40 - quantum optics · 2015-12-01 · uagh?bl ua?bdfe^a a^l `3?bdzdf?am dfe >sn ik 0dfe^i...
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%>>.=0lIKlmDFE %og ?aMIKUaUa9 P IKMG P bQLN l P GCDFECA_bIKMwmD P ACDQX dfLNE "1 M " G DI" Z-?aMbQDgDFMIKM AACDF ?aMIKU `LNEH\ ?(VACDFrxDFE^IKU lECLbFDQX P ECDQAcIrsDwmDQDFMXDFrsDFUBLNlmDQX%GCLDbF?BDFMGHUa AC?a P UVIKGCDAH7AHGCDF0A LNMI P IKMG P #bQLN l P GCDFEx`3?aGHc0ML \ML `3MDbF?BDFMG bQL P MGCDFEHlIKEHGLNMI bFUVINACAH?BbIKUbQLN l P GCDFE cD/lECLNlmL]A^IKULKd ) wE^IK0A[IKMXj UBL 7X = d LNE_AH?a P UVIKGH?aMWYIKMn wmLX7 gDFEH ? AC9AHGCDF0AS?BA P AHG_LNMDDIK lUBD] L bF?aGCD J_?BDUBA^DFM%IKMX c P IKMW vuvG ?BA rsDFEH,Ua?a\sDFUa,GHcIKGLNMD0LKd3GHcD I HLNEIKllUa?BbIKGH?BLNMALKd P IKMG P bQLN l P GCDFECA/?aM GHcDd P G P ECD `3?aUaU wmD lmDFEpdfLNEH ?aMWAH?a P UVIKGH?BLNMALKd P IKMG P 0DQb^cIKM?BbIKUAH7AHGCDF0A_GCLL X7? b P UaG[GCLYAC?a P UVIKGCDLNMI bFUVINACAH?BbIKUTbQLN l P GCDFEkIlECLNwUBDF`3?aGHc,lECLKdfL P MX,A^b?BDFMGC?H%hbIKMX GCDQb^cMLNUBLNW]?BbIKU ?a lUa?BbIKGH?BLNMA ^ cDFECD IKECDI0UVIKEHWsDrNIKEH?BDFGvLKd lECLNlmL]AJIKUaA@dfLNE@GHcD<9> @?"O BD47G'OQ;S<G*AR; AR698 48:O C76 LKdI P IKMG P bQLNnl P GCDFEW3cDQACD[?aMbFU P XD• TcLNGCLNM P IKMG P bQLN l P GCDFECA ' @o• Ir9?aGv$ @R > IH (o• J@ &% H • UsL]ACDFlcA^LNM P MbGC?aLNMA > ] o• RTUBDQbFGHECLNMA+LsIKGH?aMW LNM cDFUa? P %o• P IKMG P XLNGCA ] o• -l?aM7n %#" M P bFUBDF?k?aMIACDF ?BbQLNMX P bFGCLNE_IKGHEH?t ] o• u LNM GHE^IKlA = c?BA Ua?BAHG IKMX0GHcD3W]?arxDFM0ECDd DFECDFMbQDQAx`3c?Bb^clmLN?aMG GCL ACDF ?aMIKU`LNEH\9A ?aM0GHcDZECDQAHlmDQbFGH?arxD %DFUBXA9IKECD3wML 0DIKMA/D7cI P AHGH?arsD] cD ECD/ P ?aECDF0DFMGCAXDR%MDQX%IKwmL rxD A^DFG rxDFEHc?aW]c GCDQb^cMLNUBLNW]?BbINU XDFIKMXAIKMX bQLNMACD/ P DFMGHUa IKUaUmD9lmDFEH?a0DFMG^IKUkIKGHGCDF lGCA d INbQD_bQLNMAH?BXDFE^IKwUBD[b^cIKUaUBDFMWsDQATq_MGH?aUhECDQbQDFMGHUa7bQLNnlUBDFGCD P IKMG P IKUaWsLNEH?aGHc0A cIrsDSLNMUawmDQDFM ECDIKUa? O DQX`3?aGHc J@ GCDQb^cM?* P DQA ' IH ouvG AHcL P UBX0wmDMLNGCDQX,GHcIKG GHcDFECD?BA bQDFEHG^IK?aM bFEH?aGH?BbF?BAH IKwmL P G[G^LY`3cIKG DJ7GCDFMG lECDQACDFMG XI J@ P IKMG P bQLNnl P G^IKGH?BLNMd P U&%UBA GHcD AHlmDQbF?&%DQXECD/ P ?aECDF0DFMGCAbQLN lUBDFGCDFUa (#=%o. cDQA^D bQLNMbQDFEHMA ECDFUVIKGCDIK?aMUaGCL0ACbIKUVIKw?aUa?aGv IKMX GHcD@d INbFG3GHcIKGZGHcD AH7AHGCDF ?BA3?aMIc?aW]cUaYGHcDECIKU A^DFlIKE^IKwUBD AHG^IKGCD]uvM?BLNMGHE^IKl P IKMG P bQLN l P GCDFECA! P IKMG P ?aM7d LNEHIKGH?BLNM?BA AHGCLNECDQX,?aM,Gv`LYUBLNMWNn Ua?arxDQX,?aMGCDFEHMIKUAHG^IKGCDQA |g〉 IKMX |e〉 LKd GHE^IKllmDQX?BLNMA SuoMGHc?BA_` IDINb^c?BLNM,bQLNMG^IK?aMA[LNMD( P w?aGSq_MXDFE_GHcD/?aM P nDFMbQD LKd_IKMD7GCDFEHMIKUGHE^IKll?aMWlmLNGCDFMGH?VIKU?aMbQLNw?aMIKGH?BLNM`3?aGHcGHcD P G P IKU L P UBLNwECDFl P UBAH?BLNM GHcDY?BLNMAT?td_bQLLNUBDQXTdECDQD O DY?aMGCLIbFEH7AHG^IKUtn Ua?a\sD AHGHE P bFG P ECD] uvMI GHO6 A 47EI P U GHE^IKl TGHcDYGHE^IKll?aMWlmLNGCDFMGH?VIKUk?BA3bF?aWxIKEpnoAHcIKlmDQXIKMXYd LNE3AHGHECLNMW E^INX7?VIKU GHEJIKl bQLNM %MD0DFM9GZGHcD bFEH9ACG^IKU AHGHE P bFG P ECD ?BAZIUa?aM7nDIKE[AHGHEH?aMW LKd ?BLNMA c?BA@bQLNM %W P E^IKGH?BLNM?BA_lIKEHGH?Bb P UVIKEHUaA P ?aG^IKwUBD/dfLNE_GHcDIKM?al P UVIKGH?BLNM LKd GHcD?BLNM
P w?aGCA `3?aGHc D9GCDFEHMhIKUmUVINACDFE %DFUBXA# cD_?BLNM AHGHEH?aMWD9c?aw?aGCAMLNEHIKUm0LXDQA LKd 0LNGH?BLNM r9?awE^IKGH?BLNMIKU0LXDQA"S `LKnV P w?aG[LNlmDFE^IKGH?BLNMA IKECD0DQX7?VIKGCDQXw LNMDACDFUBDQbFGCDQXr?awE^IKGH?BLNMIKU0LXD GHcD( P IKMG P w P A 0LXD cD lECLNlmL]A^IKU LKd ?aE^INb ]mLNUaUBDFE = ECD/ P ?aECDQA GHcD$ P IKMG P w P A 0LXD GCL wmD lECDnlIKECDQX ?aM ?aGCA3UBL `DQAHGS P IKMG P AHG^IKGCD/IKMX ?BAZwINACDQX LNM ECDQACLNMIKMG_Db?aG^IKGC?aLNMA_LKdTLNlGH?BbIKU GHE^IKMAH?aGH?BLNMA -9DFrxDFE^IKULNGHcDFE ACb^cDF0DQA@cIrxDwmDQDFMlECLNlmL]ACDQX`3c?Bb^c IKUBACL P GH?aUa? O D I bQLN 0LNM,r?awE^IKGC?aLNMhIKU0LXDGCL?a lUBDF0DFMG P IKMG P WxIKG^DFA3wmDFGv`DQDFM ?BLNM, P w?aGCA = = V ==V =6'oTLNECDQL rxDFE7?BLNM GHE^IKl, P IKMG P
bQLN l P G^IKGH?BLNM ACb^cDF0DQA DJ7?BAHG`3cDFECD GHcD Gv`LKnV P w?aG bQL P lUa?aMW ?BA0DQX7?VIKGCDQX0X7?&Y DFECDFMGHUasW\_MD3DIK lUBD?BA_IKMIKEHE^I LKd ?BbFECLNGHE^IKlAhDINb^c GHEJIKlbQLNMG^IK?aM?aMWYLNMD ?BLNM =6@o uoM GHc?BAZlECLNlmL]A^IKU Gv`LKnV P w?aGSWxIKGCDQAIKECD_0DQX7?VIKGCDQX w0GHcD L P UBLNwY?aMG^DEJINbGC?aLNM wmDFGv`DQDFMY?BLNMA?aM MDF?aW]cwmLNEH?aMW ?BbFECLNGHE^IKlA uoM IKMLNGHcDFElECLNlmL]A^IKU GHcD ?BLNM^ P w?aGCA_IKECD bQL P lUBDQX r?VI0I0LXD LKdGHcD DFUBDQbFGHECLNIKW]MDQGH?Bb%DFUBXLKdTIKMLNlGH?BbIKU ECDQAHnLNMIKGCLNE = H o\_rxDFEECDQbQDFMG sDIKECA3I/W]ECDIKG XDIKU LKd lECLNW]ECDQACAchINA wmDQDFM INXD@?aM A"<0ARERO; AR68 47G O C76^8:EF4"< 132547698:2 ;BDC7;S<2 8 48:O:C76x?aM lIKEHGH?Bb P UVIKE?aMGHcDSW]ECL P lYLKd > U[T4?aMDFUVIKMX IKGZJ@u - ?aMYL P UBXDFE ) d P MXIK0DFMG^IKU P IKMG P WxIKGCD7GHcD[bQLNMGHECLNUaUBDQX9nVJ \ wmDFGv`DQDFMIAH?aMW]UBD@GHE^IKllmDQX ?BLNM IKMX Ir9?awE^IKGH?BLNMIKU 0LXD9cINAwmDQDFM XDF0LNMAHGHE^IKGCDQX =6( Gv`L ?BLNMA =6> IKMXA P wACD/ P DFMGHUad L P E ?BLNMA ?' SfGHcDUVIKGHGCDFEDF lUBL ?aMWGHcD 0DFGHcLXASLKd = QcIrxD[wmDQDFMDFMG^IKMW]UBDQX uvM L P ETW]ECL P l xACDFrxDFE^IKUlECDFECD/ P ?BAH?aGCDQATd LNE# P IKMG P (bQLN l P G^IKGH?BLNM 9A P b^cINA0LNGC?aLNMhIKUW]ECL P MX0AHG^IKGCDbQLLNUa?aMW0T?aMX7?ar?BX P IKU ?BLNMINXX7ECDQA^AH?aMWIKMX%DFUBDF0DFMG^IKEHbQLNcDFECDFMGIKM?al P UVIKGH?BLNMAcIrsD wmDQDFMlECDFr?tnL P AHUa XDF0LNMACGHE^IKGCDQX?' %V ?' V =o _DQbQDFMGHUaK`D cIrsD D7lmDFEH?a0DFMG^IKUaUa/ECDIKUa? O DQXGHcD>@D P GCA^b^c7nVUsL O A^IIKUaWsLNEH?aGHc ?' o# c?BA ECDFlECDQACDFMGCA GHcDS%ECAHG?a lUBDF0DFMG^IKGH?BL]M LKd IKM DbF?BDFMG+ P IKMG P IKUaWsLNEH?aGHc LNMIKM ?BLNM GHE^IKl$ P IKMG P *bQLN l P GCDFEIKMXGHcD%ECAHG A P b^c?a lUBDF0DFMG^INGH?BLNM L P GCAH?BXD+J_/7 cD3>[D P GCACb^c7nUsL O A^ID7lmDFEH?a0DFMG ECDFlECDQACDFMGCAGHcDYIK?aMINb^c?BDFrsDF0DFMG0LKd_GHc?BAGHcDQAH?BA ECDFUt?a ?aMINEH%ECDQA P UaGCALNMIGv`LKn ?BLNMYbQLNMGHECLNUaUBDQX9nVJ\4WxIKGCD_IKECD_IKUBACL ECDFlmLNEHGCDQX DQAH?BXDQA ACDFrxDFE^IKUGCDQb^cM?BbIKUh?a lECL rxDF0DFMGCAGHcDQACDD7lmDEC?t0DFM9GCA dfLNUaUBL `DQXGHcD 0DFGHcLXAlECLNlmL]ACDQXw ?aE^INb ]mLNUaUBDFELT D IKUBACL_ECDIKUa? O DFX0IZML rxDFU9Gv9lmD LKd P IKMG P *WxIKGCDsECDFlmLNEHGCDQX0?aM0ACDQbFGH?BLNM (7&%s`3cDFECDZl P ECDFUaX7?BAHlmDFECAH?arxD UVINACDFEpn ?BLNM bQL P lUa?aMW fL@Ymn ECDQACLNMIKMGbQL P lUa?aMW`Zc?Bb^c ?aMX P bQDFA ML/lmLNl P UVIKGH?BLNM GHE^IKMApd DFE"T` INADF lUBL xDQX GCL/?a lUBDF0DFMGZI P M?arxDFECA^IKU P IKM7nG P WxIKGCD] cD@GHcDQAH?BA ?BA3AHGHE P bFG P ECDQX INAdfLNUaUBL `ZA cIKlGCDFE ?aMGHECLX P bQDQA3GHcD_d LNEHIKUa?BAH LKd GHcD@wINAH?Bb P IKMG P LNlmDFE^IKGH?BLNMA/ECD/ P ?aECDQXd LNE/IN P IKMG P bQLN l P GCDFE ? D] AH?aMW]UBD$ P w?aG ECLNG^IKGH?BLNMAIKMX Gv`LKnV P w?aG/UBLNW]?BbWxIKGCDQA cIKlGCDFE d Lb P ACDQALNM GHcD_GHcDQLNECDFGH?BbIKUdfL P MXIKGH?BLNMA3LKdkGHcD@D9lmDFEH?a0DFMG7? D]GHcDSd P MbFGH?BLNM LKdI P UGHE^IKlA9GHcD_0DQb^cIKM?BbQAZLKd IUa?aMDIKE" ?BLNM bFEH9ACG^IKU IKMXYGHcD_?aMGCDFE^INbFGH?BLNM LKd ?BLNMA`3?aGHc Ua?aW]cGZ-lmDnbF?VIKU IKGHGCDFMGH?BLNM?BA3lIK?BX GCL 40 I + 7`3c?Bb^c?BA3GHcD[?BLNM P ACDQX ?aM L P ESD9lmDFEH?a0DFMGCAIKMX GCLGHcD DRY DQbFGZLKdMLNM7n ECDQACLNMIKMG_?aMGCDFE^INbFGH?BLNMA cIKlGCDFEL=0XDQACbFEH?awmDQAScL ` P IKMG P bQLN l P G^IKGH?BLNMbIKM wmD lmDFEpdfLNEH0DQX?aM?BLNMGHE^IKlA cIKlGCDFE '?BAbQLNMbQDFEHMDQX`3?aGHcGHcD r]IKEH?BL P ADFUBDF0DFMGCA0LKdSGHcDYD7lmDFEH?a0DFMG^IKUZACDFG P lk c?BA b^cIKlGCDFE ?aMbFU P XDQAIXDQACbFEH?alGH?BLNM LKd_GHcD XDQAH?aW]MlECLbQDQA^A LKd_GHcD b P EHECDFMG GHE^IKlwINACDQXLNMGHcDD7lmDEC?aDFMbQDQAINXD0`3?aGHc lECDFr?BL P A/GHE^IKl%bQLNMAHGHE P bFGH?BLNMA cIKlGCDFE @ XDIKUBA/`3?aGHc GHcD GCDQb^cM?BbIKUa?aGH?BDALKdLNlmDFE^IKGH?aMW GHcDGHE^IKlk/uvMGHc?BA bQLNMGCD7G I MDF`-GHE^IKl,UBLsINX7?aMW ACb^cDF0DwlcLNGCLN?BLNM? O IKGH?BLNM!k`3c?Bb^ccINA@AC?aW]M?&%bIKMGHUa?a lECL rxDQXLNMGHcDGHE^IKllmDFEpdfLNEHIKMbQD ?BA_lECDQACDFMGCDQX [uvMb^cIKlGCDFE H D7lmDFEH?a0DFMG^IKUINb^c?BDFrxDF0DM9GCA[A P b^c,INAS0LNGH?BLNMIKU W]ECL P MXAHG^IKGCDbQLLNUa?aMW0mLNlGH?BbIKUTINXX7ECDQACAC?tMW IKMXUVINACDFE_lcINACDbQLNM7nGHECLNU `3c?Bb^c ECDFlECDQACDFMGZlECDFECD/ P ?BAC?aGCDQA d LNE3L P E3?BLNM GHE^IKl^ P IKMG P bQLN l P GCDFE_IKECD X7?BACb P A^ACDQX cIKlGCDFE(,X7?BACb P A^ACDQA GHcD I HLNE0ECDQA P UaGCA0LKd[GHc?BA GHcDQAC?BA ? D]TGHcD ?t lUaDF0DFMGJIKGH?BLNMLKd P IKMG P WxIKGCDQA IKMXIKUaWsLNEH?aGHc0A/g ?aMIKUaUa b^cIKlGCDFEC> A P IKEH? O DQA GHcD`LNEH\lECDQACDFMGCDQXcDFECD IKMXW]?arxDQA IKML P GHUBLLN\LNMd P G P ECD[WsLsIKUBA
=
"( "( $ %,&
y CZC pz cD 0L]AHGZWsDFMDFE^IKU ACG^IKGCD LKd I. P w?aG3E^DQINXA
|ψ〉 = eiγ(cos
α
2|0〉+ e−iβ sin
α
2|1〉).
&%J@LNGCD GHcIKG/MLNEHIKUa? O IKGH?BL]M%G^LT%0?BAIKUaECDINX7?aMbFU P XDQX ?aM GHc?BA/ECDFlECDQACDFMG^IKGH?BLNMk uvM%I l P ECDACl?aM7n %#" AH7AHGCDF,hGHcDL rsDFE^IKUaUlcINACD
γcINA@MLYLNwACDFEHr]IKwUBDDRY DQbFG IKMX,bIKMGHc P A@wmD?aW]MLNECDQX LNMA^D/ P DFMGHUaGHcD AHG^IKGCD rsDQbFGCLNESbIKM wmD ECDF`3EH?aGHGCDFMINA|ψ〉 =
(cos
α
2|0〉+ e−iβ sin
α
2|1〉).
) ACLN0DFGH?a0DQA P ACDd P U r?BA P IKUa? O IKGC?aLNMLKdZGHcD P w?aGAHG^IKGCDLKdSD/MkZ ?BA/GHcD UBLb^c%rsDQbFGCLNE^ cDUVIKGHGCDFE?BAXDR%MDQX%w%IlmLN?aMGLNMGHcD P M?tGAClcDFECDGHcDYUBLb^cAHlcDFECDTbQLNEHE^DFAClmLNMX7?aMW,GCLGHcD >IKMW]UBDQA
αIKMX
βINAZAHcL `ZM ?aMg ?aW &%]
) AHG^IKGCD |ψ〉 = a|1〉+ b|0〉 `3?aGHcbQLN lUBDJ bQLDbF?aDFM9GCA a IKMX b LKdI. P w?aGZbIKMIKUBACLwmD[`3EH?aGHG^DMINASIrsDQbFGCLNE[ψ] =
[ab
].
P IKMG P LNlmDFE^IKGH?BLNMA ?aM,GHc?BA AH?aMW]UBD. P w?aG AHlINbQD IKECDECDFlECDQACDFMGCDQXwIKGHEH?BbQDQAdfLNE DIK lUBDGHcD P IKMG P `3c?Bb^cN?alAZGHcD P w?aGSAHG^IKGCD
NOT = X =
[0 11 0
].
cD _n IKGHEH?t0?BA ?aM0d INbFG LNMD_LKd GHcECDQD3d P MXIK0DFMG^IKUhIKGHEH?BbQDQA `3c?Bb^c IKECDZlIKEHGH?Bb P UVIKEHUa P ACDd P U?aMGHcDXDQACbFEH?alGH?BLNMLKd IGv`LKn UBDFrxDFU AH7AHGCDF,7GHcD LNGHcDFEZGv`L wmDF?aMWY =
[0 −ii 0
]and Z =
[1 00 −1
].
cDQACDSIKECD GHcD GHcE^DFDZI P Ua?IKGHEH?BbQDQA `3c?Bb^c xL P GCAH?BXD+ P IKMG P *?aM7dfLNEHIKGH?BLNM 9IKECD3MLNEHIKUaUaXDFMLNGCDQXσx = X, σy = Y
IKMXσz = Z.
cDFECDZIKECDGv`L@?a lmLNEHG^IKMGTGHcDFLNE^D0A bQLNMbQDFEHM?aMW P M?aG^IKEHGHE^IKMApd LNEHIKGH?BLNMAT?aMGHcDGv`LKn UBDFrxDFU _?aUawmDFEHGAHlINbQD&% % ) M47E RO:8:EF47E P M?aG^IKEH,LNlmDFE^IKGH?BLNM U ?aM,GHcD AH?aMW]UBDX P w?aG AHlINbQDbIKM 47G 4 @?wmDD7lECDQACACDQX INA I
'
|1
0
>
>|
α
β
ψ
g ?aW P ECD &% UBLb^cAClcDFECD[E^DlECDQACDFMG^IKGH?BLNMLKd I. P w?aGZACG^IKGCD |ψ〉
ECLNG^IKGH?BLNM%IKwmL P G/IYbQDFEHG^IK?aM%Ie7?BA n ?aMGHcDUBLb^c AHlcDFECDw,IYbQDFEHG^IK?aM%IKMW]UBD θ 0LX P UBL IYlmL]ACAH?awUBDW]UBLNwIKUklcINA^Dκ
U = eiκRn(θ). [?trxDFM,Gv`L MLNM7nobQLNUa?aMDIKEIeDQA n1 IKMX n2 kIKM47E RO:8:EF47E ECLNG^IKGH?BLNM bIKM IKUa` I7A[wmD XDQbQLN lmL]ACDQX?aMGCLIA^DFEH?BDFASLKd ECLNG^IKGC?aLNMA_IKwmL P G n1 IKMX n2 uvM4lc9AH?BbIKU@ECDIKUa? O IKGH?BLNMA LKd( P w?aGCA3GHcDGv`L%UBDFrsDFUBAYGvl?BbIKUaUa cIrsDX7?&Y DE^DM9G DFMDFEHW]?BDQA IKMXAC?aM7nW]UBD P w?aGYLNlmDFE^IKGH?BLNMA IKECDECDQACLNMIKMG D7bF?aG^IKGH?BLNMAYwmDFGv`DQDFMGHcDFA^DDFMDFEHW]UBDFrsDFUBA cDD7bF?aG^IKGH?BLNMl P UBACD0?BA Gv9l?BbIKUaUaACLN0DDFUBDQbFGHECLNIKW]MDFGH?Vb` I rsD] cDvAHG^IKGH?Bb$ P w?aG/ECDFlECDQACDFMG^IKGH?BLNM dfECLNIKwmL rsDfACDQDd LNE[DIK lUBDD/Mk# F_?aA LNwG^IK?aMDQX,wI GHE^IKMAHdfLNEHIKGH?BLNM ?tM9GCLYGHcDECLNG^IKGH?aMW dfE^IK0D]/uvM?BLNMGHE^IKlD9lmDFEH?a0DFMGCA P w?tG^A@IKECD MLNEHIKUaUaDFMbQLXDQX?aMGCLGv`L0UBLNMWNn Ua?arxDQX?aMGCDFEHMIKU UBDFrxDUBA@LKd DINb^c?BLNMk E^IKMAH?aGH?BLNMASwmDFGv`DQDFM GHcDFA^D DFMDFEHW] UBDFrsDFUBA_IKECD LNlGH?BbIKUaUa X7EH?arsDFM `3?aGHc UVINACDFEZE^INX7?VIKGH?BLNMkuvMXDFlmDFMXDFMGLKd GHcDZlECDQbF?BACDZlc7AH?BbIKU P w?aG ECDIKUa? O IKGH?BLNM 9GHcD P M?aG^IKEH GHE^IKMApd LNEHIKGH?BLNMYXDQACbFEH?aw?aMWGHcD INbFGH?BLNMLKd[Il P UBACD LKd[IKMDFUBDQbFGHECLNIKW]MDFGH?Bb ` IrxDYECDQACLNMIKMG GCLGHcDN P w?aGGHE^IKMAH?aGH?BLNM fD] WTACDQDACDQbFGH?BLNM ECDINXA
R(θ, φ) = eiθ2(eiφσ++e−iφσ−),
`3cDFECD
σ+ = (σx + iσy)/2 =
[0 10 0
],IKMX
σ− = (σx − iσy)/2 =
[0 01 0
]
IKECD GHcD E^IK?BAH?aMW IKMX UBL `DFEH?aMW LNlmDFE^IKGCLNECAECDQAClmDQbFGH?arsDFUaxR(θ, φ)
bIKM wmD `3EH?aGHGCDFMINAR(θ, φ) =
[cos θ2 ieiφ sin θ
2
ie−iφ sin θ2 cos θ2
].
uoGSG P EHMA_L P G_GHcIKG_A P b^cI ECDQACLNMIKMG@DbF?tGJIKGH?BLNM R(θ, φ) bQLNEHECDQAHlmLNMXA_GCL I ECLNG^IKGH?BLNM@LKd GHcD/UBLb^crxDQbFGCLNE3IKwmL P GSIKM Ie7?BA nc `3c?Bb^c Ua?BDQA ?aMYGHcD x n y lUVIKMDACDQD@g ?aW W cD@lIKE^IK0DFGCDFE φ XDR%MDQA GHcD! #"%$&$&('*)(+-,.)(%'/ 0)1"** .$&+2*(34657+8$.95:;+83<)=>$&*?A@*">>*+CBD)(734!$&"**+E;+C"3E@;+%F
@
|1
0
>
>|
φ
g ?aW P ECD ) ECDQACLNMIKMG fbIKEHEH?BDFE" DbF?tGJIKGH?BLNM R(θ, φ) bQLNEHECDQAHlmLNMXATGCL ISECLNGJIKGH?BLNM0LKdGHcD P w?aGTAHG^IKGCDIKwmL P G3IKM Ie9?BA nc `3c?Bb^c Ua?aDQA?aM GHcD x n y lUVIKMD 9GHcD[IKMW]UBD φ XDR%MDQAGHcD_lmL]AH?aGH?BLNM LKd nc`3?aGHc?aM GHchIKGZlUVIKMD]
lmL]AH?aGH?BLNM LKd GHc?BA Ie7?BA `3?aGHc?aMGHcDxnylUVIKMD_INA ACcL `3M0?aM g ?aW TR9lmDFEH?a0DFMG^IKUaUB φ ?aA XDR%MDQX0wGHcDSlcINACD_LKdkGHcDSDbF?aGH?aMWE^INX7?VIKGH?BLNMk R7lmDFEH?a0DFMG^IKU bQLNMGHECLNULKd
φ`3?aUaUhwmD_X7?BACb P ACA^DQX0?aMYACDQbFGH?BLNM '7
cD lIKE^IK0DFGCDFEθECDFlECDQACDFMGCAZGHcD[l P UaA^D AHGHECDFMW]GHc XDR%MDQXINAZ_IKw?dfECD/ P DFMbFYGH?a0DQA3GCcD X P E^IKGH?BLNMLKd GHcDl P UBACDNfbdpACDQbFGH?BLNM /uvMGHcDUBLb^cAHlcDFECDl?BbFG P ECD θ ?BA@GHcD IKMW]UBDw`3c?Bb^c I W]?arsDFMAHG^IKGCD rsDQbFGCLNEZWsDFGCAZECLNG^IKGCDQX IKwmL P G nc cD0lcINA^DQA
φ = πIKMX
φ = π/2bQLNEHECDQAHlmLNMX GCL ECLNG^IKGH?BLNMAIKwmL P G/GCcD x nvIe9?BAIKMX GHcD y nvIe7?BAECDQAHlmDQbFGH?arxDFUasW cD bQLNEHECDQAClmLNMX7?aMWIKGHEH?BbQDQASECDINX
Rx(θ) := R(θ, π) = e−iθ2X =
[cos θ2 −i sin θ
2
−i sin θ2 cos θ2
] = IKMX
Ry(θ) := R(θ, π/2) = e−iθ2Y =
[cos θ2 − sin θ
2
sin θ2 cos θ2
].
