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ßÙ@,EJA\ºMRY,EMRJgJA=WFoJQ\ve,YNhF3UWX8F;MOY[>QMVUWFo7,FOE8CMOJJ/nhMZ>AU^@>A@U^@,FOU^B/YbMoU^@bMRJ­MRY,E=?>§SVFVJA\@,>QMR7,93EyU^FoMOY,EyFTMR7,C*PJA\ve*YbgFOU^X§>A=F3gF3MOEGFIJ/@>\7,@,C,>A¾E8@bMR>A==^E8j/E8= <lm@¹b7[>A@NMR7,JAe*MRU^XGFMRY,EDUW@bMOEG93EGF3MyUWFU^@¦MRY*EUW@NMREG9O>AX8MOUWJ/@ÖJA\=WUWBAYbMÙSVU^MOY©·>QM3MRE89G¾_[J/=^=^JÉSoUW@*BtMRY,E>QnJÉj/E´93EGX8UWeE©=WEG>AC,FPMRJ²>A@ UW@Nj/EGF3MOUWBN>MRUWJA@ JA\#MRY*E´n[>AF3UWX©E8@bMOU^Mm JA\#MOY,E©EG=WE8X8MO9OJf¾>AB/@*E8MRU^XP»,EG=WC²U EA5>´FOU^@,B/=^EP»[EG=^CJgC,EU^@bMOEG9R>QX8MRU^@,B¦SVU^MRY¯MRY,En[>AF3UWXE8@bMOU^Mm¯JA\¾>QM3MREG9oU ÓEQa>A@>QMRJ/ J¾¾>ALAE¹b7[>A@bMO7,E8«¬EGX¤MRFIC,JA¾U^@[>A@NMo\J/9onJAMRYË[=WU^B/YNMV>A@,C¾>QM3MREG98Ë*MOY,UWFo­7,FTM;nEX§>A939OU^EGCPJ/7*M;>QMoMRY,E=WE¤j/EG=JQ\>­\E8S¹b7[>Q@bMZ>h|NI~ Pus g ,O>, ® JA@,FOU^C,EG9»[93F3MMRY,EFThFTMRE8XGJA¾eJ/F3EGCJA\h>ÀMmSJfg=^E8j/E8=N>QMRJAúMR9R>Qe,eEGCUW@>ÀY[>A9OJ/@,U^XeJQMREG@NMRUW>A=<U^@bMOEG9R>QX8MRU^@,BPSVU^MOYÇ>FOU^@,B/=WEDJgC,EJA\ÀMOY,E¾EG=^EGX¤MR9OJÉfg·>QB/@,E¤MRUWX­»[EG=^CP_[J/9MOY,E­¾J/EG@NMÙ@*JPX8J/7,e*=WUW@*BtUWFÙ>AF3FO7,EGCÖMOJ~nEe*9OEGF3EG@NM§#_[J/9IMRY,E#\9OE8EXG>¥jgUkM)»[EG=^CMOY,EyüK>Q¾U^=kMRJA@,UW>Q@ Hc

U^FoB/U^j/E8@nb6`Hc = ~ωc(b†cbc +

1

2) , Ã hkøÉÅ

SVY*EG9OE b†c >A@,C bc >A93EMRY,EDe*Y,JAMRJA@©XG93E§>QMOUWJ/@©>A@*C´>A@,@,U^Y,UW=W>MRUkJ/@ÖJAeEG9R>MRJ/93F9OE8FOeE8X¤ÆMOU^j/E8=^AË,J/nE8gU^@,BDMOY,E9OEG=W>QMRU^J/@,FZ`[bc, b

†c] = 1 ,

b†c|mc〉 =√mc + 1|mc + 1〉 , Ã hµ/Å

bc|mc〉 =√mc|mc − 1〉 ,

ù

%cb;de(+*,f wi@%.d;¡+jI(s¢¤£s*gm

¥

_vU^B/7,9OE¾hkø0`ãhXZY,E8¾E¾JA\o>MmSJfg=^E8j/E8=5F3gFTMREG MR9O>Ae,eE8CÇU^@°>Y[>Q9OJ/@,U^XDeJAMRE8@bMRUW>A=UW@NMRE89R>AX¤MRUW@*BÃÄXGJA7,e,=WU^@,BXGJA@,F3MR>A@bM g ÅSVU^MOY>FOU^@,B/=^EE8=WEGX¤MR93UWXÙ»[E8=WCJgC,EQ

\J/9ÀMOY,EEGUWBAEG@,FTMZ>QMOEGF |mc〉 Ë mc = 0, 1, 2... JQ\1MOY,EyüK>AUW=ÊMRJ/@*UW>A@ Hc Y,EüI>AUW=kMOJ/@,U^>A@ Ha

\J/9ÀMRY,EMmSJfb=WE8j/E8=F3gFTMREGUWFoB/Ukj/EG@Pnb6`Ha =

~ωa2

σz ,Ã hµ/Å

SVY,E89OE ωa/(2π) UWFÀMRY,E\93EG¹b7,E8@,X8JQ\1MRY*Ey>QMRJ/U^XKMR9O>A@,FOUkMRU^J/@P>A@*Ca`σz =

(1 00 −1

) Ã g$bÅUWFoJ/@*EJA\1MOY,Eyv>A7,=^U¾>QMR93UWXGE8FG Y*E>QMOJ/âJ/FOX8UW=W=W>MRU^@*BoUW@U^MRFY[>A93¾J/@*UWXMO9R>Ae*e,UW@,BVeJQMREG@NMRUW>A=/UWF½C,E8FOX89OU^n¬E8C#nNMRY*EFR>Q¾EoüK>AUW=ÊMRJA@,UW>A@­>AFMRY*Eo\93EGEÀ»[E8=WCn,7*M\J/993E§>AF3J/@[>An,=^EoeJQMREG@NMRUW>A=WF1MRY*EojgU^n,9R>QMOUWJ/@\9OE8¹b7,EG@,X¤ ωv

SVUW=^=KnE©­7,XZYú=^JÉSE89~MOY[>A@ MOY,E¦X§>§jhUkM)±9OE8FOJ/@,>A@,XGE¦\9OE8¹b7,EG@,X¤ ωc(ωv ¿ ωc)

¯Û U^MRYÂMOY,Ee,Y,JA@,J/@²XG93E§>QMOUWJ/@²>Q@,C>A@*@,UWY,U^=W>QMOU^JA@ÂJ/eE89R>QMOJ/9OF a†v >A@,Cav9OE8FOeE8X8MRUkj/EG=k/Ë5MOY,EüI>AUW=kMOJ/@,U^>A@°\J/9#MRY*E>QMOJ/¶ÓF#jgUWn,9O>QMRU^J/@[>A=5¾JQMRUWJA@ÇX§>A@ÂnESV9OUkMOMOEG@>QFI`

Hv =p2

2M+Mω2v2

x2 = ~ωv(a†vav +1

2) , Ã hÕùAÅ

SVY,E89OE M U^FoMOY,E¾>AFOFVJQ\1MOY,EUWJ/@P>A@,CPMRY*E9OEG=W>QMRU^J/@,FZ`x =

√

~2Mωv

(av + a†v) ,Ã h¦/Å

p = −i√

~Mωv2

(av − a†v) Ã h§!AÅ

i¨h0i©(+b;*q&sfq¢ !Y,>¥j/EÙnEGE8@P7,F3EGC Y,E>A@,@,U^Y,UW=W>MRUkJ/@·>A@,CtXG93E§>QMOUWJ/@tJAeEG9R>MRJ/93F av >A@*C a†v J/nE8MRY,EX8J/9O93EGF3eJ/@,C,U^@,BP93EG=W>QMRU^J/@,F;MRJ¦Ã hµ/Ť Y,EXGJA¾eJ/F3U^MOE#F3gF3MRE8UWFKC,E8FOX89OUWnE8C©nNMRY,EF37,àJA\IMOY,EPUW@*C,U^jgUWC*7[>A=üK>AUW=ÊMRJ/@*UW>Q@,F­e,=W7*F>A@ÂUW@bMOEG9O>AX8MOUWJ/@ÇMOEG9O H1

Â_[J/9#MRY,EChh@,>A¾U^XGF5JA\UW@bMOEG93EGF3MY,EG93EAËgMRY,EI68EG9OJ#¹b7[>A@NMR7,¡EG@*EG9OBQ¾=^E§>AC*F5MRJ#>A@¾JÉj/EG9O>A=^=ae,Y[>AF3En*7*MVC,JgE8FV@*JAMoX¿Y,>A@,B/EÙMRY*EC*g@[>AUWX8FG<_[93J/Ø@*JÉS J/@~U^MÀUWFMOY,EG93E8\J/93EJ/U^MOMOEGCt>A@,CMOY,E·J¥j/EG9O>A=W=5üI>A¾U^=kMRJA@,UW>Q@©UWFyC,E8»,@,EGCËSoU^MRY*J/7*MMOY,E·68EG93Jܹb7[>A@NMR7,£X8J/@bMO9OU^n,7*MRU^J/@U^@~MRY,EXG>¥jgU^Mm·»[EG=^C>A@*CPjhU^n,9R>MRUWJA@[>A=¼üK>AUW=kMOJA@,UW>A@¼Ë,>AFZ`H := ~ωc(b†cbc) +

~ωa2

σz + ~ωv(a†vav)︸ ︷︷ ︸

+H1 .Ã hÓûNÅ

=: H0

ª êé[ êVè`æGçèFbéñ,íAèÉîÄêçx«ñÇî7]è§ê¬çºîñ,ç Y*EU^@bMOEG9R>QX8MRU^J/@!JQ\¦MRY,E>QMRJ/U^XâFTgF3MRE8 SVU^MOY!MRY,EEG=WE8X8MO9OUWXâ=WU^B/YbM¯»[EG=^CXG>A@nEE¤he[>A@,C*EGC >AFt>´F37, JA\U^@bMRE89R>QX8MRU^J/@¯MOEG93¾FSVY,E89OE`MOY,E»,9OF3M·@,J/@gfbj>A@,U^FOY,U^@,BMOEG93¢UWFMRY*E~J/FTM93EG=^E8jQ>Q@bM­\J/9MOY,EtF3gF3MOEG´_[J/9MRY*E~E¿he¬E89OU^¾E8@bMR>A=FThFTMRE8 >QMY,>A@,CPMRY,U^FÀMREG93!UWFÀMOY,EEG=^EGX8MO9OU^X¹g7,>AC,9O7*eJ/=WEMOEG93!B/Ukj/EG@nNyWkø§/Y`

H1 =e

2

∑

i,j

rirj

(∂Ei(x1, x2, x3, t)

∂xj

)∣∣∣∣

x

, Ã hÓýNÅ

SVY*EG9OE r = (r1, r2, r3)T UWFIMRY,E­eJ/F3U^MRU^J/@j/EGX¤MRJ/9JQ\<MRY,E#jQ>Q=WEG@*XGEE8=WE8X8MR93J/@Ö9OE8=?>MRU^j/EMOJÖMRY,EP>QMRJ/U^XteJAFOU^MOUWJ/@ x = (x1, x2, x3)

T >A@,CÂMOY,EPe,>A93MRUW>A=C,EG93U^jQ>MRU^j/E8FDJQ\;MRY,EE8=WE8X8MR93UWXo»,EG=WC¾>A9OEVE8jQ>Q=W7[>QMOEGC·>M5MOY,E;eJ/FOUkMRU^J/@DJA\¼MOY,EI>QMOJ/ JX§>A=^XG7,=W>QMREyà gµýNÅ1MRY,EE8=WE8X8MR93UWX»,EG=WC°­7,F3M#nE·Lb@,JÉSV@¼Ö_[J/9#>ÜÞ>A7,FOF3U?>A@ :5á00FTMZ>A@*C,UW@,BSÀ>§j/EtJgC,ESVUkMRYÜSÀ>AU^F3M9O>AC,U^7,F w0

>A@*C©XG>¥jgUkM)`=WE8@,BAMRY L MRY*EDE8=WE8X8MR93UWX#»[E8=WC¦J/eE89R>QMOJ/9 E @,EG>A9MOY,EX§>§jgU^Mm~X8EG@NMREG9;XG>A@nEMZ>ALAE8@`>AF¬WÊø¥hËøK/Y`E(x1, x2, x3, t) = e

√

2~ωcε0Lπw2

0

(b†c + bc)e− ρ2

w20 sin(k · x + φ) , Ã g^ø¥/Å

SVY*EG9OE ρ U^FMRY,E<9O>AC,U?>Q=AC,UWFTMZ>A@*XGE\93J/ MRY*E<X§>¥jgUkM)>hU^FGË ε0 UWFMOY,E<eEG93¾UkMOMRUkjgUkMmKJQ\h\93EGEF3e[>AX8E;>A@,C φ UWF>ÙXGJ/@*F3MZ>Q@bM<e,Y[>QFOEVF3Y,U^\]MC,E8eEG@,C*EG@bM5J/@MRY*EV>QMRJ/U^XeJ/F3U^MRU^J/@­UW@#MRY,EFTMZ>A@*C,UW@,BISÀ>¥jAEA Y,EÀeJ/=W>A9OU^FR>MRU^JA@7,@,UkMvj/E8X8MOJ/9 e Y[>QFvMRJKnEÀXZY,J/F3EG@DeE89OeEG@*C,UWX87,=?>A9MOJ k ­Û°E>AF3FO7,EMRY,E>QMOJ/ÔUWF>QM x = 0 ˽U EAU^@ÖMRY,EDXG>¥jgU^MmÜXGEG@NMRE89yJ/FOX8UW=^=?>MRUk@,B>Q=WJ/@,B k lm@,FOE893MOUW@,B`à hkø¥NÅU^@´Ã hÓýNÅSE»[@,C`H1 =

e

2

√

2~ωcε0Lπw2

0

(b†c + bc)(r · e)(r · k) cos(k · x + φ) . à g^ø/ø¥Å

û %cb;de(+*,f wi@%.d;¡+jI(s¢¤£s*gm JFOU^¾e,=^Uk\]~à g^ø/ø¥Åv\7,9TMRY,E89GËbMOY,E;·>MR9OUÊE8=WE8¾E8@bMOF5JA\ (r · e)(r · k) \J/9MRY*EIE¿heEG9OUÊÆEG@bMR>A=W=k­U^@bj/E8F3MRU^BN>QMOEGC 40 ® > + ¹b7[>AC*9O7,eJ/=^EVMR9R>Q@,FOUkMRUWJA@ S1/2 Ë mj = −1/2↔ D5/2

Ëmj = −5/2 à λ = 729 @,tÅKY[>§j/EDMRJnEDX§>A=^XG7,=W>QMRE8CiIF3UW@,B~MRY,ED9OE8\E89OE8@,XGE8FWÊøÉùY>A@,C­WÊøF[Y¾ÃFOE8E´>Ae,eE8@,C,UÊ ÏžMRY,EÖ\J/=^=^JÉSVU^@*BÁ9OE8FO7,=kMU^F~J/n*MR>AUW@*EGCÃMOY,E©>A9On*U^MR9O>A93e,Y[>QFOEyU^FoXZY,J/FOE8@MRJDnE6GE89OJ,Ë e Ë k >A@*C B >Q9OE>AFOF37,¾E8C­7*MO7[>A=^=k·JA93MRY*J/B/J/@,>A=¬\J/9¾>hUW¾>A=¬XGJA7,e,=^UW@,BbÅ®`

〈S1/2,−1/2|(r · e)(r · k)|D5/2,−5/2〉 =1

e

√

5γ⊥ε0λ3~2π2

. Ã hkøÉ/ÅiIF3UW@,BMRY*EyC*E8»[@,UkMRU^J/@,FI`

|g〉 := |S1/2,−1/2〉 , Ã g^ø¥/Å

|e〉 := |D5/2,−5/2〉 , Ã g^øKNÅ

η := k

√

~2Mωv

, Ã hkøÉù/Å>A@,C°Ã h$NÅSEyJAn*MZ>AU^@a`

H1 = ~g(σ+ + σ−)(b†c + bc)× cos(η(a†v + av) + φ) . Ã h^øKNÅ

Y,E¾XGJA7,e,=^UW@,BX8J/@,FTMZ>A@NM g U^@,XG=^7,C,E8FMRY,Ee,9OJ/eE893MOUWEGFyJA\MRY,E¾>AX8MO7[>A=5>QMOJ/UWX­MR9O>A@hÆFOUkMRU^J/@ÁÃÄU^MOFIMR9O>A@,FTj/EG9OFO>A=C,E8X§>§Ü9R>MRE γ⊥ ÅK>Q@,CÜMRY,E­e,93J/eEG9TMRU^EGFÙJA\MRY*EXG>¥jgUkM)JhC*Eà L Ë w0Å¿5i;FOUW@*B`à h^øAøÉÅÀ>A@*C°Ã hkøÉ/Å¿Ë,U^MoUWFoF3EGE8@MRY,>QMK`

g =

√

5γ⊥cλ2

2π2Lw20

. Ã hkø!/Å Y,E`eJ/FOUkMRUWJA@²UW@¯MRY,E`FTMZ>A@*C,UW@,B´S>¥j/E`X§>A@±nE`C*E8MRE89OUW@,E8Cnb¯MOY,E`X8J/@,F3MR>A@bM φ _[9OJA MRY,E`U^C,EG@NMRUkM)6` cos(k · x + φ) ≡ cos(k · x) cos(φ) − sin(k · x) sin(φ) ∝H1

SE¦FOE8E¦MOY[>QM~MRY*E¦UW@NMRE89R>AX¤MRUWJA@'JA\yMRY*E©>QMRJA =^JgX§>QMOEGCú>QM x = 0 XG>A@ >A=kSÀ>§hFnEÜC,EGX8J/eJ/FOE8C UW@'>Çe,>A93MtUW@NMRE89R>AX¤MRUW@*B°SoU^MRYMRY*E¦>Q@bMRU^@,JgC,E`JQ\MRY*EÜ»,EG=WCà φ =π/2, H1 ∝ − sin(k · x) ⇒ E ∝ cos(k · x) Å>A@,CÇ>e[>A9TMUW@bMOEG9O>AX8MOUW@,BPSVUkMRY©MRY,E@,JgC,E¾Ã φ = 0, H1 ∝ cos(k · x)⇒ E ∝ sin(k · x) Å¿<lm@PMRY*E\J/=W=^J¥SVU^@,BnJQMRYX§>QFOEGF>A93EMRY*EG9OE¤\J/9OEUW@Nj/EGFTMRUWB/>QMRE8CF3EGe[>Q9R>QMOEG=^Alm@ÖMRY,EDE¿heEG9OU^¾E8@bMMRY,EUWJ/@ÖU^FÙSE8=W=XGJgJA=WEGC¦>A@,C©93EGF3UWC,E8FyU^@ÖMOY,Ed½>A­ngfhHIUWXZLAE9OE8B/UWESVU^MOYú>ÇFO¾>A=W=Idº>QnbfhHKUWXZLAEÜe[>A9O>AE8MRE89 η i;@,C,EG9tMRY*UWFtXGJ/@*C,U^MOUWJ/@MRY*EY,U^B/Y,EG9oJA9OC,E89OFVJA\ η XG>A@nEy@*EGB/=^EGX¤MREGC¼ Y*E»[9OFTMI@*J/@gfbj>A@,UWF3Y,UW@*BMRE89OF;>A93E`

H1 = ~g(σ+ + σ−)(b†c + bc)×

1 +O(η2) , @,JgC,E−η(a†v + av) +O(η3) . >A@NMRU^@,JgC,E Ã hkø¥ûNÅ

i¨h0i©(+b;*q&sfq¢ ý 9R>Q@,F3\JA9OUW@,BÜMRJ¦MOY,EPUW@bMOEG9O>AX8MOUWJ/@Áe,U^X8MR7*9OE>A@,C¯eE893\J/93¾U^@,BÜMRY,EP9OJQMZ>QMOUW@,BÜSÀ>§j/E>Qe,e,9OJ§hUW¾>QMRU^JA@©Ã@,JAMRE/` ωv ¿ ωc ≈ ωa

ÅU^MÀ\J/=W=^J¥SVFÀMOY[>QMÃ ∆ := ωc − ωa Å®`

HI := eiH0t/~H1e−iH0t/~

' ~g ×

σ+bce−i∆t + σ−b†cei∆t +O(η2) , @,JgC*E

−η[σ+a†vbce−i(∆−ωv)t

+σ−a†vb

†cei(∆+ωv)t

+σ+avbce−i(∆+ωv)t

+σ−avb†cei(∆−ωv)t] +O(η3) . >Q@bMRU^@,JgC,E

à g^ø¥ý/Å

® JA@,FOU^C,EG9Á@,J¥S MRY*E C,Uk«¼E89OE8@bMÁMOEG93¾FÇUW@MRY,U^FÁ>Ae,e,93J¥hUW¾>QMOU^J/@ lm@MRY*E @,JgC*EAËSVY*EG9OE¾MOY,E O(η) MRE89OF#>A93E·>An,F3EG@bMGËvMRY*E·JA@,=^¦X8J/7,e,=^UW@,BPMRY,>QM#E¿hUWF3MOF#UWFynE¤M)SE8EG@FTMZ>QMOEGFtSVUkMRY±MOY,EÜFR>AE`jgUWn*9R>QMRU^J/@[>Q=;¹b7[>A@NMR7, @b7,­nEG9¦Ã nv = const. Ť Y,E8FOEMO9R>A@*FOU^MOUWJ/@*F>A9OE©X§>Q=W=WE8C³µXG>A9O93UWE89PMR9O>A@,F3U^MRU^J/@,F8¶^ Y,E σ+bc MRE89O 9OE8e,9OE8FOEG@NMRFMRY,EE¿hXGU^MR>QMRU^J/@JQ\MRY,EÜ>QMOJ/ nb>An,F3J/9On,U^@,BÇ>°XG>¥jgU^Mm²e,Y*JAMRJ/@¼ σ−b†cC*EGFOX89OU^nEGF·MRY,EE8¾U^FOF3UWJ/@²U^@bMRJ©MOY,E`X§>§jgU^MmâÃÄFOE8E`»[B/7*9OEÁà hÕAÅOŤúlm@MOY,EÜ>A@NMRU^@,JgC,EAËÀJ/@²MRY,EJAMRY,E89Y,>A@,C˽MRY,EXGJA@bMO9OUWn*7*MRU^@,B~MREG93¾Fy>A93EJA\»[9OFTMJ/9OC,E89yUW@ η ·_[9OJA à hkø¥ý/ÅÙU^MUWFF3EGEG@MOY[>QMVMOY,EyJA@,=^·MR9O>A@,F3U^MRU^J/@,Fo>A=W=^J¥SEGC>A93EMR9R>Q@,FOUkMRUWJA@,FÀMOY[>QM;XZY[>A@,BAEMRY,EjgUWn,9O>QMRU^J/@[>A=¹b7[>Q@bMR7* @b7,­nEG9`nNâJA@,E'à ∆nv = ±1 Ť 9O>A@,FOUkMRU^J/@,FXZY[>A@,BAUW@,B²MOY,E´jhU^n,9R>ÉÆMOUWJ/@,>A=b¹b7[>Q@bMR7*Ú@b7,­nEG9 nv nb ±1 >A93EX§>A=^=WE8C³Ó»,9OF3M)fgJ/9OC*EG91FOUWC*EGn[>A@*C#MR9O>A@,F3U^MRU^J/@,F8¶^

¯°± ²4³

´µ¶

··¹¸2º·¼»\º

½

½ ¾ ¾ ¾σ ¿+ σ ¿−

+½ ½À

ωÁ

_1UWB/7,93EhÕ`ÑÙU^FO7[>Q=WU^FR>QMOU^J/@PJA\vMRY,EE8@,EG93BAF3MZ>MREGFIJA\vMRY,E#>MRJ/!MO9R>Ae*eEGC`UW@`>Y[>A9)ÆJ/@,U^XeJAMOEG@NMRU?>Q= Y,EV>QMOJ/¾U^XF3MZ>MREGF |g〉 >A@,C |e〉 >Q9OEÀC,9OE8FOFOE8CSVUkMRY­MRY*EoY[>A93¾JA@,UWXJAFOXGU^=W=W>QMOJ/9D=^E8j/E8=IF3MO9O7,X¤MR7,9OEQ ® >Q9O9OU^EG9MO9R>A@*FOU^MOUWJ/@*F`Ã ∆nv = 0 Å>A9OEUW@,C,U^X§>MREGC²nb>Q9O9OJ¥SVFGÀãgMZ>A9TMRU^@,BSVU^MRY>A@>A9On,UkMR9O>A93·jgU^n,9R>QMOUWJA@[>A=¼¹g7,>A@bMO7,@g7,­nEG9 nv Ë[MOY,EyC*NÆ@,>A¾U^XGF5JA\MOY,EKFThFTMREGØU^@t>y»,EG=WC·@*JhC*EKUWF59OE8F3MR93UWX¤MREGC·MOJMRY,EIe[>AU^95JA\=WE¤j/EG=^FLAEGEGe*UW@,BnvX8J/@,FTMZ>A@NM§

ø¥ %cb;de(+*,f wi@%.d;¡+jI(s¢¤£s*gm Y,Ey93EG=^E8jQ>Q@bMVMOEG9OFV>A9OE/`σ+avbc

`<E¤hX8U^MR>QMRU^J/@PJA\ºMRY,Ey>MRJ/!>A@*C ∆nv = −1 Ëσ−a

†vb

†c`<E8¾U^FOFOU^J/@~UW@NMRJMOY,EX§>§jhUkMmP>Q@,C ∆nv = +1 ËÃÄ93EGCFOU^C,EGn,>A@,CPMR9R>Q@,FOUkMRUWJA@,FGË*FOE8E»[B/7*9OE¾Ã hÓNÅ3ÅrÂ

σ+a†vbc

`<E¤hX8U^MR>QMRU^J/@PJA\ºMRY,Ey>MRJ/!>A@*C ∆nv = +1 Ëσ−avb

†c`<E8¾U^FOFOU^J/@~UW@NMRJMOY,EX§>§jhUkMmP>Q@,C ∆nv = −1 ËÃÄn*=W7,EFOU^C,EGn,>A@,CPMR9R>Q@,FOUkMRUWJA@,FGË*FOE8E»[B/7*9OE¾Ã hbÅ3Ť

ÃoÄKÅÆÃ Æ ÃÈÇqÅÆÉaÊËÍ̬Î

Ï0Ð"Ñ

Ò

Òσ ÓsÔÆ ÒÕ×Ö)Ö

σ ÓÔÆ ÒÖ Ø ωÙ

_vU^B/7,9OE;hÓw`1ÑÙU^FO7[>Q=WU^FR>QMOU^J/@JA\¼MRY*EIF3MR>QMRE8FJA\¼MOY,E;F3gF3MRE8ªU^@bj/J/=kj/EGC¾U^@9OEGC¾FOU^C,EGn[>Q@,CMR9O>A@,F3U^MRU^J/@,F·ÃUW@,C,U^X§>MREGC´nb´>A9O93JÉSVFZÅ¿ Y,EtChh@,>A¾U^XGFJA\ÀMRY*E·F3gF3MOEG¢X§>Q@ÇnE·C,E¤ÆFOX89OU^nEGC©nbÜXGJ/@,F3UWC,E89OU^@,B>A@NÖe[>QUW9ÙJA\F3MR>QMRE8F |e, nv − 1,mc〉 >A@,C |g, nv,mc + 1〉 ËSVU^MOYÁ>A9On*U^MR9O>A93ÖjgUWn,9O>QMRU^J/@[>A=<¹b7[>A@NMR7,£@g7,­nEG9 nv >Q@,CÇ»[E8=WCǹb7[>A@NMR7,¢@b7,­nEG9mc

Ú Û2ÜÞÝàß

áâ/ã

ä å^æZçèå è

åNéçè

äσ ê×ëè äì íσ êëè äí í

îωï

_vU^B/7,9OE¾hO`#ÑÙU^FO7[>Q=WU^FR>QMOU^J/@©JQ\ÀMRY,E¾F3MR>QMRE8F#JA\ÀMRY*E·FThFTMRE8¢UW@Nj/J/=^j/E8C°UW@´n,=^7,E¾F3UWC,E¿Æn[>A@*CMR9O>A@,F3U^MRU^J/@,FyÃÄU^@,C,U^X§>QMOEGCnb>A939OJ¥SVFZŤ Y,EC*g@[>Q¾U^XGF;JQ\vMRY,EyF3gFTMREGX§>A@nEC,E8FOXG93UWnE8CPnN¾XGJ/@,F3UWC,E89OU^@,B­>A@b¾e,>AUW95JA\½F3MR>QMRE8F |e, nv +1,mc〉 >A@,C |g, nv,mc+1〉 ËSVU^MOYÁ>A9On*U^MR9O>A93ÖjgUWn,9O>QMRU^J/@[>A=<¹b7[>A@NMR7,£@g7,­nEG9 nv >Q@,CÇ»[E8=WCǹb7[>A@NMR7,¢@b7,­nEG9mc

i¨h0i©(+b;*q&sfq¢ ø/ø >ALbUW@,BúUW@NMRJú>AX8XGJ/7*@bM´J/@*=^âMRY,EÂ6GEG93JAMRYÒ>Q@,CÚ»,9OF3M´JA9OC,E89©U^@ η MREG93¾F8Ë­MRY,E¾>AU^@ÒeJ/U^@bMOFÇ>A9OE@,J¥S FO7,·>Q9OUWF3EGCa` æGçàèF ç½êëÝí/êì + îÄç5î7è§êèí/ñ*ééîDbé±êç+]óðè béî7²çºê¡í/êì +Äî ç5Òè¥êÒè¤ñvé[8èZ<¼êaéë béò½î7LºéÉñ*èÉîê¬ç½ñGî ëLºñ[çºë+/ò æGç èF ñ,ç¬èÉîÄçºêë êç+]ó í/êì +Äî ç5 è§ê èóñ1é[8èZ<¼êaéQëbéòî7LºéÉñ*èÉîÄêç½ñ¬Gî ëLºñ[çºë+¦î7ºéIgç¬è"ôX8êúè bé²îÜç½êM2íNîè¥ñ*è¥îê¬çê¬çÚèí/ñ*ééîDbéò XGJA@,FOU^C,EG93UW@,B°MRY,E`\7,=^=IE¿*e,>A@,FOU^J/@ JQ\MRY,EÜY[>Q9OJ/@,UWX\7,@*X8MOUWJ/@,F8Ë;U^MU^FtF3EGE8@ MRY[>M~UW@MOY,EÖ@,JgC*EÖU^@'BAEG@,E89R>A=IMRY,E89OEÜUWF·J/@,=^²X8J/7,e,=^UW@*BÇnE8MmSE8EG@úF3MR>QMRE8FMOY[>QMtC,Uk«¼E89·nb±>Q@'E8j/E8@ @g7,­nEG9·JA\jgUWn,9O>QMRU^J/@[>A=V¹b7[>A@NMZ> à ∆nv = 0, 2, 4, . . . ÂXG>A9O93UWE89GË­F3EGX8J/@,CgfgJA9OC,E89©FOUWC*EGn[>A@*C,FGË­\J/7,9TMRYgfgJA9OC,E89¦F3UWC,E8n[>A@*C,F¯GGZÅ¿Ë­SVY,E89OEG>AF´U^@MOY,E©>A@NMRU^@,JgC,EÖJA@,=^XGJA7,e,=WU^@,BÁnE8MmSE8EG@ F3MZ>QMOEGF~MRY,>QMPC,U^«¬EG9tnb >Q@úJgC,C @g7,­nE89JQ\~jgU^n,9R>QMOUWJ/@,>A=¹b7[>A@bMR>âJgXGX87,9OF'à ∆nv = 1, 3, 5, . . . »[9OFTMTfgJ/93C,EG9´FOUWC*EGn[>Q@,C,FGËMOY,UW93CgfgJ/93C,EG9ÀFOU^C,EGn[>Q@,C,FGËGGZÅöõ¿

