C2$g27,?65K<7$g&)(+2@73&)2@03(+73( (.--3J3KE6 DE6 65D<2D ,/$J37$x,<K?$C7D<2852(+76 $x,?65=1-%6)K<$',?JK?$72@,/$#@273D ,?6)2@7n6 GN(+DE6F736)C*2 $C73732 U6)7,?2|06)* DE6)&5(+*(D<&5(+K<DE(1A$C773(x=(+DE,<K?$',<(g&/A36CB - 65KJ37 $CDU032;Q (+DE(+732*2EQ 65K<237(+7 2@7",<6)K<$C+6)7,<2 B"736)**@2=2@,?6,?65K<=(027$C=2&5(Z&)2@(

N → ∞ BV → ∞ B =>$

N/V → costjB 6)DE2DE,<6xJ37 ''$C*@(+K<6 732 %

,?(>036)*@*$ ,?65=1-%6)K<$',?J3K<$1-%6)K &)J322*H-%(C,?65738)2$C*6g&/A32@=>2@&)(D<2$C737"J3**$NDE61*$ ,?65=1-%6)K<$',?JK?$D<&56)7306FD<(C,E,?("J36)D ,?(''$C*(CK<6 &)(+73028)2@(+736 736)&56)D<D<$CK<2$- 65K<&/AF2@* 7"J3=6)KE(1032-$CKE,<2&56)**@6 DE2&)(C73D<65K0'2 F&/A36g*( DE,?$',?( (+70$C=6)7,/$C*@6 -%(+D<DE26)03$J37b7"J3=6)KE(02|(&5&)J-3$C8)2@(+736g= $'&)K<( %D<&5(-32@&)( Z&52( :06)** " (CK<032@736:06)*7"J3=6)KE( ,?(C,?$C*6032 -$CKE,<2&56)**@6kB%=6)7,<K<61*$ DE,?$',?2@DE,?2@&Q$032GN(+DE6$CD<DE6)+7$$C+*@2d$C*-,?K<2 DE,?$',?2dJ37u7"J3=6)K<(032d(&)&5J-$C852(+76F032d(+K<02736 #" 65DE,<6)73DE2(+736 $'*H&Q$'D<(p032 -3$CKE,<2&)65**@6 &5(+77$I,?6 Z&52( ;2@7 -3K<65D<65738Q$ 032HJ7 - (C,<6)73852$C*@66)D ,?65K<73( 6)0>27,?65K?$CC6)7,?20365*K<2DEJ3*@,?$',?(F02G(+D<6t6)0 @273D ,?6)2@7>7(+7 27+65736)K<$C*6tDE6)= -3*2 %&)6' 7F" 6'2@036)738)$ 036)C*2d6A 65,E,?20365**$&)(C73036)7D?$C852(+736:27uJ37;DE2D ,?6)=>$>2@7,?6)K<$C+657",<6:DE(+73(*6 -3K<(-K<26j,SR10365** " 65*2(*2E"J3203( ! B= $12@* DE2DE,<6)=>$127n&5J32F"J36)D ,?6 -3KE(-3K<2@65,R16)=6)K<C(+73(27=>$C732@6)K?$ &/A2$CK<$ tJ37$CDU032$',<(+=23032*@J32@,<( 6t&)(+77$',<(:2@7J71-%(C,?65738)2$C*6t$'K<=(+732&5( K /$C*2]&5(+73032@8)2(C732D<(C73( D0- 65K<2=6)7,?$C*=6)7,?6nK<$C++2@J373+2EQ32*@2 &5 K5&Q$D-| 127 J37D<2@DE,<6)=>$;$C0 J3*@,<K?$u$C*@,<( 'J3(C,<(;6>=>65032$C7,<6J7X&)(+77$C=6)7,?( 032U,?2?-%(u= $CC7365,<2&)( *-%(C,?65738)2$C*6 02d&)(+737$C=6)7,?(3BD&'2*J --$',<( 27,?(+KE73($C*|=2732@=>(B$CDED<J3=6g*$x (+KE= $

V =1

2m (ω2

xx2 + ω2

yy2 + ω2

zz2)

$n&)J32&5(+K<KE2D0- (+7036 J373(bD*-%65,E,?KE(u02D<2@73+(+*$ -$CKE,<2&56)**$>&)$CK?$',E,?65K<2858Q$',<(b0$ ,?K<6 7"J3=6)KE2"J$C7,?2@&)2

ε(nx, ny, nz) = h((nx + 1/2)ωx + (ny + 1/2)ωy + (nz + 1/2)ωz)

SUT QWIT3YOQP"P YM RP TO)M|VTY PZO PZ^+T

LN(+=6,'+6)0K<6)=(F$C* -$'K?$C+K<$' ( N]*$],?K<$.--%(+*$tJ,?2@*2@8)8Q$I,/$7365+*2"6)D0- 65K<2@=>657",<2"K<2?-%(+K ,/$',<22@7uDE6)+J2@,?(-K<6)DE6)7,/$J37u&)(+77$C=6)7,?($CKE=>(C732&5(127nJ373$C6)(+=65,<K<2$1&52*@27303KE2&)$:27n&5J32"J32@73032UD<2#-3J";27032E'20J$CK<6xJ37X$CDED<6032D<2@=>=65,<K<2$ Z27032&)$',?(n&5(+7 8 j J6)DE,<(;73(C7 &/A36J37u&Q$CDE(1-3$CKE,<2&)(C*$CKE6H0365*-3K<65&)6)06)7,?6:27b&)J32

ωx = ωy = ω⊥

G 65**7" $.--K<(+DED<2=>$C852(+736gDE6)=2&)*$CD<DE2&)$>032 - (C,<6)KH,<K?$CDE&)J3K<$CK<6x*$ DE6=-$CK<$C8)2@(+736x,<K?$ 2*2 %

'C6)**@2F6 ,?K?$I,<,/$'K<6 *$p036573D<2-,SR 036)C*2 D ,/$',<2F&)(+=6 J37 &5(+7,?2@7J( Z$.--3KE(+D<DE2=>$C8)2@(+736 &/A36032 '26)76n6)D?$I,<,/$ 7365*]*2@=>2-,?6

N → ∞ 6nDE6kBT hωi

Bi = x, y, z

:D<2](',<,?2@6)736 &/A362@*''$C*@(+K<6036)*@*$,?6)= -%6)K?$I,?J3K<$ $ &)J22*H-%(C,?65738)2$C*6&/A32=2&5( K<$C++2@J373+6:2*''$C*(+KE6x*@2=2@,?6

µc = 3/2h(ωx ωy ωz)1/3 g0$',<(0$C**$1&)(+7032852(+736 K

N = ζ(3)(kBTc)

3

h3 (ωx ωy ωz)

03( 'C6

ζ *$x J373852(+736F8565,?$032HPt265= $'737

ζ(3) = 0.94j

$'**$-K<6)&56)03657,?6xDE2d6'27&)6g&/A36

kBTc = 0.94 h (ωx ωy ωz)1/3N1/3 65

$'**$ 6="J$C852(+76 5 -%(+D<DE2?Q32@*6UK<2@&Q$ ''$CK<6J37$H036 732@8)2(C736N&5(+K<KE65,<,?$H032*2=2@,<6,?65K<=(D%032@7$C=2&5( -%6)K "J36)D ,?2 DE2D ,?6)=2%$x&)J2 *$036 732@8)2(C736 03$',/$27 -K<6)&56)0365738Q$-%6)K2*$CD73(+72@7",<6)K<$C+6)7,<6:73(C7 DE2$.--3*@2&Q$B30$',?$*$ *(+KE(2@73(+=(++65736)2-,SR*d*@2=2@,<6 ,?65K<=(027$C=2&5( 036 732-,?(0$C**@6 &)(C73032852(+72

N → ∞ Bωi → 0

= $N(ωxωyωz) → cost

27b=(03(&/A36*$,<6)= - 65K?$',<J3K?$ &5K<2@,<2&)$ D<2$ Q 657 036 72@,/$ 76)**@2=2@,?6F,<6)K<=(0327$'=>2@&)(3 7 "J365DE,<( *@2=2@,?6 - (+DED<2EQ32*@6:KE2&Q$ ''$CKE6:* " $C70$C=6)7,?(J3732 '+6)KED?$C*@6x0365**$ K<$C8)2@(+736&)(+7306)73D<$',/$n$C*032D<(C,E,?(0365**$x,?6)= -%6)K?$I,?J3K<$&5K<2-,?2&)$ K

Nc

N= 1 −

(T

Tc

)3 ON J365DE,?(>$C7303$C=>657,?(1,?2E-32&5(10365**6 ,?K?$'73D<2@8)2(C732 02 Z$CD<6 -J"6)D<DE6)KE6:D0-3265$',?( CJ$CK<03$C7303(2@*&5(+= - (+K ,/$C=6)7,<( "J$'7",<2D ,?2&5(H0365+*2$',<(+=2 73$ DE,<2=>$ "J$'7",<2@,?$',?2 'C$t0365**7" 6)D ,?6573D<2@(+7360365* -3$C&)&/A36j,<,<( 0F" (+730$ $',?(+=2&5( >0$I,/$b0$C**$n*J373CA36)858Q$ 0 " (+70$;$CDED<(&)2$',/$u$C*U=>(',?(,<6)K<=2&5(n036)*@*61DE273C(+*6 -$CKE,<2&56)**@6CB 0365,E,/$ I%uj<;1t?C 6036 732-,/$0$

λT =

√√√√ 2πh2

mkBT

!

SUT ^ %YRMYO)T"P YM RP %O)M P YO MPZY

65K>2 -$CK?$'=>6j,?K<2,?2?-2&)2N036)+*@26)D0- 65K<2@=>657",<2](+++6j,<,<( 02 J6)DE,<( *$ 'C(+K<(B$'**$ DE&Q$C*$ 032,?65=1-%6)K<$',?JK<6H06)* &56)7,?2@7$C2@(x032%7$C73( % F6)* '2@7|B"D<2 A$

λT ' 30073=uB"=>657",<K<6 *$02DE,?$C738)$

=6)032$b,<K?$ *@61-3$CKE,<2&)65**@6CB&/A36 -3J" 6)DED<6)KE6n&)(+7D<206)K?$I,/$ 06)** " (CK<032@736032n−1/3 Z0('C6

n27032&)$1*$1036573D<2-,SR=>65032$10365*d&Q$C= -32@(+736kB 'B736)*d73(CDE,?KE(>&)$CD<(B327 65K<2@(+K<6F$

25073=uG 6

&)(C73D<65+J36F&/A36F$ J6)DE,<6 ,?65=1-%6)K<$',?J3KE6H032 '26)76H2= -%(+D<DE2?Q32@*6,?K<$CD<&5J3K?$'K<6H*$:&5(+K<KE6)*$'8)2(C736,?K<$2H-$'&)&/A36j,<,?20F" (+730$>$',?(C=>2@&)2 B&/A36:,<6)730(+73( $D<( 'K?$D-- (CK<D<2N*$ ,<K?$C7D<2852(+7602Z$'D<6D<2 6)D0-3*2@&Q$ -3KE(-3KE2(17365* Z$',E,?(1&/A36F*$:*@J373+A3658)8)$1032 &5(65K<6)78Q$ 036)* D<2@DE,?65= $ -$CD<D<$B$',E,?K<$ %'+65K<D<$C7303(u2* ''$C*@(+K<6x032,?65=1-%6)K<$',?JK?$ &)KE2@,<2&)(B|0$C2&52KE&Q$

30073= 03652D<2@73+(+*@2 -$'&)&/A36j,<,?2

0F" (+730$$',<(+=>2@&)2 $C**@6 0365&)27602d=>2@&)K<(C7n(' 'C6)K<( $C**$x032@=>6573D<2@(+736 ,?(',/$C*@6 036)*dDE2D ,?6)=>$ 76)&)65D<D<$CK<2@(u0J37"J361,?K<$',<,?$CK<6x2*-3K<(Q*6)=>$>2@7 =(03(1-32657$C=6)7,?6 J3$C7,?2D ,?2@&)(n$

-$CK ,?2@K<60$C** " A3$C=>2@*@,<(+732$'7$>&5(+=1-*65,?$ 0365*NDE2DE,<6)=>$ &/A36'B27 D<6)&5(+730$ "J$'7",<2858Q$C852(+736'B$CDED<J3=6g*$x (+KE= $ K

H =∫

dr ψ†(r)

[− h

2∇2

2m+ Vext(r)

]ψ(r) +

+1

2

∫dr dr′ ψ†(r)ψ†(r′)V2(r − r′) ψ(r′)ψ(r)

OK

03( '+6ψ†

ψ273032@&Q$ *7" (-%6)K?$I,?(+KE6 Q%(+D<(C732&5( 032 &)KE6Q$C852(+736Z02DE,<K<J3852(+76jB

Vext

2* -%(D%,?65738)2$C*6 65DE,<6)K<7( 6

V22@*B- (',?6)78)2$'*6 0322@7,?6)K<$C8)2@(+736$b03J36&)(CK*-32 LN(+=>6'C6)03KE6)=( 2@*

-$CK<$C=65,?KE( 0F" (+K<02736:036)*@*$>,<K?$C73DE2852(+7360365D<&)KE2@,E,/$ D<(.-3K?$ -K<(-3KE2( 2@*&)$C= - ( &5*$CDED<2@&)(Z&5(+= -3*65D<D<( 〈ψ〉 ≡ Φ

s obehjr:`bs mnh r:aa y hjcegs a h5h

* -$CK<$C=65,?KE(:0 " (+KE03276 2@036)7,<2 &)$',?(gD<(.-3K?$ '26)76 &5(+=1J37365=>657,?6 0365,E,?(k I H I j/I%j)C 6'2@6)736(C,E,?657J,?(u0$C**$ OK &)(+=6 (+K<027361856)KE( 0365** " $.- -3K<(+DED<2@= $ %8)2@(+736 032 &)$C= - (:=>65032(:6)DE,<6)D<$$'* &)$CD<(12@73(+=(++65736)( Z&)2@( 6)D ,?65K<7$C=6)7,<6F&)(+737$',?( 6032E- 6573036)7,<603$C*|,?65=1-%( N"B K 7X+65736)K<$C*6'B DE6)= -3K<6 *6)&52@,<(;6)D0-$C7306)K<6 *7" (- 65K?$',<(+K<6032 GN(+D<6>27,?(CK<73(;$C*DEJ3( ''$C*(CK

S M|VT"P YM RPnW %OQO(:P TMdOdPZP >

=6)032@( NB K

ψ(r, t) = Φ(r, t) + ψ ′(r, t) +

46 2@*t7J=>65K<(p032H$',?(+=2t736)*H&)(+7306)73D<$',?(3BNc

B ,/$C*@6;0$ -%(C,?65K &5(+73D<2@036)K<$CK<6Nc ∼

Nc ± 1 ∼ N(=6)+*@2(3BD<6 ,?$C*6F03$&5(+73D<2@036)K<$CK<6g*@6&5(+7CJ3K?$C852(+72|&)(+KEK<2@D*-%(+7303657",<2$

"J365DE,?2DE,?$',?2U&5(+=>66J2?-3KE(Q$.Q32@*2 Bd$C**@(+K?$ DE2,-3J" ,<K?$CDE&)J3K<$CK<6*$ J,<,?J3$C8)2@(+73627,?(CK<73($C*;''$C*(+KE6 =6)02(1650n(C,<,<6)7365K<6 J37 " 6J3$C8)2@(+736F&/A36 C('C6)KE7$ *$:027$C=2&)$06)*F-$'K?$C=65,<K<(0F" (+KE032736

Φ(r, t)B3$ -$I,<,?(032 0$CK<6F (+KE= $165D*-3*@2&52@,/$ $'*F- (',?6)78)2$'*6F032|2@7,?6)K<$C8)2@(+736

V2

"J365DE,<(1D<&5(- (xA$:65DE,?KE6)=>$:2@=1-%(+KE,?$C738)$g2@*&)(+D065,<,<(J-3$CK?$C=65,<K<(g032%032*@J32852(+76

&52( n a3

s

&/A36;=>2@D<J3K<$ 2@* K<$.--%(+KE,<( ,<K?$ *$ 032@DE,?$C738Q$ 6&Q$'&)6 0365**@6 2@7",<6)K<$C8)2@(+732 B0365* %* " (+KE0327606)**$ *J373CA36)858Q$032032EA J3D<2@(+736

as

jB036 732@,?$>$ -$CK ,?2@K<6g0$C*@*$1D<658)2@(+7360F" JKE,?(65*$CD ,?2&)$$',?(+=(D%$',?(+=( + B6g*$102DE,?$C738)$=6)02$1,?K<$*@6 -$CK ,?2&56)*@*6CB

n−1/3 7X,<J,<,<6 *6>D<2-,?J$C852(+72D0- 65K<2@=>657",?$C*22@7X&)J2 D ,/$',?$;K?$'++2J7",?$u*$ &5(+73036573D?$C852(+76 032GN(CD<6(%4@273D ,?6)2@7 76)2]$CD$',<(+=2&)2 ''$C*6n* " $.- -3K<(+DED<2@= $C852(+76 032]$CD1 (CKE,?65=>657,?6b032*@J32@,<(-%6)Kt&)J2

n a3s 1

M

7 "J365DE,?(nKE6)+2@=>6g*@6:2@7",<6)K<$C8)2@(+732D<(+7( -3K<6(''$C*657",<6)=6)7,?61$>03J36x&)(+K0-3265D<DE6)7303(b=(+*@,<(Q$'D<D?$b*$ -K<(Q$DQ32*@2@,SR>&/A36 ,?K<6( -32 -3$CKE,<2&)65**@6 -%(+D<D<$C73(b,?K<( ''$CKED<2$n032@DE,/$'738Q$ ,/$'*6 0$2@7",<6)K<$C+2KE6 ,<K?$x*@(+K<( J36)D ,?( D0-3265$ -%6)K<&/AF736)*@*$ OK U2@*-%(C,<6)73852$C*@6 032 27,?65K?$C852(+76# &5(+73D<2@036)K<$',?(>J37 -%(C,<6)73852$C*@6g$03J6x&)(CK*-32 J36)D ,?(3B7365**@2=2@,<6H032HQ$CDED?$,?6)= -%6)K?$I,?J3K<$B 036)DE&)K<2-,<,<(xJ3732@&Q$C=6)7,<6 0$'**$g*J73+A3658)8Q$032%032EA J3D<2@(+736t27>(+730$

s7 "J36)D ,?6F$D--3K<(CD&%

DE2=>$C8)2@(+732 B &)(+=6K<2&)$ 'C$I,?(b736)* M+!n27 J37 $C*@,<K<(b&)(+7,<6)DE,<(;0$ @73K<2@&)( #6)K<=2 M BD<2-3J"$.--3KE(+D<DE2=>$CK<6 * " 2@7,?6)K<$C8)2@(+736 &)(+7nJ7 -%(C,<6)73852$C*@6H032 &)(+7,/$I,<,?(-3D<65J303(-%(C,<6)73852$C*@6&/A36:$C*@*7" (+K<032@736 -2 Q$CDED<(A$*$x (+K<=>$

V2(r − r′ = x) = U0 δ(x) =4πash

2

mδ(x)

$F&)(+D ,/$C7,<6U0

t0365,<6)K<=273$',/$HKE2&/A32@6)03657303(g&/A36 *$ DE6)8)2@(+736]0F" J3K ,?(6)*$'DE,?2@&Q$ ,<(C,/$'*6&/A36DE2dK<2&)$ 'C$ 0$'**$ DE2$ *$1D ,?6)DED?$0$',/$0$C* - (C,<6)73852$C*@6 K<6)$C*6'

! ST ^ %YRMYO5T"P %YM RP %O)MFPZYO MP Y

LN2 " - (+D ,?(3B'0$',<( &/A36N2@7xK<$.--3KE6)D<657,/$C852(+736N032 6)2@D<6)7 Q%6)KE ''$C*6K<2@+(+K<(CD?$C=6)7,?6 K

ih∂

∂tψ(r, t) =

[ψ,H

]=

=

[− h

2 ∇2

2m+ Vext(r) +

∫dr′ ψ† V2(r − r′) ψ(r′, t)

]ψ(r, t)

. 4(+D ,?2-,?J3657303( $

V2

* " 65D*-3KE6)DED<2(C736 6)027,<K<(03J3&)657303( *7" $.--3KE(+D<DE2=>$C8)2@(+736 + $C*@*7" (+K<032@736H856)KE(

ψ = ΦD<2d(',<,?2@6)736

ih∂

∂tΦ(r, t) =

(− h

2 ∇2

2m+ Vext(r) + U0 |Φ(r, t)|2

)Φ(r, t)

&/A36 -3K<6573036g2* 73(+=6F032d6="J$C852(+7602$gKE(+D<DJ%2@,?$C6'D*)"2d032E- 657303657",<6g0$C*|,?65=1-%( ' N*@2=2@,<2%02 ''$C*2@032@,Rx032 ,?$C*6F6J3$C8)2@(+736gD<(+7( "J365**@2|036)*@*6F$D--3K<(CD<D<2@= $'8)2(C732 032-D<6)J03(D%-%(C,?65738)2$C*6Q$CDED?$ ,?6)= -%6)K?$I,?J3K<$B3$CDt032@*J32-,?( 6 +

Nc ∼ NBN 1

j $C*@*7" 6="J$C852(+736 DE2KE2&Q$ ''$ *$ (+KE= $t0365**( DE,?$',?( (+730$C=6)7,?$C*6$ ,<6)= - 65K?$',<J %

K?$:8)6)KE(Z03( '+6FDE2 A$N = Nc

j= -%(+736)703( 27Z$I,<,?2J37 -3KE( *@(x032 036)73DE2@,R1DE,/$'8)2(C7$CK<2@(

Φ(r, t) =√n(r) exp(iµt)

* " 6"J$C8)2@(+736 d032E'2@6)736J37$ 6="J$'8)2(C7360324&/A3K02032@73+65K73(+7u*2@736Q$'K<6

(− h

2 ∇2

2m+ Vext(r) + U0 n(r)

)√n(r) = µ

√n(r)

LN(+=6 D<2 -3J" 032@=>(CDE,?K<$CK<6 Z&j K) K :*$ D<(+*@J38)2@(+736 $'**7" 6="J$'8)2(C7361-K<6)&56)03657,?6b=>2@732 %=28)8)$ 2*| J373852(+73$C*6F65736)KE+2$$7"J3=6)KE( 032-3$CKE,<2&)65**@6D<D<$',?(>6:*$&5(+DE,?$C7,?6

µ$CDED<J3=6

2* D<2C732 &)$',?( 02- (C,<6)73852$C*@6 &/A32@=>2@&)( 0$',<(&/A36

µ =δ〈H〉δN

=1

N(Ek + Eho + 2Eint)

5 03( '+6

Ek

BEho

BEint

27302&Q$'73( K<2@D*-%65,E,?2E''$C=6)7,<62*3'C$'*(+K=>65032( 06)*&)(C7",<K<2EQ3J,?(F&52736j,?2@&)($CKE=>(+72&)( 6F02d27,?65K?$C852(+736g$'* ''$C*(+KN=>65032(0365**7" 6)76)K<C2$ J$C730(u,<K?$',E,?6)KE6)=(b27 0365,E,/$CC*2(n2@*B-3KE(Q3*65= $ 06)**$nDE,/$DQ32*@2@,SR>032J37 &)(+7306)73D<$',?( 27J37 K<65,<2&5(+*( / N jB(CD<D<65K0'C6)K<65=>(;&/A36 0$'**7" $C7$'*2DE20365** " 6J3$C8)2@(+736 032E- 6573036*$DE,?$.Q32@*2@,R 6)7365K<+6j,?2&)$ 0365* DE2D ,?6)=>$B=>657",<K<6 *$ D<(++*@2$ 0365**7" 27DE,/$DQ32*@2@,SR 0327$'=>2@&Q$ -3J"6)DED<65K<6(C,<,<6)7"J,/$$ -$CKE,<2KE6 0$C*@*$ j

S -M|VT"P %YM RP W %OQOI :PQT3M OdPZP <

"!$#&% '(

$]D<(+*@J38)2@(+736$'**7" 6="J$'8)2(C736 -3J"t6)DED<65K<6U,?K<( ''$',?$- 65KF'2$]7J=>65K<2&)$B =>$B 736)*+&)$CD<(0322@7",<6)K<$C8)2@(+732|KE6=-3J3*@D<2 '+6

as > 0jBJ373$&5*$CDED<6032D<(+*@J38)2@(+732d$C73$C*2-,?2&/A6F032-$CK ,?2&5(+*$'K<6

J,<2*@2@,SR76)**$ '+(+KE( D0- 65K<2@=>657",?$C*6'B "J36)*@*$]&/A36D<2(C,<,<26)7676)**2=2@,<62@7:&5J322@*&)(+7,<K<2?QJ,?(03227,?65K?$C852(+76x03(+=273$>D<J "J365**@( &)2@7365,<2&)(>( '3'C6)KE(u2@**@2=2@,<6CB036j,<,?(b$D--3K<(CD<D<2@= $'8)2(C736032/]A(+= $'D&%#65K<=2 B27b&)J32

Eint

Ek' Eint

Eho' N U0 n

N hω∼ U0 N a−3

ho

hω' N as

aho 1

N 03( 'C61&)(+7

aho

D<2 27302&Q$I,/$*$ &)(+DE203065,<,?$>*@J373+A3658)8)$>032(+DE&)2*@*$',<(+K<6g$'K<=(+732&5( -$CK<2$√h/mωi

7 "J36)D ,?6b&)(+7032852(+732D<2]A$ &/A36b* " 2@7",<6)K<$C8)2@(+736 0(+=>2@7$;DEJ3**$; (+K<8)$ 032K<2@&/A32$'=>( 06)*-%(C,?65738)2$C*6 $CK<=(+732@&)(x6F02|&)(+7D<6)CJ36)738)$>+*@2 $I,?(+=2 ,<6)7303(C73($C0n6)DED<6)KE6:D0-327,<2|*(+7,?$C73(0$'*&56)7,?KE(n036)*@*$ ,?K?$D-- (C*$* " 6(A 6j,<,<( &)(C=1-3*@6)DED<2E'C( "J365**@( 032K<6573036)KE6-32 J3732- (+K<=62@* -3KE( *@(;032036)73DE2@,R;6 032@=>2@7J2K<6 (+K ,?65=>657",<6 2*,<6)K<=276 &)2@7365,<2&5(;&/A36n032E- 6573036 03$C*+K<$C032@6)7,?6032036573D<2-,SRG 6)**2@=>2-,?6 N HDE2#-3J"u036)*,<J,<,<(u,<K?$CDE&)J3K<$CK<62@*&)(+7,<K<2?QJ,?(&52736j,?2&5(6* " 6"J$C8)2@(+736 ND<2dKE203J3&56g$C0uJ37F" 6="J$C852(+736$C*@+6=QK<2&)$10$ &)J32dDE2dK<2@&Q$ ''$

n(r) =

1U0

(µ− Vext(r)) se µ− Vext(r) ≥ 0

0 se µ− Vext(r) < 0

!

H2%&)(C73D<65+J36)78Q$127>$.--K<(+DED<2=>$C852(+736t032;/]A3(+=>$CDJ%#6)KE=>23*$g (CK<=>$0365*-K<( *@(:02%06)7 %DE2@,SR CB7365*d&Q$CDE( 032d&5(+773$C=>657,?( $'K<=(+732&5(3BJ7 -$CK<$.Q%(+*(+2@036 65**2-,<,<2&)(:,?K<(C73&Q$',<(&/A36CB7365*&)$CD<(F032J37 &)(C77$C=6)7,<(F$ D<2@=>=65,<K<2$]&52*2@7303KE2&Q$B -J"F6)DED<6)KE6]D<&5K<2@,E,?( 76)**$ (+KE= $

n(r) = n0

(1 −

(r⊥R⊥

)2

−(rz

Rz

)2) K

J365DE,/$ D<2032 A 65K<65738)2$ "J$C*@2@,?$',?2 'C$'=>657",<60$'**$02DE,<K<2EQ3J38)2@(+736&/A362*C&)(+7306)73D<$',?(H$ 'KE6=Q %Q%6H2@7 $CDED<6)78Q$1032%27,?65K?$C852(+72 27 "J365DE,<( &Q$CDE(x27Z$',<,<2%65D<D<$1D?$CKE6=QQ%6 "J365**$036)*@*(DE,?$',?( (+70$C=6)7,/$C*@6F036)*-%(C,<6)73852$C*@6 6)D ,?6)KE73( 6 "J32@73032d$'J3D<DE2$C7$

7 J6)DE,?$D ,?65D<D?$$.--K<(+DED<2=>$C852(+736# - (+DED<2EQ32*@6HK<2@&Q$ ''$CK<6HJ37$1 (+K<=>$xDE6)= -3*2@&)6 -%6)K2 -$CK<$C=65,?KE2&Q$'K?$',E,?6)KE2D ,?2&520365**$ K j 7 -$'KE,?2@&)(+*$CK<6'BUD<2&5&)(+=6b*$ 032@DE,?KE2?Q3J8)2(C736

8 ST ^ %YRMYO5T"P %YM RP %O)MFPZYO MP Y

DE,<6)DED?$ D<2t$C737"J3*@*$;DEJ3**$ D<J-%6)K &)2@6Vext(r) = µ

BUD K<J,E,/$C730( *$ 73(+KE= $C*@2858Q$C852(+736032

n B%D<2(C,E,?265736xJ7F" 65D*-3KE6)DED<2(C736-%6)KH*@6 "J$'7",<2@,RR

&/A36x&)(+= -$C2@(+73(736)*@*$ K ]6x&/A36D<(C73( 036j,<,<6>E N06)**$ 032D ,?K<2EQ3J3852(+736g032H/]A3(C= $CDJ%#6)KE=>2 6)K J37 -%(C,<6)73852$C*@6g$ 'C6)7,?6D<2@=>=65,<K<2$1$CD<DE2$C*@6 D<2dA3$

Rz,⊥ =

√√√√ 2µ

mω2z,⊥

=a2

ho z,⊥

aho

(15Nas

aho

)1/5 +

03( '+6*$"J$'7",<2@,Raho

27302&Q$1*$1=>65032$

aho = (a2ho⊥ aho z)

1/3.

LN(+=6b0365,E,?( D<2tA$ &/A36 2@7 J6)DE,<6;&5(+73032@8)2(C732*6n02=6)73DE2(+732N036)*@*$ 7"J '+(+*$ &5(+7303657 %D?$I,/$XD<(C73( =>(+*-,?( D<J-%6)KE2(+KE2 $ "J36)*@*6 06)**@(XDE,?$',?(X (C730$C=6)7,/$'*6 0365* -%(C,?65738)2$C*6 032&)(C77$C=6)7,<( ∼ aho

k

2 732%0365**$g,?K<$',<,?$C8)2@(+736tD<J3&5&)65D<D<2 ''$ 736)&56)DED?$CKE2(1D0- 6573036)KE6 "J$C*&/A6 -$CKE(+*$K<2@+J$CKE03($C*@*$: (+KE= $1&/A6:2@*|&)(+7036)73D<$',?(>$CDED<J3=6g736)*@*( D*-3$C8)2@(06)2d=(+=>657,?2 &Q$CJD?$ 06)**$ D<J$ 6)D ,?6)7D<2(C7363732@,?$ 2@*&)(C73036)7D?$',<( -%(+D<DE26)06 J373$ *$CK<+A6)8)8)$ 76)**@(D*-3$C8)2@(n036)+*@22@=1-3J*D<203( 'J,/$u$C* -3K<2@73&)2E-32( 02273065,?65K<=27$'8)2(C736:6 03$',/$n0$C*=>(03J3*@( "!#$%&'!)(+*-,./%0$1,.2345!6-6-7.148%:902;*<,.$=(0!7.$12;!?>

∫

V

n(r)d3r = N

(+$@*A4CB !5%D7"*-E1,F45*HG*I7.*-"$;14@1$4902;*V(+*<J&%0!7"$K(0

V (r) ≤ µ

L-M#<#<*<%0%0!)2;$ND90!PO0,"*-@1*-2=*-%D7"*;4RQS7"7"$#FT0*1U8#<$12;*/*-@:!5(+*-%06<!)7.$V"$1W&,.&UX%0*-4Y#<'$V(+!Z!%D7.*<,F6<!5$1%&*7"7",F7"7"!5@[\4?B W&W0,"$D""!52;16<!5$1%0*](+!_^`T0$231'acb&*<,.2=!A%0$1%dGeW&W04!)#<1O0!54*f*g%06-!A4CB !%D7"*-,.16<!5$1%0*h!%&(09&#*90%d#<$12;Wi$1,"7.2;*-%7.$j$1W&WX$D7.$&>k4?B 7"7",F6-!$%0*]7",FlE4!7.$12;!m7.*<%&(0*elQS1,V1902;*<%D7.1,"*h45j(0*<%&"!n7Foj*ND90!5%&(+!/"9&@1$147.4?B !5%7.*<,F6-!$%0*1YpA90*-'7"$=2;*-#-#<%&!5"2;$/#-7.*<%&=1#-#,.*-.#*q4)W&,"*I""!$%0*mND9&1%7.!5'7"!)#<(+*-4r'!)'7"*<23kS#<!$:G;!54Z7"*-,"2;!5%0*K#!5%0*7.!5#<$DAW0,.$@1$+#<1%&(+$V4545J&%0*;!54r#$44)1."$V(+*-4r#$%&(+*-%&"7"$(+!rs $"*at !5%&'7"*<!5%8u^v14*]!5%&'7.1O0!45!7.ol(+!5%&2;!)#<k!5%wW0,.*-"*<%06Ig(0!q!%D7.*<,F6<!5$1%&*7'7.,.7'7"!5@kGf7F7Fk$1E1E*7"7"$l(+!!52=Wi$1,"7.1%D7"!'7"9&(0!v"!5=7.*<$1,.!)#!r?#xQy,IAz5[-R*,.*<4)7"!5@:!8,.!nQy*-,"!52;*<%D7"!8O&!O045!5$1E1,FJi#!|xU&"!53"Wi*<,.!2;*<%D7F45!#$2=*1(*-"*<2;W0!5$z 0xc

S -M|VT"P %YM RP ^WdRP Y M"W YH YF # :ZPZY XP T : M

"J$C0K<(x06)**$ ,?K?$'DE (+KE= $',?$F032#(+J3K<2@6)K0365**$F J738)2@(+736 0F" (C730$g736)*@*(gD*-$'8)2(0365**6t&5((+KJ%032@7$',<6C6)KgJ37 - (C,<6)73852$C*@6 $ '+657",<6>DE2==>6j,?KE2$ $CDED<2$C*6'B|0365,E,?(

A2*K?$.--%(+K ,?(

ωz/ω⊥

BdD<2A3$b2@*DE6)+J3657,?6xKE2DEJ3*@,?$',?( +

Φ(p) = h−3∣∣∣∣∫

dr e−ip·r/h Φ(r)

∣∣∣∣2

= N15

16A

(R⊥

h

)3(J2(x)

x2

)2 M

03( 'C6J2

27032&)$1*$1 J373852(+736F02HGN6)DED<6)*$C* D<6)&5(+7303(>(+K<02736F6x

*$"J$C7,?2-,SR

x =1

hR⊥

√p2⊥ + (pz/A)2

*''$C*(CK<61 0365**$ *$CK<+A6)8)8)$06)**$ 032D ,?K<2EQ3J3852(+736F76)**@(>D0-$C852(036)2=(+=>657,?2 B&)2( *$ "J$'7",<2@,R

∆pi =

√√√√∫

dp p2i Φ(p)

∫dp Φ(p)

g*65$',<($C*dK?$CC+2(102/]A3(+=>$CD&%#65K<=2Ri

0$'**$:K<6)*$C8)2@(+736

∆pi =

√21

8h/Ri

C

7 3+J3K?$ D<(C73( -3KE6)DE6)7,/$',<603J36D<658)2(C732"036)*@*$t032D ,?K<2EQ3J3852(+736U032"036573D<2-,SR 736)*@*( D*-$C852(0365**@6F&)((+K<032@7$',<6 6g736)*@*( D*-3$C8)2@(06)+*@2|2= -3J3*@D<2

s obeNh5r:`ns mnh>q nmnh)`s `nr:` y ri5h5`nr h5egi5s

J$C703( DE2$ -K<6)DE6)7,?6;J37 -%(C,?65738)2$C*6b65DE,?65K<73( -32 &)(+= -3*2@&Q$',<( 032 "J36)*@*( $CK<=(+732@&)("J$C* "J36)*@*( -%6)KE2(032&5(;27,<K<(03(C,<,<(;0$ J7 KE65,<2&)(C*(;(',<,?2@&)(3B$C73&/A36n7365*N*@2=2@,?6032032/]A3(C= $CDJ%#6)KE=>2 N jB73(+7 -%(+DED<2?Q2*6K<2@D<(+* '+65K<6 -%6)K'2$;$C7$C*@2@,?2@&Q$b*7" 6="J$C852(+76 032$FK<(+DED&%U2-,/$C6('D*)"22 kB=>$;(&)&)(CK<K<6>K<2@&)(+KEK<6)KE6 $uD<(+*@J38)2@(+732U7"J3=>65K<2@&/A36C J365DE,<6 27+65736)K<$C*6nKE2&/A32@6)03(C73( J37$ &)(+= -3*65D<DE2@,SR 02&Q$C*@&)(+*@( -32J,<,?(CDE,?( 65*6('C$I,/$ 6 J273032ND<(+73(&5(+=1J37"J36 J,?2*@2d036)*@*6:$.--3KE(+D<DE2=>$C8)2@(+732 B%D0- 65&)2$C*=6)7,?6D<6x*$>C6)(+=65,<K<2$0365*D<2@DE,<6)=>$B&5(+=>6 76)*&)$CD<(u036)C*265D*-%6)KE2=6)7,?2(C++65,E,?(n032 J6)DE,<(;*$ 'C(+K<(B 6)DED<6)78)2$'*=6)7,?6J3732 %032@=>6573D<2@(+7$C*@6CG (+7n657",<K<65K<6)=( 7365* 036j,<,/$'+*2@(x0365**6 032E'C6)KED<6 ,?65&)732@&/A36 &/A36F&5(+73DE6)7,?(+7(

ST ^ %YRMYO5T"P %YM RP %O)MFPZYO MP Y

-5 -4 -3 -2 -1 0 1 2 3 4 5

z [RTF]

0

0.5

1

Den

sità

[U. A

.]

-5 -4 -3 -2 -1 0 1 2 3 4 5

pz [ h/(2π RTF) ]

0

0.5

1

Den

sità

[U. A

.]