?') ECLNGJIKGH?BLNMIKwmL P GSGHcD z nvIe7?BA3cINA3GHcD IKGHEH?t dfLNEH
Rz(θ) = e−iθ2Z =
[e−i
θ2 0
0 eiθ2
].
?@jkL `DFE?aMX7?BbQDQA
x, y, z INA LNllmL]ACDQXGCL
x, y, zkXDFMLNGCDGHcD MDFWxIKGH?arxD Ie7DQA fLNE]D/ P ?ar]IKUBDFMGHUaNECLNG^IKGH?BLNMA?aM GHcD LNllmL]AC?aGCD ACDFMACD
) bQbQLNECX7?aMWGCL,GHcD IKwmL rxD AHG^IKGCDF0DFMG E^DWxIKE^X7?aMW GHcD XDQbQLN lmL]AH?aGH?BLNM4LKd[IKM IKEHw?aGHE^IKEHECLNG^IKGH?BLNM IKM P M?aG^IKEHAH?aMW]UBDX P w?aG[GHE^IKMApd LNEHIKGH?BLNM bQIKMwmD?a lUBDF0DFMGCDQX INA IYA^DEC?aDQA LKd Rx(θ)IKMX
Ry(θ)LNlmDFE^IKGH?BLNMA IKEHEC?aDFESUVINA^DFEZl P UBACDQAZ`3?aGHc GHcD/IKllECLNlEH?VIKGCD lchINACDQAZGHc P AZECDFlECDQACDFMG@I,B C; < G A38VAAH?aMW]UBD P w?aG3GCLLNUawmL RIK lUBD I π/2 n ECLNG^IKGH?BLNMIKwmL P GSGHcD z nvIe7?BA
Rz(π/2) =
[e−i
π4 0
0 eiπ4
],
H
bIKM wmD[ECDIKUa? O DQXINARz(π/2) = Ry(π/2) Rx(π/2) Ry(π/2).J@LNGCDGHcIKG GHcECL P W]cL P G GHc?BA GHcDQAH?BA bQLNMbIKGCDFMIKGCDX LNlmDFE^IKGH?BLNMAIKE^D`3EH?aGHG^DM XL `3M ?aM,GHcD ACG^IKMXIKECXMLNG^IKGH?BLNM LKd IKGHEH?t P UaGH?alUa?BbIKGH?BLNM k? D]b^cECLNMLNUBLNW]?BbINUaUa,ACD/ P DFMbQDQA[LKdLNlmDFE^IKGH?BLNMA Ua?a\xDGHcD IKwmL rsD P AHG wmD ECDINX/dECLNEH?aW]cG GCL_UBDdGL cDLNlmDFE^IKGH?BLNMA X IKMX
YbQLNEHECDQAHlmLNMXGCL −iRx(π)
IKMX −iRy(π)ECDQAHlmDQbFGH?arxDFUas
pzh Hz | m zmCZ LNMAH?BXDFE0GHcD bQLN lmL]AH?aGCD AC9AHGCDF LKd
n P w?aGCAuvGcINA wmDQDFMlECL rxDFMGHcIKG IKM IKEHw?aGHE^IKEH P M?aG^IKEHLNlmDFE^IKGCLNE ?aM4GHcDbQLNw?aMDQX
2nX7?a0DFMAC?aLNMhIKU _?aUawmDFEHG AHlINbQDbIKMwmDDJ7lECDQACA^DFX INA I lECLX P bFG LKd
<547OE #O?/A IKMXAH?aMW]UBD P w?aGLNlmDFE^IKGC?aLNMA&% H '.= ' H o c?BAECD/ P ?aECDQAGHcD IrNIK?aUVIKw?aUa?aGvLKd(2 60OP7ARE"?347GGv`LKnV P w?aGWxIKGCDQALNM`3c?Bb^cGHcD E^DFACGLKdSGHc?BAA^DQbFGH?BLNM`3?aUaU3bQLNMbQDFMGHE^IKGCD] ) A^DGLKd@LNlmDFE^IKGCLNECA?aMIGv`LKnV P w?aG_bQLN l P G^IKGH?BLNMIKUAHlINbQD ?BASbIKUaUaDQX P M?arsDFECAJIKUk?td IKM IKEHw?aGHE^IKEH LNlmDFE^IKGCLNE_?aM GHcIKG_AClINbQD bIKMwmD[D7lECDQACACDQX INAZIlECLX P bFGSLKd GHcDF uoGd LNUaUBL `ZAdE^LN GHcD IKwmL rxD@GHcIKGSI P M?arxDFECA^IKU @Gv`LKnV P w?aGSACDFG?BA IKUBACL P M?arsDFECAJIKUdfLNE GHcD_`3cLNUBD n nV P w?aG AHlINbQD]#J_LNGCDcL `DFrxDFEGHcIKGGHc?BAAHG^IKGCDF0DFMG AJI9A MLNGHc?aMWIKwmL P GDRY bF?aDFMbF[x? D]xcL `4IKM,lmLNUa9MLN ?VIKUaUaLNE D7lmLNMDFMGH?VIKUaUa0IKM0WxIKGCDQA P ACG wmDSbQLN lmL]ACDQX?aMLNECXDFEZGCL bFECDIKGCDIW]?arxDFM P M?tGJIKEHYGHE^IKMAHdfLNEH ) M?a lmLNEHG^IKMG/Gv`LKnV P w?aG WxIKGCD ?BA GHcD0bQLNMGHECLNUaUBDQX9n MLNGWxIKGCD, CNOT X cD CNOT LNlmDFE^IKGH?BLNM
?alALNMD@LKdkGHcDSGv`L P w?aGCA GHcD 8 47E A38 P w?aG"T?tdkGHcD@LNGHcDFELNMD GHcD.B C768:EFC7G P w?aG"T?BA ?aMYAHG^IKGCD |1〉IKMX UBDIrxDQAGHcD_G^IKEHWsDFG P w?aG P Mb^cIKMWsDQX ?td GHcD@bQLNMGHECLNU P w?aG?BA?aM |0〉 ) M DIK lUBD_dfLNE3I CNOT?BA3W]?arxDFM wYGHcD IKGHEH?t
CNOT =
1 0 0 00 1 0 00 0 0 10 0 1 0
,
`3cDFECDSGHcDZMLNG^IKGH?BLNMYLKd GHcDZIKGHEH?t P AHG Ua?t\xD_IKUaUdfLNUaUBL `3?aMWGv`LKnV P w?aG IKGHEH?BbQDQARECDd DFECA GCL GHcDSwINAH?BALNECXDFE |0, 0〉, |1, 0〉, |0, 1〉, |1, 1〉 TuvMYbINACD@LKd GHc?BA CNOT 9GHcD@ACDQbQLNMX P w?aGEH?aW]cG ?aMYGHcD_wE^Ien \xDFGMLNG^IKGH?BLNMZbQLNM9GHECLNUBASGHcD noLNlmDFE^IKGH?BLNMLNMGHcD%ECAHG uoG_bIKMwmD/AHcL `3MGHcIKG_GHcDCNOT
WxIKGCD ?aMbQLNw?aMIKGH?BLNM `3?aGHcYAH?aMW]UBDS P w?aGLNlmDFE^IKGH?BLNMAd LNEH0AI P M?arxDE^A^IKUmACDFG LKd LNlmDFE^IKGCLNECA uoM GHc?BA A^DFMACDZGHcDCNOT
?BASbQIKUaUBDQXI 2 60OQP@ARE"?347G'8 CI132 RO:8 48VAF cD
CNOT?aGCACDFUtdbIKM wmD XDQbQLN lmL]ACDQXINA
CNOT = H1 Φ H1 H
`3cDFECDHXDFMLNGCDQA [INXIKIKECX WxIKGCDQA3`3c?ab^cIKECD AH?aMW]UBD P w?aGSLNlmDFE^IKGH?BLNMAW]?arxDM w
H =1√2
[1 11 −1
].
cDUBL `DFE[?aMXD %XDFMLNGCDQA[GHcIKG@GCcDIKGHEH?t?BA[D7GCDFMXDQXGCL GHcDbQLN lmL]AH?aGCDAHlINbQD IKMXGHcIKG[GHcDWxIKGCD/INbFGCASLNM GHcD WE"?R8 P w?aG uoM GHcD nV P w?aG _?tUawmDFEHG_AHlINbQD[?aGZGHc P AZECDINXA
H1 =1√2
1 1 0 01 −1 0 00 0 1 10 0 1 −1
.
(
.
Φ?BASI<> 4@?/A48VA ECDINX7?aMW
Φ =
1 0 0 00 1 0 00 0 1 00 0 0 −1
.
?(
LNMACD/ P DFMGHUa ΦfACLN0DFGH?a0DQA@bIKUaUBDQXI0bQLNMGHECLNUaUBDQX9n
ZWxIKGCDSIKUBACL0ECDFlECDQACDFMGCA[I P M?arsDFECA^IKU Gv`LKnV P w?aGWxIKGCD]uvG AHcL P UBX0wmDZ0DFMGH?BLNMDQX GHcIKG GHcD [INXIKIKECXWxIKGCD9IKUaGHcL P W]cdfECD/ P DFMGHUa P ACDQX0?aMGHcDZUa?aGCDFE^IKG P ECD?BA MLNG lIKEHGH?Bb P UVIKEHUa`DFUaUA P ?aGCDQX/d LNETISlc9AC?BbIKU9?a lUBDF0DFMG^IKGH?BLNM uvG cINA wmDQDFM AHG^IKGCDQX IKwmL rsD GHchIKGTIKMP M?aG^IKEHLNlmDFE^IKGH?BLNM%bIKM wmD D7lECDQACACDQX INA/IYECLNG^IKGH?BLNM U = eiκRn(θ)
uvM bQINA^D LKdHkGHcD ECLNG^IKGH?BLNMIe7?BA
n?BAGHcD X7?VIKWsLNMIKU3LKd_GHcD
xnzn lUVIKMD MIK0DFUa
(x, y, z) = 1/√2(1, 0,−1) IKMXGHcD ECLNGJIKGH?BLNMIKMW]UBD
θ?BAπ LNMACD/ P DFMGHUa[?aGXLDQAMLNGbQLNEHECDQAHlmLNMXYGCLI ECDQACLNMIKMG D7bF?aG^IKGH?BLNM LKdkGHcD P w?aG`3c?Bb^c?BASIKUa` I9ASIECLNG^IKGH?BLNMIKwmL P G_IKMIe7?BA Ua9?aMW ?aM GHcD x n y n lUBIKMD fACDQD IKwmL rsD H bIKM GHcDFECDd LNECD MLNG3wmD?a lUBDF0DFMGCDQXw IAH?aMW]UBD ECDQACLNMIKMG(UVINACDFE"l P UBACD]@L `DFrsDFE?aGZbIKM wmD XDQbQLN lmL]ACDQX?tM9GCL0IA^DFEH?BDQALKd
xnyn ECLNG^IKGH?BLNMAd LNEZDIK lUBD
H = i ·Rx(π) Ry(π/2),LNE
H = −i ·Ry(π/2) Rx(π),`3c?Bb^cbQLNEHECDQAHlmLNMXGCL,IGCLNG^IKU3LKdSGHcECDQDYECLNG^IKGH?BLNMAwθ = π/2
J_LNGCD GCcIKG?tdSGHcDYIKGHEH?t%?BADnGCDFMXDQX GCLI P UaGH?tnV P w?aGSAHlINbQD GHcD[lcINACD@d INbFGCLNEZLKd i ECDFIK?aMAZW]UBLNwIKU hIKMX ?BA3GHc P ASLKd MLlc7AH?BbIKU?a lmLNEHG^IKMbQD]uvM,LNGHcDE `LNECXAkI [INXIKIKECX,WxIKGCD0INbGC?tMW LNM P w?aG M P wmDFE %?aM I S?aUawmDFEHG AHlINbQDbQLN lmL]ACDQXLKdTIKEHw?aGHE^IKEH?aUa IKM P w?aGCAZbIKM wmD `3EH?aGHGCDFMINAH1 = −i ·Ry1(π/2) Rx1(π).gLNEHG P MIKGCDFUa[TGHcECDQD π/2 n l P UBACDQA IKECD MLNGMLNEHIKUaUa MDQDQXDQXGCL?a lUBDF0DFMGI@INXIKIKECXWxIKGCD IKl7nlmDIKEH?aMW?aM ACLN0D3W]?arxDFMYIKUaWsLNEH?aGHc 9GHcDSL rsDFE^IKUaUhA^D/ P DFMbQDSbIKMGv9l?BbIKUaUawmDSbQLN l?aUBDQX?aMGCLI AHcLNEHGCDFELNMD] gLNEZDIK lUBD
ΦbIKM wmD GHE^IKMApd LNEH0DQX ?aMGCLI
CNOTIKUBACL w GHcD ACD/ P DFMbQD
CNOT = Ry1(π/2) Φ Ry1(π/2).
c P AGHcD'@INXIKIKECX WxIKGCDQASLKdD/Mk H 3bIKM wmD[ECDFlUVINbQDQX w A P ?aG^IKwUBD AH?aMW]UBD π/2 UVINACDFEZl P UBACDQAJ@LNGCDGHcIKGΦ?aGCACDFUtd?BA AH9 0DFGHEH?Bb`3?aGHc ECDQAHlmDQbFG GCL IKM Db^cIKMWsD LKdGHcDX P w?aGCA$\_MUaGHcD A P EpnECL P MX7?aMW fAH?aMW]UBD P w?aG"l P UBACDQAZAHlmDQbF?tdf `3c?ab^cLNMD ?BA3GHcD G^IKEHWsDFG P w?aGcDFECD M P wmDFE %
IKEH?BL P A P M?arsDFECA^IKU lcINACD@WxIKGCDQA3D7?BAHGM3cD −1 LKd GHcD@WxIKGCD Φ bIKM d LNE DIK lUBD7wmD IKM`3cDFECDLNM GHcD X7?VIKWsLNMIKU IKMX GHcD[WxIKGCD `3?aUaUkAHGH?aUaUkACDFEHrsD DQACACDFMGH?VIKUaUa GHcD AJIK0D[l P EHlmL]ACD u d GHcD[WxIKG^D[?BA3A^IKMX9n`3?Bb^cDQX wmDFG `DQDFM Gv`L[INXIKIKECX WxIKGCDQA_INA3?aMD/xMk! H RGHcDFM rNIKEH9?aMW GHcD[lmL]AH?aGH?BLNM?aMW LKdGHcD −1LNMGHcD0X7?VIKWsLNMIKU LKd GHcDlcINACD WxIKGCD`3?aUaU IKG 0L]AHGk?aMrsDFEHG GHcDECLNUBD0LKd |0〉 IKMX |1〉 LKdGCcD bQLNMGHECLNU P w?aG7GHc P A3ECDQA P UaGH?aMW?aMI O DFECLKnobQLNMGHECLNUaUBDQX9noMLNG
0− CNOT =
0 1 0 01 0 0 00 0 1 00 0 0 1
.
0 − CNOT IKMXCNOT
bIKMIKUa` I7A_wmDGHE^IKMApd LNEH0DQX?aMGCLYDINb^c,LNGHcDFE[w Gv`L LKlmDFE^IKGC?aLNMALNM GHcD bFLNM9GHECLNU! P w?aG0− CNOT = NOT2 CNOT NOT2.
>
) MLNGHcDFE lcINACD WxIKGCD3`3c?Bb^c0`D3cIrsDSINbFG P IKUaUa?a lUBDF0DFMGCDQX D9lmDFEH?a0DFMG^IKUaUa^ Φ(τ0) LKd ACDQbFGH?BLNM (7&%ECDINXA
Φ∗ =
1 0 0 00 1 0 00 0 i 00 0 0 −i
.
c?BA bFUBDIKEHUaXDFr9?VIKGCDQAdfECLN GHcD vbFUVINACAC?BbIKU lcINACD3WxIKGCDΦw P GT?aGT?BA AHGH?aUaUI P M?arsDFECAJIKUGv`LKnV P w?aG WxIKGCD
cD D/ P IKGH?BLNMΦ = −i ·Rz2(π/2) Φ
∗,`3c?Bb^cY?BADINAH GCL/rxDFEH?td[AHcL `ZA GHcIKGΦ∗ bIKM wmD@X7?aECDQbFGHUa0GHE^IKMAHdfLNEH0DQX ?aMGCL Φ w9I
π/2ECLNG^IKGH?BLNM LKd
P w?aG IKECL P MX GHcD z nvIe9?BA# c?BA lECL rxDQA GHcD P M?arxDFECA^IKUa?aGvYLKd Φ∗ INA ?aG3bIKM wmD@GHE^IKMAHdfLNEH0DQX ?aMGCL I`DFUaU \9ML `3M P M?arxDFECA^IKUkGv`LKnV P w?aGZWxIKGCD P AH?aMW0LNMUa IAH?aMW]UBD P w?aG3ECLNG^IKGH?BLNMk cD Φ∗ WxIKGCD bIKMIKUBACLwmD GHE^IKMApdfLNEC0DFX ?aMGCLI P M?arxDFECA^IKUkWxIKGCD C AH?a ?aUVIKEZGCL GHcD `DFUaU \9ML `3M P AH?aMW0GHcD ACD/ P DFMbQD
C = Rx1(π/2) Φ∗ Rx1(π/2) =
1 0 0 00 1 0 00 0 0 −10 0 1 0
.
uvG@?BA[?a lmLNEHG^IKMG GCL MLNGCDGHchIKG `3c?aUBDX P IKMG P IKUaWsLNEH?aGHc0AIKECD LKdGCDFM`3EH?aGHGCDFM XL `3M P AH?aMWvAHG^IKM7nXIKECX /WxIKGCDQA@Ua?a\xD LGHcDFbIKMIKUa` I7A_wmDECDIKEHE^IKMWsDQXGCL ?aMAHGCDINX?aMbQLNEHlmLNE^IKGCD, dfLNE[DIK lUBDCLNEC ′
g ?aMIKUaUa IKM?tM9GHEH?aW P ?aMW,d DIKG P ECD LKd IXEH?arsDFMG `LKn UBDFrsDFUZAH7AHGCDF,TMIK0DFUa%?aGCA 4π lmDFEH?BLX7?BbF?aGv[ AHcIKUaUwmDX7?BACb P ACACDQX ) UBLLN\IKG@D/ P IKGH?BLNMA. 3GHcECL P W]c ?'SAHcL `ZA_GHcIKG_?aGS?BA_GHcDIKMW]UBD θ2
?aMAHGCDINXLKdθ`3c?Bb^c IKllmDIKECA@?aMGHcD/d P MbFGH?BLNMA LNMACD/ P DFMGHUaGHcDF?aE@GHE P DlmDFEH?BLX?BA 4θ = 4π
@SLNG^IKGH?BLNMA@wθ = 2π
wEC?tMWYIKM AHG^IKGCD rsDQbFGCLNE_wINb^\ GCL0?aGCACDFUtd LNMGHcD/UBLb^cAHlcDFECDw P G[IlcINA^D d INbFGCLNE@LKd −1 ?BAINb/ P ?aECDQX R(2π, φ) = −I 7`3cDFECD I ?BA3GHcD ?BXDFMGH?aGv LNlmDFE^IKGCLNE c?BA −1 LNMYI%ECAHG r9?BDF`?BA I W]UBLNwIKUlcINACD3d INbFGCLNEIKMX0?aG cINA wmDQDFMYIKEHW P DQX IKwmL rxD(fbdpsD/Mk9 &%FGHcIKG?aG bIKM wmD@X7ECLNllmDQX W c?BA?BA?aMYWsDFMDFE^IKU LNMUaGHE P D_?td GHcD_Gv`LbQL P lUBDQX AHG^IKGCDQA d LNEH I l P E^D[IKMXACDFlIKE^IKGCDGv`LKn UBDFrsDFU AH7AHGCDF,mGHcD S?aUawmDFEHG AHlINbQDLKd `3c?Bb^c`3?aUaUwmDXDFMLNGCDQX,w H1 u d GHc?BA AH7AHGCDF?BA@bQLNw?aMDQX,`3?aGHc,LNGHcDFECA H2,H3, . . . `3c?Bb^c,bQL P UBXwmDGv`LKn UBDFrsDFUTAH7AHGCDF0A@INA@`DFUaU LNE IKUBACLUVIKEHWsDFEAH7AHGCDF0A" GHcD bQLN lmL]AH?aGCD AC9AHGCDF ?BAZX7?BA^bEC?twmDQX wYGHcD+_?aUawmDFEHGSAHlINbQDH = H1 ⊗H2 ⊗H3 ⊗ . . .
uvG ?BA bFUBDIKEGHcIKG GHcD −1 lcINACD_d INbFGCLNE ?tM GHc?BA bINACD_?BAW]UaLNwhIKUm?aM H1 IKMX IKUBA^L?tM IKM IKEHw?aGHE^IKEH?aUa UVIKEHWsDD7GCDFMAH?BLNM H uod NcL `DFrxDFE H1 bQLNMG^IK?aMA 0LNECD GHcIKM G `L@UBDFrxDFUBATLNET?tdmGHE^IKMAH?aGH?BLNMA IKECD X7EH?arxDFM0ACDFUBDQbFGH?arxDFUawmDFGv`DQDFMbQDFEHG^IK?aM UBDFrsDFU lIK?aECA/LKd GHcD0bQLNw?aMDQX AH7AHGCDF H kGHcDFM GHcDINb/ P ?aECDQX lcINACDQAIKECD ?aMWsDFMDFE^IKU MLNGW]UBLNwIKUkIKM0LNECD]Z- P b^c MLNM7n W]UBLNwIKU lcINACDQAZIKECD@INXD P ACD@LKd dfLNEGHcD@?a lUBDF0DFMG^IKGH?BL]M LKdL P IKMG P UBLNW]?Bb `3?aGHc GHE^IKllmDQX ?BLNMAACDQD ACDQbFGH?BLNM = WJ_LNGCD[GHcIKGZGHcD UBLb^cAHlcDFECD[l?BbFG P ECD/XLDFAZMLNGZX7?aECDQbFGHUaX7?BAHlUVIYGHcD Gv`LKn UBDFrxDFUkW]UBLNwIKUklchINACD]
%
= 4 :;0(# 6 (' = 0. 4 8 9) 64)
CT Z 3|) bQbQLNECX7?aMWGCL/GHcD_j IKlUVINbQD@D/ P IKGH?BLNM ∆Φ = 0
bQLNM %MDF0DFMG3LKd I/b^cIKEHWsDQXlIKEHGH?BbFUBD_?aM GHcECDQD_X7?a0DFM7nAH?BLNMA3`3?aGHc ?R8 48:O B DFUBDQbFGHEH?Bb%DFUBXASIKUBLNMD ?BA3MLNGZlmL]ACAH?awUBD]W IK\xD@dfLNEZDIK lUBD/IKM DFUBDQbFGHEH?Bb P INX7E P lmLNUBD%DFUBX
E = −∇Φ `3c?Bb^cLNEH?aW]?aMIKGCDQAZdECLNI. P INX7E^IKGH?Bb lmLNGCDFMGH?VIKUΦ = αxx
2 + αyy2 + αzz
2.