ötó½ç½ñÇîÄí"êç²è ©ðñ,éÉéîDgélm@PMRY*E@,JgC,ESEyY,>¥j/E/`

HI ' ~g × σ+bce−i∆t + σ−b†cei∆t +O(η2) . Ã hµANÅ

l \DMRY,EÇMOEG93¾FÜJA\FOE8XGJ/@*CÚJ/9OC,E89ÖU^@ η >Q9OEÇ@,EGB/=^EGX¤MREGC¼Ë HIX8J/7,e,=^EGFÖJA@,=^ F3MR>QMRE8FMOY[>QMyC,Uk«¼E89UW@©JA@,EX§>§jgU^MmÖJ/9y>QMRJ/ÔE¤hXGUkMZ>QMOUWJ/@¦¹b7[>A@NMR7,¾ÏªXZY,J/U^XGEJQ\Àn,>AFOU^FyU^FI`

|e, nv,mc〉 Ë |g, nv,mc+1〉 Ë/SVY,EG93EVMRY,EÀjgUWn,9O>QMRU^J/@[>A=h¹b7[>Q@bMR7*@g7,­nE89 nv UWFvMRY,EFO>AE#\J/9Ù>A=W=ºF3MZ>MREGFÃJ/@,=^X§>A939OUWE89KMO9R>A@*FOU^MOUWJ/@*FGˬX8\)¬»[B/7,93Eà hµ/ÅOÅ¿ Y,EC*g@[>AUWXGFJQ\vU^@bMRE89OE8F3MI>A93E`|ψ(t)〉I := eiH0t/~|ψ(t)〉 = C1(t)|e, nv,mc〉+ C2(t)|g, nv,mc + 1〉 , à hµhøÉÅ

SVY*UWXZYUWFoC,EG93U^j/E8C\93J/4`i~d

dt|ψ(t)〉I = HI |ψ(t)〉I . Ã hÕA/Å

Û UkMRY~MRY*E7,FOEyJQ\oà gÕ/ÅMOY,E@,J/@bfg6GE89OJ·>MR9OUÊtE8=WE8¾E8@bMOF;>A9OEJ/n*MR>AUW@,E8Ca`〈e, nv,mc|σ+bc|g, nv,mc + 1〉 = 〈g, nv,mc + 1|σ−b†c|e, nv,mc〉

=√mc + 1 , Ã hÕQNÅ

>Q@,CP\J/9ÀMRY*Ey¾>QMO9OUÊtJA\ HIU^MoUWFÀ\J/7*@,Ca`

HI = ~g(

0 e−i∆t√mc + 1

ei∆t√mc + 1 0

)

. à hÕ[bÅ÷løùZú ûgüý$þùrÿúûýý¦ùý

H1 = er ·E [ùFü û[üsû ý[ùFü ûrúýú [û ü×ý ý$úlù[þÿ$ý [û®úûú

øÉ %cb;de(+*,f wi@%.d;¡+jI(s¢¤£s*gm:¹b7[>QMRU^J/@°Ã hµ//ÅXG>A@@,J¥SnESV9OUkMOMRE8@a`

i~d

dt

(C1

C2

)

= ~g(

0 e−i∆t√mc + 1

ei∆t√mc + 1 0

)(C1

C2

)

,

à hµ/ù/ÅSVY,E89OE¦Y,EG93EGU^@'MRY,EÜMRU^E`C,EGeE8@,C,EG@,XGE©U^@±MRY,E¦>A93B/7,EG@NMRF~JQ\yMRY*EÖXGJgEZvX8UWEG@NMRFPU^FJ/U^M3MREGC¼ JtFOUWe,=^Uk\]tMRY,U^FI\7,9TMRY,E89GË>¾MO9R>A@*F3\J/93·>QMOU^J/@U^@bMRJMRY,E¾³µ93JAMZ>MRUW@*B,¶an[>AF3UWFUWFyeE893\J/93¾E8C Y,U^FUWFyC,J/@,E¾nN°>7,@,UkMZ>A9T²Ã 1 = UU † = U †U ⇒ U † = U−1 ŤËMRU^¾EÙC*EGeEG@*C,EG@NMIMR9R>Q@,F3\JA9O¾>QMRU^J/@·SVU^MRY`

U :=

(ei∆t/2 00 e−i∆t/2

)

, Ã hµNÅ(C1

C2

)

:=

(ei∆t/2 00 e−i∆t/2

)(C1

C2

)

, Ã hµ0!/Å>A@,CPMOY,E\J/=^=WJ¥SVU^@,B#\Ä>AUW=kUW>A9\JA9OUWFÀ\JA7,@,Ca`

i~d

dt

(C1

C2

)

= ~(

−∆2

g√m+ 1

g√m+ 1 ∆

2

)

︸ ︷︷ ︸

(C1

C2

)

. à hµAûNÅ=: HI Y,U^FX8J/7,e,=^EGC¾C,U^«¬EG93EG@NMRUW>A=[E8¹b7[>QMRU^J/@MZ>QLAEGF5MRY,EVSE8=W=äfgLb@,JÉSV@\J/93\JA9<MOY,EIU^@bMOEG9R>QX¤ÆMRU^J/@JQ\,>oMmSJfb=WE8j/E8=NF3gF3MOEGSVU^MRYy>V=W>AFOE89G Y,E5F3J/=W7hMRUWJA@U^FJ/n*MZ>QUW@,E8CynNÙMR9R>Q@,F3\J/93DÆUW@*BySVU^MRY·>yMRUWEVUW@,C*EGeEG@*C,EG@NMo7,@*U^MZ>Q93D¾>QMR93UkDU^@bMRJ>n[>AF3UWF5SVY,E89OEKMOY,EIEG¹b7[>QMOUWJ/@,FC,E8XGJ/7,e*=WEy>Q@,C>F3J/=W7hMRUWJA@~U^FoE§>QF3tMRJ­»[@,C 9R>A@*F3\J/93¾U^@,Bn[>QX¿LaË,MRY*EC*g@[>Q¾U^XGFoUW@MRY*E |e, nv,mc〉 Ë |g, nv,mc+1〉 n[>AF3UWF>A9OE;\J/7,@,C¼<I>Qn,UJ/FOX8UW=^=?>MRUkJ/@,F<JgXGXG7*9SVUkMRYMRY*E\93EG¹b7,E8@,X86`

Ωcarrier :=√

∆2 + 4g2(mc + 1) , à gÕAý/ÅSVY,U^X¿YUWFoMOY,Ey¹b7[>A@NMRU^FOEGCE¿*e*9OEGF3FOUWJA@\J/9oMOY,E;>An,U¼\9OEG¹b7,E8@,X8Ao_[J/9o6GE89OJ¾C,E¤MR7,@*UW@,Bà ∆ = 0 ÅI>A@,C`MRY,E>MRJ/UW@,UkMRUW>Q=^=ÊPE¤hXGUkMRE8C²Ã C1(t = 0) = 1 Ë C2(t = 0) = 0 ſˬMRY,EXGJgErvXGUWE8@bMOF;Y[>¥j/EMRY,EMOUWEÙC,EGeE8@,C,E8@,XGE`

C1(t) = cos

(Ωcarrier(∆ = 0)

2· t)

, Ã hÓ/NÅ

C2(t) = −i · sin(Ωcarrier(∆ = 0)

2· t)

. à hÓ*øÉÅ Y,E¾U^¾eJ/9TMZ>Q@bMeJ/U^@bMMRJ¾>ALAEDY,EG93E¾UWFMRY[>M#>AFyF3EGEG@´\9OJ/ à hÕQýNŤ˽E8j/E8@ÇUW@¦MRY*EX§>QFOEDJQ\@,J~e,Y,JQMRJ/@,Fà mc = 0 Å¿ËMOY,EG93E>Q9OEI>An*UvJ/FOX8UW=W=W>QMOU^JA@,F8˼e,9OJ¥jhU^C,EGCÜMRY[>QMÙMOY,E

i¨h0i©(+b;*q&sfq¢ ø¥>MRJ/ U^FUW@*U^MRU^>A=k=kÇUW@¯MOY,EE¤hX8U^MOEGC±FTMZ>QMOEA Y,UWFUWF7,F37[>A=^=^Á@,JAMMRY*EX§>AF3E\J/9¾>Q@>MRJ/ªUW@NMRE89R>AX¤MRUW@*B#SVUkMRYP>#\93EGE8=^·e*9OJ/e[>QBN>QMRU^@,B#=?>QFOEG9nE§>A 7,93@,UW@*BJA«MRY,EÙ=?>AF3EG9MOY,EE¤hXGUkMRE8CÇ>QMRJAC,EGXG>¥gF#E¿*eJA@,EG@NMRUW>A=W=k/ËSVU^MOY,J/7*M>A@N¦eEG93UWJgC,U^XD93E¿fgeJAe,7,=?>MRUWJA@JQ\½MRY,EÙ7,e,eE89o=WE¤j/EG= Y,E93E§>AF3J/@~\J/9MRY*UWFC,EGXG>¥tU^FMOY,EX8J/7,e,=^UW@,BMRJ>A@~UW@*»[@*U^Mm¾JA\j>AXG7*7, JgC,E8FG Y,EGF3E·JgC,EGFy>A93E93EGF3e¬JA@,FOU^n,=WE\J/9ÙMRY,EF3eJ/@bMR>A@,E8J/7,FE8¾U^FOF3UWJ/@>Q@,C =WEG>ACúMOJ>Q@ E¤heJ/@,E8@bMRUW>A=C,E8X§>¥ÃÛ´EGU^FOFOLQJ/e*\WfbÛ U^B/@,E89 Y,EGJ/9T*ÅWkø!Y Y,E>QUW JQ\MRY,U^F·SJ/9OL²UWF¾MRJÇ9OEG>A=WU^FOEMOY,EÖF3UW@,BA=WE`JgC,E`¾JgC,E8=Ù>Q@,C±SJ/93L²SVUkMRY´È37*F3MJA@,Ete[>A9TMRU^XG7,=W>A9yEG=WE8X8MO9OUWX»[EG=^CÁJgC,EQ JÖC,J`MRY,U^F­U^@°MOY,EtE¤heEG93UWEG@NM§ËJ/@*=^´JA@,Ee,>A93MOUWXG7*=?>A9·JgC,EÖJA\yMRY,EÖj>AXG7*7, ­7,FTMnE¦F37,e,eJ/9TMRE8C˾>ALbU^@,BÂ>A=W=IJAMRY,E89OF~MRJF3J/E©E¤gMRE8@bMÜ@*EGB/=^UWB/Ukn,=^EA:vheEG9OU^EG@bMR>A=W=k/ËIMRY*UWFU^FC,J/@*E´nb X¿Y*JgJ/FOU^@,B²FOeE8XGUW>A=nJA7,@,C[>A9T~XGJ/@*C,U^MOUWJ/@,F\J/9ÀMOY,EEG=WE8X8MO9OUWXÙ»,EG=^CPSVU^MOYPMOY,EY,EG=^eJA\v>DXG>¥jgU^MmA

ötó½ç½ñÇîÄí"êç²è 4ñ1é[8èZ<¼êaéQëbé§îÄëLºñ[çºë+® JA@,FOU^C,EG9V@,J¥SâMOY,EC*g@[>AUWX8FVU^@~MRY,Ey>A@NMRU^@,JgC,E`

HI ' ~g × −η[σ+a†vbcei(∆+ωv)t+σ−a

†vb

†ce

−i(∆−ωv)t à gµNAÅ+σ+avbce

i(∆−ωv)t

+σ−avb†ce

−i(∆+ωv)t] +O(η3) .HI

XGJ/7*e,=WE8FF3MZ>MREGF5MRY,>QMC,U^«¬EG9<nbJA@,EIX§>§jgU^Mm>A@,C·nNJ/@,EI>QMRJAE¤hX8U^MZ>MRUWJA@¾¹b7[>A@gÆMO7,Ë*>AFSE8=W=¼>AFÀnN·J/@*EKjgUWn,9O>QMRU^J/@[>A=a¹b7[>A@NMR7, nv ® JA@,FOU^C,EG9MOY,EÙX§>QFOEÙSVY*EG9OEKMRY,E=^UWB/YNMÙUWFÙ9OEGC©C*E8MR7*@,EGC´nN ∆ = −ωv + δ Ã]»[B/7,93Eà gµNÅ3Ť#ÏKBN>AU^@¦>P93JAMZ>MRUW@*BtSÀ>§j/E>Qe,e,9OJ§hUW¾>QMRU^JA@~XG>A@nE>Qe,e,=WU^EGCP>A@,CPMRY*EyüK>AUW=kMOJ/@*UW>A@tnEGXGJA¾E8FI`

HI ' ~g × −η[σ+avbce−iδt + σ−a†vb

†ceiδt] +O(η3) . Ã hÓ/NÅ

l M¾UWFDeJ/FOF3UWn,=^EPMRJ´XZY,JgJ/F3EÜ>AF¾>©n[>AF3UWFI` |e, nv − 1,mc〉 Ë |g, nv,mc + 1〉 Y,EChh@,>A¾U^XGFoJA\1UW@bMOEG93EGF3M;>A9OE/`|ψ(t)〉I := eiH0t/~|ψ(t)〉 = C1(t)|e, nv − 1,mc〉+ C2(t)|g, nv,mc + 1〉 , Ã gµNÅ

SVY*UWXZYUWFoC,E8MOEG93¾U^@,EGC~\9OJ/V`i~d

dt|ψ(t)〉I = HI |ψ(t)〉I . Ã hµ/ù/Å

Û UkMRY~MRY*E7,FOEyJQ\oà gÕ/ÅMOY,E@,J/@bfg6GE89OJ·>MR9OUÊtE8=WE8¾E8@bMOF;>A9OEJ/n*MR>AUW@,E8Ca`〈e, nv − 1,mc|σ+avbc|g, nv,mc + 1〉 = 〈g, nv,mc + 1|σ−a†vb†c|e, nv − 1,mc〉

=√nv√mc + 1, Ã hµNÅ

øK %cb;de(+*,f wi@%.d;¡+jI(s¢¤£s*gm>A@,CP\JA9ÀMRY,E¾>QMR93Uk·JA\ HI

UkMVU^FÀ\J/7,@,C`HI = −~ηg

(0 e−iδt

√nv√mc + 1

eiδt√nv√mc + 1 0

)

. Ã hÓ"!/ÅüIE8@,XGEQË1Ã hÓNù/ÅnEGXGJA¾E8FI`i~d

dt

(C1

C2

)

= −~ηg(

0 e−iδt√nv√mc + 1

eiδt√nv√mc + 1 0

)(C1

C2

)

,

à hÓ/ûNÅSVY,E89OE·\9OJ/ @,J¥SªJ/@°MOY,E·MOUWEC,EGeE8@,C,EG@*XGEPUW@´MOY,E~>Q9OB/7,E8@bMRFUWFJ/U^MOMOEGC JFOU^¾e*=WUk\?²MRY*UWFt\7,93MOY,EG98Ë>ÇMR9R>Q@,F3\J/93·>MRU^J/@±UW@bMOJÁMOY,EÁ³µ93JAMZ>QMOUW@,B*¶n[>AF3UWFPU^F~X§>A939OUWE8CJ/7*M7,F3UW@,B¯MRY,E´7*@,U^MR>A93'·>MR9OUÊ U B/U^j/E8@âU^@ EG¹b7[>QMOUWJ/@à gÕ/NŤ Y,E°93EGF37,=^MOUW@,BE¤he,93EGF3FOUWJ/@¼Ë[SVY,UWXZYUWFV>A@,>A=WJAB/J/7,FMRJà hµAûNÅ¿Ë,U^FoB/U^j/E8@nb6`

i~d

dt

(C1

C2

)

= ~(

−δ2

−ηg√nv√mc + 1

−ηg√nv√mc + 1 δ

2

)

︸ ︷︷ ︸

(C1

C2

)

.

=: HIÃ hÓ/ýNÅ

Y,EFOJ/=^7*MRU^J/@¦U^FJ/n*MZ>QUW@,E8C©UW@¦MOY,EFR>Q¾ED¾>A@,@,E89y>AF\J/9ÙMRY,EXG>A9O93UWE89MR9O>A@,F3U^MRU^J/@,F8;>An,UJ/FOX8UW=^=?>MRUkJ/@,Fo>Q9OEy>QBN>AUW@t\J/7*@,CË[n*7*MVSVU^MOYPMOY,EyI>An*U\93EG¹b7,E8@,X8PB/U^j/E8@nb6`Ωred−sb :=

√

δ2 + 4(ηg)2nv(mc + 1) . à g$N/ÅÛ U^MOYÜMRY,ED>QX8MR7,>A=1MR9R>Q@,FOUkMRUWJA@,FKJ/@¦93EGF3J/@[>A@,X8Eà ∆ = 0 Ë δ = 0 ÅIUkMU^FÙ@,JÉSeJ/F3FOU^n,=WEMRJÖX8J/¾e,>A9OEtE¿*e*9OEGF3FOUWJA@âà hµAýNÅ>A@,Cúà hNNÅ¿ Y,E~\Ä>Q¾U^=^U^>A99OE8FO7,=kM­\J/9#MRY*EPI>Qn,U\9OE8¹b7,EG@,X8UWE8FIU^FV93EGX8JÉj/E89OEGC`

Ωcarrier = 2g√mc + 1 , à h,ø¥Å

Ωred−sb = 2ηg√

nv(mc + 1) = η√nvΩcarrier .

à hb/ÅÏK@¦>A@[>Q=WJ/BAJ/7,FK>A@[>A=kgFOUWFKXG>A@ÖnEXG>A9O93UWEGCÜJ/7*MK\J/9IMRY,E­n,=^7,E­F3UWC,E8n[>A@,CÜMR9O>A@,F3U^MRU^J/@,F8 Y,En[>AF3UWFDF3MR>QMRE8F |e, nv + 1,mc〉 >A@,C |g, nv,mc + 1〉 =^E§>AC¼Ë5\J/96GE89OJ´C,E¤MR7,@*UW@,Bà δ = 0 Å¿Ë,MOJDMOY,EE¤he,9OE8FOFOU^J/@a`

Ωblue−sb := 2gη√

(nv + 1)(mc + 1) = η√nv + 1Ωcarrier .

à hNNÅ

i¨h0i©(+b;*q&sfq¢ øÉùg u J!!uàcl#"lm@°>¾J/93E93E§>Q=WUWFTMRU^X¾JgC*EG=MOY,EG93ESVUW=^=nE¾C*EGX8J/Y,EG93EG@,X8EtC,7,EMOJXGJ/7*e,=WU^@,BPMRJPMRY,EE8@bjgUW93J/@,EG@NM§_[JA9E¤*>Ae,=^EAËAMRY,EV>QMOJ/XG>A@DC,E8X§>¥­UW@NMRJÙJ/@,EoJQ\MOY,EoUW@h»[@,U^MOEÀJgC,E8FJQ\MRY,ED\9OE8E»[E8=WC˺UW@,FTMRE§>QC°JQ\ÀEGU^M3MRU^@,B#È37,FTM#U^@bMOJPMRY,EX§>§jhUkM)Ö¾JgC,EQ Y,E¾X§>¥jgUkM)UkMRF3EG=^\VXG>A@Â=^J/FOEte*Y,JAMRJA@,FnbÇ>An,F3J/9Oe*MOUWJ/@ÇJ/9#MO9R>A@,F3¾U^FOF3UWJ/@´MRY,9OJA7,B/YÇMRY,Et¾U^9O93J/9OF8lm@*XG=W7*C,UW@,B¦MRY,E8FOE`E¤«¼E8X8MOF·UW@¯MOY,EMOY,EGJ/93E8MOUWX§>Q=VC,EGF3XG93UWe*MOUWJ/@²=^E§>AC*FMOJ´MOY,EC,E8@,FOUkMmNf¾>QMO9OUk\JA9O¾>A=^U^F3 Y*UWFFOEGX¤MRU^J/@D\JgX87,FOE8F5J/@XG>A9O93UWE89MR9O>A@,F3U^MRU^J/@,F1SVY,EG93E;@,Jy>QMOMOEG@hÆMOUWJ/@DY[>QF<MOJynE;e[>AU^CDMOJMRY,EVY[>A9OJ/@*UWXMR9R>Qe,e,UW@*ByeJQMREG@NMRUW>A= Y,E;·>AFTMRE89<EG¹b7[>QMOUWJA@\JA9­>A@Á>MRJ/­fgX§>§jgU^Mm¦F3gF3MOEG U^@,XG=^7,C,U^@,BMRY,E·C,UWF3FOU^e[>QMRUkj/E¾E8«¬E8X8MRF­F3MZ>MREGCÁ>AnJÉj/E·UWFBAU^j/E8@nbyWÊø¥û[Y `

ρ =1

i~[HI , ρ] + Lρ , Ã h0bÅ

Lρ = γ⊥(2σ−ρσ+ − σ+σ−ρ− ρσ+σ−)+κ(2bcρb

†c − b†cbcρ− ρb†cbc) . Ã h$Nù/Å

ü;EG93EAË ρ UWFKMRY,ED>QMRJ/­fgXG>¥jgU^MmC,E8@,FOUkM)Ü·>MR9OUÊ¬Ë γ⊥ UWFKMRY,E­MR9R>Q@,F3j/E89OFR>Q=<C,E8X§>§¦9O>QMREJQ\VMRY,E~>QMOJ/£UW@NMRJ`\93EGEtFOe[>QXGE©Ã γ⊥ = γ‖/2ËSVY*EG9OE γ‖ UWFMRY,EtC,E8X§>§Ç9R>QMOEtJA\oMRY,EeJAe,7,=?>MRUWJA@aÅD>Q@,C κ U^F¾MOY,E`C,EGXG>¥9O>QMREJA\MRY*EÜXG>¥jgU^Mm»[EG=^Cà 2κ U^FMRY,E`e,Y,JAMOJ/@

κ$

γ

_1UWB/7,93Ehµù`IÏJ/93E­93E§>A=^UWFTMRU^XFOXZY,E8¾EJA\>¾M)SJfg=WE¤j/EG=vFThFTMRE8UW@NMREG9O>AX8MOUW@,BÖÃÄX8J/7hÆe*=WUW@*B g ÅDSVUkMRY'>´FOU^@,B/=WEX§>§jhUkM)¯¾JgC*EA κ C,EG@,JQMREGF·MRY*EÜXG>¥jgU^Mm²C*EGX§>§9R>MRE`C,7,EMOJ`UWeEG9T\EGX8MUW939OJ/93FGË γ⊥ >AXGX8J/7,@NMRF\J/9yMOY,Et>QMOJ/UWXDC,E8X§>§°U^@bMOJMRY,E\9OEGE¾FOe[>AX8En,>AX¿LbBA9OJ/7,@*C

øF %cb;de(+*,f wi@%.d;¡+jI(s¢¤£s*gmC,E8X§>¥~9R>QMOE¥Å¿5HKE¤»[@,U^@,BDMRY,E@,JA@gfgY,E89OU^MOU?>Q@PüK>Q¾U^=kMRJA@,UW>Q@ Heff

`Heff := HI +Hdis ,

à h"NÅHdis := −i~[γ⊥σ+σ− + κb†cbc] ,

à h2!/Åà g$0NÅX§>A@nESV9OUkMOMOEG@>QFI`

ρ =1

i~(Heffρ− ρH†

eff) + 2γ⊥σ−ρσ+ + 2κbcρb†c .

à hNûNÅ® J/@,F3UWC,E89JA@,=^©MRY*Et»[93F3MMOY,9OE8E~=WJ¥SE8F3MEG@,E89OBA°F3MZ>MREGF8Ë<MOY,E~68EG9OJÖ¹b7[>A@bMO7, FTMZ>QMOE|g, 0〉 >A@,C~MRY,EMmSJDF3MR>QMRE8F |e, 0〉 >Q@,C |g, 1〉 Ë*SVUkMRYPE8U^MRY*EG9V>A@>QMOJ/JA9oX§>§jgU^MmtE¤hXGUÊÆMZ>MRUWJA@·¹b7[>A@NMR7,ÏI@b¾C,UWF3FOU^e[>QMRUkj/EIe,93JgXGEGF3Fo=WEG>AC,F5MRY,EKF3gF3MOEGØMOJ#MRY,E |g, 0〉 F3MR>QMREQ Y,E89OE8\JA9OEUkMÀU^FeJ/FOF3UWn,=^EIMRJ­7,FOEKMRY,EÙFTMZ>QMOE |ψ〉 = C1(t)|e, 0〉+C2(t)|g, 1〉 >A@,C~UkMRFE8j/JA=W7*MOUWJ/@~UWFoB/Ukj/EG@Pnb6`

i~d

dt|ψ〉 = Heff |ψ〉 . Ã g$Ný/Å

Y,EÖ@,J/93 JA\ |ψ〉 C*EGX89OE§>QFOEGF>QF~C,U^FOF3UWe[>MRU^j/E`e*9OJgXGE8FOF3EGFJgXGX87,9P>A@,C±MRY*EÖF3gF3MOEGE8j/JA=^j/E8F;UW@NMRJMOY,E |g, 0〉 F3MR>QMREQ Y,Ey¾>QMR93Uk¾\J/9O­7*=?>QMOU^JA@~U^FoB/U^j/E8@nb6`Heff = ~

(−∆2

gg ∆

2

)

− i~(γ⊥ 00 κ

)

= ~(

−∆2− iγ⊥ gg ∆

2− iκ

)

, Ã hµùANÅ>A@,CPMOY,E\J/=^=WJ¥SVU^@,B­UWFoJ/n*MR>AUW@*EGCa`

d

dt

(C1

C2

)

=

(i∆2− γ⊥ −ig−ig −i∆

2− κ

)(C1

C2

)

. à hµùhøÉÅ Y,U^F#UWF#>F3gFTMREG¢JA\À»[93F3M­J/93C,EG9XGJ/7,e*=WE8CÇC,U^«¬EG93EG@NMRU?>Q=<EG¹b7[>QMOUWJ/@*FSVU^MOY°XGJ/@,FTMZ>A@NMXGJgErvXGUWE8@bMOFGPÏØF3J/=W7hMRUWJA@©UWFyJAn*MZ>AU^@,E8C´nb¦MO9R>A@,FT\J/9OUW@*BU^@bMOJ`>n[>AF3UWFSVY,E89OE¾MRY*EEG¹b7[>MRUWJA@,FVC,E8XGJ/7,e*=WEy>A@,CXG>A@nEyF3J/=^j/E8CE§>AF3UW=^AË*MOY,EFOJ/=^7*MRU^J/@~Y[>AFoMOY,E\J/9OV`

C1 ∼ eλ+t , C2 ∼ eλ−t , Ã hµù//Åλ±

>A9OEyMOY,E#E8UWBAEG@bj>A=W7*EGFIJA\MOY,E#X8JgEZvX8UWEG@NMÙ¾>QMR93Uk¬V_[JA9VMRY,E·>MR9OUÊPU^@`E8¹b7[>QMRU^J/@à gÕùhø¥ÅU^MoUWFoJ/nhMZ>AU^@,EGCa`λ± = −γ⊥ + κ

2±

√(i∆− γ⊥ − κ

2

)2

− g2 . Ã hµùANÅ

i¨h0i©(+b;*q&sfq¢ ø! Y*EyX§>AF3E\J/9o68EG93J¾C,E¤MR7,@*UW@,Bà ∆ = 0 ÅSVU^=W=¼@,J¥SnEX8J/@,FOU^C,EG93EGC¬è¥éÉêç 5°íNêì.ÄîÄç 5 î7°î]èl \ g À γ⊥

Ë g À κ Ë1MOY,E¾XGJ/7,e*=WUW@*BE¿hXGEGE8C,F­MOY,E¾C,EGXG>¥°>Q@,C´MRY,E¾FOJA=W7*MOUWJ/@©F3Y,JÉSVFC,>A¾eE8C;>An,UJ/FOX8UW=^=?>MRUkJ/@,FÀnE¤M)SEGEG@ |e, 0〉 >A@,C |g, 1〉 5_[9OJAØMRY,EE8UWB/E8@bj>A=W7,E8FI`λ± = −γ⊥ + κ

2± i

2

√

4g2 − (κ− γ⊥)2 Ã gÕù/NÅMOY,EyI>Qn,U¼\9OEG¹b7,E8@,X8PX§>A@nEUW@h\EG9O93EGCa`

Ω =

√

4g2 − (κ− γ⊥)2 ≈ 2g . à hÕùAù/Å% bñOA²í/êì.Äî ç5Äî7Çî?èlm@PMRY,U^FVXG>AFOEMOY,EEGUWBAEG@bj>A=W7*EGF λ± >A9OE93E§>A=5à 4g2 < (γ⊥ − κ)2 ÅÀ>A@,C@,JJ/F3XGUW=^=?>MRJA93nE8Y[>¥jgU^J/7,9VJgX8XG7*9OFG5lm@PMOY,E³µn[>QCX§>¥jgUkM)t9OE8B/UWEA¶hSVY*EG9OE/`

γ⊥ ¿ g2/κ¿ κ , Ã hµùNÅMOY,E\J/=^=WJ¥SVU^@,B­X§>A@nESV9OUkMOMRE8@a`

λ± = −γ⊥ + κ

2± κ− γ⊥

2

√

1− 4g2

(κ− γ⊥)2

≈ −γ⊥ + κ

2± κ− γ⊥

2

(

1− 4g2

2(κ− γ⊥)2+O

(g

κ

)4)

,

λ+ ≈ −(

γ⊥ +g2

κ

)

=: −γ⊥(1 + 2C) Ã hµù0!/Åλ− ≈ −

(

κ− g2

κ

)

, à hÕùQûNÅSVY*EG9OEMOY,EXGJgJ/eE89R>QMOU^jgU^Mm·e[>A9O>A¾E¤MRE89oUWFoC,E¤»[@,EGC`

C :=g2

2γ⊥κ. Ã hÕùQýNÅ

® JA@,FOU^C,EG93UW@,BMRY*EÙE8UWBAEG@bj/E8X8MOJ/9OFo>QFOFOJgX8U?>QMOEGC·SVUkMRY λ+ >Q@,C λ− 7,@,C,E89MOY,EKXGJ/@,C*U^MRU^J/@à hµùNŤËhMRY,EE8UWB/E8@bj/E8X8MRJA9OFI>A93E\J/7*@,CPMRJDnE`λ+ :

1√

κ2 + g2

(κ−ig

)

, Ã h¦ANÅλ− :

1√

κ2 + g2

(igκ

)