151> :*<6-!$%0*K(+*-44)](+!)7.,"!5O0906-!$%0*K(+!r(+*-%&"!n7Fol15[1x 4r,.1E1E1!5$VG3,"!5WX$,'7F7.$V!5%k90%0!7.o(+*-4Z,.1E1E1!5$(0!r^`T0$123a b0*-,"2;!xD7"*I".'*-6<!5$1%0*U2;V%0*-445$]'WX6<!5$](+*<E4!Z!52;W0904)'!qS*-Ni"x>=45f.#<145$,"!56<6-$1%D7.14*;GK!5%j90%&!n7Fo

h/RTF

f8* O&,.,"*3@*<,"7"!)#<14!r"$1%&$]!%\#<$1,.,"!)"WX$%&(+*<%&6-(+!±√

21/8 ' ±1.620S@iZ*INi`'1 D'xY*_(+90*.#<45*@1*-,'7.!5#-45!"$1%&$;(0!@*<,F'*

032K<20J3K<KE61*$032=6)73DE2(+73$C*2-,SR 036)*@*7" 6="J$C852(+76 - 65K *6 "J$C*2D<2K<2=>$C7303$ $ C B=>$x036)DE&)KE2E'+65K<65=>("J32 *7" $.--3KE(&5&)2(x$C03(C,E,/$',<(-%6)KN(',<,?65736)KE6F*6 D<2=xJ3*$C852(+723,?6)(CK<2&/A6 K<2 %-%(+KE,?$',?6F7365*d&Q$.-2@,?(C*(53/$C*6$.--K<(&)&)2@(D<2d (+730$D<J J37$D<65=1-3*@2&56&5(+73DE20365K?$C852(+736 U736)*@*7" $C=Q32-,?( 032dJ373$>+65(D%=65,?KE2$ &52*2@7303KE2&Q$ 27&)J22*&5(+773$C=>657,?(736)*@*$ 032@K<6)852(+76K?$C02$C*@6D<2$ =>(+*-,?(F=>$C++2@(+K<60365*|&)(+77$C=6)7,?(2@7 032KE6)852(+736 $CD<DE2$C*@6CBDE2F-3J" -%6)73D<$CK<6F&/A62 +K?$'032 032 *2?Q%6)K ,SR:$CD<DE2$C*@26tK?$C02$C*@2D<2$C73($C*@=>6573( -$CKE8)2$'*=6)7,?602D?$'&)&)(.--32$I,?236 "J327032&/A36 *$FD<(+*@J38)2@(+736]0365** " 6I%"J$C852(+736 ]DE2$ Z$',<,<(+K<2@8)8)$.Q32*@6 27uJ37$ -$CK ,?6:$CD<DE2$C*@6650;2@7 J373$>K?$'032$C*@6C73(+*@,<K<6'B$CDED<J3=6)730(u2@*&)$CK?$',E,?65K<6$CKE=>(+72&)(>036)*&)(+737$C=6)7,?(bK?$'032$C*@6CB D<2B-3J" $'D<D<J=>65K<6-%6)K*$ -3$CKE,<6 K<$C032$C*6036)*@*$H J373852(+76]0F" (C730$gJ37$ (+K<=>$ 032,<2?-%(F$CJ3DED<2$C73(3 J365DE,<(3B&)(+=6'+6503K<65=>(B%&5(+73DE6)7,?6032dK<2@03J3KEK<6g*7" 6="J$C852(+76 ,?K<2@032=6)7D<2(C7$C*6 $'0uJ7F" 6"J$C8)2@(+736032 A 65K<6)78)2$'*6tJ3732@032=6)73DE2(+73$C*6-%6)K*$ -$CK ,?6H$CD<DE2$C*@6 6 $C0 J7F" 6"J$C8)2@(+736H$C*+6Q3K<2@&Q$ -%6)K

S -M|VT"P %YM RP ^WdRP Y M"W YH YF # :ZPZY XP T : M 3

*$ -3$CKE,<6FK?$C032$C*6't,?2*@2858)2$'=>(:03J37"J36*$1DE6)+J3657,?6: (CK<=>$xZ$',<,<(+K<2@8)8Q$I,/$

Φ(z, r⊥, t) = φ(z, t)ψ(r⊥, t; σ(z, t))

03( 'C6:*$ -3$CKE,<6FK?$C032$C*6HD<2$CDED<J3=6CB&5(+=>6F065,<,<(3B$CJD<D<2$C7$

ψ(r⊥, t; σ) =1√π σ

e−r2

⊥/(2σ2) +

73DE6)KE6)7303( *$u (+KE= $ g736)*@*7" 6)D*-K<6)DED<2@(+736 06)** " $'8)2(C736 $'D<D<(&)2$',/$ $C** " 6"J$C8)2@(+736032$FK<(+DED&%U2-,/$C6('D*)"226 Z$C&56)7303(;*7" 2?-%(C,?65D<2U&/A36 *$ -$CK ,?6>K?$C032$C*6''$CKE2*@6)7,/$C=6)7,<6 &)(C7p*$&5((+KE03273$',/$p$CDED<2$C*6 6 "J32@73032FD<2 -%(+DED?$ $CD<DEJ3=>65K<6 ∇2ψ ' ∇2

⊥ψBHD<2F(',<,?2@6)736'BH03(.- (

* " 2@7",<6)+K<$C8)2@(+736 76)**$-$CK ,?6gK?$C02$C*@6CB*$ D<65+J36)7,<6:65D*-3KE6)D<DE2(+76

S[Φ] =∫

dt∫

dz LNPSE(z, t; σ) C

03( 'C6:*$06)73DE2@,SR 032d*$C+K<$C73+2$C7$ LNPSE

A3$*$x (+K<=>$1D<65+J36)7,<6 D5

LNPSE = φ∗

[ih∂

∂t+

h2

2m

∂2

∂z2− V (z) − gN

2

|φ|22πσ2

− h2

2mσ2− 1

2mω2

⊥σ2

]φ

D5 = - (C736)730( *$NDE,?$C8)2@(+7$CKE26j,SRN0365*+ J373852(+7$'*6 C jB 03$C**@66="J$'8)2(C732+0329@J3*@6)KE(D%$C+K<$C73+6'BDE2d(C,<,<6)73+(C73(>J7F" 6"J$C8)2@(+736$C*@+6=Q3KE2&)$- 65K]*$ *$CKE+A3658)8Q$ 0365**$-$CKE,<6FK?$C02$C*@6

σ(z, t) = a⊥(1 + 2asN |φ(z, t)|2)1/4 N 650 J37 " 6J3$C8)2@(+736032EA 6)KE6)73852$C*@6 - 65Kt*$-$CK ,?6$CDED<2$'*6

ih∂

∂tφ(z, t) =

[− h2

2m

∂2

∂z2+ V (z) +

gN

2πσ2|φ|2 +

1

2hω⊥

(a2⊥

σ2+σ2

a2⊥

)]φ(z, t)

.! J365DE, " J3*-,?2=>$BJ7$ '+(C*@,/$ D<(+D ,?2-,?J32-,/$ $

σ*7" 6)D0-3K<65D<D<2@(+736 N jB -3K<6573036 2@* 73(C=>6u032

6J3$C8)2@(+736x024&/AK*203273C6)K 73(+7 -%(+*@273(+=2$'*6 ZGH4 @ j 46 'C$'*6g2**2@=>2-,?6 02/]A(+= $'D&%#6)KE=>2 N N* " 6J3$C8)2@(+736 -3KE6)&)65036)7,<6xD<2-J" $'73&)(+K<$ D<6)= -3*@2 &)$CK<6 &5(+=6

ih∂

∂tφ(z, t) =

[− h2

2m

∂2

∂z2+ V (z) +

3

2

g√N

2πa2⊥

√2as

|φ(z, t)|]φ(z, t)

K


eFe scdh `o sh5qNhpmns]iji sta sh st`cr

4&5(- ( 032 "J365DE,<( -$CK<$C+K?$I ( ;02D<&5J,?65K<6 Q3K<6('+6)=6)7,<6 J3$C*2t0365**6 $.--K<(+DED<2=>$C852(+7320365*276Q$',<6H7365* &)(CK<D<(x036)* &Q$.-32-,?(+*@(:DE2|$.- -3*2@&Q$C73(x$C+*@2%65D*-%6)KE2=6)7,?2-3K<65D<6)7,?$',?2 27 "J365DE,?(*$ 'C(+K<(3$:D0- 65&)26 $',?(+=2&)$FDE,?J032$',?$ H2@* 87 P Q B&/A36 7(+7 * " 2@D<(C,<(-%( -2 $DQQ (C730$C7,?6H036)*%=>6I%,/$'**(B K<$C032@($',<,<2E'C(3B+=>$gD ,/$.Q2*6tDEJ D<&)$C*$g+65(+*(+C2&Q$ *&5J3732<-$CK<$C=>6j,?KE23&Q$CK<$',<,<6)KE2DE,<2&52032"J365DE,?(>$',<(+=>( DE(+73(K<2$CD<DEJ37,?2d7365**$x,?$.Q%6)**$

VP RP 9;$CDED?$ $I,?(+=2&)$

m 1.443 160 60(11)× 10−25 Kg 2-,/$ =>65032$17"J3&5*6)$CK<6

τn 4.88 × 1010 anni QQ%(+7303$C738Q$K<65*$',<2E''$

η 27.83(2)%dJ373+A3658)8)$>032d032 A JD<2(C736

as 5.77(53) nmJ37,?(032| J3D<2@(+736

TF 39.31 oC

^ROi*<4545151> (&7"!W0,.*-"*<%D7F7"!!5%VND90*I7F7FOi*<4545;"$1%0$7",F7"7"!8(0z [81(*-#-#*-6<!5$1%0*(+*<4@45$1,.*(+!as#FT0*_"!8,"!Qy*<,.!)"#<*H1445$37F7"$

|F = 2, mF = +2〉*-(G7",F7"7"$3(0Vz [c

#" $.- -$CK?$I,?( D*-%6)KE2=6)7,/$C*@6b$',<,<2E'C( -K<6)DED<( 2*]H@UG 4p6b036)DE&)KE2@,<,<( 76)*]&Q$D-32@,<(+*( &Q$'K?$',E,?6)KE2858Q$',<(03$J37 -%(C,?65738)2$C*6F032d&5(+773$C=>657,?( $'K<=(+732&5( &)(+7uD<2@=>=65,<K<2$1$CD<DE2$C*@62&)J32-$CK<$C=>6j,?KE2DE(+73( KE2?-%(+KE,?$',?2d76)**$,?$.Q%6)**$ H $1J6)DE,<2 -$'K?$C=65,<K<2D<2K<2@&Q$ ''$C73(

FO M"WIP MY S

#K<6J6)738)$>032d&5(+77$'=>657",<($CD<DE2$C*@6ωz 2π × (8.8 ± 0.2) Hz

#K<6J6)738)$>032d&5(+77$'=>657",<(K?$C032$C*6ω⊥ 2π × (90 ± 2) Hz

G J=>65K<(032$',<(+=>2 736)*&5(+73036573D?$',<(N ' 150000

H6573D<2-,SR ,?2?-2&Q$n ∼ 1014 cm−3

^ROi*<4545 1 +> (07"!_W0,.*-"*<%D7.7"!!5% ND90*I7Fj7FOi*<4545 '$%0$ Qy,"9+7"7"$w(+!_2;!)'90,.*k(+!#<1,.7'7"*-,"!56<6I6-!$%0***7"7"9&7"*(+9&,.1%7.*_N9&*-'7"$;45[@$1,.$/(+!87"*-"!C

Z$C&52*=6)7,<6]*6t*J37+A36)858)6 032(+D<&52*@*$',<(+K<6t$CKE=>(C732&5(a

(ho)⊥ = 1.14µm

Ba(ho)

z = 3.60µm

T3W T M9 WIPY|V MWIPZ^PRM3: : MO M"W'P MY

J365DE,?( &)(C73D<657",<6 032 'C$'*J,?$CK<6 DE6 *$p$.--3KE(+D<DE2=>$C8)2@(+736 DE6)=2&5*$CDED<2&)$XJ,?2@*2@8)8Q$I,/$ -%6)K''$C*@J,/$CKE6

Tc

7365**$ 65 g&)(+KEK<65,E,/$42dA$2@7Z$',E,?2

kBTc

h= 0.94(ω2

⊥ ωz)1/3 N1/3 ' 9.1 KHz ω⊥ > ωz

6 "J32@73032%*$:$.--3KE(+D<DE2=>$C8)2@(+736tD<65=>2@&)*$'D<D<2@&Q$ '+65K<2 &Q$',?$*<'C$'*(+KE6 0365**$g,<6)= - 65K?$',<J3K?$&5K<2@,<2&)$1&)(+KEK<2D0- (C73036)7,<6

Tc ' 100 nK .+

$>06 732@8)2@(+736g032d,?65=1-%6)K<$',?J3K<$&5K<2-,?2&)$ 0$',/$0$C*@*$ 5 ''$C*@676)**2@=>2-,?6H,<6)K<=(0327$%=2&5( UD<2-3J" 'C$'*J,?$CK<6>*$ Q%(+7,SR 032 "J36)D ,/$ $.--3KE(+D<DE2=>$C8)2@(+736>D0'2*@J--$C703( 27 D<6)KE26*$u,?65=1-%6)K<$',?JK?$ &5K<2-,?2&)$ KE2D*-%65,E,?(;$C* -3$CK?$C=65,<K<( -32&5&)(+*@(

1/N LN2 " &/A36 DE2N(C,E,?265736

&/A36b* " 6(A 6j,<,?(;036)*@*6 032@=>6573D<2@(+732 732-,?6 0365*NDE2DE,<6)=>$ -3K<( 'C(&)$ J7 $.QQ$CDED?$C=6)7,<( KE6)*$ %,<2E'+(n06)**$,<6)= - 65K?$',<J3K?$ &5K<2-,?2&)$ -3KE(- (CK<8)2@(+7$C*@6x$

(1/N)1/3 K : 2&)(C73D<65+J36)78Q$b*$,<6)= - 65K?$',<J3K?$&)K<2-,?2@&Q$g(+D<DE6)K0''$',?$ -32 Q$'D<D?$:032

TcB= $032 J7$ "J$C7,<2@,SR:27 65K<2(CK<6 $C**$

,<2?-32@&Q$>2@73&)65KE,<6)8)8)$ 0$',?$ 0$C*@*6 J,<,<J$C852(+732DE,?$',?2@DE,<2&/A36:036)*7J=>65K<(n032$',?(+=2d,?K<$ 03J36032 '+65K<D<6=2D<JK<6F&/A36gD<(C73(>06)** " (CK<032@736 0365*

5%

7F" $'*@,?K<$>$D--3K<(CD<D<2@= $'8)2(C736 (+730$'=>657",?$C*6gJ,<2*@28)8)$',/$12@7 =(03(1-%6)D<$C7,?6:736)*@*$1,?K<$',<,?$ %852(+736 -3K<65&)6)06)7,?6 "J365**$032 +$CDN (+K ,?6)=6)7,<6 032*@J32@,<( M j t,?2@*2858Q$C703(x*@6 "J$C7,?2-,SR2@7b,?$.Q%6)**$ DE2d(C,<,<265736 ZD ,?2=>$1&)(C73D<65K0''$',?2 ''$

na3s ∼ 2 × 10−5 1

6 "J327032d*7" $.--3KE(+D<DE2=>$C8)2@(+736 032d$CD]032@*J32-,?( $C7&/AF" 65D<D<$ *$CKE$C=6)7,?6 '+6)KE2 &)$',/$Pt6)D ,/$277603$02D<&5J,?65K<6g*7" $.--K<(+DED<2=>$C852(+736H032F/]A3(+=>$CDJ%#6)KE=>2 N

N as

a(arm)⊥

>N as

a(arm)z

' 240 1

61J273032$C73&/A6 "J36)D ,/$ $.- -3K<(+DED<2@= $C852(+76 -J" &)(C73D<2@036)K<$CK<DE2 'C6)K<2 &Q$I,/$ 76)+*@265D*-%6)KE2 %=6)7,?2(C++65,E,?(n032#"J365DE,<( *$ 'C(+K<( 2@=1-%(+KE,?$C7,?61DE(C,<,<(+*2@736Q$'K<61&/A36DE6* " $D--3K<(CD<D<2@= $%852(+736H032F/]A3(+=>$CDJ%#6)KE=>2;-3J" -%6)K<06)K<6F02 ''$C*2@032@,RB J6)DE,<($C&)&)$C036FD<(C*( 27 -K<(+DED<2=2@,R032r = R

03( 'C6F2*%+K?$C026)7,<6H032 036)73DE2@,R = $'D<D<2@=>( Z6H27n$.- -3K<(+DED<2@= $C852(+76 032/t#p$C0 %032@K<2-,<,?JK?$:02D<&5(+7,?2@7J( j#/J,<,?$ '2$B&5(+7bJ373( D&'2@*J- - (12@7nD<65K<26HK<2D0- 6j,<,<($C* -$'K?$C=65,<K<(


-32@&)&)(C*(aho/R

03$',?( 0$'**$ + jBDE2F-3J" 02=(+DE,<K?$CKE6 K &/A6F*$x032@D<&5(+7,?27"J32-,SR '2@6)736=6)032$',/$xDEJ J7$ 032D ,/$C738)$-$CK<2 $ &)2KE&Q$12@*

10%0365*dK?$C+C2(1032 /t# K<$C032$C*6 6F032d&)2@K<&Q$12@*

3%032 "J36)*@*( $'D<D<2$C*66x&/A61*$ -3K<2@= $&)(+KEK<6)852(+761$C*- (',?6)78)2$'*6&/A32=2&5(3B6x032&)(+73DE6(%

+J365738Q$:$C**$F036)73DE2@,R:032<-32@&)&)(B&/A36# J7$:0365**@6 "J$'7",<2@,R=2DEJ3K?$',<6tD*-%6)K<2@=>657,/$C*@=>657",<6Z&j K) ? K kB g27 65K<2(CK<6F$C*@*(

0.5%

#" $C8)2@(+736 032J37p&Q$'=1-%( 6)*@65,E,?K<2@&)( 02NK?$'032$C852(+76 73(+7pKE2DE(+7$C7,?6 D<JpJ37p&)$C=1-2(+736>032$',<(+=2U KE6)0302032E- 6573036>0$C*+K?$C026)7,<6>03227,<6)73DE2@,SRu032U,/$C*@6>K<$C032$'8)2(C7366 0$C*@*(uDE,?$',?(2@7",<6)KE73( 6 02 =>(',?(p0365+*2H$',?(+=2 /$'*6 2@7",<6)K<$C8)2@(+736 IIBt2@7 '2@KE, 0365**$ D<J$2@7",<6)73DE2@,R 6 0365*HK<2@03(C,E,?( KE2DE&Q$C*@0$C=6)7,?( &/A36 2703J3&)6 *( DE,?KEJ3=6)7,?(X (+70$C=6)7,/$C*@6-%6)Kb*$p=>$C732E- (+*$C8)2@(+736u73(+7032DED<2E-$',?2 ''$X036)+*@2F$',<(+=>2 K<6)0032F6)027 -$CK ,?2@&)(+*$CK<6 06)*&5(+73036573D?$',<( 032HG(+D<6I%J@273D ,?6527 .K

d(pD<&/A365= $ 032 "J365DE,?( &)$.-32@,<(+*( 2*HDE6)+J6)7,?6 76)* -3KE2=( -$CK<$C+K<$' ( '+6573+(+73(-3KE6)DE6)7,/$',<6 *@6n$.--K<(+DED<2=>$C852(+732U2@7p&)J32DE2]036 3732DE&)6n2* - (',?6)78)2$'*6 02?-%(+*$CKE6 6n*$;D<J$ (+KE= $ 65D*-3*@2&52@,/$ 27X J373852(+76 036)2 -3$CK?$C=65,<K<2D*-%6)K<2@=>657,/$C*@2 G 65*D<65&)(+730( -$'K?$C+K<$' ('265736036 3732@,<(u2@*KE65,?2@&)(+*@( (C,E,?2&5( DE2=>(CDE,?K<$ &)(+=6 03J36*$CD<65KF&)(+7,?KE(-3KE(-$C$'7",<2 B ,<K?$&5J3236)DE2D ,/$FJ37$ Q%6)7036 3732@,?$ K<6)*$C8)2@(+736032Z$CDE6CB&)(CDE,?2-,?J32@D<&5(+73(FJ37 -%(C,?65738)2$C*6 - 65K<2(032@&)(7365*&)J32D<2@DE,<6)=>$g02K<2@ 65K<2@=>657",<(F+*23$',?(+=2-%(+D<DE(+73(:6)D<DE6)KE6H=6)D<DE227=>(',?(F&)(C71'+65*(&)2-,SR&5(+7,?K<(C**$I,/$ '+6)0K<6)=( $C* -$CK<$C+K<$' (p X&)(+=6 "J36)D ,?( - (+DED?$p6)DED<65K<6 -3K?$',<2&)$C=>657,?6KE6Q$C*@28)8)$',?( jgG 65*B-$CK<$C+K<$' ( DEJ3&)&56)D<DE2E'C( '+6573+(+7(u02D<&5J3D<DE2+*@26(A 6j,<,?203( 'J,?2$C*=>(',?(0365+*2 $',<(+=>2t&/A36;&)(+= -3*2@&Q$C7( 2@* "J$C0K<(X0365*2@736Q$',<( 2@7 -3K<65&)65036)738)$ G 65* -$'K?$C+K<$' (65D<2H-3K<65D<657",?$C73(1- 65K DE(+==>2|&)$.-32d+*@2|6A 65,E,?20365*KE65,<2&)(C*( (C,E,?2&5(D<J3*@*$1032@7$C=2&Q$106)*&5(+73036573D?$',<( 032HG(+D<6I%J@273D ,?6527|7 -$CK ,?2&5(+*$'K<6 DE2-3K<65D<6)7,?$>*7" $.--K<(&)&)2@( 036)*@*6 (C73036g032GN*@(&/Ab*@2736)$CK<2 6FD<2d=(+DE,<K?$1&)(C=>6FDE2$-%(+D<DE2?Q32@*6 -3KE6=-$'K?$CKE62@* &5(+73036573D?$',<(>2@7nJ373(DE,?$',?(032dDE273C(+*$1(+70$ 032HGN*(&/A|

3>

8 ! W M9 P ^ ;:ZP# <= P ^P

*d&Q$D-32@,<(+*( DE2&5(+73&5*J3036g&5(+7;J37(>D ,?J3032@(>06)**$1DE,?$.Q32*@2@,R1036)*dDE2DE,<6)=>$#'26)76:=6)DED<(27K<2@*2@6'+(b&5(+=>665D<2D ,/$C7( 03J6>KE6)+2@=>202E'+65K<DE2022@73DE,?$.Q32@*2@,R 2@6)736 "J32@73032,-3K<65D<657",?$',?(J37 =(06)**@( J372032@=>6573D<2@(+7$C*@62@7 &)J2&Q$C*@&)(+*$CK<6*$>D<(++*@2$ - 65K *7" 273D ,/$CJK?$CKED<2032 "J365DE,?627DE,/$DQ32*@2@,SR12@7 032E- 6573036)78Q$ 0$'**$036573D<2-,SR 0365*&)$C= -32(+76:6:0$C** " 657,?2@,R06)*- (C,<6)73852$C*@6-%6)K<2@(02&)(

_i rgcst` h)egijs mbh ri)eFs

LN(+73DE20365K<2$C=>( J37$',?(+=( 736)*@*( DE,/$I,?( (C730$C=6)7,/$'*6b&/A36 DE2 =1J(''$ *6)7,?$C=>657,?6u27J37$>8)(+7$>036)*@*(D*-$C852(>27;&5J32DE2$ -3K<65D<6)7,<6 036)*@*$*J3&56x*$'D<6)K *$&)J32 K<6"J36)738)$

ωLDE2$

'2&527$>$ J6)**$>032J37$ ,<K?$C73DE2852(+736:$',<(+=>2@&Q$ω0

B =>$ *(+7,?$C7$>03$C**@6x$C*-,?KE6 $D--3K<(CD<D<2 %=>$C8)2@(+736t032 $',?(+=(:$x03J36 *2E'C6)*@*2 k LN(+73D<2@036)KE2$C=(2*&Q$'D<(12@7 &)J2 * " 2@7",<6)73DE2@,R=6)032$:0365*&Q$'=1-%(*$CDE6)K''$CK<2 &)(+7u*$-%(+D<2@8)2(C736 $'0 65D<6)= -32@( -%6)KE&/A2*|=(0( 036)*d&)$C=1-%(6)*@65,<,<K<2 %&)( +$CJ3D<DE2$C7( 61DEJ--%(+732$C=>( 032 -%(C,?65K &)(C73D<2@036)K<$CK<6 2@* -$'&)&/A36j,<,?(b0F" (+7303$n$',?(+=2&5(*(&Q$'*2858Q$',<( K<2@D*-%65,E,?(u$C**$ K?$C02$C852(+736xD<2$n2@7 ,<6)K<=272032#-%(+D<2@8)2@(+736CBDE2$n27 ,?65K<=2732032'+65*(&)2-,SR J36)D ,?( 6="J32 ''$C*6n$ KE2&/A32@6)0365K<6n&/A36n*@6 032@=>6573D<2@(+732$',?(+=2&/A6 D<2$C73( =(+*@,<(=273(+KE20365**$ *J373CA36)858Q$F0F" (+730$

∆r λL

6t&/A36t*(gD*-%(+DE,?$C=6)7,?(g03( 'J,?(:$C*@*7" 6A 65,<,<(H(- -3*65KFD<2$ -32@&)&5(+*(n2@7 &)(+7 KE(+7,?(u$C*@*$ *$CK<+A6)8)8)$n7$',?JK?$C*@61036)*@*$ ,<K?$C73DE2852(+736x&/A36>DE2DE,?$&5(+73DE20365K?$C730(3B

Γ .M

k · v Γ

* -3KE273&52?-32@( 032]270365,<6)K<=273$C8)2@(+736 2@=1-%(+736 &/A6∆r × ∆v > h/m

6 "J327032]*6n03J6&)(C73032852(+72 -K<6)&56)03657,?2-%(+D<DE(+73(\65D<D<65K<6pDE(0302DEZ$I,<,?6 &)(+7,<6)= - (+K<$C736)$C=>657,?6 DE(+*( D<6 .M

Erec =h2k2

L

2m hΓ

&)2@( D<6F*$ ''$CK<2$C8)2@(+736 032 65736)KE+2$x&52736j,?2@&Q$1$CDED<(&)2$',/$1$'**7" 6)=2DED<2@(+736Z( $'**7" $CDED<(+K0Q32=6)7 %,?( U032J37> (C,?(+76 0365**$gK<$C032$C8)2@(+736Z65736)KE+2$032K<27&)J3*@( =(+*@,<( -32&5&)(+*$KE2D*-%65,E,?(x$C**$*$'K<+A3658)8)$2@7,?K<2@73D<65&Q$ 27n65736)KE+2$ 0365**$x,?K?$'73D<2@8)2(C736 &/A36D<2dD ,/$-3K<657303657303( 2@7n&)(+7D<206(%K?$'8)2(C736C

: # MY"P T : M;RP # ;: T3W M 8

"J36)D ,?6>&)(+73028)2@(+732032E'265736 - (CD<D<2EQ32*@61036 732KE6*$ K<6"J36)738)$ 032,P $.Q3206)**$ ,?K<$C73D<2 %852(+736

hΩ0 = −dfe ·E0 > 0

03( 'C6dfe

2@73032@&Q$F*7" 6)*@6)=6)7,?(g032=>$',?KE2&56]036)*@*7" (- 65K?$',<(+K<6]02032?-%(+*(F65*6j,<,?KE2&5(6E0

*7" $C=%-32@6)858Q$ 036)* &Q$C= - (x6)*6j,<,<K<2&5(3U46H273032@&/A32$'=>(:&)(+7

δL = ωL − ω0

2*jZ >036)* *$CDE6)K0$'**$g K<6J6)738)$02|K<2@D<(+7$'738Q$x6HD<J--%(+732$C=(1032 65D<DE6)K<6g7365** " $.- -3K<(+DED<2@= $C852(+76 02 K<2@$Q%6)7uK<2@D<(+*-,/$

Ω =√

Ω20 + δ2

L Γ 65

* " 6(A 6j,<,?( 06)**$ K<$C032$'8)2(C736 DEJ3+*@21$',<(+=>2 "J365**@(\032x2@7,?K<(03J3KEK<6XJ73(\D*-%(+D ,/$C=6)7,?(03652 *2 '+6)*@*2g032165736)KE+2$ ? j C %/ b&/A36 $C+2DE&)6X&5(+=6pJ37 - (C,<6)73852$C*@6&5(+73D<65K0''$',<2E'+( Ij ZC:I-C?j7,?KE(03(',<,?(2@* -$CK<$C=65,?KE(032dD?$',<J3K?$'8)2(C736

Ωs(r) =Ω2

0(r)

2δ2L

ON DE6 ''$C*@6F$C73&/A36*7" $.--K<(+DED<2=>$C852(+736 032dD?$I,?J3K<$C8)2@(+736 ,?K<$CD<&5J3K?$DQ32*@6

Ωs 1 !

DE2dA$&/A6:2@* -%(C,?65738)2$C*6 02d032?-%(+*($'D<D<J=>6g*$x (CK<=>$ .+B.MBC

VDIP (r) =hδL2

Ωs(r) =hΩ2

0(r)

4δL' hΩ

2

OK 7 "J365DE,?(n&5(+7,?65DE,?(n* " 6(A 6j,<,<( 0365**$ (+K<8)$ 032032?-%(+*@(1-3J" 6)DED<6)KE6 036)DE&)KE2@,<,<( 27 = $'73265K?$DE6)= -3*2@&)61$>D<6)&5(+730$n0365*DE6)+73(n06)*]jZ Cg4J--%(+732$C=( &/A36 * " $',<(+=(>657",<K<2736)*@*$85(+7$>022@7,?6)K<$C8)2@(+736F7365**@(D ,/$',<( (C730$C=6)7,/$'*6CBDE6:2@*j KE(+D<DE(

δL < 0*7" $ %

,<(+=>(gKE2D<657,?2K*RF032J37 - (C,<6)73852$C*@6&/A36 =272=(H27&5(+K<KE2D*-%(+7306)738)$032J7>=>$CD<DE2=(F0322@7",<6)73DE2@,Rx6)0 "J32@73032 $',<,<2K?$I,?( 'C6)KED<(8)(+76 032|$'*@,/$27,?6573D<2-,SR D<6F2@*jZ Q3*@J '2 %&56'+65K<D<$

δL > 0*7" $',<(+=>(HK<2DE6)7,?6 032J371- (',?6)78)2$'*6]&/A36 t=>2@732=( 27&5(+K<KE2D*-%(+7306)738)$

0365**$1=>2@732=>$:2@7",<6)73DE2@,R( ' '+65K<(>D?$CK*RKE6)D*-27,?(0$C*@*6 85(+736g03( '+6*7" 27,<6)73DE2@,SR g65*6''$',?$7,?KE(0J3&)657303(;* " 65D*-3KE6)DED<2(C736>06)**$ *$CKE+6)858Q$b7$',<J3K?$C*@6 27 $.--K<(+DED<2=>$C852(+736xD<6)=2&5*$CDJ%DE2&Q$

Γ =ω3

0

3πε0hc3|dfe|2

8<8 W M9 P ^ ;:ZP# <= P ^P

6g036)*@*7" 27,?6573D<2-,SRx76)*42D ,?65= $17,?65K<7$'8)2(C7$C*6

I(r) =cε02

|E0(r)|2

D<2(C,<,<26)76xJ37F" 6)D0-3K<65D<DE2(+7360365*B-%(C,?65738)2$C*6:032032E- (C*( JD?$',?$ K<6="J3657,?6)=6)7,<6>2@7 *6j,<,?6I%K?$I,?J3K<$

VDIP (r) ' 3πc2

2ω30

Γ

δLI(r)

+ 42 73(C,?2%&/A36 2@*-%(C,<6)73852$C*@6 KE2D<J*@,/$032KE65,E,/$C=6)7,?6 -3KE(- (CK<8)2@(+7$C*@6H$'**7" 27,<6)73DE2@,SRF650n279'+65K&%D?$'=>657",<6 -3K<(-%(+KE8)2(C7$C*6 $C*UjZ C

_i sch)qr:ijr rgcdch5qNr

LN(+73DE20365K<2$C=>(03J36Z$CD<&52*$CDE6)K &)(+7,?KE(-3KE(-$C$'7",<2657,?K?$'=Q32H- (C*$CKE28)8)$',?2 *276Q$CKE=>657",<6*J73+(g*(DE,<6)D<DE( $CDED<6 6 2@73032&/A2$C=( &)(C7

z* " $'D<D<6 032<-3KE(-$C$'8)2(C736CB&5(+7

EB"&)(+7

k1

6ω1

K<2@D*-%65,E,?2E''$C=6)7,<6*7" $C= -32658)8Q$ 036)*&)$C=1-%(3B+2@* 'C65,<,<(+K<6t0F" (+7303$g6t*$ K<6J6)738)$ $'73+(+*$CK<6 032dJ37(06)203J36FZ$CDE&)2 B3$C**@(+K?$xDE2-3J -%(+K<KE6

E1(z, t) = E1 cos(k1z − ω1t) M

E2(z, t) = E2 cos((k1 + ∆k)z + (ω1 + ∆ω)t)

03( '+6bDE2 ,?K<$CD<&5J3K?$I,/$ J37$ 6('+6)7,<J$C*6b032 A 65K<65738Q$ 032Z$CD<6 &5(+DE,?$C7,?6n,?K<$ *6 0J36n(+73036'G 65*D<6)CJ32@,<(DEJ--%(+K<KE6)=(>&/A6:*@6 "J$'7",<2@,R2@73032&)$',?6F&5(+7

∆D<2$C73(=>(+*-,?( =273(CK<2 06)**@6

"J$C7,?2-,SR$ &)J32'C6)73+(C73(>DE(+== $',<6C#" 2@7,?6)7D<2@,R K<2DEJ3*@,?$C7,?6F0$'**$:D<( 'K?$.--%(+DE28)2@(+736F03652d03J36&Q$C= -32 g0$I,/$ 0$

I(z, t) = cε0 (E1 + E2)2 = I1 + I2 + cε0 (E1 ·E2) ×

× cos(∆ω t+ 2kz) + f(z) cos(2ωt)

03( '+6D<2dDE(+73(27,?KE(0(C,<,<2

2ω = 2ω1 + ∆ω ' 2ω ± ∆ω

2k = 2k1 + ∆k ' 2k1

:W M9 PZ^ : PZ^ 8

6UD<2DE(+73( 27302&Q$I,?6&5(+7Ii

*627,<6)73DE2@,SR]=6)032@6032&52$CDE&)J373( 0365203J36Z$CD<&52 cε0E

2i /2 6U&)(C7

fJ37$ J738)2@(+736g032?-%6)7036)7,<6x0$C*@*$ D<(C*$ - (+DE2852(+736' J365DE,?(,?65K<=2736gD?$'KR,<K?$CDE&)J3K<$',?(

7365*DE6)+J32-,?( -%6)KE&/A CB"$'27320320365,<6)K<=273$CK<62* - (',?6)78)2$'*66A 65,<,<2E'C(g$C+6)7,<6D<J3C*2$',?(+=2 B(&)&5(+K<KE6=>65032$'K<6NDEJ ,?6)= -32032032 '+65K<D<2&)2&5*2(C,<,<2&52&5(+K<KE2D*-%(+7306)7,?2$C**$ D<&)$C*$ ,?6)= -%(+K?$C*@60365**$x03273$C=>2@&Q$x06)+*@2|$',<(+=>2 Z06)** " (CK<032@736H0365**$x*$CKE+A3658)8Q$173$',?J3K<$C*6

ΓB736)*@*7" 2?-%(C,?65D<2%032

D<$',?J3K<$C8)2@(+736 ,<K?$CDE&)J3K<$.Q32*@6k DE6)+J2@,?( 032,/$C*@6 =6)032$;,?6)= -%(+K?$C*@6 2@*N&)(C7",<K<2EQ3J,?( 06)*,<6)K<=276

cos(2ωt)032E'2@6)736 "J32@73032|,?K<$CD<&5J3K?$DQ32*@6C

LN2 "-%(+DE,<(3BD<2d(',<,?2@6)736F&/A6:* " 2@7",<6)73DE2@,Rx,?(C,?$C*6F$'D<D<J=>6g*$x (CK<=>$

I(z, t) = IS + IM cos(∆ω t+ 2kz) 5

03( 'C6IS = I1 + I2

6IM = 2

√I1 I2

G (C,<2$C=( "J32U&/A36 "J36)D ,?603J36 J3$C7,?2@,RnD<(+73(J3CJ$C*2|$C*F-3KE2=( (+K<032@736F7365**$102EA 6)K<65738Q$

E1 − E2

Pt2&5(+K<0$'7303(g*7" 6)D0-3K<65D<D<2@(+736 K Z($C7$C*@(+$C=6)7,<6*$ O+ DE2(C,<,<26)76]&/A36CB$ =>6573(g032J37,?65K<=2736]&5(+DE,?$C7,?6CB2* - (C,<6)73852$C*@6032032E- (+*@( &)KE6Q$',<(:03$C203J36tZ$CD<&52&)(+7,?KE(-3KE(-$C$'7",<2 g0$',<(0$

VLAT (z, t) = UL cos

(∆ω t+

2π

λ/2z

) N J365DE, " J3*-,?2=>$;KE6)*$'8)2(C736 '2@6)736nD0- 65D<D<( DE&)K<2-,<,?$ 27,?KE(03J&)6)703( 2* -$CK<$C=>6j,?KE( $C032@=>657 %DE2(+7$'*6

sB3J3D<$C7303(&)(+=6FJ3732-,SR 7$',<J3K?$'*6F*7" 6)76)K<C2$1032dKE273&5J3*(

Erec = h2k2/2m

VLAT (z, t) =sErec

2cos

(∆ω t+

2π

λ/2z

) !