) llUa?aMWGHcDj IKlUVINbQD D/ P IKGH?BLNM%9?BDUBXA αx + αy + αz = 0 LNMACD/ P DFMGCUt[IKG/UBDINAHGLNMDLKd3GHcDGHcECDQD3bQLDbF?BDFMGCA P AHGTwmDMDFWxIKGH?arxD `3c?Bb^c0bQLNEHECDQAHlmLNMXA GCL I@ECDFl P UBAH?arsD lmLNGCDFMGH?VIKU?aMGHc?BA X7?aECDQbFGH?BLNMkI P UGHE^IKlA IKMXINACA %UaGCDFECA@IK\xD P A^DLKd GH?a0DnoXDFlmDMXDFMGkL]ACbF?aUaUVIKGH?aMW %DFUBXA@GCL bQLNM %MDb^cIKEHWsDQXlIKEHGH?BbFUBDQAS?aMIKUaUkGHcECDQD X7?a0DFMAH?BLNMA I P UINACAM%UaGCDFECA IKECD wINACDQX LNM0Gv`L X7?a0DFMAH?BLNMIKU P INX7E P lmLNUBD%DFUBXAs? D] αz = 0
M c?BATlmLNGCDFMGH?VIKUAHcIKlmDbIKMwmDD7lmDFEH?a0DFMG^IKUaUaECDIKUa? O DQXwIKMDFUBDQbFGHECLXD AHGHE P bFG P ECD INA@XDFl?BbFGCDQX?aM,g ?aW &%] cDj IKlUVINbQD/D/7IKGH?BLNM ?aM GHc?BAZbINACD W]?arsDQAαx = −αy
`3c?Bb^c 0DIKMAZGHcIKGΦbIKM wmD `3EH?aGHGCDFMINA
Φs(r) = Vx2 − y22r20
, &%
`3cDFECDr = (x, y, z)
) GHE^IKl%AHGHE P bFG P ECDYINA ?aM%g ?aW &%`3cDFECD0GHcD0 ?aM?a P X7?BAHG^IKMbQD dfECLNGHcDGHE^IKl4Ie7?BA GCL IKM DFUBDQbFGHECLXD ?BAr0IKMX`3cDFECD rxLNUaG^IKWsDQA
V1,2 = ±V/2 IKE^DIKllUa?BDFXGCL X7?VIKWsLNMIKUaUaLNllmL]AH?aGCDECLXA@ECDQAHlmDQbFGH?arxDFUalECLX P bQDQA DINbFGHUaGHc?BA@lmLNGCDFMGH?VIKU?td GHcD?tMMDFE[DFUBDQbFGHECLXD A P Epd INbQDFA IKECDc9lmDFEHwmLNUa?BbIKUaUa AHcIKlmDQXD gLNE3E^INX7?VIKU bQLNM %MDF0DFMG@LKdb^cIKEHWsDQX lIKEHGH?BbFUBDQAGHcD[lmLNGCDFMGH?VIKUΦs
?aAZ0LX P UVIKGCDQXAH?aM P ACLN?BXIKUaUa Φ(r, t) = Φs(r) cos(Ωt) ,
`3cDFECDΩGv9l?BbIKUaUa ?BAZ?aMGHcD E^INX7?BLKdfECD/ P DFMbF E^IKMWsDLKdTACDFrxDFE^IKU # O Z c?BAS0DIKMA_D9lmDFEH?a0DFMG^IKUaUaGHcIKG IKM_g rsLNUaG^IKWsD
V1(t) = V cos(Ωt) P AHG wmD IKllUa?BDQXGCLLNMD lIK?aE0LKd X7?VIKWsLNMIKUaUaLNllmL]AH?aGCDDFUBDQbFGHECLXDQAIKMX I
180lcINACD AHc?tdfGCDQXrxLNUaG^IKWsD
V2(t) = −V1(t)GCLYGHcD LNGHcDFE lIK?aE fACDQDg ?aW &%
c?BA bQLNM %W P EJIKGH?BLNM` INA INbFG P IKUaUa GHcD AHG^IKEHGH?aMW_lmLN?aMG LKdIKUaUxE^INX7?BLKdfECD/ P DFMbF GHE^IKlANIKMX/` INA ?aMrxDFMGCDQXw T LNUtdfWxIKMW/I P U?aM %>' INA I[INACAM%UaGCDFE \_MUa I_d DF`xDIKECATUVIKGCDFE ?aG ` INATECDIKUa? O DQX0GHcIKG GHcDZA^IK0DACDFG P lbQL P UBXACDFEHrxD INA_IGHE^IKl d LNEZb^cIKEHWsDQX lIKEHGH?BbFUBDQA ?'(Vw INXX7?aMW > noDFUBDQbFGHECLXDQA_`3c?Bb^cbQLNM %MDGHcD Ie7?VIKU0LNGH?BLNMkT4cDFM P ACDQXINAIINACA(%UaGCDFETGvl?BbIKUaUaIKMINXX7?aGH?BLNMIKUZL@YA^DFG> rsLNUaG^IKWsDY?BAIKllUa?BDQX0GCL LNMDZLKdGHcDZX7?VIKWsLNMIKUaUa LNllmL]AH?aGCD3ECLX lIK?aECAuvM GHcDZD9lmDFEH?a0DFMGCA lECDQACDFMGCDQX0cDFECDZA P b^cIKM *+$&@/ 0(. ($ +<$&$&+87 .)()= *+$ + .$-( %@;-+ :"+<; +87$1 %)4 .,.+
V1 = V($ .""*)((+?5 !.*+ +
5 .,.%* .)()=!%""%$&+ ;:5>"/ 0;$*()+ *+ 0*+<; "* .; ($ +8)5! ,.;.@*/5>"0+87 .)V2 = 0
%%
. .
g ?aW P ECD &%,-9b^cDFIKGH?Bb X7?VIKW]E^IK LKdI P INX7E P lmLNUBD[INA^AZ%UaGCDFES9lmDFEHwmLNUa?BbIKUaUaYACcIKlmDQXY?aMMDFE3A P Epnd INbQDQA W]ECDFbQLNMGCL P ECAR_W P IKE^IKMGCDQD I l P ECD$ P INX7E P lmLNUBD lmLNGCDFMGH?VIKU ?aMAH?BXDGHcD DFUBDQbFGHECLXDAHGHE P bFG P ECD] gLNE wmDFGHGCDFEZLNlGH?BbIKU INbQbQDQACA9ECLX9noAHcIKlmDQX DFUBDQbFGHECLXDQA(wUVINb^\ bF?aECbFUBDQARIKECD P A P nIKUaUa b^cL]ACDFMk
L@Y ACDFG3rsLNUaG^IKWsD ` INA3MDFrxDFE P A^DQX IKMX GHcDFECDd LNECD `3?aUaUkMLNGZwmD[G^IK\xDFM ?aMGCLINbQbQL P MGS?tM GHc?BAZGHECDIKGH0DFMG cDSbFUVINACAC?BbIKUhD/ P IKGH?BLNMALKd 0LNGH?BLNMdfLNEIKM?BLNM LKd INACA M IKMXb^cIKEHWsD
e?aMGHcDZlmLNGCDFMGH?VIKU
Φ(r, t)IKECDF = −e∇Φ =M r
# cDF IKECD@AHlmDQbF?VIKUmrxDFECAH?BLNMALKd GHcD 48Q>[O AR2,A 132548:O C765?IKMXYbQIKMYwmD_`3EH?aGHGCDFM?aM GHcD@d LNUaUBL `3?aMWAC?a lUBD[dfLNEH
d2x
dτ2− 2q cos(2τ)x = 0
d2y
dτ2+ 2q cos(2τ)y = 0
d2z
dτ2= 0
`3cDFECDτ =
1
2Ωt and q =
2eV
Mr20Ω2.
cDQACD D/ P IKGH?BLNMAZcIrxD AHG^IKwUBD A^LNU P GH?BLNMA d LNE 0 < q < 0.908
?'>o uvM GHc?BA3lIKE^IK0DFGCDFESE^IKMWsD GHcD[?BLNM?BAZbQLNM %MDQX E^INX7?VIKUaUa[? D]T?aMxIKMX
yIKMX 0L rxDQA dfECDQDFUa IKUBLNMW
zuvMGHcD D9lmDFEH?a0DFMGCA lECDQACDFMGCDQXcDFECD Ic?aW]cXDFW]ECDQD LKd_LNlGH?BbIKUSINbQbQDQACAdECLN IKM%X7?aECDQbFGH?BLNMA?BA[XDQAH?aE^IKwUBD ?aMlIKEHGH?Bb P UVIKE wmDQbI P A^DAH?aMW]UBD?BLNMINXX7ECDQA^AH?aMW IKMXECDINX,L P G[?BA@ECD/ P ?aECDQX cDFECDdfLNECDDFUBDQbFGHECLXDQA `3?aGHc AHIKUaUBDFE bFECL]ACA ACDQbFGH?BLNMA IKECD b^cL]ACDFMk cDFbQL P UBXd LNE[DIK lUBDwmDwUVINXDnSLNE[ECLX9nAHcIKlmDQX g ?aW &%_uoMGHc?BA@bINACDGHcDINACA P lGH?BLNMGHcIKG_GHcDDFUBDQbFGHEH?BblmLNGCDFMGH?VIKU ?BAS P INX7E P lmLNUBDn Ua?a\xD?BALNMUarNIKUa?BX ?aMI ECDFW]?BLNM%MDIKEGHcDGHE^IKlIe9?BA g P EHGHcDFEI` I,dfECLNGHcDGHE^IKlIe9?BA c?aW]cDFELNECXDFE P UaGH?almLNUBDbQLNMGHEH?aw P GH?BLNMA P AHG wmD G^IK\xDM ?aMGCLINbQbQL P MG/GCL XDQACbFEH?awmD0GHcD0INbFG P IKU lmLNGCDFMGH?VIKU cDQACD P UaGH?almLNUBDlmLNGCDFMGH?VIKUBA UBDINXGCL_MLNMUa?aMDIKE ECDQA^LNMIKMbQDQA ?@ 9`3c?Bb^c bIKMbI P ACD GHE^IKll?aMW[?aMAHG^IKw?aUa?aGH?BDQA d LNEr]IKU P DFA 0.5 . q < 0.908
M cDSDINbFG lIKGHGCDFEHMYLKdkMLNMUa?aMDIKE ECDQACLNMIKMbFDQA IKMXGHcDF?aE ACGHECDFMW]GHcYAHGHECLNMW]Ua
%
% - .
g ?aW P ECD ) Ua?aMDIKEI P U GCE^IKl dfLNEb^cIKECWsDFX%lIKEHGH?BbFUBDQA_g n rxLNUaG^IKWsDQA V1 IKMX V2 lECLX P bQD E^INX7?VIKUbQLNM %MDF0DFMG_`3c?aUBD GHcD DFMX9nobIKl rxLNUaG^IKWsDQAVcap
bQLNM %MD GHcD/Ie9?VIKUk0LNGH?BLNMk
XDFlmDFMXA_LNM GHcD ?aMX7?ar?BX P IKU GHE^IKlXDQAC?aW]MkW cD ECDQACLNMIKMbQDQAZwmDQbQLN0D 0LNECD/XDFMACDFUa AHlINbQDQX GCL ` IKECXAUVIKEHWsDFEqn r]IKU P DQASMDQIKE ?> ( ?@ o cD/c?aW]cDFE_GCcD/LNE^XDFE@LKd I P UaGH?almLNUBDbI P AC?aMW I ECDQA^LNMIKMbQDhGHcDAHIKUaUBDFE/GHcDDRYDQbFG/MDIKE/GHcD0GHE^IKl bQDFMGCDFE LNMACD/ P DFMGHUa[ GHcD ?aMACG^IKw?aUa?tG ,lECLNwUBDF ?BA ECDQX P bQDQX ?tdFINA_?aML P E@D9lmDFEH?a0DFMG^AhGHcD?BLNMA[IKECDbQLLNUBDQXbFUBL]ACD/GCLGHcDF?aE_0LNGH?BLNMIKUW]ECL P MXAHG^IKGCD/wmDQbI P ACDGHcDFMGHcD?BLNMA#h` IrsDd P MbFGH?BLNMA LNMUaD7GCDFMX,IKwmL P G[GCDFMMIKMLN0DFGCDFECA[dfECLN;GHcDGHE^IKlIe7?BA[u dIGHE^IKlUa?a\xDL P ECAK`3?aGHcMLNM7n c9lmDFEHwmLNUa?Bb DFUBDQbFGHECLXDQA]?BA AHGH?aUaUlmDFEpd DQbFGHUa INb^c?aMDQX?aMI3d L P Epn dfLNUBX ?aEHECLNEAH9 0DFGHEHIKMXlmLNGCDFMGH?VIKUBA
+VIKMX −V IKECD0IKllUt?BDQX GCL X7?VIKWsLNMIKUaUaLNllmL]AC?tG^D lIK?aECAkGHcDFM GHcD.%ECAHG UBL `DQACG" P UaGH?almLNUBDbQLNEHECDQbFGH?BLNM`3c?Bb^c,LbQb P ECA@?BA I %@en lmLNUBDNcDINXDQbIKlmLNUBD?@ o+[DQLN0DFGHEH?Bb?a lmDFEpd DQbFGH?BLNMALNM GHcD LNGHcDFEZcIKMXIKUBACL UBDINX GCLUBL `DFE_LNECXDFESbQLNMGHEH?aw P GH?BLNMAgLNEd P UaU >bQLNM %MDF0DFMGZGHcD I P U INACA%UaGCDFE3?BA G P EHMDQX ?aMGCL0I GHO6 AF4E $ 472 G 8:EF4"<w INXX7?aGH?BLNMIKUDFUBDQbFGHECLXDQA]bIKUaUBDQX vDFMX9nobIKlA DNGCLS`3c?Bb^c> n rsLNUaG^IKWsDQATIKECD IKllUa?BDQX g ?aW AHcL `ZA IZlmL]ACAH?awUBD Ua?aMDIKEGHE^IKl bQLNM %W P E^IKGH?BLNM `3cDFECD@GHcD@DFMX9nobIKlAZIKECD@ECLX9noAHcIKlmDQX IKMX lUVINbQDQX LNM GHcD_GHE^IKl Ie7?BAW- P b^c DFMX9nbIKlWsDQLN0DFGHEH?BDQA cIrsDwmDQDFM D9lmDFEH?a0DFMG^IKUaUa,ECDIKUa? O DQX?@ % w P G/IKUBACL LNGCcDFE DFMX9nobIKl bQLNM %W P E^IKGH?BLNMAA P b^c INAYEH?aMWNnoAHcIKlmDQX DFMX9nobIKlA IKECL P MX GHcD,Sg n ECLXA ?@ LNE ACDFW]0DFMGCDQX Sg n ECLXA ?@ @'o*uvG`3?aUaU wmD AHcL `3MGHcIKG0Ie7?VIKU3bQLNM %MDF0DFMG w%> n rxLNUaG^IKWsDQAcINA0IXDd Lb P AH?aMWDRY DQbFG LNMGHcDYE^INX7?VIKUbQLNM %MDF0DFMGZ cD MDF` D/ P IKGH?BLNMA_LKd 0LNGH?BLNMMDIKEZGHcD GHE^IKlbQDFMGCDFE_IKECD W]?arxDFM w ?@@
d2x
dτ2+ (b− 2q cos(2τ))x = 0
d2y
dτ2+ (b+ 2q cos(2τ))y = 0
= d2z
dτ2+ 2bz = 0
?'`3cDFECD
b =eαVcapmL2Ω2
?@fACDQDg ?aW 3`3?aGHcI M P 0DFEH?BbIKU d INbFGCLNE α `3c?Bb^c,XDFlmDFMXA@LNMGHcD/WsDQLN0DFGHEHLKd GHcD/GHE^IKlbQLNM %W P nE^IKGH?BLNMk@uvG_?BA@AHGHE^IK?aW]cGpd LNEH` IKECXGCL ACcL ` GHcIKG@GHcDIe7?VIKU 0LNGH?BLNM,?BA_cIKEH0LNM?Bb`3?aGHcI dfECD/ P DFMbF ?aMECDIKUkGH?a0D G"
ωz =
√b
2Ω =
√eαVcap2mL2
. H
%
. .
) M IKMIKUa9AC?BA?'>V ?@@ LKd GHcD[,IKGHc?BD P D/ P IKGH?BLNMA K = dfLNE GHcD@E^INX7?VIKU0LNGH?BLNM AHcL `ZA GHcIKGIKM ?BLNM AZ0LNGH?BLNM?aMGHc?BAZlmLNGCDFMGH?VIKU ACDFlIKE^IKGCDQAS?aMGCLI ACUBL `=?A B32 G 47E(;$C8:O:C76 IKECL P MX GHcD GHE^IKlbQDFMGCDFEIKMXYI[d INAHGCDFE X7EH?arxDFM L]A^bF?aUaUBIKGC?aLNM bIKUaUBDQX^;.O B3EFC7;$C8:O C76IKG GHcDZdE^D3 P DFMbF ΩIKECL P MXGHcDSGHE^I HDQbFGCLNEH0LKdGHcDSACDQb P UVIKE 0LNGH?BLNMkTLNECDQL rsDFEsd LNE b, q ¿ 1
GHcDSACDQb P UVIKE 0LNGH?BLNM ?BA cIKEH0LNM?Bb]TuvMLNGHcDFE `LNECXA GHcD?BLNMASbIKMGHcDFMwmD/bQLNMAH?BXDFECDQXGCL 0L rxD ?aMIcIKEH0LNM?Bb( P INAH?tn lmLNGCDFMGH?VIKU `3c?Bb^c?BAZlECLX P bQDQX w GHcDSg noX7EH?arxD]# c?BA GHECDIKGH0DFMG3cLNUBXAZDFrsDFM ?aM GHcD P IKMG P 0DQb^cIKM?BbIKUkUa?a ?aG ?@ H V7? D]TGHcD@cIKEH0LNM?BbL]ACbF?aUaUVIKGCLNELKd3GHcD0A^DQb P UBIKE/0LNGH?BLNMbIKM wmD P IKMGH? O DQX P AHGINA ?aM%I MLNEC0IKU MLNM7nV P INAC?[lmLNGCDFMGH?VIKU cD[dfECD/ P DFMbF LKdGHc?BAZACDQb P UVIKEZE^INX7?VIKU 0LNGH?BLNM ?BA
ωr = ωx = ωy =Ω
2
√1
2q2 − b
c?BA AHcL `ZA GHcD_lECDFr9?BL P AHUa00DFMGH?BLNMDQX XDd Lb P ACAH?aMWDRY DQbFG LKdkGHcD[Ie7?VIKUmbQLNM %MDF0DFMGZLNMYGHcDSEJINX7?VIKUGHE^IKll?aMW k`3cDFMLNlmDFE^IKGH?aMW@I Ua?aMDIKE GHE^IKlIKG I3bQDFEHG^IK?aMSg noX7EH?arsDTlmL `DFEe? D] I3bQDFEHG^IK?aMSg nvIK lUa?aG P XDVIKMXIKGZrsDFEHYUBL ` DFMX9nobIKl rxLNUaG^IKWsDQA
Vcap⇒ b ≈ 0
LNMD %MXAZGHcDol P ECD[E^INX7?VIKUkdECD/ P DFMbF
ωr0 =qΩ
2√2
?(IKMX w?aMbFECDINAH?aMWGHcDDFMX9nobIKlrxLNUaG^IKWsDQA3? D] ?aMbFECDINAC?aMWGHcDIe9?VIKUSdfECD/ P DFMbF ωz GHcDEJINX7?VIKUdfECD/ P DFMbFY?BA3ECDQX P bQDQX GCL
ωr =
√ω2r0 −
1
2ω2z .
uvG bIKMwmD ACDQDFMdfECLN GHc?BA0D/ P IKGH?BLNMGHcIKG dfLNE ωz ¿ ωr0GHcD XDdfLb P A^AH?aMW?BAMDW]Ua?aW]?awUBD]uvMGHcDLNGHcDFEZUa?a ?aGZcL `DFrxDFE`3cDFECD
b¿ 1MLUBLNMWsDFEZcLNUBXA?aG3cINAZDRY DQbFGCAZLNM GHEJIKll?aMWAHG^IKw?aUa?aGv IKMX GHcDbQLNM %W P E^IKGH?BLNM LKdS?BLNMbFEH7AHG^IKUBANfACDQD IKUBA^LACDQbFGH?BLNM uvM%d INbFGTGHcD b n q noACG^IKw?aUa?aGv%X7?VIKW]E^IK ?@@AHcL `ZAZGHcIKGZ`3?aGHcIe9?VIKU bQLNM %MDF0DFMG? D]9d LNE3MLNM7n O DFECL ωzI
ωrbIKM ML UBLNMWsDFEZwmD/IKEHw?aGHE^IKEH?aUa UBL ` w
qbIKMwmD,IKwmL rxDGHcDUa?a ?aG d LNE I Ua?aMDIKE INACA %UaGCDFE `3?aGHcL P G DFMX9nobIKlA ?> ( LNECDQL rxDFEd LNE[c?aW]c DFML P W]c ωz ?aMlEH?aMbF?alUBDIKEHw?aGHE^IKEH?aUa,c?aW]cE^INX7?VIKUTdfECD/ P DFMbF?BDQA/bQL P UBX,wmDECDINb^cDQX uvM,GHc?BAUa?a ?aGH?aMWYbINACDcL `DFrxDFEGHcD lIKE^IK0DFGCDFE@E^IKMWsD dfLNE_AHG^IKwUBD GHE^IKll?aMWYwmDQbQLN0DQA_?aMbFECDINAH?aMW]Ua MIKEHECL `IKMXd P EHGHcDFEH0LNECD3GHcDZIe7?VIKU7dECD/ P DFMbFIKUa` I7ATcINATGCL[wmD UVIKEHWsDFETGHcIKM0GHcD E^INX7?VIKUdECD/ P DFMbF`3c?Bb^c0?BAMLNG3A P ?aG^IKwUBD@dfLNE GHE^IKll?aMW Ua?aMDIKE ?BLNM ACGHEH?aMWsA3INA MDQbQDQACA^IKEHdfLNEGCcD@GvlmD[LKd D9lmDFEH?a0DFMGCA3lECDQACDFMGCDQXcDFECD]
CT 7Z zhZ ! cD lmLNGCDFMGH?VIKU7DFMDFEHW] LKd
N?BLNMA LKdINACA
MIKMXb^cIKEHWsD
ebQLNM %MDQX?aM/GHcD noX7?a0DFMAH?BLNMIKU7cIKEH0LNM?BbGHE^IKl lmLNGCDFMGH?VIKU `3?aGHc IKMW P UVIKE"dfECD/ P DFMbF?BDQA ωx, ωy IKMX ωz ?BA3W]?arxDFM w ?@ H
U =M
2
N∑
n=1
(ω2xx2n + ω2yy
2n + ω2zz
2n) +
e2
8πε0
N∑
n,m=1m6=n
1
|rn − rm|,
?>
%*=
% 4
`3cDFECDrn = (xn, yn, zn)
?BAYGHcDlmL]AC?aGH?BLNMrxDQbFGCLNE LKd GHcDnGHc4?BLNMk= cD^%ECAHG GCDFEH ?BA GHcDGHE^IKllmLNGCDFMGH?VIKU `3c?aUBDGHcDA^DQbQLNMX XDFA^bFEH?awmDQA GHcD, P G P IKU L P UBLNw ?aMGCDFE^INbFGH?BLNMAuoM IUa?aMDIKE GHE^IKl`3cDFECD
ωzXDFMLNGCDQAGHcDYIe9?VIKU dfECD/ P DFMbF GHcDGv` LE^INX7?VIKU dfECD/ P DFMbF?BDQAIKECD XDFWsDFMDFE^IKGCDY?tdZGHcDGHE^IKlbQLNMAHGHE P bFGH?BLNM?BA3lmDFEpd DQbFGHUa AC 0DFGHEH?Bb
ωx = ωy = ωr .
cDTdfLNUaUBL `3?aMW_GHECDIKGH0DFMG LKdD/ P ?aUa?awEH? P lmL]AH?aGH?BLNMA INACA P 0DQA I3Ua?aMDIKE ?BLNMb^cIK?aMbQLNM %W P E^IKGH?BLNMkM c?BA?BAGHE P D[LNMUa?td GHcD@E^INX7?VIKU bQLNM %MDF0DFMGZ?BA A P bF?BDFMGHUaYAHGHECLNMWsDFE GHcIKM GHcD Ie7?VIKUbQLNM %MDF0DFMG_IKMX ?tdGHcD M P wmDE[LKdT?BLNMASJ?BAZMLNG_GCLL0UVIKEHWsDhLNGHcDEC`3?BACD/GHcD ?BLNMAS`3?aUaUINACA P 0DI O ?aWNn O IKWYLNE_DFrxDFM0LNECDbQLN lUa?BbIKGCDQX noX7?a0DFMAH?BLNMIKUTWsDQLN0DFGHEH?BbbQLNM %W P E^IKGH?BLNMA 3cDDINbFG[bQLNMX7?aGH?BLNMA@d LNE_GHcDGHE^IKMAH?aGH?BLNMdfECLNUa?aMDIKE@GCL O ?aWNn O IKW bQLNM %W P E^IKGH?BLNM,`3?aUaU wmDX7?BACb P ACA^DFX?aMA^DQbFGH?BLNM ) Ua?aMDIKE[bQLNM %W P E^IKGH?BLNM?a lUa?BDQA GHcIKGx(0)n = y
(0)n = 0
]`3cDFECD3GHcD P llmDFE ?aMXDJ (0) XDFMLNGCDQA GHcD ?BLNMA#]D/ P ?aUa?awEH? P (lmL]AH?aGH?BLNMA cD/Ie7?VIKUkD/ P ?aUa?awEH? P lmL]AH?aGH?BLNMA
z(0)nIKECD GHcDFMXDFGCDFEH ?aMDQX wYGHcD[dfLNUaUBL `3?aMW D/ P IKGH?BLNM
[∂U
∂zn
]
zn=z(0)n , xn=yn=0
= 0 .
gLNEN = 2
IKMXN = 3
7GHcDF bIKM wmD ACLNUarxDQX IKMIKUa9GH?BbIKUtUa ?@ H @( N = 2 : z1 = −(1/2)2/3 l, z2 = (1/2)2/3 l
% N = 3 : z1 = −(5/4)1/3 l, z2 = 0, z3 = (5/4)1/3 l ,
`3cDFECDl =
(e2
4πε0Mω2z
)1/3 &%%ECDFlECDQACDFMGCA GHcD3MIKG P E^IKUUBDFMW]GHc A^bIKUBDZLKdGHcD3lECLNwUBDF # cDSD9lmDFEH?a0DFMGCA lECDQACDFMGCDQX0cDFECD3`LNEH\`3?aGHcIe7?VIKUmGHE^IKlYdfECD/ P DFMbF?BDQA3LNMYGHcD[LNECXDFE3LKd ωz = 2π · 1 O gLNE 40 I + ?BLNMAGHc?BAZIK0L P MGCA GCL?aMGCDFEpn?BLNM7noACDFlIKE^IKGH?BLNMA/LKd 4z = 5.6 µ
IKMX 4z = 4.8 µ;dfLNE
N = 2IKMX
N = 3ECDQAHlmDQbFGH?arxDFUax/gLNE0LNECD[GHcIKM GHcECDQD[?BLNMA GHcD@lECLNwUBDF P AHG wmD[ACLNUarxDQX M P 0DFEH?BbIKUaUax cD[AHlINbF?aMW wmDFGv`DQDFMINX CINbQDFMG?BLNMA GHcDFM ?aMbFECDINACDQA ?aM AH? O DdECLN GHcD0bQDFMGCDFE/GCL GHcD0L P GCAH?BXD0LKd GHcD0AHGHEH?aMW0kcDFMbQD0GHcD ?aM?a P ACDFlIKE^IKGH?BLNM 4zmin ?aM GHcD[bFEH7AHG^IKU LbQb P ECA wmDFGv`DQDFM GHcD bQDFMGHE^IKU ?BLNMA#J P 0DFEH?BbIKUkAH?a P UVIKGH?BLNMA3?BDFUBX?@(
4zmin ' 2.0 lN−0.57 . c?BAZIKllECL9?aIKGH?BLNM ECDFIK?aMAZINbQb P E^IKGCD@d LNEGHcD[DFMGH?aECD@?BLNM M P wmDFE3E^IKMWsD[LKd ?aMGCDFECDQAHGZIKMX DFrxDFM P lGCLN ' 1000
gLNE P IKMG P bQLN l P GH?aMW l P EHlmL]ACDQA[?aMX7?ar9?BX P IKU?BLNM INXX7ECDQACAH?aMW ?BA[ECD/ P ?aECDQX ( c?BAINXX7ECDQACAH?aMW?BAZLNwr?BL P ACUa 0L]ACGZX7? b P UaGZd LNE3GHcD 0L]AHGZbFUBL]A^DUa AHlINbQDQX bQDFMGHE^IKUk?BLNMA 9gLNEACIKUaUSL]ACbF?aUaUVIKGH?BLNMA IKECL P MX GHcD D/ P ?aUa?awEH? P lmL]AH?aGH?BLNMA
(y(0)n = 0, x
(0)n = 0, z
(0)n )
GHcDj IenW]E^IKMW]?VIKML = T − U LKdGHcD AH9ACGCDF
TwmDF?aMW0GHcD \?aMDFGH?Bb DFMDFEHW]0G^IK\xDQA3GHcD[dfLNEH ?@>
L =M
2
∑
i=x,y,z
N∑
n=1
(%2i|n)− ω2zN∑
n,m=1
(Cin,m%i|n%i|m)
,
%'
. .