. Ã h¦høÉÅ

ø¥û %cb;de(+*,f wi@%.d;¡+jI(s¢¤£s*gm_[9OJA à h¦ANÅÀUkMVUWFoFOEGE8@MRY[>MVMRY,EF3MZ>QMOE>AF3FOJgX8U?>QMOEGCSVUkMRY λ+ Y[>AFI>DFTMR9OJA@,B¾X8J/@bMO9OUÊÆn,7*MOUWJ/@à κÀ g Åv\9OJA |e, 0〉 λ+ U^FMOY,EG93E8\J/93E;93EG=?>MRE8CDMRJMRY,E;>QMRJA¾U^XÀC,EGXG>¥>Q@,CU^MUWFFOEGE8@¾\9OJ/ÒEG¹b7[>QMOUWJ/@`à gÕù!/ÅvMOY[>QM<MOY,UWF<C,E8X§>§U^F<U^@,XG93E§>AF3EGC¾nbMOY,EI7,93XGEG=^=[\Ä>QX8MRJA9F := (2C + 1) Wkø¥ý[Y Y,E>QMRJ/U^X#C,EGXG>¥ÖJgXGX87,9OFe,93E8\E89OEG@NMRUW>A=W=kUW@NMRJtMRY*EDX§>§jgU^MmJgC,E#SVY*UWXZYÖMOY,EG@Ü\7,93MOY,EG9ÙC,E8X§>§gFG Y,E­\9O>AX8MOUWJ/@`JA\5E8¾U^FOF3UWJ/@`EGU^M3MREGC`UW@bMOJ·MRY*EX§>§jgU^Mm~JgC,EUWF β = 2C/(2C + 1) Y*E­E8UWB/E8@bj/E8X8MRJA9KUW@Áà h$*øÉŤËJA@`MOY,E#JAMOY,EG9IY[>A@,C¼Ë¼F3Y,JÉSoFKMRY[>QMIMOY,E#FTMZ>QMOE­>AFOF3JQÆXGUW>QMRE8C·SVUkMRY λ− U^FC,J/UW@[>MREGC¾nb |g, 1〉 λ− UWF5MRY,E89OE¤\J/9OEK9OE8=?>QMOEGC¾MRJMRY,EKC,E8X§>¥¾JA\MRY*EX§>§jgU^Mm¾»[E8=WC<_[93J/ªEG¹b7[>MRUWJA@©Ã hµùAûNÅ<SEF3EGEÙMRY[>MÀMRY,U^FÀC*EGX§>§tUWFC,EGX89OEG>AFOE8CnNMRY*E\Ä>QX8MRJA9 (1− g2/κ2) Ë,U EA*MRY*EyXG>¥jgUkM)tUWFÙ³Ó9OE¤»[=W=^EGC¶*nbtMRY*Ey9O>AC,UW>QMRU^@,B>QMRJA5ÁXZY,JgJ/F3UW@,BÖFOeEGX8U?>A=ÀnJ/7*@,C[>A9TÁX8J/@,C,UkMRUWJA@,F#\J/9#MRY,E~EG=^EGX8MO9OJfb·>QB/@,E8MOUWX¾»[E8=WC>A@,CMRY*EG9OE8nbUW@h¸[7,EG@*XGUW@*BIMRY,EJgC,EF3MO9O7,X¤MR7,93Eo>QF1SEG=W=g>AFvE8@,Y[>A@*XGU^@,BKj>AXG7,7*¸[7,X¤ÆMR7,>QMRU^J/@,FG˺³ÓFOeJ/@NMZ>A@,E8J/7,F8¶C,EGXG>¥`X§>Q@ÖnE>A=^MOEG9OE8C`U^@bMO9OJgC,7,X8UW@,B¾MOY,E­XG>¥jgU^MmSEY[>§j/EÙC,U^F3MO7,9OnE8CtMOY,EKUWF3JAMR93J/ebDJA\½F3e[>AX8EA Y,U^F\J/93XGEGFMOY,EÙ>QMRJAMRJ#e,9OE8C,J/UW@[>Q@bMR=kEGU^M<UW@NMRJMOY,EIXG>¥jgU^MmDJgC,EVF3g@bMOY,EGF3UWFOU^@,B­>D³ÓJ/@,EZfgC,UWEG@*FOUWJA@[>A= ¶N>QMOJ/ WµQ/Y lm@¾MRY*EFOJÜX§>Q=W=WE8C²³µn,>ACÂ>MRJ/ =WU^UÊM§¶ κ >A@,C γ⊥ >A9OE·U^@bMOEG9OXZY[>A@*B/EGC Ã κ ¿ g2/γ⊥ ¿ γ⊥

ŤË>A@,CPMOY,EX§>§jgU^Mm~C,E8X§>§gF;\Ä>AFTMREG9ÀjhUW>­MRY,EX8J/7,e,=^EGC>QMOJ/|N | &('*) uÀs,+8Í uVqºx=q -Òw/.*+¿ 0 &21 3@!"ww5476¼,al8 Y,EMRY,E8J/9OE¤MRU^X§>A=bXGJ/@,X8EGehME¤he,=W>AUW@*EGC#UW@yMRY,E=?>AFTMvF3EGX¤MRUWJA@ËAEGFOeE8XGUW>A=W=kMRY,E>A=^MOEG9O>QMRU^J/@JA\½FOeJ/@NMZ>A@,E8J/7,FoE8¾U^FOFOU^J/@·9R>QMOEGF\J/9À>QMOJ/¾FFO7,ngÈ3EGX¤MRE8CPMRJ­EG=WE8X8MO9OJfg¾>ABA@,E8MOUWX;»[E8=WC,FY[>§jhU^@,BtnJ/7*@,C[>A9T`XGJA@,C,U^MOUWJA@,FIC,U^«¬EG93UW@,B\9OJ/MRY,E­X8J/@,C,UkMRU^J/@,FIJA\\93EGE­F3e[>AXGEQËS>AF»[93F3M1XGJA@,XGE8U^j/E8C­nNy7,9OX8EG=^=bUW@Dø¥ý"XWkø§ý/Y 1lm@Y,UWF½F3Y,J/93Mº@,JAMRE57,9OX8EG=^=b¾E8@bMOUWJ/@,E8CMRY*EeJ/F3FOUWn*=WEJgC,U^»,X§>QMOUWJA@ÙJA\hFOeJ/@NMZ>A@,E8J/7,FºC*EGXG>¥Ù\J/9½>o¾>AB/@,E¤MRU^XFOe,U^@MR9R>A@*FOU^MOUWJA@XGJA7hÆe,=^EGCÖMRJ>~93EGF3J/@[>A@NMyXGU^9OX87,U^MG Y,E8J/9OE¤MRUWXG>A=vSJ/93LÜJA@©¹b7[>Q@bMR7*fgE8=WE8X8MR93JfgChh@,>A¾U^XGFUW@PX8J/@*»[@*EGCFOe[>QXGEGFV\J/=^=WJ¥SE8CWÓhøIY Y*E»[9OFTM E¤heEG93UWEG@NMZ>A=j/EG93U^»[XG>QMRU^J/@JQ\©MOY,UWF±X8J/@,XGE8e*M SÀ>QFú>QX¿Y,U^E8j/E8CÝnbÐ[ü[HK9OE¿*Y,>AB/E\WÓ/[YU^@¦ø¥ý0/ûh<ü;EFOY,J¥SEGCMRY,>QMoMRY,E¸[7*J/9OE8FOXGE8@,XGEC,E8X§>§~MOUWEÙJA\>¾JA@,JfgJ/=kEGX87,=?>Q9<=?>§/EG9JQ\½E¿hXGU^MOEGCtC*/EK¾J/=^EGX87,=WE8F5UWFU^@*¸[7,E8@,XGE8C~nbDMRY,EKnJA7,@,ChÆ>A9TUWeJ/FOE8C`nN`>¾UW9O93J/9VF37,93\Ä>QXGEAKVEGB/>A9OC,U^@,B¾FOU^@,B/=^E>QMRJAF3gFTMREGFG˼E8@,Y[>A@,X8EGCEGUWF3FOU^J/@±S>AF~»[93F3MPJ/n,FOE893j/E8C nb ã¬oüK>Q9OJgXZY,E:9;=<?>A@XWÓA[Y Ë;SVY,E89OEG>AFH#oÐ=WE8e,ehÆ@,E89B9;<?>A@9OE8e¬JA93MRE8C`J/@UW@,Y,U^n,UkMRE8CFOeJ/@NMZ>A@*EGJ/7,FIEGUWF3FOUWJA@ Wµ/*ˬ/ùY oüK>Q9OJgXZY,E#>A@,CÐ=WE8e,e,@,E897,FOE8CÁVgC,nE89OBfbF3MZ>MREGFSVUkMRY´¾U^XG9OJ¥SÀ>§j/EDMR9O>A@,FOUkMRU^J/@,F¾Ã\J/9FO7,e,e*9OEGF3FOUWJA@JA\VFOeJA@bMZ>Q@,EGJA7,FEGU^FOFOU^J/@°UW@´MRY,E·@,E§>Q9­UW@h\9R>A93EGC°X8\)×Wµ//Y?ŤËvSVU^MRY°S>¥j/E8=WE8@,BAMRY*FUW@MRY*EIUW=^=kU^DE8MOEG9C,J/¾>AUW@¼Ë/SVY,E89OEIFO7,eE89OXGJA@,C,7,X8MOUW@,B#XG>¥jgU^MOUWE8F5XG>A@·nE;7,F3EGCÏúJgC,Ej/J/=^7,¾E¾JA@ÇMOY,EtJ/93C,EG9#JA\ λ3 U^F#9OE8=?>QMOU^j/E8=^¦EG>AF3´MOJÜ9OE§>Q=WUWF3EMRE8X¿Y*@,UWXG>A=W=k/ËvF3MO9OJ/@*B/=^UW@h¸[7,EG@*XGUW@*BPMOY,EDJgC,EC,E8@,FOUkM)¦>Q@,C¦MRY*EG9OE8nbÖEG@,Y,>A@,XGU^@,BPMOY,EjQ>A93UWJA7,F ® ¨:5HªE¤\?Æ

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3/2 ⇔ H 3/2

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τnat@,F 7.7(2) 7.4(3) 94.3 901 101

λair@* 396.847 393.366 866.214 849.802 854.209

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τnatF 1.045 1.080

λair@, 729.147 732.389

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H5/2

gj 2 2/3 4/3 4/5 6/5

Û Y,EG@'>Ae,e,=kgUW@,B°>´·>QB/@,E¤MRUWX~»[EG=^C G MmSJ°MRY*UW@,BAF¾­7,F3M·nEMZ>ALAE8@±UW@bMOJÇ>QX¤ÆX8J/7,@NMK`J/@tJ/@,EÙY,>A@,CtMRY*EÙB/JN>A=UWFMOJC*UWF3MOUW@,BA7,UWF3Y~>Q@,C~9OE8FOJ/=kj/E>A=^=FOeEGX¤MR9O>A==^UW@,E8FJA\U^@bMOEG9OE8F3MGDüK>§jhU^@,BPXZY,J/F3EG@°>~e[>A9TMRU^XG7,=W>A9KX§>A939OUWE89ÙMO9R>A@*FOU^MOUWJ/@¼ËU^MU^FUWeJ/93MR>A@NMKMRY[>QM>Q=W=>AFOF3JgXGU?>MREGC©F3UWC,E8n[>A@,C¦MR9O>A@,FOUkMRU^J/@,F>A9OESEG=^=FOE8e[>A9O>QMRE8C´\93J/>A=^=JQMRY,E89X§>A939OUWE89JA9IFOU^C,E8n[>A@,CMR9R>Q@,FOUkMRUWJA@,FG Y*UWFIUWFVMRY*E#XG>AFOE#U^@F3MR93J/@,B·¾>AB/@,E¤MRU^X»[E8=WC*FISVY*EG@`MRY,E=^E8j/E8=WF>Q9OEIJ/9OE;B/93E§>MR=^FOEGe,>A9R>QMOEGC~>A@,C·Y,E8@,X8EKMRY,EIFOeE8X8MO9R>A=¬=^UW@,E8F>A9OE;¾J/93E;FOE8e[>Æ9O>QMRE8CUW@\9OE8¹g7*EG@,X¤/<ß@yMOY,EJQMRY,E89ºY[>A@,CyMRY,E5F3MO9OJ/@,BI¾>ABA@,E8MOUWX»[EG=^C#>A=WF3J;n,9OJ/>AC,EG@*F

/ %cb;dàe(+*gfZjixf;*dlkj-d;( jr&s'=&m@(+b*(E>&mà*#k

D5/2

D5/2

D5/2

S1/2

S1/2

S1/2

1/30

2/30

3/30

4/30

5/30

1/30

2/30

3/30

4/30

5/30

1/30

2/30

3/30

4/30

5/30

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

(a)

(b)

(c)

1/2

--

- -

3/2

1/2

--

- -

1/2

1/2

---

1

/2

1/2

--

- 3/2

1/2

--

- 5

/2

-1/2

--

- -

5/2

-1/2

--

- -

3/2

-1/2

--

- -

1/2

-1/2

--

- 1/2

-1/2

--

- 3/2

_vU^B/7,9OE¾hÕ` Y,E S1/2 ↔ D5/2¹g7,>AC,9O7*eJ/=WEMR9O>A@,FOUkMRU^J/@´F3e,=WUkMRFyUW@NMRJMRE8@Á>A=W=^J¥SEGCXGJA¾eJ/@*EG@NMRF¾ÃÄ>NÅKUk\>~¾>AB/@*E8MRU^X#»[E8=WC¦U^F>Ae,e*=WUWE8C¯ÃÄB/Ukj/EG@¦U^@¦7,@*U^MRFÙJA\ µBB Ť Y,E9OE8=?>QMOU^j/E=^UW@,EFTMR9OE8@,BAMOYÁU^Fe,93J/eJ/93MOUWJ/@,>A=MOJ`MOY,EGU^9F3¹g7,>A9OE8C ® =WEGn*FOXZYgf*ÞJ/9OC[>Q@°\Ä>AX¤ÆMRJA9OFWÊøÉùgËøF[Y ÖßÙe*MRU^X§>Q=5e,7,e,U^@,BMOJÜJ/@*E·JA\oMOY,E S1/2 =WE8jAEG=WF9OE8C,7,XGE8F­MRY*UWF@g7,ÆnEG9#MOJ`»,j/EA©5´MZ>QLgU^@,B`MOY,EtB/E8J/E8MO9OUWX¾X8J/@*»[BA7,9R>QMOUWJ/@°JA\VeJ/=W>A9OU^FR>QMOU^JA@ e ˾>ABQÆ@,E¤MRUWXK»[E8=WC B >A@,CPnEG>AC,U^9OE8X8MRU^J/@ k UW@NMRJ>AXGX8J/7,@bMGË,\7*93MRY*EG9oF37,e,e,93EGFOFOU^J/@JA\º=WU^@,EGFX§>Q@ nE´>AXZY,U^E8j/EGCÃnËÓX§Å¤ªlm@ÃÄnaÅ¿Ë e ⊥ B ⊥ k ËÙ>Q\]MRE89PJ/e*MOUWXG>A=Ùe,7,e,UW@*BÂJ/@,=kMRY*EtF3MO9OJ/@,BAEGF3MD>Q@,CÇMOY,E·SE§>ALAE8F3MMR9O>A@,FOUkMRU^J/@°93EG¾>AU^@Ë1MRY*EGFOEtnEGU^@,BÜFOE8e[>A9R>MREGCÁnN

≈ 4.8× 1.4 á`üI6/Þy>Q7,FOF8<lm@ÖÃX¥Å¿ËgMOY,EIXGJ/@h»[B/7,9O>QMRU^J/@¾U^F>AF<\J/=W=^J¥SVFI`ºSÀ>§j/E8j/E8X8MRJ/9 k>A@,C`·>QB/@,E8MOUWX»[E8=WC B FO7,nhMREG@*C©>Q@Ö>A@,BA=WEyJA\ 45 Y,E­eJA=?>A93UWFO>QMRU^JA@ e U^FIXZY,J/F3EG@MRJ=^UWEUW@©MOY,E¾e,=W>A@,EF3e[>A@,@*EGCÇnb k >A@*C B Y,U^FX8J/@*»[B/7*9R>QMOUWJ/@©UWF>Qe,e,9OJAe,9OUW>QMREU^\<·>QB/@,E8MOUWX»[E8=WCܸ[7,X8MO7[>QMOUWJ/@,F>A9OE­XG93U^MRU^X§>A=ºnEGXG>A7,FOEMRY,E8UW9ÙU^@*¸[7,E8@,XGEJ/@ÖMRY*E=WUW@*EFOY*U^\]MVUWFo93EGC,7,X8EGCnbPû/ i XGJ/e[>A93EGCPMOJXG>AFOE¾ÃÄnaÅ¿5_vU^B/7,93EÙMZ>QLAEG@\93J/ WÊø¥[Y

j2i¨h0i@%.d + jr&'¤dE-£Nn d'N(nIEE *q%cbqdà';jr%.dlk - ¢-(+*gE /ùMOY,E S1/2 ↔ P1/2

HKJ/e,e*=WEG9yXGJgJA=WU^@,B~MR9R>A@*FOU^MOUWJA@¦MRY*EG9OE8nb©UW@,X89OEG>AFOU^@,BMOY,E·HKJ/e*e,=WE89=^UWUÊM õ l \½J/@*EKXGJA@,FOU^C,EG93FMRY,EK¹b7,n*U^MMR9O>A@,F3U^MRU^J/@¾>QM 729 @,ª>A@,C¾MRY,EI¹g7,>AC,9O7*eJ/=WEIFOE8=WEGX¿ÆMOUWJ/@9O7,=^E ∆mj = 0,±1,±2 Ë>MRJAMR>A=ºJA\MOEG@ÜMO9R>A@*FOU^MOUWJ/@*FK>Q9OE­>A=^=WJ¥SEGCÇÃÄFOE8E#»[B/7,93EÃÄ*µ/ÅOÅ¿ßehMRUWXG>A=<e*7,¾e*UW@,BPSVU^MOY σ± fg=WU^B/YNMyJA@´MRY,E 397 @,MR9O>A@,FOUkMRU^J/@¦MRJJ/@,E¾JQ\MOY,E S1/2 F37,ngfg=^E8j/E8=WF9OE8C,7,XGE8F#MRY,E¾@b7,­nEG9JA\ÀJ/n,FOE893j/E8CÇMR9R>Q@,FOUkMRUWJA@,FMOJ»,j/EQXZY,JgJ/F3UW@,B©MRY,EÜ>Q@,B/=WEnE¤M)SE8EG@±MRY,EeJ/=?>A93UWFO>QMRUkJ/@Âj/EGX¤MRJ/9 e JA\ÙMOY,E 729 @, =WU^B/YNM>Q@,C´¾>AB/@,E¤MRUWX­»,EG=WC©C,U^9OEGX¤MRU^J/@ B >AFSEG=^=5>AFMRY,E¾>Q@,B/=WEnE8MmSE8EG@Ǿ>AB/@,E¤MRU^X­»[EG=^C>Q@,Cn¬EG>AC,UW93EGX¤MRUWJA@ k ËN\7,9TMRY,E895F37,e,e,93EGFOF3UWJ/@¾JA\¬MO9R>A@,F3U^MOUWJ/@,FX§>Q@¾nE;>AX¿Y*UWE8jAEGCÜÃÄF3EGE»,B/7,9OE8F#à hÕAÅÀ>A@,C°Ã hµNÅ3ŤÏI@E¤gMRE8@,FOUkj/EC,U^FOX87,FOFOU^J/@X§>Q@nE\J/7*@,CUW@WÊø¥[Y

20 40 60 80

10

20

30

40

50

60

70

80

90|∆ m| = 0

γ (Pol.,B)

φ (k

,B)

20 40 60 80

10

20

30

40

50

60

70

80

90|∆ m| = 1

γ (Pol.,B)

φ (k

,B)

0

0.1

0.2

0.3

0.4

20 40 60 80

10

20

30

40

50

60

70

80

90

γ (Pol.,B)

φ (k

,B)

|∆ m| = 2

_1UWB/7,93E5*µ`1ÞE8J/E8MR93UWXG>A=NX8J/@,F3UWC,E89R>QMOUWJ/@yJA\*MRY,E59OEG=W>QMRUkj/E<XGJ/7*e,=WU^@,B;F3MO9OEG@*BAMRY#J/@yMRY,ES1/2 ↔ D5/2

¹b7[>QC,9O7,eJA=WEMR9O>A@,F3U^MRU^J/@~U^@BAEG@,E89R>A=\J/9ÀMO9R>A@,F3U^MOUWJ/@,FSVUkMRY |∆m| = 0 Ë|∆m| = 1 >A@,C |∆m| = 2 φ C,EG@,JQMREGF#MRY,E~>Q@,B/=^E·nE¤M)SE8EG@Â=?>AF3EG9#nEG>A k >A@,C¾>AB/@*E8MRU^X#»[E8=WC B γ U^FÙMRY*E>A@,B/=^EnE¤M)SEGEG@´eJ/=?>A93UWFO>QMRUkJ/@ e >A@,C¦MOY,EDe,93JQÈ3E8X8MOUWJ/@JQ\ B JA@bMRJMRY*EÙe*=?>A@*EK@,JA9O¾>A=[MRJ k <HÙ>A93LFOY[>QC,UW@,BAFXGJ/939OE8FOeJ/@,CtMOJ­=^JÉS'XGJ/7*e,=WU^@,B,_1UWB/7,93EÙMR>ALAE8@\9OJ/ WÊø¥[Y

÷ ùD [û8 #þ ú ûrÿùþùaù S1/2 ↔ P1/2

ú ûý ý$ùû®ú,ùúùrÿü

%cb;dàe(+*gfZjixf;*dlkj-d;( jr&s'=&m@(+b*(E>&mà*#k[\ | w|^_ svw ) x Y,E­MR9R>Qe,e,UW@*B~JA\5XZY[>A9OBAEGC¦e[>A9TMRUWX8=WE8FÙUW@ÜC*g@[>AUWX­E8=WE8X8MO9OUWX»[EG=^C,FÙSÀ>AF93E§>A=^UWFOE8CÖnNÛ1>A7,=>Q@,CÖX8JÉSJA9OLAEG93FUW@°ø¥ýNùAû W/ý/Y yãhU^@,XGEMRY,E8@¦v>A7,=äfbM)geEMR9R>Qe,FÙY,>¥j/E­nEGE8@7,F3EGCUW@>­jQ>Q9OUWE¤M)tJQ\vE¿he¬E89OU^¾E8@bMOFG Y,UWFoFOEGX¤MRU^J/@9OE¤jgUWE8SVFoMRY,Ee,93UW@,X8UWe,=^EGFÀJA\vv>A7,=MR9O>Ae,F8 Y,E·¾>QMRE89OU?>Q=XG>A@°nE¾\JA7,@,C°U^@°E¤j/EG9T´C,JgX8MOJ/9R>A=MOY,EGF3UWFe,9OE8e[>A93EGCÇU^@´J/7,9B/93J/7,e WÊø¥hËaýhËø/øAËøÉgËaû[Yº>AFoSEG=^=½>AFVU^@PFOE¤j/EG9R>Q=½e[>AeEG93FWÓùAhËùhøZY bs ¡ U";aOgBVw"h4ul M;U^FoJ/nbjgUWJA7,Fo\93J/ØMRY*E»,9OF3MKá`>ÉhSEG=W=E8¹b7[>QMRU^J/@U^@~jQ>AX87,7,V`

∇ · E = 0 , ÃÄ*µ/ÅMRY,>QMV@,JDFTMZ>QMOUWXÙeJAMRE8@bMOU?>A=¼E¿*U^F3MOFoMRY[>MVXGJA@*»[@,E8FVMRY,EJAMRU^J/@tJA\1>X¿Y,>A9OBAEGCe[>Q93MRU^XG=^EUW@¦\93EGEFOe[>AX8E¾UW@´>A=W=C,U^9OEGX¤MRU^J/@,FG·üIJ¥SE¤j/EG9GË>XZY[>A93B/EGC´e[>A93MOUWX8=WEX§>A@´nEMO9R>Ae*e¬E8CU^\IMOUWEPC,E8eEG@,C*EG@bM·»[E8=WC,F¾>A93E7*FOEGC¼ ® JA@,FOU^C,EG9DMRY,EP\J/=W=^J¥SVU^@,BÖMOUWEPC,E8eEG@,C*EG@bM¹b7[>AC,937,eJ/=^EeJAMRE8@bMOU?>A=`

Φ(~x, t) = Φ0(t)3∑

i=1

αi

(xir

)2

, ÃÄ*µ/ÅSVY,E89OE

Φ0(t) = U + V cos(ΩRF t) ,ÃÄ*bÅ

>A@,CMOY,E αi, r >A9OEe[>A9R>Q¾E¤MREG93FVMOJ¾nEyC,E¤MRE89OUW@,E8C JFR>QMOUWF3\]~E8¹g7,>QMRU^J/@ÇÃÄ*µ/ÅSEC,E8·>A@*C ∑αi = 0 Y,E·E8¹b7[>QMRU^J/@,FJA\oJAMOUWJ/@¦\JA9#>XZY[>Q9OB/E8CÇe[>A9TMRUWX8=WEDUW@©MRY,U^FeJAMOEG@bMOU?>A=>A9OEB/U^j/E8@nN `xi +

2αiQ

mr2(U + V cos(ΩRF t))xi = 0 , ÃÄ*ÕùAÅ

SVY,E89OE m U^F;MOY,Ee[>A93MOUWX8=WE·>QFOFK>Q@,C Q U^MOFIXZY[>Q9OB/EQ Y,U^FIC,U^«¬EG93EG@NMRU?>Q=½E8¹b7[>QMRU^J/@`U^FX§>Q=W=WE8C©>á`>QMOY,UWE87©E8¹g7,>QMRU^J/@©>A@,C©UWFÙSEG=^=Lb@,J¥SV@´UW@¦¾>QMOY,EG¾>QMRU^X8FG­l MUWF7,F37[>A=W=kSV9OUkMOMOEG@>QFI`d2u

dζ2+ (a− 2q cos(2ζ))u = 0 . ÃÄ*¦/Å

Û U^MOY~MRY,EFO7*n,F3MOU^MR7hMRUWJA@,FI`a := ai = − 8αiQU

mr2Ω2RF

, ÃÄ*§!AÅq := qi =

4αiQV

mr2Ω2RF

, ÃÄ*µû/Å

j2i¦wixe6dÈnIk (+fsdàe,- 0!

q

a

Stable

solutions

Unstable

solutions

0 2 4 6 8 10

10

8

6

4

2

0

2

_1UWB/7,93E*O`¦ãgMR>An,U^=WUkMmÁC,UW>AB/9R>Qà\JA9MOY,EÖá>QMRY,U^EG7E8¹g7,>QMRU^J/@à *$NÅ¿ v>Q9R>AE8MOEG9XZY,J/U^XGE8F;U^@~MRY,EF3Y[>AC,E8C`>A9OEG>AFV=WEG>AC~MRJDF3MZ>Qn,=WEFOJA=W7*MOUWJ/@,F8_vUWB/7*9OEKMZ>ALAE8@\9OJA Wkø¥[Y

>Q@,C ζ = 12ΩRF t

EG¹b7[>QMOUWJ/@úà *µù/ÅUWF­J/n*MR>AUW@,E8C Y*EF3J/=^7*MRU^J/@,F#MOJ²Ã h¦NÅXG>A@¯nEE¿he,9OE8FOFOEGCÜ>AF¬WÓùA/Y`u(ζ) = A

∑

n∈ZC2n cos((2n+ β)ζ) +B

∑

n∈ZC2n sin((2n+ β)ζ) , Ã *ÓýNÅ

SVY*EG9OEMOY,E C2nFO>QMRU^F3\]>KX8EG9TMZ>AU^@­93EGXG7*9OFOU^J/@­9OE8=?>QMOUWJA@#>A@,C β = β(a, q) UWF1U^@#B/EG@*EG9R>Q=>DXGJA¾e,=^E¤~@g7,­nE89G5Û°E>A93EyU^@bMRE89OE8F3MRE8C`U^@nJ/7,@,C,E8C`FOJ/=^7*MRU^J/@,FÀMRY,>QM;XGJ/939OEGF3eJ/@,CMOJMRY,EFTMZ>An*=WEMR9O>Ae,e,U^@,BJA\e[>A9TMRUWX8=WE8FGo_1UWB/7*9OE¾Ã *bÅÀFOY,J¥SVFIMRY,EyF3MZ>An*UW=^U^MmtC,UW>AB/9O>A93EG=W>QMRE8CPMRJEG¹b7[>QMOUWJ/@°Ã h¦NÅ¿ãgMZ>An*=WEMO9R>Ae,e*UW@,BDUWFVJ/@*=^teJ/FOF3UWn,=^EU^\ºMRY,EyF3J/=W7hMRUWJA@~\J/9>Q@b©C,U^9OE8X8MRU^J/@ i SVU^MRY´B/Ukj/EG@°e,>A9R>AE8MOEG93F ai, qi U^FynJ/7,@,C*EGC_[J/9>9OJAMR>QMRU^J/@[>Q=^=kU^@bj>A9OUW>A@bMI¹b7[>AC*9O7,eJ/=^E#eJAMRE8@bMRUW>A=à z nEGU^@,B·MRY*E­FThE8MR9T`>hUWFRÅVMOY,E­e,>A9R>AE8MOEG93F

ai>A@*C qi >A9OEB/U^jAEG@nN `

ax = ay = −8QU

m(r02 + 2z02)ΩRF2 , az = −2ax ,

à h^ø¥/Åqx = qy =

4QV

m(r02 + 2z02)ΩRF2 , qz = −2qx ,

Aû %cb;dàe(+*gfZjixf;*dlkj-d;( jr&s'=&m@(+b*(E>&mà*#k

0 0.2 0.4 0.6 0.8 1 1.2 1.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

q z

a z

¢r

¢z

0.2

0.4

0.6

0.8

0.1

0.20.4

0.60.8

_vU^B/7,9OE<*µù` Y,E=^JÉSE8F3MºF3MZ>Qn,UW=^UkM)K6GJA@,E5UW@>;v>A7,=NMO9R>AeSVU^MOYy9OJAMR>QMRU^J/@[>A=AF3g¾E¤MR9T/ Y,EX8J/@bMOJ/7,9=WUW@*EGFBAU^j/E­MRY,Ej>A=W7,E8FJA\ ωr >A@,C ωz UW@¦7*@,U^MOFJA\ ΩRF/2 Y,EDC,JAMUW@*C,UWXG>QMRE8FÀMRY,E9OE8B/U^J/@PJ/7,9ÀMR9O>AeU^FoJ/eEG9O>QMRE8CU^@<_vU^B/7,93EÙMR>ALAEG@P\9OJ/ WÊø¥[Y

SVY,E89OE¾SE·Y[>§j/E·X¿Y*J/FOE8@a` αx = αy = 1 Ë αz = −2 >A@,C r2 = r02 + 2z0

2 Y,UWFXZY,J/UWX8EE8@,FO7,93EGFKMOY[>QM Φ(r0, 0, 0, t) − Φ(0, 0, z0, t) = U + V cos(ΩRF t) Y,E#H ®MRU^ej/J/=kMZ>AB/E5UWF1B/Ukj/EG@­nN U >A@,C#MRY*EÀ9O>AC,U^JI\93EG¹b7,EG@*X8PÃÄI_Ž>A¾e*=WUkMO7,C,E5UWF1C,E8FOX89OUWnE8Cnb V _vUWB/7*9OE­Ã *µù/Å5F3Y,JÉSVFÀMOY,EÙ9OE8=WE¤jQ>A@NMoe[>Q93MÀJA\MRY,EÙFTMZ>An,U^=WUkMmDC,U?>ABA9R>AJA\º>#MR9O>AeSVU^MOYÖ93JAMZ>MRUWJA@[>A=½F3g¾E¤MR93Ay_[J/9 ai ¿ qi ¿ 1 MRY,E­FOJA=W7*MOUWJ/@MOJtMOY,E­EG¹b7[>QMOUWJ/@ÜJA\JAMRU^J/@©ÃÄ*µù/ÅX§>A@nEy>Ae,e*9OJ¥hU^·>QMOEGC~nb6`

ri(t) = r0i cos(ωit+ φi)(

1 +qi2cos(ΩRF t)