$'**$p (+KE=1J3*$ -3KE6)&)65036)7,<6 $.--$C2@(+73( &/A32$'K?$C=6)7,?6 03J36 $CD*-%65,E,?2&Q$CK<$',<,<6)KE2DE,<2&5206)*DE2DE,<6)=>$ &)(+73DE20365K?$',<(

• * - (C,<6)73852$C*@6]&)KE6Q$',<(:0$'203J36tZ$CD<&52 -3K<65D<6)7,?$xJ37$ - 65K<2@(032@&)2-,SRH7365**@(FD*-$C852(K<6)$C*6-3$CK<2$

λ/2*J37+( *$032@K<658)2(C736g032 -K<(-$'$C8)2@(+736 %03$1"J32*$036)73(C=>2@7$C8)2@(+736g032

?j /InI / Z j I%Cj

• $CK<2$C7303(>(--%(+KE,<J37$C=6)7,?6x*$>02EA 6)K<65738Q$>032 KE6="J36)78Q$ ,?K<$ 20J361Z$CD<&52DE2 -3J"(',<,?65736)KE6&/A36]2@* -%(C,?65738)2$C*6 -%6)K<2@(02&)( D<2=1J3( ''$H&)(C7'+6)*@(&52@,R DED?$',?$ 736)*D<2D ,?65= $

8 W M9 P ^ ;:ZP# <= P ^P

032dKE2@ 65K<2=6)7,<(1036)*d*$.Q%(+K?$',<(+K<2@(

vLAT =dz

dt

∣∣∣∣∣V

=∂t VLAT

∂z VLAT=

∆ω

2π

λ

2=λ

2∆ν

K

032d&5(+73DE6)+J365738Q$,/$C*@6 K<6j,?2@&)(+*@( (C,<,<2&)( F036j,<,<( j

G 65+*26)D0- 65K<2=6)7,<26A 65,E,?J$',<2D?$'KRnJ,?2@*2@8)8Q$I,?( J7 DE2D ,?6)=>$ 032KE2@ 65K<2=6)7,<( 27365K<852$C*@6x$'+65*(&)2-,SR

vLAT

2@7,?$C*6]DE2D ,?6)=>$g032K<2@ 65K<2@=>657",<( - (C,<K<65=>(g&)(C73D<2@036)K<$CK<6 2*KE65,?2@&)(+*@(F&)(+=6J37F" (+7303$D ,/$C852(+7$'K<2$:6C*2d$',<(+=2 27b=(C,?( &5(+7 '+6)*@(&52@,R −vLAT

z scdch a uon

LN(+=6 +2 R $C&5&)6)77$',?(Bg$C*=6)73(p0J36 6(A 6j,<,<22@7,?6)K&'+6573+(+73( $ &)(+= -3*2@&Q$CKE6 2* "J$'03K<(0365*276Q$',<(n27 -3KE6)&)65036)78Q$g*6 J,<,<J$C852(+732036)*B-%(C,<6)73852$C*@61032?-%(+*$CK<603( 'J,?6$C*@*7" 6)=2D&%D<2@(+736 D0- (C7",?$C736)$6 * " 6(A 6j,<,<('2@D<&5(+D<(0365**7" 27,<6)K?$'8)2(C736 $',<(+=(D%&)$C=1-%(*$CDE6)K)7 "J365DE,?(-$CK<$C+K<$' ( 032DE&)J,<6)KE6)=(>2@* K<J(+*( 032"J365DE,<20J36g6A 65,<,<2

e l

G 65*-$CK<$C+K<$' ( 2* ''$C*@(+K<6N=>65032( 06)*-%(C,?65738)2$C*6N0323032E- (+*@( K tK<2&)$ 'C$I,?( 7365** " 2E- (D%,?65D<2|&/A6F*7" $',?(+=(xDE2$: 65K<=(16F&/A36F2@*-%(C,?65738)2$C*6 032?-%(+*$CK<6 73(+7n032E- 65730$10$'**$ '+65*(&)2-,SR0365+*2d$I,?(+=2 46g*$-K<2=>$ $CDED<J373852(+76 -J">65D<DE6)K<6&)(C73D<2@036)K<$',/$ &)(CK<K<6j,<,?$DE6 ''$C*@6F*$ jB3*$DE6)&5(+7 %0$n$.- -3K<(+DED<2@= $C852(+76 1,/$'7",<( -32 &5(+K<KE65,E,/$ "J$C7,?(b=6)73(nKE6=-%6)7,?2@7$ *$1''$CKE2$C852(+76D*-3$C8)2$C*6 0365*,- (C,<6)73852$C*@6C7Z$',E,?27365**@2=2@,<6x032,-%(C,?65738)2$C*6 &5(+DE,?$C7,?6 Z6 J2730322@73032 %-%6)7303657,?60$C**$ -%(+D<2@8)2(C736 * " 2@=1-J3*DE( J37$n&5(+DE,?$C7,?6036)*=(C,<(3x*=(+=6)7,?(n06)+*@2$',<(+=>2 :$C*-,?6)K<$',?(0$C*@*$-3KE6)D<65738Q$ 06)**$ K?$C032$C8)2@(+736 -%6)K 6A 65,<,<( 0365*KE273&5J3*(>$CDED<(&)2$I,?($C0 (+C7323$CDED<(+K0Q32=6)7,<( (F6)=2DED<2(C736 D*-%(+7,/$C736)$ 6DE,<2=(+*$I,/$ |032 (C,?(+72 7 &)(+KEK<2D0- (C7 %0365738Q$n032&)2$'D<&)J73(n032 "J365DE,<26'+657,?2 B-%6)KH*$ &5(+73D<65K0''$C852(+736x036)*@*$ "J$C7,<2@,SR>032=(C,?(B*7" $',<(+=>($'&="J32D ,/$J37;=(+=6)7,?( -$'K<2$C0

hk0('C6

k27302&Q$ 2@*F'+65,E,?(+KE6:0 " (+70$06)*d (D%

,?(C736x$'D<D<(CK*Q32-,?( (>*7" (--%(+DE,<(>06)*H'+65,E,?(+KE6x0F" (+7303$>0365* (C,<(+7366)=6)DED<(3 J36)D ,?( 6(A 6j,<,<(3B

UM9 PO VW 8

-%6)KgJ37 273DE26)=61032U$I,?(+=2 BdDE2#-3J"b,?K?$I,<,/$'K<6 7365**6$D--3K<(CD<D<2@= $'8)2(C7320$b73(+2JD?$',<6Z$C*-3KE2=(1(+KE032736F2@7

Ωs

N&5(+=6 J37n 6)73(C=>6573(032EA J3D<2 '+( &5 K) .+ 27 -$CK ,?2@&)(+*$CK<6 ? 1 @] 6 / @] 5 &6 .MB+BC ]2@*|&)J2d&)(6&)2@6)7,?6 g0$',<(03$ +

2Dp =∂

∂t(〈|p|2〉 − (〈|p|〉)2) =

h2

2Γ Ωs

(k2

L +∣∣∣∣∇r Ω0

Ω0

∣∣∣∣2) +

03( 'C6 &)(+=6 $C* D<(+*@2@,<( *6 -$CKE6)7,?65D<2g$C73C(+*$',<6 2@73032@&Q$C73( =>650326 "J$C7,?(+=6)&5&Q$C72&/A36 6D ,/$',<2DE,<2&/A36' $'**$036 3732852(+736:K<2@D<J3*-,/$>&/A2$CKE( &/A3612@*&5(6>&52657",<6

2DpK?$D--3K<65D<657",?$n*$ 'C6)*@(&52@,SR032

6('+(+*@J38)2@(+736g036)*@*$ ''$CKE2$C738)$1036)*d=(+=6)7,?(0365+*2|$',?(+=2 76)*d&Q$'=1-32@(+736

σ2p(t) = σ2

p(0) + 2Dp t M

#" 6="J$C852(+736 + :&5(+7,?265736n03J36 ,<6)KE=>2@732 2* -3K<2@=>( nK<2&5(+7303J&)2?Q2*6>$C*@*7" 6)=2D<DE2(+76D0- (+7,?$C736Q$X6;D<2 ,<K<( ''$ D*-%6)DED<(pDE&)K<2-,<,<(X27,<6)KE=>2@732t032 ,?$CD<DE(X032H02EA J3D<2@(+736u032 (C,<(+732 46 2@7Z$',E,?2HDE2H,?2@6)736;&)(+7,?( &/A36 2@* "J$C03K<$',?(p06)*F=(+=>657,?( ,<K?$CD 6)K<2-,?(

h2k2L

D<2FA$2@=>=6)032$',/$'=>657",<6H&/A36

Γsc =2Dp

h2k2L

=Γ

2Ωs =

1

h

Γ

∆VDIP =

3πc2

2hω30

(Γ

∆

)2

I C

03( 'C6 736)*@*7" J3*@,<2=( -$CDED?$C+C2(D<2 HJ3D<$',/$x*$g (CK<=>$0365*-%(C,<6)73852$C*@6 02?-%(+*$CKE6t27 &5J32%&)(+=%-$'K<6g6)D0-3*2@&)2@,?$C=6)7,?6 * " 2@7,?6)7D<2@,R + k* D<65&)(+730(X,?6)KE=>2@736u036)*@*7" 6="J$C852(+736 + &5(+7,?2@6)736 2@79'+6)&56 *7" 27 (+K<=>$C8)2@(+736 DEJ3** " 6j % 6j,<,?(n06)** " $'D<D<(CK*Q32@=>657",<( 6 0365**7" 6)=2DED<2@(+736DE,?2@=>(C*$',?$( ' '+65K<(u036)*@*6 3J,<,<J$C8)2@(+732036)*@*$-3KE6)DED<2(C736x032K<$C032$'8)2(C736F6:06)**$1 (+K<8)$>032032E- (+*@( ]65K J7F" (C730$ D ,/$C852(+7$'K<2$"J365DE,<(DE6)&)(C7303(>,<6)KE=>2@736F$CDED<J3=6g*$1 (+K<=>$ +

∣∣∣∣∣∣∇r Ω

(LAT )0

Ω(LAT )0

∣∣∣∣∣∣

2

= k2L tan2(kLz)

%;,"*I47.oH%0*<4&%0$'7",.$HQy$1,.2;14!)"2=$_#FT0*`T&*-4!52;!%X7"$_4)HQS1"*(+*<454)HQy,"*IN9&*<%06IH(0!q1O0!0E145!+**7"7"!i(+!W0,.*-.'!5$1%0*`(+!:,F1(+!)6-!$%0*r%0$%=#<$12;W&1!$%0$&U[23m*I""!+'$%0$A7",F1.#90,FO0!545!D"* 45$G 45$'Wi$'7.12=*-%D7"$m$W0W045*<,?0[

8 W M9 P ^ ;:ZP# <= P ^P

= Q 6503J36F2|&5(+7,?K<2EQ3J,<2d$C*|&5(6>&56)7,?6032|02EA J3D<2@(+736 DE(+73(032?-%6)7036)7,?2d03$C**$ -%(+D<2@8)2(C7360365**7" $',<(+=>(x736)*@*7" (+730$1D ,/$C852(+7$'K<2$B=>$1*$ *(+KE( D<(+==>$173(+7b*( C

2Dp =h2k2

L

2Γ Ω(MAX)

s cos2(kLz)(1 + tan2(kLz)) =h2k2

L

2Γ Ω(MAX)

s

C

LN(+=6 D<2 6'27&)6 0$'**$ + kBKE2&5(+K<0$'7303(1*$x036 3732852(+736 032Ωs

B2@*%,?$CD<DE( 032 032 A JD<2(C736032 (C,?(+72 12@7 'C6)KED?$C=6)7,?6-K<(-%(+K<852(+73$C*6:$C* "J$'03K?$',<(n036)*j 6 032@K<6j,<,/$'=>657",<6-3KE(- (CK<8)2@(+7$C*@6H$'**7" 27,<6)73DE2@,SR J36)D ,?(1R1D&/A36$-$CKE2@,SR:032|$C*-,?658)8Q$106)*|KE65,?2@&)(+*@(

I/∆DED?$',<( $'J3=>657,/$CKE62@*tjZ K<2@03J3&Q$b2*K<2DE&Q$C*@0$C=6)7,?(b6 "J327032$CJ3=6)7,?2*$ '2-,/$=6)032$1036)*d&5(+73036573D?$',<(3

8

$ &)(+= - (C736)7,?6 '2DE&)(+D<$B3&52( F032?-%6)7306)7,?6g0$C*@*$ 'C6)*@(&52@,SRB0365**$x (CK<8Q$ 65D<6)KE&)2-,/$',?$03$C**$'D<6)K]DEJ3+*@2d$',?(C=>2F-J"6)D<DE6)KE6x&Q$'*&)(C*$',?$1$-$CKE,<2KE6F0$ .M

F =dp

dt=i

h[H, p ] = −∇rH = −∇rHINT

'

46n*$ '+6)*@(&52@,R $',?(+=2&)$B&5(+=>6 $.QQ32$C=>( - (+D ,?J3*$',?(BD<(03032D Z$ $C*NKE6="J32DE2@,<( jBU2@*''$C*(CK:=6)032@(;032,/$C*@6 (CK<8Q$ D<2NDE&)(+= -%(+736 2@7X03J36 -$CKE,<202N&)J2NJ37$ &5(+73DE6)K0''$',<2E''$;+2?R,?K<$',<,?$',/$27 -3K<65&)65036)738)$n6)0 J373$>K<2@+(+KE(+D?$C=6)7,<6 '2DE&)(+D<$B%&)2( *276Q$CKE676)**$ 'C6)*@(&52@,SR#" 65D*-3KE6)DED<2(C736]K<2@D<J3*-,/$C7,?6 - 65K*$ &)(+= - (C736)7,?6 '2@D<&5(+D?$ 'B$C* -K<2=(H(CK<032@736NDE2+732 &Q$I,?2E'C(27

Ωs .MBC

fv = hk2 δ

Γ(Ωs/2)3 v

5 /K?$CDE&)J3K<$CK<6F,/$'*6F&)(C=1-%(+73657",<6 '2@D<&5(+D?$ g*6)&52@,<( D<6

fv

fl=

fv

∇rVLAT 1

&)2@( uD<6 *$ (+K<8)$ 6)DE6)KE&)2@,?$',/$ 0$C* K<6j,?2&5(+*( fl

=>(+*-,?( D<J - 65K<2(CK<6 $'**$ &5(+=1-%(+73657,?6'2DE&)(+D<$4(+D ,?2-,?J3657303(*7" 6)D*-K<6)DED<2@(+736 D5 -%6)Kt*$ &)(C=1-%(+73657",<6 '2@D<&)(CD?$6:* " 65D*-3KE6)D<DE2(+76 N -%6)Kt2*- (C,<6)73852$C*@6F032|KE65,?2@&)(+*@( D<2d(C,E,?2@6)736'B3K<2@&)(+KE0$C7303(* " $.- -3K<(+DED<2@= $C852(+76

fv

fc Ω3

s

.N

:^ %YRMYO)T XRP %O)M*FPZYOQMPZY YM3:NW M9 PZ^ : PZ^ 8

H2&)(+7D<6)CJ36)738)$n736)*@*7" $.--3KE(+D<DE2=>$C8)2@(+736F032D?$',<J3K?$C852(+76,<K?$CDE&)J3K<$.Q32*@6 x*6)&52@,<(,<K?$CDE&)J %K<$CK<6$C73&/A6C*2|6(A 6j,<,?2F'2@D<&)(CD<2d0365**$x (CK<8Q$ 032d02?-%(+*(3

_%ipqrx`bmnst`baeFcr mnh vwrxadsNy3zh5`ba|cs]h)` `ns]iXsch)qr:ijr

LN(+=6t6)=6)KE+6t0$C** " $'7$C*2@D<2"D0'C(+*@,?$F736)*<-3KE2=(F&Q$.-32-,?(+*@(3BC*$F&Q$CK<$',<,<6)K<2@DE,<2&Q$ -%6)&5J3*2$CK<6]06)*&5(+73036573D?$',<(>02GN(+DE6(%J@2@73DE,<6)27 "J365**$1032|&)(CDE,?2-,?J32@K<6 J37bD<2D ,?65= $1=>$C&)KE(+D<&5(-32@&)(3B&)2( &5(+DE,<2@,?J2@,?(b0$b=(+*@,<6-$CKE,<2&56)**@6CB|02&5J32UD<2#-3J";036 732KE6J37$n J3738)2@(+7360F" (+730$u&/A36D<265DE,?6573036uD<J3*@*$ DE&Q$C*$ 06)**@6 0365&)276u02]=>2@&)K<(C=>6j,?K<2 J6)DE,<( &5(+= - (+K ,/$ &/A36'BNKE2D*-%65,E,?($C0pJ7pK<65,<2&5(+*( (C,<,<2&)(;$C** " 65DE,<K<6)=(;KE(+D<DE( 036)*@*( D*-%65,<,<K<( '2DE2?Q32@*6

λ/2 ∼ 400nmkB2*

&5(+73036573D?$',<(D<2 65DE,<6)73036gDEJb$C*&5J3736H&56)7,?2@7$C2$x032 DE2@,<2 KE65,?2@&)(+*$CK<2 J6)DE,<(&)(+KEK<2D0- (C73036J373$:*$'K<+A3658)8)$:7365**(gD0-$C8)2@(x0365+*22@=1-J3*DE2=>(+*-,?( -32@&)&)(C*$F6 ,<K?$CDE&)J3K<$.Q32@*6t27>&)(+7 KE(+7,?($C*@*7" 2= -3J3*@D<( 032KE273&5J3*( D?$C703(;*$ C 6 -3K<6573036)703( &)(C=>6''$C*(CK<2,<2?-32@&)2

RTF '50µm

6λ ' 800nm

D<2dA$27Z$',<,<2

∆p

hk=

√21

8

h λ

hRTF' 0.3

λ

RTF' 4 × 10−3 .!

J365DE,?203J361Z$',E,?2&5(+73DE6)7,?(+7(u02,?K<$',<,?$CK<6x27 -3KE2=>$ $.--K<(+DED<2=>$C852(+7362*&)(+7036)73D<$',?(7365*UK<65,<2&5(+*(n&5(+=>6 J7F" (C730$ -32$C73$b2@=>=6)KED?$ 2@7 J37 -%(C,<6)73852$C*@6- 65K<2(032@&)(3 J365DE,<( ,?$C7,?( -32 '+65K<(D<6 *$F032KE6)852(+736]2@7&)J32 -3K<65D<657",<6H2@*3K<6j,?2&5(+*( "J36)*@*$F=6)73(g&)(C77$',?$2@7"J36)D ,/$ 032@K<6)852(+762@7Z$',<,<2"*7" 6)DE,<6)73DE2(+76D*-$'8)2$'*6036)*&)(+7036)73D<$',?( N= $CDED<2@= $t6 "J32@73032=273(CK<6 *$*$CK<CA36)858Q$ 736)*@*( D0-$C8)2@(036)+*@2|2= -3J3*@D<2

(!

465+J3657303( *7" $.--3KE(&5&)2@( 0365*276Q$',<( 7365**7" 27,<K<(03J38)2@(+736b2* D<2@DE,<6)=>$ &)(+D ,?2-,?J32-,?( 03$C*H&5(+7 %036573D?$I,?(p032 GN(+DE6(%J@2@73DE,<6)27736)*@*( D ,/$',<( (+7303$C=>657,/$C*@6 0365* -%(C,?65738)2$C*6 $CKE=>(+72&)(i>62@=>=6)KED<( 7365*dK<65,<2&5(+*( (C,E,?2@&)(3BK<2657,?K?$ 7365**@6F2?-%(C,?65D<2|0365*|,?65(+K<65= $ 032HGN*@(&/A .5 42<-%(+D<DE(+73( - 65KE,?$C7,?(036 72K<6 J3727302&)6]02;Q3$C730$g032DE&)KE65,?(

n6)0J37 2@=1-3J*D<(FKE6)*$',?2 '+(

L %W&,.7"!)#<;'!8#<$1%&"!)(+*<,F45$K'7.7.$=Qy$1%&(012;*<%D7.14*H!%]1W0W0,.$.'!5236<!5$1%&*A(+!^`T&$1231'acb&*<,.2;!?

8 W M9 P ^ ;:ZP# <= P ^P

$C* K<65,<2&5(+*(3B"036j,<,<(J3$CD<2 %2= -3J3*@D<(3Bhq

&)(C7",<27"J3( 6 -%6)KE2(032&5(x032- 65K<2@(03(2πh/(λ/2) =

2hk 65K>=6)858)( 032 "J36)D ,?6 "J$C7,?2-,SR *6b$CJ,?(C J378)2(C7320365*]D<2D ,?65= $BU036j,<,<6pI%u

t B3DE2- (CD<D<(C73(D<&)KE2E'C6)K<67365**( D0-$C852(K<6)$C*6F&5(+=>6

〈z |n, q〉 = eiqz un,q(z) K

03( '+6u J37$ J738)2@(+736 - 65K<2(032@&Q$ 032H- 65K<2@(03(

d = λ/26:DE(+*J8)2(C736g036)*-3K<(.Q3*65= $ 032

4&/A3K020273+65K

(h2

2m(−i∇z + q)2 + VLAT (z)

)un,q(z) = En,q un,q(z)

.+

G 65*DE6)+J2@,?(>27302&/A365K<6)=( &5(+7hqB = hk

2@*"J$'D<2 % 2= -3J3*@D<(&)(+KEK<2@D*-%(+7303657",<61$C*Q%(+K<03(0365**$8)(+73$ 032BGK<2*@*(+J27| LN(+=6xDE6)+J6 0$C**$ K jB%*(>D*-%65,E,?K<(n06)**@6:(C730361032 GN*(&/A7365**( D0-$C852(036)C*2|2@=1-3J*D<2 g032DE&)K<6j,?(6gD<2-%(C,?K*R D<&)KE2E'C6)K<6 N

|n, q〉 =∞∑

j=−∞

an,q | q + 2jk〉 .M

03( '+6 | q + 2jk〉 2@73032&)$n*7" (+730$ -32$'7$exp(−i(q + 2jk)z)

J6)DE,?$ -%6)KE2(032&52@,SR 76)* %*(D0-$C8)2@(>06)+*@2|2= -3J3*@D<2F-3J" $C7&/A36:65D<DE6)K<6 '2DE,?$&5(+=6F&)(+73DE6)+J6)738)$ 036)*@*7" 27,?65K?$C852(+76&)(6)KE6)7,?6n,<K?$ +*2N$',?(C=>26)0 2]03J36 Z$CDE&)2]*$CD<65K *@( DE,/$I,?( 036)C*2$I,?(+=2 bJ37$ DE('K<$.- %-%(+D<2@8)2(C736 &)(6)KE6)7,?6032032 '+6)KED<2UDE,?$',?2U0322@=1-J3*DE(n0365,<6)K<=273$',?203$C*Z$',<,<(b&/A36 * " $',<(+=(-3J":&)(+7,<27"J$C=6)7,?6 D<J Q32KE6 J371-3K<(&)65D<DE(1$C*D<65&)(+703(1(+KE032736t2@7>&5J32%$CD<DE(+K*Q%6 J7> (C,<(+7360$ J37( 036)20J36 Z$CD<&52N&5(+DE,<2@,?J6)7,?22@*KE65,<2&)(C*(;6n*@( K<2@6)=65,E,?6 76)** " $'*@,?KE(3 6)K ,/$C7,?( 2@*D<2@DE,<6)=>$$I,?(+=2&5( D<23,<K<( ''$2@7J7$FD<( 'K?$.--%(+DE28)2@(+736t&)(6)KE6)7,?6t032D ,/$',<2032=(+=>657,?(F&/A36032 A 65K<2DE&)(+7(02J37$ J3$C7,?2@,RF032=>(',?( -$CKE2$

2hkB/(+KE736)KE6)=(DEJ "J365DE,/$$C7$'*(++2$ $C*

-$CK<$C+K<$' ( 53 "J$C703( 036)DE&)KE2E'+65K<65=>( 2@*%=65,<(0(xJ3D<$',?( D0- 65K<2=6)7,?$C*=6)7,?6 -%6)KN,?$CK?$CKE6*7" $C*-,?6)858Q$10365*dK<6j,?2&5(+*(

+732F-%6)K ,?J3K0Q$C8)2@(+736F65DE,?65K<7$B273&5*J3DE6 'C$'K<2$'8)2(C732 76)*-%(C,?65738)2$C*6H$CK<=(+732@&)(3BD<J %&)2@6)7,?65=>657,?6036=Q%(+*20$n6)D<&5*J306)K<6,?K<$C73DE28)2@(+7322@7,?6)K0Q$C7303$ -%(+DED<(+73(u6)DED<65K<6>,<K?$',E,/$',<6

pA9&45!7.7.!@12=*-%D7"*;90%k#,.!n7.*<,.!$VG3#FT0*;45V@,.!516<!5$1%0*=!5%e*-%0*<,.E1!)"904545.#<145V(+*<4Y#<$1%&(0*<%&.7.$V"!5W0!)#<#<$14)/,.!)'Wi*7"7"$34EDW7",F/4)=W0,.!23=*_45;"*-#<$1%&(0/O&1%&(00

:^ %YRMYO)T XRP %O)M*FPZYOQMPZY YM3:NW M9 PZ^ : PZ^ 8 >

2@7b$.--3KE(+D<DE2=>$C8)2@(+736HDE6)=2&5*$CDED<2&)$ .5 7 "J365DE,<(&5(+7,?65DE,?(*$ '+6)*@(&52@,Rx036)* &)6)7,<K<(032=>$CD<D<$06)*d&)(C73036)7D?$',<(>KE2D*-%65,E,?($C*|KE65,<2&)(C*( g0$',?$03$

vn,q =1

h

∂En,q

∂q

+

=6)7,?KE6H2@*&)(+7306)73D<$',?( D ,?6)DED<(1DE2 &5(+= - (+K ,/$gK<2@D*-%65,<,<(1$C*@*$ -%6)KE,<J3K*Q3$C8)2@(+736 $C+6)7,<6H*J73+(* " $CDED<6g0365*dK<65,<2&5(+*( &5(+=>6FDE6x$ 'C6)D<DE6:J7$ )n </F0$',/$ 0$

m∗n,q = h2

(∂2En,q

∂q2

)−1

LN(+7 "J36)D ,?6u036 3732852(+732 BU27 $.--3KE(+D<DE2=>$C8)2@(+736 DE6)=2&5*$CDED<2&)$B2*N=>(C,<( 036)* -3$C&)&/A36j,<,<(0F" (+7303$032HG*(&/A -J"6)D<DE6)KE6xK<2@&Q$ ''$',?($-$'KE,?2@K<6F0$'**6H6="J$C852(+732dDE6)+J6)7,?2 .5

z = vn,q

hq = Fz

03( 'C6Fz

27032&)$1*$x (+KE8Q$ 6)D ,?6)KE7$>&/A6:$'+2DE&)6g&)(+=6 -%6)KE,<J3K*Q3$C8)2@(+736C

(' V !

$x&5(+=1-%(+DE28)2@(+736 036)*@*$F J373852(+736H0F" (C730$x0365* &5(+73036573D?$',<( 27,?6)KE=>2@732032 (C73036 032GN*@(D%&/A -3J" 65D<DE6)K<6n(',<,?657"J,/$ 0$'**$ .M g&)(+7D<206)K?$'7303( &)(C=>6>+2 R;065,<,<( 2*&)(+7036)73D<$',?(&5(+=>6J37 " (+70$ -32$C7$>H2U&)(C73D<65+J36)78Q$B$'&)&)657303657303( 2@=1-3KE( '3'2@D?$C=6)7,?62@*K<6j,?2&5(+*(B2@* &5(+73036573D?$',<( D?$CKE6=QQ%6 -3KE(+26j,<,?$',?(:D<Jn(+7306 032GN*(&/An$.--3$CKE,<6)73657",<2|$:,<J,<,<6 *6 Q$C7306CB0$I,?(&/A36F7365**$ .M 7(+7b&)(+= -$C2@(+73(1KE6)+(+*@6 032|DE6)*@6)8)2@(+736 -%6)K

n J36)D ,?(K<6)7036)KE6=Q %

Q%62= - (+DED<2EQ32*@6xJ373(uDE,<J3032@( DE2DE,<6)=>$',?2@&)(n0365**@6-3K<(-K<26j,SRn0365*D<2D ,?65= $b6 J27303206'+665D<D<65K<6:6'2@,?$',?(0$C* -3J37,?(032F'2D ,/$ D*-%6)KE2=6)7,/$C*@6C6)K -3KE6=-$'K?$CKE6 2*t&)(C73036)7D?$',<(p27\J37D<27+(+*( DE,?$',?(X02 GN*@(&/A6)D<2@DE,<(+73(X03J6 032E'C6)KED<2$.- -3K<(&)&52 2*-3KE2=( :&5K<6Q$'K<62*d&)(C73036)7D?$',<( 032KE65,E,/$C=6)7,?6g7365*H- (C,<6)73852$C*@6F&)(+= Q327$I,?($CKE=>(C732&5( 6-%6)K<2@(02&)(Z&j K):$C0 6)DE6)= -32( j LN(+DZ$C&56)7303(b2@7Z$',<,<22@*DE2D ,?6)=>$ -%6)K&5(+DF032@K<6 73$CD<&56b7365**@( DE,/$I,?( (+730$C=6)7,?$C*6n&/A36 $.--3$CKE,<26)76b$C**$ -3K<2@= $ Q3$C730$ 032GN*@(&/A

! W M9 P ^ ;:ZP# <= P ^P

G 65+*265D*-%6)KE2=6)7,?2U(++C65,<,<( 02 "J365DE,<( *$ '+(CK<(u279'+65&)61'2@6)736J,?2@*2@8)8Q$I,/$ J37F" $C*-,?K?$ -3KE(D%&)6503J3K<$2* &)(+7306)73D<$',?('265736 -3KE(03(',<,?(x736)*F-%(C,<6)73852$C*@6H$CKE=>(C732&5(x6 "J327032 2@* KE65,?2@&)(+*@('2657361$C&5&)65D<( 2@7 J37 ,?6)= -%(

δt 46g,/$C*@6,<6)= - (D<(03032@DEZ$$C*K<6="J32@D<2-,?( 02$'032$.Q3$',?2@&)2@,R

!

∣∣∣∣∣〈n, q |δV

δt| 0, q〉

∣∣∣∣∣1

h∆E2

n,q

C

03( '+6∆En,q

27032&)$X*$X032 A 65K<6)78Q$En,q − E0,q

BHDE2 A$p&/A36 *$ -K<(Q$DQ32*@2@,SR &/A36 J37$ K?$'8)2(C7361032B-%(-%(+*$'8)2(C736 'C6)73$n,?K<$CDE 65K<2-,/$ 7365**( DE,/$I,?( |n, q〉 1,<K?$CDE&)J3K<$.Q32*@6161,?J,<,/$*$ -%(-%(+*$'8)2(C7361D<2,?K<( 'C6)KRu736)*@*7" (+730$ 02#GN*(&/A | 0, q〉 4(0302DEZ$'K<6 $n,/$C*@61&)KE2@,<6)K<2@(n032$C032$.Q$',<2&52@,SR &)KEJ3&)2$C*6 -%6)K +*2d65D*-%6)K<2@=>657,?2K<2E- (+K ,/$',<2|27 "J36)D ,?( *$ '+(+KE( 6D<J 03265D<D<(,?(CK<7365K<6)=(D<J3&5&)65D<D<2 ''$C=>657,?6C#2@D<2&)$C=6)7,?6 "J36)D ,/$ -3KE(&56)03JK?$1&)(CK<K<2@D*-%(+73036 $C0 J7$x036j (+K<=>$C8)2@(+736t*657,/$x0365**@(:D0- 6j,0%,?KE( 0365*tD<2D ,?65= $ &/A36 -3$CD<D<$ 0$ "J365**@( 032t(+70$ -32$C7$

Eq = h2q2/2m$ "J36)*@*( 032

(+7303$032HG*(&/AEn,q

LN(C=>6 FZ$C&)2@*6FKE6)730365K<DE2&5(+7,?((+D<DE6)K&'C$'7303( *@(D*-%65,<,<K<(032GN*(&/A7365**(DE&/A36)=>$&5(+D<2@03036j,<,?(0328)(C7$KE2?-%65,<J,/$ K<2E- (CKE,/$I,?( 27CJ3K?$> B2*|DE2D ,?6)=>$'+65K<KR,?K<$CDE 65K<2-,?( 7365**( D ,/$',<( | 0, q〉 D<6

q < qB27 "J36)*@*( | 1, q〉 DE6

qB < q < 2qB6&5(+D '2$

7K<6Q$'*@,SRB9-$CKE,<6)730(x0$gJ37F" (+730$gDE,?$C8)2@(+7$CKE2$BC*7" J3732&5( 'C$'*(+KE6]032<J3$CD<2 %2= -3J3*@D<(F$'&)&)65D&%D<2EQ32*@6

q = 0&)(C=>60365K<2E''$b2==>65032$I,/$C=6)7,?6:0$C*@*$ .M j /J,<,/$ '2$ D<J &)2@6)7,?6

K<2@&)(+KE0$CKE6 &/A36CBU7365*D<2@DE,<6)=>$ 032K<2- 6)KE2=6)7,?( D<(+*@20$C*@6>&5(+7 2*K<6j,?2&5(+*(;=>(Q2*6'B2@*&5(+7 %036573D?$',<( nK?$.--K<6)DE6)7,/$I,?( 0$ J37F" (+730$ -32$C7$ $ '+657,?6u=>(+=6)7,<(

p = −mvLAT

H2&)(C73D<65+J36)78Q$B 3D<D?$'7303( *$ '+6)*@(&52@,R 036)*tK<6j,?2@&)(+*@(3B - (+DED<2EQ32*@6 ,?K?$'DE 6)KE2KE6b$C032$DQ$',?2@&Q$ %=6)7,?6b*7" 27,<6)K?$ -%(-%(+*$C852(+76 $',<(+=>2@&Q$ 7365** " (+70$ 032 GN*@(&/A | i,mvLAT/h〉

03( 'C6CB-%6)K"J$C7,?(n065,<,<( D<(.-3K?$B

i 1*$1-$'KE,?6:27,?65K?$ 02

q/qBHGH$',<J3K?$C*@=>657,?627 -3KE(+D<DE2=2@,R0365*

Q%(+K<03(>,?K?$>03J36Q$C7036CB|0('C6 *$032EA 6)KE6)738)$ 27;65736)KE+2$,?6)7036$ 8)65K<(n&5(+7n

B|D<$CKR>2=%-%(+D<DE2?Q32@*6HDE(0032D Z$CK<6F$'* K<6"J32DE2@,?( 02d$C032$.Q$',<2&)2-,SRg6F032 &5(+73DE6)+J365738Q$ "J36)D ,?2|D ,/$',<2|73(+7D?$'K?$C737( $C&5&)65D<D<2EQ32*@2 0$'*-3J37,?(032F'2D ,/$ D*-%6)KE2=6)7,/$C*@6C

:^ %YRMYO)T XRP %O)M*FPZYOQMPZY YM3:NW M9 PZ^ : PZ^

+51> 0#FT0*<23K(+*-!P4!5@1*-445!R(0!Rs 45$:#FThWX*-,A90%eWX$17"*-%06<!)45*Wi*<,.!$+(+!)#$V#$1%s = 3.5

HYB *<%0*-,"E!5En,qG/#-4)#$14)7F;W0W0,.$."!23%X(+$;4*$%&(+*(0!Rs 45$:#FT]#<$12;*90%& '$2=233(+! 3$%&(+*_W0!)%0*S#!5$:G!5%&"*<,.*<%&(+$

jMAX = 4%0*-44)hC+ "*I(]G/#<$1,.,"*<7'7.;*-%D7",.$K4$V'Wi*-."$1,.*(+*<454*/4!5%0*-*13W0,.!52;K*4)37"*-,"6I3O&1%&(0

'$%0$A#<$145$1,F7.*Z!%/O049U45A'*I#$1%X(0*r4)mND9&1,'7F!5%,.$.'$X8q45!5%0*-q7",F7'7.*<EE1!)7.`G 45$A"WX*<7'7",.$qW&1,.1OX$4!)#$(+!PW&,"7"!)#*-44)K45!5OX*-,.&Ui4)K45!5%0*-$1,.!6-6<$%7F45*_!%0J&%0*3#$,",.!5"Wi$1%&(+*/454?B 47"*-6<6- (+*<4PWi$7"*-%06<!)45*Wi*<,.!5$:(0!5#<$& 4v,.*-ND90!)"!n7.$ (+!P1(+!)O&7"!)#!7.ofC+ D`GW0!]QS#!54*=(0 "$+(0(+!)QS1,"*4)1(&(+$@1*_%0$%h#!P"$1%0$345!@*<454!R@:!5#<!%&!RND90*<454$/#FT0*q'!X@:90$145*,.1E1E!90%&E1*<,.*1 m!i#$1%X'*-E190*-%06-!5407.,.Qy*-,"!52;*<%D7"$_1(0!51O&7.!5#<$HGmW0!KQS1#!545*q#*-%7.,"$O&%X(0V*=W&!l(+!K#!54*K14ZOX$,.(+$Xm*<4Y#-1"$V!54549&'7",F7.$!5%gJ&E19&,.V1(e*-"*<2;W0!5$WX*-,

q = 3qB

@*<%0E$1%0$WX$WX$457"*_%0*I#*I".,.!512;*<%D7"*H'!)=45/7"*-,"6I;'!)=45;ND9&,"7.=O&1%&(0&

<8 W M9 P ^ ;:ZP# <= P ^P

s h hpmnhph)`ba|cefbhji5hkcl

65Kg279'+65DE,?2@$CKE6* " (CK<2C2736:036)*@*7" 273D ,/$.Q32@*2-,SR (&5&)(+KEK<67365&)6)DED?$CKE2$C=6)7,<6&5(+73DE20365K?$CKE6* " 6I%"J$C852(+736u032 $FK<(+DED&%2-,/$C6('D0)"22 j $ DE,/$DQ32*@2@,SR 02t,?$C*6b6J3$C8)2@(+736 -3J" 65D<DE6)K<6270$C$',?$ 0$nJ37 -3J37,?(u032B'2D ,/$ (+KE= $'*6 $C0(C,<,?$C7303(nJ37 $.--K<(&)&)2@( - 65KE,<J3K*Q$I,?2E'C(b27$.--K<(+DED<2=>$C852(+736H*276Q$CKE6C@=>65K<+(C73(n&)(+D03J361,<2?-3202273D ,/$.Q2*2-,SR]J7$n032,?2E- ( 65736)KE+65,<2&5(u&5(+**@6)$I,/$ $.- -3J37,?($C*@*7" 6)7365K<+2$0365* DE2DE,<6)=>$x6 "J32@73032|$'**$gDE,<K<J,E,?J3K<$x0365**7" A$C=2*-,?(+72$C7$ 650nJ37$1032@7$C=2&)$0$',?$>0$6)&5&)2@,?$C8)2@(+732*$&)J2$C= -32658)8)$&5K<6)DE&)6165D*-%(+7365738)2$C*=6)7,?6:736)*,?65=1-%(6 "J327032*65$',?$n$C**@6x6="J$'8)2(C732032=(C,<(3 -3K<(.-3K<2@(n*$ -3KE6)D<65738Q$b0365**$ 7(+7 *276Q$CKE2@,SR03( 'J,/$$C*@*6]27,<6)K?$'8)2(C7323&/A36 0365,<6)KE=>2@7$F*$F032EA 6)KE6)738)$,<K?$ J6)DE,<203J36t 6)73(+=6)72 42;-J":2@7Z$',&%,?2F'C6)KE2 &)$CK<6 5 &/A36*7" 273D ,/$.Q2*2-,SRg03273$C=>2@&Q$12@=1-3*@2&)$ "J36)*@*$ 6)76)K<C65,?2@&Q$B=6)7,<K<6g73(+7''$C*6F2@*'2&56'C6)K<D<$

$ x 'R

7 $.- -3K<(+DED<2@= $C852(+76 032 G(++(+*@2(+JQ%( ' O+ jBF0$C*@*$pA$C=2*-,?(+72$C7$ OK jBF2@73&)*@J303657 %03(p2@* -%(C,?65738)2$C*6 - 65K<2@(032@&)(3BND<2 -J" &)(+D ,?K<J2K<6 2*t J373852(+7$'*6;65736)KE+2$

E[Φ] H6j,<,/$

Φ0(z, r⊥) exp(iµt)*$ D<(+*@J38)2@(+736n032t&5J32tD<2'J3(+*@6uD ,?J302$CKE6u*$ DE,/$DQ32*@2@,SRB2@=1-%(+73657303(

J37$ ''$CKE2$C852(+7606)**$ J378)2(C736x0365*,<2?-%(Φ = Φ0 + δΦ

B D<2(C,<,<265736xJ37 " 65D*-$C7D<2(C736 032@\$C* D<6)&5(+7303(>(+K<02736F2@7

δΦ

E = E[Φ0] + δE1 + δE2 D5

03( '+6δE1

N7"J3*@*( D<6Φ0

ND<(C*J3852(+7360365**7" 6="J$'8)2(C7367(+71*2@736Q$'K<6# k$ ''$CKE2$C852(+76

δE2

J37 J738)2@(+7$C*@6J3$C03K?$I,?2&5(b&/A361-J" 65D<DE6)K<6D<&5K<2-,<,?(u27 (+KE= $ &5(+= -$',<,?$ &)(+=6 M

δE2 =1

2

∫δΦ†AδΦ dr

9N 03( '+6D<2dDE(+73(27,?KE(0(C,<,<2 2*'+65,E,?(+KE6

δΦ =

δΦ

δΦ∗

.!