`3cDFECD%x|n, %y|n
IKMX%z|n
IKECDGCcDX7?BAClUVINbQDF0DM9GCA LKd GHcDnGHc?BLNMdfECLN;?aGCA D/ P ?aUa?awEH? P lmL]AC?aGH?BLNM?aM
x, yIKMX
zX7?aECDQbFGH?BLNM ECDQAHlmDQbFGH?arxDFUasW cD bQL P lUa?aMWIKGHEH?BbQDQA@IKECD[W]?arxDFM w
Czn,m =
1 + 2N∑p=1p6=m
l3∣∣∣z(0)m −z(0)p
∣∣∣3
?tdn = m ,
−2l3∣∣∣z(0)m −z(0)n
∣∣∣3
?tdn 6= m
IKMXCxn,m = Cy
n,m =
(1
α+
1
2
)δn,m −
1
2Czn,m ,
`3cDFECDδn,m
?BAGHcD [ECLNMDQb^\xDEZXDFUaG^I IKMXα = (ωz/ωr)
2 ?BA3IlIKE^IK0DFGCDFEZXDQACbFEH?aw?aMW GHcD IKM?BACLNGHECLNlLKdmGCcD GHE^IKl0lmLNGCDFMGH?VIKU uoG bIKM0wmDZACDQDFM GHcIKG GHcD30LNGH?BLNMA ?aMx, y
IKMXzX7?aECDQbFGH?BLNMYIKECDZbQL P lUBDQX # cDE^INX7?VIKUX7?aECDQbFGH?BLNMA
xIKMX
yIKECDZINACA P 0DQXGCL wmD3XDFWsDFMDFE^IKGCD] ) A I[bQLNMACD/ P DFMbQDxGHcD3A^LNU P GH?BLNMA LKdmGHcDlECLNwUBDF, LNwG^IK?aMDQX w,X7?VIKWsLNMIKUa? O IKGH?BLNMLKd GHcD Cx, Cy IKMX Cz IKGHEH?BbQDQA IKECD Gv`L ?aMXDFlmDFMXDFMGACDFGCA3LKd MLNEHIKUk0LXDQAZ\_MD ACDFGZbQLNMG^IK?aMAZMLNEHIKU 0LXDQAZ?aM GHcD Ie9?VIKUkX7?aECDQbFGH?BLNM GHcD LNGHcDFE3?aM GHcDE^INX7?VIKUZX7?aECDQbFGH?BLNM GHcD UVIKGHGCDFEwmDF?aMWGv`LKn d LNUBXXDFWsDMDEJIKGCD bQLNEHE^DFAClmLNMX7?aMW GCL
xn/IKMX
ynoX7?aECDQbFGH?BLNMA
cDFECD/IKECDNIe9?VIKU IKMX
2NE^INX7?VIKUk0LXDQA
cDlEH?aMbF?alIKU 0LXD LKdGHcD,4 7O 47GM;$C#[A"?_?BA[GHcD bQDFMGCDFEpnoLKdn INACA 0LXD ?tM`3c?Bb^c IKUaUT?BLNMA 0L rsDlIKE^IKUaUBDFU ?aM GHcDIe7?VIKU X7?aECDQbFGH?BLNM `3?aGHc GHcD0A^IK0DIK lUa?aG P XD] c?BA `3?aUaU wmD0bIKUaUBDQX GHcDIe7?VIKU \@0LXDxIKMXcINAGHcD dfECD/ P DFMbF ωz GHcD3Ie7?VIKU9dECD/ P DFMbFLKdI@AH?aMW]UBD?BLNM _?tW]cDE LNECXDFE Ie7?VIKU70LXDQAIKUa` I7AScIrsD c?aW]cDFE_dECD/ P DFMbFs cD/ACDQbQLNMXIe9?VIKU 0LXD/?aASGHcD/wECDIKGHc?aMW 0LXDh`3cDFECD/GHcD ?BLNMAL]ACbF?aUaUVIKGCD`3?aGHcIK lUa?aG P XDQA lECLNlmLNEHGH?BLNMIKUTGCL GHcDF?aE X7?BAHG^IKMbQDdECLN;GHcDGHE^IKlbQDFMGCDFE@n ?aMIKW]M?aG P XDINAS`DFUaUINAS?aMAC?aW]Mk c?BA_0DIKMASGCcIKGZd LNE P MDFrxDFM?BLNMM P wmDFECA@GHcD/bQDFMG^DE@?BLNMAHG^IKMXA_AHGH?aUaU ?aMGHcDwECDIKGHc?aMW 0LXD] 3cD/dE^D3 P DFMbFLKd GHcDwECDIKGHc?aMW 0LXD?BA √3ωz m?aMXDFlmDFMXDFMG LKd GHcDM P wmDFELKd_?BLNMA ?aMGHcD b^cIK?aMk cDY0LNGH?BLNMIKUZwmDFcIr9?BL P E0LKd@c?aW]cDFE0LNECXDFE00LxXDFA ?BA XDQA^bFEH?awmDQXw%0LNECDbQLN lUa?BbIKGCDQXDF?aWsDM9rsDQbFGCLNECA cD@dECD/ P DFMbF?BDQAZLKd GHcDQACD@0LXDQA3cIrxD[GCLwmD XDFGCDFEH ?aMDQX M P 0DFEH?BbIKUaUaIKMX XDFlmDFMXAHUa?aW]cGHUa LNM GHcD GCLNG^IKUk?BLNM M P wmDFEZ c?BASXDFlmDFMXDFMbQDcL `DFrxDFE?BASACL`DIK\ GHcIKG7d LNEDIK lUBD7dfLNESIKUaUN ≤ 10
GHcD/Ie9?VIKU 0LXD[dECD/ P DFMbF?BDQASIKECD XDQACbFEH?awmDQX `3?aGHc rxDFEH c?aW]cINbQb P E^INbF wGHcD[Ua?BAHG 1, √3, √29/5, 3.051, 3.671, 4.272, 4.864, 5.443, 6.013, 6.576 ?aM P M?aGCAZLKd ωz ?@(o cDAHlIKGH?VIKUTD9GCDFMAH?BLNM,LKd GHcD?BLNMA#` IrxDFlINb^\xDFGCA@?BA@XDQACbFEH?awmDQXw GHcDAHG^IKMXIKECX,XDFr9?BIKGC?aLNMLKd GHcD I P ACAC?VIKM W]ECL P MXAHG^IKGCD[lECLNwIKw?aUa?aGv X7?BAHGHEH?aw P GH?BLNMLKdGHcD Ie7?VIKU \@ 0LXD`3c?Bb^c?BA3W]?arsDFM w
δzcom =
√~
2NMωz.
c?BASDQAHGH?aIKGCD ?BAZLNMUaYECDINA^LNMIKwUBD?tdGHcD ?BLNMASIKECD bQLLNUBDQX rsDFEH bFUBL]ACD[GCL GHcD W]ECL P MX ACG^IKGCD7`3c?Bb^c cL `DFrxDFE?BAZlmL]ACAH?awUBD ?aML P E_D9lmDFEH?a0DFMG INAZ`3?aUaU wmD AHcL `3M ?aMb^cIKlGCDFE H gLNEZGv`L 40 I + ?BLNMAIKMXωz = 2π · 1 O LNMD.%MXA δzcom = 7.9
M `3c?Bb^c ?BA GHcECDQD0LNECXDE^A LKd IKW]M?aG P XDACIKUtUBDFEGHcIKM GHcD ACDFlIKE^IKGH?BLNMwmDFGv`DQDFMGHcD Gv`L?BLNMA cD lEH?aMbF?alIKU EF4 7O 47G ;XC#[A ?BA_IKWxIK?aM GHcD bQDFMGCDFEpnoLKdn INACAS0LXDhIKUaU ?BLNMASL]ACbF?aUaUVIKGH?aMWE^INX7?VIKUaUa ?aMlcINACD IKMX`3?aGHcD/ P IKUTIK lUa?aG P XD0IKG@GHcDdECD/ P DFMbF ωr c?BA[0LXD?BA_ECDd DFEHECDQXGCL INA@GHcDEJINX7?VIKU \@ 0LXD]SuvMbQLNMGHE^INACG_GCLGHcDIe7?VIKU 0LXDQAhcL `DFrxDFEmDFrxDFEH c?aW]cDFE[LNECXDFE_0LXD?BA[IKG[ING C ZAREdfECD/ P DFMbF ?@>oM cDDF?aWsDFMrsDQbFGCLNECATLKdGHcD E^INX7?VIKU70LXDQA lmLN?aMGE^INX7?VIKUaUa/?aMAHGCDINXLKd?aMGHcD z noX7?aECDQbFGH?BLNMw P GZIKECD[LNGCcDFEH`3?BACD@dfLNEHIKUaUa ?BXDFMGH?BbIKU G^LGHcD[DF?aWsDFMrsDQbFGCLNECASLKd GHcD Ie9?VIKU0LXDQA ) A`3?aGHc GHcD Ie9?VIKU
%@
% 4
g ?aW P ECD 0J@LNEHIKU0LXDQA LKdkI[Gv`LKn ?BLNM bFEH7AHG^IKU L cD3dECD/ P DFMbF?BDQA IKECDIe7?VIKU \@ ωzswECDIKGHc?aMW
√3ωz7E^INX7?VIKU \@ ωr
ECLb^\?aMW √ω2r − ω2z
2π
g ?aW P ECD = LXDdfECD/ P DFMbF AHlmDQbFGHE P LKdIZbFEH7AHG^IKUxLKdGHcECDQD 40 I + ?BLNMA ?aMI Ua?aMDIKE GHE^IKl/`3?aGHc ωr =2π·4.5 # O IKMX ωz = 2π·1 # O `3c?Bb^c IKECD Gv9l?BbIKUlIKE^IK0DFGCDFECA d LNE GHcD D7lmDFEH?a0DFMGCAlECDQACDFMGCDQX cDFECD cDdECD/ P DFMbF?BDQAIKECD ωaxialCOM = 2π · 1 # O ωbreathing = 2π ·1.73
# O ωwobble = 2π · 2.41 # O ωzig−zag = 2π · 4.22 # O ωrocking = 2π ·4.39
# O IKMX ωradialCOM = 2π · 4.5 O M3cD@0LXDQAZIKECD@UVIKwmDFUBDQX `3?aGHc ACb^cDFIKGH?Bbl?BbFG P ECDQASLKdGHcDF?aESDF?aWsDFMrxDQbFGCLNECA
0LXDQAsGHcDdfECD/ P DFMbFLKdmGHcD3ACDQbQLNMXxLKdfGCDFM0bIKUaUBDQX ECLb^\?aMW0s0LXDZbIKM wmD3XDFGCDFEH ?aMDQX IKMIKUa9GH?BbIKUaUaIKMX XLDQAZMLNGZXDFlmDFMXLNM GHcD M P wmDFESLKd?BLNMA3?aM GHcD b^cIK?aM
ωrock =√ω2r − ω2z .
cDSdECD/ P DFMbF?BDQALKd c?aW]cDFE LNECXDFE0LXDQA P AHG wmD@bIKUab P UVIKGCDQX M P 0DFEH?BbIKUaUaYIKMX IKWxIK?tM XDFlmDFMXYLNMUarxDFEHYAHUa?aW]cGHUa LNMN
cD MLNEHIKU_0LXDQA LKd Gv`L?BLNMAYIKECDXDFl?BbFGCDQX ?aM4g ?tW `3c?aUBDg ?aW =%AHcL `ZAGHcD 0LXDAHGHE P bFG P ECD@LKd I/Ut?aMDIKE GHcECDQDn ?BLNM bFEH9ACG^IKUh?aM I GHE^IKlYlmLNGCDFMGH?VIKUGv9l?BbIKUhdfLNEL P ElECDQACDFMG D7lmDFEH?a0DFMG< *+E;+ $-!3<)+8 0; 38. +87(% .;-*+* (*,>,.*+<;-.;45+E; :5:+8$1*+* +8$C34*.$&+8 *+<;+ ( "/ ;&3<@*) 0;
.'*'*)(+ 5:+ (, ., 5+ 0;+-5:+8$&3<;(": %+C *+ 0./ 0)'+</ .@;?F
% H
. .
0 \_MD 0DFGHcLXGCLb^cIKE^INbFGCDFEH? O D GHcD GHE^IKM?BAH?aGH?BLNMdfECLN IUa?tMDQIKE?BLNMb^cIK?aMGCL,I O ?aWNn O IKWbFEH9ACG^IKU IKECDM P 0DFEH?BbIKU 0LNUBDQb P UVIKEX7MIK ?BbQA AC?a P UVIKGH?BLNMA LKdGHcDD/ P ?aUa?awEH? P bFEH9AHGJIKUtUa?aMD bQLNM %W P E^IKGH?BLNMAINAXLNMD3w$-9b^c?&Y DFE HN o ) MIKUaGCDFEHMIKGH?arxD_IKllECLsINb^c 9L@YDFEH?aMWI@0LNECDZlc9AH?BbIKUl?BbFG P ECDSLKdGHcD3lECLbQDQACA?BAIAHG^IKw?aUa?aGv IKMIKUa7AH?BALKdZGHcDE^INX7?VIKUL]A^bF?aUaUVIKGH?BLNMA?@> X cDE^INX7?VIKU 0LXD LKdZc?aW]cDQAHGLNECXDFEN IKMXGHcDFECDd LNECD UBL `DQACGZdECD/ P DFMbF0 ?aA@IKUa` I7ASGHcD O ?aWNn O IKW 0LXD `3cDE^D/DINb^c?BLNM0L rxDFAS?aMGHcD/LNllmL]AC?aGCDX7?aECDQbFGH?BLNM4LKd[?aGCA MDF?aW]cwmLNEfA" cD dECD/ P DFMbFLKd[GHc?BA 0LXDXDQbFECDINACDQA0`3?aGHc?aMbFECDINAH?aMW N IKMX`3?aGHc?aMbFECDINAH?aMW IKM?BACLNGCECLNl lIKE^IK0DFGCDFE
α ) M?aMAHG^IKw?aUa?aGvLbQb P ECA`3cDFM ωzig zag = 0
T? D] GHcDbQLNEHECDQAHlmLNMX7?aMWYL]ACbF?aUaUVIKGH?BLNM dECDQD O DQASL P G_IKMX GCcD[MDF` D/ P ?aUa?awEH? P bQLNM %W P E^IKGH?BLNM?BASI O ?aWNn O IKWgLNE GHcD bFEH?aGH?BbIKUIKM?BACLNGHECLNl lIKE^IK0DFGCDFE`3cDFECD GHcD GHE^IKMAH?aGH?BLNMLbQb P ECAIZlmL `DFEpn UVI`ACbIKUa?aMW αcrit =cNγ ?BAZINbQb P E^IKGCD dfLNE3?BLNM M P wmDFECASLKd P l GCL % ]] ?@(oZ\Sd GHcD Gv`L0IKllECLsINb^cDQAS0DFMGH?BLNMDQXIKwmL rxD0LNUBDQb P UVIKE X7MIK ?BbQAAH?a P UVIKGH?BLNMA cI rsDZwmDQDFM lmDFEpdfLNEH0DQX 0LNECD_D9GCDFMAH?arxDFUa[s? D]LNMYI W]ECDIKGCDFEACDFG LKdr]IKU P DFAIKMX GHcDFECDd LNECD GHcDF?aEZECDQA P UtG^A d LNE3GHcD bQLNMAHG^IKMGCA c IKMX γ IKECD lECLNwIKwUa GHcD[0L]ACG3ECDFUa?VIKwUBD
(ωzωr
)2
crit
= 2.53N−1.73. &%
c?BA[A^bIKUa?aMW d LNE@GHcDGHE^IKMAH?aGH?BLNMdfECLN;Ua?aMDIKE[GCL O ?aWNn O IKW bQLNM %W P E^IKGH?BLNM cINA@wmDQDFM,?aMrxDQAHGH?aWxIKGCDQX,D9nlmDFEH?a0DFMG^IKUaUa wYR M O DFE A38 47G ?@>o P IKMG P bQLN l P G^IKGH?BLNMINU IKllUa?BbIKGH?BLNMA P AHG cIrxDZGHcDZ?BLNMA ?aMYI Ua?aMDIKEb^cIK?aM AH?aMbQDZ?aM I O ?aWNn O IKWbQLNM %W P E^IKGH?BLNMGHcDF `L P UBXDFMbQL P MGCDFEGCLL P b^c ?BbFECLN0LNGH?BLNMk 3c?BA`L P UBXd LNEDIK lUBDTUBDINXGCL D7bQDQACAC?arsD 0LNGH?BLNMIKU_cDIKGH?aMW l?BbIKU_lIKE^IK0DFGCDFECAYLKd L P E D9lmDFEH?a0DFMG IKECD d IKE dfECLNA P b^c4IWsDQLN0DFGHEH?Bb/GHE^IKMAC?aGH?BLNMk c?BA_bIKMwmD/ACDQDFM dfLNE_DIK lUBD/?aMg ?aW =0dfECLNGHcD d INbFGSGHcIKGSGHcD O ?aWNn O IKW0LXD dfECD/ P DFMbF ?BASAHGH?aUaUkd IKE3dfECLN O DFECLRWMk' &% 3bIKMwmD P A^DQX GCL0bIKUBb P UVIKGCD/GHcD Ie7?a P ?BLNMM P wmDFE wmDd LNECD_GHcD[bFEH7AHG^IKUmwmDQbQLN0DQA O ?aWNn O IKWNnoAHcIKlmDQX dfLNEGHcL]ACD@GHE^IKl dfECD/ P DFMbF?BDQAZIKMXYGHcD_ECDQA P UaG ?BA
N < 10 uvM0lE^INbFGH?BbQDxcL ` DrxDFELNMDZ?BA Ua?a ?aGCDQX0GCL ACIKUaUBDFE ?BLNMM P wmDFECA# c?BA ?BA IK?aMUa ECDFUVIKGCDQX0GCLGHcD[d INbFGZGHcIKGZGHcD j IKw7no>@?Bb^\sD lIKE^IK0DFGCDFE_XDFbFECDINA^DQAZ`3?aGHc GHcD M P wmDFESLKd?BLNMA(fbdpA^DQbFGH?BLNM = =
8 *+ . 0)@+8$D+9"* 0;4 +E+<;c( :34* +<;$D"* ."+<; 0"*"+? ;D '+ $&":;(7+?5 .$ *+;+838(":; 38 .)($*+8;D&;@+. .)(@*+<$8F
%(
y[| z m 3| + 0 40 +
! "
#$ " g ?aW P ECD ?' RTUBDQbFGHECLNM?Bb UBDFrxDUBA LKd 40 I + K cD,GHE^IKMAH?aGH?BLNMA ?aMX7?BbIKGCDQX wACLNUa?BX Ua?aMD IKEHECL `ZAIKECDINbFGH?arxDFUtX7EH?arxDFMw UVINACDFE E^INX7?VIKGH?BLNM?aMGHcDD9lmDFEH?a0DFMGCA LJ@LNGCD GHcIKGTIKUaU7AHG^IKGCDQA ?aMX7?BbIKGCDQXcDFECD AHlUa?aG P l ?aMGCL0A^DFrsDFE^IKUL]mDFDFIKM7noA P wAHG^IKGCDQA
cD ?BLNMLKd[b^cLN?BbQD dfLNE0L P E$ P IKMG P bQLN l P GH?aMW%D9lmDFEH?a0DFMGCA0?BA 40 I + g ?aW ?',ACcL `ZA0IKUaUUBDFrxDFUBA `3c?Bb^c DFrxDFE WsDFG lmLNl P UVIKGCDQXY?aMYGHcD@D7lmDFEH?a0DFMG uvM GHcDSd LNUaUBL `3?aMW09GHcDS%ECAHGGv`LGCDFEH0A LKd GHcDAHlmDQbFGHECL]ACbQLNl?Bb MLNG^IKGH?BLNM fDFUBDQbFGHECLNM?BbYAHcDFUaUZ P IKMG P M P wmDFEIKMX GHcD0 P UaGH?alUa?abF?aGv0IKECDLN ?aGHGCDQX cD HN >MGHE^IKMAH?aGH?BLNM`3?aUaUwmD P ACDQX dfLNEbQLNcDFECDFMGIKM?al P UVIKGH?BLNMA,uvMLNGHcDFE`LNECXAGHcD UBDFrxDFUBAS1/2
IKMXD5/2
`3?aUaU wmD INACACLbF?VIKGCDQX `3?aGHc%GHcD0UBLNW]?BbACG^IKGCDQA/LKdSI, P w?aG L wmD00LNECD AHlmDQbF?&%b LNMDLKd3GHcD]mDQDFIKMA P wAHG^IKGCDQA m = −1/2,+1/2 LKd S1/2 ?BAACDFUBDQbFGCDQXINA/GHcD0UBLNW]?Bb |0〉 IKMX LNMD LKdGHcD]mDQDFIKMA P wAHG^IKGCDQA m = −5/2,−3/2,−1/2,+1/2,+3/2,+5/2 LKd D5/2 INAGHcD UBLNW]?Bb |1〉 cD LNlGH?BbIKUSGHE^IKMAC?aGH?BLNM wmDFGv`DQDFM GHcD Gv`L ACG^IKGCDQA ?BA IKM DFUBDQbFGHEH?Bb^ P INX7E P lmLNUBD GHE^IKMAH?aGH?BLNM fX7?almLNUBDnd LNEHw?BXXDFM ) bQbQLNECX7?aMW]Ua[3GHcDD5/2
AHG^IKGCDcINA I E^IKGHcDFE UBLNMWUa?tdfDFGH?a0D ≈ 1A"`3c?Bb^c DFMA P ECDQAbQLNcDFECDFMbQDLKdTGHcD( P w?aG@L rxDFE_GHc?BA_GH?a0DACbIKUBD] cDlIKEHGH?Bb P UVIKE_d DIKG P ECDQA@LKd GHc?BA_GHE^IKMAH?aGH?BLNM`3?aUaU wmDX7?BACb P ACA^DQX?aM ACDQbFGH?BLNM ?'7 cD P w?aG ?BA/bQLNcDFECDFMGHUaIKM?al P UVIKGCDQX w9MIKEHECL `(wIKMX7`3?BX7GHc%UVINACDFEE^INX7?VIKGH?BLNMIKG HN >M fbdHb^cIKlGCDFEL=
) P IKMG P bQLN l P GH?aMWlECLNGCLbQLNU IKUBACL/ECD/ P ?aECDQA I P w?aGS; A 4@?"2 E AR; AR68TlE^LxbQDQX P E^DN gLNE GHc?BAxGHcDX7?almLNUBD GHE^IKMAH?aGH?BLNMIKG > H M-?BADF lUBL xDQX`3c?Bb^c0bQLNMMDQbFGCATGCL[LNMUaLNMD3LKdhGHcD P w?aGTAHG^IKGCDQANMIK0DFUaS1/2
( cDFECDd LNECD`D bIKM IKllUa,IKM ARG*AFB8:EFC76 ?D> A3G POQ6GCDQb^cM?* P D = GCL 0DINA P ECDGHcDX P w?aG AHG^IKGCD- P llmL]ACDGHcD?BLNM,?BA@?aM IYA P lmDFEHlmL]AH?aGH?BLNM LKd S1/2 IKMX D5/2 j ?aW]cG/IKG > H M?BA IKllUa?BDQXkX7EH?ar9?aMWGHcD
S1/2 ↔ P1/2GHE^IKMAH?aGH?BLNM IKMX,GCcDFECDFwlECL HDQbFGH?aMW GHcDIKGCLN ?Bb0AHG^IKGCD GCL DF?aGHcDFE
S1/2LNED5/2
J@L$ P LNECDQA^bFDFMbQD?BA[XDFGCDQbFGCDQX?td GHcD?BLNM,?BA_?aMGHcD D5/2 AHG^IKGCDm`3c?aUBD?aMGHcDLNllmL]AH?aGCDbINACDUa?aW]cG[?BAACbIKGHGCDFECDQXLNM,GHcD
S1/2 ↔ P1/2GHEJIKMAH?aGH?BLNMkECDFlmDIKGCDQX7UalECDFlIKEH?aMW GHcD?BLNM,?aM,GHcD A^IK0DAHG^IKGCDIKMX0DINA P EH?aMWGHcD. P LNECDQACbQDFMbQD INA XDQA^bFEH?awmDQX GHcD D5/2 ACG^IKGCD LbQb P lIKGH?BLNMbIKM wmD XDFGCDFEH ?aMDQX >[DFG^IK?aUBA_LKdTGHc?BA_DFUBDQbFGHECLNMAHcDUar9?aMWlECLbQDQX P ECDcIrsD wmDQDFMX7?BACb P ACACDQX lECDFr9?BL P AHUa =o 3cD/XDFGCDQbFGH?BLNMDbF?BDFMbF ?BASUa?a ?aGCDQXLNMUa w GHcD%M?aGCD/Ua?td DFGH?a0DLKdTGHcD
D5/2AHG^IKG^DIKMXbIKMwmDINASc?aW]c,INA >>7?>&%`3?aGHc GHcD lECDQACDFMGSD7lmDFEH?a0DFMG^IKU ACDFG P lk
%>
. .