)

, Ã *kø/øÉÅSVY,E89OE ωi U^FVBAU^j/E8@nb6`

ωi = βiΩRF

2>Q@,C βi =

√

ai +qi2

2. Ã *køÉ/Å

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/ %cb;dàe(+*gfZjixf;*dlkj-d;( jr&s'=&m@(+b*(E>&mà*#k¹b7,EG@*XGUWE8F ωi X§>A@tnEKUW@*XG9OEG>AFOE8C~nN\EGEGC*UW@,BMRY,EIMO9R>Ae¾SVU^MRYt>#Y*UWB/Y*EG9I_©\93EG¹b7,E8@,X8ΩRF

ËY,UWBAY,EG9;_'eJ¥SEG9>A@,C¦nbÜC,E8XG93E§>AF3UW@,BPU^MOFF3UW68EÃ]MZ>ALbU^@,B~X§>A93E­MRJP93EG¾>AUW@ÜUW@>­F3MZ>Qn,=WEKMR9O>Ae,e,U^@,B­9OEGBAUWJ/@tJA\½MRY,EFTMZ>An,U^=WUkMmC,U?>QB/9R>A·Å¿vlm@~MRY*EE¤heEG93UWEG@NM;>­XGJ/Æn,U^@[>QMRU^J/@¯JA\>A=W=ÀMRY,93EGE`eJ/F3FOUWn*UW=WUÊMRU^E8FY[>QF¾MRJ°nE`n[>Q=?>A@*XGEGC²MZ>QLgU^@,B©MREGXZY,@,U^X§>Q=K>A@,Ce,9O>AX8MOUWX§>Q=XGJ/@*FOUWC*EG9R>MRU^J/@,FoUW@NMRJ>QXGX8J/7,@bMG_[JA9·UWJA@MO9R>Ae,F¾MRJÇnEÜJ/eE89R>QMOEGC±SVUkMRY©È37,F3MtJ/@,E`UWJ/@²MRY,EÜF3UWe,=^EGF3Mt>Qe,e,9OJ/>AXZYUWFyMRY,E¾\J/=W=^J¥SVU^@,BO`>`9OU^@,B`JA\oSVU^9OE·UWFXGJ/@*@,EGX¤MRE8CÂMOJÖ>`9OEGF3J/@[>A@NMD;_âXGU^9OX87,U^MGË1M)SJEG@*C,X§>Ae*F¾>AC,EJA\MOY,E#FR>AESVU^9OE#>A9OE#>AC,C,E8C¦>QMIMRY,E#F3gE8MR9T`>hU^FKJA\MRY,E#9OU^@,B>A@,C¾X8J/@,@,E8X8MOEGC·MRJyB/9OJA7,@,C¾J/9<FOJA¾EVXGJ/@*F3MZ>Q@bMJA«¼F3E8M5j/J/=kMZ>ABAEAvÏ MO9R>Ae¾JQ\¼MOY,UWFM)geEUWFDF3Y,J¥SV@UW@»[BA7,9OE°ÃÄ*$NŤ Y,E»,EG=WC¯e[>QM3MRE89O@²B/E8@,EG9O>QMRE8CnbÇMRY,U^FDFOE8MO7,eU^FD@,JAM>~e,7,93ED¹b7[>AC,937,eJ/=^ED>AFXG>A@©nED>AXZY,UWE¤j/EGC¦\J/9ÙYbgeEG93nJ/=WU^X§>A=^=kFOY,>AeEGC©EG=WE8X8MO9OJgC,E8FGË@,E¤j/EG93MOY,EG=^EGF3F¾MRY,EP¹g7,>AC,9O7*eJ/=WE~MRE89OàUWFDnNÁ\Ä>A9­MRY,EP=WE§>QC,UW@*BÖMRE89OàUW@ÁMRY,EX8EG@NMREG9>QMMRY,EeJ/FOUkMRUWJA@©JA\MRY,EUWJ/@¼ Y,E·>AChjQ>A@NMZ>AB/E8FyMRY*EGFOE·v>A7,=MO9R>Ae,FyJA«¬EG9>Q9OEDMRY*EGUW9FOU^¾e*=WU^X8UkMmt>A@*CPMRY,E8UW9o=?>Q9OB/E>AJ/7,@NMVJA\1J/e*MOUWX§>Q=>AX8XGE8FOFG[\[ ¥¦) +8Àw§_©¨uVxqPwsxÏK@DJAe*MRU^X§>A=h9OE8FOJ/@[>MRJ/9U^F7,FOE8CDMRJXGJ/7*e,=WEMRY,EoU^J/@­MOJy>ÙFOU^@,B/=WEJgC,EÀJA\aMRY,EoE8=WEGX¤MR93Jf¾>AB/@,E¤MRUWXy»,EG=WC¼ J~X8J/@*»[@,E#MRY,E­E8=WEGX¤MR93UWXG>A=º»[E8=WC`MRJ~>tnJ/7,@*C,EGC¦93EGB/U^J/@ÜU^@ÜFOe[>QXGEAË>ÖFTMZ>An,=^E~9OE8FOJ/@,>QMRJ/9­7*F3MDnE~X¿Y*J/FOE8@Â_[7,93MOY,EG93¾J/93E·MRY,E~¾JgC*Etj/J/=^7,EtFOY,JA7,=WCnE´>AFPFO¾>A=W=Ù>AFeJAFOFOU^n,=WEÜMRJ²>AXZY,UWE¤j/E´>ÂY,UWB/Y EG=^EGX¤MR9OU^XÖ»[E8=WCúeEG9e,Y*JAMRJ/@¼ Y,EeJ/F3U^MRU^J/@´JA\ÀMRY,E·U^J/@°­7,FTM­nE·U^@´MRY,E·XG>¥jgU^Mm¦SÀ>AU^F3M§Ë1SVY,E89OE·MRY,E¾»,EG=WC°UWFBA9OE§>MREGFTMU EAhMOY,Ee,93J/n[>An*UW=^UkM)DMRJ»,@,C>e,Y*JAMRJ/@>Q@,CPMOY,EG93E8\J/93EMOJDUW@NMRE89R>AX¤MoSVUkMRY~MOY,EUWJ/@~U^FB/93E§>QMOEGF3MGbsbs c ªöw

τ#";f«^"U!¬! F J>AXZY,UWE¤j/EKMOY,EK9OE8¹g7*UW9OE8EG@bMOFÀe*9OE8jgU^J/7,F3=^F3MZ>MRE8CË*U^MUWFX8J/@bj/E8@,UWE8@bMMRJ#7,FOEÙ>#F3gÆE8MR93UWX9OE8FOJ/@,>QMRJ/9¾>AC,EÀJA\aMmSJX8J/@,X§>§j/EVUW9O93J/93F1SVU^MOYEG¹b7[>A=,9R>QC,UW7,FJQ\¬X87,93j>QMR7*9OEA_[9OJA¢>`F3e¬E8X8MO9R>A=eJ/U^@bM#JA\ojgUWE¤SËMRY,E·e[>A9O>A¾E¤MRE89OF#UWeJ/9TMZ>A@NMUW@°X¿Y[>Q9R>AX¤MRE89OUWF3UW@,BMRY*Ey93EGF3J/@[>QMOJ/9;>A93EMRY*EyU^9O9OJA9ÀC*UWF3MR>A@,X8E L >A@,CPMOY,E»[@,E8FOFOE F C,E8»,@,EGCnb6`

F :=νfδν

, Ã *kø¥NÅSVY,E89OE νf UWFoB/Ukj/EG@nN6`

νf :=c

2L, Ã *køKbÅ

>A@,C`SVY,E89OE δν U^FIMOY,E­_1ÛúüÙá¢JA\MOY,E#MO9R>A@*FOUWFOF3UWJ/@eEG>ALa νf UWFIX§>A=^=WE8CMRY*E·³ \9OE8EFOeE8X8MO9R>A=9O>A@,B/EQ¶^ ® J/@,F3UWC,E89V>FOU^@,B/=^EÙUW939OJ/95SVUkMRY~»,EG=WC~9OE8¸,EGX¤MRU^jgUkM) r Ë*»[E8=WC~MO9R>A@,F)ÆUWFOF3UWJ/@ t >A@*C»[E8=WC=WJAFOFOE8F l Y,E<XGJA@,FOE893jQ>QMOUWJ/@JA\hEG@*EG9OBQC*EG¾>A@,C,F R+T +L = 1

j2i­ji@&e( jr%.dlk>f;*,-a&';d;(×&sf;- *øSVY*EG9OE R = r2 Ë T = t2 >A@,C L = l2 Ë[U ÓEQ,MOY,E\9O>AX8MOUWJ/@*FVJQ\v9OE¤¸[EGX¤MREGC¼ËaMR9R>Q@,FOU^M3MRE8C>Q@,CP=^J/F3MÀUW@NMRE8@,FOUkMRUWE8F­7,F3MoF37,ØMRJJA@,EAl \ºMOY,EG93E>A9OEÙ@,J=WJAFOFOE8FoUW@·MRY,EÙX§>§jhUkM)j/J/=kÆ7*¾E>A@,C´BAEGJ/E8MO9OU^X§>A=^=k`eE893\EGX¤M#F37,n,F3MO9R>QMOEGF#>A9OE>AFOF37,¾E8C˺MOY,ED»[@,E8FOFOE·C*EGeEG@*C,FJA@,=^~J/@MOY,Ey9OE8¸,EGX8MOU^jgU^MmPJQ\vMRY,EyUW939OJ/93FG_[JA9I>DF3MZ>Qn,=WEX§>§jhUkM)~·>QC,EyJQ\vM)SJ¾¾U^9TÆ93J/9OFyY[>§jhU^@,BMOY,E»[E8=WC´9OE8¸,EGX¤MRU^jgUkM) r1 >A@*C r2 ˺MRY,E»,@,EGF3FOE F UWFB/U^j/E8@°nNÃ@,JAMRE/`ri / 1 ÅàWµù//Y`

F =π√r1r2

1− r1r2' π

1− r1r2. ÃÄ*^ø¥ù/Å

_[JA9IU^C,EG@NMRU^X§>A=½U^9O9OJA9OF ri = r ËMOY,E#e,93JgC,7,X¤MÙJA\MOY,E»,EG=WC`93E8¸[E8X8MRUkjgU^MOUWEGF r1 · r2 XG>A@nE;FO7,n,FTMRUkMR7*MOEGC·nbMRY,E;UW@NMREG@*FOU^Mm9OE¤¸[EGX¤MRU^jgUkM) R JA\J/@,E;JA\¼MOY,EIU^@,C,U^jgU^C,7[>A=,UW939OJ/93F>Q@,CPMRY,EE¿*e*9OEGF3FOUWJA@\7,93MOY,EG9VF3UWe,=WUÊ»[EGFMRJO`F ' π

1− r2 =π

1−R =π

T + L , ÃÄ*^øKNÅSVY*UWXZY¦U^FKFOJ/E8MOUWEGFI7,FOE8C©U^@`MRY,E­=^U^MRE89R>QMO7,9OE^WÓù/ùY Y,E­»[@,E8FOF3EDUWFK9OE8=?>QMOEGC`MRJ·MRY,EXG>¥jgUkM)tEG@,E89OBA~F3MOJ/9R>ABAEMOUWE¯®VnN `

F =πc

L· τ . ÃÄ*^øF!/Å

lm@*F3MREG>ACJA\ºMRY,E=WUk\E8MOU^E τ JA\]MRE8@PMRY,E¾Ã]»[EG=^Cgf*ÅC,E8X§>§P9R>MRE κ C,E¤»[@,E8CÜ>AFZ`κ =

1

2 τà h^ø¥û/Å

U^FD7,F3EGC²l \IMRY,EP»,@,EGF3FOE F U^FDC,UkjgUWC,E8C¯nb π >ÖjQ>A=^7,EPUWF­\J/7,@,CÂ\J/9­MRY,E>¥j/EG9O>AB/E@b7,­nEG9JA\IMRUWEGF>¦e,Y,JAMOJ/@¯MR9R>§j/EG93FOE8F¾MRY,EX§>§jgU^MmÂnE8\JA9OEU^MU^FDMR9O>A@,F3¾UkMOMRE8C¯J/9>Qn,FOJ/93nEGCÖnNÜ>¾¾U^9O9OJA9G Y*UWFK@g7*­n¬E89KUWFIJA\B/9OEG>QMÙU^@bMOEG9OE8F3M>AFKUkMKXGJA9O9OE8FOeJ/@,C*FÙMRJMOY,E@g7*nE89VJA\ºMRUWE8FÀMRY*Eye*Y,JAMRJA@e[>AF3FOEGFVMRY,EMO9R>Ae,eE8CU^J/@bs°bs ± g5!"

w0#";S 2² g5³

Y*Ey»[@*EGFOF3E F U^FV@,JAMVMOY,EyJA@,=^t9OE8=WE¤jQ>A@NMI@b7,­nEG9VXZY[>A9R>QX8MOEG9OU^FOU^@,BDMOY,EyXG>¥jgU^Mm~FThF)ÆMOEGKl MÙUWFK>A=WF3JtU^¾eJA93MZ>Q@bM;MOJ·\JgXG7*FKMRY*E­X§>§jgU^Mm=^UWB/YNMIJ/@`MRY*E­UWJA@`U EAaMRJ~>QX¿Y,U^E8j/E>·FO¾>A=W=½SÀ>QUWF3M´>AFKSEG=^=>AF>tF3·>Q=W=º¾JgC*E#j/J/=^7,E L · πw20 ® J/@*FOUWC*EG9KMRY,E#\7,@hÆC,>A¾E8@bMR>A= :5á

00¾JgC,EIUW@~>#FTg¾E8MO9OUWXIXG>¥jgU^Mm¾JA\½=WE8@,BAMOY L Y*EÙSÀ>QUWF3MFOUW68E ω0Ã9R>AC*UW7,FoJA\ºMRY,ES>AUWFTMI>MVMOY,EX§>§jhUkM)tXGE8@bMOEG9RÅoUWFoB/Ukj/EG@nNyWÓù/Y`

w20 =

λ

2π

√

L(2R− L) , à *kø¥ýNŵ¶8 ¸·[ÿ$ÿüû¹Ký»ºXüûº,¼Fþù ý¦ûrÿ$ÿºXûlùrúüFý/ ù

I(t) = I(0) · e− tτ I(t) ½ ý¾ ÿ$ý ¯ý5ý»ºû¿8 ýL

tÀ ¶8 D ù° ùÿ§ü¾ù ½ , ùKù°û®ÿ¦ÿÁ ùúÿ¦ý¦û ½ ÿº/ù¿cû[ü¾Âþ¾ ý$ùýàý5¯ú

N %cb;dàe(+*gfZjixf;*dlkj-d;( jr&s'=&m@(+b*(E>&mà*#k

∆ Ã

Ä Ä

_vU^B/7,9OEo*¦!`.]IEG>A9<X8J/@,X8EG@bMO9OU^X;XG>¥jgUkM)A Y,EV¾U^9O9OJA9OF>Q9OEVF3EGe[>Q9R>QMOEGC¾nb­FO=WU^B/YNMR=k#=^EGF3FÃ ∆L ÅMRY[>Q@ 2R

∆ L

R R

∆ xθ

_vU^B/7,9OED*Óûw`­áU^FR>A=^UWBA@,E8@bMJA\ÀJA@,E¾U^9O9OJA9nN ∆x Y,E¾FThE8MR9T©>hUWFJA\MOY,EUW9O93J/9XG7,9TjQ>QMO7,9OE8FK>Q@,CPMRY,E89OE8\JA9OEMRY,E=^UWB/YNMoJgC,EU^FoMOUW=kMREGC~nN θ

SVY,E89OE R C,EG@*JAMRE8FKMRY,E¾U^9O9OJA9V9R>QC,UW7,F;JA\XG7,9TjQ>QMO7,9OEQÙ_[J/9V»*hEGC L ÃC,EG¾>A@,C*EGCÜnNMRY*EF3UW6GEJQ\ºMRY*EMO9R>Ae[ŤËhMRY,E9OE8FOJ/@,>QMRJ/9ÀJA\1XZY,J/U^XGEU^FMRY,E@,EG>A9oXGJ/@*XGE8@bMR93UWX9OE8FOJ/@[>MRJ/9XGJA¾eJ/F3EGCÁJA\VMmSJÖX87,93j/E8C¯¾U^9O93J/9OFySVUkMRYÁ>Ü9R>QC,UW7,F#JA\VXG7,9TjQ>QMR7*9OE~FO=^UWB/YNMR=k¦n,UWB/BAEG9MRY,>A@'Y,>A=^\ÙMOY,EÜUW939OJ/9DC,UWFTMZ>A@,X8E L >AF·F3Y,JÉSV@'UW@²»[B/7,93EÂÃÄ*§!AŤHKE8»,@,UW@,B R :=(L+∆L)/2 Ë,EG¹b7[>MRUWJA@°Ã h^ø¥ý/ÅX§>Q@nESV9OUkMOMOEG@>QFI`

w20 =

λ

2π

√L ·∆L =

L · λπ

√

∆L

4L. Ã *µANÅ

_[J/9ÀMOY,EFOeJAM;F3UW6GE>QMVMOY,EUW9O93J/93FMRY,EE¿*e*9OEGF3FOUWJA@a`w2mirr =

L · λπ

√

4L

∆LÃ *µhøÉÅ

UWF5jQ>Q=WU^CWÓù//Y _[9OJ/MRY,E8FOEÙE¤he,93EGFOFOU^J/@,FÀU^MUWFXG=^E§>A95MOY[>QMÀ>#F3·>A=^=[SÀ>AU^F3M@,E8EGC,FÀ=?>A93B/EUW9O93J/93F>Q@,CMRY,EonE§>AÒC,Ukj/EG9OBAEGF<¹b7,UkMREoFTMR9OJA@,B/=^AvÏIe[>A93M\9OJ/ÚMRY*UWFGË/¾U^FR>A=^UkB/@,E8@bM

j2i­Åixfq*dlkj-w*gm %&/E eq&s-jI(+*J- ¢g-(+*gE /JQ\oMOY,E·UW939OJ/93FyJQ\ O(∆L) =^E§>AC©MRJ`C,U^FR>AFTMR9OJA7,F#X8J/@,FOE8¹b7,EG@,XGEGFGÜÏIFOFO7,Et>FO¾>A=W=F3Y,U^\]MJA\ ∆x JA\¬JA@,Eo¾U^9O9OJA9>QnJ/7*MMOY,EoF3g¾E¤MR93­>hUWFFOY,J¥SV@¾U^@­»[B/7,93EÃÄ*ÓûNŤ Y,E=^UWB/YNMÀ¾JgC,EÙSVU^=W=¬MOUW=^MMRY*9OJ/7,BAY>A@>Q@,B/=^E θ `

θ ≈ tan θ =∆x

∆L, ÃÄ*ÕA/Å

>Q@,C E¤j/EG@NMR7[>A=^=^²e[>A93MOF~JQ\MRY,EÜnE§>A SVUW=^=IY,UkM·MRY,EÜEGC,B/EÖJQ\MRY,EÜY,UWBAY'9OE¤¸[E8X8MRU^J/@X8JN>QMOUW@,B/FÀJA\1MOY,E¾U^9O93J/9OF8 JBN>AUW@©>PFO¾>A=W=v¾JgC,Ej/JA=W7,EMOY,EX§>§jgU^Mm¦=^EG@,BQMRY©FOY,JA7,=WC©nEX¿Y,JAFOEG@´MOJnE>QFFO¾>A=W=v>AFeJ/FOF3UWn,=^EADVEGC*7,X8MOUWJ/@¦JA\MOY,EDX§>§jgU^MmÖ=^EG@,BQMRY L ≈ 2R U^F¾J/FTMF3MO9OUW@gÆBAEG@bMO=^­=WU^UkMOEGCnb­MRY*E;FOU^6GEVJA\¬MOY,EVMO9R>Ae¾>Q@,C¾X8J/@NMRUW@*B/EG@NMR=^#nb­MRY,EVFOY,U^EG=^C,UW@*BJA\MRY,E@*J/@,XGJA@,C,7,X¤MRUW@*B¾U^9O9OJA9OF5ÆI>AFoSEG=^=½>AFoMOY,E@,EGX8EGF3FR>A93>AXGX8EGFOF;JA\v=W>AFOE89onE§>AFG[\Ç ¨!uVw§_+Gx½uVr -­Í ) #xU+¿½u aÈ·x½uVÍ Y*E­e[>A9R>Q¾E¤MREG93FIMRY[>M>A93E#9OE8=WE8j>A@bMK\JA9IMOY,E­FTgF3MRE8ÔXGJA¾eJ/F3EGCÜJA\<UWJA@ËMO9R>AeÜ>A@,CXG>¥jgUkM)~>A93EMOY,Ey>QMOJ/UWXKC,E8X§>§P9O>QMRE γ⊥ = 2π · 0.076 ü;6AË,MOY,EX§>§jgU^MmtC,EGXG>¥~9R>MRE κ>Q@,CMRY,E;XGJ/7,e*=WU^@,B g n¬E¤MmSEGE8@~XG>¥jgU^Mm»,EG=^C¾¾JgC,E;>A@,C¾UWJA@ κ >Q@,C g >A9OEV9OEG=W>QMRE8CMRJE¿heEG9OU^¾E8@bMR>A=aC,E8FOUWBA@·e[>Q9R>AE8MOEG9OF L Ë w0

>Q@,C F ËbMRY,E;X§>§jhUkMmD=WE8@,BAMRY¼ËbS>AUWFTM9R>AC*UW7,F>Q@,CP»[@,E8FOFOE93EGF3eEGX8MOU^j/E8=^AËanb6`κ =

πc

2FL , ÃÄ*ÕQNÅ>Q@,C°Ã\J/9ÀMOY,E¹b7[>AC,937,eJ/=^EMR9O>A@,FOUkMRU^J/@ S1/2 Ë mj = −1/2↔ D5/2

Ë mj = −5/2 Å

g =

√

5cλ2γ⊥2π2Lw2

0

. ÃÄ*Õ[bÅlm@ÖB/E8@,EG9O>A= κ >A@,C γ⊥ ÃÄJ/9Ù>QMÙ=WEG>AF3MÙJA@,E­JA\<MOY,EG·Å;FOY,JA7,=WCÜnEF3·>Q=W=ºX8J/¾e,>A9OE8CÜMRJg Ö_[9OJ/MRY,E·E¤heEG93UWEG@NMZ>A=eJ/U^@bM#JA\ojgUWE¤S γ⊥

>Q@,CÇMOY,E·MR9O>A@,F3U^MRU^J/@©SÀ>¥j/E8=WE8@,BAMOYλ >Q9OE¾»*hEGCÁ\J/9MOY,EtF3gFTMREGà>QM­Y,>A@,C κ X§>Q@ÁnE·MR7,@,E8CÂU^@,C,EGeE8@,C,E8@bMR=k°nb©MRY,E»,@,EGF3FOE F SVY,U^X¿Y°C,E8eEG@,C*FF3J/=^EG=^ÖJ/@´MOY,E¾¹g7,>A=WUkM)ÖJA\MRY,E¾UW939OJ/93F¾ÃÄF3EGE·EG¹b7[>MRUWJA@ÃÄ*køÉù/ÅOÅ¿#Û U^MOYÖMRY,EDXG>¥jgU^MmÜB/E8J/E8MR9TÜ9OE8¹g7*UW9OE8C¦MRJPUW@,X8J/9OeJA9R>QMOED>A@¦UWJ/@ÜMR9O>Ae©>A@,CMOY,E93EGF37,=^MOUW@,B;F3eJAM1FOUW68EGF1JA@MOY,E5¾U^9O9OJA9OF½>V»[@,E8FOF3EÀJQ\[///A/;>QM 729 @,U^F1>;9O>QMRY,E89JAe*MRU^¾U^FTMRU^XDj>A=W7,EQ ® J/@,F3UWC,E89­@*JÉS¡MOY,E·C,EGeE8@,C,EG@*XGEPJA\ g `DC,E8XG93E§>AF3UW@,B`MRY,E¾SÀ>AU^F3M9O>AC,U^7,F w0

U^@,XG93E§>AF3EGF g SVU^MOY,J/7*MXZY[>A@*B/UW@,B κ ÏIFÀF3Y,JÉSV@~U^@·FOEGX¤MRU^J/@¦Ã *ÓNŤËh>#F3·>Q=W=XG>¥jgUkM)#SÀ>AU^F3M<9OE8¹g7*UW9OE8F<j/E893B/JgJgC>A=WU^B/@*EG@NMJA\MRY,EVXG>¥jgU^Mm­¾U^9O9OJA9OFGºÏoC,C,U^MOUWJ/@,>A=^=kUkMU^F¾JA9OEC,U vXG7,=kMMRJPXGE8@bMOEG9MOY,EDUWJ/@ÖUW@ÖMOY,ESÀ>AU^F3M§ßÙe*MOUWU^F3MOU^X§>A=k=k`>~SÀ>QUWF3MJA\È ùlù[üýÿ5LûrúU #úû®þþ?üàý$ùàù­5É ùZÿ¦üÉ» ú û¹º¾ /û®úUûýþ[úù ½ÊÿLlùú[ýG #lù×þ? û¿ý¦ù;ùËýúù×ùý¦ù¬

%cb;dàe(+*gfZjixf;*dlkj-d;( jr&s'=&m@(+b*(E>&mà*#kw0 ≈ 10 µ ª¾>¥nEK·>Q@[>AB/EG>An,=^EAvÏâXZY[>A@,BAEÙUW@¾MRY*EÙXG>¥jgU^Mm¾=^EG@,BQMRY L UW@*¸[7*EG@,X8EGFκ >Q@,C g ~Û Y*EG9OEG>AF­>=?>Q9OB/E g ∝ 1/

√L 93EG¹b7,U^9OEGF#>FO¾>A=^= L U EAº>F3·>Q=W=JgC,Ej/J/=^7,¾EQË,F3·>Q=W= κ ∝ 1/L C,E8·>A@*C,F;>D=WJA@,BX§>¥jgUkM)A

bs°Ìq d×O#"¾ #³;àl"hlm@#MOY,EF3MO9OJ/@*BÙX8J/7,e,=^UW@,B;9OE8B/UWE5UkM1UWF193EG¹b7,U^9OEGCMRY[>QM g À κ >Q@,C g À γ⊥

1lm@,F3EG93MOUW@,Bà hÕA/Å>Q@,C·ÃÄ*µ/bŬUW@NMRJÀMRY*E»[93F3MºU^@,EG¹b7[>Q=WU^Mm;MRY,E\J/=^=WJ¥SVU^@,BÀ93EG=W>QMRU^J/@,FOY*UWe,F>A9OE\JA7,@,Ca`√

5cλ2γ⊥2π2Lw2

0

À πc

2FLÃ *µ/ù/Å

⇒ L ·( Fw0

)2

À π4c

10γ⊥λ2∼= 1022

1 . Ã *µNÅlm@,F3EG93MOUW@,BDMOY,EJ/e*MOUWU^F3MOU^XKjQ>Q=W7,E8Fo\J/9 w0

>A@,C F >D=WU^UÊMoJA@ L U^FÀ\J/7,@,Ca`L À 10 . Ã *µ0!/Å

Y,U^FUWF>A=W93E§>ACh­7,X¿Y·MOJgJ­=W>A9OBAE;\J/9>yM)ge,U^X§>A=¹b7[>A@bMO7,ªJ/e*MRU^XGFE¤heEG93UWEG@NM§<_[J/9E¤*>Ae,=^EAË[MOY,EF3eJAMIFOUW68EtÃ9R>AC*UW7,FRÅoJA@MOY,Ey¾U^9O9OJA9OFV>QM 5 C,UWFTMZ>A@*XGEyUWFV¾J/93EMRY[>A@10 XG>AFOF37,UW@,BP> 10 µ SÀ>QUWF3MG#lm@,F3EG93MOUW@,B´ÃÄ*Õ[bÅIUW@NMRJtMOY,EFOEGX8J/@,C©UW@,E8¹g7,>A=WUkM)Ã g À γ⊥

ÅgUWE8=WC,FZ`√

5cλ2γ⊥2π2Lw2

0

À γ⊥Ã *µAûNÅ

⇒ Lw20 ¿

5cλ2

2π2γ⊥∼= 100 XG 3 . Ã *µAýNÅ

iIF3UW@,BMRY*Eye*9OE8jgU^J/7,FojQ>A=^7,EÙ\J/9 w09OE8FO7,=kMRFVUW@>9OE8¹g7*UW9OE8¾E8@bMVE§>AFT~MOJ¾>AXZY,U^E8j/E`

L ¿ 1000 Lb . Ã *Ó//Å® J/­n,U^@,UW@*BynJAMRY·XGJA@,C,UkMRUWJA@,F5U^MXG>A@·nEKFOE8EG@tMOY[>QMUkMU^F>9O>QMRY,E89X¿Y,>A=W=^EG@,BAUW@,BMR>AFOLMRJJAn*MZ>AU^@­F3MR93J/@,BXGJA7,e,=WU^@,BKMOJ>ÙX§>¥jgUkM)»,EG=WCJ/@­MOY,E S1/2 ↔ D5/2

® > + MR9O>A@,FOUkMRU^J/@ Y,EyXG93U^MRU^X§>A=¼XGJA@,C,U^MOUWJA@~\J/9VJ/nhMZ>AU^@,UW@*BDFTMR93J/@,BXGJ/7,e*=WU^@,BMRJDMRY*Ey¹b7[>AC,937,eJ/=^EMR9O>A@hÆFOUkMRU^J/@UWF g À κ Ë[SVY,EG93E§>AFIMRY,EX8J/@,C*U^MRU^J/@ g À γ⊥UWF;EG>AF3PMRJ·>AXZY,U^E8j/EC,7,EyMRJMRY*EFO¾>A=^=½MR9O>A@,F3U^MRU^J/@=WUW@*E8SVUWChMRY Y,U^FÙU^FK@,JAMKMR937,E­Uk\5>·C,U^eJ/=WEMR9R>A@*FOU^MOUWJA@`UWFIXGJ/7*e,=WE8C

j2i­Åixfq*dlkj-w*gm %&/E eq&s-jI(+*J- ¢g-(+*gE NùMOJ°MOY,E`X§>§jhUkM)»[E8=WC ® J/@,F3UWC,E89¾\J/9¾E¤*>Ae,=^EMOY,E D5/2

Ë mj = −5/2 ↔ P3/2Ë

mj = −3/2 C,UWeJA=WEÖMR9O>A@,FOUkMRU^J/@ >QM 854 @, Y,E´MR9O>A@,FTj/EG9OFO>A=C,E8X§>¥ú9R>QMOE´UWF2π · 790 LgüI6QË<MOY,EtXGJA7,e,=WU^@,B gdip JA@ÇMRY,U^FMR9R>Q@,FOUkMRUWJA@°UWF gdip = 2/