M %PXPURPNP YO TP: P

6g*$ =>$',?KE2&56

A =

L U0Φ2

U0(Φ∗)2 L

9K

2@7u&5J32$D<J$'+(C*@,/$ L 27032&)$

L = − h2

2m∇2 + V (r) + 2U0|Φ0|2 − µ

+ 61*$1"J$C7,<2@,SR

µB &/A61A$ 2@*DE2+732 &Q$I,?(032 -%(C,?65738)2$C*6&/A32@=>2@&)(036)*D<2D ,?65= $B;-3KE('2@6)736

0$'*F'2@73&)(C*( 032d7"J3=6)KE(02$',?(+=2 3D<D?$I,?(3$;=>$',?KE2&56

A >A36)KE=>2-,?2$C7$b6 "J327302A$ D<(+*@( $CJ,<(''$C*@(+K<2UKE6Q$C*@2 $ &)(+73028)2@(+736032

D ,/$.Q32@*2-,SR 65736)KE+65,<2&)$ &/A36n,?J,E,?2 "J365DE,<2t$CJ,?( ''$C*@(+K<2D<2$C73( -%(+D<2-,?2E'2 "J365DE,<( 2@=1-3*@2&)$&/A36 *$F&)(C7+J3K<$C8)2@(+736tDE,?$C8)2@(+7$CKE2$FD<&56)*-,/$ J373$&5(+7+JK?$C852(+736 032=2732@=>(H036)*@*7" 6)7365K&%+2$ 46 '2@&)6'C6)KED?$;65D<2D ,?(+7( $CJ,<( 'C$'*(+KE2736)+$',?2 '2 B$C**@(+K?$ 65D<2D ,?6 - 65Kg*(uDE,/$I,?(

Φ0

J37$-%6)K ,?J3K0Q$C8)2@(+736g&/A36g&)(C7303J3&56 'C6)K<DE( D ,/$',<2$=27(+K<6 65736)KE+2$ 6 "J327032-32bD ,/$.Q32@*2 #" 6="J$C852(+736t032$gKE(+D<DJ%2@,?$C6'D0)2@2,?J,<,/$ '2$B+036)DE&)KE2E'+6 -%6)K2E- (C,<6)DE2J37$F6('+(+*@J38)2@(+736]J3732 %,?$CK<2$ 2@7x&)J2DE(+73( &5(+73D<65K0''$',<2D<2$ 2@*7J=>65K<( 032$',<(+=>2

NB+DE2$ * " 65736)KE+2$

EG 6&)(+73DE6)+J6

&/A36*$x,<6)(+KE2$ 73(+7 -3KE6'C6)036,?K?$'73D<2@8)2(C732'C6)K<DE( $C*-,?KE2 DE,?$',?2d&/A6 -%(+DED<(+73( '+65K<2 &Q$CKED<2dDE(+*(2@7 -3K<65D<6)78Q$032 6)7(+=>65732 02D<DE2?-$I,?2E'2 H2|&5(+73DE6)+J365738Q$2*%K<65+2=6 032 273D ,/$.Q2*2-,SRg6)7365K&%+6j,?2@&Q$ 032 &)2*@=>657,?6N=(036)*@*2858Q$.Q2*60$C* -J37,?(g032 '2@DE,/$H,<6)(+KE2&5( D<6]73(C7 D<23&5(+73(+DE&)(+73(65D?$',E,/$C=6)7,?6g2d=6)&5&Q$C732@D<=2|032@D<D<2E-$',<2E'2 &5J32d2*|DE2D ,?6)=>$ gD<(C,E,?(-%(+DE,<(3

$ 8 F

6)K ''$C*J,/$CKE6;*$ DE,/$DQ32*@2@,SR 032@7$C=2&Q$ 036)*HD<2D ,?65= $ &52( ;*$ &Q$.-3$C&)2-,SR 0365* D<2@DE,<6)=>$ 032KE6)DE,?$CK<6 2@7J72@7,?(+KE73(p032FJ37$p&5(+7+JK?$C852(+736 02 6="J32@*2?QK<2(X$ DE6)+J2@,?(p02 - 65KE,?JK&%Q$'8)2(C732032E- 657303657",<20$'*,?65=1-%( (&)&5(+K<KE6-$CK ,?2KE6x0$'**7" 6="J$'8)2(C736x032B$gKE(+D<DJ%2@,?$C6'D0)2@2032E- 657303657",<6;0$'* ,<6)= - ( k42$

Φ0(z, r⊥; t)J37$ D<(+*@J38)2@(+736b032t,/$C*@6u6"J$C8)2@(+736

6&)(+7D<206)K<2$C=(nJ37$ - 65KE,?JK*Q$C852(+76δΦ(z, r⊥; t)

#" 6="J$C852(+76 -3J"u6)DED<6)KE6 $"J365DE,?(1-3J37,?(*2@736Q$'K<2858Q$',?$1&)(+7uJ373(D0'2@*J--%($C*F-3KE2=( (+K<02736F7365**$ -%6)K ,?J3K0Q$C852(+736'#" 6="J$C852(+736g032d=(C,<( &/A36:DE2d(C,<,<265736 M

ih∂ δΦ

∂t= σz AδΦ

M

W M9 P ^ ;:ZP# <= P ^P

03( '+62|DE2=Q%(+*2 D<(+7(>+*@2 D ,?6)DED<20365*d&Q$CDE(1-3KE6)&56)03657",<6x6σz

2@73032@&Q$1*$1=>$',<K<2&56F032d$CJ3*2

σz =

1 0

0 −1

$ = $I,?K<2@&)6σz A

73(C7 A6)K<=2@,<2$C73$>6"J327032B-3J"b$ 'C6)K<6$CJ,?( ''$C*(+KE2&5(+= -3*65D<D<2 142$

λ = h(ωr+iωi)J37( 032,?$C*2$CJ,?( ''$C*(+KE26

δΦλ(z, r⊥, t)2@*&5(+K<KE2D0- (+7036)7,?6N$CJ,<('C65,E,?(+KE6C

#" 6"J$C8)2@(+736 M -3J"6)DED<6)KE6xDE&)K<2-,<,?$ -%6)K,?$C*6 -%6)K ,?J3K0Q$C852(+736F&5(+=6

ih∂ δΦλ

∂t= λ δΦλ

5+

&/A36D<2dKE2D<(C*E'+6 Z$'&)2*@=>657,?6 (C,E,?65736)730(>&5(+=>6FDE(+*J8)2(C736

δΦλ(z, r⊥; t) = δΦλ(z, r⊥; 0) exp(ωit) exp(−iωrt) 5

42,-3J" 032=(+D ,?K?$'K<6 5 &/A36 DE6>65D<2@DE,?6>*7" $CJ,<(''$C*@(+K<6 &5(+= -3*65D<D<(λ

$C**@(+K?$ 65D<2-,?6>$C73&/A362*D<J( &5(+= -3*65D<D<( &5(+732@J3$',<(

λ∗ J365DE,?( A$ &5(+=6 &)(+7D<6)CJ36)738)$ &/A6CBD<6n6)DE2D ,?6nJ37

$CJ,<( 'C$'*(+KE6 &)(+= -3*65D<DE( ωi 6= 0

jB$C*@*(+K<$ 6)DE2DE,<6>J7$ -%6)K ,?J3K0Q$C8)2@(+736*$n&5J32$C=1-26)858Q$&)KE6)DE&)6165D*-%(+736)78)2$'*=6)7,?6

ωi > 0 -%(+K ,/$C730( 2*DE2DE,<6)=>$>*@(+7,/$C73(0$C*@*$D<(+*@J38)2@(+736032

6="J32@*2EQ3K<2@(3*DE2DE,<6)=>$ "J327302 %< j j HDE6H*@(:D0- 6j,<,?KE(x032σz A

&5(+7,?26I%736gD<(C*($CJ,<(''$C*@(+K<2 K<6)$C*2

ωi = 0k

LN(+7,?K<$CK<2$C=6)7,?6n$'*&Q$CDE( -K<6)&56)03657,?6u032]273D ,/$.Q2*2-,SRb6)7365K<+6j,?2@&Q$B "J36)D ,?2]=(03203273$ %=2&Q$'=>657",<6 27DE,/$DQ32*@2-%(+D<DE(+73(6)DED<65K<6036)DE&)K<2-,<,<2|736)*@*7" $C=Q32-,?(:036)*@*7" 6="J$C852(+736H032$gKE(+D<DJ%2@,?$C6'D*)"2@2 0$',<(1&/A36 "J365DE,?6g65&)&)2-,/$C852(+72 DE(+73(1&5(+7,?6)7"J,<6g736)*@*7" 6="J$C852(+736 DE,<6)D<D<$,/J,0%,/$ '2$ "J$C7303( * " $'=1-32@6)858Q$b032 "J365DE,<2N=(032U&5K<6)DE&)6n* " $.- -3K<(+DED<2@= $C852(+76 *276Q$CKE6 .M 73(+7 -32 ''$C*@20$16 "J32@73032|2@*|=>(0365**(x0$ 73(+2$C03(',<,/$I,?('265736$&)$C036)KE6CLN(+=6g+2 R 6'2@036)73852$',<(27 $C*&5J3732d*$ 'C(+K<2 ,<6)(+KE2&52 +BMB<5 J6)DE,?$ 27DE,/$DQ32*@2@,SR03273$ %=2&Q$ 73(C7 65D<2D ,?67 - 65K 2@*|&)(+7036)73D<$',?(>27 $CD<DE6)738)$ 032-%(C,?65738)2$C*6 -%6)KE2(032&5(7 -%6)Kt2@*&)(C73036)7D?$',<( 2@7 K<6)C2=610322@7",<6)K<$C8)2@(+732,<K?$CDE&)J3K<$.Q32@*2 27 "J365DE,?(u&Q$'D<(3B&)(CK<K<2@D*-%(+7303657,?6$U0 = 0

B*$;=>$',<K<2&56σzA

27Z$',E,?2U032$C+(+7$'*66 K<6)$C*6 6)0 J37X 6)73(C=>6573(;,?2E-32&5(0365**7" 6="J$'8)2(C736H0F" (+7303$173(+7n*2@736Q$'K<6 0$',/$:0$C**$:&5(+=Q32@7$C8)2@(+736 036)*@*6 27,?65K?$C852(+732%6 0365*-%(C,?65738)2$C*6 -%6)KE2(032&5(3


F 8 F

6)Kt(C,<,<6)7365K<6FJ373$D ,?2@= $B$C7$'*2@,<2&)$:( 7"J3=6)K<2@&Q$B036)*@*$xDE(++*2$ -%6)K]* " 2@73D<(+KE+6)KE6 03652|03J36KE6)+2@=>2032273D ,/$.Q2*2-,SR (&5&)(+KEK<6t736)&56)DED?$CKE2$C=6)7,?6 27,<K<(03J3K<KE6 "J$C*&/A6 $D--3K<(CD<D<2@= $'8)2(C736-%6)K ''$C*J,/$CKE6F+*2 $CJ,?( ''$C*@(+K<2 036)*@*$1=>$',?KE2&)6

A 9K N6g036)*@*$1=>$',?KE2&56

σzA

@D<2@DE,<(+73( (+7303$C=>657,/$C*@=>657",<6>0J36 032 A 65K<6)7,<2N$.- -3K<(&)&52N$C* -3K<(.Q3*65= $x736)2*(+KE( *$ 'C(D%KE2032 J 6bG 2J 65BB5 A$C7373( D ,?J3032$',?(;2* -K<(Q3*@6)=>$;0365**$ DE,/$DQ32*@2@,SRu2+7(+K?$C703( 2+K<$C032 032|*2EQ 65KE,R:,?K<$CD0'C6)K<DE2 Z6J3$C8)2@(+736 032$gKE(+D<DJ%2@,?$C6'D*)"2@2 J3732@032=6)7D<2(C7$C*6 U650b2=%-%(+73657303(-%6)KN* " $'J,?(C J738)2@(+736t036)* &)(+7036)73D<$',?( J7$: (+KE= $F,?2E- (:(+730$x02 GN*@(&/A K k73DE6)KE6)7303( J6)DE,<( C)C ;736)*@*7" 6="J$C852(+736032 $FK<(+DED&%U2-,/$C6('D*)"22 jBD<2U(C,E,?6)7+(+73(0365**@6DE(+*J8)2(C732&/A36 -%(+D<DE(+73( 6)D<DE6)KE6 &)(+7D<206)K?$I,?6036)*@*6 (+7036>02 GN*@(&/A 73(+7 *2@736Q$CKE2 2@*N&)J2ND*-%65,E,?KE( 02?-%6)73036'B#-3K<(.-3K<2@( &)(+=6 7365*N&)$CD<( *2@736Q$'K<6CB0$ J37p27032&56 032 Q3$C730$ 6J37 "J$'D<2 % 2= -3J3*@D<(3B&)(C7",<27"J3(n6-%6)KE2(032&5( &5(+7 *$ - 65K<2(032@&)2@,R06)*KE65,<2&)(C*( KE6)&52?-3KE(&5(3$;032 A 65K<65738Q$; (+70$C=6)7,/$C*@6,<K?$ "J365DE,<6 (+73036 6 2*U*(CK<(;&5(+K<KE2D0- 6j,<,?2 '+( *276Q$CKE6 &)2( 2@7 $CDED<65738Q$u032U27,?65K?$C852(+732 # &/A36>C*2D ,/$',<2(',<,?657"J,?2U0$C*@*7"C)C K H7(+7 DE(+73(u*6J372&/A36FDE(+*J3852(+72%06)** " 6"J$C8)2@(+736 032$FK<(+DED&%U2-,/$C6('D*)"22%032?-%6)7036)7,?6F03$C*%,<6)= - (B=>$x73665D<2D ,?(+7( $C*-,?KE6b$C73&/A36u032]7$',<J3K?$ *(&Q$'*2858Q$',?$ ZDE(+*2-,?(+72&Q$ 9K B73(C7 &5(+=>=6)7D<J3K<$',/$&5(+7 2@*|K<6j,?2@&)(+*@( 65" ($ 'C6)7,?2H-%6)KE2(032&52@,SR:=1J3*@,<2?-3*$:032"J365**$06)*dK<6j,?2@&)(+*@(1(C,<,<2&5( M $ D ,/$.Q2*2-,SR 0365**6b(+7036;032 GN*@(&/A 73(+7 *276Q$CKE2'2@6)736;$C7$C*@2858Q$',?$ JD?$C730(XJ37$ - 65K&%,<J3K*Q$'8)2(C736 $ 'C6)7,?6b$uD<J$ '+(+*-,/$ *$ -%6)KE2(032&52@,R 036)*K<6j,?2&5(+*(u6&Q$CK<$',<,<6)K<2@8)8)$',/$u0$ J372@=1-3J*D<(

ke

K<2D0- 6j,<,<($C*|K<6j,?2@&)(+*@(3G 65** " $C= Q32@,<(:032"J365DE,<( (+KE= $'*2DE=>( '+6573+(+7(>&)$C*&5(D%*$',?2|7"J3=6)KE2&)$C=>657,?6F2d*2@=>2-,?2<-%6)K]*7" 27DE,/$DQ32*@2@,SR03273$C=>2@&Q$x650 65736)KE+65,<2&)$ 7$.--3KE(&5&)2(HD<2=2*@6U$ "J36)*@*(H02 J16]G 2@J DE6)+J32-,?( 03$F9;$C&/A3A(+*=uB+65,?&/A2&S)164=>2 %,<A M &/A6 (&Q$C*@2858Q$C73(H*$F*(+KE(g$',<,<6)73852(+76]D<J3*@*$ DE,<K<J,E,?J3K<$06)**@6Q$C7036 65736)KE+65,<2&/A36t6D0- 65&)2$'*2858Q$C7(1*( D ,?J3032@( 036)*@*$xD ,/$.Q2*2-,SR:$$'+*2%DE,?$',?2|&/A67(+7bA$C7373( J7u&5(+K<KE2D0- 6j,<,?2 '+(*@2736)$CK<6' 7 $.--3KE(&5&)2@(032 '+6)KED<( -K<6)7036 *@6 =(+D<DE6 0$'**7" 6="J$'8)2(C736 03214&/A3K*20327+6)Ku032DE&)KE65,/$ 9KBC+BD5 6NK<2&)$ 'C$ *(HD0- 6j,<,<K<(F0365**66)&5&)2@,?$C8)2@(+73227 $D--3K<(CD<D<2@= $'8)2(C736N032 GN(++(C*2(CJQ ( '(C,E,?65736)730( $C7&/A36:*@6F&)(+7032852(+732 - 65K]*7" 273D ,/$CJ3K<$CK<DE2|03652d03J36gK<65+2=2 032d2@73DE,?$.Q32@*2@,R

7p*2@736Q$u032 -3KE273&52?-32@( -%(+DED<2?Q2*6D0'C(+*C6)K<6 2*&Q$C*@&)(+*@(;7"J3=6)K<2@&)( 02K<6j,<,?$C=>657,?6n$

W M9 P ^ ;:ZP# <= P ^P

-$CK ,?2@K<6]0$C*@*7" 6="J$C852(+76032;$FK<(+DED&%2-,/$C6('D0)"22 /J,<,?$ '2$B&)(+=6t+2 RF$'&)&)65737$',<(x$'*-$CK<$ %+K<$' ( B3*$1=(+*@6 032d&Q$'*&)(C*(1KE2&/A32@6)DE,?$ F73(C,?6('+(+*@6g6 - 65K "J365DE,<(>DE2dK<2&5(+K<KE6g$ =>(036)*@*2J3732@032=6)7D<2(C7$C*2 /K<$ "J36)D ,?2 *7" 6="J$C852(+736 032 4&/A3K020273+65K7(+7 %4-%(+*@273(+=2$'*6 -K<6)DE6)7,/$2* 'C$'7",?$C++2@(3B|KE2D*-%65,E,?(u$C*=(036)*@*( 2@7 &5J322+K<$C032032*@2?Q%6)K ,SR K?$'032$C*@2DE(+73(u&)(+= -3*@65,/$%=6)7,?6 2+7(+K?$',<2 7$g,@B% kB02%(',<,?65736)KE6CB - 65K "J3652-$'K?$C=65,<K<2&)(+7>&)J32%D<2&5(+7 K<(C7",?$C73(2KE2D<J*@,/$I,?2D*-%6)K<2@=>657,/$C*@2 B ''$C*@(+K<2 -32X 65036)*@2]$ "J36)*@*2036)*]&)$CD<( ,?KE2032@=>6573D<2@(+7$C*@6C 6)KD<65=1-*2&52@,R ,?J,E,/$ '2$ -3K<65D<657",<2$C=( J2* " $D--3K<(&)&52(b0365**6 (C7303673(+7 *276Q$CKE2032 GN*(&/A7365*H&Q$'D<(pDE,<K<6j,<,/$'=>657",<6 J3732@032=6)7D<2(C7$C*6' /K<$CD<&5J3K?$C703(p2H+K<$C032 032H*@2?Q%6)K ,SR K?$C032$C*2*7" (-%6)K<$',?(+KE6 L &/A36g&)(+= -$CKE6F736)*@*$ + $CD<DEJ3=>6g*$x (+K<=>$

L = − h2

2m

(d

dz

)2

+ V (z) + 2U0|Φ0|2 − µ 65"

46$"J36)D ,?( -J37,?(D<2d2?-%(C,<28)8)$- 65K]*$1 J373852(+76Φ0

*$x (CK<=>$

Φ0(z) = F0(z) eiqz 5+

6gD<2|,<K?$',E,/$ *$-%6)K ,?J3K0Q$C8)2@(+736F&5(+=>6

δΦ = eiqz[uke(z) eikez + v∗ke

(z) eikez] 5.5

03( '+6u

6v

A$C737( *$ -%6)KE2(032&52@,SRg0365*|K<6j,?2&5(+*(x032KE65,<,<(16ke

F2@* =(+=6)7,?(106)** " 65&)&52@,/$%8)2@(+736 76)*|DE2DE,<6)=>$x032|KE2@ 65K<2=6)7,<(x0365* K<6j,?2@&)(+*@(3BDE2|A$1&/A36F2@*'C65,<,<(+K<6F02 -%6)KE,<J3K*Q3$C8)2@(+736$CDED<J3=6g*$x (+KE= $

δΦ =

uke

vke

59N

D<J$'+(C*@,/$1*$ = $',<K<2@&)6 032 '2@6)736 M

A =

L+ U0Φ2

U0(Φ∗)2 L−

5!

03( '+6 L±

27032&)$

L± = − h2

2m

(d

dz+ i(±q + ke)

)2

+ V (z) + 2U0|Φ0|2 − µ 65 K


"J36)D ,?( -3J7",<(;DE2D&'2@*J- -$C73(u

6v

D<J3*@*$ Q$CDE6>06)**@6(+730361-32$C766'B&5(+73D<65K0''$C730(DE(+*( J7p7J=>65K<( 72@,?(;032 "J365DE,<6n(+73036 -2$C736'BD<2K<2@03J3&)6 2*,-3K<(.Q3*65= $ $C*N&)$C*&5(+*(u032$CJ,?( ''$C*(+KE2032J7$u=>$',<K<2&5617365*&Q$C= - (n06)27"J3=6)K<2&5(+= -3*65D<D<2 1HKE2DEJ3*@,?$',?27J=>65K<2&52-%(+D<DE(+73( 6)D<DE6)KE6 J,<2*2@8)8)$',?2 - 65K(',<,?65736)KE6>2@7 (+KE= $C852(+72D<J**$ DE(++*@2$ - 65Kg203J36K<6)C2=20322@73DE,?$.Q32@*2@,R>6D<J**$ 'C6)*@(&52@,SR 02&)K<65D<&52@,/$b03652=>(0322@73DE,?$.Q32@*2 B ( '3'C6)KE(;D<J*B'C$'*(+KE61032

ωi

&/A36;&)(+= -$CKE6;7365**$ 65 k J365DE,/$ "J$'7",<2@,RBt&)(+=6;DE2H6'2@73&)6 03$C**$ 5 jB *@6)$',?$:$'*,?$CD<DE(1032;- 65K<032-,/$g032 $',?(+=20$ -$CKE,<6 06)*%&)(+7306)73D<$',?( &/A36'B&5(+=>6 '+6503K<65=>(1$C*&)$.-32@,<(+*(5B g*$ "J$'7",<2@,R&/A36 gDE,?$',/$=>2@D<J3K<$',/$ D0- 65K<2=6)7,?$C*=6)7,?6'46:DE2d&)(+7D<206)K?$J37 &5(+73036573D?$',<( &)2*@2730K<2&5(3B&52( DE2N,?K<$CD<&5J3K?$ 2*&)(C77$C=6)7,<( $'D<D<2$C*6'B7365**2@=>2-,?6032

s → 0B]*$ J373852(+736 0F" (+730$ 06)* &)(+7306)73D<$',?(3BtDE(+*J3852(+76;0365**7" 6="J$'8)2(C736 02 $FK<(+DED&%

2@,?$C6'D0)2@2J732032@=>6573D<2@(+7$C*@6CB * " (C730$ -32$C73$ Z&j K)k653 Φ0(z) =

√n exp(iqz)

B03( 'C6

n27032&)$ *$ 036573D<2-,SRb=>65032$ 0365*&Q$C= -32@(+736C16)K "J$C7,?(uK<2@+J$CKE0$ *7" 27DE,/$DQ32*@2@,SR

65736)KE+65,<2&Q$ "J365DE,?$u$D--3K<(CD<D<2@= $'8)2(C736:2@7,?K<(03J3&561J37$ CK?$C73061D<65=1-3*@2 &)$C8)2@(+736 - 65K<&/A*$ = $',<K<2@&)6

AD<2KE203J3&56n$C0pJ373$ =>$',<K<2&56

2 × 20$ &5J32D<2N(C,<,<26)76 &/A36n2*D<2@DE,?65= $

032@7$C=2&)$C=>657,?6 D ,/$.Q32@*6det(A) 6= 0

NDE6 M

q2 <k2

e

4+(mvs

h

)2 659+

03( 'C6vs

27302&Q$1*$ '+65*(&)2@,R1036)*DEJ3(+73(&/A36 g0$',/$ 0$ K

vs =(nU0

m

)1/2 659M

03( 'C6U0

N03$',?( 0$C*@*$ k J365DE,<( D*-32@6)$ -%6)K<&/A]$C*&5J3732"$CJ,?(+KE206 732@D<&5(+73( "J365DE,<(,<2?-%(u022@73DE,?$.Q32@*2@,R &)(C=>627DE,/$DQ32*@2@,SR032$C730$CJ :2@7 $'D<D<65738Q$ 032&)(+77$C=6)7,?(B -%6)K"J3652 ''$C*@(+K<2]032

s6q-%6)K&)J32tD<2 -3J"X$CDED<J3=6)KE6

v ∝ qB*$ &)(C73032852(+76u02 D ,/$.Q2*2-,SR

0365*=(C,<(uDEJ-%6)K3J3203( 036)*&)(C73036)7D?$',<(;7365**@2=2@,?6:203KE(027$C=2&5( ke → 0

>&/A36>*$'C6)*(&)2-,SRg036)*%&)(C73036)7D?$',<(1D<2$2@7 6)KE2(+KE6t$C**$ 'C6)*@(&52@,SRg0365*%DEJ3(+73(B73(C,<(:&5(+=6 &5K<2@,<6)KE2(g032$C7303$CJ|6)K,"J$C7,?(gK<2@+J$CKE0$ 279'C6)&)6t* " 2@73DE,?$.Q32*@2@,Rt027$C=2&)$H&/A36'B&5(+=60365,E,?(3B ]J37 6573(+=6)73(&/A36x273DE(+K<+6gDE(+*(27 -3K<65D<6)78Q$ 032 Z 036)*H-%(C,?65738)2$C*6 -%6)KE2(032&5( Z 0365**@62@7,?6)K<$C8)2@(+732 B$C7&/A36:2@7 "J365DE,<6x$.- -3K<(+DED<2@= $C852(+72|73(+7 Q$C7$C*@6FK<2@&Q$ ''$CK<6g6)D0-3K<65D<DE2(+732$C73$C*2-,?2&/A6C LN2

W M9 P ^ ;:ZP# <= P ^P

*2@=>2-,?2$C=>(u$ K<2E- (+K ,/$CKE6 *7" 6)D0-3K<65D<DE2(+736b&/A36uD<2 -J" 06)K<2 ''$CK<6b7365*]*2=2@,<6>02 - (C,<6)73852$C*@6-%6)K<2@(02&)(g036Q (C*6

s 171J6)DE,<(1K<65+2=6]2*DE2D ,?6)=>$ 032@7$C=2&)$C=>657,?6]DE,?$.Q32@*673(

$ &/A36 MB5

|q| < qB

√nU0

2Erec+

1

4

N'

F

7 "J365DE,<(1&Q$.-2@,?(C*( '26)76H0365D<&)KE2@,E,?(x* " $D--$CK<$',?( - 65K*$ -K<(03J38)2@(+736 036)*%&)(C73036)7D?$',<(1032GN(CD<6(%4@273D ,?6)2@7n032 87 P Qn6F*$xC6)7365K?$C852(+736H036)*|KE65,<2&)(C*(1(C,E,?2@&)(1J372032@=>6573D<2@(+7$C*@6 $I,<,?2 '+(-3KE6)DED<(p2* H@UG 4 02H#2@K<65738)6' *@,?65K<2@(+K<2]036j,<,?$C+*2tDEJ32 02E'+65K<DE2H$CD0- 6j,<,<2 0365** " $.- -$CK?$I,?(-%(+D<DE(+73(6)DED<6)KE6:,<K<( ''$',?2|2@7 655B 5 N

rmbo Nh5r:`ns mns]ipqNr:`umnst`baegcr

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>

! S T T3W T O M"WIP MY; T : M

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( 3

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W dRVH"P YM RM3:N^ %YRMYO)T

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25 l/s$CK<2$C7 $C&5(+7>U*@J3D (.- 65K?$C7,?6tDEJ3**$ -3K<2@= $ &)6)*@*$BC6)0J37$F0$

55 l/s$CK<2$C7

M ! %+C K (-%6)K<$C7,?6 D<J3*@*$gD<6)&5(+730$ - (+DED?$C7(x=>$C7,?6)76)K<6 03J36 K<6)C2=2032 'J(C,?(&)2KE&Q$

10−9 ,<(+K<K]76)**$-3K<2@= $9 / 6g&)2KE&Q$10−12 ,<(+K<K]76)**$1D<65&)(+7303$

"J365DE,<60J36 - (C=1-%6 F$ $'73&Q$',?$1J37$-%(+= -$1$1D<J Q3*2@= $C852(+76 032%,?2-,/$C732@(x&/A36 '2@6)736$C&5&)65D?$ 032NDE(+*2-,?( (++732%;DE65,E,?2=>$C736>27X=(03( 0$ KE2=1J3( 'C6)K<6 2*K<JQ32@032( 27X65&)&56)D<DE($C06)D<(>$C*@*6 -$'K<65,<2d036)*@*6 &56)*@*6CBK<20J3&)657303(&)(+DJ3*@,<6)K<2@(+K<=6)7,<6 *$-3KE6)D<DE2(+76C

P / l '

6)Kg2*K<$DA K<6)00$C=6)7,?(b6 *$ ,<K?$.- - (+*$ = $'+7365,<(D%(',<,?2@&Q$D<2J,<2*2@8)8)$ *$ ,<K?$C73DE2852(+736D2

0365* 87 P Q $C**$g*@J373+A6)8)8)$:0F" (+7303$:032&)2@K<&)$780.25

73=u J365DE,?$:K<2@$ -3K<65D<6)7,?$x*$gDE,<K<J,&%,<J3K?$2?-%6)K 736F2*@*J3D ,?K<$',/$x2@7+J3K<$ $ &)(+= - (C736)7,?6bDE&)6)*-,/$ - 65K>2@*K?$ A KE6)030$'=>657",<( b*$ |F = 2〉 → |F ′ = 3〉 6)K&%&/Ab2@*K<$DA KE6)0303$C=>657,?( D<2$ 6&)657,?6 Z&j K) / (&)&5(+K<KE6 - 65K " &/A36n*$ ,?K<$C73D<2@8)2@(+736DE2$ )CB]&)2@( u&/A36 +*@2 $I,?(+=2 - (-%(+*@273( D<(+*@( 2 03J36 *@2E'C6)**@2]D<(-K?$ 036j,<,<2 7 KE6Q$C* %,RX2*H*$CD<65K 032 K<$DA K<6)00$C=6)7,?(3B -%6)K "J$C7,<( 7(+7 27 KE2D<(C7$C738)$X&)(+7 *$ ,?K<$C73D<2@8)2@(+736

|F = 2〉 → |F ′ = 2〉 B# &5(+=1J37"J36 2@7 +K?$C0( 02 - (.- (+*$CK<6;*(pDE,?$',?( |F ′ = 2〉0$ &5J32NC*2$',?(C=>2 - (CD<D<(C73( 036)&)$C0365K<6n7365**( DE,?$',?( |F = 1〉 J365DE,<( &)(+D ,?2@,<J32DE&)6>J37

8 S T T3W T O M"WIP MY; T : M

&Q$'7$C*6 02#- 65K<032-,/$n&/A36 -3J"u&)(+= -3K<(C=>6j,<,?65K<6*7" 6&526)78Q$n036)*@*7" 27,?K<$.--%(+*$'=>657",<(316)K6'2-,/$CKE6 "J36)D ,?( $C*N*$CD<65K 032NK<$DA KE6)0303$C=>657,?( 736 '265736nD<( 'K?$.- - (+D ,?( J37 $C*-,?K<(B0365,E,?(032K<2?-%(+= -$C+C2(3B&/A36n*$ 'C(+K?$;DEJ3**$;&)(C=1-%(+73657",<6 |F = 1〉 → |F ′ = 2〉 =6)032$C7,?62*K<2@*$CDED?$C=6)7,<( |F ′ = 2〉 → |F = 2〉 6)DED<( K<2E- (CKE,/$u+*2$',<(+=2 -%6)KED<27365*N*@2E'+65**@( 032K?$ A KE6)030$'=>657",<(3B3032|Z$I,<,?(&/A32@J303657303(*$x,?K<$C73DE28)2@(+736'$D ,/$.Q2*2-,SR 0365** " 65=>2@D<D<2@(+736]0365**@6]D<(+KE+6)7,?2 $CK?$'7",<2@,?$g=6)02$C7,?6]J7 D<2@DE,<6)=>$65*65,E,?KE(D%732@&)(n032$C+$C7&)2( &/A36 = $'7",<265736x*$> K<6J6)738)$u0365**$'D<6)KH$C0 J37$n032@DE,?$C738Q$ 3D<D?$I,/$ 0$J37$,?K<$C73D<2@8)2@(+736g036)* 87 P Q| 73$1-$CK ,?6036)*@*$ *J&)66)=6)D<D<$ $',E,?K?$ 'C6)K<D<$ J37$&)65**$ &5(+7 %,?65736)7,<6H0365* PtJQ32@032(g6 '2@6)736 K<6j,?K<(CK<236)D<D<$:KE6Q$C*@28)8)$C7303(J373(:D<&/A365= $032D*-%65,E,?K<(CD<&)(.-32$036j,<,?( 27 D<$',?J3K<$C8)2@(+736b&/A36u6)*@2=27$;* " $'**$'K<$C=6)7,<(;0365**$ KE2$ -%6)K6A 65,E,?( H(--*6)K5 $C* -3K<( *( 02D?$',<J3K?$'8)2(C736xDE26)D ,?K?$'6 J37 D<65+7$C*@6103265K<K<(CK<61&/A36'2@6)7361J,?2*@2858Q$',<( -%6)K&)(CK<K<65++65K<6*6036)KE2E'C6>0365*U*$CDE6)KgD<J J37$ Q3$C730$bD*-%65,E,?K?$'*673( $b KE6="J36)78)6 032$C*@&)J372 8'd60J36:DE(+K<C6)7,?2*$'D<6)Kt7365&)65D<D?$'K<26x$C** " 65D*-%6)KE2=6)7,?( DE(+73( &5(+DE,<2@,<J32@,<6F0$02(032d*$CDE6)Kt$

D<65=>2@&)(+703J,<,<(+K<6=>(C7",?$',?227x&Q$ '2@,SR 6)DE,<6)D<$g2@71=(03( 0$ =2+*@2(+K<$CK<736*@6N&)$CK?$',E,?65K<2D ,?2@&/A36D*-%65,E,?K<$C*2 $u*@J3&)6032UKE2?-%(+=1-3$C++2@( '26)76 (+KE732@,?$ 0$ J7 032@(0( 0365**$ /(-,?2&)$ A3(',?(+732@&)D=>( %0365**(x4 %JND5 %J$ $ '+657,?6 J373$x&)(+KEK<657",<6H(-%6)K<$',?2 ''$032&52K<&)$

60= 6)0 J7$ -%(C,?65738Q$

,?2E-32&)$H27 JD<&)2-,/$ 0$'**$ &Q$ '2@,SR 65DE,<6)D?$g0323&52KE&Q$20

= .LN(+=6036j,<,?(gDE(-3K?$FJ373$ -$CKE,<6]032"J36)D ,/$*J3&56 '265736 -3K<65*6('C$I,/$-%6)K]6A 65,E,?J$'K<6F*7" $C++$C73&)2@(x2@7 K<6="J365738Q$ =6)7,?KE6F2* K<6)D ,?('265736gD<J30302E'2DE(2@7b03J36FZ$CD<&52d&/A36+2J37+(+73($C*@*6 0J36:9 /F*t*$CDE6)K J,<2*@28)8)$',?( DEJ3**$ ,?K?$'73D<2@8)2(C736n032tK?$DA K<65030$C=6)7,<( uJ374$C7+( K% 5+b*$&)(CK<K<657,?6F(-%6)K<$',?2 'C$ -$'K<2|$

70= 6 *$ -%(C,?65738Q$12@7 J3D<&52@,?$ 27,<(+K<73(x$

25= /$C*6

-%(C,?65738Q$F73(+7 ]DEJ &)6)7,<6 -%6)K* " 65D*-%6)K<2@=>657,?(F6 "J3270323*$ *J3&56CB03(-%(g$ '+65K<736 -3K<65*6('C$I,?(J37$ -3$CKE,<6- 65K* " $CC$C73&52(H2@7x K<6"J36)738)$B'265736]$C= -3*2 &Q$I,/$t=>65032$C7,<6NJ37$C= -3*@2 &)$',?(D%K<6$D<65=>2@&)(+730J,<,<(+K<6 /(-,<2&)$A3(',?(+732@&)D /$.- 65K<650 = -3*@2 65K/ + k$ -%(C,?65738Q$&)(C=1-3*@6)DED<2E''$n&/A36D<2(C,E,?2@6)736 2@7 JD<&)2-,/$ 032&)2@K<&Q$

620= &)(+7 J37F" $C= -3*2 &Q$'8)2(C7361032

(+*-,?K<6FJ7u(CK<032@736F032d+K<$C7303658)8Q$*Z$CDE&)2@( 27 J3DE&)2-,/$ 0$C*@*7" $C= -3*2 &Q$',<(+K<6 '2@6)736 KE2?-$CK ,?2-,?( 2@7 &)2@7"J361Z$CDE&)2032@DE,<27,?2&)(+=6


051> 0#FT0*<233(+*-44)37.,"907'7"9&,.;!5WX*-,'JX%0*(+*-44);,"!5E ;(+*-44CB 7"$2=$K(0!87O8 `4!5@1*-445!(+!R#!)1.#90%

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2@73032&)$',?(u27 +JK?$u xJ37 Z$'D<&)2@( -%6)K2*#-%(+= -$C+C2(n(C,E,?2&5( 06)+*@2U$',?(+=2#-3KE2=>$b06)*,<K?$CD 6)K<2@=>657,?( 7365**$b,?K<$.--%(+*$;=>$C+765,?2@&Q$ &5 K5 ? N kBJ373( -%6)K *$ -3K<2@= $ 6)0pJ373(-%6)K1*$uDE6)&5(+730$ 9 / 6)0p2@7736>J373( -%6)K:2*UZ$CD<&52( 032D0-327,?$;6)0pJ37( - 65K "J365**( 032KE2E'+65*$C852(+76C/J,<,<6 *@6>DEJ303032 '2@D<2@(+732$ '3'C6)73C(+73( =6)032$C7,?6J37$;*$C=27$

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8)$',?(+KE6 3CK?$C8526F$C*@*7" J3D<(x032F"J$',E,?K<( g9 27302?-%6)7303657,?2 -%(+DED<2?Q2*6 &)(+7,<K<(+*@*$CKE6 Z657,?K<($C*@&)J37606)&)2@736g0329 8*$x KE6="J36)78Q$>02d&)2$'D<&)J73(036)2dZ$CDE&)2