cDS1/2 ↔ P1/2
GHE^IKMAH?aGH?BLNM?BA INXX7?aGH?BLNMIKUaUa P ACDQX dfLNE >[LNllUBDFEbQLLNUt?aMWXfbdH ACDQbFGH?BLNM H &% w ECDQX9nXDFG P M?aMW GHcD@UVINACDFEZIKG > H M wYIKwmL P G cIKUtdIUt?aMDF`3?BX7GCck#TcDFMDFrsDFE ECDQA^LNMIKMG Ua?aW]cG ?BA ACbIKGHGCDFECDQXLNMGHc?BASGHE^IKMAH?aGH?BLNM dfLNE_AHG^IKGCD/XDFGCDQbFGH?BLNMLNEZd LNES>[LNllUBDFE@bFLLNUa?aMW ZGHcD ?BLNMACLN0DFGH?a0DQA_XDQbI7AZdECLNP1/2
?aMGCLGHcDD3/2
AHGJIKGCD]M cDFECDd LNECD I ECDFl P l?aMW UVINACDFE3IKG (@@/M ?BA IKUa` I7A IKllUa?BDQX IKUBLNMW`3?aGHcGHcD > H MUt?aW]c9G cD wE^IKMb^c?aMW E^IKGH?BL ?BAC%' &% wmDFGv`DQDFMGHcDP1/2 → S1/2
IKMX GHcDP1/2 → D3/2XDQbIxg ?aMIKUaUa IYUVINACDFE/IKGC('.= M bIKMwmD P ACDQX,GCL ECDFl P l%IYlmL]A^AH?awUBD D5/2 lmLNl P UVIKGH?BLNM GCL GHcD S1/2AHG^IKGCD LNESGCL$ P DFMb^cGHcD D5/2 UBDFrsDFUk?tdTECD/ P ?aECDQX , P DFMb^c?aMW ?BA3MDQbQDQACAJIKEH d LNESAH?BXDFwIKMXbQLLNUa?aMW LNMGHcD
S1/2 ↔ D5/2GHE^IKMAH?aGH?BLNMfbdp ACDQbFGH?BLNM H uvMGHcDYdfLNUaUBL `3?aMW0 GHcDY?aMGCDFE^INbFGH?BLNM LKd IcIKEH0LNM?BbIKUaUaGHE^IKllmDQXGv`LKn UBDFrxDFUSIKGCLN fLNE?BLNM`3?aGHcUa?aW]cGZ`3?aUaUkwmD X7?BACb P ACACDQXIKMX GCcD S1/2 ↔ D5/2
P w?aGZGHE^IKMAH?aGH?BLNM`3?aUaU w9 IKMIKUa7ACDQX + 0 ' 0 cD d LNUaUBL `3?aMWGv`L,A^DQbFGH?BLNMA0IKECDYwINA^DQXLNMGHcD X?aA^b P ACAH?BLNMLKd_UVINACDFEpn ?BLNM?aMGCDFE^INbFGH?BLNM?aMGCcD Tc>GHcDQAH?BA@LKd cEH?BAHGH?VIKM,SLL]A =o ) IK?aMX7?&YDFECDFMbQDcDFECD/?BASGHcIKG_GHcD/ECDFUVIKGH?arxD/lcINACD φ wmDFGv`DQDFMGHcDUVINACDFE%DFUBXIKMX GHcD/IKGCLN ?Bb lmLNUVIKEH? O IKGH?BLNM?BA3?aMbFU P XDQX GHcECL P W]cL P GSGHcD X7?BACb P A^AH?BLNMk) M?aLNM GHE^IKllmDQX ?aM IYcIKEH0LNM?Bb0lmLNGCDFMGH?VIKU `3?aGHcdECD/ P DFMbF ω k?aMGCDFE^INbFGH?aMW`3?aGHc GHcD GHE^IrsDFUaUa?aMW` IrxD LKdI@AH?aMW]UBD0LXD UVINACDFEG P MDQX bFUBL]ACDGCL I_GHE^IKMAH?aGH?BLNM GHcIKGdfLNEH0A IKM DRY DQbFGH?arsDG `LKn UBDFrsDFUAH7AHGCDF,?BAZXDQACbFEH?awmDQX w GHcD+[IK ?aUaGCLNM?VIKM H % HN
H = H0 +H1 &%
H0 =p2
2m0+ 12m0ω
2x2 + 12~ωaσz
&%*= H1 = 1
2~Ω(σ+ + σ−)
(ei(kx−ωlt+φ) + e−i(kx−ωlt+φ)
),
&%'`3cDFECD
k?BAGCcDY` IrxDYM P wmDFE ωl GHcDYdECD/ P DFMbFIKMX φ GHcDYlcINA^D LKd@GHcDYUVINACDFE E^INX7?VIKGH?BLNM 3A^DQDACDQbFGH?BLNM &% dfLNESI XDR%M?aGH?BLNMLKd GHcD IKGHEH?BbQDQA σz, σ+ IKMX σ− g ?aMIKUaUa m0 ?BA3GHcD INA^AZLKdGHcD ?BLNMk cD lIKEHGH0
LKdTGHcD'[IK ?aUaGCLNM?VIKMXDQACbFEH?awmDQASGHcD/AHG^IKGCD/LKdGHcD ?BLNM`3c?aUBD GHcD UVINACDFEpn ?BLNM?aMGCDFE^INbFGH?BLNM?BAbQLNMG^IK?aMDQX?aMH1?aGCAACGHECDFMW]GHcwmDF?aMW,W]?arsDFMw GHcD bQL P lUa?aMWbQLNMAHG^IKMG Ω _DFECD ?aGcINAwmDQDFMINACA P 0DQXGHcIKG/LNMUa,I AH?aMW]UBDGHE^IKMAH?aGH?BLNM ωa wmDF?aMW GHcDIKGCLN ?Bb GHE^IKMAH?aGH?BLNM dECD/ P DFMbF0_?BA bFUBL]ACDGCLECDQACLNMIKMbQDIKMX GHcIKG_GHcD/UVINACDFES?BA_X7?aECDQbFGCDQX,IKUBLNMW GHcD nvIe7?BASGCL0GHcD/?BLNMk cDI P Ua?LNlmDEJIKGCLNECA[INbFGLNM GHcD[?aMGCDFEHMhIKU IKGCLN ?Bb/AHG^IKGCDQA |S〉 IKMX |D〉 >[DR%M?aMW0GHcD j IKw7no>@?Bb^\xD lhIKE^IK0DFGCDFE[
η = k
√~
2m0ω,
&%@GHcD ?BLNM A_D7GCDFEHMIKU XDFW]ECDQDQA_LKd dECDQDQXLN,? D] ?aGCAZcIKEH0LNM?BbL]ACbF?aUaUVIKGH?BLNM hbIKM wmD/D9lECDQACA^DQX ?aM GCDFEH0ALKdTbE^DQIKGC?aLNMIKMXIKMM?ac?aUVIKGH?BLNMLNlmDFE^IKGCLNECA@INA
H0 = ~ω(a†a+ 12) +
12~ωaσz
&% H H1 = 1
2~Ω(eiη(a+a
†)σ+e−i(ωlt+φ) + e−iη(a+a†)σ−ei(ωlt+φ)
).
&%( *+9) .$&+<;($ .> .*,.)(+
β *+9.$&38()) 0(%6 :$ *+25:+/./ .$D ' + ;+8") .38+?5'7=
η = k cosβ√
~/2mωF
]
@DFECDGHcD_ECLNG^IKGH?aMW` IrsD[IKllECL7?aIKGH?BLNM HN cINAwmDQDFMYINXD] uoM GHcD_?aMGCDFE^INbFGH?BLNM l?BbFG P ECD@XDR%MDQXwU = eiH0t/~ GHcD'[IK ?aUaGCLNM?VIKM
HI = U †HUG^IK\xDFAZGHcD[dfLNEH
HI =12~Ω
(eiη(a+a
†)σ+eiφe−i∆t + e−iη(a+a†)σ−e−iφei∆t
),
&%>`3?aGHc
a = aeiωtIKMX
∆ = ωl − ωa cD UVINACDFEbQL P lUBDQA0GHcD AHG^IKGCD |S, n〉 GCL IKUaUSAHG^IKGCDQA |D,n′〉 `3cDFECD
n, n′IKECDr?awE^IKGH?BLNMIKU P IKMG P M P wmDFECA u d GHcDUVINA^DEY?BAG P MDQX4bFUBL]A^DGCL ECDQACLNMIKMbQDLKdIGHE^IKMAH?aGH?BLNM |S, n〉 ↔ |D,n + m〉 `3?aGHc %DQX m IKMX
n = 0, 1, 2, 3, ...`3c?Bb^c bQLNEHECDQAHlmLNMXA0GCL
(ωl − ωa) ≈ mωRTbQL P lUa?aMWGCLLNGHcDFEUBDFrxDFUBAbIKMwmDMDFW]UBDQbFGCDQXTlECL r9?BXDQX GCcDUVINACDFE?aMGCDFMAC?aGv ?BAA P bF?BDFMGHUaUBL ` Ω¿ ω
L c?BATINA^A P lGH?BLNM?BA r]IKUa?BX/d LNEIKUaUD9lmDFEH?a0DFMGCAlECDQACDFMGCDQX?aMGHc?BA GHcDQAH?BA]uvM/GHcIKG bINA^DKGHcD UVINACDFE ?aMX P bQDQATI3lhIK?tEC`3?BACD bQL P lUa?aMW[wmDFGv`DQDFMGHcD UBDFrxDFUBA |S, n〉 IKMX |D,n+m〉 ' cDGH?a0D DFrxLNU P GH?BLNM0LKdhGHcD AHG^IKGCD Ψ(t) =∑
k(ck(t)|S, k〉+dk(t)|D, k〉)?BAWsL rsDFEHMDQXw/GHcD+-9b^cE (X7?aMWsDFED/ P IKGH?BLNM i~∂tΨ = HΨ
`3c?Bb^c?BAZD/ P ?tr]IKUBDFMGZGCL GHcD ACDFGZLKdTbQL P lUBDQX D/ P IKGH?BLNMAcn = −i(1−|m|)eiδte−iφ (Ωn+m,n/2) dn+m
] dn+m = −i(1+|m|)e−iδteiφ (Ωn+m,n/2) cn .
%δ = ∆−mω INbQbQL P MGCA3d LNESIXDFG P M?aMW LKdGHcD UVINACDFE3dECLN GHcD[GCE^IKMAH?aGH?BLNM GHcD bQLNMAHGJIKMG
Ωn+m,n := Ω‖〈n+m|eiη(a+a†)|n〉‖ ] ?BA bIKUaUBDQX @IKw? dfECD/ P DFMbFx$-9LNU P GH?BLNMA GCL GHcDQACD D/ P IKGH?BLNMA/AHcL `(IKM L]ACbF?aUaUVIKGCLNEHIKMXbQLN lUBDFGCD0D9nb^cIKMWsDLKd lmLNl P UVIKGH?BLNM wmDFGv`DQDFM GHcD0bQL P lUBDQX UBDFrxDFUBA \_M ECDQACLNMIKMbQD GHc?BA/L]ACbF?aUaUVIKGH?BLNM G^IK\sDQA lUVINbQD`3?aGHc,IdfECD/ P DMb D/ P IKU GCL0GHcD/_IKw?kdECD/ P DFMbF Ωn+m,n
T4cDFMGHcD UVINACDFES?BASXDFG P MDQX dECLNECDQACLKnMIKMbQDGHcD lmLNl P UVIKGH?BLNMGHE^IKMApd DFEZ?BA3ML UBLNMWsDFE_bQLN lUBDFGCD IK lUa?tG P XD Ω2n+m,n/(δ2 +Ω2n+m,n)RxDFG
?aGZG^IK\sDQA3lUVINbQDIKGSIc?aW]cDFE3dfECD/ P DFMbF =√δ2 +Ω2n+m,n =o
E^IKMAH?aGH?BLNMAGHcIKG XL/MLNG b^cIKMWsDSGHcDSM P wmDFELKdkr?awE^IKGH?BLNMIKU P IKMG^I ↔ m = 0 IKECDSbIKUaUBDQX BD47E
ERO ARE/GHE^IKMAH?aGH?BLNMA ) GHE^IKMAH?aGH?BLNM ?BA GCDFEH0DQX 3G 20A ?"O [A 476@?td IKM IKwACLNEHlGH?BLNM lECLbQDQACA ?BA INbQbQLN lIKM?BDQXw IKM ?aMbFECDINACD ?aM GHcD[0LNGH?BLNMIKU! P IKMG P M P wmDFE↔ m = +1 `3c?aUBD[?aG3?BA GCDFEH0DQX E A $?"O [A 476?td ?aGSXDQbFECDINACDQA P lmLNMIKwACLNEHlGH?BLNM↔ m = −1
+
gLNE I W]?arxDFM j IKw7no>@?Bb^\xDlIKE^IK0DFGCDFEηZGHcD,bQL P lUa?aMWAHGHECDFMW]GHcA Ωn,n
Ωn−1,n
IKMXΩn+1,n
LKdbIKEHEH?BDFE3ECDQXIKMX wU P DAH?BXDFwIKMX3ECDQAHlmDQbFGH?arxDFUaSINAYI,d P MGH?BLNMLKd n bIKM4wmDbIKUBb P UVIKGCDQX = P AH?aMWD/Mk ] 9ACDQD@g ?aW ?@7# c?BA bIKUBb P UVIKGH?BLNM bQLNMAH?BXDFE^IKwUa AH?a lUa?&%DQA ?aM GHcD@ACLKnobIKUaUBDQXYj IKw7no>_?Bb^\xDECDFW]?a0DYXDR%MDQX wGHcD bQLNMX7?aGH?BLNMη2(2n + 1) ¿ 1
uoM GHc?BAECDFW]?a0D GHcDYIKGCLN ?Bb ` IrxDlhINb^\sDFG?BAbQLNM %MDQXGCL I AClINbQD P b^cAHIKUaUBDFESGHcIKMGHcD ` IrsDFUBDFMW]GHcLKdTGHcD GHE^IKMAH?aGH?BLNMk ) %ECAHG_LNECXDFES I UBLNED7lIKMAH?BLNM ?aM D/Mk ] ?BA3GHcDFMIrxDFEH WsLLXIKllECL9?aIKGH?BLNM eiη(a
†+a) = 1 + iη(a† + a) +O(η2) . ":;(*38("*)(+ %@:;>$&+E@*" 38%*5(%$38%@) 5 ' + ;+? .34*+85 *+E;+
Ω 0/5
ω ;+ % *+6$ +60;45+<;?F +
"/ ;&3<@*) 0;";%"+E;&+<$1$ ;+<,% +C ;+ .5+9@*$&+2 + ":;%".$ .)'7=%%* 0* . )(+8*( C,.7 .; #$&"+838 0) ="+27@/ .7@ , +%F
%
. .
0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
1
!"$#%&!&'#(*)#'%&#
+ ,-/.01/2,3456-1/7890 :,780;
g ?aW P ECD ?@ YSDFUVIKGH?arxDSbQL P lUt?aMW AHGHECDFMW]GHc Ωn+m,n/ΩdfLNE
η = 0.05LNMYbIKEHEH?BDFE3IKMX %ECAHG IKMX ACDQbQLNMXAH?BXDFwIKMX dECLN =QM cDZr]IKU P DSLKd η = 0.05
E^DlECDQACDFMGCAI Gvl?BbIKUhLNECXDFE LKd IKW]M?aG P XDdfLNEL P ETD7lmDEC?t0DFM9GCAW\_MGHcD ACbIKUBDLKdGHc?BAW]E^IKlcGHcDX7?&Y DFECDFMbQDwmDFGv`DQDFM ECDQXIKMXwU P DAH?BXDFwIKMX ?BA3A^LACIKUaUGHcIKG3?aG ?BA MLNG3r9?BAH?awUBD]# cD j IKw7no>@?Bb^\sD ECDFW]?a0D bQLNEHECDQAHlmLNMXA3GCLGHcDSECDFW]?BLNM LKdkUBL `n . 100
`3cDFECD_GHcD_bIKEHEH?BDFE bQL P lUt?aMW?BAbQLNMAH?BXDFE^IKwUa AHGHECLNMWsDFEGHcIKMIKUaU AH?BXDFwIKMX bQL P lUa?aMWsA
R7lUa?BbF?aGHUaGHcD bQL P lUa?aMW AHGHECDFMW]GHcLNM GHcD bQIKECEH?BDES?BA3`DFUaU IKllECL7?aIKGCDQX wΩn,n = (1− η2n)Ω ]
\_M ECDQACLNMIKMbQDGHcD[?aMG^DEJINbGC?aLNM#[IK ?aUaGCLNM?VIKM &%>ECDQX P bQDQAZGCLHI =
12~Ωn,n(σ
+eiφ + σ−e−iφ) .
J@LNGCD GHcIKG ?aM0GHcD3ECDQACLNMIKMG bINACD3GHcD3GH?a0DZXDFlmDFMXDFMbQDSLKdHI
rNIKM?BACcDQA uodmGHcD UVINA^DFET?BATl P UBA^DQXsGHcDFMGHc?BA3?aMGCDFE^INbFGH?BLNM?BAZLNMUaYLNM d LNESIbQDFEHG^IK?aMl P UBACD X P E^IKGH?BLNM t # cD INbFGH?BLNMLKdA P b^c IbIKEHEH?BDFEZl P UBA^D[?BAXDQACbFEH?awmDQX w GHcD P M?aG^IKEH LNlmDFE^IKGCLNER(θ, φ) = eiHI t/~ = ei
θ2(eiφσ++e−iφσ−),
`3cDFECDθ = Ωn,nt
' c?BA?BAGHcD P M?aG^IKEHLNlmDFE^IKGCLNETE^DlECDQACDFMGH?aMW/IKM DFUBDF0DFMG^IKEHAC?aMW]UBD P w?aGECLNG^IKGH?BLNM`3c?Bb^c cINA_IKUaECDINX7YwmDQDFM?aMGHECLX P bQDQX ?aMACDQbFGH?BLNM &% fD/xMk! F-?BXDFwIKMX L P lUa?aMW?BA3W]?arxDFM w
Ωn−1,n = η√nΩ
= LNM GHcD[ECDQX AH?BXDFwIKMXIKMX
Ωn+1,n = η√n+ 1Ω
'LNMGHcD/wU P DAH?BXDFwIKMX \_MGHcD/ECDQXAH?BXDFwIKMX |S, n〉 ↔ |D,n − 1〉 RhGHcD&[IK ?aUaGCLNM?VIKMG^IK\xDQASGHcDd LNEH
HI =12~ηΩ(aσ
+eiφ − a†σ−e−iφ) , @
]
`3c?Bb^cbIKMIKUBACL wmD[`3EC?tGCGCDFMINAHI =
12~Ωn−1,n(arσ
+eiφ − a†rσ−e−iφ)`3?aGHcGHcDoMLNEHIKUa? O DQX /bFECDIKGH?BLNMIKMXIKMM?ac?aUVIKGH?BLNMLNlmDFE^IKGCLNECA ar = a/√nIKMX
a†r = a†/√nW c?BAXDR%M?aGH?BLNM DFMA P ECDQA GHcIKG GHcD3LNlmDFE^IKGCLNECA arσ+ IKMX a†rσ− XL MLNG b^cIKMWsDZGHcD MLNEH(LKdkI[AHG^IKGCD rxDQbFGCLNE
) bQbQLNECX7?aMW]Ua9GHcDSIKGHEH?tLKd I ECDQA^LNMIKMG UVINACDFE l P UBACD_LKd X P E^IKGH?BLNM t LNM GHcDSECDQX AH?BXDFwIKMX ?BA W]?arsDFM wR−(θ, φ) = ei
θ2(eiφσ+ar−e−iφσ−a†r),
IH `3cDFECD
θ = Ωn−1,nt
\_M GHcD wU P D AH?BXDFwIKMX |S, n〉 ↔ |D,n+ 1〉
HI =12~ηΩ(a
†σ+eiφ − aσ−e−iφ) . GHcD A^IK0D/IKEHW P 0DFMG@INA d LNE3ECDQX AH?BXDFwIKMX bQL P lUa?aMW0GHc?BA [IK ?aUaGCLNM?VIKMbIKM wmD[ECDF`3EH?aGHGCDFMINA
HI =12~Ωn+1,n(a
†bσ+eiφ − abσ−e−iφ) ,
`3cDFECDab = a/
√n+ 1
IKMXa†b = a†/
√n+ 1
) ECDQACLNMIKMG/UVINACDFE l P UBACD0LNM GHcD wU P DAH?BXDFwIKMX ?BAW]?arxDFM wR+(θ, φ) = ei
θ2(eiφσ+a†
b−e−iφσ−ab),
(`3cDFECD
θ = Ωn+1,nt
0 Z &
+C:.!6! 1T! "M /F! M gLNESGHcD/A^IK\xD LKdTMLNG^IKGH?BLNMIKU AH?a lUa?BbF?aGv[GHcD lE^Dr9?BL P A_X7?BACb P AHnAH?BLNM` INAUa?a ?aGCDQX GCL@GHcD AC?a lUBD bINACD LKd IKM?BLNM bQLNM %MDQX?aM0I@LNMDnoX7?a0DFMAH?BLNMINUcIKEH0LNM?Bb3lmLNGCDFMGH?VIKU [DFMDFE^IKUa? O IKGH?BLNM GCLYGHcD bINACD LKd I GHcECDQD X7?a0DFMAC?aLNMhIKUTcIKEH0LNM?Bb lmLNGCDFMGH?VIKU ?BA AHGHE^IK?aW]cGpd LNEH` IKECX IKMXLNMUa IK0L P MGCA GCL ECDFlUVINbF?aMW kx wGHcDZACbIKUVIKE lECLX P bFG k · r M cDZLNlmDFE^IKGCLNE exp(iη(a†+a)) ?BA GHcDFMECDFlUVINbQDQX w
exp(ik · r) =∏m exp(iηm(a†m+ am))
GCcDFECDFwYIKUaUBL `3?aMW lECLbQDQACA^DQAGHcIKG3b^cIKMWsD[GHcD P IKMG P M P wmDFECA_LKd Gv`L0LNESDFrsDFMIKUaU GHcECDQD/L]ACbF?aUaUVIKGCLNECASAH?a P UaG^IKMDQL P ACUas M9 I G"M ) AH?a ?aUVIKE bIKUBb P UVIKGH?BLNM bIKM,wmDlmDFEpd LNEH0DQX,?aMGHcDbINACDLKd IKM,bFEH7AHG^IKUTLKd N?BLNMA uoM GHc?BASbINACD GHcD[?aMGCDFE^INbFGH?BLNM#@IK ?aUaGCLNM?VIKM
H1?BA3W]?arsDFM ?aM GHcD(-9b^cE (X7?aMWsDFESl?BbFG P ECD w
H1 =12
∑
j
~Ωjσ+j exp
(i∑
m
ηjm(a†m + am)
)exp(−i(ωlt− φj)) + h.c. ,
>`3cDFECD
a†mIKMX
amIKECD/GHcDbFECDIKGH?BLNMIKMXIKMM?ac?aUVIKGH?BLNMLNlmDFE^IKGCLNECA_dfLNE_GHcD/MLNEHIKU 0LXDQA@LKd L]ACbF?aUtnUVIKGH?BLNM,UBIKwmDFUBDQXwmh`3c?aUBDGHcD?aMXD
j = 1, . . . , N ECDdfDFECA@GCL GHcD?aMX7?ar?BX P IKU?BLNMAJ@LNGCD/GHcIKGGHcD lcINACDQAφjIKMX bFL P lUa?tMWACGHECDFMW]GHcA Ωj
d LNE3GHcD X7?&Y DFECDFMGZ?BLNMA_IKECD[MLNGZMDQbQDQACAJIKEH?aUa D/ P IKU uvM GHcD^8 CVO C76 BD4@?/A/ GHcD00LX P U P ALKd3GHcDj IKw7no>@?Bb^\sD0d INbFGCLNECA |ηjm| =: ηmXDFlmDFMXALNMUaLNMGHcD 0LXD
mR`3c?aUBD GHcD AH?aW]M LKd
ηjm?aM WsDFMDFE^IKU XDFlmDFMXA_LNM
jW cD
ηmIKECD W]?arxDFM w
ηm = k cosβ
√~
2mcωm,
]
]
. .