√5 · g Wkø¥ù[Y i;FOU^@,BDMOY,UWF8Ë*MRY,EE8¹g7*U^jQ>Q=WEG@NM;U^@,EG¹b7[>Q=WU^Mm¾MOJX8J/@,C,UkMRU^J/@©Ã *µNÅnEGXGJA¾E8FI`

L ·( Fw0

)2

À 10151 , ÃÄ*µhøÉÅ

SVY*UWXZYE¤j/EG@\JA9ve[>Q9R>AE8MOEG9OFºMREG@#MRUWE8FºSJ/9OF3EMRY[>Q@e*9OE8jgU^J/7,FO=ktÃÄ>I»[@,E8FOFOEoJA\ 30000>Q@,C>#SÀ>QUWF3MoJQ\ 100 µ ·Å93EGF37,=^MOFoUW@·MRY,EÙ9OEG¹b7,U^9OE8¾E8@bMÀMRY[>M LÀ 11 VÂh@,JAMÀMRJgJC*U¨vXG7,=kMÀMRJ­\7,=^»,=W= Y,EF3EGXGJA@,CX89OU^MOEG93UWJ/@ËgMRY*EE8¹g7*U^jQ>Q=WEG@NMVXGJA@,C,UkMRUWJA@tMRJEG¹b7[>QMOUWJA@ÃÄ*µAýNÅX8J/@,X8EG9O@*UW@,BDMOY,E¾JgC,EÙj/J/=^7,EAË*UWFI`Lw2

0 ¿ 9× 103 ·µ 2 . ÃÄ*µ//ÅßÙnNjhU^J/7,F3=^ÖFO¾>A=W=XG>¥jgU^MOUWEGFy>A9OE@,E8EGC,E8C°MRJ`>AXZY,U^E8j/E¾FTMR9OJA@,B`X8J/7,e*=WUW@*BJ/@°>C,UWeJA=WEMO9R>A@*FOU^MOUWJ/@¼ºl MvU^Fv>AF3FO7,EGC#MOY[>QM1MOY,E=WEG@*BAMRY#JQ\a>IXGJA@bj/E8@bMRU^J/@[>A=gXG>¥jgUkM)yn*7,UW=kM1>A9OJ/7*@,C>~UW@,UW>MR7*9OU^F3EGCÖUWJ/@ÜMR9O>Ae¦UWFK\9OJ/Ô>A9OJ/7*@,C©F3E8j/E89R>A=¾ Wµù//Y7,e¦MOJ>t\E8SÒXG˽>AF7*FOEGC'U^@²MRY,U^FSJ/9OLaâlm@,F3EG93MOUW@,B L = 2 X8 UW@NMRJ'ÃÄ*ÓN/žEG>A@,Ft>©SÀ>AU^F3M·9R>AC*UW7,Fw0 ¿ 21 µ UWF;93EG¹b7,U^9OEGC¼;ÐEGE8e,UW@,B¾U^@¾U^@,C>A=^=MRY,EJAMOY,EG9;9OE8¹g7*UW9OE8EG@bMOFIJA\vMRY,EE8@bMOUW9OE>Ae,e,>A9R>MR7,FoMOY,UWFoUWFoFTMRU^=W=>­j/EG93~C,EG¾>A@,C*UW@,BD>A@,CXZY[>A=^=WEG@*B/UW@*BX8J/@,C,UkMRUWJA@bs»Ìq ± ÎÍ #³;"¾l \MRY*E#¹b7[>AC,937,eJ/=^EMO9R>A@*FOU^MOUWJ/@U^FIXGJA@,FOU^C,EG93EGCÖ>AFK>¾¹g7*n,U^MIMO9R>A@*FOU^MOUWJ/@PMOJ·F3MRJA9OE#>A@,Ce*9OJgXGE8FOFt¹g7,>A@bMO7, U^@*\J/93·>MRUkJ/@@W/Y ËVUkM·U^F¾SJ/93MOYbSVY,U^=WEMRJ°=WJgJ/LÇ\J/9¾E8MOY,JgC,F¾MRJX8J/@bjAEG93M¾MRY,EF3MR>QMRU^J/@[>A9T¹b7,n*U^M¾UW@*\JA9O¾>QMRUkJ/@ÂXGJ/Y*EG9OE8@bMO=^ÂMRJ°>©e,Y,JAMOJ/@²MRY,>QM·XG>A@nEFOE8@bMtJÉj/E89·=W>A9OBAEC,U^F3MZ>Q@,XGE8FG üIEG93EAËÀMRY,ESEG>AL¯XGJA7,e,=WU^@,B©JA\¹b7,n,UkM¾MR9R>Q@,FOUkMRUWJA@>Q@,CÁX§>§jgU^Mm´»[E8=WCÇUW@°MOY,E`³µn[>QCÁX§>§jhUkM)´9OE8B/UWEQ¶1X8J/EGF­U^@bMRJ`e,=W>¥A Y,Et\J/=^=WJ¥SVU^@,BX8J/@,C,UkMRU^J/@,FÀ7*F3MVnEyE8MI`

γ⊥ ¿ g2/κ¿ κ . ÃÄ*Ó/NÅi;FOU^@,BÇÃÄ*ÕQNŤËà *µ/bÅ>A@*C°93E§>A=^UWFTMRU^Xe[>A9O>AE8MRE89OF\JA9»[@*EGFOF3E F Ë1XG>¥jgUkM)Ö=WEG@*BAMRY L>Q@,C°SÀ>AU^F3M#9R>AC*UW7,F w0

ËvSEtFOE8E·MRY[>M#MOY,E·=WJ/@*Bfg=^Ukj/E8C°¹b7[>AC*9O7,eJ/=^E¾MR9O>A@,FOUkMRU^J/@´U^F­>BAJgJgC²X§>A@*C,UWC[>MREUW@ÂMRY*E©³µn,>ACXG>¥jgU^MmÂ9OE8B/U^¾EQ¶^ Y*E`93EG=WE¤jQ>A@NM¾¹g7,>A@bMOU^MmÁ\J/9MRY,EFTgF3MRE8U^FÀMRY,EXGJgJAeEG9R>MRU^jgUkM) C Ë[C,E¤»[@,E8C`U^@´Ã hµùAýNÅ>AFI`C :=

g2

2γ⊥κ=

5Fλ22π3w2

0

. ÃÄ*µ/bÅ Y*EX§>§jgU^MmP=^EG@,BQMRY L U^FVFOEGE8@`MOJ¾XG>A@,X8EG=ºJA7*MK>Q@,CX§>A@nEyXZY,J/F3EG@\9OEGE8=^~e,93JÉjgUWC*EGCMOY[>QMXGJ/@*C,U^MOUWJ/@ÃÄ*µANÅ93EG¾>AUW@,FjQ>A=^UWC·ÏØY,U^B/Y´n,7*M#@*J/@,E8MOY,EG=^EGF3F­93E§>Q=WUWFTMRU^Xj>A=W7,E

0 %cb;dàe(+*gfZjixf;*dlkj-d;( jr&s'=&m@(+b*(E>&mà*#k\J/9#MRY*E~»[@*EGFOF3EUWF 200000 Ë<\JA9­MOY,E~X§>¥jgUkM)´SÀ>AU^F3MD>Ö9R>QC,UW7,F­JA\ 25 µ UWBAYbM­nE>AXZY,UWE¤j/EGC¼ Y*EGFOEj>A=W7,E8FoUWe,=k C ≈ 14 Y,EE8¾U^FOF3UWJ/@t9R>QMOEUWFÀEG@*Y[>A@,X8EGCnNtMOY,E7,9OX8EG=^=º\Ä>AX8MOJ/9 F = (2C + 1) ≈ 29 >A@,C`MOY,E\9R>AX¤MRU^J/@`JQ\<E8¾U^FOFOU^J/@EGU^M3MREGCUW@NMRJMRY*EyXG>¥jgU^Mmt¾JgC*EU^F β = 2C/(2C + 1) ≈ 96 i

Y*E·9OEG¾>AU^@,U^@,BXG>AFOE·7,@*C,EG9yMRY,E¾XGJ/@*C,U^MOUWJ/@©JA\ÀSE§>AL¦XGJ/7,e*=WU^@,BU^FC*E8»[@,E8CÁnNMRY*EyXG93U^MOEG9OU^J/@a` κ¿ g2/γ⊥ ¿ γ⊥Ë[MOY,UWFonEGU^@,BLb@,JÉSV@`>AFoMRY,E³µn,>AC`>QMOJ/9OEGBAUWEA¶k_[J/9yMRY,E·¹b7[>AC,937,eJ/=^E·MO9R>A@*FOU^MOUWJ/@©MOY,UWFX§>AF3EtUWFyMREGXZY,@,U^X§>Q=W=^¦@,JQM­9OE§>Q=WU^FR>An,=^EC,7,E¾MRJMRY*Ey=^J/@,B=WUk\E8MRUkEÙJQ\1MOY,E¾E¤MZ>AFTMZ>An,=^EFTMZ>QMOEA

_[JA9VMRY,EE¤heE89OUWEG@NMKe,9OE8FOE8@bMRE8CÖY,EG93EMRY*E#e,>A9R>AE8MOEG93FIFR>QMOUWFT\]hU^@,BMOY,EXG9OUkMRE89OU?>JA\ºMRY,ESEG>AL~X8J/7,e*=WUW@*BD93EGB/U^¾E>Q9OE`>QMOJ/UWXKC,EGXG>¥P9O>QMRE γ⊥ = 2π · 0.076 üI6ËXG>¥jgU^MmtC,EGXG>¥P9O>QMRE κ = 2π · 102 Lgü;6yËU^J/@gfgXG>¥jgU^Mm·XGJA7,e,=^UW@,B g = 2π · 89 ü;6

Y,EyX8JgJ/eEG9O>QMRUkjgU^Mm C >A@*C<7*9OXGE8=W=\Ä>QX8MRJA9 F >A9OEX§>A=^XG7,=W>QMRE8C~MRJDnE`XGJgJ/eE89R>QMOU^jgU^Mm C ≈ 0.51 Ë7,9OX8EG=W=¼\Ä>AX8MOJ/9 _ ≈ 2

Y,E8FOEe[>Q9R>AE8MOEG9OFV>A9OEC,UWF3XG7,F3FOE8CMRY,JA9OJ/7,BAY,=^·UW@PXZY[>AehMREG9Ã7NŤ

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diode laser system854 nm

diode laser system866 nm

diode laser system423 nm / 390 nm

diode laser system785 nm

Ar -ion laser

frequency doubling397 nm

Ti:sapphire 793 nm

Ti:sapphire 729 nm

fixed referenceresonator

lock lock

CCD PMT

+

experiment table

laser table

lock

vacuumapparatus

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785ç+

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866ç+ ñ[çºë

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D3/2>A@,C~MRY,E D5/2

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423ç ñ,ç1ë

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shieldingcaps

PTZ

bellow

micrometerscrews

glass cell

to pump unit

feedthrough

integrated spring element

cavitymirror

electrongun

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)

Time (µs)

_1UWB/7,93E^,¦!`·:vheJ/@,EG@NMRUW>A=C,EGXG>¥ÇJA\VMOY,Et»,=W=WE8CÇX§>§jhUkM)©SVU^MOYÇMRUWE¾XGJ/@*F3MZ>Q@bM τ =(5.38 ± 0.004) µ FE§>QFO7,9OE8Câ>Q\]MOEG9tMRY,EÖ»[9OFTMn[>QLAEGJ/7hM§ Y,E¦MRY*UW@ =WU^@,EÖU^F~MOY,EE§>QFO7,93EGC±MO9R>A@,F3¾U^FOF3UWJ/@ Y,E`MRY,U^XZL¯=WU^@,EUWFMRY,E`E¤heJ/@,E8@bMOU?>A=;»,MG ® >A=^XG7,=W>QMRE8C»,@,EGF3FOE` F = 241300± 170 ÷°À Þß ºý û®ÿJÚ°úL5Ô|ú ûý " Ê !K ×ØGàäüú ù #Dúû¯º

Nû %cbqdàe(+*gfåÅ2i¹*DCEeg*gfsj5E *g'N(Odhkk-w*(n+ee,U^EG6GJAFGÛ U^MOY~MRY,EY,U^B/Y~j/J/=kMZ>AB/EÙFOJ/7,93XGEye*9OJÉjgU^C,UW@,B7,ePMRJ ≈ 330 Ñ#Ë[U^MoUWFoeJ/FOF3UWn,=^EMRJFOY*U^\]MEG>AXZYUW939OJ/9v>AnJA7*MJ/@,EÀ\93EGEoFOeE8X8MO9R>A=,9R>A@*B/Eyà ≈ 400 @*tÅ¿ Y,U^FU^F>ÙMOY,UW93CJA\¼MOY,E;9R>A@,BAE;J/@,EVSJ/7,=^C¾E¿*eE8X8M\93J/ÒMRY,E;n[>A9OE;e*UWEG68JyMR7,nE8F>A@,C¾UWF<e,9OJAn[>An,=kC,7,EMRJMRY,E93EG=W>QMRUkj/EG=k·Y,U^B/YPFOe,93UW@,BMOEG@,F3UWJ/@5~E§>AF37,9OU^@,B­MRY,EC,EGXG>¥~JA\ºMRY*E»[=^=WE8CPXG>¥jgU^Mmõ»Æ;SEX§>A=^XG7,=W>QMRE8C~MOY,E»[@*EGFOF3E FMRJnE ≈ 240000 >Q\]MOEG9MOY,Eo»[9OFTM5n[>ALAE8J/7*MKû,B/7,9OEyÃ7,¦!/ÅOÅ¿HK7*EVMRJF3E§>A=^UW@,Be,93J/n,=^EGFÃÄn,>AFOE8e,=?>QMOEQBA=?>AF3FOX8EG=W=?ÅMOY,Eyn[>ALAE8J/7*MIY[>ACMOJnE93EGeEG>QMRE8CÜ>A@,C7*@*\J/93MO7,@[>QMOEG=ktMRY,U^F=WEG>AC©MOJ`>FOE¤j/EG9OE·C,E8B/9R>QC[>QMRU^J/@©UW@¦»[@,E8FOFOEQËC*JÉSV@´MRJ ≈ 35000 tÏI=W=E¤heEG93UWEG@NMRFC,E8FOXG93UWnE8CÂUW@´MRY*UWFMOY,EGF3UWF#SEG9OEteEG9T\J/9OE8CÇSoU^MRY°MOY,UWF=WJ¥SEG9y»[@,E8FOFOEQ¦Û UkMRYÇ>»*Æ@,E8FOFOE F = 35000 >Q@,CÜS>AUWFTMÙ9R>AC,U^7,F w0 = 54 µ MRY*E­e[>A9O>AE8MRE89OF g >A@,C κ >A9OEX§>Q=WXG7*=?>QMOEGC~MOJnE¾ÃÄX¤\)[FOE8X8MOUWJ/@h$bÅ®`

κ =πc

2FL = 2π · 102 LgüI6 , Ã7,køÉÅ

g =

√

5cλ2γ⊥2π2Lw2

0

= 2π · 89 ü;6 , Ã7,µ/Å\9OJA¢SVY*UWXZYÂMOY,E~E¤heEG93UWEG@NMZ>A=^=^´U^¾eJ/9TMZ>A@NM­e,>A9R>AE8MOEG93FXGJgJ/eE89R>QMOU^jgU^Mm C >A@*C7,9OX8EG=^=\Ä>AX8MOJ/9 F X§>Q@nEC,E8C,7,X8EGCa`

C :=g2

2γ⊥κ= 0.51 , ÃD,ÓNÅ

F := 2C + 1 ≈ 2 , Ã7,bÅ>A@,C~MOY,EÙ\9R>AX¤MRUWJA@~JA\º=WU^B/YbMEGU^M3MREGC~U^@bMOJ­MRY*EXG>¥jgU^Mm·¾JgC,EQË*B/U^j/E8@Pnb β Ë*nEGXGJA¾E8FI`

β := 2C/(2C + 1) ≈ 0.5 . ÃD,µù/Å® J/e[>A93EGC¦MOJPMOY,EDe[>A9O>AE8MOEG9OFe,=?>Q@,@,EGC©>A@,C©C*UWFOX87,FOF3EGC°U^@¦XZY[>Ae*MOEG9¾Ã NÅ¿ËSEXG>A@E¤heE8X8M#J/@,=kÖ>PFO¾>A=^=E¤«¼E8X8MJQ\MRY,EDX§>§jgU^Mm/È]IE8j/E893MOY,EG=^EGFOFU^MUWFFO7wvX8UWEG@NMR=kÖ=?>A93B/EMRJDnE¾EG>AFO7*9R>An,=^EAÌq>L M , JuOU46l#"h!k#" ##"42«\llm@tJ/7*9E¤heEG93UWEG@NMSEKB/EG@,E89R>A=^=^e,9OE8e[>A9OE;MRY,EIUWJ/@¾UW@MRY*EKEG=^EGX¤MR9OU^X§>Q=BA9OJ/7,@*CtFTMZ>QMOEnbPe*7,¾e*UW@,BDUW@NMRJDMOY,E S1/2 Ë mj = −1/2 =WE8j/E8=ÃÄX¤\)XZY[>Ae*MOEG9#à NÅÀF3EGX8MOUWJ/@ÇÃÄ*kø/ÕAÅOŤ۴EyX¿Y*JgJ/FOE>X8J/@*»[BA7,9R>QMOUWJA@JA\v¾>AB/@,E¤MRUWXK»[EG=^C G Ë[nEG>AC,U^9OE8X8MRU^J/@ A >A@*CeJ/=W>A9TÆUWFO>QMRU^J/@ SVY,E89OE#SE­B/E¤MÙY*UWB/Y,E8F3MKXGJA7,e,=WU^@,B¾EZvX8UWEG@*X8MOJ·MOY,E#XG>¥jgU^MmJ/@`MOY,E S1/2 Ë÷ È ¶8 #ûKý»ºsý û üh7ÿ¦ù¿ÿºqù¹úaúù[ûcû[üM·[ÿ¦ÿ Uý àÿ¦ý Ihû¿¼FýgQ ú û Êýý$ù/ cÿ¦ý ý 8ý üqù ½ º;û Ü û/ü/ D5¼Fþùý§û®ÿ2üû¹ºù¬ ,úûýüÿ¦ý ýúlùZú ü üsùàûL°N/ýÿº/ û»þ ù ùFüFý¦ùFü

Å2i¦wi@%.dq¡.jI(s¢¤£*,m jr&' (+fqde¹dàege dfsd(n.- Nýmj = −1/2 ↔ P5/2

Ë mj = −5/2 MR9R>Q@,FOUkMRU^J/@U EA e ⊥ B ⊥ k VÏI=W=½JQMRY,E89;MR9O>A@,FOUÊÆMOUWJ/@*FºE¤hX8EGe*MvMRY,E S1/2 Ë mj = −1/2↔ P5/2Ë mj = 3/2 MR9R>A@*FOU^MOUWJA@>A93EF37,e,e,93EGFOF3EGCU^@°MOY,UWFX§>AF3E©ÃX8\)1XZY[>Ae*MOEG9PÃÄNŤËv»,B/7,9OE8FPà *µ/Åy>A@,CúÃÄ*ÓNÅOÅ¿ Y*E~E¿*>AX8MB/E8J/E8MR93UWXJA9OUWE8@bMR>QMRU^J/@,FJA\v>A=^=¼J/ehMRUWXG>A=¼nEG>A¾FV>A@*CC,E¤MRE8X8MRU^J/@PC,E8jgUWX8EGF;>A9OE>AFÀ\J/=^=WJ¥SVFZ`X§>¥jgUkM)>É*U^FGËvF3g¾E¤MR9T°>É*U^FJA\ÀMRY,E¾MO9R>AeÇ>A@*C°MRY,E·e*Y,JAMRJA­7,=^MOU^e,=^UkEG9C,E8MOEGX¤MRUWJA@ÇX¿Y,>A@,@,E8==^UWEtU^@ÂMOY,EPe,=?>A@*EPJA\;MRY,E~MZ>Qn,=WEQ Y,EC,E¤MRE8X8MRU^J/@¯XZY[>A@,@,E8=o\J/9­MRY,E ®o® HVfgXG>AEG9O>F37,n*MOEG@,C,FK>Q@>A@,BA=WEJQ\ ≈ 9 SVU^MOYPMOY,Ee,=?>Q@,EJA\1MOY,EMZ>An*=WEQg]IE8B/=WE8X8MOUW@,BMOY,UWFoF3·>A=^=>Q@,B/=WEQË5á >A@*C ®o® H\Ä>AXGEEG>AXZY©JAMRY*EG9G Y,EDXG>¥jgU^MmÖ>É*U^FU^FeE89OeE8@,C,UWX87,=?>Q9ÙMRJ5á >A@,C ®o® H# Y,EÜFThE8MR9T>hU^F·JA\ÙMRY,EMR9R>Qe±U^MOFOE8=^\=^UWE8F·>QMt>A@'>A@,BA=WEJA\

≈ 45 MOJMOY,E¾XG>¥jgUkM)¦>hUWF¾Ã >A@,C©MOY,EG93E8\J/93E·>A=^FOJPMOJ`5á >A@,C ®o® H=^UWB/YNMXZY[>A@gÆ@*EG=]Å¿ÏK=W==WU^B/YbMonE§>Q¾F;E¿hXGE8e*MKMRY*E 397 @, σ fgnE§>A!=WU^EUW@>De,=?>A@*EyF3e[>A@,@*EGC`nbCCD

PMT

397 nm866 nm854 nm

729 nm

397 nm

s397 nm

objective

objective

B-field

cavity mode

ring electrode

tip

tip

symmety axisof trap

normal to planeof table

_1UWB/7,93EÈ,Óûw` JAe°jgUWE¤S¡JA\ÀnE§>AF#>A@,C´MO9R>Ae Y,E·X§>§jhUkMm¦SÀ>AU^F3M#>AFySE8=W=>QF9OUW@*B>Q@,CÇEG@,C*X§>Ae,F­JA\oMOY,E¾MR9O>AeÁ>A93E·FOLAE¤MRXZY,EGCÂFOXZY,E8·>QMOUWXG>A=W=k/ÏI=W=<n¬EG>AF#E¤hX8EGe*M­MRY,E397 @, σ fgnEG>A=^UWE#>Ae,e*9OJ¥hU^·>MREG=kU^@Ö>te,=W>A@,EA Y,E¾>AB/@,E¤MRU^X»[E8=WCÖU^FKX¿Y,JAFOEG@eE89OeE8@,C,UWX87,=?>Q9oMOJDMRY*EX§>§jhUkM)~>hUWF8

ùA %cbqdàe(+*gfåÅ2i¹*DCEeg*gfsj5E *g'N(Odhkk-w*(n+e

PMT

CCD

s

729 nm

397 nm866 nm854 nm

397 nm

397 nm

objective

objective

B-fieldplane of table

_vU^B/7,9OE ,Óýw`ÑKUWE¤SJA\ÙMO9R>Ae>A@,C²nE§>AF¾U^@²C,UW93EGX¤MRUWJA@²JA\KMRY*EX§>§jhUkM)¯>hU^FG Y,E®o® HVfgXG>AEG9O>=^UWB/YNMXZY[>A@,@*EG=¼UWFF3=WU^B/YbMO=kMRUW=kMRE8C·JA«MRY,EKe,=?>A@*EKJA\MOY,EKMR>An,=WE;MRJ­FO>¥j/EUW9O93J/93FÀJ/@PMRY*ESÀ>¥tMOJDMOY,EX§>AEG9O>­SVY,UWXZYUWFoJ/7,@NMREGCPJ/@>Y,UWB/Y*EG9o=WE¤j/EG=

MRY*EFTg¾E8MO93`>hUWFKJA\<MOY,E#MR9O>Ae¦>Q@,CÜMRY,E­@,JA9O¾>A=½MRJ·MRY,E­e,=W>A@,E#JA\MRY,E#MR>An,=WEQÏXGJgJA=WUW@*BPnEG>A >QM 397 @, >Q@,C©MRY*E=WUWBAYbM>QM 866 @, >A@*C 854 @*UWFFOEG@NM\93J/>AnJ¥j/EIMRY,93J/7,B/Y¾MRY*EIMR9O>AeË*>MÀ>A@t>A@,B/=^EIJQ\ ≈ 30 MOJMRY,EKe*=?>A@*EIJA\MRY,E;MZ>An,=^EA_[9OJ/nEG=^JÉSËv>F3EGX8J/@,CÇXGJgJ/=^UW@,BPnE§>Q UWF\JgXG7*FOEGCÇU^@bMOJMRY*E·>Ae,e[>A9O>QMR7*F#>QM#>A@Ç>Q@,B/=WEDJA\≈ 36 MOJDMRY,Ee,=?>Q@,EJA\1MOY,EMR>An,=^EA Y,E 729 @,!nE§>QE8@bMOEG9OFV\9OJAMOY,EyJ/e*eJ/FOUkMREC,U^9OEGX¤MRU^J/@t>A@,C~F37,n*MOEG@,C,FV>A@~>A@,B/=^EKJQ\ ≈ 31 SVUkMRY·MRY*EÙe,=?>Q@,EKJA\MRY*EKMZ>An*=WEQ J­B/E8MXGJA7,e,=WU^@,B·JA\<MRY*Eã

1/2Ë mj = −1/2↔ H 5/2

Ë mj = −5/2 MR9O>A@,FOUkMRU^J/@`UW@*C,EGeE8@,C,E8@bMJA\MOY,E#eJ/=W>A9OU^FR>QMOU^JA@JQ\MRY,E#XG>¥jgUkM)JgC,EQËMOY,E#¾>AB/@,E¤MRU^X»[E8=WC B UWF;X¿Y,JAFOEG@ÜMOJ·nEeEG93eEG@,C*UWXG7*=?>A9­MRJ¦MOY,EPX§>§jhUkM)Á>hUWFD>QM>A@²>A@,BA=WE~JQ\ ≈ 27 MRJ¦MOY,EPe,=?>Q@,EPJQ\IMRY*EMZ>Qn,=WEQ Y,UWFoUWFoE¤*>AX¤MR=ktMRY,EC,U^9OE8X8MOUWJ/@~JA\1MOY,E 397 @* σ fgnEG>A Y*E¾>AB/@*E8MRU^XI»[E8=WC B >QMMRY,EÙeJ/FOUkMRUWJA@tJA\½MOY,EU^J/@tUWFB/EG@*EG9R>MREGCPnNtJ/@*E¾>AUW@XGJAUW=½>Q@,CMmSJ¾FO¾>A=^=WE89oX8J/UW=^FG Y,E»[EG=^CJQ\vMRY*E¾>AU^@PXGJAUW=UWF;>A=^9OE§>QC*P>A=^UWBA@,EGC~SVUkMRYMRY*EC,U^9OE8X8MRU^J/@JA\1MRY,E 397 @, σ fbn¬EG>A Y,EF3·>A=^=^EG9oX8J/UW=^FVFOE893j/EyMRJ¾X8J/eEG@,FO>QMREF3MR>QMRU^X¾>AB/@*E8MRU^X­FTMR9R>§`»[EG=^C,FFO7,XZY°>AF8Ë\JA9E¿,>Q¾e*=WEA˼MRY,EE§>A9TMRY¶ F¾>AB/@,E¤MRU^X#»[E8=WCSVY,U^X¿Y²UWFD@,JQM¾e[>A9O>A=W=^E8=MRJ¦MRY,EP»[E8=WC²BAEG@,E89R>QMOEGCnNÂMOY,E·>QUW@ÂX8J/UW= ²HK7,EMOJ©MRY*EXGJA¾e[>QX8MR@*EGFOFKJA\vMRY,EB/=W>AFOF;XGE8=W= Ë[MOY,EXGJ/U^=WFVXGJ/7,=^C`nE¾JA7,@bMOEGC`@,EG>A9;MRY,EyMR9O>AeÜ>A@,CMRY*EG9OE¤\J/9OE>·XG7,939OE8@bMJA\<=WEGF3FKMRY[>Q@ 100 ÏÒU^FKFO7v¾X8UWE8@bMMOJtB/EG@,E89R>QMOE>t¾>AB/@*E8MRU^X»[E8=WC¾JA\ ≈ 3 Þy>A7*FOFG ® J/@*F3MZ>A@NMX87,9O93EG@bMFOJ/7,93XGE8F\J/9MRY,E8FOEI=WJ¥S±X87,9O93EG@bMOFÀ>Q9OE;EG>AFOU^=^>¥j>AUW=W>An,=^EQ Y*E~·>ABA@,E8MOUWX»[E8=WCÁF3Y,J/7,=^CÁnE~XZY,J/F3EG@ÂUW@ÇFO7*X¿Y¯>`SÀ>§´MRY[>M­MRY*E~¹b7[>Q=WU^Mm©JA\

Å2i­ji©(+fsdà'-Ëmc*gf©mjr&mà*(kdà-w*gf ùhøMOY,E­HKJ/e,e*=WE89KXGJgJA=WUW@*B¾J/@MRY,Eã

1/2↔ 1/2 MR9O>A@,F3U^MRU^J/@U^FI@,JAMÙ>Q«¬EGX8MOEGCÖnNMRY*E)¼EGE¤Æ¾>A@PF3e,=WUkMOMRU^@,B­JA\1MOY,E=WE8jAEG=WFoUW@Nj/J/=^j/E8C<ÏÀMVMOY,EFR>AEMOUWEÙY,JÉSE¤j/EG98ËaMOY,E¼EGEG¾>A@F3e,=WUkMOMOUW@*BJA\oMRY*EPã1/2↔D5/2

=WE¤j/EG=WFFOY,J/7*=WCÇFOe,=^U^MyMRY,E·9OE8=WE8j>A@bM­F3eEGX8MO9R>A==^UW@,E8FFOJMOY[>QM~MOY,E8 >Q9OEÖXG=^E§>A93=^²9OEGF3J/=^j/E8C¡_[J/9·MR9O>Ae'\9OE8¹b7,EG@,X8UWE8F ωx = 2π · 2.9 á`ü;6AËωy = 2π · 3.9 á`ü;6Ü>Q@,C ωz = 2π · 7.4 á`ü;6`>QM ≈ 1 Û I_\E8EGC,U^@,B´eJÉSEG9>¾>AB/@*E8MRU^X²»[E8=WCJA\ ≈ 3 Þy>Q7,FOFǾE8E8MRF°MRY,E8FOE'XG9OUkMRE89OU?>hË·F3e,=WUkMOMOUW@*B MRY,E ã

1/2Ë

mj = −1/2↔D5/2Ë mj = −5/2 X§>Q9O9OU^EG9MR9R>Q@,FOUkMRUWJA@nb ≈ 5 á`ü;6Ü\9OJA MRY*Eã

1/2Ë mj = −1/2↔D5/2

Ë mj = −3/2 XG>A9O93UWE89G

0 50 100 150 200 250 3000

2

4

6

8

10

B (

Gau

ss)

I (mA)