S T T3W T O M"WIP MY; T : M

Diodo Laser TA-100

Spettroscopia in saturazione

Divisore di fascio e

AOM


AOM


AOM


AOM

2

4

AOM

2

Otturatore

Otturatore

Diodo ottico Diodo ottico

& 0> +#FT0*<23(+!Z'9&(&(+!@:!)'!5$1%&*(+*<!RQS"#<!vWi*<,H!Z(+!@*<,F'!R'*<7'7.$1,.!R(+*-44CB *-"WX*-,"!52;*<%D7"$X>45 49&#<*@:!5*<%0*,.!W&1,'7.!n7FK#$%]90%h"!5'7"*-2334512;!%&:#<90OX$ WX$451,"!56<6I7"$,"*>r,.90$7F%&(0$K45K4512=!5%&3G/WX$D""!5O0!45*@1,"!),.*_45Qy,F6<!5$1%&*A(+!!5%D7"*<%X'!7.o=#FT0*H@D!5*<%&*H"9&#<#<*-.'!5@2;*<%D7.*q7",F1"2;*-."/(04#<90OX$XY^v14*mQy,.16<!5$1%0*HW0,.$"*<E90*qWi*<,45*V"9&#<#<*-.'!5@1*V(+!5@D!)"!$%0!?Ur2=*-%D7",.*4)fQy,F6-!$%0*V,"! *-..h7"7",F[@1*-,..!% (+$W0W0!5$eW&""$h9&%\2;$+(+904)7"$,"*1#<9&'7"$:$7"7"!)#$h?Mh*/W09K*I""*<,.*!5%+J&%0*=#$1%:@$1E145!)7.=@*<,F'$34CB *I'Wi*<,.!52=*-%D7"$&H8*(090*45*<%D7"!P"*<,.@1$1%&$=Qy$+#<14!56<6I,.*A!548QS"#<!$3#$445!52;7"$;%0*-44CB M Wi*<,q23""!2;!56<6I,.*A4CB * K#*<%&6-3(+!82;$+(+904)6-!$%0*1 (+!5$+(+!$7"7"!)#!"*<,.@1$%0$/=W0,.$7.*<EE1*<,.*H!48(+!5$+(+$34)1"*<,`*_4?B 2;W045!JX#<7"$1,.*(048#<$"!5(0(0*7'7.$ $7'7.!5#<$&


( # _8

#" 27,?K<$.--%(+*$C=6)7,<( = $CC7365,<(D%(C,E,?2@&)( 0365+*2d$I,?(+=2 g2*-3KE2=(-$CDED<(1'C6)KED<( *$ &)(+7306)7 %D<$C8)2@(+736 6 &)(+D ,?2@,<J32DE&)6>*7" 6)*65=>657,?( (C730$C=6)7,/$'*6036)2=>(0365K<7326)D0- 65K<2=6)7,<2032 D<2@&Q$$',<(+=2&Q$ 7 D<2@7",<6)DE2 DE2 D K<J,E,/$ *$ -3KE6)D<DE2(+76 032tK<$C032$'8)2(C736n6)DE6)K<&52@,?$',/$ 0$ DE6)2]Z$CDE&)2*$CD<65K1&)(+7,<K<(-3KE(-$C+$C7,?2 - 65K1K?$C*@*657",?$CK<6>+*2$',<(+=2 B=6)7,<K<6 J37pCK?$C032@6)7,?6 032N&)$C=1-%(=>$C+736j,?2@&)(1 (+KE732DE&)6g*$x (+KE8Q$ 032dKE2&/A32$C=>(x736)&56)D<D<$CK<2$>$'**7" 27,<K?$.--%(+*$C=>657,?( 59! 71=>$C732@6)K<$ -32 "J$C7,<2@,/$I,?2E''$t2* -3KE273&52?-32@( 02 J378)2(C7$C=6)7,?( N2@*DE6)+J6)7,?6'UHJ36 Q%(Q32 %736F2@7 &)(C7+J3K<$C8)2@(+736F$'7",<2 % 6)*@=>A(+*@,<8CB&)2@( 27 &5J32|DE&)(+KEK<6F*$=>65036)DE2=>$x&)(CK<K<657,?6CB=>$2@71'+65K<DE((.-- (CDE,?(B&5K<6Q$'73(1J37 &)$C= - (:= $CC7365,<2&)(7"J3**@(1$C*%&)657,?K<(x036)* &)2*@2730K<(g0$x65D<D<62@73032E'2@03J$',<(3B&5(+7XJ37pCK?$C032@6)7,?6>$.--K<(+DED<2=>$',<2E''$C=6)7,?6&)(CDE,/$'7",<6 7365**6,?KE6 032@K<6)852(+720365**@(:D0-$C8)2@( ,?K<$CD<&5J3K?$C703(x+*@26A 65,E,?2032<Q (CK<03( kB/$C*6t&Q$'=1-%(:=>$C+765,?2@&)(gD<2%$C&5&)(- -32$&5(+7 2*=(+=>657,?( =>$C+765,?2@&)(1-3K<(-K<2(>036)C*2$',?(+=20$C730( *J3(C+( $C0;J37$ DE6=-$CK<$C8)2@(+736-%6)Kt6A 65,E,?( 6)65= $C7 -$'K<2d$

∆E = gFµB mFB(r)

03( 'C6gF

2*|Z$',E,?(+KE6g032$'730:$',?(+=2&5('% ] jBµB

:2@*d= $CC7365,<(+736g032GN(+A3Kt6

mF

g*$-3KE(+2658)2(C736F036)*d=(+=6)7,?($'73+(+*$CK<6 DEJ3** " $'D<D<6g0365*d&Q$C= -%(3G 6)**$ 85(+7$ 27x&)J322*&Q$C= - ( = $'+7365,<2&5( *@2736)$CK<6D<22@7,?6)KED<65&Q$C73(FDE6)2Z$'D<&)2*$CDE6)K&)(+7,?KE(D%-3KE(-$C+$C7,?2 B $C&)&5(+K<03$',?2&5(+7 J37 jZ KE(+D<DE(bK<2D0- 6j,<,<(n$C0 J37$ KE2D<(C7$C738)$n$',?(+=2&)$ 736)* 73(+D ,?K<( &Q$CDE( *$ ,<K?$C7D<2852(+76 |F = 2〉 → |F ′ = 3〉 16;$ '+657,?2 -%(+*$CK<2@8)8Q$'8)2(C736&52K<&5(+*$'K<6 (--%(+DE,?$x27>=>(03(x&/A6 2*Z$CDE&)2( - (C*$CKE28)8)$',?(

σ+ DE2 -3KE(-$CCA32 76)**$02K<658)2@(+7362@7u&5J32d2* &Q$C= - ( g&5K<6)DE&)657",<6C#" 6A 65,<,<(:DEJ3+*@2$',?(C=>2 -3J"6)D<DE6)KE6H$C73$C*2@8)8Q$I,?(F27J373$gDE2@,<J$C8)2@(+736]DE6)= -3*2 &Q$',?$ 27&)J2*$,<K?$C73DE2852(+736]DE2$ 0365*3,?2E- ( |F = 0〉 → |F ′ = 1〉 6

gF = 1 /$C*@6]D<2-,?J$C852(+76 ]2*@*J3D ,?K?$I,/$

7365*d&Q$CDE(>J732032@=>6573D<2@(+7$C*@6H2@7 3+J3K?$ 7 J6)DE,<(\&Q$'D<( J37 $',<(+=( &/A6 DE2g,<K<( '2g27J37$ 8)(+7$ 27 &)J22@*F&Q$C= -%( 7365$',?2 %'C(3Bg$ &)$CJ3D<$ 0365*>j \K<(+DED<(3BFA$ J37$ -3KE(Q$.Q32@*2-,SR =>$C++2@(+K<6;032g$CD<DE(+K*Q2K<6 J37 (C,<(+736>0$C*UZ$CD<&52(;&5(+7-%(+*$CKE2858Q$C852(+736

σ+ BB-%6)K<&/An27 KE2DE(+7$C738)$ &5(+7p*$n,?K<$C73D<2@8)2@(+736

|F = 0, mF = 0〉 → |F ′ = 1, mF = +1〉 &Q$CJD?$ 0365*K<2@73&)J*( "J32@73032D<J3*@*7" $',?(C=>(


$C+2@D<&56F=>65032$C=6)7,<6FJ37$1 (+K<8)$02K<6j,<,?$ '+6)KED<(>2*|&56)7,?KE(>0365**$x,?K?$D-- (C*$ *@*(uDE,<6)D<DE( =>(03( J37p$',<(+=>(u7365**$b8)(C7$;032&Q$C= -%( -%(+D<2-,?2E'C(;DE2,<K<( ''$;27 K<2DE(+7$C78Q$&)(C7X*$n,?K<$C73D<2@8)2@(+736 |F = 0, mF = 0〉 → |F ′ = 1, mF = −1〉 6 "J32@73032$CD<DE(+K*Q2KR&)(C7 =>$C++2@(+K<6 -3K<(Q3$.Q32*@2@,RxJ37b (C,?(+7606)*|Z$CDE&)2(

σ− 73&/A3627 J6)DE,<( &)$CD<(*7" 6A 65,E,?(

=6)032@( D?$CK*RJ373$ (CK<8Q$ &/A36D*-273+6g* " $',<(+=( '+65K<D<(>2* &)6)7,<K<(036)*@*$x,?K<$.--%(+*$G (',?2$'=>( &/A6 $'0(C+732 (C,<(+736t$CDED<(+K0Q32@,<(g&)(+KEK<2D0- (C73036]J37$FDEJ3&)&56)D<DE2E''$:65=>2@D<DE2(+736ND*-%(+7 %,/$'736Q$B',?J,E,/$ '2$H* " 6(A 6j,<,?(H032 "J365DE,/$FDEJ3**$ (+K<8)$H=6)032$ $C+657",<6D<J**7" $',<(+=>( 7"J3*@*( 0$I,/$*7" 2DE(C,?KE(-32$x032 ,?$C*6F65=>2@D<D<2@(+736' "J365DE,<( 6A 65,<,<(x032%K<2@&/A32$'=>(D<2 $C++2@J373+6 $C7&/A36 J37 6(A 6j,<,<( '2D<&5(+D<(x0$',?(:0$C*@*$ -3K<65D&%D<2@(+736u032 K?$C032$C8)2@(+736 * " $CDED<(+K0Q32=6)7,<( DE6)*6j,<,<2E'+( 032tJ373(X06)2t03J36uZ$CD<&52# 27Z$',E,?2t$C&I%&)KE6)DE&)2J,?(u0$C**@(nD*-%(+DE,?$C=6)7,?( H(--*6)Kg&/A36>$ ' '2@&)2@7$b$C**$ K<2@D<(+7$'738Q$b2*Z$'D<&)2@(b&/A36>DE2-3KE(-$C$27 032KE6)852(+736g(- - (+D ,/$>$C**$'C6)*@(&52@,SR0365**7" $',<(+=>(7 $'D<D<65738Q$>032&)$C= - (= $%+736j,?2@&)( "J36)D ,?( 6(A 6j,<,<( '2@D<&)(CD<( KE6)D0- (+7D?$.Q32@*6F0365**@6F&)(+DE203036j,<,<6 =>65*$CDED<6 &5*$CDED<2@&/A36?42'C6)K<2 &Q$D*-%6)KE2=6)7,/$C*@=>657,?6]&/A36 "J36)D ,?2%K<2DEJ3*@,?$',?2 'C$'*+(+7(g$C73&/A36 D<6 *$FDE,<K<J,E,?J3K<$:03652*2 '+65**2&)(+2@79'+(+*-,?2 -2 &)(+= -3*65D<D<$u6D<6>D<2,-$CDED?$ $C0 J37$u+6)(+=65,<K<2$ ,?K<2@032=6)7D<2(C7$C*6=6)032$C7,?6F*7" $C+C2J37,/$:032$C*@,<K<6F03J6&5(--32@6F032|Z$CDE&)2|,?K<$ *(+KE( (+KE,<(++(+7$'*2

SUT WIP TL

LN(+=6C2 R $C&)&56)737$I,?(3B+*$ -K<2=>$ 9 /pA$H*( D<&5(-%( 032"&Q$',E,?J3K<$CK<6J371+K<$C730367"J3=>65K<( 032$',<(+=>2%$ -3$CKE,<2K<6 0$C*;''$.- (CK<6H02 PtJQ32@032( 65 K J365DE,=" J3*-,?2@=>( H(C,E,?6)7"J,<( =>6j,<,<6)7303(x27&)(C=1J3732@&Q$C852(+736t$I,<,?K<$ '+6)KED<(1J7$ ''$C* '+(+*$FJ37F" $C=1-%(+*@*$H&5(+7,?6)76)7,?6 J7 &Q$'=1-32@(+736]DE(+*2@03(0365*|=>6j,/$C*@*(:&)(+7b*$1&56)*@*$x=>$C7,?657J,/$B3&5(+=6FD<2 '2D ,?(3B3$Q$CD<D<$-3K<65D<D<2@(+736 2@* =65,?$C**@(D<J Q3*2@= $16FD<$',?J3K<$*$1&)65**$1&5(+7u2dD<J3(+2F''$.-%(+KE2 * &Q$C= - ( =>$C+736j,?2&5( "J$'03K<J-%(+*$CK<6 '26)76;+65736)K<$',?( 0$ 03J36 Q (.Q32736u&)(CDE,?2-,?J32-,?6b0$$ ' '+(+*@+2=6)7,?2032*(b032K<$C=6 032$'=>6j,?K<(

0.8== - 65KgJ37 032$C=65,?KE(n032&)2@K<&)$

15&)=

6 -%6)K<&5(+K<DE6 0$XJ37$p&)(CK<K<657,?6 032 &52K<&)$3

* +K?$'032657",<6 032H&)$C= - (p=>$C+736j,?2@&)(X0$6)DED<6>+6)7365K?$',<( 032&52KE&Q$

7$$'J3D<D

/&)=u*K<$DA K<6)00$C=6)7,?(b0365**6Q%(Q32@736 (C,<,<6)7"J,?(

-%6)K &5(+7,/$',E,?( ,<6)KE=>2@&)( ,?K?$ 2*tD<J- - (+K ,?( 032 $C*@*J3=272( 0365**$ Q%(Q32@7$ 650\J37,<JQ ( 032K?$'=>6 2@7p&)J32]DE&)(+KEK<6b*7" $C&J3$ 032]K?$DA K<65030$C=6)7,<(3X$ -3K<65D<65738Q$ 032]6'C6)7,?J3$C*2]&)$C=1-2


& &> +#FT0*-2; (+!HQy90%06-!$%&2;*-%7.$ "*<2;W045!nJi#<7.$ (0!H9&%&j7.,.1W0Wi$14)j23E%0*7.$ac$7'7.!5#-0 (+90*OX$O0!5%0*]'$%0$g(0!5"WX$D7.*!% #$%+J&E19&,.16<!5$1%0* 1%D7"!am*<452=T&$147"6]#!5$:GV"!2;2;*7.,"!)#<12;*<%D7"*V*]#$1% #$,",.*<%D7"*90E19X45* 23f!5%j@1*-,."$]$1W&WX$D7.$e#$2;*K!%X(+!5#-7.$h!5%jJ&E90,.& %lJ&E90,F]G ,.!Wi$1,"7.7"$h1%&#FT0*V7"!7"$4$g(+!*-"*<2;W0!5$\4CB 1%&(012=*-%D7"$ (0*<454$ 'Wi$'7.12=*-%D7"$i*-*<23% (+*<454*k#$2;WX$%0*<%D7"!(+!_(+90*g45!@*<454!

F = 0*-(

F = 1!%\Qy90%06-!$%0*V(0*<4545hWX$D'!56<!5$1%0*lYB %&(012;*<%D7"$h(0*<4#<12=Wi$g2;1E1%0*<7"!)#$fGVW0,.$1Wi$1,.6<!5$1%&14*K144)

4!5%0*I+1v45!5%0*-7.,.7'7.*<E1E!57.,"$D".,.1W0W0,.*-"*<%D7.V4CB *-%0*<,.E1!)(+!Y90%kQy$17"$1%&*345'*-,W&,"7"!5,"*3(01445$

7F7.$3Qy$1%&(&2;*<%D7.14*7"$2=!)#$X v;6<$1%X3(+!v#-2;WX$ 4!5%0*-1,"*_%&$1%G!5%f"#-4);,"!)'Wi*7"7"$K1445*(+!52;*<%&"!$%0!(+*<4545*_OX$O0!%&*1

=>$C+736j,?2@&)2CD0-3J3K&C Z2@73&)*@J3D<( "J365**(,<6)KEK<6)D ,?KE6 6)DE&)*@J3D?$t=>65032$'7",<6J37gD<6j,032C,<K<6#Q%(Q3276032&5(+=1-%6)7D?$C852(+736t,?K<$*@(+K<((+K ,?(++(C7$C*227&)J32%D<&5(+K<KE6 J7$:&5(+K<KE6)7,?6 032<J3$C*&/A36 &)657",<27$'2(032d= * 7"J3=6)KE( 032F$',<(+=>2 &Q$I,<,?JK?$',<2 0$XJ37$ 9 / 032?-%6)7036 0$p032E'C6)KED<2HZ$',E,?(+KE2 Bt=>$ 27-$'KE,?2@&)(+*$CK<60$C*032$C=>6j,?KE( 06)2+Z$CDE&)2"&/A3627 J2D<&56UD<2$]DEJ3*9'+(+*@J3=>606)**$,?K?$D-- (C*$DE,<6)DED?$BDE2$HDEJ3**$H&)(+DE203065,<,?$ '+65*(&)2@,R 032&Q$',E,?J3K<$B+&52( N*$ = $CDED<2@= $ '+6)*@(&52@,R &)(C7&5J32J7$',?(+=(036('+6]657",<K?$CKE67365**$tK<65+2(C736N032D<( 'K?$D-- (CD<2852(+76N03652Z$CDE&)23- 65KK<2@= $C76)K0'2"27,<K?$.--%(+*$',?(36)K"J365DE,/$:K?$C+2@(+736t2Z$CD<&52 06)**$ -K<2=>$9 / A$C737(1J37 02$C=65,<K<(032&)2KE&Q$

5&)= &)(C7

J373$ -%(C,<6)738)$ &5(+=1-*6)DED<2 'C$02+&)2@K<&Q$60

= 6)0gJ37 j H032+&52K<&)$ −2Γ ' −129 8'

6)KK<6Q$'*2858Q$CKE6*$-3KE2=>$t9 /XD<2JD?$C73(t,?KE6Z$CD<&52K<65,<K<(+KE265D<DE2 |,?K<$ *$ 7365DE,?K<$ 032JD<&)2-,/$6]*@(FD*-%6)&)&/A2( ]273DE6)KE2@,/$ -%6)K&52$CDE&)J373$ 032KE6)8)2@(+736J7$F*$C=27$

λ/427 =(0( &/A36t*$ *@J3&)6


-%(+*$CKE2858Q$',?$σ+ B30(- ($ '+65K $',<,<K?$ '+65K<D?$I,?(*$1&)65**$B -$'D<D?$'7303(03J36 '+(+*-,?6g$',E,?K<$ 'C6)KED<(*$

*$'=>2@7$nDE,?65D<D<$BB'2KE2@,?(CK<732U&)(C7 - (C*$CKE28)8)$C8)2@(+736σ−

LN(+DZ$C&56)7303( ,?J,E,/$ '2$ 2@*Z$CD<&52(K<2 6)DED<( $I,<,?K<$ '+6)KED?$103J6 '+(C*@,?6 *$ 7365DE,?K<$1&)(+7 J37$ -%6)KE032@,?$:032%27,<6)73DE2@,SRB/$'*6 - 65K<032-,/$'265736 &)(+= -%6)73D<$',/$XJ,<2*@28)8)$C7303( - 65K *$XKE65,?KE(+K<2 6)DED<2@(+736 0365+*2 D*-%6)&5&/A32g*6)C+6)KE=>657",<6&)JK0'2

R ∼ 2= j **$;*J&)6n0365*]*$CDE6)K-3KE273&52?-$'*6 '265736nD<( 'K?$D-- (CDE,/$ &)(C=>6n036j,<,<(

"J36)*@*$10365**$CDE6)K]032|KE2?-%(+= -$C++2@(3$ -3KE2=>$ 9 / K<$C&)&5(++*2@6g&)2KE&Q$

1010 $',<(+=>2$C*@*$1,?65=1-%6)K<$',?JK?$>02&52K<&)$200µK

&)(C7J37$036573D<2-,SR 036)*@*7" (+K<02736 02

109 cm−3

: W TO MWIP MY W T :ZM;RVM L

*,<K?$CD 6)K<2@=>657,?(n0$C*@*$ -K<2=>$ $C*@*$D<6)&5(+730$ 9 /w$ ' '26573627 =(03( 2= -3J3*D<$',?( &)(+7J37u&)2@&)*(x032d&)2@K<&Q$

220=>D5$ D<6="J365738Q$032|&52$CDE&)J37u2= -3J3*@D<( g*$1DE6)+J6)7,?6 *$-3K<2@= $

9 / '26)76 *$CDE&)2$I,/$>&)$CK<2@&Q$CKE6 -%6)K &52KE&Q$200

=>D5B "J32@730322@*&)$C= - ( =>$C+765,?2@&)( '2@6)736D*-%6)7,<( 6bD<65+J36bJ37$ Q3KE6'+6bZ$CDE6u02=>65*$CDED?$ (C,E,?2&)$ &/A36bK?$DA K<65030$ *$ -3KE2=>$ 9 /F**$CDE6)K032K<$DA K<6)00$C=6)7,?( '2@6)736>D*-%6)7,?( 6>D<J3&5&)6)DED<2 'C$'=>657",<6CBJ3D<$C7303( *@61Q%(Q32@736032&)(C=1-%6)73D<$C8)2@(+736'BB'2@6)736b$'&)&)65D<( J37p&Q$'=1-%( = $'+7365,<2&5( 736)*@*$;02K<658)2@(+736 06)*NZ$CDE&)2@( 032D*-27,/$ 6'B DEJQ32-,?( 03(.- (3B%2*dZ$CDE&)2@( 032D*-32@7,/$>D ,?65D<D<( J36)D ,?( A$>J37$ -%(C,?65738Q$>032&52KE&Q$

4= BJ37 jZ u0$C*@*$1,?K<$C73DE28)2@(+736 |F = 2〉 → |F ′ = 3〉 &)2KE&Q$J3+J$C*@6x$ −Γ

650 -%(+*$'K<2858Q$',<(

σ+ 27X=(03(;0$ - (+= -$CKE6>(',<,?2@&Q$C=6)7,<6 +*@2$I,?(+=2U736)*@*(;D ,/$',<( |F =

2, mF = 2〉 * Z$CDE&)2@(p032HD0-327,/$ $C**@2736)$',?( *J37+(X2* ,?JQ%(X032H&)(+*@*6)+$C=>657,?( ,<K?$*6 0J36 &)6)*@*6 &/A36'B -%6)KnZ$C&)2@*2-,/$CKE6;2* =>(',?( 036)C*2 $I,?(+=2 6)06'2-,/$CKE6 2* -32 - (+DED<2EQ32*@6*7" $C03DE(+K*Q2=6)7,?(uD<J3*@*61-$CK<6j,?2036)*U,?J Q (3B &)2@K<&)(C730$',<(;0$;DE6)2DE,?KE2DE&)6 KE65,E,/$C73C(+*$CKE2U032=>$',?65K<2$'*6x=>$C+765,?2@&)(3 J36)D ,?6&)KE6Q$C73(uJ37 &Q$'=1-%(n6)D<$.- (C*$CKE6 &)(+7 =2732@=>( $C*&)657,?K<(0365* ,?JQ%( 27 '2KE, 036)* Z$',E,?(+KE6 032g$C730 -%(+D<2-,?2E'C( +*2F$I,?(+=2 ,<6)7303(C73($ D<65+J32@K<6,/$'*6u=>2@732=( =>2@+*2@(+K?$'7303( &5(+Dg*7" 6&)65738Q$ 032 ,<K?$CD 6)KE2=6)7,?(3 * &)2@&)*@( DE2 &5(+=1-*65,?$&)(C7X*( D*-%6)+732@=>657,?( 0365*Z$CDE&)2@(;032D*-32@7,/$;6 *$ KE2$C&5&)6)7D<2(C736 03652Z$'D<&)26 0365*N&)$C=1-%(=>$C+736j,?2&5(x0365**$ -3KE2=>$x9 /F$:D<6)&5(+730$9 / '265736 &Q$'K<2&)$',/$&)(+7n&52KE&Q$

300&52&5*2

032dKE2?-%65,?2@8)2@(+736C

W dRVH"P YM RM3:N^ %YRMYO)T >

SUT O5M^ %YRT L

$ D<65&)(+7303$>9 /FB&)(+=6F0365,E,?( 7365*-$CK<$C+K?$I ( B '265736FK<6)$C*2@8)8Q$I,/$x7365** " $C= Q32657",<6032UJ3*-,?K<$ $C*-,?( 'J3(C,?( $C*@*$ -3K<65D<DE2(+736>032U&)2@K<&Q$

10−12 ,<(+K<K5 @D<D?$ (C,E,?2@=>2@8)8Q$I,/$ -%6)K*$K<$C&)&5(+*@,?$02$',?(+=2F-3KE6)K<$DA KE6)0303$',?2d650;2* K?$C+C2J373C2=6)7,?(102|,?6)= -%6)K?$I,?J3KE6:$'73&)(+K<$-32Q$'D<D<6 #- 65K "J365DE,?( '+6)7+(+73(1JD?$',<2 DE6)2Z$CD<&52 &5(+7 J37>032$C=65,<K<(032%&)2KE&Q$

2&)= *6t&5J32-%(D%

,<6)738)$ 6 -%(+*$CKE2858Q$C852(+736#-%(+DED<(+73(F65D<DE6)K<6N&)(+7,?KE(+**$',?627:= $'73265K?$t273032E- 657303657",<6C 73&/A36*$1"J$C*@2@,SRD*-$'8)2$'*6036)*=>(03(036)**$CDE6)K '2@6)7361(',<,?2@=>2@8)8)$',/$$C&5&)(--2$C730( 2*dZ$CDE&)2(>27J3DE&)2-,/$ 0$C*=(03J3*$I,?(+KE6 $C&5J3DE,<((',<,?2@&)( &5 K) x$C0 J37$ Q3K<$ (C,<,<2&Q$ $ 'C6)7,?6bJ37/I? 032

5µmB0(C,/$',?$ 032&)(C**2@= $I,?(+KE66]2@7CK?$C03(F02&)(+73DE6)K&'C$'K<6t*$ -%(+*$'K<2858Q$C852(+76CBC&/A6

:DE,/$I,/$>2@73DE,?$C**$',/$ 03JK?$C7,?6 "J36)D ,?(n*$ 'C(+K<(>032d,?6)DE2 $ -%(C,?65738Q$27;JD<&)2-,/$0$C**$ Q3K?$ g&)2@K<&)$

80= &)(+7uJ37$6&)2@6)738)$02d&)2KE&Q$ 2*

50%HJ3K<$C7,?62* =(+7,/$'++2($DQQ32$'=>(

'C6)K<2 &Q$I,?( &/A36'BF-%6)KH(',<,?65736)KE61J37F" 6A 65,E,?2 'C$&)(+7D<6)K&''$C8)2@(+73610365**$ -%(+*$'K<2858Q$C852(+762@7;2@7 %+KE6)DED<(3B 76)&)65D<D<$CK<2@( $C*@*276Q$CKE6"J36)D , " J3*@,<2=>$bDEJ J37( 0365+*2U$'D<D<2U(C,E,?2@&)20365*U= $',<6)KE2$C*@6032&)J32*$ Q3K<$ &)(+= -%(+DE,?$ "J36)D ,?(bD<&5(- ( D ,/$',?$ 273DE6)KE2@,/$>J37$ *$'=>2@7$

λ/2-%6)K

KEJ3(C,/$'K<6;*$ -%(+*$CK<2@8)8Q$'8)2(C736CB=>657,?K<6 - 65K =2+*@2(+K<$CK<6>*7" $C&)&5(--32$C=6)7,?( 036)* Z$CD<&52( 272@73+K<65D<DE(&5(+7n2* &)(+*@*2@= $',<(+K<6 D ,/$',<(273DE6)K<2-,?(1J7b,<6)*@6)D<&5(-32@(1&)(+=6 2@**JDE,?K<$',?(:27 CJ3K?$653*U&Q$'=1-%(b= $CC7365,<2&)(b032,"J$C0K<J-%(+*( +6)76)K?$I,?(;03$u03J6>0365**@6 Q%(Q32@736 J3D<$',?6>$C73&/A36-%6)K1*$ ,?K<$.--%(+*$u= $'+7365,<2&)$ 33 ON j $;&5(+K<KE6)7,?6n(-%6)K<$',?2 ''$ 03J3K<$C7,?6n*$uZ$CDE6 0329 / 02d&)2KE&Q$

12 -%6)KtJ37b+K<$C032657,?6g032d&Q$'=1-%( 032d&)2@K<&)$

10$$CJD<D

/&)=u

t&Q$'=1-32=>$C+736j,?2@&)2D*-3JK0CDE(+73(n&5(+7,?K<(C**$I,?2$',E,?K?$ 'C6)K<DE( ,?KE6x&)(.--326:032 Q%(Q32@73627 &5(+7 %CJ3K?$C852(+76 65*=A3(+*@,<8 (+K ,?(+C(+7$C*@2U,<K?$ *@(+K<(3 *N&)(+7,?KE(+**@(;DEJ3**$;&)(+KEK<657",<6n27"J365DE,?6Q%(Q32@736 &/A36 ''$CKE2$ ,?K<$

06

2 &)2@K<&Q$7365**6 ''$CKE26gZ$CD<20365**7" 6)D0- 65K<2@=>657",<( x76)&)65D<D<$CK<2@(

-%6)K (C,<,<2=2858Q$CKE6 2*F-K<(&)6)DED<(>032|,?K<$CDE 65K<2=6)7,<(06)+*@2$',?(+=2|0$'**$1DE6)&5(+730$ 9 / $C**$,<K?$.--%(+*$1= $CC7365,<2&Q$*|7"J3=6)KE(032d$',<(+=>2%27,<K?$.--%(+*$',?2%$C* ,<6)KE=>2@736H0365**$xZ$CD<6 02|&Q$CKE2&)$C=>657,?( 2= -3J3*@D?$',<( 2@7",<(+K<7(>$

109 $C0bJ373$,<6)= - 65K?$',<J3K?$-%(&5(DEJ-%6)K<2@(+K<6g$'2100µK

Xar^ Kra a' a :[ aZaF1 a 0 M :a [a a a


& X> M#-#$1W&W0!512;*<%D7"$K!5%]J&O&,.(+*<4vQS1.#!5$K9+7.!45!6-6-7"$KWi*<,H45'*I#$%&(0 H^8*=(+90*45*<%D7.!R#<$1%+a@1*-,"E*<%D7"!+#<$'7"!7"90!)"#<$1%0$90%=7.*<45*-.#$1W&!$%0*I#*-..,.!$AWX*-,r1(07'7F,.* 45*(+!52=*-%&"!$%0!&(+*-4+QS1.#!5$W0,"$@*<%0!5*<%D7.*(0144CB M 4P#$14545!237.$1,.*2;$1%D7.7"$ "904545KJ&O0,F08*;(+!)7F%06-*/"$1%0$ '7.7"*/$7'7.!2;!56<6I7"*=Wi*<,H$7'7.*<%0*-,"*4)/23""!23=* K#<*<%06I/!5%9&.#!7.0

( 8

73$ '+(C*@,/$ &)(+= -3*@65,/$I,/$t*$tK?$C&5&)(+*-,/$t036)+*@2$',?(+=27365**$tD<65&)(+70$F9 /pD<227,<6)K0'2@6)736&)(+703J36(-%6)K<$C8)2@(+732dD<J&)&)65D<DE2E'+6x2*d&)J2DE&)(-%( g0322@73&)KE6)=6)7,/$CKE6g*$036)73DE2@,R0365*&Q$C= -32(C7366nK<2@03J3KE736 J3*-,?6)KE2(+KE=>657,?6*$;,?65=1-%6)K<$',?JK?$ J365DE,?( n7365&)65D<D?$'K<2( -%6)KE&/An2*K?$DA K<650 %0$C=6)7,<( 6''$.-%(+K?$',<2E'C( K<2@&/A3265036CB&5(+=>61'+6)0K<6)=(3BU&/A36b*$ 036)73DE2@,R D<2$ *$ -32 p+K?$'73036-%(+D<DE2?Q32@*6Cg#" $CJ3=6)7,?(n02036)7D<2@,R '26)76 K<6Q$'*2858Q$',<( $',E,?K<$ 'C6)KED<(uJ37$ Z$CDE6 736)*@*$1"J$'*6'265736 ''$CKE2$',<(2*d+K<$C032657,?6g032&Q$C= -%(>=>$C+736j,?2@&)( - 65K =>$CDED<2=2858Q$CKE6F*$ 036573D<2-,SR>06)+*@2$',<(+=>2d76)**$9 /F $03J3K?$I,/$>02"J365DE,?$>Z$CD<6 x02&52KE&Q$

20=>D5t4J3&5&)65D<D<2 ''$C=>657,?6x2@*

&Q$'=1-%( =>$C+736j,?2@&)(1'2@6)736:D*-%6)7,?(n650 2*j u06)2Z$'D<&)20329 / '2@6)736 - (+K ,/$',<( $

−229 8C J6)DE,?$&5(+7CJ3K?$C852(+76 -K<6)7036:2@*|73(+=6F032d=6)*$CD<D<$(',<,?2@&Q$ D<J Q %H(.--3*65K)

7 '2@KE, 0365**$n2@7",<6)73DE2@,RuKE203(C,E,/$b6>06)*U0365,<J3732@73 65*6('C$I,?(3B*$bD?$',<J3K?$'8)2(C736 ON 73(+7 x,<K?$CDE&)J3K<$.Q32*@6:6x*7" 6A 65,<,<( '2DE&)(+DE(n036)*@*$ (CK<8Q$>032032E- (C*( Z&5 K5 k kB &)(C=Q3273$',?( $C*+K<$C032657,?6 032 - (+*$CK<2@8)8)$C8)2@(+736+6)7365K?$',<( 0$C*@*$ &5(+7CJ3K?$C852(+76

σ+ σ− K n6&)$C&)6

7365*NK<2@03J3KEK<6 *$ 'C6)*(&)2-,SR;06)+*@2N$',<(+=2 373( $ -%(+KE,?$CK<6>2*U&Q$C= -32@(+736 $'**$n,?65=1-%6)K<$',?JK?$032&52KE&Q$

50µKx$ Z$CDE61032=6)*$CD<D<$ A$ J37$n03J3K<$',/$ 02&)2@K<&Q$

5=DH,<K?$CDE&)(+KED<22B"J$'*2

2*$CD<65K '+6)7+(+73(uD*-%6)7,?2U650 A3$b2@732852( 2@*,?K<$CDE 65K<2@=>657",<(n036)C*2$',<(+=227 ,<K?$.- - (+*$ = $%


+765,?2@&Q$ J6)DE,<( DE&)(-%( '26)76D0- 657,?( -3K<2@= $ 2@**$CD<65KF032K?$ A KE6)030$'=>657",<(b6750µs

03(.- ( "J36)*@*(n032K<2?-%(+= -$C+C2( 2@7 =>(03(n&/A36+*@2$',<(+=2D<2,?K<( '27(u76)**@( DE,?$',?(n (+70$ %=6)7,/$'*6 &5(+7

F = 2 7pKE6Q$C*-,SR $C**@(;D*-%6)C732=6)7,?(;036)2Z$CD<&52N0329 / C*2$',?(+=2UD<2

,<K<( ''$C73(032D ,?K<2EQ3J32-,?232@7,<J,<,<22D<(C,E,?(+*@2E'+65**@2 656)=>$C7>06)**2 '+6)*@*( |F = 2〉 (C2&/A'B&5(+=>6'C6)03KE6)=( $C* -3$CK?$CCK?$' (; NBD<2 '+(++*@2(+7( 2@7",<K?$.- - (+*$CK<6=>$C+736j,?2&)$C=6)7,?6+*2$',<(+=27365**@( DE,?$',?( |F = 1, mF = −1〉 BD<2KE6)7306 76)&)65D<D<$CK<2@(6(A 6j,<,<J$CK<6]J7- (+= -$CC+2( (C,<,<2&)('C6)K<DE( "J36)D ,?(nDE,?$',?( J365DE,?(1'2@6)736:K<6)$C*2@8)8Q$I,?(2**@J3=27$C703(- 65K

200µs+*2$',<(+=203$

J373$x*J3&56 -%(+*$CK<2@8)8Q$I,/$σ−

KE2D<(C7$C7,?6 &)(+7>*$,<K?$C7D<2852(+76 |F = 2〉 → |F ′ = 2〉 #" $CD<DE6032 J3$C7,?2858Q$C852(+76 '26)76F036 72@,?( 2@7 "J36)D ,/$ Z$CD<6H&)K<6)$C7303(J37b036Q (C*6H&)$C= - ( =>$C+736I%,<2&)( *@J373+( * " $'D<D<6n06)**$;,?K<$.--%(+*$;=6)02$C7,?6 *6 Q%(Q32@736n032&5(+=1-%6)7D?$C852(+736' J365DE,?(-3KE(&56)032@=>657,?(0365,<6)KE=>2@7$ J37 -%(+= -$C+C2(x(',<,?2@&)('+65K<DE(>2@*|*2 '+6)*@*(106)D<2@036)K<$',?(

(

4J -- (C736)730( 02 -%(C,?65K $CDED<2=2*$CK<6 * " $I,?(+=(;$'0pJ37X032E- (+*@(;=>$C+736j,?2@&)( -3J37,?2- (+K<=6µ

6>032 -%(C,<6)Kg,?K?$I,<,/$'K<62*U&Q$'=1-%( =>$C+765,?2@&)(u27 =>(03(b&)*$CD<DE2&)(BD<2A$u&/A36 * " 65736)KE+2$b0322@7",<6)K<$C8)2@(+736F0365**7" $',<(+=>(x&)(+7uJ37u&Q$C= - ( =>$C+765,?2@&)( 65DE,?65K<73( F0$',?$0$

UM(r) = −µ ·B(r)Z

46F2@* &)$C=1-%( ''$CKE2$xDEJb,<6)= -32|=(+*@,<(x*@J373+A32 K<2@D*-%65,<,<( $C*F-%6)KE2(03(1032 $CK<=(+K)B*$ -K<6)&56(%03657,?6K<65*$C852(+736xD<2B-3J"u65DE,<6)730365K<6>$C*&)$CD<( "J$C7,?2@DE,<2&)(n6'Bd0$',<(u&/A6* " $I,?(+=( DE6)+J32@KR$C02$.Q$I,?2&)$C=6)7,?6 *$b02K<658)2@(+736036)*U&)$C=1-%(3BD<2,-3J";$CDED<J3=6)KE6>&/A6 *7" $CDED<6032,"J$C7,?2@8(%8)$C8)2@(+7361&5(+273&520$2D ,/$C7,?6 - 65K 2D ,/$C7,?6x&)(+7 ,?$C*6032KE6)852(+736'gH2&5(+73DE6)+J365738Q$n* " 2@7",<6)K<$ %852(+736 ,?K<$1$',?(C=>(:6H&Q$'=1-%(1=>$C+736j,?2&5(:02?-%6)73036HD<(+*@(x0$C*%=(03J*(036)* &Q$C= - (:D<65&)(+730(* " 65DE,?6573D<2@(+736g0365**$1KE6)*$C8)2@(+736 -3KE6)&56)03657",<6

Hmag = gFµB mF |B(r)|

D<2@73+(+*@2D ,/$',<2 6)65= $'7 D<2032@DE,<273+J(+73(n$ "J365DE,?( -3J37,<(u2@7 03J36 &5*$CDED<2 +*@2DE,?$',?2&)(+7

gFmF > 0A$C737( 6)7365K<+2$ =273(+KE6:2@7;&)$C=1-2=6)73(>27,?6573D<2 j- / 61+*@2

D ,/$',<2N&)(C7gFmF < 0

&/A36 A$C773( 6)7365K<+2$;=273(+KE6>2@7 &)$C=1-2 -32 X27,<6)73DE2 j-


/ kX*N,?65K<(+KE6)=>$ 032 273 59+ -K<(+2EQ32DE&)6n27pJ7$ 85(+7$ 0365**( D0-$C852( *(+7,?$C7$0365**6DE(+K<+657,?2U*7" 6)DE2DE,<6)738)$ 032U=>$CD<DE2=22DE(+*$I,?20365*U&Q$C= - (u=>$C+736j,?(+D ,/$',<2&)( H2U&5(+7 %D<65+J365738Q$ DE(+*(u+*2UD ,/$',<2 j- / -%(+D<DE(+73( 65D<D<65K<6 =>$C+765,?2@&Q$C=6)7,<62@7,?K?$D- %-%(+*$',<2 @D<$C=>2@7$C730( *$3+J3K?$ DE273(C,?$ &/A36n736)*N73(+DE,<K<( &)$CD<( +*@22@7",<K?$.- - (+*$.Q32@*2D<(C73( |F = 1, mF = −1〉 B |F = 2, mF = 1, 2〉 )B/K<$1032%6)D<DE2|*(xDE,/$I,?(1&5(+7

mF = 2

=>$C++2@(+K<=6)7,?6D<6573D<2EQ32*@6]$C*3&)$C= - (F=>$C+736j,?2@&)( 6 "J365DE,?(D<2,?K?$'03J3&)6t0$FJ37*$',?( 2@7 J37$=>$C++2@(+K<6-K<(C (+7032@,Rb06)**$ ,?K?$D-- (C*$Bd=>$n0$C** " $'*@,?KE( 27 J373$;$C&5&)K<65D<&52J,?$ D<6573D<2EQ32*@2@,R0365**7" $.--3$CK?$',<($C*K<J3=(+K<6gDEJ3*&)$C= - (= $'+7365,<2&5(D ,?65D<D<(H6)K "J365DE,<( *(DE,?$',?(>0$>7(+227,<K?$.--%(+*$',?(1=>$C+765,?2@&Q$C=6)7,<6 |F = 1, mF = −1〉 $1,?K?$D-- (C*$1=>$C+736j,?2&)$1J,?2@*2@8)8Q$I,/$ g032|,<2?-%( (.A 6I%K<2-,?&/A$'K<0 65M 6)0 g&)(+D ,?2-,?J32-,/$10$"J$',E,?K<( Q (Q2736&)(+=6K?$.--K<6)DE6)7,/$I,?(F27 +J3K<$HON603J36 Q (Q2736&)(+71$CD<DE6 '+65KE,?2@&Q$C*@6036j,?6)KE=>2@7$C73(>J37;&)$C= - (1J3$C03K<J - (+*$CK<6&)(C=>6 "J36)*@*( J,?2*@2858Q$',<( -%6)K *$>9 /FB *$>&5J326)D0-3K<65D<DE2(+736>6)D0-3*2@&)2@,?$ 'B$'*DE6)&5(+7303( (CK<032@736K<2D0- 6j,<,<( $C*&56)7,?KE( 036)*U&52*2@7303KE(u03$ 6)DED<627032E'20J$',?(659+

BQ(r) = B′Q y y +

1

2B′

Q ry ey

Z65 03( '+6

ry =√x2 + z2

Bey = −(z/ry) x + (x/ry) z

6B′

Q

-%6)Kt036 732852(+76∂y|BQ|

U*&Q$'=1-%(g+6)7365K?$',<(:03$C**$ Q (Q27$ L 0365,E,/$ 032&5J3K0''$',<J3K?$ J37&Q$'=1-%(g032?-%(+*$'K<6]*$F&5J326)D0-3K<65D<DE2(+736'B%$C*dD<65&)(+730(>(+KE03276FK<2D0- 6j,<,<(>$'*|&)657",<K<(>036)*d&)$C=1-%( 032"J$C0K<J-%(+*(B

BC(r) =[B

(0)C − B′

C z +B′′C

(z2 − 1

2r2z

)]z +

[B′

C

2rz − B′′

C (z rz)

]ez

ZON

03( '+6$C73$C*(++$C=>657,?6:$ -3K<2@= $rz =

√x2 + y2

Bez = −(x/ry) x + (y/ry) z

6B

(0)C =

|BC(0)| B B′C = |∂z|BC ||

BB′′

C = |∂2zz|BC ||

;46 *@6>,<K<6 Q%(Q32@736D<(+73( - 65K<&)(CK<D<6 0$C**$=6)0365D<2=>$ &)(CK<K<657,?6t6tD<23,?KE( 'C$'73(g$C**$H=6)0365D<2=>$ 032@DE,/$'738Q$F0$C*&56)7,?KE(:DE23A$g&/A36

B′Q =

2B′C

&5(+D<2@&)&/A2@*=(0J3*( 06)*&)$C=1-%( KE2D<J*@,/$'7",<61$CDED<J3=6CBdDE6)= -3K<6$'*D<65&)(+703( (CK<032@736K<2@D*-%65,E,?($C*|&56)7,?KE( 0365*d&Q$C= - ( 02J3$C03K<J - (+*@(3B3*$x (+K<=>$ 65.5

|B| = B(0)C +

(9

32

B′2Q

B(0)C

− 1

2B′′

C

)r2z +B′′

Cz2 Z !