`3cDFECDmc = 2m0
ωm
XDFMLNGCDQA GHcDdECD/ P DFMbF LKd/GHcDECDQAHlmDQbFGH?arxD0LXDSbdH D/Mk &%@ ) A0DFMGH?BLNMDQXβ?BA GHcD IKMW]UaDwmDFGv`DQDFM%GHcDYINXX7ECDQACAH?aMWUVINACDFE/wmDIK IKMX GHcDL]ACbF?aUaUVIKGH?BLNMIe9?BA −→e LKdGHcD?BLNMk J_LNGCDGHcIKG@?aMIUa?aMDIKE[bFEH7AHG^IKU −→e lmLN?aMGCA@?aMIe7?VIKU X7?aECDQbFGH?BLNM,dfLNE IKUaUTIe9?VIKU 0LXDQA IKMX?aME^INX7?VIKUTX7?aECDQbGC?aLNM,dfLNE IKUaU E^INX7?VIKU 0LXDQAm?aMXDFlmDFMXDFMG LKd `3c?Bb^c,?BLNM?aMGHcDAHGHEH?aMWY?BA[INXX7ECDQACA^DQX @uvMWsDFMDFE^IKU GHcD/j IKw7no>_?Bb^\xD d INbFGCLNECA_ACbIKUBD `3?aGHcGHcD/A P IKECD E^LxLNG@LKdTGHcD GCLNG^IKU INACA mc = N ·m0
LKdGHcD bFEH9ACG^IKU ( c?BA lECLNlmDFEHGv,bIKM,lmL]A^DlECLNwUBDF0A `3cDFM IKM?BLNM,GHE^IKl P IKMG P lECLbQDQA^ACLNE ?BA[GCLYwmDACbIKUBDQX P lY?aMY?BLNMYM P wmDFE uoMYGHcD_Gv`LKn ?BLNM D7lmDFEH?a0DFMGCA lECDQACDFMGCDQXYcDFECD9Gvl?abIKUaUa LNMUaLNMD@LKd GHcDGv`L?BLNMA3?BA3LNlGH?BbIKUaUa INXX7ECDQACA^DFXIKG3LNMD[GH?a0D] LNMA^D/ P DFMGHUa[7GCL0IWsLLXIKllECL9?aIKGH?BLNM GHcD A P L rxDFEjbIKM wmD X7ECLNllmDQX
J@LNGCD/GHcIKG@dfLNE@0LNECDGHcIKMGv`L ?BLNMA@GHcD0LX P U P A |ηjm| ?aMWsDFMDFE^IKUTXDFlmDFMXA[LNMwmLNGHc,GHcD0LXDm IKMX GHcD ?BLNM
j?@ H VXDFlmDFMX7?aMWYLNM GCcD DF?aWsDFMrxDQbFGCLNE_LKd GHcD 0LXD fbdHA^DQbGC?aLNM
W cD 0LXDFU d LNESUVINACDFEpn ?BLNM?aMGCDFE^INbFGH?BLNM,XDFrxDFUBLNlmDQX?aMGHcD lECDFr9?BL P A@ACDQbFGH?BLNMA GCL = 3?BASML `IKllUa?BDQX GCL GHcD$ P w?aG GHE^IKMAH?aGH?BLNM uoM GHcD0D7lmDFEH?a0DM9GCA/lECDQACDFMGCDQX cDFECDIN P INX7E P lmLNUBD0GHE^IKMAH?aGH?BLNM dfECLN GHcD3W]ECL P MX0AHG^IKGCD3GCL[GHcD D5/2 AHG^IKGCD3?aM 40 I + IKG HN >@M,s` INA b^cL]ACDFM dfLNE bQLNcDFECDFMGZ P IKMG P IKM?al P UVIKGH?BLNMA N cD?aMX P bQDQX%DFUBDQbFGHEH?BbnV P INX7E P lmLNU IKE0LN0DFMG Q bQL P lUBDQAGCL GHcDW]E^INX7?BDFMGLKd3GHcDDFUBDQbFGHECLNIKW]MDFGC?Bb%DFUBX
HI = Q∇E(t) c?BAGvlmD LKdS?aMGCDFE^INbFGH?BLNMG^IK\xDQAGHcDYAHcIKlmD LKd_D/xMk &%' `3cDFMGHcDY_IKw? dfECD/ P DFMbF ?BAXDR%MDQXINA ?@ H
Ω =
∣∣∣∣eE02~〈S,m|(ε · r)(k · r)|D,m′〉
∣∣∣∣ , %
`3cDFECDE0
?BAGHcD3DFUBDQbFGHEH?Bb %DFUBX0IK lUa?aG P XD r ?BAGHcD3LNlmDFE^IKGCLNE XDQACbFEH?aw?aMW GHcD lmL]AH?aGH?BLNM0LKdmGHcD r]IKUBDFMbQDDFUBDQbFGHECLNMECDFUVIKGH?arxDGCL GHcDIKGCLN ?BbbQDM9GHECDLKd INACA IKMXmm′ ?aMX7?BbIKGCDGHcDIKW]MDFGH?Bb P IKMG P M P wmDFEsTuoM I P INX7E P lmLNUaD[GHEJIKMAH?aGH?BLNM b^cIKMWsDQA ?aM GHcD@IKW]MDFGH?Bb P IKMG P M P wmDFEdfECLN ∆m :=
m′ −m = 0 P l GCL ∆m = ±2 IKECD/IKUaUBL `DQX uoML P ESbINACDWsLN?aMW dfECLN GHcDS1/2
W]ECL P MXAHG^IKGCD `3?aGHcGv`L,]mDQDFIKM7noA P wUBDFrsDFUBA m = ±1/2 GCLYGHcD D5/2 ACG^IKGCD`3?aGHcAH?tA P wUBDFrxDFUBA m′ = −5/2, . . . ,+5/2GHc?BAACDFUBDQbFGH?BLNME P UBD UBDIrxDQATISGCLNG^IKU7LKdGCDFM. P INX7E P lmLNUBD3IKUaUBL `DQXGHE^IKMAH?aGH?BLNMA uvMlEH?aMbF?alUBDZIKM/LKdGHcDQACDGHE^IKMAH?aGH?BLNMATbQL P UBXwmD P ACDQX/dfLNEbQLNcDFECDFMGIKM?al P UVIKGH?BLNMA LKdGCcD ?BLNM AM P IKMG P -ACG^IKGCD]'3cDQACD rNIKEH?BL P AGHE^IKMAH?aGH?BLNMAcIrxD_X7?&YDFECDFM9GbQL P lUa?aMW AHGHECDFMW]GHcA Ω `3c?Bb^c XDFlmDFMXYLNMYGHcD
(m,m′)bQLNw?aMIKGH?BLNM IKMXLNMGHcD WsDQLN0DFGHEH ? D] LNMGHcD ECDFUVIKGH?arsD0X7?aECDQbFGH?BLNMALKdGHcDIKW]MDFGH?BbX%DFUBXkGHcDlmLNUVIKEH? O IKGH?BLNM ε IKMXGHcD_` IrsD_rsDQbFGCLNE
kLKdkGHcD_UVINACDFE# cDSdfLNUaUBL `3?aMW D/ P IKGH?BLNMYdfLNE Ω ECDFlECDQACDFMGCAZI/A P ?aG^IKwUBDSd LNEH P UVIKGH?BLNMd LNEZL P ESbINACD$fXDFEH?arxDQX ?aM ) llmDFMX7?t ) &%
Ω =
∣∣∣∣eE02~〈S1/2,m|(ε · r)(k · r)|D5/2,m′〉
∣∣∣∣
= κE0
√A
k3Λ(m,m′)g(∆m)(φ, γ) ,
] `3?aGHc
κ =e
2~
(15
cα
)1/2= 3.089× 1012
−1 fAH −1/2 ,.; + $ + .4 0(./ .):$& "*)((38 = %"%$&$&('*)(+)(? +<; (/5(38+<$
j
%(/5(3? (*,2* 0 + 0,%*+E3 7@/ .7@ @ #'+E;$;+ +E; #+2.4 .) .*,.@*) 0; +87@
J/ ;+-5:;%""+?5 ;,
m 0/5
m′($ +? %5>
mj
.*5m′j
F
=
g ?aW P ECD H ]mDQDFIKMA P wAHGHE P bFG P ECDLKd GHcD S1/2 ↔ D5/2GHE^IKMAH?aGH?BLNM dECLN =QX ISuoM,IMLNM7n O DFECLIKW]MDFGH?Bb%DFUBX GHcD
S1/2 ↔ D5/2 P INX7E P lmLNUBD GHE^IKMAH?aGH?BLNMAClUa?tG^A ?aMG^LGCDFMbQLN lmLNMDFMGCA
cD bQLNEHECDQAHlmLNMX7?aMWAHlmDQbFGHE^IKUTlmL]AH?aGH?BLNMA/IKECD X7E^I`3M?aM P M?aGCA LKd µBB/~ `3cDFECD µB ?BA LNcE A3IKW]MDFGCLNMZ cD[cDF?aW]cGSLKdGHcD[wIKECAZECDFlECDQACDFMGCAZGHcD AHlmDQbF?&%b UBDFwACb^c7n LNECXIKMbQLDbF?BDFMGCAΛ2w ) UaUaLe`DQX0GHE^IKMAH?aGH?BLNMA ?tMGHcDSbINACD
ε ⊥ B ⊥ k (φ = γ = 90)M cDcDF?aW]cG@LKd GHcD/wIKECAS?aMX7?BbIKGCDQA@GHcD/ECDFUVIKGH?arsDUt?aMDACGHECDFMW]GHcA
(Λg(∆m))2fbS-7IK0DINA ww P G3dfLNEZGHcD[WsDQLN0DFGHEH φ = 45, γ = 0
`3cDFECDA?BA/GHcDAHlmLNMG^IKMDQL P AXDQbIE^IKGCD LKd3GHcD D5/2 AHG^IKGCD]N cD AHlmDQbF?&%b UBDFwACb^c7n[LNECXIKMbQLKnDbF?BDFMGCAΛ(m,m′)
IKECDACcL `3M ?aM4g ?aW H I,ECDQAClmDQbGC?trxDFUaGHcDF?aEYA P IKECDQA" IKMX Ua?BAHGCDQX?aM GHcDIKllmDFMX7?t ) &%]N cD0d P MbFGH?BLNMA g(∆m) bFLNM9G^IK?aM%GHcDWsDQLN0DFGHEHxnoXDlmDQMXDFMbDYLKd Ω `3cDFECD φ ?BA/GHcDIKMW]UBDZwmDFGv`DQDFMGCcD3UVINACDFE wmDIK IKMX0GHcD3IKW]MDFGH?Bb+%DFUBX IKMXγGHcDSIKMW]UBDZwmDFGv`DQDFM GHcD3lmLNUVIKEH? O IKGH?BLNMIKMX GHcDIKW]MDFGH?Bb %DFUBX%rsDQbFGCLNElECL HDQbFGCDQX%?aMGCLGHcDlUVIKMD LKdZ?aMbF?BXDFMbQD] cDF IKECDUa?BAHGCDQX ?aM%GHcD
) llmDFMX7?t ) &%] uoG ?BA[GHc P A lmL]ACAH?awUBDGCL A P llECDQACA ACLN0D LKdGHcD GHE^IKMAH?aGH?BLNMA w,IYlECLNlmDFE/b^cLN?BbQD0LKdGHcDWsDQLN0DFGHEHs ) XX7?aGH?BLNMIKUaUa[m?aMGHcDD7lmDFEH?a0DFMGσ−
lmLNUVIKEH? O DQXUa?aW]cG[LNMGHcDX7?almLNUBDGHE^IKMAH?aGH?BLNMIKG > H M ?BA P ACDQXYGCLl P l GHcD[?BLNM ?aMGCLGHcD m = −1/2 A P wAHG^IKGCD[lEH?BLNE3GCL bQLNcDFECDFMGZIKM?al P UVIKGH?BLNMA
'
. .
LNM GHcD P INX7E P lmLNUBD GHE^IKMAH?aGH?BLNM]# cD[d LNUaUBL `3?aMW0Gv`L0bQLNM %W P E^IKGH?BLNMA@IKECD lIKEHGH?Bb P UVIKEHUa P A^Dd P U =oφ = 90, γ = 90
@uod GHcDlmLNUVIKEH? O IKGH?BLNM GHcDwmDIKIe7?BA IKMXGCcDX7?aECDQbFGH?BLNM LKd IKW]MDFGH?Bb.%DFUBX IKECD P G P IKUaUa LNEHGHcLNWsLNMIKU hLNMUaYGHcD ∆m = ±2 bQLN lmLNMDFMGCA@IKECD D7bF?aGCDQX φ = 45, γ = 0
uvMGHc?BA bQLNM %W P E^IKGH?BLNM!xGHcD ∆m = 0GHE^IKMAH?aGH?BLNMA IKECD3DbF?aGCDQX0L]ACGTACGHECLNMW]Ua`3c?aUBDGHcD
∆m = ±1 GHE^IKMAH?aGH?BLNMA_XLMLNGSbQL P lUBD GCLGHcD UVINACDFEg ?aWsA H w XfbSAHcL ` GHcD ECDFUVIKGH?arxD Ua?aMD/AHGCECDFMW]GHcA ∝ Ω23LKdTGHcD rNIKEH?BL P ASGHE^IKMAC?aGH?BLNMAS?aMGHcD Gv`LbINACDQA uvMd INbFGkIKUaUTD7lmDEC?t0DFM9G^IKUTECDQA P UaGCA[lECDQACDFMGCDQX,?aMGCc?BA@GHcDQAH?BA[cIrxDwmDQDFMLNwGJIK?tMDFX?aM,LNMDLKdGHcDQACD3Gv`L bQLNM %W P E^IKGH?BLNMA# cD %ECAHG bQLNM %W P E^IKGH?BLNMYL@Y DFECATGHcD3AH?a lUBDQA AHlmDQbFGHE P *`3?aGHc0GHcDZAHIKUaUBDQAHGlmL]ACAH?awUBDM P wmDFELKd_IKUaUBL `DQX%GHE^IKMAH?aGH?BLNMAuoGcINA cL `DFrxDFE GHcDX7?BAJINX7rNIKMG^IKWsD GHcIKGGHcDYIKUaUBL `DQXGHE^IKMAH?aGH?BLNM0IKECD d IKEL P GCAH?BXD?aMGHcDAClmDQbFGHE P ,]cDFMbQD GHcDF?aE dfECD/ P DFMbF?BDQATXDFlmDFMXAHGHECLNMW]UaLNMGHcDIKWNnMDFGH?Bb%DFUBX SIKW]MDFGH?Bb%DFUBXMLN?BACDAHc?tdfGCASGHcD E^DFA^LNMIKMbQD dfECD/ P DFMbF?BDFA@E^IKMXLN UahGHc P A_bQLNMGHEH?aw P GH?aMWGCLXDQbQLNcDFECDFMbQD] c?BA_DRY DQbFGS?BA%rxD GH?a0DQAS` DQIK\xDFE_LNMGHcDS1/2(m = −1/2)↔ D5/2(m
′ = −1/2)GHE^IKMAH?aGH?BLNMk ) A_IKW]MDFGH?Bb(%DFUBX^ P bFG P IKGH?BLNMA IKECDIKM?BACA P D/?aMGHcDlECDQA^DFMG[ACDFG P l fACDQDA^DQbFGH?BLNM H RIKUaU DJ7lmDFEH?a0DFMGCA X7?aECDQbFGHUa ECDFUVIKGCDQXGCL P IKMG P bQLN l P GH?aMWDF lUBL GHcD AHG^IKGCDQA S1/2(m = −1/2)IKMXD5/2(m
′ = −1/2) GCL E^DlECDQACDFMGGHcD ?tM9GCDFEHMIKU P w?aGgLNE/MLNG^IKGH?BLNMIKU AH?a lUa?BbF?aGv[ GHcDQA^D0Gv` LAHG^IKGCDQAZ`3?aUaU P A P IKUaUaYwmD XDFMLNGCDQX|S〉 := S1/2(m = −1/2)|D〉 := D5/2(m
′ = −1/2)GHcECL P W]cL P GSGHcD ECDQAHGSLKd GHc?BAZGHcDQAH?BA) M D7lmDFEH?a0DFMG^IKU b^cIKUaUBDFMWsD0dfLNE P IKMG P bQLN l P G^IKGH?BLNM%?BA GHcDIrNIK?aUVIKw?aUa?aGvLKdZIKM%IKllECLNlEH?VIKGCDUa?aW]cG ACL P ECbQD] uoML P E bINACD GHcDYUa?tdfDFGH?a0D 1/A
LKdSGHcDY0DFG^INAHG^IKwUBDD5/2
UBDFrxDFU?BA %]&% H A?@ %o cDbQLNcDFECDFMbQDGH?a0DLKd3L P ED7lmDFEH?a0DFMGCA cL `DFrxDFE ?BA MLNGINA UBLNMWINA GCcIKG `3c?Bb^c ?BA/X P D0GCLACDFrxDEJIKU0DQb^cIKM?BAH0A0A P b^cINA( P bFG P IKGH?BLNMALKd_GHcDYIKW]MDFGH?Bb %DFUBX uvML P ElECDQACDFMG0DJ7lmDFEH?a0DFMG ?aGG P EHMAL P G/GHcIKGGHcDA P LKdSIKUaUbQLNMGHEH?aw P GH?BLNMA IK0L P MGCAGCLI bQLNcDFECDFMbQDGH?a0D LKdSIKGIKwmL P G 600 µ A fACDQDACDQbFGH?BLNM H M cDSUVINACDFEIKG HN > M# P AHG cIrxDSI Ua?aMDF`3?BX7GHc LKdF% ] O LNE UBDQA^A ?aM LNECXDFE GCL IK?aMG^IK?aMbQLNcDFECDFMbQD/L rxDFE3GHcIKGZGH?a0D ACbIKUBD] R W! ! 0
-9Ld IKEIKUaUMLNM7n ECDQACLNMIKMGZlECLbQDQACACDQA cIrxD@wmDQDFM MDFW]UBDQbFGCDQXINACA P ?aMWGHcIKG GHcDF?aEZbQL P lUa?aMW0ACGHECDFMW]GHcΩi
?BA P b^cAHIKUaUBDFEGHcIKMGHcD XDFG P M?aMWuoGG P EHMA L P G GHcIKG?aML P E0D9lmDFEH?a0DFMGCA DQAHlmDQbF?VIKUaUa%?aMGHcD@j IKw7no>@?Bb^\sD@ECDFW]?a0D[IKMX `3cDFMYGHcDSUVINACDFE D7bF?aGCDQA AH?BXDFwIKMX GHE^IKMAH?aGH?BLNMA7L@Ymn ECDQACLNMIKMG bQL P lUa?aMWGCLGCcD bIKEHEH?BDFE0cINAGCL,wmD bQLNMAH?BXDFECDQX `LX7?&Y DFECDFMG lECLbQDQACACDQA0bIKMwmD X7?BAHGH?aMW P ?BACcDQXIKMX`3?aUaUwmDX7?BACb P ACA^DQX?aMGHcDdfLNUaUBL `3?aMW @lmLNl P UVIKGH?BLNM GHE^IKMApd DFE IKMXUa?aW]cG AHc?tdfGCA IK\9?aMW GHcL]ACDDRY DQbFGCA[?aMGCLINbQbQL P MG ?BA lIKEHGH?Bb P UVIKEHUa0?a lmLNEHG^IKMG ?aMGHcD vXDQAH?aW]M ZLKd P IKM9G P IKUaWsLNEH?aGHc0A IKMX0?aMGHcDSDFrNIKU P IKGH?BLNMLKdGHcDF?aEZUa?a ?aGCA 4/ M +-." !6 cDFECD@?BA IKUa`3I9A3IAHIKUaUmGHEJIKMApdfDFE LKd lmLNl P UVIKGH?BLNM GCLGHcD_MLNM7n ECDQACLNMIKMGHUabQL P lUBDQXUBDFrxDFUBA u dFd LNE[DJIK lUBDGHcDAH?BXDFwIKMXGHE^IKMAH?aGH?BLNM |S, 0〉 ↔ |D, 1〉 ?BA[X7EH?arxDFM`3?aGHc IKUaUGHcDlmLNl P UVIKGH?BLNM?aM?aGH?VIKUaUa?aM |S, 0〉 GHcDFM GHcDlmLNl P UVIKGH?BLNMAYLKd GHcD UBDFrxDFUBA |D, 0〉 IKMX |S, 1〉 L]ACbF?aUaUVIKGCDQA @ "*(, 7
m = +1/2
σ+ )((,%7 ($+ 7@/ .)()=>".$&$&')(+%F
@
P ?Bb^\Ua IKGZI dECD/ P DFMbF LKd √Ω2 +∆2fL@Ymn ECDQACLNMIKMGZ_IKw?L]ACbF?aUaUVIKGH?BLNMA" `3?aGHc IKM IK lUa?aG P XD LKd HN
p =
(Ω
∆
)2
d LNE∆À Ω
@DFMbQDp?BA_I0DINA P ECD[d LNEZL@Ymn ECDQACLNMIKMG_D7bF?aG^IKGH?BLNMA uod GHcD AH?BXDFwIKMX GHE^IKMAH?aGH?BLNM?BAX7EH?arxDFMECDQACLNMIKMGHUahGHcDXDFG P M?aMW ∆
dfECLNGHcD/bIKEHEC?aDFE@?BA∆ = ωm
`3cDFECDωm
?BASGHcD dfECD/ P DFMbF LKdGHcD bQLNEHECDQAHlmLNMX7?aMW r?awE^IKGH?BLNMIKU 0LXDGHc P ASL@Ymn ECDQACLNMIKMGSbIKEHEH?BDFE_D7bF?aG^IKGH?BLNMA@IKECD W]?arsDFM w
p =
(Ω
ωm
)2.
) bQbQLNECX7?aMWYGCL D/MkL 'RmGHcD@IKw? dECD/ P DFMbF ΩS = Ω1,0LKd GHcDAH?BXDFwIKMXGHE^IKMAC?aGH?BLNM |S, 0〉 ↔
|D, 1〉 ?BAΩS = ηΩ .uvM P IKMG P bQLN l P G^IKGH?BLNM`3?aGHc,GHE^IKllmDQX?BLNMAAH?BXDFwIKMXGHE^IKMAC?aGH?BLNMA IKECD IKM,DQACACDFMGH?VIKU?aMW]ECDQX7?BDFMG
fbdH ACDQbFGH?BLNM &%YIKMX%b^cIKlGCDFE = uoM r9?BDF` LKdSI Ua?a ?aGCDQX%bQLNcDFECDFMbQDYGH?a0D `3c?Bb^c%?BAIKUa` I9AW]?arsDFM?aMECDIKU AC9AHGCDF0AfbdH ACDQbFGH?BLNM H RT?aG?BALKdS?aMGCDFECDQAHGGCLX7EH?arsD GHcDQA^DYAH?BXDFwIKMXGHE^IKMAH?aGH?BLNMA0INA/d INAHGINAZlmL]A^AH?awUBD?aMLNECXDFESGCL INbQbQLN lUa?BAHc,INASIKM WxIKGCD/LNlmDFE^IKGH?BLNMA[INASlmL]ACAH?awUBD `3?aGHc?aMGHcD/bQLNcDFECDFMbQDGH?a0D] cD P llmDFE_Ua?a ?aGZd LNESGHcD/AH?BXDFwIKMX@IKw?kdECD/ P DFMbF ΩS?BA_ACDFGSw L@Ymn ECDQACLNMIKMG@D7bF?aG^IKGH?BLNMA uvMLNGHcDFE`LNECXAKd LNETI@A P bF?BDFMGW P IKUa?aGvLKdhGHcD P IKMG P -WxIKGCD3LNlmDFE^IKGH?BLNMAsL@Ymn ECDQACLNMIKMG D7bF?aG^IKGH?BLNMAIKECD3ECD/ P ?tE^DFX0GCL[wmDZAHIKUaUBDFE GCcIKMACLN0D r]IKU P D p0 s`3c?Bb^c0?BATGv9l?BbIKUaUaLNM0GHcDZLNECXDFE LKdkI_d DF`lmDFECbQDFMG
p < p0# cD Ua?a ?aG3d LNE
ΩS?BA3GHcDFECDdfLNECD W]?arsDFM wYGHcD ?aMD/ P IKUa?aGv
p =
(ΩS
ηωm
)2< p0 ,
`3c?Bb^cbIKM wmD ECDF`3EH?aGHGCDFMINAΩS < ηωm
√p0 .
] LNECDQL rsDFE7GHcD j IKw7no>@?Bb^\xD@d INbFGCLNE ?BA lECLNlmLNEHGH?BLNMIKU GCLGHcD[?aMrsDFECA^D[A P IKECD@ECLLNGZLKd GHcD@r9?awE^IKGH?BLNMIKUdfECD/ P DFMbF[ η ∝ (ωm)
−1/2 fACDQD0D/ P IKGH?BLNMA &%@ IKMX ] FF LNMACD/ P DFMGCUt[ ?td `DINA^A P 0D GHcD?BLNMM P wmDFEmGHcD/wmDIK;WsDQLN0DFGHEHβSIKMXGHcDIe7?a P IKUaUBL `DQXL@Ymn ECDQACLNMIKMG[DbF?aG^IKGH?BLNM
p0GCLwmD %7DQX fbQLNMAHG^IKMG"R?aMD/ P IKUa?tG ] bIKM wmD `3EH?aGHGCDFMINA
ΩS <bQLNMAHG · √ωm .
= c?BA[0DIKMA[GHcIKG[GHcDIe9?a P IKUaUBL `DQXAH?BXDFwIKMX,@IKw? dfECD/ P DFMbF?aMbFECDINACDQA[`3?aGHc √ωm LNM7nACD/ P DFMGHUa ?tdGHcDFECD ?aA/IYb^cLN?BbQD wmDFGv`DQDFM ACDFrxDFE^IKUTr?awE^IKGH?BLNMIKU 0LXDQA `3c?Bb^c bIKMwmD DF lUBL xDQXd LNE P IKMG P bQLN l P G^IKGH?BLNM ?aG3?BASINXrNIKMG^IKWsDQL P ASGCL0b^cL]ACD[GCcD[0LXD `3?aGHc GHcD c?aW]cDQAHG3dECD/ P DFMbF ωm ;<2: ": " uvM WsDFMDFE^IKU SIKM9Gv`LIKGCLN ?BbUBDFrxDFUBA`3?aGHc IKMDFMDFEHW] X7?&YDFECDFMbQD
E = ~ωIKECDAHc?tdfGCDQX w,GHcDX79MIK ?Bb-G^IKEH\ DRYDQbFG?tdfL@Yhn ECDQA^LNMIKMG" Ua?aW]cGLKd dECD/ P DFMbF ωl ?BAAHcLNMDLNM9GCL GHcDIKGCLN ECDQAHlmDQbFGH?arxDFUa?BLNM HN ouvM,GHc?BA bINACD GHcD DFMDFEHWsDFGH?BbIKUaUaUBL ` DFE UBDFrsDFUT?BA AHc?tdGCDQXw Ω2/4∆IKMXGHcD P llmDFEUBDFrxDFUw −Ω2/4∆ `3cDFECD ∆ = ωl − ω
IKMXΩXDFMLNGCDQAGHcDY@IKw? dfECD/ P DFMbF LKd
9)())(+ +8)$&* $9 0;+C,% +<!( @*$ .,%@) 0; ;+ 7@*+83<=7F
IH
. .