_1UWB/7,93Ep,^ø§w`vá`>ABA@,E8MOUWX»[EG=^C B >QMvMRY,EÀe,=W>AXGEJA\[MRY*EÀUWJ/@j/EG9OF37,F<X87,9O93EG@NM I MOY,9OJ/7*B/Y¾>AU^@ÖX8J/U^= Y*ED»,EG=WCÖY[>AFnE8EG@©¾EG>AFO7*9OEGC¦nbÜJ/n,F3EG93jgU^@,BPMOY,EDF3e,=WUkMOMOUW@*BtnE8MmSE8EG@MOY,EÜX§>A939OUWE89OFtJQ\MRY,E S1/2 Ë mj = −1/2 ↔ D5/2Ë mj = −5/2 >Q@,C±MRY,E S1/2 Ë

mj = −1/2↔ D5/2Ë mj = −3/2 MO9R>A@*FOU^MOUWJ/@*FG

Ço[ svwqPxÎQus 1 +Zr~uPO´wxuÀsw785

qPÍlm@ÁJ/9OC,E89#MRJÖF3MO7,C*ÇF3MR>QMRU^J/@[>A9T´XGJAY,EG93EG@bMDUW@NMRE89R>AX¤MRU^J/@ÇJA\I>A@Â>MRJ/¢>A@,CÂ>ÜX§>¥jgUkM)JgC,EoUkM<U^F@,E8XGE8FOFR>A9TDMRJFTMZ>An,U^=WU^F3EMRY,EVXG>¥jgU^Mm­9OE8FOJ/@[>Q@,XGEV\93EG¹b7,E8@,X8MRJMRY,EV>QMRJ/U^XMO9R>A@*FOU^MOUWJ/@Â\9OEG¹b7,E8@,X¤/lm@±J/9OC*EG9¾MOJÁ>§j/J/U^C²MRY[>QM·9OE8FOJ/@,>A@bM 729 @, =^UWB/YNM¾nE8UW@,Be*9OEGF3EG@NMÙU^@PMRY,EyXG>¥jgUkM)tSVY,UWXZYSJ/7*=WCeEG93MO7,9OnMRY,EyX8J/@bMO9OJ/=^=WE8C>MRJ/­fgXG>¥jgU^MmtU^@bMOEG9TÆ>QX8MRU^J/@ËSE­7,FOE> 785 @*C,U^JgC,E=?>AF3EG9I\JA9IMOY,E­F3MR>An,U^=WU^FR>MRU^JA@FOU^@,XGE#U^MRF;\9OE8¹b7,EG@,X¤

ù/ %cbqdàe(+*gfåÅ2i¹*DCEeg*gfsj5E *g'N(Odhkk-w*(n+eUWF\Ä>Q9JQ«Æ 9OE8FOJ/@[>Q@,XGEP\9OJ/¢MR9R>A@*FOU^MOUWJA@,FD>QM 729 @, Y,U^FC,UWJgC*E=W>AFOE89>A@,C¯MOY,EMR9O>A@,FT\EG9=WJgXZL>Q9OEKE8FOF3EG@bMOU?>A=\JA9E¤heEG93UWEG@NMRFJ/@tX8J/Y,E89OEG@NMoUWJA@gfgX§>§jgU^MmDUW@NMRE89R>AX¤MRUWJA@>A@,C>Q9OEMOY,EG93E8\J/93EyC*EGFOX89OU^nEGCMRY,JA9OJ/7,BAY,=^·UW@~MOY,E\J/=^=WJ¥SVU^@,B­FOE8X8MRU^J/@Ìqbs d³;Ï£FOLAE¤MRXZYJA\#MOY,E©FOE¤MR7,eâJA\#MOY,E 785 @, =W>AFOE89U^FPFOY,J¥SV@U^@ »[B/7,93E±ÃD,kø/øÉŤØÏFOU^@,B/=^E¾JgC*E<=?>AF3EG9½C*UWJgC,E õ) UWF¾JA7,@bMOEGCyUW@y>VY,J/7,F3UW@,B õ 1XGJ/@NMZ>AU^@,U^@,BoMOY,E5X8J/=W=^U^¾>MRUkJ/@J/ngÈ3EGX¤MRUkj/EAËK1E8=^MOUWEG9·E8=WEGEG@NMRF~>A@,C >°MREGeEG9O>QMR7*9OEÜFOEG@*FOJ/98 Y,E¦=W>AF3EG9tC,U^JgC,EÜU^FMRE8¾eE89R>QMO7,9OE#F3MR>An,U^=WU^FOE8CMRJ·SVUkMRY,U^@Ü>·\E¤SÚ¾U^=^=^UkLAEG=kjgUW@¼IÏK@ÜE¤gMRE89O@[>A=vY,JA=WJ/B/9O>Ae,Y,U^X

EOM

PBS

PBS

PBS

fixedreference

cavityl/2

l/4

l/4AOM

PZT

fibre to exp. table

729 nmlock

PD

opt.diode grating

1800/mm

collimator

laser diode785 nm

16 MHz

(270 60) MHz+

PI-controller

16 MHz

_vU^B/7,9OE),kø/ø`#ãhE¤MR7,e´JA\MRY,E 785 @,C,UWJgC*ED=?>AF3EG9G¾d½>AFOE89yC,U^JgC,EA˽XGJ/=^=WU^>QMOJ/9>A@*CB/9O>QMRU^@,B©ÃÄdU^MOMO9OJ¥SÚXGJ/@*»,B/7,9R>MRUWJA@ W$NgË/[Y?ÅÙ>A9OEJ/7,@NMRE8C¦UW@¦>~F3MR>An,=WE>A=W7*¾U^@,Uk7,nJ¥¬ Y,E`nE§>A à ≈ 35 KÛ ÅU^F¾FOe,=^U^MUW@NMRJ°>´=WJgXZLÁnE§>A à ≈ 2 KÛË<C[>AFOY*EGC=WU^@,E¥Å>Q@,C>nE§>QnEGUW@*B\93EG¹b7,EG@*X8PF3Y,U^\]MOEGCPU^@P>C,J/7*n,=WEÙe[>AF3FVXGJA@*»[B/7,9O>QMRU^J/@P>A@,CFOE8@bMIMRJMOY,EE¿*eE89OU^¾E8@bM­ÃÄFOJA=WU^C=WU^@,E¥Å¿ Y,E=^JgX¿LtnE§>Ae[>AF3FOEGFIMOY,9OJ/7*B/YÜ>Q@Ü:5ßáC,93U^j/EG@>QM 16 á`ü;6>A@,CU^FoXGJ/7,e*=WEGCU^@bMOJMRY,EFO>AE9OE¤\EG9OE8@,X8EX§>§jgU^MmtMOY[>QMVUWFo7,FOE8CMRJFTMZ>An,U^=WU^FOEÙMOY,E 729 @,=W>AFOE89G5Ï1J/7*@,CgfhHK93E8j/E89)fhüI>A=W=½E89O9OJA9;F3UWB/@,>A=U^FVJ/n*MR>AUW@,E8C>A@,C°e,9OJgX8EGF3FOEGCÂU^@Ç>`l fgX8J/@bMO9OJ/=^=WE89>AX8MOUW@,BJ/@°=?>QFOEG9yC,UWJgC*E¾XG7,939OE8@bM>A@,C°B/9R>QMOUW@,Be,U^EG6GJ*5d½>AF3EG9V=WU^@,E8SoUWC*MOYa` ∆ν785 = (1.8± 0.1) Lgü;6A÷)",à Ê K Û - Ê Q Û 80 â û¿¼Fýgùþþù¿|ú ¶6 IÞ ¶Ô Ü Þ ù ùý 5 Ü # ûrú° ýú ý ü÷)Ý R­úTS ? ý$ú ùQ<U#, ½ +û ½ [ú

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_1UWB/7,93E;,køÉ`:9O9OJA9FOU^B/@[>A=J/nhMZ>AU^@,EGC·nb¾FOXG>A@,@,U^@,B#MRY,EK=?>AF3EG95jgU?>yMRY,EKB/9R>MRUW@*B#J¥j/EG9MOY,EI93EGF3J/@[>A@,X8EKJA\¼MOY,EI93E8\EG93EG@,X8EKX§>§jgU^Mm/HK7,E;MRJMRY,E;\Ä>QF3MFOXG>A@·9R>QMOE#à >AF5X8J/¾e,>A9OE8CMOJMOY,EX§>¥jgUkM)ÖC,EGXG>¥¦9R>QMOE¥Å¤ËºU^@,XGJA¾U^@,BP=^UWBAYbMUW@bMOEG9T\EG9OE8FSVU^MRY¦MRY,E=^UWB/YNMFTMRJ/93EGC°U^@MOY,EX§>§jgU^Mm/Ë[=^E§>QC,UW@,B­MOJDMOY,EJ/n,F3EG93j/E8C´³Ó9OUW@*B/UW@,B*¶^÷ 0?ý 6 +þý 2s 5[û÷)*àJ Ý Þß ùù[ýt#, ½ #Tu° ý5÷)/? "v¹Þ àJ Ý Þß ù ùý 56#, ½ #Tuý5

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785"à

J·C,E¤MRE89OUW@,EMRY,Ey=WU^@,E8SVU^C*MRYPJA\vMOY,E=W>AFOE89;UkM;UWFVEGF3FOE8@bMRUW>A=½MOJ¾Lb@,JÉSMRY*E=^UW@,E¤SVUWC*MOYJA\MRY,EKX§>§jgU^Mm¾UkMU^F=^JgXZLAEGC·MRJ, Y,EKX§>¥jgUkM)=WU^@,E8SVU^C*MRY¾XG>A@tnEÙXG>A=WX87,=?>MREGC¾\93J/MRY*EX§>§jgU^Mm#»[@*EGFOF3E F >QFS>AF<E§>AF37,9OE8CU^@VWÊø/øZY,MRJnEV93J/7,B/Y,=k 14000 ÃFw λ ≈ 785 @,·ÅnbÙC,E¤MREG93¾U^@,UW@*BÀMOY,E<C,E8X§>§ÙMRUWEJQ\gMRY,EX§>¥jgUkM)A1lm@MRY,U^FMRY,E8FOU^Fº>A@y>AX8XG7,9O>QMRE<E8MOY,JgC\J/9#¾EG>AFO7*9OUW@*BÜMOY,EtX§>¥jgUkM)°=^UW@*E8SVUWChMRYÇ7*FOUW@*BÜMRY,EtEG939OJ/9­F3UWB/@[>Q=UWF­C*EGFOX89OU^nEGC°MR7*@,UW@,BÖMRY,EPE¤gMRE89O@[>A=VB/9O>QMRU^@,B,ËMRY,EP=?>AF3EG9DU^FDFOXG>A@,@,E8CJÉj/E89DMRY,EP9OE8\E89OE8@,XGEXG>¥jgU^Mm9OE8FOJ/@,>A@,XGEK>A@,CMRY,E;EG939OJ/95F3UWB/@[>Q=[UWF593EGX8J/9OC,E8C<DXGJ/e[>A93UW@,BMRY*EIX8EG@bMO9R>A=aeE§>ALJA\MRY*EyE89O93J/9VF3UWB/@[>Q=¼SVU^MOY>@g7,E89OUWXG>A=F3UW­7,=W>QMOU^JA@yW$*Ë"!Y Ë*»[@,EGF3FOE F >Q@,CFOXG>A@,9R>MREνω

W$0hË+"!YÀ>A93E·C,E¤MRE89OUW@,E8C ÃÄX¤\)º»,B/7,9OEÖÃD,kø¥NÅOÅÙMRJ`nE F = 16500 ± 1000 >A@,C

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yz|yy

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_vU^B/7,9OE),^ø§w` ® E8@bMR9O>A=5eEG>AL¦JQ\MRY,EEG9O93J/9yFOU^B/@[>Q= ·_[9OJAMOY,E¾E¿heJ/@,E8@bMRUW>A=<C,EGXG>¥ÃMOY,UWXZLÁFOJA=WU^C¯=WUW@*E¥Å­JA\KMRY*EJ/FOX8UW=^=?>MRUkJ/@,FMOY,E»[@*EGFOF3E F = 16500 ± 1000 X§>A@nEX§>Q=WXG7*=?>QMOEGC¾Ã Y,UWFÙY[>QFnEGE8@©C,JA@,E­\J/9KFOE¤j/EG9O>A=F3X§>A@,FÙMRJ~JAn*MZ>AU^@`MRY,E­B/Ukj/EG@ÜjQ>A=^7,E>A@,CÜU^MOFKEG939OJ/9RŤ Y,E­FOU^7*=?>QMOU^JA@ÇÃMRY,U^@Ü=^UW@,E§ÅV7,F3EGFÙMRY*E>AnJ¥j/E#»[@,EGF3FOED>A@,C¦>·F3X§>A@9R>MREÀJA\ νω = 15.8 Y*EV>Ae,=WUkMO7,C,EY[>AFnE8EG@D@,J/9O¾>A=^U^F3EGC#MOJÙ»,MMOY,EÀE¤heEG93UWEG@NMZ>A=C[>QMR>*;HKE¤jhUW>QMOUWJ/@,F;JA\E¿heEG9OU^¾E8@bMÙ>A@*CÜF3UW­7,=W>QMOU^JA@>A9OEe*9OJ/n[>Qn,=^PC,7*EMRJjQ>A93gU^@,BFOXG>A@j/E8=WJgX8U^MmtX§>Q7,FOE8C`nNtMRY,EB/9O>QMRU^@,Be,U^EG6GJ*

Å2i­Åi@%.dq¡.jI(s¢ -(wd;×jkj-d;( jZ&' ù/ùνω = 15.8 ± 0.7 Y,Et@*J/9O¾>A=^U^FOE8C´FOX§>Q@,9R>QMOE νω U^FC,E¤»[@,E8CÂ>AFMOY,E·F3X§>A@,9O>QMRE·U^@7*@,U^MOF­JA\;Y[>A=k\;JQ\VMRY,EtX§>¥jgUkM)°=^UW@,E¤SVUWChMRY à ∆νcav = c/(2FL) ÅeEG9­X§>¥jgUkM)°E8@,EG93BAFTMRJ/9O>AB/EÙMRUWE τ `

νω :=2FLωτπc

, Ã7,$NÅ

SVY*EG9OE L U^FMRY,EXG>¥jgUkM)·=WE8@,BAMOY<_[93J/ªMOY,EÙ»[@*EGFOF3E F >­XG>¥jgU^Mm·=WU^@,E8SVU^C*MRY ∆νcav =(45± 2.5) Lgü;6yU^FoXG>A=WX87,=?>QMOEGC_[9OJAØMOY,EX8J/e[>A9OU^FOJ/@~JQ\1EG9O93J/9oFOU^B/@[>A=>Ae,=^UkMR7,C*ESVY*EG@PF3X§>A@*@,UW@,B#MOY,EÙ=W>AFOE89FO=^JÉSV=kJÉjAEG9MOY,EÙXG>¥jgU^Mm¾93EGF3J/@[>A@,X8E>A@*C~93¾F>A¾e*=WUkMO7,C,ESVY*EG@#MRY,E=W>AFOE891U^Fº=WJgXZLAEGCSEXG>A@#EGFTMRUW¾>QMOE<MRY,E=W>AFOE89º=WUW@*E8SVUWChMRYynb>Ae*e,9OJ§*U^·>MRU^@,BMOY,EoXGE8@bMR9O>A=,e[>A9TM<JA\aMRY,EoE89O9OJA9FOU^B/@[>A=hnb>ÙF3MR9O>AUWBAYbM=^UW@,EQvi;FO7[>A=^=^yMRY,EV>AX¤MR7[>Q=,EG939OJ/9F3UWB/@,>A=,U^FE8j/EG@¾F3MOEGE8eEG95UW@MRY*EV9R>A@,BAEoSVY,E89OEVMRY*E;=W>AFOE89UWF=^JgX¿LAE8C·>Q@,CDMRY*EG9OE¤\J/9OEoMOY,UWF>Q@[>A=^gF3UWFÙ9OEGF37,=^MOFUW@¦>~XGJ/@*FOEG9TjQ>QMOU^j/EjQ>A=W7*E­\J/9ÙMOY,ED=WU^@,E8SVU^C*MRY¼­_[J/9KMRY,E­MR9R>A@*F3\EG9mf=^JgX¿L¾=?>AF3EG9ÀSEJ/n*MZ>QUW@a` ∆ν785 = (1.8± 0.1) Lgü;6A

ÇoÇ -w/.*+¿ aDºwyR+_+Gx½w+I#q Y*E©=W>AFOE89P>QM 785 @* >A=^=WJ¥SVF·MRY,EÜMR9R>Ae XG>¥jgU^MmMRJÂnE¦9OE8FOJ/@[>Q@bMPSVUkMRY =WU^B/YbM~>QM729 @, SVU^MRY*J/7*M­Y[>§jgUW@,B`MOY,UWF#=^UWB/YNMe,9OE8FOEG@NMDUW@°MOY,EtX§>§jgU^Mm/ Y,Etn[>AFOU^X·FOXZY,EGEJQ\5MOY,E­MR9R>Q@,F3\E89=WJgXZLUWF>QFÙ\J/=W=^J¥SVFI`;SE\9OE8¹b7,EG@,X¤bÆ =WJgXZL`MOY,E 785 @,ÔC,U^JgC,E­=?>AF3EG9MOJ·MOY,E#FO>A¾E7,=kMR9R>ÉÆmF3MR>An,=^E9OE8\E89OE8@,XGE­X§>§jhUkMmÇÃX8\)+W$[Y]Å;>AFIMOY,E 729 @, U`ÓFO>Ae,e,Y*UW9OE=W>AFOE89G·FTMZ>An*UW=WU^F3U^@,ByMRY,EKMR9O>AetX§>§jhUkM)MOJ­MOY,EÙC,U^JgC,EK=?>AF3EG9\9OE8¹b7,EG@*X8/Ë,MRY*EÙ=WE8@,BAMRY*FJQ\V93E8\E89OEG@*XGE~>A@,C´MR9R>QeÇXG>¥jgU^Mm´>A93E¾»h*E8CÁ9OE8=?>QMOU^j/EDMRJ`EG>AXZYÁJQMRY,E89G`_[7,9TMRY,E89OJ/9OEQËMOY,E;=^EG@*BAMRYDJA\¬MRY,EoMR9O>AeX§>§jgU^MmUWF»*hEGC¾9OE8=?>QMOU^j/EÀMOJMRY*EVSÀ>§j/EG=^EG@,BQMRY¾JA\¬MOY,E 729 @, U`ÓFO>Ae,e,Y*UW9OEo=W>AFOE89G­X¿Y*JgJ/FOU^@,By>X8J/@Nj/EG@,U^EG@NM=^J/@,B/UkMR7,C*U^@[>A=hJgC,EoJA\MRY*E 785 @,=W>AFOE89>A@,C¯>Ae,e,=kgUW@,BÖ>A@¯>Ae*e,9OJ/e*9OU?>MRE·\93EG¹b7,E8@,X8ÂF3Y,U^\]M­MOJÖMOY,E~C,UWJgC,Et=W>AFOE89=^UWB/YNM§ËMOY,EÙ=^EG@*BAMRY·JA\MRY*EKMO9R>AetXG>¥jgU^Mm¾U^FÀ>QCÈ37,F3MOEGC~7,@NMRU^=MOY,EKX§>¥jgUkM)UWF9OEGF3J/@[>Q@bMÀSVUkMRY¾MRY,E729 @,¢=^UWB/YNM§ >ALbUW@,BÜUW@NMRJÜ>AXGX8J/7,@NM>MRY,E89O¾>A=E¿*e,>A@,FOU^J/@ÁXGJgEZvX8UWE8@bM\JA9MRY,EMO9R>Ae¦XG>¥jgU^MmܾJA7,@bMÃÄF3MOEGE8= ˽e*UWEG68JPXGEG9O>AUWX8FZÅKJA\ 10−5/ ÐË>tMRE8¾eE89R>QMO7,9OEXZY[>A@,B/EJQ\=WE8FOF<MRY[>Q@ 2 Ð>A=W93E§>ACh9OEGF37,=^MOF5UW@¾>y9OE8FOJA@[>A@,X8EKF3Y,U^\]M5JA\¼J/@,EV\9OE8EIF3eEGX8MO9R>A=a9R>A@,BAEà ≈ 7 ÞüI6§ÅÀJA\1MRY,EMO9R>AeX§>§jgU^Mm/lm@MOY,UWFoX§>AF3EAˬ>DC,Uk«¼E89OE8@bM;=WJ/@*B/U^MO7,C,U^@,>A=¼JgC,E\J/9MOY,EIC,U^JgC,E;=?>QFOE895Y[>AF5MOJn¬EV\J/7,@,CMRJF3MZ>Qn,UW=^U^FOE;>AB/>AUW@7*FOUW@*BMOY,UWF<C,J/7,n,=^E;9OE8FOJ/@[>Q@,XGEF3X¿Y*EGEAy];E8j/E893MRY*EG=WE8FOFU^MMO7,9O@,E8CÁJ/7*M#MOY[>QM>`XG>¥jgUkM)´C*9OU^\]MJA\VJ/@*E·\9OE8EtF3eEGX¤MR9R>Q=9O>A@,B/EC,JgE8Ft@,JQMtJgX8XG7,9¾j/E893²JA\]MRE8@à >QnJ/7*Mt©MRUWEGFU^@âø¥´E¤heE89OUWEG@NMZ>A=;937,@,FZÅ=^E§>§jgUW@,BD>D=^J/@,BEG@,JA7,B/YPMRUWEKMRJDeEG9T\J/9OE¤heEG93UWEG@NMRFG

ù %cbqdàe(+*gfåÅ2i¹*DCEeg*gfsj5E *g'N(Odhkk-w*(n+eÌq°Ìq d³;ÏFOLAE¤MRXZYJA\;MOY,E~MO9R>AeÂXG>¥jgU^MmÇFOE¤MR7,e¯UWF­B/U^j/E8@¯U^@Á»,B/7,9OE©Ã7,køKbÅ¿ Y,EFTMZ>An*UW=WU^F3EGC785 @,£=WU^B/YNM=^E§>§jhU^@,BMRY,E¾»[n*9OE·UWFe,Y,>AFOEtJgC,7*=?>QMOEGC´nb°>A@Ç:ßá ® ( C,93U^j/E8@Á>QM16 á`üI6¾>A@,C´UWFX8J/7,e,=^EGC´UW@NMRJPMOY,EDMR9R>Qe´X§>§jgU^MmÖjgU?>>Pe¬JA=?>A93UWF3U^@,BPnEG>AFOe,=^U^M3MREG98 Y,E 729 @, =WU^B/YbMDU^FXGJA7,e,=^EGCU^@bMOJ¦MRY,EXG>¥jgU^MmÁ\93J/àMOY,EJ/e*eJ/FOUkMREF3UWC,EQ Y,E9OE¤¸[EGX¤MRE8C 785 @*à=^UWB/YNM­UWF­C,U^FOeEG93FOEGC¯nbÁ>ÖB/9R>MRUW@*B¯Ã 1200/ tÅMRJÖFOEGe,>A9R>QMOEPU^M\9OJAÚMRY,E 729 @,MR9O>A@,F3¾U^FOFOU^J/@­=WU^B/YNMnE8UW@,BJ/@,U^MOJ/9OE8C#SVU^MOY> ®o® HúXG>AEG9R>>A@*C>Ùe*Y,JAMRJgC*UWJgC,EQ Y*EÀ»[9OFTM<JA9OC,E89B/9R>MRUW@*BK9OE¤¸[EGX¤MRUWJA@>M 785 @,U^F1\JgXG7,F3EGCDJ/@D>I\Ä>AF3Me,Y,JQMRJgC,U^JgC,E¯® õ MRJtJAn*MZ>AU^@Ü>t1J/7,@*CgfhHK9OE¤j/EG9mfhüK>Q=W=vE89O9OJA9KFOU^B/@[>A=1>Q\]MOEG9KUkhU^@,BSVUkMRYMRY*Et¾JgC,7*=?>QMOU^J/@©\9OE8¹b7,EG@,X¤/ Y,EtEG939OJ/9#FOU^B/@[>Q=U^F#e,9OJgX8EGF3FOEGC¯U^@Á>Ül ÆmXGJA@bMR93J/=W=^EG9

EOM

PBS

trap cavity

l/4

fibre 785 nm

729 nm

PD785 nm

grating

CCD/PD 785 nmtransmission

l/4

CCD/PD 729 nmtransmission

grating

PBS

16 MHz

PZT PZT

PI-controller

16 MHz

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0 1 2 3 4 5

Time ( ms )

397

866

854

729

397 s

cavity

counter

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#

τ = 5 µ

τ = 7.5 µ

τ = 12.5 µ

τ = 15 µ

τ = 17.5 µ

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/h4; -#@4+# 8!MN"o" ¡ 4#V%&!"['# 9W #j#e!9./0!4 #e( !?Æ'8+! AA"4;*'8F6 !4 #L ^"!4N#''A#' 4;,9 L8/F?+!4#';1\i#10# -?+ ! AA" G 8% P !#w# 8" O 4"*l %"*dA"l\;A"WL(E4# 8A!"u(!`%4n'#w!4(C 8@ !4;# 8(=^9 8#^;A"W4`9&-8W/0!^#w*# %'-*d9[1l# -%#FMN8- q!"q!(_ 8q%A P !#~ ?D%#^!46'#@#' O * \'!l?D 84" ;_( '!4"g"g'8\ A*\!(N'8\?+A !4 # %&6 8%#(!4;;A!M_-*o9.17~ ?D% ¡k' "(! " -;A1gAi#i"!4 O "!MN"3MN86"$!@MN8 83%A P !9&68/6#j -4 ;] ;1u 6%j(!F#%, 4; # -#eMN86 D'8+49&!/4 !4 !"d84#9&6-":4%%;A6*c9.1c4## ?D"iV;A"W4H9&68/0A!K!4(J'8`!4'8-K%A P ! n'86 ?D! ^#w' 8-^* ;w'!R 6 ! ".F(!4w'8q6 !#F* i'!< 8qA"*/A* ;'86 ?+4;* A(= #`!4(H9&!'8l?D !L?D! ". #-,'8@*A(=`!4(C 8 W/0Ar1u?D! "[^4"*x 8@* (=w!4( 8 /0Ar1u 6;]h'A/V !\'8j''% eG O A":4;A;'86# q4 ?+6".'#wA". !: - ! ".e"

; Â`ÃÄ! E>:$Ä+"CÄ! $#uÄ!%qÃ&(')%*#+')%Â,?<-#-Ã$=¨Â)'.X/#0%P- !U(!U'8 ! '6*f/!4;Aw/h4; 6#N!4( ±2 1 4"R9&~ #' -*

G 8N A''A!"@% !949;Ar1F/46#X#%'4;;~1FMNA'8@ 8QA".'6"# 21V!4(´'8Q#"*75"eMK/ |W '86! k' 4;uK;! 80v06. 'A!"+4"4;10# #H4#H*-# A9&-*\% 6/0A! #;A1q%+5* '#+)"W4;A17%

cos2# % ]4;U/4]h'!4"$4#D# 8!MN"5"3 8R'!%$*]'4?Ì!4(

¡lG 8<#!;A*g;A"+#V'8< %& '-*3 %&-?D-"[4;_ Ah'A!"p%!959A;r1)4# # ?D"lxrMK!v0;k/-;K#10# -?Å A'6*39[1ox# A";Ac#"*"xMUW/ 6;*?D!0* E ∝ sin(k · x) ¡Ym K84#_9&6-" 4; ;]h'-*D9.1DerMK!v0;k/-;fK;! 80v06. 'A!"4";A10# A#_4# # ?D"

νL = 0.6 YG 8S '6"# A!"\!4(b'8SMK/S% O 6KMU#K#?@ ; -*9.1c4%%;10"@j;A*A"j/6'43N!/-K'8w%&!# 'A!" G 8w"0 ?@9&6U!4(B%&!A". #K # 6*(!e 8#V/6'4DMU4# 8!#-"o# 8p 8V 8c?D"A?j ?Æ Ah'!4"p 6-#VMN'8'8N %&- A?+6".'4;* G 8N?+ ?@ ? T 49( 60 -" 1 Ω0

!"i 8U' 4"#A'A!"@A#*k'-?+A"-*R( !4? S'!D 8V %&-?D-"[4;I/; e!4('8V?+ ?@ ?ë A''A!"Q 8F"!0*F!4('8F#'4"*A"iMU/4 _1\"!?c4;AA#A"j'8F A''A!"<% !949A;r14?D%;~' *'8L#?@ ;]h'!"i#X/-n1D !49 #K[4"#X/4 'A!"#CA"+4;A;%4'4?D6 -# G 8^- !U!4(B'8^ %&-?D-"[4;b*'@#UA"* -*c(!U 6% 6# 6".h'/F%&!"['# ( ¡ ¡YG ! "*+'8L*6/0'A!"D!4(b'8S?D! !4?+%;A ?D!0*-;( !?¥'8L# A?5%;A

cos2v0?D!0*-;J'8V# A?@ ;]h'-*:*'D# '6*fMNA 8u

cos2v.( " 'A!"J' O "q'8

"*A/0* 4;0- !#X!4(´ 8Q*'w%&!"['#YA". !F - ! ". YG 8Q 6# ;'"cos2

v[( " !"#A"* -*@A" 4 !% ¡ 4#Yw*# 86*q;A" G 8N9# !;A 'K*92,- 6" U!4('8?D!0*-;A# *4# 86*R;A" ¡ #UMK-;A;J4#U'8^#n #n' 4;,- !U!4(B'8F 4;J %&- A?D-". #!;A*R;A" ¡ A#L# 8!MN"xA" 9&!4'!? ¡k V#-V 8L(!L!4 S %&- A?D-".'8c#n # 4;X- !e#Vj4"[1d%&!A".Vj;AW#j4;?D!#nF'8 -D'A?+6#e;4 6^'84"'8+*6/0]h'A!")!(U 8DrMK!u?D!0*6;# cG 8-6(!+'8 "R!4(K'8+ %&- A?D-".'4;*'i !q 8F# A?+%;A

cos2v0?D!0*-;b#-~ # 6* !?Æ 8

cos2v e'!R'8+*hR%&!"['#F'8 !"[''#j'h'! /0#9A;Ar1 V ¡ !4(

'8L%&!4# A !"R5r*6%&-"*6".C !"i#Y( "*q !e9&V = (96.3±2.6)% YG 8#X/61

8A8x/0#9A;Ar1R 6# ;'#w(!? 8i#n' !4":;A! 4;AA#'h'A!4"u!4(C'8iA!"|È#w# %'4;MU/J5( " 'A!" RG 8c;]4#-2v !0!;A-*)!"b!4# A;; A":MN'8gA #V# ;4e( -. 6" A-#i4"*MNA 8g'86 ?+4;;1>*#n'9 -*o4?D%;A' *-#-Y84#i4"p ?D#j# % ]4;H -"#!"74;!4"'8 W/0Ar1l A#^!4(

wc@G 8A#F '-"# !4");AW*#w'!R\ -* !">!4(H 8q A''A!"