L A14X2;$2=*-%D7"$#FT0*m!4iWX$2;W&EE1!5$_$7'7.!5#<$%0*-445$'7.7.$| 2, 1〉

%0$1%3W09 *-."*<,.*q#$12;W045*7.$&U7.14*m'7.7"$%0$%G_!5%D7"*<,.*-..%D7"*(04W&90%D7"$K(+!8@:!)7F;'Wi*<,.!2;*-%7F45*1


J365DE,?(BKE2&)(CK<0$C703( *7" 6)D*-K<6)DED<2@(+736n0365* - (C,<6)73852$C*@6 =>$C+736j,?2&5( kBU2= -3*2@&Q$ &/A6u2@*&5(+77$'=>657",<( $CKE=>(C732&5( &)(+7n K<6"J36)73856x0$I,?6F0$

ω⊥ '√√√√gFµB

m

(B′Q)2

B(0)C

ωz =

√gFµB

mB′′

C

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B0

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IMAX ' 250

D<2$ &)(+=6F+6)76)K?$I,?(+KE602|,?6)7D<2(C736 VMAX ' 11

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|F = 1, mF = 0〉

hνRF = 1/2µB B(r) +

H6j,<,?$ K<6="J365738Q$n032 (+730( νfondo

"J365**$> K<6J6)738)$ - 65K &)J32D<(+7( 27 KE2DE(+7$C738)$ +*@2$',<(+=2|D<J3*| (C7303(036)*@*$x,<K?$.--%(+*$

νfondo =µB

2hB0 ' 7 GHz

B0

1T' 1.5 MHz

ZOM

03( 'C6 DE2 -%(+D ,?(B0 ' 2

$CJ3DED)B+*@2$',<(+=22@71K<2@D<(+7$'738Q$ $'**$t K<6J6)738)$νtaglio > νfondo

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BDED?$',?$03$C**$1&)(+73028)2@(+736

η =∆ν

νfondo

=B(rrms)

B0

− 1

03( 'C6∆ν = νtaglio − νfondo

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η ' 1

2

mω2〈r2〉µBB0

Z 0$ &)J32 BN0$',<( &/A36 - 65K>J373$ ,?K<$.--%(+*$ $CKE=>(+72&Q$ 〈r2〉 ∝ 〈p2〉 ∝ T

D<2 A$ &/A36'BD&%D<$C7303( 2*B''$C*(CK<6 032

νtaglio

BDE2U(- 65K?$b27 = $'73265K?$nD<65*65,E,?2 'C$n2@7 ,<6)= - 65K?$',<J3K?$ J$C7,?($C*@*$ - (+DED<2EQ32*@2@,SR 032(C,<,<6)7365K<61J7$ K<2-,?6)KE= $'*2858Q$C852(+76F6&)2@6)7,?6'Bd2*-3J37,?(n&5K<J3&52$C*@6 12@*K<$.--%(+KE,<(b,<K?$n*$ -3KE(Q$.Q2*2-,SR 02&)(+*@*2DE2(+76x6)*$CDE,<2&)$

γel ∝ n√TjB&/A360365,<6)K<=273$C73(

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η61"J32@73032KE203J3&56)730(

*$,?65=1-%6)K<$',?JK?$>06)+*@2$I,?(+=2d&/A36 'C6)73+(C73(n6)*@2=27$',<2 NG 6)*&)(CK<D<( 032 "J365DE,<( -3K<(&)65D<D<(,<J,<,?$ '2$2@*3,/$CDED<(x02%&5(+**@2DE2(+7326)*$CDE,<2&/A36t032@=>2@7"J32DE&)6]6 *7" 6&)2@6)738)$:0365*%K<$DA K<6)00$C=6)7,?(-3J" 0365+K?$'0$CK<6'6)KE&/A;&52 " 73(+7 $C&)&)$C0$ *7" 6''$.-%(+K?$'8)2(C736 b*(++$CK<2-,?=2&Q$u736)*,<K?$',E,?(


2728)2$C*6 30 →∼ 1.8

9 8g2743

D 6*276Q$CKE676)*d,?K<$',<,<( 73$C*6 ∼ 1.8 →∼ 1.79 8

2713

D k * ,<6)KE=>2@736 06)*F-3KE(&56)DED<(*$K?$C032@(C K<6"J36)738)$ '26)76 D*-%6)7,/$ 6 C*2 $',?(+=2%736)*@*$,?K<$.--%(+*$ D<2,?K<( ''$C73(g$ ,?6)= -%6)K?$I,?J3K<$H27 65K<2@(+K<6$

Tc

BC736)*@*$ Z$'D<6032&)(+7036)73D<$',?(g032 GN(+DE6(%@273D ,?6527|B &)(+7 J7 7J=>65K<( 032$',?(C=>2032&)2@K<&Q$

1.8 · 105 FG (C,?2$C=(>&/A36'Bd$ &)$CJ3D?$ 06)**@6''$CK<2$C8)2@(+732032,?6)= -%6)K?$I,?J3K<$B&)2@K<&Q$'032$C76]6 -K<(-3KE26t0365*%&52&5*(F032K<2E- 6j,?2@8)2(C7360365**7" 6)D0- 65K<2 %=6)7,?(3B

νfondo

73(+7>D<2%= $C7,<26)76 &5(+DE,?$C7,?6CH2%&)(C73D<65+J36)78Q$1D<2%27,<6)K0'2@6)736 'C$'K<2$'7303(*$ K<6"J36)738)$n032,?$C+*@2(7$C*@6x27;=>(03( 03$ = $C7,<6)7365K<61*$,?65=1-%6)K<$',?JK?$ Q%6)7 $'*032D<(C,E,?(032H"J36)*@*$&)KE2@,?2@&Q$ 6 J273032=>$C7,?65736)KE6 -3JK<( 2@*d&)(+7036)73D<$',?( /]2?-32@&Q$C=6)7,?6g*$032$C+73( %DE,<2&)$;036)*@*7" 6)D0- 65K<2=6)7,<( '265736 6(A 6j,<,?J3$',/$;(CD<D<65K0''$C730( '% ? K g*$u7J Q 6 02N$',<(+=2,?65K<=2&52&/A36HDE2%(',<,?2@6)736 $CKEK<6)D ,/$C703(x* " 6('C$D- (+K<$C8)2@(+736t$C0>J37$g K<6="J365738Q$:032

39 8'46 2@*

7"J3=6)K<(g032$I,?(+=2 -3KE6)DE6)7,?2%$ J6)DE,<( -J37,?( 2@7 6)KE2(+KE6]$1.7 ·106 *7" J3*@,<6)KE2(+KE6N6''$.-%(+K<$ %

8)2@(+736:&)(+730J3K<K*Rn$C**@(>D&'J(C,/$C=6)7,<( 036)*@*$,?K?$D-- (C*$>DE6)738)$ - (CD<D<2EQ32*@2@,R032K?$'++2J73+6)KE6*$g,?65=1-%6)K<$',?J3K<$x&)KE2@,<2&Q$g&)(C7nJ37$x036573D<2-,SR $.- -3K<658)8Q$DQ32*@6Cd6 &5(+73032@8)2(C732(C,<,<2=>$C*23DE(+73('2&56'+65K<D<$ J37$b7"JQ%6,<6)K<=2&)$ &)(+7 J37 7"J3=>65K<(u032U$',<(+=>227,?(+KE73(u$

2.1 · 106 6)0 J37$036573D<2-,SR (C,<,<2&)$ '% ? K N032- (&)(DEJ- 65K<2@(+K<6g$

3

/

$;032$C+73(+D ,?2@&Q$ 06)*N&5(+73036573D?$I,?( (3BB-32 X27 +6)7365K?$C*@6CB0365*N&5(+7,?6)7"J,<( 036)*@*$u,<K?$.- - (+*$=>$C+736j,?2&)$B '2@6)7366A 65,E,?J$',?$ =>65032$C7,<6J7F" 2@=>=>$C+2@736H2@7u$CD<DE(+K*Q32@=>657,?(036)*@*$17"J 'C(+*$$',<(+=>2@&Q$J7;2= -3J3*@D<(0365**$ *@J3&)6032dK<2E- (C=1-$'++2('% 3 ,<K?$CD 6)KE2D<&56:C*2d$',<(+=20365*d&)(+7306)73D<$',?(>0$C**@(1DE,?$',?( |F = 1〉 $C**@(1DE,?$',?( |F = 2〉 B32@=>=6)02$',?$C=>657,?6H0(- (*$7"J '+(C*$n$',<(+=>2@&Q$ '2@6)736x2**@J3=27$',?$10$ J37 Z$CDE&)2@( 032*@J3&)6xK<2DE(+7$C7,<61D<J3*@*$,?K?$'73D<2@8)2(C736032K?$DA K<65030$C=6)7,?( Z&/A36'B$'*&5(+7,?K?$'K<2(;032 "J365**$;032NKE2?-%(+= -$C++2@(3B J373$;,?K?$'73D<2@8)2( %736&/A32JD?$ 6)0 2*#-3KE( *@(u022@7,?6)7D<2@,Rn036)*Z$CDE&)2(b,?K<$CD<=6)DED<( '2@6)7362@7",<6)+K<$',?(u0$bJ37$,?65*65&Q$C=6)K<$ L Lxd*&5(+73036573D?$I,?( '2@6)736x032D ,?K<J,<,?(n03$ J6)DE,?$ -3K<(&)6503J3K?$n6"J32@73032DE2$C&J2D<2@D<&56J7$DE(+*$12@=>=>$C+2@736 -%6)K](C+732d&)2@&)*@( 032dK<2E- 6j,?2852(+76H0365**7" 6)D0- 65K<2@=>657",<(3G 65** " 2@=>=>$C+2@7361K?$'&)&)(C*@,/$u0$C*@*$ L L *7" 27,?6573D<2-,SR >=>2@73(+KE6>2@7 &)(CK<K<2@D*-%(+730365738Q$ 06)**$7"J '+(C*$ $',<(+=>2@&Q$B3-%6)K<&/AF27 "J36)D ,/$ 8)(C7$ +*2 $',?(C=>2%A$C737($CD<DE(+K*Q32-,?(1 (C,<(+732 0$C**$:K<$ %032$C8)2@(+7366"J3273020$ J6)DE,?$ (+=QK?$ DE2 -%(+D<DE(+73(FKE2&Q$ ''$CKE6N27 (+K<=>$C8)2@(+732"J$'7",<2@,?$',?2 '+6

W dRVH"P YM RM3:N^ %YRMYO)T <>

DEJ3**$u7"J 'C(+*$uDE,?65D<D<$b65K:,?65736)KE6 &5(+7,?(;06)**@6 J,<,<J$C852(+732022@7,?6)7D<2@,Ru76)**$CDE6)Kg032KE2E'+65*$C852(+76CB06)**$ *@J3&)6>D*-3J3KE2$;&/A6 279'+65DE,<6n*$u,<6)*@6)&Q$'=>65K?$ 6 036)*@*$ &5(+K<KE6)7,?6 032 QJ32(0365*DE6)73DE(+K<6 L L:BF- 65KH(+C732&5(+73036573D?$I,?( '+6573+(+7(nK?$C&5&)(+*-,?6g,?K<6:2== $'+2732U*$1-K<2=>$ "J36)*@*$1036j,<,?$DE(-3K?$ 2@7u&5J32dD<(C73(1-K<6)DE6)7,?2+*@2|$',<(+=2 B*$1D<65&)(+7303$K<$C&)&5(++*2@6 2* -3K<( 3*(03227,<6)73DE2@,SR>036)*Z$CD<&52(>27 $'D<D<65738Q$n032$',<(+=>2 B=>657,?K<6:*$,?6)KE8Q$ K<$C&)&5(++*@26g2*H-3KE( *(>0322@7",<6)73DE2@,RH036)*@*$ *J3&56032 (C7303(3B'&)2( NDE6)738)$F*J3&56N032K<2E'C6)*$C8)2@(+736C71,?J,<,?6N6N,?KE6*62== $ %+2@732'265736HJ3D<$',?(:2*=6)0365D<2@=>(g,?65=1-%(:02%65D*-%(+D<2@8)2@(+736 27 =(03(g0$:&5(+=1-%6)7D?$CKE6 -%6)K*$&5(+K<KE6)7,?6:032HQ3J32(]$036573D<2-,SR(C,<,<2&Q$ 0365*&)$C= -32(+76 -3J" 65D<DE6)K<6xK<2&)$ 'C$I,/$ &)(+D ,?KEJ36)730(*$=>$',<K<2&56

OD (z, y) = ln

(I2(z, y) − I3(z, y)

I1(z, y) − I3(z, y)

) Z 03( 'C62@*7"J3=6)KE(n27 Q$'D<D<(b2@73032&)$ *$ (',?( DE6)&)(C7303(u*$ DE6="J36)78Q$b0365,E,/$nD<(-K?$4- (',?28I%8)$C7303(n&/A36x2*H- (C=1-$'++2((C,E,?2@&)( DE2$ &5(+= -3*6j,?(61&/A361*$,?K<$C73DE28)2@(+73673(+7 DE2$ 65&)&56)D<DE2 %''$C=6)7,?6 D?$',<J3K?$I,/$B3"J36)D ,/$ HJ7$x=2D<JK?$g036)*@*$g036)73DE2@,Rx7"J3=>65K<2@&Q$:2@7,?6)CK?$',?$*@J373+(:*$032@K<658)2(C736>06)*UZ$CD<&52(u&/A36CB76)*73(+DE,<K<( &Q$CDE(3B&5(+273&52036&)(C7 J37$u032KE6)852(+736K?$'032$C*@6'% N N+

OD (z, y) = σ0

∫n(x, y, z) dx

G 6)**$ (CK<=1J3*$ -K<6)&56)03657,?6

σ0

27032&)$ *$ D<658)2@(+736;0 " J3K ,?(X2@7\$CDED<(+K0Q32=6)7,<( &/A36'B 76)*&)$CD<(2* &Q$C= -32(C736F$.QQ2$xJ373$036573D<2-,SR1DEJ &)6)7,<6)=6)7,?6 Q$CD<D<$03$16)D<&5*J306)K<6032 A J3DE2(+72=1J*@,?2E-3*6 032| (C,<(+732 B736)*@*6F2E- (C,<6)DE2 '2DE,<6 -3K<2@= $B3''$C*6

σ0 = 3λ2/(2π)

$>=>$',<K<2&56OD

K<2@&Q$ ''$',/$0$C*@*6F,?KE6 (C,<( '2@6)736:J,?2@*2858Q$',?$- 65K J373, Q32032@=>6573D<2@(+7$C*@60365**$:032@DE,?KE2?Q3J8)2(C736 $',<,<6)D?$ - 65KNJ37$:7J Q 6H,<6)KE=>2@&Q$ $CJ3DED<2$C7$ ( -%6)KJ37n&)(+7036)73D<$',?(2@7uKE6)+2@=>6F02/]A3(+=>$CD&%#65K<=2 ! jU $C*, 'C6)73C(+73( KE2&)$ ''$',<2|*$ 036573D<2-,SR032-32@&)&)(6*@6x032@=>6573D<2@(+732

σ0365**$>$CJD<D<2$C7$ (nK<$C++2@( 032/]A3(+=>$CDJ%#6)KE=>2 + jFG (C,?2"J36)D ,?2

-$'K?$C=65,<K<2DE2(C,E,?26573612@*7"J3=6)K<(n02$',?(C=>26CBd7365*&Q$CDE(n032J373$b+$CJ3D<DE2$C73$B|*$,?6)= -%6(%K<$',?J3K<$u06)**$n7J '+(+*$u=6)032$C7,?62*#-K<27&)2?-2(n032U6="J32E-$CK ,?2852(+76C J$C7303( 1-K<6)DE6)7,?6J37;&)(+7306)73D<$',?( DE( 'K<$.--%(+DE,<( $C0 J37$ 7"JQ%6:,<6)KE=>2@&Q$ D<2-3J">6A 65,<,<J$CKE6xJ373, $'0;J37$032@DE,<K<2?QJ38)2@(+736 Q32@=>(0$C*@6H$CJD<D<2$C7$16 /]A3(+=>$CDJ%#6)KE=>2

$F,<6)*@6)&Q$'=>65K?$FJ,?2@*2@8)8Q$I,/$ tJ37$ /]A365,?$14 D ,?6)= 4 4 %D?.+9NB&/A36 DE,/$I,/$F273D ,/$C*@*$',?$03JK?$C7,?6 "J36)D ,?(X*$ 'C(+K<( 032t,?65D<2 u,/$C*@6b032D0- (+DE2@,<2E'+( =(+7,/$ J37\DE6)73DE(+K<6 L L 4FG


&LgD+9N B"*6]&5J32%&Q$CK<$',<,<6)K<2@DE,<2&/A36tD<(C73(:KE2?-%(+KE,?$',?6]76)**$ ,/$.Q%6)*@*$F *D<6573D<(CK<6 L L

MYO= WIM 8 nSH2@=>6573D<2@(+732 036)*D<6573D<(CK<627 -3265*

1392 × 1040H2=6)73DE2(+72 0365*dD<27+(+*(-26)*

6.45µm × 6.45µm@ &)6)78Q$1"J$'7",<2&)$$

78073=

40%LN(+KEK<6)7,<602Q3J32($

15oC 2 e−/pixel/sKE(C (+73032-,SR10365*L

14Q32@,

(0 − 16383)

^v1OX*-44) 051> Z,F2;*7.,"!(+*-4"*<%&"$1,.* 9+7"!54!56<6I7.$=Wi*<,45*_!52=231E1!5%0!!%]""$1,.O0!52=*-%D7"$

-3KE(C,?6j,<,<(n0$ J37 (C,E,?J3K<$',?(+KE616)*@65,E,?K<(C732&5( 6)0 1=>$C7,?657"J,?(b$ ,<6)= - 65K?$',<J3K?$ &5(+DE,?$C7,?60$uJ37$;&56)**$ 6)*-,?265K)n*,?65=1-%(;022@7,?6)CK?$C852(+7366*7" $.- 65KE,<J3K?$b036)*@*7" (C,<,<J3K?$I,?(+KE6DE(+73(&)(C7",<K<(+*@*$',<2 03$J7 2@=1-3J*D<( / /t 279'2$I,?( 0$C*| L &/A36g&)(+7,<K<(+*@*$x*$x,?6)= -32@DE,?2@&Q$10365** " 6I%D*-%6)KE2=6)7,?(* ,<6)= - (x,<(C,/$'*6 736)&56)D<D<$CK<2@($C*%,<K?$CD 6)KE2=6)7,?(106)2 03$',?2 0$C**$g,<6)*@6)&Q$'=>65K?$$C*@*$xDE&/A36)0$032$C&J2D<2@8)2@(+736 g032|&52K<&)$

250=D)

$',<6 *@6n032=6)73DE2(+72N03652 -26)*]KE2?-%(+KE,?$',?6 DE(-3K<$ b7365&)6)DED?$CKE2( 27,?KE(03JK<K<6nJ7 2@7 %+K<$C73032@=>657",<( 736)*ND<2D ,?65= $ 032KE2E'C6)*$'8)2(C736C J6)DE,<( uKE6Q$C*@2858Q$',<( 27,?KE(0J3&)657303( J37$&)(.--32$g032%*657",<2&)(+79'+65K<+657,?2 &/A6 -3KE(+26j,<,/$'73(:DEJ3*;-32$C7(:0365* DE6)73DE(+K<6 L L J37F" 2== $'+2 %736t27+K?$C7032@,?$ 036)*&)(C73036)7D?$',<(3 $',<(:&/A6CB&)(+=6t2**@J3DE,<K?$',<( 27 CJ3K?$OKB2*&)$C=>=27(0365*Z$CDE&)2@(>032D<(C730$ &5(+273&52036g2@7 -$CK ,?6&)(+7;2*&Q$C==273(032J373( 036)2dZ$CDE&)20329 /FB 7365&)6)DED?$CKE2(bK<2@D*-%65,E,/$CKE61$C*@&)J3732'273&5(+*22= - (CDE,?20$C*@*6&Q$CK<$',<,<6)KE2DE,<2&/A36x&/A36 2Z$CDE&)206)**$9 / 036'C(+73(b$ '+65K<6'g$ -3K<2@= $036)*@*603J361*@6)7,?2 -%(+D ,/$ 2@7 -%(+D<2@8)2(C736,?$C*6:0$ $ '+65K<62*&5(+73036573D?$',<(n032BG(+D<6I%J@273D ,?6527 $.- -3K<(+DED<2@= $',<2E''$C=6)7,?6g7365* J3(&)( $>,?65*65&Q$C=6)K<$ DE,?$',/$-%(+DE28)2@(+7$',?$:7365* J3(&)( 0365* ,?65*65D<&)(.-32(1&5(+DE,<2@,<J32@,<(10$C*@*6H0J36F*657",<2 6 *$ -%(+D<2@8)2(C736 DE,/$I,/$$C++2@J3DE,?$',/$ 7365=>657,?6:$C&="J32@D<6)703( J37 " 2@=>=>$C+2@736 0365**$1,<K?$.--%(+*$=>$C+765,?2@&Q$ J$C730( *@6n032=6)73DE2(+72N0365**7" 2==>$C+276DE(+73( =>2@732=6 6b*$ 06)73DE2@,SR (C,E,?2@&Q$ b= $CDED<2 %=>$;2*UD<2@DE,<6)=>$;D<2 -3J" &)(C73D<2@036)K<$CK<6n$u J3(&5(3 #" 273+K<$C73032@=>657,?(;KE6Q$C*@6 0365*NDE2DE,<6)=>$ DE,?$',?(u=2D<JK?$',<(u(CD<D<65K0''$C730(;*$b&Q$'03J,/$b*2EQ 65K?$n032J37 &)(+7036)73D<$',?( *#'C$'*(+KE6 (C,<,<6)7"J %,?(

2.22 ± 0.03&/A6> (+KE732DE&)6CBJ3D<$C7303( *$n032=6)7D<2(C736 036)2,-326)*U27 ,/$.Q%6)*@*$n BJ37

Z$',E,?(+KE6F032dD<&)$C*$-%6)K]*6F2@=>=>$C+2@732<-$CKE2$(2.96 ± 0.02)µm/px

:NW M9 PZ^ : PZ^ P:ZMuVYPZRP MYOQP %YT : M

4Fascio

di sonda

Fascio di MOT(sin)

Fascio di MOT(dx)

f = 60 mm

4

BEC

f = 100 mm

Filtro interferenziale

Telecamera CCD

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hqBB36g0$',?$B - 65K

λ = 820.573=uB0$

vB =hqBm

=hk

m' 5.60 mm/s

5


Pt2&)(CK<0$C703( * " 65D*-3KE6)D<DE2(+76 K -%6)K *$ 'C6)*@(&52@,SR0365*K<6j,?2@&)(+*@(>2@7 J373852(+76:0365**$>032- % 6)KE6)738)$ 032 K<6="J365738Q$B D<2(C,<,<265736:&/A6 $ "J36)D ,/$ 'C6)*(&)2-,SR&)(+KEK<2@D*-%(+73036:J37$ 032 A 65K<65738Q$032| KE6="J36)78Q$,<K?$ 2|03J6Z$CDE&)2-$'K<2d$

∆νB =2vB

λ=

2h

mλ2' 13.6KHz

N

7 *@2736)$g02-3KE273&52?-32@( J273032<-%(C,?KE6=QQ%6H65D<DE6)K<6HD<J &)2@6)7,?6H=>(03J3*$CK<6]J37(xD<(C*(x06)2%03J36Z$CDE&)2t&)(+7 K<6J6)73856 032 $C*&5J3732 8' 7,?K<(03J3KEK<6uJ37$ =>(03J3*$C8)2@(+736n$ J6)DE,?$ K<6I%"J36)78Q$x&5(+7 $'&)&)JK?$',<6)8)8)$xD<J >&52657",<6 -%6)KU273(+D ,?K<2DE&)(-3273(C7 ],?J,E,/$ '2$gD<65=1-3*@2&56C6)K"J36)D ,?(:D<2=(03J*$C73(j E 23Z$CD<&526 DE227,?KE(0J3&)6]J373$032 A 65K<65738Q$g0323 K<6"J36)738)$,<K?$*6:03J361=(03J3*$'8)2(C732 LN(+DZ$C&56)730(3B *@( D*-%(+D ,/$C=6)7,?( 032 KE6="J36)78Q$ -3J"n65D<DE6)K<6 (',<,?6I%7"J,?(uZ$C&)2@*=6)7,?6=6)032$C7,?603J36>=>(03J3*$',?(CK<2$'&)J3D ,?((C,<,<2&)2&/A6 -%6)K "B -%6)K -3K<65D<6)K&''$CK<6*$1KE6)*$C8)2@(+736F032|Z$CDE6 ,?K<$2|0J36FZ$CD<&52 B3036('+(+7( J,<2*2@8)8)$CK<6 *$1DE,<6)D<D<$1Q$'D<6g036)2|,<6)= -32

R / '

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8 2@71D<2@73+(+*@(H=(03( $C**$ *J73+A3658)8Q$

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1100D<2@73(1$

60073= =>$:* " 65=>2@D<D<2@(+736t0365*

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0.573= ]63736 ∼ 10

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hq6g032 '+6)KED<2''$C*(CK<2 06)** " $'*@,?658)8)$1036)*dKE65,?2@&)(+*@( DE,<6)D<DE(3 J365DE,?(xA$ -%6)KE=>65D<DE(3B -%6)K*$ -3KE2=>$ 'C(+*@,?$B"032(C,<,<6)7365K<6 J37$:&)$CK?$I,<,?65K<2@8)8Q$'8)2(C736tK<2DE(+*@,?$2@7X,?6)= -%( 036)*@*7" 273D ,/$CJ3K<$CK<DE206)** " 2@73DE,?$.Q32@*2@,Rn0327$'=>2@&Q$ 7 -$CK ,?2@&)(+*$CK<6*6 =2DEJ3K<6 65 % 6j,<,?J3$',?6nDE(+73( 27X$C&)&5(+K<03( &5(+7 2*&Q$C*@&)(+*@( ,<6)(+KE2&)(;736)*KE6)+2@=>6>*2@736Q$CKE6 -3KE6)DE6)7,/$',<( $C*-$'K?$C+K<$' ( N 6)0 6)D ,?6)DE6 $C*@*6 -3KE2=6 ,?K<6u8)(C736;032 GNKE2*@*(+J32@7| 65K "J$C7,<( K<2@+J$CKE0$* " 2@73DE,?$.Q32*@2@,R 6)7365K<+6j,?2&)$:DE2 (+D<DE6)K0''$',<(1&/A36'B&)(C=>6 -3K<6('2DE,<(10$C*@*$ ,?65(+K<2$B"2 DEJ3(+2%6A 65,E,?2032E- 657303(+7(x0$C*@*$ -3K<65D<65738Q$1032%=6)&)&)$C732@D<=2032DED<2?-3$',?2 '2&5(+=>6 $C0 65D<65=1-32@(x*$ -3KE6)DE6)738)$032 $',<(+=2N73(+7 &)(+7036)73D<$',?2 $C*@*7" 27,?65K<73( 06)**$;,?K<$.--%(+*$p7(+*@,<K<6bD<2 u(+D<DE6)K&'C$I,?( &/A36CB$ &)$CJ3D?$ 06)**$>73(C7 (+=>(C+6)73652@,R>0365*D<2@DE,?65= $B 73(+7 65D<2@DE,?6xJ37$ ,?K<$C73DE28)2@(+736:7365,E,/$B|=>$-32@J,<,<(+DE,<( 2@*d7J=>65K<( 02$I,?(+=2dD<2K<2@03J3&56 -3KE(++KE6)D<DE2E''$C=6)7,?6=>$C7 =>$C73(&/A36:J37$> K<$ %852(+736g=>$C++2@(+K<6 036)*d&5(+73036573D?$I,?(032E'265736g6)7365K<+6j,?2@&Q$C=6)7,?6F2@73DE,?$.Q32@*6C7 "J36)D ,?( &)$.-32-,?(+*@( -3K<65D<657",<6)KE6)=( 2KE2D<J*@,/$I,?2N(',<,?657"J,?2 B032@D<&5J,?657303( 27p036j,<,/$'+*2@( *$

<>

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6)7(+=>6573(+*@(++2$:6)=6)K<D<$ 736)*d&5(+K<DE(>032"J365DE,<( DE,<J3032@(3

cdch h ]eNh5r:`ns mns]i ah5a|cs e

65K1&Q$'K?$',E,?6)KE2858Q$CKE6>* " 6(A 6j,<,?(;036)*@*6273D ,/$.Q32@*2-,SR D<J*,/$'D<D<(;032 -%6)KE032@,?$ 0365+*2$',<(+=203$C*&)(C73036)7D?$',<( (&)&)(CK<K<6*2@=>2-,/$CKE6 $C*d=>$CDED<2=(*6F (+7,?2d02KE2DE&Q$C*@0$C=6)7,?(D*-J3K<2@6C]d603J36 (+7,?2 -3K<2@73&)2E-$C*2032]KE2D<&)$C*03$C=>657,?( 6b032-%6)KE032@,?$ 032]$',<(+=>20$C*&5(+73036573D?$I,?( 27 27,?6I%K?$'8)2(C736 &5(+7\2@* KE65,<2&)(C*( D<(+73( 2*tK<J3=(+KE6 0365* &)$C= - ( = $CC7365,<2&)( &/A36 &5(+73032@8)2@(+7$ *$DE,?$.Q32@*2@,RH0365*;-%(C,<6)73852$C*@6032&)(C77$C=6)7,<(:650 2*K<2DE&Q$C*@0$C=6)7,?(F2@7,?K<(03(C,E,?(F0$'*K<65,<2&5(D%*(:DE,<6)D<DE(3 J36)D , " J3*@,<2=(x$CC2D<&56 $x0J36 *2 '+65**2d0$xJ37n*$',?(&)(C7n*6 J,<,<J$C852(+732;-3KE(-3K<2@60365*- (',?6)78)2$'*6 -%6)KE2(032&5( &/A36b036j,?6)KE=>2@7$C73(;J37$ 032EA J3DE2(+736 036)*=(+=6)7,?( $',?(+=2&5(Z&j K)/ 6 03$C** " $C*-,?KE( &)(+7*@6 3J,<,<J$C8)2@(+7323D ,?KEJ3=>657,/$C*@2303( 'J,?6 $C*3KEJ3=>(CK<6]D<J*%*$CD<65K KE6="J365738Q$n6 $C= -32@6)8)8)$ ]6 $C*KEJ3=(+K<6:=>65&)&Q$'732&5( 27,?KE(0(C,<,<( 0$C*@*$'2?Q3K<$C8)2@(+736x06)+*@2D*-%6)&5&/A32 ;7 ,?J,E,?22&)$CD<2*(uD0'J3(C,?$C=6)7,?( 0365*&5(+73036573D?$I,?( $C*@2=6)7,/$bJ37$;7"J 'C(+*$ 032$',<(+=>2%,?65K<=2&)2 &/A36CB&5(+**@203657303(1&5(+7b+*2 $',?(+=2 KE2=>$CDE,<2 7365*d&)(+7036)73D<$',?(B0365,<6)K<=273$C73(J3*-,?6)KE2(+KE2 -%6)KE032@,<6C

65K"J$C7,?( KE2+J3$CK<0$b*6 J,E,?J$'8)2(C7322@7,?K<2@73D<65&/A36 06)* - (C,<6)73852$C*@6>02?-%(+*$CKE6CB6)D<$ %=27$C703(>* " 6"J$C8)2@(+736 + kBD<2 x+2?RDE(C,<,<(+*2@736Q$I,?(&/A3612@*dK<2DE&Q$C*@0$C=6)7,?(27,<K<(03(C,<,<(0$C*KE65,?2@&)(+*@(-3J">6)D<DE6)KE6x=2732@=>2@8)8Q$I,?(12@7",<6)K&'+6)76)7303( D<J3*Nj K<2@D*-%65,<,<( $'**$1KE2DE(D%7$C78Q$ J3=6)7,/$'K<62*UjZ nK<2@&/A3265036 - 65K "032K?$CC+2J37+6)KE6F27,?6573D<2-,SR6)*6(''$',?6 -%6)K=>$C7,?6)76)K<6g&5(+DE,?$C7,?6*7" $C*@,<6)8)8)$1036)*dKE65,<2&)(C*(3B&)(+=6FD<2d6('2@73&)6g0$'**$ + j#" 2@7,?6)7D<2@,R

I 032UJ37 Z$'D<&)2@( *$CD<65K / @9

00

0365&)K<65D<&56n$C**@(+7,/$C73$C7303(+DE20$'*&56)7,?KE(;0365*Z$CDE&)2(DE6)&)(C7303(*$ K<65*$C852(+736

I(r) =2P

πw20

exp

(−2r2

w20

) 53

03( '+6P

27302&Q$u*$ -%(C,<6)738)$ ,<(C,/$'*66w0

2*&)(+DE203036j,<,<( j:0365*NZ$CDE&)2(B&)2@( *$ 032 %DE,?$C738)$ $ &)J2]*7" 27,?6573D<2-,SR 032@=>2@7J2D<&56 032J7 Z$',E,?(+KE6

e2KE2D0- 6j,<,?( $'* ''$C*(+KE6n032-32@&)&5(3

-3$CK<2-,SRb032 -%(C,<6)738)$B J273032 -%(+D<DE2?Q32@*6&)(+= -%6)73D<$CK<6>*7" 6A 65,<,<(;DEJ3** " $C*-,?658)8Q$b0365*KE6(%,?2@&)(+*@( 03( 'J,?(u$C*@*7" $CJ3=6)7,?( 036)*tjZ CBdK<20J3&)657303(b*6102=6)73DE2(+7320365*Z$CD<&52(3B|&52(

= P P9T"P %YM RM3:OQPZO M T

$CJ=>657",?$C7303(C736*$ (&Q$C*@28)8)$C8)2@(+736':**@2=2@,?627 6)K<2@(+K<6x$C**@61032@=>6573D<2@(+732J,?2*@2858Q$.Q32@*2 DED?$',<(b0$n$C*&5J3736 &5(+73D<2@036)K<$C8)2@(+732032 -3KE273&52?-32@(t27 -3KE2=( *@J3(++( 76)&)65D<D<$CK<2@(u&/A6 *6032@=>6573D<2@(+7320365*%Z$'D<&)2@(xDE2$C73(=>$C++2@(+K<2306)**@6 02=6)73DE2(+73206)*&)(+7036)73D<$',?( 2@7>02K<658)2@(+736$CDED<2$C*6