GHcD_GHE^IKMAH?aGH?BLNM wmDFGv`DQDFM GHcD_Gv`LUBDFrxDFUBA?td GHcD_GHE^IKMAH?aGH?BLNM ` INA X7EH?arsDFMYECDQACLNMhIKMGHUa0`3?aGHc Ua?aW]cG LKd GHcDA^IK0D?aMGCDFMAH?aG IKMX,lmLNUVIKEH? O IKGH?BLNMu dFhd LNE[DIK lUBD GHcDUa?aW]cG ?BA[ECDQX9noXDFG P MDQXk? D] ∆ < 0GHcDFMGHcD[UBL `DFEZUBDFrsDFU ?BA3ACc?tdGCDQX XL `3M `3c?aUBD[GHcD P llmDFEZUBDFrxDFU?BAZAHc?tdfGCDQX P lkW c?BA3l?BbFG P ECD cLNUBXAZLNMUa ?aMIYECDFW]?a0D`ZcDFECDGHcD XDFG P M?aMW ∆
?BA[UVIKEHWsD0DFML P W]c ACLYGHcIKG rsDFEHUa?aGHGHUBD lmLNl P UVIKGH?BLNM GHE^IKMAHdfDFE G^IK\xDQAlUVINbQD? D]7d LNE∆À Ω
uvM4GCcDbINACD,LKd/GHE^IKllmDQX ?BLNMAY?aM GHcDj IKw7no>_?Bb^\xD,ECDFW]?a0D P A P IKUaUa4LNMUa Ua?aW]cG AHc?tdfGCA X P DGCLbQL P lUa?aMWGCLbIKEHEH?BDFE GHE^IKMAH?aGH?BLNMA P AHGwmD_G^IK\xDFMY?aMGCL INbQbQL P MG `3c?aUBD[AH?BXDFwIKMXYGHE^IKMAH?aGH?BLNMA lECLX P bQDLNMUa ?aMLNE Ua?aW]cG/AHc?tdGCA INA[GHcDF?aEΩ?BA ECDFUVIKGH?arsDFUa,AHIKUaU _Le`DFrsDFE/IKG AClmDQbF?BIKUaUa,b^cL]ACDFM XDFG P M?aMWsAIKMX ?tdkGHcD ) nI-G^IKEH\AHc?tdfGX P DSGCLGHcD_bIKEHEH?BDFECA?BAbQLN lmDFMA^IKGCDQX d LNEAC?BXDFwIKMX9n ?aMX P bQDQX UBDFrsDFUmACc?tdGCAbIKM wmD D7lmDFEH?a0DFMG^IKUaUa XDFGCDQbFGCDQX cD bQLNEHECDQAHlmLNMX7?aMW D9lmDFEH?a0DFMGCA_IKECD[lE^DFA^DFMGCDQX ?aM A^DQbFGH?BLNM (7&%]uvM I_Gv9l?BbIKUAHlmDQbFGHECL]ACbQLNlD7lmDFEH?a0DFMG LNM0L P E# P w?aGGHE^IKMAH?aGH?BLNM!fA" GHcD ) -G^IKEH\DRY DQbFG IKllmDIKECAd LNESDJIK lUBD?aMI AHc?tdfGSLKdGHcD/LNwACDFEHrxDQX AH?BXDFwIKMX ECDQACLNMIKMbQD dfECD/ P DFMbF j DFG P A_INACA P 0D GHcIKGS`DcIrxDbQLLNUBDQX IKM?BLNM4GCL%?aGCA 0LNGH?BLNMIKU[W]ECL P MXAHG^IKG^DIKMX4GHcDFMGHEHGCLX7EH?arsDGHcDwU P DAC?aXDwhIKMX
|S, 0〉 ↔ |D, 1〉 ECDQACLNMIKMGHUawACDFGHGH?aMWGHcDXDFG P M?aMW ∆ = ωl − ωaGCL GHcDr?awE^IKGH?BLNMIKU[0LXDdfECD/ P DFMbF ωm
MLNG^DGHcIKGωa
?BA GHcDdfECD/ P DFMbFLKdI BD47ERE"O ARE%GHE^IKMAC?aGH?BLNMK cD |S, 0〉 UBDFrsDFU[?BAGHcDFM -G^IKEH\AHc?tdfGCDQX 2<%X P DYGCL,L@Yhn ECDQA^LNMIKMG bQL P lUa?aMWGCLGHcD bIKEHEH?BDFE GHE^IKMAH?aGH?BLNM |S, 0〉 ↔ |D, 0〉wδ1 = Ω20,0/4∆
T`3c?aUBD bQL P lUa?aMW,GCLGHcDYbIKEHEH?BDFE |S, 1〉 ↔ |D, 1〉 ACc?tdGCAGHcD |D, 1〉 UBDFrsDFU C#6wδ2 = Ω21,1/4∆
%uvMGHcD j IKw7no>@?Bb^\sD ECDFW]?a0DΩ1,1 ≈ Ω0,0 = Ω
IKMXGHcDYGCLNG^IKUSAHc?tdG LKd@GHcDAH?BXDFwIKMX GHE^IKMAC?aGH?BLNM wmDQbQLN0DQAδ0 = −(δ1 + δ2) = −Ω2/2∆ uvMWsDFMDFE^IKU AH?BXDFwIKMXAIKECD IKUa` I9AAHc?tdfGCDQX 8 C 47E @?X8Q> A BD47ERE"O ARE3 0DIKM?aMW GHcIKG ECDQXAH?BXDFwIKMXA/IKECD AHc?tdfGCDQX P l`3c?aUBDwU P D AC?BXDFwIKMXAIKECDAHc?tdfGCDQXXL `3M,?aMdfECD/ P DFMbFs@uvMGHcDDIK lUBD IKwmL rxD7mGHcDUVINACDFE@Ua?aW]cG[`L P UBXwmDXDFG P MDQXdECLNGHcDZAC?BXDFwIKMX0ECDQACLNMIKMbQDZw
δ0IKUaGHcL P W]c ∆
?BA ACDFG DINbFGHUaGCL GHcDZr?awE^IKGH?BLNMIKUh0LXD3dECD/ P DFMbF ωm -9Ld IKELNMUa GHcDYAHGHECLNMWsDQAHGACL P ECbQDYLKdSUaDFrxDFUACc?tdGCA ? D] bQL P lUa?aMW,GCLGHcD MDIKECDQAHGbIKEHEH?BDFE LKdZGHcD]mDQDFIKM7n IKM?td LNUBX0?BAG^IK\xDFM?aMGCL INbQbQL P MGTuoM ECDIKUa?tG xDFrsDFEHbIKEHEH?BDFE bQLNM9GHEH?aw P GCDQA GCL@GHcD3L rxDFE^IKUaUUBDFrsDFUAHc?tdfG/`3?aGHc?aGCAECDQAHlmDQbFGH?arsDYbQL P lUa?aMW,ACGHECDFMW]GHc Ωq
IKMXXDFG P M?aMW ∆qLNECDQL rxDFETbQL P lUa?aMWGCLGHcD
S1/2 −D5/2 P INX7E P lmLNUBD GHE^IKMAH?aGH?BLNM!fAR ?BA3MLNGSGHcD LNMUa ACL P ECbQD LKdTUBDFrsDFU AHc? dfGCA X7?almLNUBD GHE^IKMAH?aGH?BLNMAGCL`3c?Bb^cGHcDYXDFG P M?aMW?BA P b^cUVIKEHWsDFE,`3cDFM`LNEH\?aMWMDIKE"/ECDQACLNMIKMGHUa%LNMGHcD S1/2 − D5/2GHE^IKMAH?aGH?BLNMTbQLN lmDFMA^IKGCDZGHc?BA w IKllECL9?aIKGCDFUa0ACDFrsDFMLNECXDFECA LKdmIKW]M?aG P XD_AHGHECLNMWsDFE bQL P lUa?aMW ?aMGCDFEH0ASLKd
Ω2uvMGHcD D7lmDFEH?a0DFMG `D bIKM P ACD I_IK0ACDFn GvlmD AHlmDQbFGHECL]ACbQLNl?Bb 0DFGHcLX fA^DQDYA^DQbFGH?BLNM H ?@/GCL0DINA P ECD ) nI-G^IKEH\,AHc?tdfGCA( c?BA 0DFGHcLXlECLNwmDQA GHcDECDFUVIKGH?arxD AHc?tdfG δAC LKdGv`L IKGCLN ?BbUBDFrxDFUBA )IKMX `3c?Bb^c?BAZlECLX P bQDQXw L@Ymn ECDQACLNMIKMG_Ua?tW]c9G@IKG_IUVINA^DFE3dfECD/ P DFMbF ωs AHcLNMD/LNMGCL0GHcD ?BLNMk uvMLNGHcDFEZ`LNECXA
δAC = 1~[(Es
A − EsB)− (EA − EB)
],
'`3cDFECD
EAIKMX
EBIKE^D/GHcDDFMDFEHW]?BDQA LKd GHcD/Gv`L UBDFrsDFUBASd LNE_GHcD/?BA^LNUVIKGCDQX P MlmDFEHG P EHwmDQX IKGCLN "e?BLNM `3c?aUBD
EsA
IKMXEsB
IKECDSGHcD ) nI-G^IKEH\AHc?tdGCDQX DFMDFEHW] UBDFrxDFUBA P MXDFE GHcDS?aM P DFMbQD@LKd UVINACDFE E^INX7?VIKGH?BLNMIKG3dECD/ P DFMbF ωs uvM%GHcDd LNUaUBL `3?aMW0 I d LNEH P UVI d LNE δAC `3?aUaUwmDYXDFrxDFUBLNlmDFXk,uvM%GHcDYD7lmDFEH?a0DFMG0INA?aGcINAwmDQDFMbIKEHEH?BDQXL P G GHcD %ECAHGUaDFrxDFU ) [` INA |S〉 = S1/2,m = −1/2 uoG`3?aUaU wmDYINACA P 0DQX GHcIKG ωs ?BAMDIKEECDQACLNMIKMGGCLGCcDS1/2 ↔ D5/2
GHE^IKMAC?aGH?BLNM!fA" w P G IKGGHcDYA^IK0DGH?a0Dd IKEDFML P W]c%dfECLN IKMECDQACLNMIKMbQDA^L0GHcIKG@MLlmLNl P UVIKGH?BLNM,GHE^IKMAHdfDFE_G^IK\xDQA_lUVINbQD]_LNECDQL rsDFEmGHcD/WsDQLN0DFGHEH?BbbQLNM %W P E^IKGH?BLNMφ = 45, γ = 0
fbdH A^DQbFGH?BLNM ?'0` INA b^cL]ACDFMk= c?BAY0DIKMAYGHcIKG LNMUa GHE^IKMAH?aGH?BLNMAYGCL%GHcDAHG^IKGCDQAD5/2,m
′ = −5/2,−1/2,+3/2 `3c?Bb^c `3?aUaUwmD XDFMLNGCDQX cDFECDZINA |D−5〉, |D−1〉IKMX |D+3〉
(
g ?aW P ECD ?( ) nI-G^IKEH\AHc?tdfGCA LKd3?aMX7?ar?BX P IKU IKGCLN ?Bb0UBDFrxDFUBA ?aM 40 I + fACDQDIKUBACL GCD9G"[?aMX P bQDQX wL@Ymn ECDQACLNMIKMGSUa?aW]cG@IKG3dECD/ P DFMbF ωs uoMX7?ar9?BX P IKU AHc?tdGCA_IKECD ?aMX7?BbIKGCDQXwYW]ECDF IKEHECL `ZAR INb^cW]ECDFxnoACbIKUBD/bQLNEHECDQAClmLNMXASGCL0GHcD/AHc?tdfGCA3?aMX P bQDQXw bQL P lUa?aMW GCL0LNMD/LKdTGHcD GHcECDQDLNlGH?BbIKU GHE^IKMAC?aGH?BLNMAZuoMGHcDD9lmDFEH?a0DFMGhGCcD ECDFUVIKGH?arxD/UBDFrsDFU AHc?tdGδAC
wmDFGv`DQDFM |D−1〉UBDFrxDFU ) IKMX |S〉 UBDFrsDFU_+?BA0lECLNwmDQX LNMACD/ P DFMGHUa GHcDD9lmDFEH?a0DFMG ?BAXL P wUBDACDFMAH?aGH?arxD GCL GHcDACc?tdGCA ?aMX7?BbIKGCDQX%w,0DQX7? P W]ECDF IKEHECL `ZA `3c?Bb^cINbQbQL P MGCA/d LNE GHcDd INbG^LNE3LKd ?aM D/Mk @ uvM lE^INbFGH?BbQD[GHcDFECD IKECD IKUBACL0LNGH?BLNMIKU AH?BXDFwIKMXA3GCL DQINb^c LKdGHcD@GHcECDQD bIKEHEH?BDFE3GHE^IKMAH?aGH?BLNMA3w P G GHcDQACD[bIKM wmD@MDFW]UBDQbFGCDQX cDFECD INAGHcDF IKECD@ECDFUVIKGH?arsDFUa`DIK\
ECDQAHlmDQbFGH?arxDFUaIKECD[IKUaUBL `DQX W cD[AHG^IKGCD |D−1〉` INA b^cL]ACDFM INAGHcD@A^DQbQLNMX UBDFrxDFU#f+ cD@AH?aG P IKGC?aLNM?BAXDFl?BbFGCDQX?aM g ?aW ?(7L cDMLNG^IKGH?BLNM Ωq
?BA P ACDQX/d LNE GHcD bQL P lUa?aMW ?aMGCDFEH0ATLKdh_IKw?dfECD/ P DFMbF0 LKdGHcDZL@Yhn ECDQACLNMhIKMG Ua?aW]cGSωs GCL[GHcD |S〉 ↔ |Dq〉
GHE^IKMAH?aGH?BLNMk ) bQbQLNECX7?aMW]Ua∆q
XDFMLNGCDQA GHcDZXDFG P M?aMWLKdωsdfECLN(GHcD |S〉 ↔ |Dq〉
ECDQACLNMIKMbQD@IKGωqxGHc P A ∆q = ωs−ωq
M cDSUa?aW]cGLKddfECD/ P DFMbF ωs ML `AHc?tdfGCA IKUaUUBDrxDFUBAT?aMrxLNUarsDQX ?aM0X7?&YDFECDFM9GT` I 9A L@Ymn ECDQACLNMIKMG bQL P lUa?aMW/GCL[GHcD |S〉 ↔ |D−5〉GHE^IKMAH?aGH?BLNM d LNEDIK lUBD AHc?tdfGCA |S〉 w Ω2−5/4∆−5
IKMX%GHcD AHG^IKGCD |D−5〉w −Ω2−5/4∆−5
J@LNGCD cL `DFrxDFEGHcIKGL P E/lECLNwmD00DFGHcLX ?BA/A^DFMAH?aGH?arsD0LNMUa,GCLAHc?tdfGCA LKd3GHcD |S〉 ↔ |D−1〉ECDQACLNMIKMbQD] LNECDQL rxDFEbQL P lUa?aMW%G^L rNIKEH?BL P A X7?almLNUBDGHEJIKMAH?aGH?BLNMA `3?aUaU@IKUBACL AHc?tdfG0GCcD UBDFrsDFUBA |S〉 IKMX |D−1〉
DINb^c4X?tlmLNUBDGHE^IKMAH?aGH?BLNM bQLNMGHEC?tw P GH?aMW`3?aGHcYI[ACc?tdG lECLNlmLNEHGH?BLNMIKUhGCL Ω2dip,i/∆dip,iM cDZ?aMXD
ibQL P MGCAS4GGNX7?almLNUBDGHE^IKMAH?aGH?BLNMA + cD/ECDQAHlmDQbFGH?arxDbQL P lUa?aMW AHGHECDFMW]GHcA Ωdip,i
IKECD P AHGSUa?a\xD/DINb^c ΩqLKd GHcD P INX7E P lmLNUBDGHE^IKMAH?aGH?BLNMAklECLNlmLNEHGH?BLNMIKU GCL GHcD Ua?aW]cG ?aMGCDFMAH?aGvx LNMACD/ P DFMGHUa DINb^c Ω2dip,i?BA lECLNlmLNEHGH?BLNMIKU GCL5d LNE DIK lUBD
Ω2−1q_MXDFE GHcD bQLNMX7?aGH?BLNMA/LKd L P E/D9lmDFEH?a0DFMG `3cDFECD `D0IKUa` I9A `LNEH\MDIKE GHcD
S1/2 − D5/2ECDQACLNMIKMbQDfARGCL`Z?tGCc?aM I,ACDFrxDFE^IKUZ O `D bIKM INACA P 0DYGHcD XDFG P M?aMWsA ∆dip,i
GCLwmDbQLNMACG^IKMGS IK\?aMW IKUaU GHc?BA@?aMGCLYINbQbQL P MGm`DbIKM`3EH?aGCDGHcD 8 C8 47GTUa?aW]cG[AHc?tdG_wmDFGv`DQDFM |S〉 IKMX|D−1〉
INAδAC =
Ω2−14
(d− a−5
∆−5− 2
∆−1− a+3
∆+3
),
@`3cDFECDGHcD
aqIKECD GHcDA P IKECDQALKd GHcD ECDFUVIKGH?arxDbQL P lUa?aMWAHGHECDFMW]GHcA `3?aGHc E^DFAClmDQbFGGCL GHcDlECLNwmDGHE^IKMAH?aGH?BLNM |S〉 ↔ |D−1〉
`3c?Bb^c d LNE/GHcD0W]?arxDFM WsDQLN0DFGHEH ECDINXa−5 = 0.278
IKMXa+3 = 0.0556
20+* 0 *+$ 4 0+|D〉
(3E / 0$ ' +<+8 5+ /*+85# $&+83E. F :F :($ 5:+8*0+?5|D−1〉
*+E;+ 0;D*+$ + 38) ;( =7F
>
. .
a−1
wmDF?aMW LNMD/wXDR%M?aGH?BLNMRbdHhg ?aW H fb cDbQLNMAHG^IKM9G d bQLNMG^IK?aMA@GHcDfA P IKE^DFXE ARG 48:OQP@AbQL P lUa?aMWAHGHECDFMW]GHcAZLKdFINA `DFUaU INA GHcD XDFG P M?aMWsA3GCL5 47GQG X7?almLNUBD[GHE^IKMAH?aGH?BLNMA WJ_LNGCD[GHcIKG d ECDIKUaUaY?BAI0bQLNMACG^IKMG_wmDQbI P ACD/GHcDA P ;LKd ECDFUBIKGC?trxD/Ua?aMDAHGHECDFMW]GCcA@LKd I0GHE^IKMAH?aGH?BLNM A]mDQDFIKM7n IKM?tdfLNUBX,XLDQA6C8XDFlmDFMXLNMGHcD WsDQLN0DGCEHs cDFECDd LNECD
d?aMlEH?aMbF?alUBD ?BA0I,X7?aECDQbFGd P MbFGH?BLNMLKd_d P MXIK0DFMG^IKUbQLNMAHG^IKMGCA3nIbQLN lUa?BbIKGCDQXLNMDcL `DFrsDFEuvMlE^INbFGH?BbQDGHcD wmDIK-WsDQLN0DFGHEH/?BA MDFrsDFE lmDFEpd DQbFGHUa/ECDIKUa? O DQX]? D] a−5 IKMX a+3 `3?aUaU7X7?&Y DFEAHUa?aW]cGHUadfECLNGHcDF?aETGHcDQLNECDFGH?BbIKUr]IKU P DQAL cD _IK0ACDFn Gv9lmD 0DINA P ECDF0DFMGTdfLNE δAC 0DFMGH?BLNMDQX0IKwmL rxD]`3cDFECDML `(GHcD dECD/ P DFMbF ωs ?BA rNIKEC?aDQX L rxDFEI `3?BXD0E^IKMWsDIKUaUBL `ZALNMD0GCL IKl%D/Mk @ ) A 0DFM7nGH?BLNMDQX GHcD0bQLNEHECDQAClmLNMX7?aMWD7lmDFEH?a0DFMG?BA/XDQACbFEH?awmDQX ?aM ACDQbFGH?BLNM H ?@7 ) %G/GCL GHcDQACDXIKG^I 9?aDFUBXAr]IKU P DFA dfLNE d a−5 IKMX a+3 [ML `3UBDQX7WsDLKdGHcDQACDGHcECDQD lIKE^IK0DFGCDFECA/GHcDFM%IKUaUBL `ZA LNMDGCL lECDQX7?BbFGIKMXvDFMW]?aMDQDFE UBDFrxDFUkAHc?tdfGCA
c?BA?BAlIKEHGH?Bb P UVIKEHUa?a lmLNEHG^IKMGTdfLNEGHcD UBDFrsDFUBA |S〉 IKMX |D〉 = |D−1〉`3c?Bb^cIKECD P A^DQXGCL[DFMbQLXDZI
P w?aG ) UBDFrxDFU`3c?Bb^c0?BA ) nI-G^IKEH\ AHc?tdfGCDQX?aM dECD/ P DFMbFw δ d LNE I@GH?a0D t INb3 P ?aECDQA I@lcINA^Dd INbFGCLNEeiδ·t
uoM GCcD0?a lUBDF0DFMG^IKGH?BLNMLKd+ P IKMG P IKUaWsLNEH?aGHc0AA P b^c lcINACD d INbFGCLNECAbIKMMLNGwmD0GCLNUBDFE^IKGCDQX P MUBDQACA GHcDFIKECD `DFUaUZbQLNMGHECLNUaUBDQX9 INAGHcD P IKMG P lcINA^DY?BAGHcD DQA^ACDFMGH?VIKU3bIKEHEH?BDFELKd P IKMG P ?aM7d LNEHIKGH?BLNMkuod Nd LNE DIK lUBDxGHcD3wU P DSAH?BXDFwIKMXLKdGHcD |S〉 ↔ |D〉 GHE^IKMAH?aGH?BLNM?BA X7EH?arxDFM0ECDQACLNMIKMGHUa[sGHcDFMGHc?BAAHc?tdfGCA GHcD |S〉 ↔ |D〉 E^DFA^LNMIKMbQD@INbQbQLNECX7?aMWGCL/D/xMk @T ECLNwUBDF0AX P DSGCL P M` IKMGCDQX ) nI-G^IKEH\AHc?tdfGCA D7?BAHG IKUBACL ?aM0lECDQbF?BAH?BLNM AClmDQbFGHECL]ACbQLNl H =IKMX `3?aGHc LNlGH?BbIKUhbFUBLb^\7A H ' H @o \ P E ACLNU P GH?BLNM0d LNEGHcD IT/F! " E A P b^c P M` IKM9GCDQX B N 0 ": " bQLNMAH?BACGCA3LKdAH?a P UaG^IKMDQL P ACUa IKllUa9?aMWIKM INXX7?aGH?BLNMIKUUa?aW]cGW%DFUBX M cDZXDFG P M?aMWIKMX0?aMGCDFMAH?aGvLKdGHcD3UVIKGHGCDFE ?BA b^cL]ACDFMA P b^c GHcIKG GHcDSINXX7?tnGH?BLNMIKU %DFUBXlECLX P bQDQAGHcD LNllmL]AC?aGCD ) nI-G^IKEH\AHc?tdfG? D]NLNllmL]AH?aGCD ?aMAC?aW]M IKMXD/ P IKU7?aMIKW]M?aG P XDIKMX GHc P A/M P UaUBA/GHcD L rxDFE^IKUaU AHc?tdfG cD XDFG P M?aMW?BA/b^cL]A^DFM d IKE dECLNIKM9,ECDQACLNMIKMbQDYACL GHchIKG/MLINXX7?aGH?BLNMIKUhlmLNl P UVIKGH?BLNM GHE^IKMApdfDFE LbFb P E^AW\ P E 0DFGHcLX?BATECDFUVIKGCDQX0GCL HH Vx`3cDFECDZGHcD3?aMcLN0LNWsDFML P AwECLsINXDFM?aMWLKdkI c9lmDFE %MDZGHE^IKMAH?aGH?BLNMYLKd 85 Zw7nvIKGCLN0A ` INA A P llECDQACACDQX P AH?aMWI w?Bb^cECLNIKGH?Bb_X7?almLNUBDGHE^IKlk
]
"( "( < 4 "4 8 9) :;0#' = 4 (&
TmZ Z |z ) A 0DFMGH?BLNMDQX wmDd LNECD9?aM L P E D7lmDFEH?a0DFMG3Gv`LX]mDQDFIKM7noA P wUBDFrsDFUBA |S〉 IKMX |D〉 fACDQD@ACDQbFGH?BLNM ?'IKECDZb^cL]ACDFM INATGHcD3Gv`L ?aMGCDFEHMIKU P w?aG ACG^IKGCDQA LKdkI 40 I + ?BLNMkW-9DQbFGH?BLNM &%ZX7?BACb P ACA^DQXcL `4IKEHw?aGHE^IKEHECLNG^IKGH?BLNMALKdGHc?BAZOQ698VARER647G 132 RO:8kbIKMwmDINbQbQLN lUa?BAHcDQXw[ECDQA^LNMIKMG DbF?aG^IKGH?BLNMA
R(θ, φ)IKMXACDQbFGH?BLNM
AHcL ` DQX cL ` GHcD UVIKGHGCDFE_bIKM wmD ?a lUBDF0DFMGCDQXw INXX7ECDQACACDQX9 bIKEHEH?BDFESl P UBACDQA D3Mk! F cD GHEH?Bb^\ GCLIK\sDYDbF?BDFMG?BLNMGHE^IKl P IKMG P bQLN l P G^IKGH?BLNMlmL]A^AH?awUBDlECLNlmL]A^DQX%w ?aE^INb IKMX]mLNUaUBDFE = ?BASGCL D lUaLe LNMDLKdTGHcD/r?awE^IKGH?BLNMIKU 0LXDQAh0LNECD/lECDQbF?BACDFUa ?aGCASUBL `DQACGSGv`L P IKMG P AHG^IKGCDQA |n = 0〉 IKMX |n = 1〉 INA IKMINXX7?aGH?BLNMIKU+ P w?aGGHcD PO REF48:O C7647G132 RO:8 cD AH?aG P IKGH?BLNM?BA?aUaU P AHGHE^IKGCDQX?aMg ?aW=&%]uvM GHc?BA `3IDFrxDFM I ?"OQ67G*AZGHE^IKllmDQX?BLNM0?aMbQLNw?aMIKGH?BLNMY`3?aGHc I[r9?awE^IKGH?BLNMIKU0LXDECDFlECDQA^DM9GCA I 8 CNnV P w?aG P IKMG P ?aM7dfLNEHIKGH?BLNMlECLbQDQACA^LNE(-9DQbFGH?BLNM3= X7?BACb P ACACDQA@?aMXDFG^IK?aUcL ` P M?arsDFECA^IKUZGv`LKnV P w?aG0WxIKGCDQA0wmDFGv`DQDFM?aMGCDFEHMIKU_IKMXr?awE^IKGH?BLNMIKUS P w?aGbIKMwmD ?a lUBDF0DFMGCDQXP AH?aMW_UVINACDFE l P UBACDQA LNMGHcD ECDQX/AH?BXDFwIKMX R−(θ, φ)
fD/xMk IH F IKMX/LNMGHcD wU P D AH?BXDFwIKMX R+(θ, φ)fD/Mk! (F3?aMINXX7?aGH?BLNMGCLbIKEHEH?BDFE_l P UBACDQA-9DQbFGH?BLNM = X7?BACb P A^ACDQAZGHcD >[D P GCACb^cnVUsL O A^I IKUaWsLNEH?aGHcIKMX cL ` ?aGZbIKM wmD ?a lUBDF0DFMGCDQXLNM A P b^cIGv`LKnV P w?aGS P IKMG P bQLN l P GCDFE-9DQbFGH?BLNM = =ACcL `ZAcL ` GHcD P IKMG P bQLN l P GJIKGH?BLNMACb^cDF0DYbIKMwmDYD9GCDFMXDQXGCLIACGHEH?aMWLKdGHE^IKllmDQX ?BLNM^ P w?aGCA
>||
>|| >|||
>||
>|| >||
>|| >||
ν
>||
>|||
g ?aW P ECD =&%0 cDDFUBDF0DFMG^IKEHGv`LKnV P w?aG[AH7AHGCDFnGHcD?aMGCDFEHMIKUfDFUBDQbFGHECLNM?Bb@XDFW]ECDQDLKd dECDQDQXLNLKdIKM4?aMX7?ar9?BX P IKU@?BLNM SbQLNw?aMDQX4`3?aGHc I r9?awE^IKGH?BLNMIKU@0LXD] cD0LXDECDFlECDQA^DFMGCA IcIKEH0LNM?BbL]ACbF?aUaUVIKGCLNE dfECD/ P DFMbF ν uoGCA Gv`L UBL `DQACG UBDFrxDFUBA |n = 0〉 IKMX |n = 1〉 IKECD?BXDFMGH?&%DQXYINA GHcDPO REF48:O:C76 47G 132!3OQ8 # cDSbQLN l P G^IKGH?BLNMIKUmwINAH?BA AHG^IKGCDQA LKdGHcDSbQLNw?aMDQXAH7AHGCDF fX7E^I`3M ?aM wUVINb^\ LNM GHcDZEH?aW]cG cIKMX AH?BXD`3?aUaUwmD_XDFMLNGCDQXUa?a\xD |S, n〉 > INAHcDQXIKEHECL `ZATECDQXIKMX wU P D AH?BXDFwIKMX GHE^IKMAH?aGH?BLNMA -9LNUa?BXIKEHECL `ZA bIKEHEH?BDFESGHE^IKMAH?aGH?BLNMA
%
4 .
y CZC 9Z -3zh Z _;| pzm ~@ c?BA ACDQbFGH?BLNMXDIKUBA@`3?aGHc IYAH?aMW]UBD?aLNMbQL P lUBDQX,GCL I r?awE^IKGH?BLNMIKUT0LXD]J@LNGCDmcL `DFrxDFEGHcIKG[GHcDA^IK0D/dfLNEHIKUa?BAH IKllUa?BDQA@?td GHcD/?BLNM?BA_lIKEHG[LKdIAHGHEH?aMW IKMXGCcD/0LXD?BA[I bQLN 0LNMr9?awE^IKGH?BLNM,LKdGHcD DFMGH?aECD bFEH7AHG^IKUnINA3UBLNMW INAZLNMUaYGHc?BASLNMD ?BLNM ?BASLNlGH?BbIKUaUa INXX7ECDQACACDQX q@AH?aMWI$?"OQ67G*A@GHE^IKllmDQX ?aLNM INA IbQLN lUBDFGCD@Gv`LKnV P w?aG P IKMG P bQLN l P GCDFE3ECD/ P ?aECDQA3I P M?arxDFECA^IKUACDFG@LKd P IKMG P ;WxIKGCDQA.fbdHhACDQbFGH?BLNM S c?BA_?aMbFU P XDQA AH?aMW]UBD( P w?aG_ECLNG^IKGH?BLNMA[LKd GHcD PO REF48:O:C764G P w?aG - P b^cECLNG^IKGH?BLNMA0LKd_GHcD r9?awE^IKGH?BLNMIKU+ P w?aGlmL]ACD bQDFEHG^IK?aMlECLNwUBDF0A IKEH?BAH?aMWdfECLN GHcDYd INbFGGHcIKGGHcD0LXDcINAIKM A 132 O 7O?R8 47698/L]ACbF?aUaUVIKGCLNEUBDFrxDFUBA MLNG P AHG/GHcD P w?aGUBDFrsDFUBA |n = 0〉 IKMX|n = 1〉 cDQACD ECLNG^IKGH?BLNMA@bIKMMLNG_wmD/X7?aECDQbFGHUa X7EH?arxDFMw ECDQACLNMIKMGSE^INX7?VIKGH?BLNM `3c?Bb^c` L P UBX wmD ?aMGHcD E^INX7?BLKdfECD/ P DFMbF E^IKMWsD/LKd Id DF` # O - P b^cIECDQA^LNMIKMG_D7bF?aG^IKGH?BLNM`L P UBXdfLNESDIK lUBDMLNGLNMUaDbF?aGCDGHcD |S, 0〉 ↔ |S, 1〉 GHE^IKMAH?aGH?BLNM kw P G GHcD |S, 1〉 lmLNl P UVIKGH?BLNM `L P UBX AH?a P UaG^IKMDQL P AHUawmDGHE^IKMApd DFEHECDQXLNM GCL |S, 2〉 |S, 3〉 IKMXACL0LNMk uoMLNGHcDFES`LNECXAGHcD/bQLN l P G^IKGH?BLNMIKUAHlINbQD `L P UBX MLNGwmD bQLNMACDFEHrsDQX -?aMW]UBD P w?aG LNlmDFE^IKGH?BLNMATLNMGHcD OQ698VARER647GAHG^IKGCDNLNMGHcDLNGHcDFE cIKMX]bIKMwmD DbF?BDFMGHUa/lmDFEpdfLNEH0DQXwbIKEHEH?BDFEl P UBACDQA c P AI,MIKG P E^IKU_IKllECLsINb^c ?BA GCLAH` IKlGHcD ?aMGCDFEHMIKU@IKMXr9?awE^IKGH?BLNMIKUS P w?aG?aM7d LNEHIKGH?BLNM GHcDFMlmDFEpdfLNEHGHcD XDQAC?aECDQX LNlmDFE^IKGH?BLNMLNM GHcD ?aMGCDFEHMIKU' P w?aG_IKMX%MhIKUtUa AC` IKl wINb^\m
cD XDQAH?aECDQX AH` IKl LNlmDFE^IKGH?BLNMAHcL P UBX ?aMGCDFECb^cIKMWsD GHcD lmLNl P UVIKGH?BLNMASLKd |D, 0〉 IKMX |S, 1〉 IKMX UBDIrxDGHcD lmLNl P UVIKGH?BLNMALKd[GHcD LNGHcDFE0Gv`L bQLN l P G^IKGH?BLNMIKUSwINAC?BA0AHG^IKGCDQA^ |S, 0〉 IKMX |D, 1〉 P MI Y DQbFGCDQX ) bQbQLNECX7?aMW]UaGHcD bFLNECECDQAHlmLNMX7?aMWIKGHEH?t ?aM GHcD wINAH?BASLNECXDFE |S, 0〉, |D, 0〉, |S, 1〉, |D, 1〉 ECDINXA
SWAP =
1 0 0 00 0 1 00 1 0 00 0 0 1
.