!4".' 4#9[1eN(E '!B!(exp(−2(2πwc/λ)

2) !4?+% -*F'!N'8H A''A!" !". '4#n %&6 6" 6*p9[1>:%&!"[nv0;A O i'!?ÆF 6#; W /h4; q!4(wc = 16+5−7

"?ÆA#*k'-?+A"-*D( !4?¢'8S?+-4# -*+/0#9A;Ar1

V YG 8`#?c;;/h4; S!(J 8S -"#!"wc

# 8!MN#U'8U"+'8A#U %&- A?+6".K'8w!4"\# !0!;6* ;A!# ` !j'8^a`!%%;6U;A?D~ 4Ë "? ¡W " -##'41 !"*A'A!"¤(!R4;A;^ %&- A?D-". #f-;1A"7!" !4"0v /0Ar15?D!0* !4 %;A"F#X'8S49A;r1j !V%; Q'8L!"+hK -'4"D%&!# 'A!"D!(b'8L"[' W/r1b ´o¬²ª¶y¬Áª³¸ IJª· º+ ¯²±yª+ª ¼ ³¸*¯Á¼ ¯Á¸*´o¹¶y¬²´ ½ ¼ ½ ¸ · ¸÷¯ º ´o¬)¯²±-ª+¶¡´¸ ¹o¯G]· ¸ª¶-¼¬ ¯Á¸Å³· ª-)´µy·Å¿á±-¼ªÂ¯Á´ ½ ª³´o¹O´o·*µÀ¯Áª¿B¸*¯²±¯²±-ª¦Ä²¶-¼ ¯Á¸ ¼·¶y¬Á´ ½ ¼ ½ ¸ ·*¸*¯ º ¿À¸ÅÄ ¯²¬Á¸ ½ µy¯²¸ ´¹´ ¯²±-ªh¯²¬Ã¼f¶-¶¡ª¿+¸ ´¹ Ç7~±yªNºo·*¸Å¿À¸ ¹yºÂ¼dª¬Á¼ºªµ-IJªÄ¼á¬Áª³¯Á¼¹yºµy·Å¼f¬¿À¸ Ä ¯²¬Á¸ ½ µÀ¯²¸ ´¹ B±-¸ ³Ã±V¸ÅÄ+Ä µb³¸*ª¹o¯ ´¬Â¯Á±y¸ Ä|¼f¹-¼· º IJ¸ Ä ¼Ä¼f¹ ¼¶y¶y¬Á´ ¼ ¸ «½¼ ¯²¸ ´o¹¯²´m¯²±yª|«m´o¬²ª|³´«m¶y· ¸Å³¼ ¯Áª¿á¯²±-ª¬Á«m¼·>)¼ª¶¼³Eª¯Ç

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0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00

0.01

0.02

0.030.0 0.5 1.0 1.5 2.0

Relative position (units of λ

Err

orE

xcit.

pro

b.

6@G !4% 6 '8+# !;A*);A"D#F'8+% 6* '!4"g!(L¡_;! 80v060 'A!"g4"4;¢510##@4# # ?D"xRrMK!hv0;A6/6;_#10# -? "pf#'4"*A"fMU/ -;A* fG 8\?+ ?@ ? A''A!"DMK4# 8!4# -"c'! !A" *NMN'8D'8S;k/-;&?DW# -*\A"D'8S %&- A?+6". é !C94#_4 N"* '6*q(!X'8S 4; %&-?D-"[4;*' G 8S*# 86*c;"Q#cos2

v L'!D'8V?+!0*-;I O A"+6 ! #SA". !c 6 ! ". K!4 !? 6 '8V*4# 8-*x;A"#8!MN#Y 8Q49# !4; U*92,- 6" U!4(´% 6* '!4"@9.1n_;! 80v060 'A!"@# A?@ ;]h'!"j4"*#?D%;

cos2v0?+!0*-; G 8#K6 !K?@ #nU9& !?+%4 6*c'!V'8` %&6 A?+6".'4;,#n #05

4;&- !_(!H'8w 4;, %&-?D-"[ # !;A*c;A" ¡k£ !4'S'8`*92,--"[Q# ;-#U!4" 8F! *"h' !Q*k4A;#U# 6F'

#n4"*"iMK/ -;A*<MNA'8:88<% A# !4"R4"*:88< 6% !0* 9;A21¡Í8 Xm "R! %&- A?+6".-0'8`% # A!"+!4(I%&!#A !"A"F'8 -". -K!4(b'8S!"¤|#CMUW/6( " 'A!"J # "wS?DW# -?D-"[# 8q#'8" 4 ¡ #;A?D~'6*V9[1F'8_ " -'4A".r1A"F'8H?DW4# 6*j A''A!"e% !99;Ar1 BG 8A# " -'4A".r1F#B* H'!SL#n # 4;6 !- 8 "'S". ?@9&-_!4(I#h'`*6' !"c?+-4# -?D-"['#-4"*c#10# -?+' 6 ! #L# 8l#$¥ ' h'!4"#L!4(;]# 6LA". -"#Ar1:4"*:MU/6;6"4'8b&* A(=L4"*¦~A'-!(X# 4"x% P !ck !?ë 8j " 6"['6#S"f A''A!"f%!949;~21 # 66 !_94 #_" ¡¡ 8`*6* 6*:#%'4;´% A# A!"c# "? ≈ λ/100 ¡ 8F%&!# '!4"<!4(;4 6#N#;!%& 4 "? Q'8F%&!A".Q!4(?D"A?j ?£ !":4"*Ë <"?Éh^'8q%&!4".^!4(_?+ ?@ ? A''A!" ê !MK6/6-B'8q% #!" 4"d9&6"84" -*@9.1V/4-'"S!/-S;4 4-B"0 ?@9&6!4(#n K*6' !"j?DW# 6?D-". #

Â`ÃÄ! E>:$Ä+"CÄ! $#uÄ!%qÃ&(')%*#+')%Â,?<-#-Ã$=¨Â)'.X/#0%P$§!§G¨ ©«ª!¬­0®@¯°ª²±³ª´7®0ªX?µ!­cÊ´7µ´7¶Bz 4"[1i# 8-?D-#C(!Y0 4". ?¢"(!?ch'!"@% ! -# #"FMN'8i''%%&-*D!4"#X 6;A1@!" !48--".QA".'6' 'A!"+"!4U!";A1qMN'8c 8wA". -"4;&-;A ' !4" #''`9 Q4;A# !VMN'8'8w '6 "4;b?+!'!4"4;´*6 6-#Q!4(I( 6-*!? YG 8 !". !;A;6* ! %;A"j!( !?D #''6#j* -## 6*o9[1>/09 'A!"4;?+!0*6#F'!R'8 W/0Ar1 -;A*)#VR% --. A# A D(!4 -4;A# A"@# 8f# 8-?D-# K!4?+% "@'8 4 A- ∆n = 0

"! 84"4F!4(/A9'h'!4"4;,. 4"[' ?ë". ?59&- ¡ 4"* #nv0!4 *6K/A9'h'!4"4;´# *-94"* ∆n = −1 ¡ ' 4"#A'A!"c%!949;~'-#"5g#n4"*")MUW/4L9&!4 8¤#8!MËg%&!4# A !"$*6%&-"*-". ! %;A")* u !g'8#'4"*A"qMUW/V% - "f4#L#8!MN"f":# 'A!" 44¡k ê !M_6/6-´MN86 e 8 !4 75%;A"D'!c'8 4 6`#S#'!"6#-I"! ! %;A"+ !\ 8 #nv.! *6 4;A;!0** ¡# A*-94"*#j! - #i"*g/0 +/-#'Y 4 6j4"*p!0**p# A*-9"*#j !"#?+4%+ 8`#n4"*"VMK/`# %h']; -;A*c/h4 'A!"J[9 _'8L'' -#U4S# 8(='-*c9[1c%8# ^(E '!U!4(

π XG 8A#Q%8# F# 8A(=N4#-#Q* ^'!i#10?D?+k'n1 84 - A# #U!4('8N' 4"#A'A!"i?ch' -;A-?D-"['#C!4( 4 6C4"*D# *-94"*D' 4"# 'A!"#Y"DF#"*75"uMK/ -;A*7#@*A# # #-*7" 8%'6 ¡ "*· 4ÀË 4 6 8\#%'4;H%4n@!4('8e. 4* %&!;^ '4"#A !"<?ch' -;A-?D-"[LA#N% !4%&! !";b ! 〈n′| exp(ikx)|n〉(!HF' /6;;A"^MUW/w"* 〈n′| cos(kx)|n〉 (!He#'4"*A"eMK/LMNA 8D 8S6; ' 6;*M¸ E ∝ sin(kx) ê 6

n4"*

n′4 j 8@/09 'A!";. ". ?Ë"0 ?@9&6 #^"

'8 1/2

"*>a5/2

;A6/6;-#%& A/6;A14 k A#S'8jMUW/-". ?@9&-^4"*x

#S 8@A!"|È#%&!#A'A!"f"f 8 -;A* !4`+ '/6;;A"DMK/4J4;A;B/09' !"4;B#n -#

(n, n′) 4"9& !4 %;A-*c#exp(ikx) !4"."#Ck/-"\4"*+!0**D%&!MK6 #r¹U!4(

kx Ym " !". '4#nW.(!4@#'4"*A"@MUW/ 〈n′| cos(kx)|n〉 8#U'!i9&w %4"*-*R". !@6/6"»º¼j!0**<%&!MK6 #!4(kx

*-%&-"*"+!4"R 8eA!"|È#N%&!#A'A!"J x = 0 ;!#F !+q"!0*e!4x = λ/4 ;A!# u'!74"t4". "!0* ZG 80 #6S''"# '!4"# 84"A"p'8l%8!"!4"t"0 ?@9&6<9[1

6/6"x!N!0**f"['6-# 4"f!";1<9&V?+ ?+4;;1c '6*xS*92,- 6".L%,!4# '!4"#N"'8q#n4"*"\MU/ iG 8 #2v0!*-^# A*-94"*> '4"# A !" ∆n = ±1 ¡` ! %;A-#?+ ?+4;;1\hS4"['"!*-#N!4( 8e#n4"*"iMU/44MN8-W4#L 8 4 6Q '4"#A !" ∆n = 0 ¡K ! %;6#L?+ A?c;A;~1\hL"!0*-#

m " ! *6l !t*-?D!"#' 'p'8A#l?D!4'A!";v.?D!0*p*6%&-"*-". !4 %;A"eMK !*-*f !":%!949;A'-#Q!( 8eA!"RS -* /021<# 4"f'h' νL =−0.23 ¡ 4(!*92,--".Y%&!4# A !"#MNA 8"V'8U#'4"*A"`MUW/4."*@MN'8j 8U;4# 6Y j"?£"!Mt' "6*<'!@-'86K'8 4 6K!_'8w 6*R 4;,#*694"* ∆n = −1 ;]# -B*k' "6*@9.1 −ωz ¡ !('8

S1/2mj = −1/2↔D5/2

mj = −5/2

' 4"#A'A!" m "@9&!4 8 # -#6'8_". -"#Ar1e!( 8K;]# -MK4#Y4*~ #n'6*i# 8@ 8'8_ !"#!4( 4 A-V4"*g# A*-9"*)MK- !?D%4 49;AD4"*)M_- O -%VMK-;A;H9&6;!M #' '½5'A!" Dm "d 8#F %&6 ?D6".eM_+*k'6 ?D"i'8D"['-4'4;Y Ah'!4"JB 8+4WÀ ´o¬¼m¾]µ-¼¿À¬Áµy¶¡´· ªÂ¯Á¬Á¼¹-Ä ¸*¯²¸ ´o¹¾r¿ ´f¯Áª­

kx = η(av + a†v)¼ ¯¼¹y´À¿Àª½¼f¹-¿

kx = η(av + a†v) + π/4¼ ¯¼¹P¼f¹]¯²¸ ¹y´À¿Àªm´ ¯²±-ª· ª· ¿ Ç

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mj = −1/2 ↔ D5/2

mj = −5/2

' 4"#A'A!"R4#U( " !"^!4(0'8C%A P !N!82b#6I/!;A'4 J/4! #J%&!# 'A!"#JA"^ 8CA". ' /0Ar1#n4"*"`MUW/ -;A* G 8U# !;A*j;A"-#-%-# 6". '#Y!4(&#" 2 ( " 'A!"# !w 8U*%&!4". #

!( 8^%' ;4_ !"\# %& ?R#" w'8^# %& ' q# 8!M "R4#n1?D?D6 ;A"w#84%& ( 9b W¡ ¡kG 8K8480v !". '4#n!4'8!4!"; ! %;A"S!4( 4 64"*j# *-94"*j' 4"# 'A!"#

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S1/2mj = −1/2 ↔ D5/2

mj = −5/2 4 6j''"# '!4"g( -. 6" 13#

*6# 9&6*uA":# 'A!" ÷ ÷ ¡$§§ ©«ª@¶M¶B´';ª?©«µ )®E´¡ ©«ª!¬­0®@¯ K '_'8K!"V%; 6* ;A!#_'!wS"!0*_MN'8j 6# !4"4".Y;4# 6% ;A# -#!4(*2b6 6".* 'A!"f"½~ '-*u"['!D 8e''% W/21 K84"4"D'8V% ;A# V;6"4 8f( !4? P -!'!:\* ''A!"x!4(

3 µ#`'8qW/- 4i A''A!"x% !4949A;r1RA#w*6 -?+A"-*x( !4?

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J 212 6DC1=3? > J

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M_e #F'8F6. 4;'A-#UA/6"RA"« 4 ͽ 6

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σ :=

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4

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' 4"#A'A!"<A" 40 U + MK "* 6

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√

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γ‖ = 2γ⊥

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MK "*:'8F 6# ;L4A/6"R":60 'A!" 4¡+6

|〈S1/2,−1/2|(r · e)(r · k)|D5/2,−5/2〉| =1

e

√

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Coupling a Single Atomic Quantum Bit to a High Finesse Optical Cavity

A. B. Mundt, A. Kreuter, C. Becher,* D. Leibfried,† J. Eschner, F. Schmidt-Kaler, and R. Blatt

Institut fur Experimentalphysik, Universitat Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria(Received 5 February 2002; published 13 August 2002)

The quadrupole S1=2–D5=2 optical transition of a single trapped Ca ion, well suited for encoding aquantum bit of information, is coherently coupled to the standing wave field of a high finesse cavity. Thecoupling is verified by observing the ion’s response to both spatial and temporal variations of theintracavity field. We also achieve deterministic coupling of the cavity mode to the ion’s vibrational stateby selectively exciting vibrational state-changing transitions and by controlling the position of the ion inthe standing wave field with nanometer precision.

DOI: 10.1103/PhysRevLett.89.103001 PACS numbers: 32.80.Pj, 03.67.–a, 42.50.Ct

The coherent coupling of a single atom or ion to onemode of the electromagnetic field inside a high finesseoptical resonator is of major interest for the implementa-tion of quantum information processing schemes: Singleatoms and ions are well suited for storing quantum infor-mation in long-lived internal states, e.g., by encoding aquantum bit (qubit) of information within the coherentsuperposition of the S1=2 ground state and the metastableD5=2 excited state of Ca [1]. On the other hand, fast andreliable transport of quantum information over long dis-tances is most easily achieved by using photons as qubitcarriers. The interface between static and moving qubits isrepresented by the controlled interaction of a single atomand a single cavity mode, being the basic building block fordistributed quantum networks [2]. Deterministic ion-cavitycoupling was demonstrated recently by using a singletrapped ion as a nanoscopic probe of an optical field [3].A second application of atom-cavity coupling within thefield of quantum information processing is the realizationof a deterministic source of single photons [4,5] or sequen-ces of entangled single-photon wave packets [6]. Moregenerally, trapped ions that are cooled to their lowestvibrational state [7,8] and interact with a quantized cavityfield might allow for entangling three quantum subsystems[9,10], i.e., internal electronic states, quantum vibrationalmode, and single-mode cavity field. Another application ofcoupling a trapped ion to a cavity mode is utilizing thecavity internal standing wave (SW) field [11] or the cavity-modified spontaneous emission [12] and coherent scatter-ing [13] for cooling the ion’s vibrational state well belowthe Doppler limit.

In this paper we demonstrate coherent coupling of thequadrupole S1=2–D5=2 qubit transition of a single Ca ionto a mode of a high finesse optical cavity. The ion istrapped and placed with high precision at an arbitraryposition in the SW field of the cavity for several hours ofinteraction time. We also achieve deterministic coupling ofthe cavity mode to the ion’s vibrational state by selectivelyexciting state-changing transitions with the cavity lighttuned to a vibrational sideband of the S1=2–D5=2 resonance.These demonstrations are important steps towards realiza-tion of the experiments discussed above.

The Ca ion is stored in a spherical Paul trap [14] placedin the center of a near-confocal resonator. The ion is lasercooled to the Lamb-Dicke regime, confining its spatialwave packet to a region much smaller than the opticalwavelength. We detect the coupling of ion and fundamen-tal TEM00 cavity mode by injecting an external light fieldat 729 nm into the cavity and recording the excitation onthe S1=2–D5=2 transition. The excitation probability ismonitored via the electron shelving technique [8,15], i.e.,by probing the fluorescence on the S1=2–P1=2 dipole tran-sition (see Fig. 1).

The experimental setup is schematically shown in Fig. 1.The three-dimensional rf-Paul trap consists of an ellipticalring electrode with average diameter of 1.4 mm and two

5/2 D

3/2 P

1/2 P

3/2 D

1/2 S

729 nm

397 nm

866 nm

854 nm

to CCD

toPMT

PZT 1PZT 2

FIG. 1. Schematic experimental setup (left) and Ca levelscheme (right). PZT1 denotes the offset piezo, and PZT2 thescan piezo (see text). A photomultiplier tube (PMT) is used torecord fluorescence on the S1=2–P1=2 transition, and the CCDcamera monitors the ion’s position. The following stabilizedlaser sources are used in the experiment: two cavity-lockeddiode lasers at 866 and 854 nm with linewidths of 10 kHzand two Ti:Sa lasers at 729 nm ( 1 kHz linewidth) and 794 nm( < 300 kHz linewidth), where the 794 nm laser is resonantlyfrequency doubled to obtain 397 nm. The whole laser system isdescribed in more detail elsewhere [1].The dotted arrows indi-cate directions of laser beams: 729 nm excitation laser along thedirection of the cavity axis and 397 nm cooling laser, 854 and866 nm auxiliary lasers at a certain angle to the trap axis (shownonly schematically).

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end caps with a spacing of 1.2 mm (material: 0.2 mmmolybdenum wire). The secular frequencies !x; !y; !zare 2 2:9; 3:9; 7:4 MHz at a rf drive field power of1 W. Here z denotes the direction of the trap axis, which is45 to the cavity axis. The x and y radial directions bothinclude an angle of 45 with the plane spanned by thecavity and trap axes. A magnetic field of 3 G perpendicularto the cavity axis provides a quantization axis and a fre-quency splitting of Zeeman components of the S1=2–D5=2

transition. Calcium ions are loaded into the trap from athermal atom beam by a two-step photoionization processusing diode lasers near 423 and 390 nm [16]. The trap isplaced in the center of a near-confocal resonator withfinesse F 35 000 at 729 nm, waist radius !0 54 m, mirror separation L 21 mm, and radius of cur-vature RM 25 mm. Cylindrical piezoceramics (PZT) al-low fine-tuning of the cavity length across approximately1.5 free spectral ranges.

The coherent coupling of the ion to the cavity field ismeasured in three steps as follows.

(i) Preparation: First we use Doppler cooling on theS1=2–P1=2 transition at 397 nm (see Fig. 1) to cool the ioninto the Lamb-Dicke regime. A repumper laser at 866 nminhibits optical pumping into the D3=2 level. From coherentdynamics (Rabi oscillations) on the carrier and first mo-tional sidebands [8] we determine typical mean vibrationalquantum numbers after Doppler cooling nnx; nny; nnz 20 5; 4 1; 6 1. From these mean phonon numbersand the secular frequencies given above we calculate anrms extension of the ion’s motional wave packet of 255 nm in the direction of the cavity axis, much smaller thanthe wavelength of 729 nm. After cooling, the ion is pre-pared in the S1=2 (m 1=2) substate by optical pumpingwith radiation at 397 nm.

(ii) Interaction: The laser at 729 nm is set to a fixeddetuning from the S1=2–D5=2 (m 1=2 to m0 5=2) qubit transition. We inject the laser light into theTEM00 mode of the cavity and scan the cavity with avoltage ramp applied to one of the PZTs (scan PZT).When the cavity reaches resonance with the laser fre-quency, it fills with light and the ion is excited. A constantvoltage is applied to the other PZT (offset PZT) thatdetermines the ion’s position relative to the SW field.

(iii) State analysis: The final step is state detectionby electron shelving. Fluorescence on the S1=2–P1=2 di-pole transition at 397 nm is used to discriminate withhigh efficiency ( > 99%) between excited state (elec-tron shelved in D5=2, no fluorescence) and ground state(fluorescence).

In order to obtain an excitation spectrum, the 729 nmlaser is tuned over the quadrupole transition in steps ofabout 1 kHz. For any given laser detuning the sequence(i)–(iii) is repeated 100 times to determine the excitationprobability.

In our first experiment we probe the ion’s response totemporal variations of the intracavity field by placing it

close to a node of the SW field [17] and varying the cavityscan rate. The sign of the voltage ramp applied to the scanPZT determines whether the scan mirror moves towardsthe offset mirror or away from it. For a negative (positive)scan rate, i.e., mirrors moving towards each other (apart),the intracavity field is Doppler blueshifted (redshifted), andthus the excitation spectrum will be redshifted (blue-shifted), as the excitation laser detuning has to compensatefor the Doppler shift. The scan velocity is expressed inunits of a normalized scan rate L [18,19], correspondingto the frequency shift in units of HWHM cavity linewidthsper cavity storage time: L 2F! _LLs=c, with laserfrequency !, cavity length variation _LL, and cavity (en-ergy) storage time s FL=c. The experimental resultsfor scan rates 0:16 > L > 0:69 are shown in Fig. 2. Theexcitation spectra show the expected blueshift (redshift) forincreasing positive (negative) scan rates and a broadening

0.1

0.2

0.3

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0.6

0.7

Excitation laser detuning ∆ (MHz)

Excita

tio

n p

rob

ab

ility

(a)

(b)

(c)

FIG. 2. Excitation spectra of the S1=2–D5=2 transition for differ-ent cavity scan rates: L 0:16 (a), 0:23 (b), and 0:69(c). Blueshifted excitation spectra are drawn as gray lines on theright hand side of the diagrams, and superimposed solid linesshow the theoretical simulation. The parameters used for thesimulations are excitation laser bandwidth Laser 6 kHz,natural linewidth of the S1=2–D5=2 transition SD 0:17 Hz,maximum Rabi frequency at the transition center wavelengthmax 15:5 kHz (a), 11 kHz (b), and 25 kHz (c), and the cavityparameters given in the text.

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due to the Doppler effect. An excitation probability ofmore than 0.5 as in Fig. 2(a) clearly demonstrates thatthe ion is coherently interacting with the intracavity field.We theoretically model the excitation for different laserdetunings by numerically integrating two-level Blochequations using the time-dependent intracavity field calcu-lated from the pertaining differential equations [18,19].The results of the theoretical simulation for positive scanrates are shown as solid lines superimposed on the blue-shifted spectra in Fig. 2. The calculated and experimentalspectra show good agreement for small scan rates[Figs. 2(a) and 2(b)]. For larger scan rates [Fig. 2(c)] thecenters of the spectra are slightly shifted. We assume thatthis shift is caused by nonlinearities and hysteresis effectsof the PZT motion. Although we kept the excitation powerconstant in the experiments, we left the Rabi frequencymax as a fit parameter to account for variations in excita-tion due to thermal drift of the cavity shifting the node ofthe SW away from the ion’s position.

The second type of experiment probes the ion’s responseto spatial field variations. For this, we leave the scan ratefixed at a small value, allowing for stable scans with onlylittle perturbations of the excitation spectrum, as inFig. 2(a). The intensity of the 729 nm laser is adjustedsuch that the excitation is kept well below saturation. Theoffset voltage of both scan PZT and offset PZT is thenvaried simultaneously in such a way that the SW in thecavity is shifted longitudinally with respect to the locationof the ion. The position-dependent excitation probability isdetermined by fitting each excitation spectrum with aLorentzian and adopting the peak value. Figure 3 displaysthese values as a function of the PZT offset voltage. Errorbars given for representative data points in Fig. 3 are due toPZT hysteresis (abscissa) and the errors of the fits to theexcitation spectra (ordinate). The excitation probabilityvaries spatially with the intensity of the SW [17]. A theo-

retical Bloch-equation analysis as described above predictsa nearly pure ‘‘sin2’’ spatial variation, deviating by lessthan 1%. From a sin2 fit to the data points we obtain thecontrast ratio (visibility V) of the position-dependent ex-citation, V 96:3 2:6%. This very high visibility re-sults from the strong confinement of the ion’s wavefunction. The laser-cooled ion, oscillating with its secularfrequencies and with thermally distributed amplitudes, hasa rms spatial extension along the cavity axis of ac, leadingto a reduction of the excitation contrast by a factor ofexp22ac=

2. From the measured visibility V wefind ac 165

7 nm. The small value of the extension acshows that in this experiment we cool the ion close to theDoppler limit (13 nm).

A necessary condition for all experiments relying onion-cavity mode coupling is the ability to place the ion ata certain position of the intracavity SW field with highprecision and high reproducibility [3]. In our experiment,the precision of positioning the center of the ion’s wavefunction, using a measurement as in Fig. 3, is limited by theuncertainty in the measured excitation probability. Thisuncertainty is due to a statistical error, i.e., the finitenumber of state detection measurements, and systematicerrors such as fluctuations of laser intensity and wave-length, drift, and jitter of scan PZT, etc. [20]. From theuncertainties in excitation probability (error bars in Fig. 3)we deduce a spatial precision between 7 nm ( =100) atthe position of largest slope and 12 and 36 nm at minimumor maximum excitation, respectively. We note, however,that the precision can be enhanced by averaging over alarger number of state detection measurements.

Many schemes for quantum information processing withtrapped ions rely on coherent interaction not only with theinternal state but also with the motional degrees of free-dom. A controlled coupling to the motional quantum stateis a precondition for realizing such schemes. In order todemonstrate this motion-dependent coupling, we recordedexcitation probabilities of the ion at a fixed cavity scan rate(L 0:23), for different positions within the SW, andwith the laser at 729 nm now tuned to either the carrier (nochange of vibrational quantum number, n 0) or the redaxial sideband (n 1, laser detuned by !z) of theS1=2–D5=2 transition. In both cases, the intensity of the laserwas adjusted such that the excitations of carrier and side-band were comparable and were kept well below satura-tion. In this experiment we determine the integralexcitation, i.e., the area of the respective excitation spectra,as the spectra show an asymmetric line shape [cf. Figs. 2(b)and 2(c)]. As displayed in Fig. 4, carrier and sidebandexcitations both map the SW spatial field variation, butthe traces are shifted by a phase factor of . This phaseshift arises due to symmetry characteristics of the transi-tion matrix elements of carrier and sideband transitions in aSW field [11,21]: The spatial part of the quadrupole tran-sition matrix element is proportional to hn0j expikxjni fora traveling wave (TW) and hn0j coskxjni for a SW with

120 140 160 180 200 220 240

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Piezo offset voltage (V)

Excitation p

robabili

ty

FIG. 3. Excitation probability on the S1=2–D5=2 transition of asingle trapped Ca ion as a function of the PZT offset voltage,i.e., at various positions in the intracavity standing wave field.The solid line represents a sin2 function fitted to the data points.

VOLUME 89, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 2 SEPTEMBER 2002

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;

the electric field E / sinkx [17]. Here n and n0 are thevibrational quantum numbers in the S1=2 and D5=2 levels,respectively, k is the wave number, and x is the ion’s posi-tion in the field. For a TW, all vibrational states n; n0 canbe coupled as expikx contains even and odd powers of kx.On the contrary, for a SW hn0j coskxjni has to be ex-panded into even or odd powers of kx depending on theion’s position, e.g., x 0 close to a node or x =4 closeto an antinode. Thus, transitions changing the phononnumber by even or odd integers can be excited only atdifferent positions in the SW. The red sideband transition(n 1) couples maximally at antinodes of the SW,whereas the carrier transition (n 0) couples maximallyat nodes.

The high-contrast orthogonal coupling of carrier andsideband transitions to the cavity mode facilitates applica-tions such as cavity-assisted cooling in a SW [11] or theCirac-Zoller quantum-computing scheme [22]. Bothschemes rely on driving vibrational sideband transitionsand benefit from suppressing off-resonant carrier transi-tions which induce motional heating or impose a limit onthe attainable gate speed [23], respectively. Furthermore,the ion-cavity coupling allows for demonstrating and uti-lizing cavity-modified spontaneous emission: we calculatethe cooperativity parameter C g2=2!" 0:52, with ourexperimental ion-field coupling constant g 2134 Hz, cavity decay rate ! 2 102 kHz, and spon-taneous emission rate " 2SD 2 0:17 Hz.Because of the coupling, the spontaneous emission rate isenhanced by the Purcell factor F 2C 1 2:04, andthe fraction of spontaneous emission from the D5=2 levelemitted into the cavity mode is $ 2C=2C 1 51%.This is sufficient for demonstrating cavity-assisted coolingvia destructive quantum interference of heating transitions[12], or triggered single-photon emission from the D5=2

level. Improvement of the latter scheme and realization ofthe atom-photon interface [2] can readily be achieved byemploying an adiabatic transfer technique [4,5] and using acavity with higher finesse ( * 100 000).

In summary, we have demonstrated coherent coupling ofelectronic and motional states of a single trapped ion to asingle field mode of a high finesse cavity. The position ofthe ion within the standing wave can be determined with aprecision of up to =100. As the electronic quadrupoletransition in Ca is one of the candidates for implementinga quantum bit, our experiments are a key step towardsrealization of quantum-computing and communicationschemes with trapped ions that require a controlled inter-action of ion and cavity field.

This work is supported by the Austrian ’’Fonds zurForderung der wissenschaftlichen Forschung’’ (SFB15),by the European Commission [TMR networks ’’QI’’ and’’QSTRUCT’’ (ERB-FRMX-CT96-0087 and -0077), IHPnetwork ’’QUEST’’ (HPRN-CT-2000-00121), and IST/FET program ’’QUBITS’’ (IST-1999-13021)], and by the‘‘Institut fur Quanteninformation GmbH’’.

*Electronic addresses: Christoph.Becher@uibk.ac.at,http://heart-c704.uibk.ac.at/Welcome.html

†Present address: Time and Frequency Division, NationalInstitute of Standards and Technology, Boulder, CO80305.

[1] H. C. Nagerl et al., Phys. Rev. A 61, 023405 (2000).[2] J. I. Cirac et al., Phys. Rev. Lett. 78, 3221 (1997).[3] G. R. Guthohrlein et al., Nature (London) 414, 49 (2001).[4] C. K. Law and H. J. Kimble, J. Mod. Opt. 44, 2067 (1997).[5] M. Hennrich et al., Phys. Rev. Lett. 85, 4872 (2000).[6] K. M. Gheri et al., Phys. Rev. A 58, R2627 (1998).[7] D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996).[8] Ch. Roos et al., Phys. Rev. Lett. 83, 4713 (1999).[9] V. Buzek et al., Phys. Rev. A 56, 2352 (1997).

[10] F. L. Semiao, A. Vidiella-Barranco, and J. A. Roversi,Phys. Rev. A 64, 024305 (2001).

[11] J. I. Cirac et al., Phys. Rev. A 46, 2668 (1992).[12] J. I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 51,

1650 (1995).[13] V. Vuletic, H. W. Chan, and A. T. Black, Phys. Rev. A 64,

033405 (2001).[14] W. Paul, Rev. Mod. Phys. 62, 531 (1990).[15] H. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975).[16] S. Gulde et al., Appl. Phys. B 73, 861 (2001).[17] Note that the atomic quadrupole moment couples to the

gradient of the electric field. The maximum excitation thusoccurs at the nodes of the standing wave.

[18] H. Rohde et al., J. Opt. Soc. Am. B 19, 1425 (2002).[19] M. J. Lawrence et al., J. Opt. Soc. Am. B 16, 523 (1999).[20] J. Eschner et al., Nature (London) 413, 495 (2001).[21] M. Sasura and V. Buzek, quant-ph/0112041.[22] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).[23] A. Steane et al., Phys. Rev. A 62, 042305 (2000).