R⊥ ∼ 10µmkB273(+*-,?KE6 73(+7pDE2 'J3(+*@6n27,?KE(03JK<K<6>J37 &)(+77$C=6)7,?( $CDED<2$C*6

&)$CJ3D?$I,?(036)*d+K<$C032@6)7,?6g032d2@7",<6)73DE2@,R6 "J32@73032d(&)&)(CK<K<6F=>$C7,?65736)KED<2$C*@=>6573( J37u(+K<032@736032U+K<$C7303658)8)$ D<(.-3K?$u2**@2=2@,?6 -3K<65&)6)06)7,?6' "J365DE,?6>&)(+7D<206)K?$'8)2(C732D<2$C+C2J373C(+73($C7&/A36g03J36FK<2@*2@6'2032 (+KE03276 D*-%6)KE2=6)7,/$C*@6 2@7 -3KE2=(1*J3(C+(12*Z$',E,?(1&/A36gKE203J3&56)730( *6032@=>6573D<2@(+732|06)*|Z$CDE&)2(DE2KE203J&)6$C73&/A36*$-K<(C (+7032@,R02| J3(&5(6DE6)&5(+730$CKE2$C=6)7,<62@*Z$',E,?(u&/A36CB=>2@73(+KE2D<(+7( *@602=6)73DE2(+732036)2Z$CDE&)2 B=>$C++2@(+K<2D<(+7(;*6032 &)(+*-,SR &/A36>D<22@73&)(+7,<K?$C73( 03J3K<$C7,?6 * " $C*@*2@736Q$C=6)7,<(n036)*K<6j,?2&5(+*(b6)K:,?J,E,?6 "J365DE,<6 &)(+7D<206)K?$'8)2(C732DE2 xDE&)65*@,?( J37 032$'=>6j,?K<(032&)2KE&Q$

200µm$>&)J32 B 7365**7" 2?-%(C,<6)D<2d02$ '+65K<6 J37 Z$CD<&52(n$C*@*$

*@J373+A3658)8)$ 0 " (+70$ 032820

73= *2=2@,?$',?( -%6)K1032 A K<$C8)2@(+736'B&)(+KEK<2D0- (C73036 J373$ 02DE,?$C738)$$C*@*$"J$C*@6F2* jN$CJ3=6)7,/$ 032dJ7uZ$',E,?(+KE6 √

2-$CK?$'=>6j,?K<(032FP $ *652+A -3$CK<2d$

zR =πw2

0

λ' 15

&5= 5

J365DE,?(''$C*@(+K<6 KE2DEJ3*@,?$ -32 n&/A36$C&)&565,<,?$.Q32@*6CB0$',<6F*6H,<(+**@6)K<$C738)6 736)*@*$x065,?65K<=27$'8)2(C7360365**$ -%(+DE28)2@(+736106)*U&)(+7036)73D<$',?(;$C** " 2@7,?6)KE73( 0365**$ &)6)*@*$n6736)*,-%(+D<2@8)2@(+7$C=6)7,?( 036)*@*6(C,E,?2@&/A36g032 (&Q$C*@28)8)$C8)2@(+736'*-3K<2@=>( -$CDED<(-%6)K2*<- (CD<2852(+73$C=>657,?(036)*@*6 (C,<,<2&/A36# HDE,?$',/$:*$:&)$CK?$I,<,?65K<2@8)8Q$'8)2(C736 06)2Z$CDE&)227;JD<&)2-,/$ 03$C**@6 Q3K<6' J36)D ,?( xD ,/$',<( Z$',E,?( =2DEJ3K?$'7303( 2@*032$C=65,<K<(>036)*Z$CD<&52(2@7 032E'C6)KED<6-%(+D<2@8)2(C7326:Z$C&)657303(nJ37 3,g$'**7" 6)D0-$C73DE2(+76:0365*Z$CDE&)2( $',<,<6)D?$ -%6)KH2*=>(03(/ @9

00

6g0$I,/$ 0$C*@*$xKE6)*$'8)2(C736

w(z) = w0

√1 +

(z − z0

zR

)2 53

03( 'C6 2* -$CK?$'=>6j,?K<(z0

273032@&Q$ *$ -%(+DE28)2@(+736 0365* J3(&5( KE2D0- 6j,<,?(u$C*@*7" (+K<2@+276:0365*UK<2@ 65K<2 %=6)7,?(K<2DEJ3*@,?$',?2(C,<,<6)7"J,<2;-%6)K20J36 Z$'D<&)2DE(+73(K<2E- (+K ,/$',<22@73+J3K?$ 53 -3$CKE,<2K<6]03$ "J365DE,?6=2DEJ3K<6'B3$ ' 'C$'*6)703(+D<2 032|D<2@=1J3*$C8)2@(+732<Q$CD<$',?6FDEJ3* (+KE= $C*@2DE=>(g0365**6H= $',<K<2@&)2 G L:BD<2 -3K<(C+65,E,/$',<(J7 DE2DE,<6)=>$ (C,<,<2&)(2@73D<65K<6)703(>0J36:*@6)7,?2 &)(+79'+65K<+657,?202| (&)$C*6 KE2D0- 6j,<,?2 %''$C=6)7,?6

f1 = 300== 6

f2 = 500== D<J3*-3KE2=( 6

f1 = f2 = 500== D<J3*dDE6)&)(C7303(

8 VRP O M"WIP MYQT :ZM RMPWIM%PXPRPNPZYO TP :ZP

0 20 40 60 80 100

Posizione [cm]

0

500

1000

1500W

aist

del

fas

cio

[µm

]

w0 = (148.3 ± 8.7) µm

z0 = (-4.14 ± 0.66) cm

zR = (9.89 ± 0.50) cm

FASCIO 1

0 20 40 60 80 100

Posizione [cm]

0

500

1000

1500

Wai

st d

el f

asci

o [µ

m]

w0 = (172 ± 7) µm

z0 = (2.0 ± 0.8) cm

zR = (12.5 ± 0.4) cm

FASCIO 2

&51> 1,.7'7"*-,"!56<6I6-!$%0* (+*-!:QS1.#!:!5%=9X"#<!n7FA(01445* J&O0,.*_S#xQy,IvJ&E& 0 :$%0$,"!5WX$,'7F7.* 45*`2=!)"90,"*(+*-4(0!512=*<7",.$/(+*-4iQS1.#!5$WX*-,(+!@*<,F'*A(+!5'7.1%06<*(014i#<$1454!5237"$,"*AWX$D7.$!5% 9&.#!7./(+!#!)1.#90%X_J&O0,F0 4"!E%0!nJi#<7.$=(0*<!"!2/OX$4!G"W0!5*<ED7"$=%0*-47"*I7.$&


Z$CDE&)2@(n736)*@*6 -%(+DE28)2@(+73227302&Q$I,?6x2@7 +JK?$ 53 g*KE2D<J*@,/$I,?( (',<,?657"J,?(n&5(+73D<657,?6CB*$ 'C(D%

FASCIO 2

f=500 mm

45.0 cm47.0 cm

f=500 mm

55.7 cm45.0 cm153.3 cm25.0 cm

f=500 mmf=300 mm

FASCIO 1

X 0>+#FT0*-2;A(+*<4545* $17'7"!)#FT0* WX*-,v4)mQy$:#-45!6-6-16<!5$1%0*r(0*<4:,"*<7"!)#$4$X>84?B *<4545!5.'* W0!/.#90,Fq!5%&(+!)#<A45mWX$D'!a6<!5$1%0* W0W&,"$D""!237"!5@[`(+*-4D#<*<%D7",.$m(+*-44)`7.,.1W0WX$45q2;1E1%0*<7"!)#<& ,.1E1E!$7"7"*<%:9+7.!D"$1%&$q,"!)"WX*<7'7"!5@2;*<%D7.*1>(192.4± 2.5) µm

Wi*<,m!4QS"#<!$V*(212.8± 1.5) µm

Wi*<,!54QS"#<!$K

K<$C7303(u$C*@*$>*@J373+A6)8)8)$n0F" (C730$n032820.5

73= Bd&5(+K<KE2D0- (+7036)7,?6$'0 J7jZ ;0$C**$KE2$

λ0 ' 79573= 032 J3$CD<2

100/ 8CB032dK?$CC+2J37+6)KE6CB&5(+7uJ37$ DE,<2=>$1K?$C+2@(+736I%

'C(+*6t03211

= -%6)KZ$'D<&)2@(g032 - (',?6)78Q$gD<J3*@*$ &)65**$B+J37 ''$C*(+KE60365* -$CK?$'=>6j,?K<(s

032&)2@K<&Q$

26)0 J37;&)(+KEK<2D0- (C73036)7,<6:,/$'D<D<( 03202EA J3D<2@(+736g032 (C,?(+72-3$CK<2$

Γsc ' 0.02 sec−1 B%&)(CK&%KE2D*-%(+7306)7,?6x$C0;J37u,?65=1-%(>0206)&Q$'032=6)7,?(

τsc ' 50D<65&CB%=>(+*-,?(=>$C++2@(+K<6 02 "J365**(

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τmag ' 100D<6)&' /J,<,?$ '2$BCKE2D0- 6j,<,?( $C01J37$ 7J '+(+*$ 032$I,?(+=2 ,?65K<=2&52 /B 2@*&5(+73036573D?$',<(

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820%02 (0*<E1,F1(&e,FW&!5(012;*<%D7"*

,"!)(+9&#<*<%&(0$=4);Wi$7"*-%06-;(+!)"WX$%0!O&!45*'9&44);#*<454)0

VRP O M"WIP MYQT :ZM RMPWIM%PXPRPNPZYO TP :ZP

-%6)K<02@,?6 0$C* &)(+7306)73D<$',?(3 J365DE,?( DE2 KE26j,<,?6H736)*%Z$',<,<(1&/A36F*$ '2@,/$:=>65032$036)*|&5(+7303657 %D?$I,?(736)*DE(+*(-%(C,?65738)2$C*6H= $CC7365,<2&)( -$'K<2$ D<(+*@2

(5.0 ± 0.3)D<65&C

423-3J" =2C*2(CK?$CKE6 "J36)D ,?(FKE2D<J*@,/$I,?( D<6NDE23$CC2D<&562@7x=(0( 0$ K<2=xJ3( '+6)KE6N&5(+7,?27"J$'=>657 %,?6 +*@2$',?(C=>2 -%6)KED<20$C*&)(C73036)7D?$',<(x&/A36 A$C7373(:J37$ ,<6)= - 65K?$',<J3K?$ 032<- (&)(D<J - 65K<2(CK<6$1"J365**$&5K<2@,<2&)$ J36)D ,?( -J" 6)DED<65K<6xZ$',E,?( $.--3*@2&Q$'7303($C**$1,?K?$D-- (C*$= $CC7365,<2&Q$J37&)(CD<200365,E,?( DE&)J303( $nK?$C02(C KE6="J36)78Q$ OND5 7 -3K<$',?2@&Q$ *$ K<$C032@(C K<6J6)738)$ '2@6)736*$%D<&52$',?$ $C&5&)65D?$ $C7&/A36103(-%( 2@*d,?6)KE=>2@736g036)*@*7" 6''$.-%(+K?$'8)2(C736 Z&5 K5 / ! ]&5(+DZ$C&)657303(2*H-3KE(&56)DED<( 0326('C$D- (+K<$C8)2@(+736g (+K<8)$',/$ -3K<(CD<6)CJ36DED?$C703( *$ ,<6)= - 65K?$',<J3K?$ 036)*D<2@DE,<6(%=>$ON #" J3D<(n0365**@( D<&5J303(u$ K?$'032(C KE6="J365738Q$nA$ &5(+73DE6)7,?2-,?(b032$'J3=>657,/$CKE61*$1'2-,/$=6)032$g06)*%&)(+7306)73D<$',?(:736)*@*$gD<(+*$F,<K?$.- - (+*$F= $CC7365,<2&Q$g&)(C=>6t2*@*J3D ,?K<$',?(F2@7 CJ3K?$ 53 *KE2D<&)$C*03$C=>657,?( &5(+737365D<DE(;$C* - (',?6)78)2$'*6x=>$C+765,?2@&)( KE2=>$C736:03(+=27$C7,<6xK<2@D*-%65,E,?(n$"J36)*@*(F03( 'J,<(:$'*3K<6j,?2&5(+*(BC=>$H*7" 27,?KE(0J38)2@(+736N06)**@(FD<&5J303($FK?$C032@(C K<6"J36)738)$ =>2@+*2@(+K?$032d=(+*@,<(1*$1DE6)73DE2?Q32@*2-,SR10365*dD<2D ,?65= $$C0u$C*-,?K<6 (C7",<2|032-%6)KE032@,?$

0 10 20 30 40 50 60

Tempo [sec]

0

20000

40000

60000

Num

ero

di a

tom

i nel

con

dens

ato

Scudo a radiofrequenza OFF

Scudo a radiofrequenza ON

X &> ]!)"90,.l(0!A@:!7.j2;*I(+!5\(+*<4#$%&(+*<%X"7"$l%0*-4H"$145$lWX$17"*-%06<!)45*h23E%0*7.!5#<$j#$%d"#<9&(+$\,F1(+!5$Qy,.*-ND90*-%06-"WX*-%D7"$H* 1#-#*I'$X 4:@[14$,"* (+!:@D!7.m2;*-(+!)qW&".q(01!

(5.0±0.3)"*-#Z!

(22.7±2.7)'*I#


P 65DE,?$ 0$ 'C$'*J,?$CK<62*&5(+7,?KE2?Q3J,<(F03( 'J,?(g$C*@*6 J,<,<J$C852(+732DE,<K<J3=6)7,/$'*206)*-%(C,?657 %852$C*@6 - 65K<2(032@&)(3 J365DE,?6D<(C73(>&5(+=6F0365,E,?(K<2&5(+7303J&)2?Q2*2 $C*|KEJ3=>(CK<6g032$C= -32658)8)$06)**$CD<65K)B$C*@*$xDEJ$ *$CKE+A3658)8Q$ D0- 6j,<,?K<$C*6F02|6)=2DED<2(C736 6g$C*@*6 '2?QK?$C852(+732%=6)&)&)$C732@&/A36gD<J3+*@2D0- 65&)&/A32 65K J3$C7,?(uK<2CJ$CK<03$ 2*K<J3=(+K<6x032Z$CDE6 D<J3**$CD<65K)Bd$DQQ32$'=>( C2 R 73(',/$',<(b$C*-$'K?$C+K<$' (: &/A36 "J365DE,<(x73(+7>&)(+7,<K<2?QJ32DE&)6 $C**@6 J,<,<J$C852(+73206)*;- (C,<6)73852$C*@6 - 65K<2@(D%032@&)( -%6)KE&/At73(+71DE2-3KE(-$C$H$C**$tZ$CDE6K<65*$',<2E''$ ,<K?$ 203J36Z$CDE&)2&/A36 (+K<=>$C73( 2*K<6j,?2&5(+*(*KEJ3=>(CK<6 27 $C= -32@6)8)8)$n279'+65&)6 -3J"b6)DED<6)KE6 2@73&)*@J3D<(b7365*73(+DE,<K<(b=(0365**@( 036)*KE65,<2&)(C*(&5(+=>6HJ37nK<J3=(+KE6 D<J3*-$CK<$C=>6j,?KE(

s J6)DE,<(- 65K " 73(C7n27 J2D<&56 D<J**6 =>2@D<J3KE6 -%6)KE&/ACB

&5(+=>6 '2@DE,<($C*F-$CK<$C+K<$' ( B*$ - (',?6)78Q$1279'2$I,/$xDEJ3+*2 $',?(C=>2<'26)76F&)(+7,?KE(+**$',/$27(+C732|&)2@&)*@(1032|KE2?-%65,?2@8)2@(+736H06)** " 65D*-%6)KE2=6)7,?(3B&)(+7D<6)7,<6)7303(>&)(+D032|KE2+6j,<,?$CK<6 *@6 =2D<JK<6&5(+7 65&)&56)D<DE2E'C6 J,E,?J$'8)2(C732|032d2@7",<6)73DE2@,R $ (+7,?6 -3KE273&52?-$C*@6 032KEJ3=(+K<6nD ,?K<J=>657",?$C*6 "J32@73032&5(+DE,<2@,?J2@,/$ 03$C**$ '2?Q3K<$C8)2@(+736=6)&5&Q$C732@&Q$ 0365**@61(C,<,<2&/A36': 03J361Z$'D<&)2 - 65K<&)(CK<K<(C73(n27Z$I,<,?2&Q$'=>=2732(C,E,?2&52=(+*-,?( 032 %'C6)K<DE2 B+&5(+=6N27 -$CK ,?6NDE2 -J" 6'2@73&)65K<60$'**$+JK?$ 5 6'B+$H&)$CJ3D<$H0323J6)DE,?$ 032EA 6)KE6)738)$B$C&"J32D ,/$C73(uJ37 K<J3=(+KE6 032Z$CD<6K<65*$',<2E'C(n&/A36>76*@2=2@,?$>*$n&5(65K<6)78Q$B036)CK?$C0$'7303(u*$"J$C*@2@,R1036)*-%(C,?65738)2$C*6 -%6)KE2(032&5(3 J365DE,?( 6A 65,E,?( >D ,/$',<( 0$b73(+2#"J$'7",<2 &)$',?(u=2D<JK?$C730(u*$ *$CKE+A3658)8Q$b0365**$n73(',/$n032Q$I,<,?2@=>657,?( ,<K?$ 203J361Z$CDE&)2 g6)K J7 K<65,<2&5(+*( 2@036Q$'*6&)(+=6"J36)*@*(1-3K<65D<6)7,?$',?(u$C* -$ %K<$C+K?$I (n B *@( D0- 6j,<,?KE( 032 - (C,<6)738)$ 1&5(+DE,<2@,?J2@,?(0$ J7 -32&5&)(n$

δ032H2K<$C&x&56)7,?K<$',?(

2@7",<(+K<7($C**$g KE6="J36)78Q$∆ν = ∆ω/2π

&)(CK<K<2@D*-%(+7303657,?6g$C*Q$',<,<2=6)7,?(,?K<$12 03J36 Z$'D<&)2*$CD<65K) $1-3KE6)D<65738Q$ 02KEJ3=>(CK<6032Z$'D<6xKE6)*$',?2 '+(Z$>DU&/A36 J6)DE,<( -32&5&)(1-3K<65D<6)7,<2J37$*$CK<+A6)8)8)$72@,/$ J6)DE,?$ >DE,/$I,/$ =2DEJ3K?$I,/$b(+D<DE6)K0''$C703( *(uD*-%65,E,?KE(;032 - (',?6)78Q$ 06)*DE6)+7$'*6F032|J7b (',?(032(03( 'C6)*@(&56FD<Ju&)J32|DE2|D<(C73(279'2$',<2 2 0J36FZ$CD<&52 U42 F(C,<,<6)7"J,?(J37-3KE( *@( 032K<2+$>$CJD<D<2$C73( &5(+7 J37$>*$CKE+A36)858Q$

σν

02(160 ± 20)

8x&)(C7 J37;,?65=1-%(032d2@7,?6)CK?$C852(+736F02

50=D]6 -%6)K]J7$ 'C6)*@(&52@,SR1&5(+K<KE2D0- (+7036)7,?6$

∆ν = 8 8C

L pA90*-'7"$m@[14* $@:@:!512=*-%D7"*YWi*<, 9+7"7"9&16<!5$1%0!DJ&%0$HQy,.*-ND90*-%06<* (+!+N9X4)#FT0* (+*I#!5%&m(0! !545$1T0*-,"6UI(07.45/Qy,.*-ND90*-%06-;(+!8#-2;W0!5$1%&12=*-%D7"$;(+*<48"*<E%&45*H(0*<!Qy$17"$+(+!5$:(0!? %0$4n7.,"*U+%0*<4545*H2;!)'9&,"*_/7"*-2=W&!849&%0E1T0!CU'!P"$1%0$K$D""*<,.@7.*(+*<,.!@*(+!R!%D7.*<%&"!n7Fo3%0$%f*<45!2;!5%&O&!45!vWi*<,F#FT (0$@D907"* E1,F1(+!5*<%D7"!v(+!7.*<2;Wi*<,F7"9&,.'9&E145!8M ?#xQy,I & 0 xU:233#$12/90%&ND90*#$%7.*<%:9+7.*1


#" 6(A 6j,<,<(n032 "J365DE,?(bKEJ3=>(CK<6x=6)&5&Q$C72&)( 27 J6)738)$ +*26)D0- 65K<2=6)7,<227;03J361=(02 ]0$J37 *$',?(&)(CDE,?2-,?J32@D<&56HJ37$273065,?65K<=27$'8)2(C736tD<J3*@*$ '+6)*@(&52@,R:0365* KE65,?2@&)(+*@(:&/A6CB&5(+=6HD<2 '2D ,?(3B"032E- 6573036t0$

∆ωB6 0$C*@*7" $C*@,<K<( &)(+D ,?2@,<J32DE&)6tJ37$g (+7,?6t032KE2D<&)$C*03$C=>657,?(3B9-$'K?$C+( %

7$.Q2*6g$'**6g$'*@,?KE6 J,<,<J$C852(+732d06)*- (C,<6)73852$C*@6 - 65K<2@(032@&)( ,?K<$',<,?$',?6$C*-3$CK?$CCK?$' ( *3-3K<2@=>( 032 "J365DE,<2Z$',<,<(+K<2"&5(+K<KE2D*-%(+7306$C01J7$ 273065,?65K<=27$'8)2(C736UK<6)*$',?2 ''$ 0365** " (+KE03276032

∆v

v=

σν

∆ν' 2%

*DE6)&5(+7303(g$CD0- 6j,<,<(F279'+65&)6]27 J32DE&)6271=>$C732@6)K?$ 73(+7xZ$'&)2*@=>657,?6 -3K<65032&52?Q32@*6U= $ D*-%6)K<2 %=6)7,/$C*@=>657,?6 6('2@036)7,<6&5(+=6 =>(CDE,?K<$',?(x0$C**@6H=2DEJ3K<6 KE2?-%(+KE,?$',?6 27 CJ3K?$53653,LN(+=6

Trappola Fascio 1 Fascio 2 Fascio 1 + 2 Fascio 2 + 20

5

10

15

20

Vit

a m

edia

[se

c]

& &> m!7"*_2;*-(0!*(+*-48#$%&(+*<%X"7"$=WX*-,q(+!@*<,F'*_#<$1%&(+!56<!5$1%&!"Wi*<,.!2;*<%D7F45!?>q?(0;'!5%8 !5%W0,.*-"*<%06I(+*-4"$145$Wi$7.*<%06-!514*H23E%0*7.!5#<$&UD!%W0,"*I'*-%06-/(+*<!"!%&E1$145!XQS1.#!(+!,"*<7"!)#$4$XU!5% W0,.*-"*<%&6-(+*-!8(+90*AQS1.#!(+!q,.*7.!5#<$145$l#$1%D7.,"$W0,"$W&ED%D7"! *-( !5%+J&%0*]!5% W0,"*I'*-%06-g(+!q9&% "$145$gQS1.#!5$k,"*<7",.$1,.! *I'$X % 7.9+7'7.*]4*2;!5"90,.*G;W0W&4!)#<7"$ 45$.#9X(+$V ,.(+!5$Qy,.*-ND90*<%&6-3*I(]!RQS1.#!Z(+!P,"*<7"!)#$4$XUX$@*W0,.*-"*<%D7"!CU'$%0$ #-45O0,.7"!Wi*<,

s = 1.15

D<23'+65036]2* ''$C*(+KE606)**$ '2@,?$H=6)032$ 0365*&)(C73036)7D?$',<(2@7x,?K<$.--%(+*$t=>$C+736j,?2&)$ 6"J365**@2(C,&%,?657"J,?2%2*@*J3=27$'7303(F2*&5(+73036573D?$I,?(1&5(+7 &)2$CD<&5J373(106)2Z$CD<&52%02 KE65,?2@&)(+*@(:DE273C(+*$CKE=>657,?6CBD<(C73( J3CJ$C*2d657,?K<( *7" 6)K<KE(+K<6' J36)D ,?( 27;$C&)&5(+K<0( &)(C7;2*&)$C*&5(+*( % I - 65KH&5J32*$032 A J3DE2(+760323=(+=6)7,?(F2@7303(C,E,/$ 0$F&)2$CD<&5J373(g03652Z$CD<&52 ,?K<$CD<&5J3K?$.Q2*6NK<2D0- 6j,<,<(g$C2-3KE(D%

T :ZPW T"P %YMuRM :WIM9 PZ^ ;: MRPZT3Y M RP UWIT "P %YM 1W T

&56)D<DE2&/A36 $ ' '+6573+(+7(u76)**$,<K?$.--%(+*$=>$C+736j,?2@&Q$ 7,?K<(03J3&56)730(b$C=Q%6)03J36:2Z$CDE&)2032KE65,?2@&)(+*@( $C*@*$:D ,?65D<D?$ K<6"J36)738)$

v = 0jBD<2d$'D<D<2@DE,<6g$C0nJ37$ 736j,<,?$1K<20J38)2@(+736 0365**$ '2@,?$

=6)032$:73(C7n&)(+= -$',<2?Q32@*6]&)(C7n*$:DE(+== $036)+*@2 6(A 6j,<,<2 D<JnDE273C(+*(Z$CD<&52(3 J365DE,?$1K<20J %852(+736 &Q$CJD?$',?$u0$'**6 '2EQ3K?$'8)2(C732=6)&)&)$C732@&/A36>DEJ3+*@2D*-%6)&)&/A2 27Z$',E,?2 BdD<6D<26A 65,E,?J$*$:=2D<JK?$gJ3D?$C703(1J37 DE273C(+*(FZ$CDE&)2(:K<6j,?K<(CK<236)D<DE(1$C78)2&/AH03J36 Z$CD<&52 2@73032?-%6)7036)7,?2 B"*$'2@,?$ =>65032$(C,<,<6)7"J,?$BQ 6573&/A 2@7 6)KE2(+KE61$C*'C$'*(+KE6x0365**$D<(+*$>,?K<$.--%(+*$>= $CC7365,<2&Q$B 032=(+*@,<(DEJ-%6)K<2@(+K<6:$C*&Q$'D<( 032dZ$CDE&)227302?-%6)7303657,?2 J365DE,<( - 65K<&/A'B 7365* -K<2=(&Q$CDE(3B*@6J732&/A36 '2?Q3K<$C8)2@(+732d=6)&5&Q$C732@&/A36&/A36:A3$C7373(>6A 65,<,<(>DEJ3**$ 73(C,/$032Q$',E,?2@=>657",<(D<(+73("J365**60365**(D*-%6)&5&/A32( K<65,<K<(+KE26j,<,<6)7,?6:6:DE(+73( "J32@73032=>(+*-,?(=273(+KE2|K<2@D*-%65,E,?( $'*&)$CD<(03652 0J36 &)$C==>2@732327032?-%6)7306)7,?2 4 (+K ,?J373$',/$C=6)7,<6 2@71"J365DE,/$:&)(+73+J3K?$'8)2(C736 ''$CKE2$CKE6*$ '+65*(&)2-,SRb036)*K<6j,?2&5(+*(b73(C7 ,<6)&5732&)$C=>657,?6 -%(+DED<2?Q2*66032&)(+73DE6)+J6)738)$ 6)D<D<$;73(+7 uDE,?$',/$ $.--3*@2&)$',/$ $C+*@26)D0- 65K<2@=>657",<2t6A 65,<,<J$',<2 42 ;$C7&/A36 (CD<D<65K0''$',<( &/A36 *7" 6)7,?2-,SR0365*K<2DE&Q$C*@0$C=6)7,?(F2@7,?K<(03(C,E,?(F0$'**6 '2EQ3K?$'8)2(C7323=6)&)&)$C732@&/A36t032?-%6)7306 03$C** " $C*-,?658)8Q$ 06)*-%(C,?65738)2$C*6 -%6)KE2(032&5(16g032E'2@6)736F2@* Z$',<,<(+K<6F0(+=>2@7$C7,?6 - 65K

s ≥ 1

egi5hjfne hjrx`ns mns]iXsch5qNri5r

LN(+=6 $.Q Q32$C=(1'2@DE,?(n76)*B-$'K?$C+K<$' (b >2@* -$CK<$C=65,?KE(s

&/A36 036 3732DE&)6 *7" $C*-,?6)858Q$ 06)*KE65,?2@&)(+*@( (C,E,?2@&)( -J"n6)DED<6)KE6 &Q$C*@&)(+*$',?(D<6xD<2&5(+73(+DE&)(+7(n*6x2@7,?6)7D<2@,R 0365203J36:Z$CD<&52 H7-3K<$',?2@&Q$ - 65K "3B'0$',/$ *7" 2= - (+DED<2EQ32*@2@,SR032&)(+7,?KE(+**$CK<6U* " 6(A 6j,<,?2 '+(H$C**@2736)$C=>657,?(t036)203J36Z$ %DE&)2 DEJ3*%&)(+7306)73D<$',?(16 *7" 27,?6573D<2-,SRg6A 65,E,?2E''$:$C+6)7,<6HD<J+*2%$',<(+=>2 B&)(+7 J6)DE,<( =>6j,?(03(D<2-%(C,?KE6=QQ%6)KE(>&5(+==>6j,<,?65K<6F65K<K<(CK<2|DE2DE,<6)=>$',?2@&)2|76)**$10365,<6)KE=>2@7$C8)2@(+736 06)*-$CK<$C=65,?KE(

s

6)K ( ' '2$CKE6x$ "J365DE,<( 273&5(+79'+65732657",<61*$&Q$C*@2?Q3K<$C8)2@(+736F0365** " $C*-,?658)8Q$0365*KE65,?2@&)(+*@( '2@6)7366(A 6j,<,?J3$',/$ D0- 65K<2=6)7,?$C*=6)7,?6&)(C7pJ37p=>6j,?(03( Q3$CD?$',<( D<J**$ &)(+DE203036j,<,?$;032EA K?$'8)2(C736GNK<$C+t0365*&)(+7306)73D<$',?( 032 GN(+DE6(%4@U2@73DE,<6)2@7 5 J365DE,<( =65,<(03(]DE29Q$CD<$tDEJ:J7 6573(+=6)73(DE2&5(u&/A6 -J"b6)D<DE6)KE6 '2D ,?(b27 0J36=>(032&5(+= -3*65=>657",?$CK<2 Pt2=>$C73657303( 76)** " $'=Q32@,<(0365**$ ,<6)(+KE2$ 032 G*(&/A|BDE6 DE2]2== $'+27$u032]&Q$'K<2&)$CK<6b2@7 =>$C732@6)K<$ 73(+7 $C032$DQ$',?2@&Q$;2@*&5(+73036573D?$',<( &5(+7

q = qBBd&52( $C*BQ%(+K<0(n036)*@*$ -3KE2=>$ 8)(+73$b02#GNK<2@**@(+J327 6"J327302032

D0- 65+736)KE6x2@*dK<65,<2&5(+*(D<65=1-K<627u= $C726)K<$ 73(+7 $'032$.Q3$',?2@&Q$B32@*dK<2@D<J3*-,/$',<(D<$CKR "J36)*@*(032


-%(- (C*$CKE6 J377"J3=6)KE( =>(+*-,?( 65*6''$',<( 032 Q$C7306CB&52$CDE&)J37$ 0365**6 J3$C*2]6('+(+* '+6)K*R 7365*K<6j,?2@&)(+*@( (C,<,<2&)( &5(+7uJ37 Z$',E,?(+KE6F6)D0- (+76)73852$C*@6F0$',?(0$

exp(−iEn,qBt/h)

U*dD<J3&5&)6)DED<2 %'+(xD*-%6)+732@=>657,?(2DE,?$C7,/$C76)( -K<(+2@65,<,<6)K*RD ,/$',<2%02"J$CDE2=(+=6)7,?(27DE,?$',?2032=(+=>657,?(D<6-$CK<$',?202

2hkBd&)2$CD<&5J373(b$ '+657",<6$C=1-26)858Q$ 0$I,/$ 0$C*@*7" 27,?65KE 65K<6)78Q$ 0365** " 6('+(+*@J38)2@(+736

,?65=1-%(+K<$C*6 036)2 D<27+(+*2D ,/$',<2|032FG*(&/A &)(+D&)(C=>6 &)$C*&5(+*$DQ32*@6]0$C**$ DM j6)K]J37 " $C* %,?658)8)$>0365*KE65,<2&)(C*(D<J &56)7,?65=>657",<6Q$CDED?$6 -%6)K],?65=1-202d6'+(C*J3852(+736$C*@*7" 27,?65K<73(0365*K<6j,?2@&)(+*@( 0365**7" (+KE032736032$C*&5J37361&56)7,?2@7$C2$>032=2&5K<(+DE6)&)(C73032K<2DEJ3*@,<6)K<$C7373( -%(-%(+*$I,?627=>$C73265K?$xDE2+72 &)$',?2 'C$gD<(C*( *6 &)(+= -%(+736)7,<2d$ '+6)7,<2|2= -3J3*@D<(1KE2D*-%65,E,?2 'C$'=>657",<6

06

2hk

$C*@*$; KE6="J36)78Q$ 032](CD<&)2@**$C8)2@(+736 0365**$ - (-%(+*$C8)2@(+736n032 "J365DE,?6 03J36u&)(+= -%(+736)7,<2]27 J373852(+7606)*,?65=1-%( 022@7,?6)K<$C8)2@(+736&)(C7 2@*K<65,<2&5(+*(3Bd=2DEJ3K?$I,/$n*$CDE&)2$C7303(b6)D0-$C7306)K<6*$7"J '+(+*$n$',?(C=>2@&Q$B032=>(03( &/A61*6:03J36 &)(C=1-%(+73657",<2 -%(+DED?$C73(n65D<DE6)K<6 D0-$C852$C*@=>657",<6K<2@D<(+*-,?6'B - (CD<D<2EQ32*@6 K<2@D?$C*@2KE6 $C*@*7" $C*@,<6)8)8)$x0365*dK<65,<2&5(+*(

s

J36)D ,?( D ,?6)DED<( 6)7(+=>6573( -J"$C73&/A36H6)D<DE6)KE6 '2@DE,<(1&)(+=6HJ7$x,?K<$C73D<2@8)2@(+736 P $C=>$C7>DE,?2 %=(+*$',?$N,?K<$ 03J6D ,/$',<20F" 2@=1-J3*DE( 02EA 6)K<657,?2"&)(+=6U27302&Q$I,?(]27 +JK?$ 53 N#" $',?(+=(3B-%6)K,?65=1-203227,?65K?$C852(+76xD<J >&56)7,?65=>657,?6 Q3K<6('2 B '265736x,<K?$CD 6)K<2-,?(>0$ J373( 06)20J361Z$CD<&52*$'D<6)K271J73(FDE,?$',?( '2K ,?J$'*6N03$C* "J$'*6KE2@,?(CK<7$ $C**@(HDE,?$',?( (C730$C=6)7,/$'*6 - 65K6)=2DED<2(C736DE,<2=(+*$I,/$10$C*@*7" $C*@,<K<(xZ$'D<&)2@(36)K *$ &)(+7D<6)K&''$C8)2@(+736036)*@*$ J3$C7,?2@,R02=(C,<(3B303J3K<$C7,?62* -K<(&)6)DED<(1* " $I,?(+=(g$C&="J32@DE,?$:J37>2= -3J3*DE(

2hk6 032&5(+73D<65+J365738Q$xJ7F" 65736)KE+2$&)2@7365,<2&Q$

-$CKE2%$4Erec

B=6)7,?KE6 -%6)KE036HJ37 " 65736)KE+2$0$',/$g0$C*@*$F032 A 65K<6)78Q$x03265736)KE+2$g,?K?$g2 (C,?(+7203652d03J36FZ$CD<&52

h∆ωUG 6&5(+73DE6)+J36&/A362*F-K<(&)6)DED<( gK<2DE(+7$C7,<6FD<6

∆ω = 4hk2

2m

5365

( ' '+6)KE(3B&)(+=6FD<2d(',<,?2@6)736FJD?$C730(>*$ K jBD<6g*$ 'C6)*(&)2-,SR 0365*dK<65,<2&5(+*(

v =∆ω

2k=hk

m

53ON

&)J2&)(+KEK<2D0- (C73036J37 J3$CD<2@=>(+=6)7,<(hq = hk = hqB

-$CKE2U$ "J365**@(u02#Q (+KE03(b8)(+73$ ' '2$C=6)7,?6>D<6 * " 2@7,?6)7D<2@,Ru06)2*$CDE6)K =(+*@,<( 65*6''$',?$u(;2* jZ =>(+*-,?(uK<2@03(C,<,<(-%(+D<DE(+73( $ '+6)KE6bJ37 6A 65,<,<( =>2@D<J3K<$.Q32*@6$C73&/A36 -3KE(&56)D<DE2t$ "J$',<,<K<(3BU(C,E,?( 65&)&C (C,?(+722*N&)J32t6A 65,E,?( uZ$'K &)(+= -$CKE2K<6nJ*@,?65K<2@(+K<2&)(+= - (C736)7,?2]0 " 2@=1-3J*D<( &5(+=>6b6('2@036)78)2$I,?(

T :ZPW T"P %YMuRM :WIM9 PZ^ ;: MRPZT3Y M RP UWIT "P %YM 1W T >

| >

| >

& +> +#FT0*<23(+*-44)7",F%&"!6-!$%0*;s ,FEE&>H4?B 7.$12;$'90O&!5.#*;90%kW0,.$+#*-."$V(+90*=Qy$17"$1%&!r#FT0*;%0$1%@,.!5=45$;'7.7"$;!5%7.*<,.%0$&U+23;#<12O0!)=4?B !52=W&9045"$3(0

0

2hk

2k4k6k8k 2k 4k 6k 8k0k10k 10k

& 0> t "*<2;W0!5$/(+!(+! i,F6-!$%0*ms ,FE1E_(+!i$1,F(+!5%0*q*-4*-@[7"$XU45*A#<$1%&(+!56<!5$1%&!X"WX*-,"!52;*<%D7.14!'$%0$_90%8B !5%+a7"*-%&'!7.og(+!`#!5,.#-

10 W/cm2*-(j90% (+*<7"90%&!%0Eg(+!#!5,.#-

2π × 1^ q6V(04545h,"!5E He(+*<4

87qO8 +$a

%0$f#$!%:@$147"!ZW0,.$+#*-."!Y10

Qy$7"$%0!Y#$2;*;7"*-'7"!52;$1%0!)7.$f(0454)Wi$1Wi$14)6-!$%0*;W0,.*-"*<%D7"*;%0*-445$f7F7.$(0!2;$12;*<%D7"$

±10 h

! VRP O M"WIP MYQT :ZM RMPWIM%PXPRPNPZYO TP :ZP

27 +JK?$ 53O! J365DE,?( -3K<(&)65D<DE( D<2 -3J" ,<K?$',E,/$CKE6u&5(+=6uJ7$ 'C6)K?$ ,?K<$C73D<2@8)2@(+736 P $ %=>$C7nDE,<2=(+*$',?$*$x&)J2 K<6J6)738)$032FP $.Q2 FD*-%6)KE2=6)7,/$'*=6)7,?6 =>2@D<J3K<$.Q32*@6 DE6)+J6)7303(*7" 6'C(+*J8)2(C736g036)*@*$ - (-%(+*$C8)2@(+736gK<65*$',<2E''$ 036)203J36 -32&5&/A32$'*H''$CKE2$CKE6g036)*,?65=1-%( 0322@7 %,?65K?$C852(+76;,?K<$ 2* &)(+7036)73D<$',?( 6502@* KE65,<2&)(C*(3 J365DE,?$ K<6="J365738Q$p032 P $.Q2# *@6)$',?$$C*@*7" $C*@,<6)8)8)$ 036)*H-%(C,?65738)2$C*6 -%6)K<2@(02&)(0$C*@*$ D<65=1-*2&56gK<6)*$C8)2@(+736 NN

νRabi = V0/h0$

&)J2 B3KE2&)(CK<0$C703(*$1036 732852(+76F032s ! jB3DE2d(C,<,<26)76

s =2νRabi

νrec

53 !