20 40 60 80 100

Red axial sideband

Inte

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l excitation (

a.u

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Piezo offset voltage (V)

Carrier

FIG. 4. Integral excitation on the carrier (triangles) and the redaxial sideband (circles) of the S1=2–D5=2 transition as a functionof the PZT offset voltage, i.e., at various positions in the intra-cavity standing wave field. The solid lines represent fits of sin2

functions to the data points.

VOLUME 89, NUMBER 10 P H Y S I C A L R E V I E W L E T T E R S 2 SEPTEMBER 2002

103001-4 103001-4

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DOI: 10.1007/s00340-002-1073-7

Appl. Phys. B (2003)

Lasers and Optics

Applied Physics B

a.b. mundt

a. kreuter

c. russo

c. becher

d. leibfried∗

j. eschner

f. schmidt-kaler

r. blatt

Coherent coupling of a single 40Ca+ ionto a high-finesse optical cavityInstitut für Experimentalphysik, Universität Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria

Received: 23 August 2002

Published online: 8 January 2003 • © Springer-Verlag 2003

ABSTRACT We demonstrate coherent coupling of the quadru-pole S1/2 ↔ D5/2 optical transition of a single trapped 40Ca

+

ion to the standing wave field of a high-finesse cavity. Thedependence of the coupling on temporal dynamics and spa-tial variations of the intracavity field is investigated in detail.By precisely controlling the position of the ion in the cavitystanding wave field and by selectively exciting vibrational state-changing transitions the ion’s quantized vibration in the trapis deterministically coupled to the cavity mode. We confirmcoherent interaction of ion and cavity field by exciting Rabioscillations with short resonant laser pulses injected into thecavity, which is frequency-stabilized to the atomic transition.

PACS 32.80.Pj; 03.67.-a; 42.50.Ct

1 Introduction

Laser-cooled trapped atoms or ions are ideallysuited systems for the investigation and implementation ofquantum information processing [1]. The combination ofa Paul-type ion trap [2] with laser cooling leads to uniqueproperties of trapped cold ions, such as localization of thesingle particle to less than a few tens of nanometers, con-trol of the motional state down to the zero-point of thetrapping potential and a high degree of isolation from theenvironment and thus a very long time available for ma-nipulations and interactions at the quantum level. The verysame properties make single trapped atoms and ions wellsuited for storing quantum information in long-lived inter-nal states, e.g. by encoding a quantum bit (qubit) of infor-mation within the coherent superposition of the S1/2 groundstate and the metastable D5/2 excited state of Ca+ [3]. How-ever, this stored information is ‘spatially static’ with respectto transport of quantum information over distances exceedingthe trap dimensions. A much better suited carrier of quan-tum information for fast and reliable transport over long dis-tances are photons. An interface between static and movingqubits requires the controlled coherent interaction of a sin-gle atom and a single cavity mode, this being the basic

E-mail: christoph.becher@uibk.ac.at∗Present address: Time and Frequency Division, National Institute ofStandards and Technology, Boulder, CO 80305, USA

building block for distributed quantum networks [4]. For anexperimental implementation of such a scheme one needsto understand and control the dynamics of the atom–cavitycoupling. Thus, the demonstration of deterministic coherent

coupling of a single ionic qubit to one mode of the elec-tromagnetic field inside a high-finesse optical resonator andthe detailed investigation of this coupling [5] are of majorinterest for the implementation of quantum information pro-cessing with trapped ions. Deterministic excitation of cavity-induced fluorescence on the short-lived S1/2–P1/2 dipole tran-sition in Ca+ was demonstrated recently, thereby utilizinga single trapped ion as a nanoscopic probe of an opticalfield [6].

Another application closely related to the atom–photon in-terface is the triggered emission of single photons from thecoupled atom–cavity system [7, 8] or sequences of entangledsingle-photon wave packets [9]. Efficient deterministic emis-sion of single photons into a well-defined spatial mode isa prerequisite for quantum key distribution [10] and linear op-tics quantum computation [11].

The ability to precisely place an ion at an arbitrary positionwithin a standing wave (SW) field and the highly orthogonalcoupling of different vibrational-state-changing transitions tothe SW field can be utilized for cooling the ion’s vibrationalstate well below the Doppler limit [12]. If, in addition, thecoupling of atom and cavity mode is strong enough to inducea modification of the atom’s spontaneous emission proper-ties [13], one can demonstrate cavity-assisted cooling via de-structive quantum interference of heating transitions [14].

Finally, the strong coupling of trapped ions that are cooledto their lowest vibrational state [15, 16] with a quantized cav-ity field might allow for entangling three quantum subsys-tems [17, 18], i.e. internal electronic states, quantum vibra-tional mode and single-mode cavity field.

In this paper we demonstrate important first steps towardsthe above experiments by coherently coupling the quadrupoleS1/2 ↔ D5/2 transition of a single trapped Ca+ ion to a modeof a high-finesse optical cavity. We investigate the coupling byprobing the ion’s response to dynamical and spatial variationsof the intracavity field. The ion can be placed with high preci-sion at an arbitrary position in the SW field of the cavity. Weachieve deterministic coupling of the cavity mode to the ion’svibrational state by selectively exciting state-changing transi-tions with the cavity light tuned to a vibrational sideband ofthe S1/2 ↔ D5/2 resonance. We prove that the interaction of

4 Ë

Applied Physics B – Lasers and Optics

ion and cavity field is coherent by exciting Rabi oscillationsthrough the field in the frequency-stabilized cavity.

2 Experimental setup

2.1 Trap and cavity setup

The 40Ca+

ion is stored in a spherical Paul trap [2]placed in the center of a near-confocal resonator. The ion islaser-cooled to the Lamb–Dicke regime, confining its spa-tial wave packet to a region much smaller than the opticalwavelength. We detect the coupling of ion and fundamentalTEM00 cavity mode by injecting a light field at 729 nm into thecavity and recording the excitation on the S1/2 ↔ D5/2 transi-tion. The excitation probability is monitored via the electron-shelving technique [16, 19], i.e. by probing the fluorescenceon the S1/2– P1/2 dipole transition (see Fig. 1).

The following stabilized laser sources are used in theexperiment: two cavity-locked diode lasers at 866 nm and854 nm with line widths of ≈ 10 kHz and two Ti:Sa lasersat 729 nm (≈ 1-kHz line width) and 794 nm (< 300-kHz linewidth), of which the 794-nm laser is resonantly frequency-doubled to obtain 397 nm. The whole laser system is de-scribed in more detail elsewhere [3].

The experimental setup of trap and cavity is schematicallyshown in Fig. 2. The three-dimensional RF Paul trap consistsof an elliptical ring electrode with average diameter of 1.4 mmand two end caps with a spacing of 1.2 mm (material: 0.2-mm

molybdenum wire). The secular frequencies (ωx, ωy, ωz) are2π × (2.9, 3.9, 7.4) MHz at a RF drive field power of 1 W.Here, z denotes the direction of the trap axis, which is at 45

to the cavity axis. The x and y radial directions both includean angle of ≈ 45 with the plane spanned by the cavity andthe trap axis. A magnetic field of 3 G perpendicular to the cav-ity axis provides a quantization axis and a frequency splittingof Zeeman components of the S1/2 ↔ D5/2 transition. Cal-cium ions are loaded into the trap from a thermal atom beamby a two-step photoionization process using diode lasers near423 nm and 390 nm [20]. The trap is placed in the center ofa near-confocal resonator with finesse F = 35 000 at 729 nm,waist radius ω0 = 54 µm, mirror separation L = 21 mm andradius of curvature RM = 25 mm. Cylindrical piezoceramics(PZTs) allow fine tuning of the cavity length across approxi-mately 1.5 free spectral ranges. In the following this cavity isdenoted as the ‘trap cavity’.

FIGURE 1 40Ca+

level scheme

FIGURE 2 Schematic experimental setup. PZT1 denotes the offset piezo,PZT2 the scan piezo. A photomultiplier tube (PMT) is used to record fluo-rescence on the S1/2–P1/2 transition and the CCD camera monitors the ion’sposition. The dotted arrows indicate directions of laser beams: a 729-nm ex-citation laser along the direction of the cavity axis and a 397-nm coolinglaser, 854-nm and 866-nm auxiliary lasers at a certain angle to the trap axis(only shown schematically)

The ion can be excited by a traveling wave using a free729-nm laser beam at a certain angle to the cavity axis or bythe SW built up in the cavity after injecting a 729-nm laserbeam into the cavity mode. The position of the ion relative tonodes and anti-nodes of the SW can be controlled by adjustingthe offsets of both PZTs.

2.2 Frequency stabilization of the trap cavity

In order to study stationary coherent interaction ofan atom and a cavity mode it is necessary to stabilize the cav-ity resonance frequency to the atomic transition frequency. Inorder to avoid that resonant light present in the cavity per-turbs the controlled atom–cavity interaction, we use a 785-nm

diode laser for the stabilization, this being far off-resonant totransitions at 729 nm. The basic scheme of the transfer lockis as follows: we frequency lock the 785-nm diode laser tothe same ultra-stable reference cavity (see [3]) as the 729-nm

Ti:Sa laser. By stabilizing the trap cavity to the diode laserfrequency, the lengths of the reference and the trap cavity arefixed relative to each other. Furthermore, the length of the trapcavity is fixed relative to the wavelength of the 729-nm Ti:Salaser. By applying an appropriate frequency shift to the diodelaser, the length of the trap cavity is adjusted to achieve reson-ance for 729 nm.

Figure 3 illustrates the setup of the diode laser at 785 nm

and its stabilization to the reference cavity. The single-modelaser diode (80-mW maximum output power) is temperature-stabilized to a few millikelvin. An external holographic grat-ing with 1800 lines/mm in Littrow configuration creates anextended tuneable resonator. The grating is mounted on a pre-cision mirror holder; fine tuning is achieved by a PZT. Theelements described so far are enclosed in a stable aluminumcase kept at a constant temperature of (20 ±0.1) C by cir-culating water. After passing through an optical diode with60-dB isolation the light is split by a polarizing beam split-ter. The part of the light being used for the Pound–Drever–

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MUNDT et al. Coherent coupling of a single 40 Ca+ ion to a high-finesse optical cavity

FIGURE 3 Setup of the 785-nm diode laser with extended cavity (Littrowconfiguration) and frequency stabilization to the reference cavity. The laserbeam (≈ 35 mW) is split into a lock beam (≈ 2 mW, dashed line) and an ex-periment beam that is frequency-shifted in a double-pass configuration (solidline). Laser line width: ∆ν785 = (1.8±0.1) kHz. For more details see text

Hall lock [21] to the reference cavity (FSR = 750 MHz) isphase-modulated with an electro-optical modulator (EOM) at16 MHz. The error signal is fed to a PI controller acting onthe laser diode current (fast branch) as well as on the grat-ing PZT (slow branch). The light not used for the lock isdirected through an acusto-optical modulator (AOM) (dou-ble pass), allowing for a continuous frequency shift of ap-proximately ±120 MHz, and is sent to the trap cavity via anoptical fiber.

Figure 4 shows the setup of the trap cavity lock to thediode laser at 785 nm. The stabilized 785-nm light leavingthe optical fiber is phase-modulated by an EOM at 16 MHzand is coupled into the trap cavity (from the right in Fig. 4)via a polarizing beam splitter. The 729-nm light is coupledinto the cavity from the opposite side (left in Fig. 4). Onthe right-hand side of the cavity, the light is dispersed bya grating (1200 lines/mm) to separate the 785-nm reflectedlight from the 729-nm transmitted light, which is monitoredby a CCD camera and a photodiode. The first-order grat-

FIGURE 4 Setup of the trap-cavity lock. The light at 785 and 729 nm iscoupled into the cavity from opposite directions. Gratings (1200 lines/mm)on both sides separate the beams and ensure clear control of the transmissionson both wavelengths without perturbation by reflections of the other light. Formore details see text

ing reflection at 785 nm is focused onto a fast photodiode toobtain a Pound–Drever–Hall error signal after mixing withthe modulation frequency. The error signal is processed ina PI controller acting on one of the trap-cavity PZTs. On theleft-hand side, again a grating is used to separate the trans-mitted 785-nm light from the reflected 729-nm light. The785-nm transmission is monitored on a CCD as well as ona photodiode.

The experimental procedure to achieve double resonanceof the trap cavity is as follows: by scanning the trap cavity,transmission on the 729-nm TEM00 mode is recorded. Nowthe wavelength of the diode laser has to be chosen in sucha way that the 785-nm light is also resonant for both the refer-ence and the trap cavities within the range of ±120 MHz, thisbeing covered by the double-pass AOM. Usually, this can beachieved by current tuning the diode to one of its longitudinalmodes.

3 Experiments on ion–cavity coupling

In the following, we report four different experi-ments exploring various aspects of the ion–cavity coupling:First, we probe the ion’s response to temporal variations ofthe cavity internal field by scanning the cavity over the res-onance with the incident laser light, thus imprinting a phaseand amplitude modulation on the cavity field [22, 23]. Second,using the same scanning excitation technique we detect theion’s response to spatial variations of the intracavity SW field(mapping of the longitudinal field distribution). Third, we in-vestigate coupling of the ion’s vibrational modes to the cavityfield by comparing the excitation on the carrier transition andon the first vibrational sideband at different positions in theSW. Finally, we observe coherent interaction of ion and cav-ity field by exciting Rabi oscillations with short resonant laserpulses coupled into the frequency-stabilized cavity.

3.1 Measurement procedures

The general procedure for all experiments is thefollowing:

– The ion is Doppler-cooled and prepared in the electronicground state.

– A sequence of laser pulses is applied to the ion.– The final electronic state is detected.

These three steps are described in detail in the following (cf.Fig. 5).

First, the ion is Doppler-cooled for 1.9 ms on the S1/2–P1/2 transition. During cooling the 854-nm laser is switchedon to remove any population in the metastable D5/2 state. Inaddition, we irradiate the ion with a laser beam at 866 nm at alltimes to prevent shelving in the metastable D3/2 state. Fromcoherent dynamics (Rabi oscillations) on the carrier and firstmotional sidebands [16], we determine typical mean vibra-tional quantum numbers after Doppler cooling (nx, ny, nz) =(20 ±5, 4 ±1, 6 ±1). From these mean phonon numbers andthe secular frequencies given in Sect. 2.1 we calculate a rmsextension of the ion’s motional wave packet of (25±5) nm inthe direction of the cavity axis, much smaller than the wave-length of 729 nm. At the end of the cooling sequence the ion isoptically pumped into the S1/2 (m = −1/2) Zeeman sub-level

4 Í

Applied Physics B – Lasers and Optics

FIGURE 5 Example of a 5-ms pulse sequence.In the first 2 ms a defined ‘starting’ state is pre-pared. Then this state is manipulated (2 ms to2.5 ms). Finally the outcome is detected (excita-tion or not) for the remaining time. The 10-msand 20-ms pulse sequences differ only in a longermanipulation window (up to 3 ms) and a longerdetection time

by applying a 0.1-ms σ−-polarized pulse at 397 nm. Afterthat, all light at 397 nm is switched off.

Second, the ion is excited on the S1/2 ↔ D5/2 (m = −1/2to m ′ = −5/2) transition by a free laser beam or by light res-onant in the trap cavity. A time window of 0.5 to 3 ms isassigned to this process. Interaction of the ion and light withinthe trap cavity can be achieved by either frequency lockingthe cavity to the quadrupole transition or by a cavity scanningtechnique. In the first case we inject short resonant laser pulseswith variable pulse length into the frequency-stabilized cav-ity in order to study coherent ion–field interaction. The secondtechnique is used for investigating the ion’s response to dy-namical or spatial variations of the intracavity field. Here,the laser at 729 nm is set to a fixed detuning ∆ from theS1/2 ↔ D5/2 transition. We inject the laser beam into the cav-ity TEM00 mode (cf. Fig. 4) and apply a voltage ramp to thescan PZT in order to scan the cavity length. When the cavityreaches resonance with the laser frequency it fills with lightand may excite the ion to the D5/2 level. Then the light at729 nm is switched off and the cavity mirror is ramped downto its initial position.

Third, we determine the excitation probability via theelectron-shelving technique. The blue light at 397 nm isswitched on (866-nm light is on anyway) and the fluorescenceis counted by the PMT for a time interval between 2.3 ms and17 ms depending on the duration of the total pulse sequence(5, 10 or 20 ms). The PMT count rate for the fluorescent ion is≈ 16 kHz; the signal-to-background ratio ∼ 10. By compar-ing the number of counts in the time interval with a thresholdwe can discriminate with very high probability (> 99%) [16]whether excitation happened (electron shelved in D5/2, nofluorescence) or the ion remains in the ground state (fluores-cence). Note that although state detection happens a few msafter the ion–light interaction the ion’s state is well preserveddue to the long lifetime (1 s) of the D5/2 level.

All three steps are typically repeated 100 times to yield theaverage excitation probability. After that, changes in the exci-tation window may be applied, such as a change in the laserfrequency (frequency scan) or a change in the excitation pulselength (pulse-length scan). For example, to obtain an excita-tion spectrum, the 729-nm laser is tuned over the quadrupoletransition in steps of 1 kHz.

3.2 Temporal variation of the intracavity field

In our first experiment we probe the ion’s responseto temporal variations of the intracavity field by placing it closeto a node of the SW field [24] and varying the cavity scan rate.

Figure 6 shows the excitation probability on theS1/2 ↔ D5/2 transition for different scan rates. For negative(positive) scan rates the mirrors move towards each other(apart) and thereby Doppler shift the light to the blue (red).The atomic transition then seems to be red (blue)-shiftedas the excitation laser detuning has to compensate for theDoppler shift. Increasing scan rate increases the shift from theline center (detuning zero), which is found by excitation witha free laser beam. In addition, faster scan rates broaden the linebecause more and more frequency components are added tothe light confined in the cavity.

FIGURE 6 Excitation spectra for different cavity scan rates νL = ±0.16(a), ±0.23 (b) and ±0.69 (c). On the blue-shifted spectra (gray lines), the-oretical simulations are superimposed (solid line). The parameters for thesimulations are: laser bandwidth ∆νlaser = 6 kHz, natural line width of theS1/2 ↔ D5/2 transition ∆νSD = 0.17 Hz, Rabi frequency for zero detuning:Ω0 = 15.5 kHz (a), 11 kHz (b) and 25 kHz (c)

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MUNDT et al. Coherent coupling of a single 40 Ca+ ion to a high-finesse optical cavity

To put these observations in a more quantitative formit is convenient to define a normalized scan rate νL =2F ωLτ/(πc). It corresponds to the resonance frequencyshift of the cavity in units of the HWHM cavity line widthc/(4LF ) per cavity energy storage time τ = F L/(πc). Tomodel the coupling dynamics we first have to know the elec-tric field acting on the ion. The electric field amplitude Ecav ina cavity swept over resonance is a solution of the differentialequation [22, 23]:

dEcav

dt= −(1 − iνL t)Ecav + i

√TF

πE0 ,

where t = t/τ , T ≈ π/F is the transmission of the input coup-ling mirror and E0 is the amplitude of the input field. For slowscan rates (νL 1) we obtain the quasi-static case, whereasfor fast scan rates (νL 1) the incoming light interferes withthe light stored in the cavity. Figure 7 shows the normalizedintracavity field amplitude for three different scan rates.

The interaction of this time-dependent electric field withthe ion can now be modeled by numerically integrating theBloch equations for a two-level system. The result for a scanrate of νL = 1 and a typical laser intensity is shown in Fig. 8.The electronic transition (Fig. 8, top) is blue-shifted andbroadened compared to the excitation with a free laser beam,as expected.

Figure 8 (bottom) illustrates, for five different detun-ings ∆, the time evolution of the excitation to the D5/2 levelas the cavity is swept over resonance. For all detunings, theexcitation probability shows a steep rising edge at the mo-ment the cavity fills with light and subsequent Rabi oscil-lations with a damping depending on excitation power anddetuning. This excitation characteristic remains qualitativelyunchanged over a large range of excitation powers and scanrates. The final stationary values of the excitation probabilitiesare taken to compose the excitation spectrum (Fig. 8, top).

The results of the theoretical simulation for positive scanrates are shown as solid lines superimposed on the blue-shifted spectra in Fig. 6. The calculated and experimentalspectra show good agreement for small scan rates (Fig. 6a,b).

FIGURE 7 Calculated time-dependent normalized electric field amplitudein the cavity for three different scan rates νL . The solid line is the real part;the dashed line the imaginary part of the electric field amplitude. For largescan rates the interference of incoming and stored light becomes visible

FIGURE 8 Simulated excitation spectrum for νL = 1 (top) and time evo-lution of the excitation probability for different detunings ∆ (bottom). Moredetails are given in the text

For larger scan rates (Fig. 6c) the centers of the spectra areslightly shifted. We assume that this shift is caused by non-linearities and hysteresis effects of the PZT motion. Althoughwe kept the excitation power constant in the experiments, weleft the Rabi frequency Ω0 as a fitting parameter to accountfor variations in excitation due to thermal drift of the cavityshifting the node of the SW away from the ion’s position.

The investigation and theoretical understanding of theion–cavity field coupling dynamics as discussed above areimportant for realization of the atom–photon interface inquantum networks. The experimental implementation of sucha scheme requires control over the shape of the light pulseemitted from the cavity and thus control over the dynamicatom–cavity coupling [4].

3.3 Spatial variation of the intracavity field

The trapped single ion is confined to a region inspace of less than 30 nm, which is much less than the excita-tion wavelength (729 nm). It thus can be used as a nanoscopicprobe of the SW field spatial variation as first shown in [6, 25].To probe this spatial variation, we again use the cavity scan-ning technique and leave the scan rate fixed at a small value,allowing for stable scans with only a little perturbation of theexcitation spectrum as in Fig. 6a. The laser is scanned overthe quadrupole transition in steps of ∼ 1 kHz to obtain an ex-citation spectrum at a certain position in the SW. To use theexcitation probability as a measure of the field intensity, it isimportant to adjust the laser power such that the excitationis kept well below saturation at all positions in the SW. Theoffset voltage of both scan PZT and offset PZT is then var-ied simultaneously in such a way that the SW in the cavityis shifted longitudinally with respect to the location of theion. Repetition of the procedure yields a number of excitationspectra shown in Fig. 9. The position-dependent excitationprobability is determined by fitting each excitation spectrumwith a Lorentzian and adopting the peak value. Figure 10 dis-plays these values as a function of the PZT offset voltage.

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Applied Physics B – Lasers and Optics

FIGURE 9 All 48 excitation spectra measured at different positions in the intracavity SW. Every box (width: 600 kHz) contains a spectral line as in Fig. 6;the offset voltage increases from left to right. The SW pattern is already visible; the total shift is ∼ 1.5 half-waves

FIGURE 10 Excitation probability on the S1/2 ↔ D5/2 transition of a singletrapped Ca+ ion as a function of the PZT offset voltage, i.e. at various pos-itions in the intracavity standing wave field. The solid line represents a sin2

function fitted to the data points

Error bars given for representative data points in Fig. 10 aredue to PZT hysteresis (abscissa) and the errors of the fits tothe excitation spectra (ordinate). The excitation probabilityvaries spatially with the intensity of the SW [24]. A theoreticalBloch-equation analysis as described above predicts a nearlypure sin2 spatial variation, deviating by less than 1%. Froma sin2 fit to the data points we obtain the contrast ratio (visibil-ity V ) of the position-dependent excitation, V = 96.3±2.6%.This very high visibility results from the strong confinementof the ion’s wavefunction. The laser-cooled ion, oscillatingwith its secular frequencies and with thermally distributedamplitudes, has a rms spatial extension along the cavity axisof ac. This extension leads to a reduction of the excitationcontrast by a factor of exp(−2(2πac/λ)2) compared to theexcitation contrast experienced by a point-like atom at rest.From the measured visibility V we find ac = 16+5

−7 nm. Thesmall value of the extension ac shows that in this experimentwe cool the ion close to the Doppler limit (13 nm).

A necessary condition for all experiments relying on ion–cavity mode coupling is the ability to place the ion at a cer-tain position of the intracavity SW field with high precisionand high reproducibility [6]. In our experiment, the preci-sion of positioning the center of the ion’s wavefunction, usinga measurement as in Fig. 10, is limited by the uncertainty inthe measured excitation probability. This uncertainty is dueto a statistical error, i.e. the finite number of state-detection

measurements, and systematic errors such as fluctuations oflaser intensity and wavelength, drift and jitter of the scanPZT, etc. [25]. From the uncertainties in excitation probability(error bars in Fig. 10) we deduce a spatial precision between7 nm (≈ λ/100) at the position of largest slope and 12 nm and36 nm at minimum or maximum excitation, respectively. Wenote, however, that the precision can be enhanced by averag-ing over a larger number of state-detection measurements.

3.4 Coupling of motional states

Many schemes for quantum information process-ing with trapped ions rely on coherent interaction not onlywith the internal state but also with the motional degrees offreedom. A controlled coupling of atomic states dressed byvibrational modes to the cavity field is a precondition for re-alizing such schemes.

If one compares carrier (no change of vibrational quan-tum number, ∆n = 0) and first vibrational sideband (∆n =−1) transition probabilities in a SW, both show a position-dependent coupling due to the SW pattern as shown inSect. 3.3. But where the coupling to the carrier is strongest,no coupling to the first (i.e. all odd) sidebands occurs and viceversa, i.e. carrier and odd sideband excitations map the SWspatial field variation, but the traces are shifted by a phasefactor of π. This phase shift arises due to symmetry char-acteristics of the transition matrix elements of carrier andsideband transitions in a SW field [1, 12]: the spatial part ofthe quadrupole transition matrix element is proportional to〈n′| exp(ikx)|n〉 for a traveling wave (TW) and 〈n′| cos(kx)|n〉for a SW with the electric field E ∝ sin(kx) [24]. Here n andn′ are the vibrational quantum numbers in the S1/2 and D5/2

levels, respectively, k is the wavenumber and x is the ion’sposition in the field. For a TW, all vibrational states (n, n′)can be coupled since exp(ikx) contains even and odd powersof kx [26]. In contrast, for a SW 〈n′| cos(kx)|n〉 has to be ex-panded into even or odd powers of kx depending on the ion’sposition, e.g. x = 0 close to a node or x = λ/4 close to an anti-node. Thus, transitions changing the phonon number by evenor odd integers can only be excited at different positions in theSW. The first-sideband transition (∆n = ±1) couples maxi-mally at anti-nodes of the SW, whereas the carrier transition(∆n = 0) couples maximally at nodes. In order to demonstratethis motion-dependent coupling, we recorded excitation prob-abilities of the ion at a fixed cavity scan rate (νL = −0.23),for different positions within the SW, and with the laser at729 nm now tuned to either the carrier or the red axial side-

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MUNDT et al. Coherent coupling of a single 40 Ca+ ion to a high-finesse optical cavity

FIGURE 11 Integral excitation on the carrier (triangles) and the red axialsideband (circles) of the S1/2 ↔ D5/2 transition as a function of the PZT off-set voltage, i.e. at various positions in the intracavity standing wave field. Thesolid lines represent fits of sin2 functions to the data points

band (∆n = −1, laser-detuned by −ωz) of the S1/2 ↔ D5/2

transition. In both cases, the intensity of the laser was adjustedsuch that the excitations of carrier and sideband were compa-rable and were kept well below saturation. In this experimentwe determine the integral excitation, i.e. the area of the re-spective excitation spectra, as the spectra show an asymmetricline shape (cf. Fig. 6b,c). The high-contrast orthogonal coup-ling of carrier and sideband transitions to the cavity mode isdemonstrated in Fig. 11.

This coupling facilitates applications such as cavity-assisted cooling in a SW [12], the Cirac–Zoller quantum-computing scheme [27] and entanglement of motional andphotonic states when coupling to the cavity vacuum field [17,18]. In particular, cavity-assisted cooling in a SW fieldmeans that sideband cooling [16] on a red-detuned vibra-tional sideband is facilitated by suppression of off-resonantcarrier transitions which induce motional heating. In a similarfashion, unwanted off-resonant carrier excitations are sup-pressed when driving sideband transitions in the Cirac–Zollerquantum-computing scheme. Here, off-resonant carrier ex-citations impose a limit on the attainable gate speed [28].As electronic excitation probabilities of the ion depend onboth the vibrational state and the position within the cavityfield, the internal electronic degrees of freedom are coupledto both the (in general: quantized) single-mode cavity fieldand the vibrational degrees of freedom. This coupling allowse.g. for generation of an entangled state of a subsystem thatcan store quantum information (vibration, electronic states)with a subsystem that can be used for propagation of quantuminformation (light).

3.5 Coherent ion-cavity coupling

The application of the ion–cavity coupling for im-plementation of quantum network schemes requires a con-trolled coherent interaction. To demonstrate such coherentcoupling to the cavity field we stabilize the trap cavity tothe S1/2 ↔ D5/2 carrier transition frequency as described in

FIGURE 12 Rabi oscillations driven by the light field in the stabilizedcavity. The parameters for the theoretical simulation are: Rabi frequencyΩ0 = 8.95 MHz, η2n = 0.049 and laser detuning ∆ = 0

Sect. 2.2. We excite the ion with resonant laser pulses of dif-ferent pulse lengths injected into the trap cavity. Starting froma pulse length of zero up to a pulse of 3-µs duration we deter-mine the average excitation probability from 100 experimentsper pulse length. The resulting coherent dynamics (Rabi os-cillations) are shown in Fig. 12 together with a numericalsimulation (solid line). The theoretical curve is calculated as-suming a thermal distribution over harmonic oscillator states|n〉. The Rabi frequency of any particular |S, n〉 ↔ |D, n〉transition depends on nη2, where η is the Lamb–Dicke pa-rameter. The thermal mixture of different oscillation frequen-cies washes out the Rabi oscillations, which therefore appeardamped.

4 Summary and outlook

In summary, we have demonstrated coherent coup-ling of electronic and motional states of a single trapped ionto a single field mode of a high-finesse cavity. The position ofthe ion within the standing wave can be determined with a pre-cision of up to λ/100. As the electronic quadrupole transitionin Ca+ is one of the candidates for implementing a quantumbit, our experiments are a key step towards realization of quan-tum computing and communication schemes with trappedions that require a controlled coherent interaction of ion andcavity field. Future experiments aim at demonstrating and uti-lizing cavity-modified spontaneous emission for applicationssuch as triggered single-photon emission and realization ofthe atom–photon interface. Another route is the extension ofthe current configuration towards trapping of two or more ionscoupled to a common cavity mode, thus allowing for imple-mentation of quantum logical gate operations.

ACKNOWLEDGEMENTS This work is supported by the Aus-

trian ‘Fonds zur Forderung der wissenschaftlichen Forschung’ (SFB15),

by the European Commission (TMR networks ‘QI’ and ‘QSTRUCT’

(ERB-FRMX-CT96-0087 and -0077), IHP network ‘QUEST’ (HPRN-CT-

2000-00121) and IST/FET program ‘QUBITS’ (IST-1999-13021)) and by

the ‘Institut fur Quanteninformation GmbH’. C.R. acknowledges support

by Fundacao para a Ciencia e a Tecnologia (Portugal) under the grant

SFRH/BD/6208/2001.

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is the Lamb–Dicke parameter and a† and a are the phonon creation andannihilation operators, respectively

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Kaler, R. Blatt: Phys. Rev. A 62, 042305 (2000)

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