03( '+6νrec = Erec/h

:4 -%6)KE2=6)7,/$'*=6)7,?6:*$ -3KE(&56)03JK?$ 032=2DEJ3K?$ &)(+= -3*@2&Q$I,/$0$C*Z$',E,?(u&/A36CB03$',?6*6K<2@03(C,E,?6032=6)73DE2(+72036)2UZ$CDE&)2#Z&j K)5 kB >736)&56)DED?$CKE2(;6(A 6j,<,<J$ %K<6>*$b&Q$C*@2?Q3K<$C8)2@(+736$C*@*7" 27,?65K<73(b0365**$b,<K?$.--%(+*$uD ,?6)DED?$ LN(+DtZ$C&56)730(3B+*@2U$',?(+=2&/A36&)(C=1-32@(+73(u*$b,?K?$'73D<2@8)2(C736 GK?$C+ D<(+7( 27 =>(C,<( $C*@*7" 27,?65K<73(b0365**$b,<K?$.--%(+*$ DE,<6)DED?$ 6&)(C**2@03(+73(>&)(+7 +*@2$',<(+=2 6)KE=>2736)*DE2D ,?6)=>$ 032KE2@ 6)KE2=6)7,?( 036)**$.Q%(+K?$',<(+K<2@(3(+2@&/A$"J36)D ,/$165736)KE+2$1*@6H&)(C**2@D<2@(+732D<(C73( 27 (C730$

s6 -3KE6''$C*657,?6)=6)7,<6F6)*$'DE,?2@&/A36CB*7" 6A 65,E,?(

&)(C=1-3*@6)DED<2E'C( >&)K<6)$CK<6>J37$u7J '+(+*$ 032$',?(C=>2U$ '+657,?2N2@=1-J3*DE(u=6)02( -$CK<2$hk

B&/A36"J327032NKE203J3&56 *$ '2DE2?Q32@*2@,Rn036)203J36 -32@&)&/A32&)(+7 =>(+=6)7,<(

06

2hk&5(+=>6>=>(CDE,?K<$',?(

27 +J3K<$ 53OK J36)D ,?( 273&5(+79'+6)726)7,<6 '26)76 D<J - 65K?$',<( 27 03J36>=>(032 :0$ J7X*$',<( DE2

& :>k^v,"*]!52;2;1E1!5%0!m$17'7.*<%:9+7"*gW0W&4!)#<1%&(+$j4H#$%&(+*<%X"7"$l90% !52;W0904)'$ s ,FEEl(+*<4545\(+90,F7.,.!5"WX*<7'7.!@2;*-%7.*H(0!

83U167

*625 µs

%K7.9+7'7.*H*m7",.*A45*H!2;23E!%0!i!54#<$1%&(0*<%&.7.$$1,.!5E1!5%&,.!$_GND90*<4545$]"!%0!)'7",F0UR2=*-%D7",.*Kf(+*-'7",FGK@:!)'!5O0!54*3!54rW0!)#<#<$]Qy$1,.237.$f(0E4!r7"$2=! #$1%l2;$2=*-%D7"$

2h >;"!Y%0$7.!

4CB 14$%0*_#!5,.#<$14),.*qQy$1,.237.$;(&E145!87.$12;!8#FT0*T&1%0%0$3#$2=W&!9+7.$3(+! 9&"!$%0*_!5%V$%&(0s

&)65K<&)$1032 =2732@=>2@8)8)$CK<6t2*,<6)= - (1&/A6F27,?65K<&)(CK<K<6 ,?K?$:*(1D0- 65+732=6)7,<(1036)* K<6j,?2&5(+*(:6)0n2@*K<2@*$CDE&)2@( 0365+*2$',<(+=20$'**$],?K<$.--%(+*$t60$'**7" $C*-,?K<(tDE26A 65,E,?J$ J37 3,06)*&56)7,?KE(F032= $CDED?$

T :ZPW T"P %YMuRM :WIM9 PZ^ ;: MRPZT3Y M RP UWIT "P %YM 1W T

0.1 0.2 0.3 0.4 0.5 0.6

Durata dell’impulso [ms]

420

440

460

480

500

520

540

Pos

izio

ne d

el c

entr

o di

mas

sa [

pixe

l]

& 0> t '*-2;W0!$H(+!7.,F7.90,Fq(+*<4D,.*7"!)#$4$m2=*I(+!)%D7"* (+! ,F6<!5$1%&*r(+!:s ,FEE& 4DJ07P1(_90%B $D"#<!454516<!5$1%0*'!5%:9&"$1!)(045*HQy$1,.%0!5.#*_4)=Qy,.*-ND90*<%&6-0>

νRabi = (2.782 ± 0.036) m61m.%&(+$;4)] X D`'!8$17'7.!*-%0*Wi*<,

λ = 820.45(1)%&2VU:!548@45$1,.*1>

s = 1.631± 0.021

0365**$u032@DE,<K<2?QJ38)2@(+736 032U036573D<2-,SR;$'&="J32DE2@,?$u7365** " 2@=>=>$C+2@736 LN(CDZ$C&)657303(3B0$',<( &/A36*$ -%(+DE28)2@(+736 06)+*@2d$',?(C=>2 032?-%6)7306F0$C** " 2@=1-J3*DE(x&/A366)DED<2$ '+6('C$'73( $C* =>(+=6)7,<( 036)*@*(D0- 65+732=6)7,<(06)**$ ,?K<$.--%(+*$BC2*&)657",<K<(0323=>$CD<D<$ 036)*@*$ 032@DE,<K<2?QJ38)2@(+736N023036)7D<2@,RF736)*@*(D0-$C8)2@( K<6)$C*6 (+K<732@D<&56 27 (CK<=>$C8)2@(+732D<J*|&)657",<K<(032 = $CDED?$17365**@(xD*-3$C8)2@( 036)2 =>(+=6)7,<2 6"J32@73032DEJ3**$1-%(- (C*$C852(+736K<65*$',<2E''$ 032$',?(C=>2$ '+6)7,<2=>(+=6)7,<(

2hk 7 6)D<65=1-2( 032

&)$C*2EQ3K?$C852(+76x&)(C7 J6)DE,<( =65,<(0( 2*@*J3D ,?K?$I,?(27+JK?$ 53 + 7$ 'C(+*@,?$ (C,<,<6)7"J,?(n2@*''$C*@(+K<6:032

sB 7(C,?(n2@*DE6)+7$'*6xDEJ32 (C,<(02(032 '% 3FO+ D<2 -J"n2= - (CDE,/$'K<62*K<65,<2&5(D%

*@(;$C*@*7" $C*@,<6)858Q$ 0365D<206)K?$I,/$bK<2&5(+K<03$C7303(u&/A36CB&)(+=627302&Q$I,?(b0$C*@*$ O+ kB* " $'*@,?658)8)$b06)*-%(C,?65738)2$C*6 -K<(-%(+K<852(+73$C*6F$'**7" 27,<6)73DE2@,SR

m!5*<%0*;#<145#<$14)7.$K!4P2;$12;*-%7.$ W0,.!52=$V(+*<454)(+!)'7",.!O09&6<!5$1%0*=(+!Z(+*<%&"!7.o$7'7.!5#- #FT0*U#$12;*=@:!5'7"$V4W&,FE,.Qy$ & G_90E19X45*;ND90*-445$3(+*<4545;(+*-%&"!n7Fo3(+!231."&>

xCM =∑

z,y z · OD(z, y)/A

yCM =∑

z,y y · OD(z, y)/A

(+$@1*A!5%&(+!)#<=!54QS7"7"$1,.*_(+!8%0$,"2314!56<6I6<!5$1%&* ∑

z,y OD(z, y)

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hkce stmbh5e mns]ipqNr:`bmnst`uaeFcr

h5`ba|cefbhji5hkcl mnh5`ue h5qe

( '( /

$ -3K<(&)6503J3K<$ D*-%6)K<2@=>657,/$C*@6 $C03(',<,/$I,/$ *$;D<65+J36)7,<6 n03(-%( $ '+65K-3K<(03(C,E,?( 2@*&5(+7 %036573D?$',<( &5(+=>60365D<&)KE2@,E,?( 76)*&Q$.-2@,?(C*( -3K<65&)65036)7,?6'B|2*dKE65,<2&)(C*((C,<,<2&5( '265736:$C&)&56)D<( 27=>$C73265K?$$C032$.Q$',<2&)$1&)(+7uJ37$K?$C= -$ 03203J3K<$',/$

tRAMP

6 "J327302dD*-%6)7,?(>27u= $C726)K<$$C032$.Q$',<2&)$b0(- ( J37 2@7,?6)K&''$C**@(u02U,<6)= - (

∆t LN(C7",<6)= - (CK?$C736)$C=6)7,?6>$C**@(bD*-%6)+732 %

=6)7,?( 0365*K<6j,?2@&)(+*@( '265736 D*-%6)7,<( $'73&/A36n2* - (',?6)78)2$'*6 $'K<=(+732&5(;6'B03(-%(28

=>D:0326)D0-$C73DE2(+76CBF'265736 =>2@D<J3K<$',?(>2*7J=>65K<(n032$I,?(+=2 -K<6)DE6)7,?276)*&5(+73036573D?$',<(b=>65032$C73%,?6nJ7F" 2@=>=>$C+2@73627 $CDED<(+K0Q32=6)7,?( &5(+=>6 036)DE&)K<2-,<,<( $'*-$CK<$C+K?$I ( K P 2E- 6j,?657303(*7" 6)D0- 65K<2@=>657",<( -%6)K 032 '+6)KED<2 ''$C*@(+K<2032

∆t032 '2@6)736 - (+DED<2EQ32*@6 K<2@&)(+D ,?KEJ32KE6 *7" $C7303$C=>657,?(

0365*7"J3=6)KE(>032$',<(+=>2dKE2=>$CDE,<2|7365*&5(+73036573D?$',<( 2@7 J373852(+76:0365*d,?6)= -%(,<K?$CDE&)(+KED<( 7365*K<6j,?2@&)(+*@(3 7n,<2?-32@&)( K<2@D<J3*-,/$',<( 032"J36)D ,/$ -3K<(&)6503J3K<$ gK<2E- (CKE,/$I,?(12@7 3+J3K?$53 M LN(+=6D<2H'+65036x2KE2D<J*@,/$I,?2dD<(C73( 2@7 $C&5&)(+KE03( &5(+7;J37 036)&)$C032@=>657",<( 65D*-%(+736)78)2$'*6g,?2?-2&)(032J37-3KE(&56)D<DE( 27X&5J32*$u K?$C852(+736>032$I,?(+=2 -%6)KED<276)** " J732@,Ru032,?6)= -%( 73(+7p032E- 6573036 03$C*7"J3=6)K<(032$I,?(+=2

N/N = −γ ⇒ N(t) = N0 e−γt 53OK

$:&5(+DE,?$C7,?6γ

27302&Q$g2*3,?$CD<DE(x032<-%6)KE032@,?$ -%6)KJ3732-,SRg032,?6)= -%(3B"=>657",<K<6 $C*D<J3(:279'+65K<DE(

τ = 1/γDE2-3J"0$CKE62@*|D<2@+732 &Q$',<( 032 <Z j /I%j)CI

d( D*-%6)C732=6)7,?( $C032$DQ$',?2@&)(036)*K<6j,?2&5(+*( 1J,<2*@6:$'*,<2?-%( 032=>2@D<J3KE6:0365D<&5K<2@,E,?61DE(-3K<$B-%6)K<&/Ax&)(CDZ$C&)657303( DE2KE2?-%(+KE,?$C73(*6g032 '+6)KED<6:&)(+= - (C736)7,?20F" 2= -3J3*@D<(&)(+KEK<2@D*-%(+7303657",<2$C*@*( DE,?$',?(>032GN*(&/Au27b&)J322* &)(+7306)73D<$',?( DE,?$',?(&Q$'K<2&)$',?(B$C0 J7F" J732&)$>&)(C=1-%(+73657 %,?6'BK<6)7036)730( &)(+D#-32uD<65=1-3*@2&56F2*d&)(C7",<6)+C2(036)+*@2$',?(C=>2 #" $C032$.Q$',<2&)2-,SRx2@7uZ$'D<6032$C&5&)6)7D<2(C736n036)*NK<65,<2&5(+*( n(',<,?657"J,/$ $.--*2&)$C7303(;J37$ K?$C= -$ $D--3K<(CD<D<2@= $I,?2E''$C=6)7,<6*2@736Q$'K<66 03203J3K<$',/$ ''$CKE2$.Q32@*6N$C2CCj 036)20J36t=>(03J3*$',?(+KE23$C&5J3DE,<(D%(',<,?2@&)232@73D<65K<2-,?2DEJ32

P*<,Z45*`(0*<%&"!n7FoA(&A%&$1!+,FE1E!9&%7.*r!540#$%7.,"!5O09+7.$H1445* WX*-,.(+!7"*(+$@:9+7.$HH,.!5#<$12/O0!%X6<!5$1%&!DA7",.*`#$1,.W0!N ∝ N3

U0GH7",F1.#90,FO&!45*/z

P T MRP T RM3:^ %YRMYO)T :P YO TP: P RPZYT P ^+T

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0.5=D)

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0 5000 10000 15000 20000 25000 30000

∆t [ms]

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s = 0.2*@1*-4$+#<!n7Fo

v = 0.3 vB

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0.26

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(0.56 ± 0.01) qB

6

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0 0.5 1 1.5 2

q/qB

0.000

0.002

0.004

γ [

ms-1

]

0 0.5 1 1.5 2 2.5 3

q/qB

0.00

0.02

0.04

γ [

ms-1

]

0 0.5 1 1.5 2 2.5 3

q/qB

0.00

0.04

0.08

γ [

ms-1

]

0 0.5 1 1.5 2

q/qB

0.0

0.4

0.8

| ωi |

[m

s-1]

0 0.5 1 1.5 2 2.5 3

q/qB

0.0

1.0

2.0

| ωi |

[m

s-1]

0 0.5 1 1.5 2 2.5 3

q/qB

0.0

2.0

4.0

| ωi |

[m

s-1]

A

B

C

D

E

F

I banda II banda

I banda II banda III banda




I banda II banda

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s


0 0.2 0.4 0.6 0.8 1

q/qB

0

0.005

0.01

0.015

0.02

0.025

0.03

γ [m

s-1]

0 0.2 0.4 0.6 0.8 1

q/qB

0

0.01

0.02

0.03

0.04

0.05

γ [m

s-1]

X[+>Mq%&(&2;*<%D7"$H(+*<47.""$q(+!DWX*-,.(0!n7Fm(+!:7.$12;!(014D#<$1%&(0*<%&.7.$%0*<4545mW0,.!236-$1%&q(0!Ds ,.!454$90!5%Wi*<,

s = 0.2 ;*

s = 1.15 vk,"!5Ee@1*-,'7.!5#-45*V!5%&(+!)#<g4)l'$E145!5g#<145#<$14)7FhWi*<,

4CB !5%&'7.9&,.1,."! (+*-44CB !%&'7.1O0!54!7.og(+!5%&2;!)#<\q = 0.56(1) qB

WX*-,s = 0.2

*q = 0.54(1) qB

Wi*<,s =

1.15xd*f,"!5E1T&*V$,"!56<6-$1%D7.14!!5%w#!)1.#90%XgJ&E190,Fg*<@:!)(+*<%06-!51%0$k90% 9&2=*-%D7"$l%0*-47.""$j(+!qWi*<,F(+!7.

,.!5"WX*<7'7.!@2;*-%7.*/(+!P90%fQS7'7.$1,.*4WX*-,

s = 0.2*/(+!P90%fQS7'7.$1,.*

10WX*-,

s = 1.15!5%h#$,",.!)'Wi$1%&(0*<%06I

(+*-44CB *-%D7",F7.=%0*-4,"*-E1!52;*(0!8!%X7FO0!54!7.o;(+!5%&12;#-0

(0.52 ± 0.07) qB

jB2*,/$CDED<(u032#- 65K<032-,/$ A$bJ37$b=>$CK<&)$',/$ ''$CKE2$C852(+7361$CJ3=6)7,/$C703(b032&)2@K<&)$bJ7 (+KE03276 032+K?$'73036)858Q$B&5(+=>6 6('20365738)2$',?(b27 CJ3K?$ 53 17 ,/$C*@6 +J3K<$n*$K<2@$ '+65KE,?2@&Q$C*@6&)(+KEK<2D0- (C73036$C**$ D<(+C*2$ DE(-3K<$u*$ "J$'*6*$ ,?65(+K<2$ -3K<65032&56>* " 65D<2D ,?65738Q$032=>(032&5(+7 |ωi| 6= 0

6)0X2@73032&)$u2@*,-3K<2@=>( -3J37,?(;032U032DE&)(+7,?2@7"J32@,R 7365**7" $C7303$C=>657,?(032 |ωi|

K<2?-%(+K ,/$',<(;7365**$ D<6)&5(+730$ &5(+*(C737$;032 +J3K<$ 53 LN(+=6 D<2 '+65036b27 "J365DE,?(&Q$'D<(b*7" $C&)&5(+K<0( "J$'7",<2@,?$',?2 '+(n$C0 J3*-,?65K<2(CK<61&5(+7 6)KE= $ &/A62@* 6573(+=6)73(b(+DED<6)K&''$',?( -3KE(-3K<2@( *7" 273D ,/$.Q2*2-,SR0327$'=>2@&Q$73(+*-,?KE6CB+(+*-,?KE6 "J365DE,/$FDE(++*2$BI736)* -3K<( 3*( 032036)7D<2@,R 036)*&)(+7036)73D<$',?(F&5(+= -$C2(C73(HD ,?K<J,0%,?JK<6 $ K?$C73C6C ,<2@,?(C*( 03265D<65=1-32@(gD<2K<2E- (CKE,/$ 2@7 +J3K<$ 5 FJ373$DE6="J36)78Q$x0322== $ %+2@732K<$C&)&5(+*@,?$ - 65K

s = 1.152@7 &)J32*$ -3KE6)D<65738Q$b032B"J365DE,<6 K<$C73+6 -$CK ,?2&5(+*$'K<=6)7,?6

P T MRP T RM3:^ %YRMYO)T :P YO TP: P RPZYT P ^+T >

6('2@036)7,<6C$-K<6)DE6)738)$02 "J36)D ,?(,?2E- (102|DE,<K<J,E,?J3KE6 K?$C73C2$',<6H7365**$:032D ,?K<2EQ3J3852(+736 032

RE

TIC

OL

O2 ms 5 ms 10 ms 20 ms 32 ms 35 ms 55 ms

&5 &> 2=231E1!5%0!q!5% 1.'$,"O0!52;*<%D7"$l(+*-4m#$%&(+*-%&"7"$j(+!As $D'*<a t !%&'7"*-!% Wi*<,V(+!@*<,F'!`7.*<2;W0!H(+!!%D7.*<,F6<!5$1%&*#$%\!54`,"*<7"!)#$145$e[@*<%D7"*

s = 1.15*-(\!% 2=$17"$k#$%\@1*-4$+#<!n7Fo

v = 0.55 vB

:!%0$7.!`45W0,.*-"*<%06If(+!`7.,"9+7"7"90,.*KQy,F%&E1!)7"* Y,.!23e(+! #<ND90!)'!5,.*K4*!2;23E!%&!r!54 #<$1%&(+*-%&.7"$h@D!5*<%&* 4)1.#!)7"$*<@$145@1*<,.*HWX*-,

5023`%0*<48"$145$;WX$17"*-%06<!)45*2;1E1%0*<7"!)#$X

036573D<2-,SR KE6)73036n=6)7( $ 0$.Q32@*62*&)(+7,<6)++2@( 0365+*2$',?(C=>2U6)0 03$;=>6j,<,<6)K<6 27X&5(+7 %7365D<DE2(+736;&)(+7 032@D<(+=(++65736)2-,SR 736)*@*$ Z$CD<6 036)*@*$ J8)2(C736;0F" (+7303$ &/A36 D<(C73(p$CDED<(&)2$I,?6$C*@*7" 6)&5&)2@,?$C8)2@(+736>036)2=(032027$C=2&)$C=6)7,?6>273D ,/$.Q32@*2 42-3J" "J32@730322E- (',?2858Q$CKE6 &/A36D0- 65+736)703(n2*K<6j,?2@&)(+*@(>(C,E,?2@&)(3B( ' '+6)KE(nK<2E- (+K ,/$C703(>2@*dD<2@DE,?65= $27 J37$ 85(+7$ 032D ,/$.Q32@*2 %,RB*7" $C= -32658)8Q$:032"J36)D ,?66)&)&52@,?$C8)2@(+732|036QQ$K<2@03J3K<DE2-3K<(C+K<65D<D<2 ''$C=>657,?6F6)0 2* D<2D ,?65= $KE2@,?(CK<732d$'**(xDE,/$I,?( (+730$'=>657",?$C*6'LN(+=6gD<2d6('2@73&)6g0$'**$ 3+J3K?$ 53 5D<2 g27b6(A 6j,<,?2d(CD<D<65K0''$',<(>&/A36'B*$CDE&)2$'7303( 2* D<2D ,?65= $6('+(+* '+6)KE6n$C*@*7" 27,?65K<73(b0365**$n,?K<$.--%(+*$b=>$C+736j,?2&)$u0(- ( *(uD*-%6)+72=6)7,?( 036)*K<6j,?2&5(+*(B*@6xDE,<K<J,E,?J3KE6x K?$'73+2$I,?6D<&)(C=1-$'2(+73(>6)0 2@*&5(+73036573D?$I,?(nK<2$C&="J32@DE,?$ 2*H-3KE( *(>03206)73DE2 %,R;K<65+(+*$'K<6K<6)7036)730( &)(+D 7"J3( ''$C=6)7,?6n$D--3*2@&Q$.Q2*6 2@* 3, $C0 J37$;032@DE,<K<2?QJ38)2@(+736032/]A3(C= $CDJ%#6)KE=>2 42 73(C,<2 &/A36 "J365DE,?( -3KE(&56)DED<($ '3'2@6)736 DEJn,?6)= -32 032|$C*@&)J376 &)657",<27$'2$032=2*@*2@D<6)&5(+73032 B'=(+*-,?(F=>$C++2@(+K<2032 "J36)*@*23&)$CK?$I,<,?65K<2@DE,?2@&)2306)** " 2@73D<(CK<+65K<60365** " 2@73DE,?$.Q32*@2 %,Rx03273$C=>2@&Q$G 652 &)$CD<2 27>&)J32;"J36)D ,?6FDE,<K<J,E,?J3KE6 K<$C73+2$',?6tD<(C73(-$CK ,?2@&)(+*$CK<=6)7,?6 -3KE(D%7"J373&52$',<6]D<2 ]=(032 &)$',/$ *$ -3KE(&56)03JK?$FD*-%6)KE2=6)7,/$C*@627x=>(03(F0$ *$CD<&52$CKE6N2*6'C(+*E'C6)K<6-%6)K J7 D<65&)(+730(u2@*&)(+7306)73D<$',?(n7365*DE(+*( -%(C,<6)73852$C*@6:=>$C+736j,?2@&)(3 J36)D ,?(nA$ &5(+73DE6)7 %,<2@,?( 02]6)DE,<6)7306)K<6u2* -32 -%(+D<DE2?Q32@*6>*7" $C7$C*@2D<27365**6 K<6)C2(+732N27p&)J2]*7" 273D ,/$.Q32@*2-,SR -32 (+K ,?6'BK<6)D ,/$C703(D<(+D ,/$C73852$C*@=>657,?6F*2@=>2-,/$',<20$C*|,<6)= - ( 7365&)65D<D?$'K<2(>$C0u6A 65,E,?J$CKE6F2*|&)$ %KE2&Q$'=>657",<($C032$DQ$',?2@&)(F0365*&)(+7306)73D<$',?(:&/A36CB27 "J36)D ,?6 K<6)C2(+732 B+032 '+6)7,?$:&)(C7 K<(+7,?$.Q32*@6&5(+7 *$ '2@,/$ =6)032$ Z&j K) 53 j

> ! VRP O M"WIP MYQT :ZM RMPWIM%PXPRPNPZYO TP :ZP

RE

TIC

OL

O

50 ms 100 ms 150 ms 200 ms 300 ms 500 ms 1000 ms 2000 ms

&5 X> 2=231E1!5%0!8!5%""$1,.O0!52;*<%D7"$;#FT0*_!54459&'7",F%0$;4CB *-@1$145906-!$%0*_(+*<4R#$1%X(+*<%&.7.$/%&*<4545=7.,.1W0WX$4523E%0*7.!5#- (+$WX$V90%e7"*<2;Wi$](+!Z!%D7.*<,F6<!5$1%&*/#$%h!54P,"*<7"!)#$145$(0!

323< 4R7.*<2;WX$!5%&(+!)#<7"$'$W0,.#!)a

.#90%&/!52=231E1!5%0*A"!,"!Qy*<,.!5.#*H4i7"*-2;WX$=7",F1.#$,."$(014#<$1%&(0*<%&.7.$%0*-4"$145$Wi$7.*<%06-!514*H23E%0*7.!5#<$& W&1,.12=*<7",.!(+*-4,.*7"!)#$4$K"$1%&$

s = 1.15*

v = 1.3vB

+!v%0$17"!v45K.#$2;W&,F"=(+*-445*Qy,F%0E*_WX*-,q7.*<2;W0!"9 K#!5*<%D7"*-2=*-%D7"*_4590%0E1T&!?

8

G 65**@6 $'7$C*2@D<23,<6)(+KE2&/A36 K<2?-%(+K ,/$',<6 76)*;-$CK?$'+K?$' ( -3KE6)&)65036)7,<6CB3&5(+7 &)J2 DE2 DE(+73(1&5(+7 K<(C7 %,/$I,?22 KE2DEJ3*@,?$',?2D0- 65K<2=6)7,?$C*2 BD<2;-3K<6573036 2@7>&5(+73D<2@036)K<$C8)2@(+736 J37 &)(C73036)7D?$',<(1&)2@*2703K<2@&)(3B&)2@( D<2,?K<$CD<&5J3K?$ 2@*&)(+737$C=6)7,?(n$'K<=(+732&5(>76)**$>032@K<658)2(C736:06)*KE65,<2&)(C*(3 LN(+=6x+2?R$C&5&)6)77$',?(b2@7 -3K<65&)6)06)738)$BJ7 &5(+7 KE(+7,?( ,?K<$n*$ DE(+*J3852(+7617"J3=>65K<2@&Q$ 0365**7" 6="J$'8)2(C736032$gKE(+D<DJ%2@,?$C6'D*)"2,?K<2@032=6)7D<2(C7$C*6 650p2NKE2D<J*@,/$I,?2U(C,<,<6)7"J,<2$ -$'KE,?2@K<6>0$C**$ G 4 @Z&j K) >=>(CDE,?K<$ &/A36 "J365DE,<2F03J36 $.- -3K<(&)&52HD<(C73(p&)(+7D<2D ,?657",<2H,?K<$ *(CK<( -%6)K J6)*&/A36uK<2@+J$CKE0$ 2 -$CK?$'=>6j,?K<2N(+D<DE6)K0''$',<2t736)+*@2]6)D0- 65K<2=6)7,<2 J2 KE2?-%(+KE,?$',?2 5 /J,<,?$ %'2$B$b&Q$CJ3D<$;0365**$ -3KE(&56)03JK?$ 02&)$CK<2@&Q$C=6)7,<( 2@7 DE273+(C*2D ,/$',<202 G*(&/A|B2@**@J3DE,<K?$',?$$C* -$CK?$'+K?$' ( 53 B2@* -%(C,?65738)2$C*6 $CK<=(+732@&)(;2@7,?K<(03J3&56nJ37$ 032@7$C=2&)$;736)*@*( D0-$C852(0365* "J$CDE2 %2@=1-J3*DE( 0365**$ "J$C*6xD<206'+6 ,<6)7365K &)(+7,<(3$ -K<(&)6)0J3K?$ 032&Q$CKE2&)$C=>657,?($C032$.Q$',<2&5( 27Z$I,<,?2 -3K<(+2@65,E,/$ (+7306 -32$C736]2@7(C73036t032<GN*@(&/A DE6 DE2(+D<DE6)K&'C$g*7" $C7303$C=>657,?(0365**$ '+65*(&)2@,R>0365**@6:(+70361032 GN*(&/A 0$I,/$ 0$'**$ + 6xKE2?-%(+KE,?$',/$27+J3K<$ 53 NB DE273(C,?$ &/A36t27-K<(+DED<2=2@,RH0365*Q (CK<03( Q3$C730$ "J36)D ,/$ '+65*(&)2-,SR 73(+7 &5(+27&)206&)(+7 "J365**$ 032(+7303$ -2$C7$H2|&5(+73D<65+J365738Q$ - 65K "J36)D ,?2F''$C*(CK<2%032

q2@* &)(C73036)7D?$',<(DE2 ,?KE( 'C$ 2@7 =>(',?(

7365*DE2D ,?6)=>$ 032K<2- 6)K<2@=>657,?( 0365**$1,<K?$.--%(+*$1= $CC7365,<2&Q$ &)$CJ3D?$032"J36)D ,?( 6(A 6j,<,<(>2@*&)(C73036)7D?$',<(1D<2D*-%(+D ,/$:03$C**$ - (+DE2852(+736t0326J2*2EQ3K<2@(F6 KE2DE6)7,?6 036)*@*$ (+KE8Q$:02K<2&/A2$C=(0365*&5(+77$'=>657",<( =>$C+736j,?2@&)(3*=(C,?(0365*&56)7,?KE( 032= $CDED?$03( 'J,?( $ "J36)D ,/$ (+KE8Q$-3J"6)DED<65K<6xKE2&5(+DE,<K<J32-,?( 27 $.--K<(+DED<2=>$C852(+736HD<6)=2&5*$CDED<2@&Q$$-$CKE,<2KE6F0$C**@6 N&/A6CB

P T MRP T RM3:^ %YRMYO)T :P YO TP: P RPZYT P ^+T >

-%6)KF*$ (+K<8)$ 032K<2@&/A32$'=>(n$'K<=(+732&)$B 76)*DE2DE,<6)=>$ 032KE2@ 6)KE2=6)7,?( 036)**$.Q%(+K?$',<(+K<2@(3B%D<2DE&)K<2 '+(+7(

z = vn(q) − vLAT = vn(q) − hq0/m

hq = −mω2zz

53 +

&5(+7 &5(+73032@8)2@(+732|2@732852$C*@2q(0) = q0

6z(0) = 0

0.4 0.5 0.6 0.7 0.80.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3

Quasi-impulso [qB]

0

0.5

1

1.5

2

2.5

3

Vel

ocità

[vB]

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

&5 +> Z*<45$+#!7.o_(+*-445*$1%&(0*q(+!&s 4$+#FT;!5%/Qy9&%06<!5$1%0*A(+*<4&ND9&'!ac!2;W0904)"$HWi*<,r90%8B 47"*<6-6-_(+*-40,.*7"!)#$4$W&,.!q

s = 1.15%0*<4545$k.#FT0*-2;k(+!m6<$1%Xh*I7.*-. 4)e,.!5Ee%0*<,Ff7.,.7'7.*<E1E!57.e!%X(+!5#-g45g@1*-4$+#<!n7Foe(+!

W&,"7"!)#*-44)H4!5Oi*<,F0PHJ&E90,.HW0!)#<#<$14)H,"!5WX$,'7FH90%;!%0E,.1%&(+!52;*<%D7"$_(+*-44)6<$%&H!%&(0!5#-7._(&4+,.*7'7F%0E$145$&

Pt2DE(+*E'C6)7303(7"J3=6)K<2@&Q$C=6)7,<6*@6 53 + D<2(+DED<6)K&''$C73( 0J36,<2?-32d02DE(+*J8)2(C732 $C0303( 'C6 ''$C*@20$F*7" $.--K<(+DED<2=>$C852(+736 "J$'03K?$',<2&)$:0365**$FKE6)*$'8)2(C736t032032D0- 65K<D<2@(+736 6HDE2;-J" "J32@73032$CDED<J3=6)KE6

vn(q) = (h/m∗n,q0

) (q − q0) + vn,q0

B2@*tD<2D ,?65= $ 53 + -3J" $C73&/A36 65D<DE6)K<6KE2D<(C*@,?( 2@7b=>(03($C7$'*2@,<2&5(x6g&)(+703J3&)6:$C*@*$xDE(+*J3852(+76

z(t) = −

√m∗

mv0

ωzsin

(√mm∗ ωz t

)

q(t) = q0 + m∗ v0

h

(1 − cos

(√mm∗ ωz t

))

03( 'C6:DE2 -%(+D ,?(v0 = vn,q0 − (h/m) q0

7 "J36)D ,?( K<6)C2=6 2*&5(+73036573D?$I,?( &)(+= -326 (+D<&52**$C8)2@(+732UD<2$ 736)*@*( D0-$C8)2@(

qBD<2$ 736)*@*(

> 8 VRP O M"WIP MYQT :ZM RMPWIM%PXPRPNPZYO TP :ZP

D*-3$C8)2@(nK<6Q$'*6 $C*@*$ K<6="J365738Q$n032,?K?$D-- (C*$ KE2D<&)$C*$I,/$1-%6)KH=>658)8)(b0365**$ = $'D<D?$ 6&Q$C&56C9 657",<K<6*7" $C=1-26)858Q$]0365**6U(+DE&)2@**$'8)2(C732C7365**@( D0-$C8)2@(

q-3J"H6)DED<65K<6DE2+732 &Q$I,?2E''$]$C*"*@2E'+65**@(

032F"J$'*&/A36 -$'KE,?6 -%6)K]&56)7,?(B*7" (+D<&52**$C8)2@(+736t7365**(xD*-$C852(K<6)$C*6 $ ' '265736g&)(+7u$C= -32@6)8)85627 6)K<2@(+K<23$C*=>2@&)K<(C76 73(+7 "J32@73032=2D<JK?$.Q32@*6&5(+7 2@*73(+D ,?K<(x$.--$'K?$',<(:032KE2E'+65*$C852(+76Z&j K) ? K k J$C730( 279'+65&)6g*7" $.--3KE(+D<DE2=>$C8)2@(+736 "J$C03K<$',?2@&Q$10365**$:K<6)*$C8)2@(+736 032|032@D*-%6)K<DE2(+76 73(+7 -32 ''$C*2@0$B"*7" $C730$'=>657",<(173(+7 -32 "J365**@(x(+DE&)2*@*$',<(+K<2@(3B+=>$xD<2 $CD<DE2D ,?6F$C*@*$&)(+= -$CKED?$032dDE(+*J3852(+72 27b&5J32|2* &)6)7,<K<(032d= $'D<D?$ 0365*d&)(+7036)73D<$',?(D<2$'**(C7",?$C7$x06 732-,?2 'C$'=>657",<60$C*<''$C*(CK<6t2732@8)2$C*6'B+,/$C7,?(:736)*@*(:D*-$C852(

qB3"J$C7,?( 7365**@(:D0-$C8)2@(xKE6Q$C*@6CB&)(+=6 =(+D ,?K?$I,?(

27CJ3K?$53 !

-200

-150

-100

-50

0

Pos

izio

ne d

el c

entr

o di

mas

sa [

µm]

Posizione del centro di massa

0 0.1 0.2 0.3 0.4 0.5

Tempo di evoluzione [sec]

0

20

40

60

80

100

120

140

Qua

si-i

mpu

lso

[qB]

Quasi-impulso

&5 0> :$145906-!$%0* (0*<454*u*-ND9&16<!5$1%0!]"*<2;!)#4)1.'!)#FT0* (0!]2=$17"$ X D\Wi*<,s = 1.15

Uq0 =

0.6 qB

>/,.!)'9047.*-@:!5(+*-%D7"* #FT0* !4 #$1%X(+*<%&.7.$]%0$1%lG3W&!\#<$1%+JX%&7.$]*-(k!54 "90$hND9&1"!aC!52;W0904)'$]#,.*-.#*!5%&(+*JX%0!n7F2;*<%D7.*

Q Q32$C=( ,?657"J,?( &)(+7,?(u032,"J36)D ,?( 6A 65,<,<( 2@7,?K<(03J3&56)730( J37 6)KEK<(+KE6DE2D ,?6)=>$',<2&)(nDEJ3*''$C*(CK<6

q0D<6)*@6)852(+7$I,?( 03J3K<$C7,?6;2@*&Q$'K<2&)$C=6)7,?( $C032$.Q$',<2&)( -3$CK<2 $C*@*7" $C=1-26)858Q$;06)**@6

(+DE&)2*@*$C852(+72|736)*@*(D*-$'8)2(q d6:=2DEJ3K<6K<2E- (CKE,/$I,?6D<2K<2@ 65K<2@D<&)(C73(>DE(+*( $C*&Q$CDE( (+DE&)2*@*$ %

,?(CK<2(>0$',<(n&/A36 *7" $C*@,<K<(&)(+= -%(+KE,?$C=>657,?( DE2K<2- 6)KE2D<&561$ 85(+736-3KE(+D<DE2=61$C*Q%(+K<03(n06)**$

:WIV ;: XRM3: : T W T"P YM QMW XPZ^+T FPZYO TP :ZP MYM"W M9 P ^+T >

85(+7$x032 GNK<2@**@(+J327 02 032 &)2*@6H$'&)&)65D<DE( 0$C*;-3J37,?( 032<'2D ,/$:D0- 65K<2@=>657",?$C*6 $x&)$CJ3D?$:036)*@*673(C7 %$C02$.Q$I,?2&52@,R:27nZ$CDE6g032d&Q$CKE2&)$C=>657,?(3

_%i onr:ijr mns]i5i5e IeNh5r:`ns cs h5qe

h)`ba|cefbhji5hkcl st`ns:sch5qe

42 H(CD<D<65K0''$',<( &/A36 -%6)KU,?J,E,?2%2;''$C*@(+K<2032s

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∆t = 14Dx6 02E'+65K<DE2 ''$C*(CK<2U036)* J3$CD<2 %2= -3J3*@D<(3 G 65**$uDE6="J36)78Q$ D<J-%6)KE2(+KE6 *7" 6(%

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1770 KHz

1795 KHz

0.00 qB 0.01 qB 0.02 qB 0.03 qB 0.04 qB 0.05 qB 0.07 qB 0.10 qB 0.20 qB

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100000

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N(q, t) = (N0 − b(t)) fq0(q) + b(t)

53

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0 0.1 0.2 0.3 0.4 0.5

q / qB

0

20000

40000

60000

80000

100000

120000

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mi n

el c

onde

nsat

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t = 150 mst = 300 mst = 600 ms

150 300 450 600

∆t [ms]

0.28

0.32

0.36

0.4

q 0 / q B

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q0 = (0.32±0.01)qB

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q0 = 0.28(1) qB

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