) ECDQXAH?BXDFwIKMXπn l P UBACD R−(π, φ)
`L P UBX?aMGCDFECb^cIKMWsD GHcD lmLNl P UVIKGH?BLNMA0LKd |D, 0〉 IKMX |S, 1〉 INAXDQAH?aECDQX 0LX P UBL/bQDFEHG^IK?aMlcINA^D d INbFGCLNECA `3c?Bb^c XDFlmDMXLNM φ _L `DFrxDFExA P b^cYIKMLNlmDFE^IKGH?BLNM`L P UaXIKUBACLXlhIKEHGH?VIKUaUt0 GHE^IKMApd DFE I_lmL]A^AH?awUBD |D, 1〉 ACG^IKGCDlmLNl P UVIKGH?BLNM `3c?Bb^c0?BATA P llmL]ACDQXGCL[wmD P MI Y DQbFGCDQX9GCL GHcD |S, 2〉 ACG^IKGCD] ) WxIK?aMGHcDZlECLNwUBDF#?BA GHcDSbQLNMA^DFEHrNIKGH?BLNMYLKd bQLN l P G^IKGH?BLNMIKUAHlINbQD]M cDZG^INAC\LKd?a lUBDF0DFMGH?aMWISWAP
LNlmDFE^IKGH?BLNM 7`3c?Bb^cYIACDQDF#?a lmL]A^AH?awUBD[IKG %ECAHG9bIKMY?aMd INbFG wmDSd P U&%UaUBDQXwGHcDGCDQb^cM?* P D0LKdBDC7;S< C@?ROQ8VA <2 G ?/AR?Rk`3c?Bb^c?aMGHcDdfE^IK0DF`LNEH\,LKdJ@-cIrxDwmDFDFM ?aM P ACDd LNEACDFrxDFE^IKU3sDIKECA H ( H >o c?BA GCDQb^cM?* P D G^IK\xDQA0INX7r]IKMG^IKWsD LKd@GHcDYd INbFG GHcIKG GHcD _IKw? dECD/ P DFMbFLKdIAH?BXDFwIKMXGHE^IKMAH?aGH?BLNMXDFlmDFMXA[AHGHE^LNMW]UaLNMGHcD0LNGH?BLNMIKU# P IKMG P M P wmDFEbdHhD/MkL = uvM lIKEHGH?Bb P UVIKEGHcD |D, 1〉 ↔ |S, 2〉 _IKw? dECD/ P DFMbF?aA/UVIKEHWsDFEw I d INbFGCLNELKd √2 GHcIKM%GHcD@IKw?dfECD/ P DFMbF LKd GHcD |D, 0〉 ↔ |S, 1〉 GHE^IKMAH?aGH?BLNM fbdH9g ?aW#= q_AC?aMWGHc?BA?aG ?BA lmL]A^AH?awUBD_GCLbQLNMAHGHE P bFGIA^DFEH?BDQAZLKd l P UBACDQA3`3c?Bb^c d P U&%UaUBAZGHcD IKwmL rxD ECD/ P ?aECDF0DFMGCA Rswap(φ0) = R−(
π√2, φ0) R
−(2π√2, φ0 + φswap) R
−(π√2, φ0),
=&%`3?aGHc
φswap = arccos(cot2( π√2)) ' 0.303π
?( oM cD 0DIKM?aMW LKd φ0 `3?aUaU9wmD D7lUVIK?aMDQXwmDUBL `/L cDECLNG^IKGH?BLNM IKMW]UBDQAθcDFECDECDd DFE@GCLYGHcD |D, 0〉 ↔ |S, 1〉 GHE^IKMAH?aGH?BLNMk cDA^IK0DUa?aW]cG[l P UBACDACD/ P DFMbQDbQLNEHECDQAHlmLNMXASGCLGHcD LNlmDFE^IKGH?BLNMR′swap(φ0) = R−(π, φ0) R
−(2π, φ0 + φswap) R−(π, φ0),
]
>||
>|| >|||
>||
>||
>|| >||
>|| >|||
>||
>||
>||
g ?aW P ECD = ! IZU P DAC?BXDFwIKMXbQL P lUa?aMW.wZSDQXAH?BXDFwIKMXbQL P lUa?aMW cD_IKw? dECD/ P DFMbF?BDQA@LKdGHcDUBL `DQACG AH?BXDFwIKMX,GHE^IKMAH?aGH?BLNMA % IKMX, %IKECDD/ P IKU `3c?aUBDGHcD _IKw? dECD/ P DFMbF?BDQALKdT IKMX IKECD w?aW]WsDFEZw Id INbFGCLNESLKd √2 fACDQD ACDQbFGH?BLNM
LNM |D, 1〉 ↔ |S, 2〉 N`3cDFECD GHcD ECLNG^IKGH?BLNM IKMW]UBDQA θ IKECD UVIKEHWsDFETw √2 L cD3UBLb^c0AHlcDFECD GHE^I pDQbFGCLNEH?BDQALKd GHcD[AHG^IKGCD@rsDQbFGCLNEZIKECD[XDFl?BbFGCDQX ?aM g ?aW = TuvG?BA?a lmLNEHG^IKMGZGCLMLNGCD[GHcIKGZIAC?BXDFwIKMXYGHE^IKMAH?aGH?BLNMXLDQAMLNG bQLNEHECDQAHlmLNMXGCLIAC?aMW]UBD P w?aGECLNGJIKGH?BLNM E^IKGHcDFEIlIK?aEH`3?BACD bQL P lUa?aMWwmDFGv`DQDFMbQDFEHG^IK?aMUBDFrxDFUBAZLKdGHcD bQLNw?aMDQXAH9ACGCDF H bQLN lmL]ACDQXLKdGHcD ?aMGCDFEHMIKU' P w?aG_IKMX GHcD 0LNGH?BLNMIKU ACG^IKGCDQA |n〉 SDQX AH?BXDFwIKMXLNlmDFE^IKGH?BLNMA_wmDFGv`DQDFM |D, 1〉 ↔ |S, 2〉 INASXDFl?BbFGCDQX ?aMg ?aW0= EH?aW]cG"Rd LNEZDIK lUBDb^cIKMWsDwmLNGHc 0LNGH?BLNMIKU IKMX ?aMGCDFEHMIKU XDFW]ECDQD LKddfECDQDQXLN ) bQbQLNECX7?aMWGCL GHcDbQLNMbFU P X7?aMWECDFIKEH\LKdTACDQbFGH?BLNM ?tdGHcD P IKMG P X7MhIK ?abQA_LKdTA P b^cI A P wAH7AHGCDF |D, 1〉 IKMX |S, 2〉 ?BASbIKUBb P UVIKGCDQXlcINACDd INbFGCLNECAY`3c?Bb^c IKllmDIKE W]UBLNwIKU[?aM GHcD,A P wAH7AHGCDF bIKM MLNG wmD,X7ECLNllmDQX INAYGHcDFIKECDMLNGW]UBLNwIKU3?tMGHcD bQLN lUBDFGCD AH7AHGCDF H cDYUBLb^cAHlcDFECD l?BbFG P ECD X7?aECDQbFGHUaX7?BAHlUVI7ALNMUa ECDFUVIKGH?arxDlcINACDQA cDbQLN lUBDFGCD$ P IKM9G P lcINACDDFrxLNU P GH?BLNM X P DGCL IYACDFEH?BDQA LKd ECLNG^IKGH?BLNMA bIKM,wmDECDFrxDIKUBDQXwY P UaGH?alUa?aMWGHcD IKGHEH?BbQDQA_LKd GHcD ?aMX7?ar9?BX P IKUkECLNG^IKGH?BLNMA uvM GHcD bINA^D LKdSL P EAH` IKll?aMWl P UBACD A^D/ P DFMbQDrNIKEH9?aMWLNMUa GHcDYIKMW]UBD φ0 ECLNG^IKGCDQAGHcD `3cLNUBDGHE^I HDQbFGCLNEH?BDQAIKwmL P G GHcD z nvIe7?BA LKd GHcDSUBLb^c AHlcDFECDQAM c?BA b^cIKMWsDQA MLNGHc?aMW/?aMGHcD |D, 1〉 ↔ |S, 2〉A P wAHlhINbFD GHcDYL rxDFE^IKUaU3ACD/ P DFMbQD R′swap
IKUa` I9A0INbFGCAUt?a\xD IKMDRY DQbFGH?arsD4π
l P UBACD cDFMbQDYUa?a\xD GHcD?BXDFMGH?aGv LNlmDFE^IKGH?BLNMI uoM d INbFGIKMl P UBACDACD/ P DFMbQD R′ = R−(π, φ1)R−(2π, φ2)R−(π, φ1)
`3?aGHcIKEHw?aGHE^IKEHφ1
IKMXφ2
0DQDFGCAGHc?BAECD/ P ?aECDF0DFMG \_MGHcD |D, 0〉 ↔ |S, 1〉 GHE^IKMAH?aGH?BLNM φ0 lUVI7A IAH?a ?aUVIKEZECLNUBD/INAφ?aMIAH?aMW]UBD
πn l P UBACD R−(π, φ)
) A3?aM GHcIKGSbINACDML0IKMW]UBDφD9?BACGCA `3c?Bb^c `L P UBXbQLNEHECDQAHlmLNMX GCL I,lmDFEpd DQbFG AH` IKll?aMW%LNlmDFE^IKGH?BLNMk cLL]AH?aMW φ0 = 1√
2π ' 0.707
9?BDFUBXA I oWsLLXIKllECL7?aIKGH?BLNM ?( V? D]
|D, 0〉 7→ −|S, 1〉 IKMX |S, 1〉 7→ |D, 0〉 ,ACLGHcIKGZGHcD bQLN lUBDFGCD IKGHEH?t LKd GHc?BA $ LNlmDFE^IKGH?BLNMECDINXA
SWAP (1√2π) =
1 0 0 00 0 1 00 −1 0 00 0 0 1
.
J@LNGCD3GHcIKG GHcD3MLNG^IKGH?BLNM $ (φ0)d LNETGHcDZL rxDFE^IKUaULNlmDFE^IKGH?BLNM INA I@d P MbFGH?BLNMLKd φ0 cINA wmDQDFM?aMGHECLKnX P bQDQX cD P M`3IKMGCDQX ?aM P A_AH?aW]M?aM $ ( 1√
2π)bQL P UBX?aMlEH?aMbF?alUBDhwmD/bQLN lmDFMA^IKGCDQX d LNEZw
]
4 .
-10
1 -1 0 1
-1
0
1
yx
z
-1
0
1
-1
0
1-1
-0.5
0
0.5
1
xy
z
g ?aW P ECD = , UBLb^cAHlcDFECDGHE^I pDQbFGCLNEH?BDQA@d LNE@GHcDbQLN lmL]AH?aGCD $ LNlmDFE^IKGH?BLNM`3?aGHc φ0 = 0.707π
;F! UBLb^c AClcDFECD[LKd GHcD P INAC?tn Gv`LKn UBDFrsDFUtnoAH7AHGCDF |D, 0〉 ↔ |S, 1〉 M cD@?aM?aGH?VIKU ACG^IKGCD?BA[INA^A P 0DQXGCL wmD |D, 0〉 h`3c?Bb^c,?BA_IKEH\xDQXw GHcDwUVINb^\IKEHECL ` P UBACDM P wmDFE%LKdGHcD[ACD/ P DFMbQD. R−( π√2, 0.707π)
ECLNG^IKGCDQA GHcD[AHG^IKGCD@rxDQbFGCLNE 0LNECD@GHcIKM cIKUtd ` I XL `3Mk P UBACD M P wmDFE wEH?aMWsA ?aG wINb^\ P lGCL ?aGCA ?aEHECLNE ?aIKWsDIKwmL P G GHcD x n y n lUVIKMD]$ cDIKMW]UBDφswap = arccos(cot2( π√
2))?BAlECDQbF?BACDFUab^cL]ACDFMGCLINbQbQLN lUa?BAHcGHc?BA UBDIKEHUa[l P UBACDM P wmDFE GHcDFMECLNG^IKGCDQA GHcDAHG^IKGCDrsDQbFGCLNE/IKUaUGHcD` IXL `3MGCLYGCcDwmLNGHGCLN LKdGHcDYAHlcDFECD] cD$%MIKU AHGJIKGCD W]ECDF%IKEHECL `S/GHc P A?BA |S, 1〉 GH?a0DQAACLN0DlcINA^D0d INbFGCLNE`3c?Bb^c?BAMLNGr9?BAH?awUBDY?aMGHcD UBLb^cAClcDFECDYl?BbFG P ECD] 92: cD A^IK0DYUVINACDFEl P UBACDACD/ P DFMbQD IKMX?aGCA INbFGH?BLNMLNM,GHcD |D, 1〉 ↔ |S, 2〉 A P wAHlINbQD] ) UaU ?aMX7?ar9?BX P IKUTl P UBACDQA LKd
R′swap(φ0)
IKECD P UaGH?alUBDQA/LKd π ?aM,UBDFMW]GHckuvG@?BA DINAHGCL P MXDFECAHG^IKMX,GHchIKG[GHcDL rxDFE^IKUaUACD/ P DFMbQDSwEH?aMWsAGHcD@AHG^IKGCDSrxDQbFGCLNE wINb^\0GCL?aGCAAHG^IKEHGH?aMWlmLN?aMGwUVINb^\ IKEHECL ` 7AJI |D, 1〉dfLNE_DIK lUBDSIKMXGHcIKGSGHcDINb/ P ?aECDQXlcINA^D d INbFGCLNECA@bIKMbQDFU L P G3uoMLNGHcDFE_`LNECXAGHcDACD/ P DFMbQD cDFECD bQLNEHE^DFAClmLNMXAZGCLIGHE P D ?BXDFMGH?aGv LNlmDFE^IKGH?BLNMk
IKMIKllECLNlEH?VIKGCD ACDFEH?BDQA[LKd AC?aMW]UBD P w?aG@ECLNG^IKGH?BLNMA INXXDQXwmDd LNECDIKMX,IedfGCDFE $ (φ0) ) A_?aMGHcDbINACDSLKd ?a lUBDF0DFMGH?aMW [INXIKIKECXWxIKGCDQAfA^DQDSACDQbFGH?BLNM &%RGHcDFECDZ?BAI AH?a lUBDFE vbQLN l?aUBDQX ZrxDFECAH?BLNM uvMGHcD_bQLN lUBDFGCD@ACD/ P DFMbQD_LKd IKM IKUaWsLNEH?aGHc,9GHcD $ LNlmDFE^IKGH?BLNM `3?aUaUhGv9l?BbIKUaUaIKllmDIKE?aM lIK?aECA cD_lECLNwUBDF LKdkGHcD P M` IKMGCDQXYlcINACDSd INbFGCLNE −1 ?BAACLNUarxDQXwAH` IKll?aMWGHcD_Gv`L P w?aGCA %ECAHG`3?aGHcACLN0DIKEHw?aGHE^IKEHφ0 d LNE_DIK lUBD $ (0) ZIKMXAH` IKll?aMW wINb^\ UVIKGCDFE_`3?aGHcI φ0 X7?&YDFECDFMG_w π $ (π) T4?aGHc GHcD0cDFUal%LKd3GHcDbQLN lmL]AH?aGCD $ LNlmDFE^IKGH?BLNM%?aG/?aA/lmL]ACAH?awUBD0GCL lmDFEpdfLNEH 4GGAH?aMW]UBD P w?aG3ECLNG^IKGH?BLNMAS?aM L P ESbQLN l P G^IKGH?BLNMINU AClINbQD]
cD UVINAHG0 ?BACAH?aMW w P ?aUBX7?aMW wUBLb^\GCLIK\xD L P EAC?tMW]UaD ?aLNM P IKMG P WxIKGCD GCLLNUawmLbQLN lUBDFGCD ?BAI 2 60OQP@ARE"?34G8 C I132 RO:8 48VA wmDFGv`DQDFM?aMGCDFEHMIKUZIKMX%r?awE^IKGH?BLNMIKU P w?aG 3c?BA`3?aUaUG l?BbIKUaUa wmDYIP M?arxDFECA^IKUlcINACD WxIKGCDUa?a\xD ΦLKd_ACDQbFGH?BLNM ) GlECDQACDFMG?aML P ED7lmDFEH?a0DFMGTGHcDFECD IKECD GHcECDQDX7?&Y DFECDFMGZ0DFGHcLXA_IKGZcIKMX GCL ?a lUBDF0DFMG P M?arxDFECA^IKUklcINACD WxIKGCDQA
=
>||
>| >|||
>||
>||
>|| >||
>|| >|||
>||
>||
>||
>| >|||
>||
g ?aW P ECD = = `LYX7?&Y DFECDFMG[0DFGHcLXA[GCL ?a lUBDF0DFMG/IlcINACDWxIKGCDD9lmDFEH?a0DFMG^IKUaUax I cD I P n?aUa?VIKEH UBDFrsDFU 0DFGHcLXECD/ P ?aECDQA@IKM,INXX7?aGH?BLNMIKU ?aMGCDFEHMIKU UBDFrxDFU D∗ ) AC?aMW]UBD 2π l P UBACD/LNMGHcD/ECDQXAC?BXDFwIKMXLKdTGHcDI P 7?aUa?VIKEH GHE^IKMAH?aGH?BLNMIKllUa?BDQA_GHcD/lcINACD d INbFGCLNE@LKd −1 GCLGHcDAHG^IKGCD |S, 1〉 w ) bQLN lmL]AH?aGCD ACD/ P DFMbQD LKdSl P UBACDQAbIKMwmD dfL P MX%`3c?ab^cDRY DQbFGH?arxDFUaINbFGCAZUa?a\sD I2π
l P UBACD LNM C8Q> wU P D AC?BXDFwIKMX GHE^IKMAH?aGH?BLNMA_ %/IKMX bdHg ?aW =
• cD 0DFGHcLX DF lUBL xDQX?aMGHcD ?aE^INbnI]mLNUaUBDFE lECLNlmL]A^IKU =wINACDQXLNM P AH?aMW IKM^472 7OQG O 47E G*ARP@ARG&`3c?Bb^c `3?aUaUkwmD X7?BACb P ACACDQX ?aM GHcD[d LNUaUBL `3?aMW• cD B C; <5C@?ROQ8VA<2 G ?A GCDQb^cM?* P D`3c?Bb^c `3?aUaU IKUBACL wmD XDFG^IK?aUBDQX wmDFUBL ` • cD30DFGHcLX LKdW132547698:OA 8 47E ?D>[O R8Q?Z`3c?Bb^c`3?aUaUMLNG wmDZX7?BACb P ACA^DQXcDFECDZw P G ?aMACDQbFGH?BLNM(7&% IKUBLNMW`3?aGHc ?aGCAZD7lmDFEH?a0DFMG^IKU ECDIKUa? O IKGH?BLNMk
cD K1M190 M!! M A^b^cDF0D ?BAXDFl?BbFGCDQX?aMg ?aW= =9 I cD −1 lcINACDYLNMLNMDYLKdSGHcD wINAC?BAAHG^IKGCDQA[?BA INb/ P ?aECDQX,wI 2πECLNG^IKGH?BLNM LNM,GHcDECDQX,AC?BXDFwIKMX,LKd IKMI P 7?aUa?VIKEHGHE^IKMAH?aGH?BLNMk \_MUaGHcD
|S, 1〉 AHG^IKG^D lmLNl P UBIKGC?aLNM P MXDFEHWsLDQA I GHEJIKMAH?aGH?BLNM h`Zc?aUaD |S, 0〉 |D, 0〉 IKMX |D, 1〉 IKECD/MLNG[I Y DQbFGCDQX?tdTL@Ymn ECDQACLNMIKMG_D7bF?aG^IKGH?BLNMA@IKECD MDFW]Ua?aW]?awUBD cD 0DFGHcLX ECD/ P ?aECDQALKdTbQL P ECACDGHcDIrNIK?aUVIKw?aUa?aGv LKdIKM I P 7?aUa?VIKEHUBDFrxDFU /uoM,L P E bINA^DkI P 9?aUa?VIKEHUBDFrsDFUBA IKECDECDINX7?aUa,IKG[cIKMX,?tMd LNEH LKdLNGHcDFE]mDQDFIKMA P wUBDFrxDFUBA_LKdTGHcD D5/2 ACG^IKGCD] J@LNGCDcL `DFrxDFEGHcIKG_IKW]MDFGH?BbA P wUBDFrsDFUBA_LNGHcDFE_GHcIKM |D〉 IKECD/0LNECDACDFMAH?aGH?arxD[GCL IKW]MDFGH?Bb%DFUBX P bFG P IKGH?BLNMA@INAZXDFG^IK?aUBDQX ?aMACDQbFGH?BLNM ?'7@L `bIKM I@lcINACD3WxIKGCD3wmD ?a lUBDF0DFMGCDX0`3?aGHc0GHcD3cDFUalLKd IT/F"9! /M9"!6" g ?aW= = ACcL `ZAGHcIKGI l P UBA^DACD/ P DFMbQD`3c?Bb^cDRYDQbFGH?arxDFUa bQLNEHECDQAHlmLNMXAGCLI 2π
ECLNG^IKGH?BLNM ? D] GCL −1 · I LNM%GHcD|S, 0〉 ↔ |D, 1〉 476 YLNM GHcD |S, 1〉 ↔ |D, 2〉 GHE^IKMAH?aGH?BLNM fXDQAHl?aGCD[GHcD X7?&Y DFECDFMbQD[?aM _IKw?dECD/ P DFMbFwI[d INbFGCLNE √
2` L P UBX XL/GHcDSGHEH?Bb^\m# cD_AHG^IKGCDQA |S, 0〉 |S, 1〉 IKMX |D, 1〉 GHcDFM INb/ P ?aECD@I −1 AH?aW]M `3c?aUBD |D, 0〉 ECDFIK?aMA P MI YDQbFGCDQX cD bQLNEHECDQAHlmLNMX7?aMW IKGHEH?t ECDINXA
Φ′ =
1 0 0 00 −1 0 00 0 1 00 0 0 1
.
J@LNGCD3GHcIKG I[lcINACD d INbFGCLNE LKd −1 s`3c?Bb^c?BA W]UBLNwIKU?aM0GHcDZbQLN l P G^IKGH?BLNMIKUAHlINbQDscINA wmDQDFM X7ECLNllmDQX ) bQLN lmL]AC?aGCD l P UBA^D ACD/ P DFMbQD `3c?Bb^c 0DQDFGCAZGHcDQACD ECD/ P ?aECDF0DM9GCAZ?BARphase = R+(π,
π
2) R+(
π√2, 0) R+(π,
π
2) R+(
π√2, 0)
=
'
4 .
-101
-10
1
-1
0
1
xy
z
-101
-10
1
-1
0
1
x
y
z
g ?aW P ECD =?' UBLb^c AHlcDFECDGHE^I HDQbFGCLNEH?BDQA d LNEGHcDbFLN lmL]AC?aGCDlcINACDLNlmDFE^IKGH?BLNM Φ′ ;<! UBLb^cAHlcDFECD LKdGHcDZ P INAH?tn Gv`LKn UBDFrxDFUtnoAH7AHGCDF |S, 0〉 ↔ |D, 1〉 ' cD ?aM?aGH?VIKU7AHG^IKGCD ?BA INACA P 0DQX/GCLwmD |S, 0〉 s`3c?Bb^c?BATIKEH\xDQX wGHcD3wUVINb^\0IKEHECL ` M cD3l P UBACDZACD/ P DFMbQD3?BA Rphase P UBACDM P wmDFE%LKd3GHcDACD/ P DFMbQD R+( π√
2, 0)[ECLNG^IKGCDQA/GHcD AHG^IKGCD0rxDFbFGCLNEIKwmL P G/GHcD x nvIe7?aAw π√
2
P UBACDM P wmDFE INbQbQLN lUa?aACcDQA/I π n ECLNG^IKGH?BLNM%IKwmL P G[GHcD y nvIe9?BA uvG@GHcDFECDdfLNE^DGHE^IKMApd LNEH0A3GHcD AHG^IKGCD[GCL?aGCA3 ?aEHECLNEZ?aIKWsD/IKwmL P G3GHcD x n y n lUVIKMD] LNMACD/ P DFMGHUaUVINACDFEl P UBACDM P wmDFE ]`3c?Bb^c ?BA?BXDFMGH?BbIKUGCL@M P wmDFEL%NECLNG^IKGCDQATGHcD3AHG^IKGCD rsDQbFGCLNE IKUaU7GHcD ` IXL `3MYGCLGHcD_wmLNGHG^LN LKd GHcD@AClcDFECD]T P UBACD@M P wmDFE =0 P AHG Ua?a\sD_M P wmDFE ECDFlE^DFA^DFMGCAIπn ECLNG^IKGH?BLNM IKwmL P G@GHcD y nvIe9?BA S3cD%MIKUAHG^IKGCDGHc P A@?BA_?BXDFMGH?BbIKUGCL GHcD?aM?aGH?VIKUTLNMD0LX P UBLA^LN0DYlcINACD]uoGG P EHMA0L P G GHcIKG GHcD INb/ P ?aECDQXlcINACDYd INbG^LNE LKd@GHcD L rxDFE^IKUaULNlmDFE^IKGH?BLNM ?BA −1 scDFMbQDZGHcDSACD/ P DMbFD_INbFGCA Ua?a\xDSI AH?aMW]UBD 2π n ECLNG^IKGC?aLNM |S, 0〉 7→ −|S, 0〉IKMX |D, 1〉 7→ −|D, 1〉 2: cD0A^IK0D UVINA^DFE l P UBACD0ACD/ P DFMbQDIKMX?aGCAINbFGH?BLNM%LNMGHcD |S, 1〉 ↔ |D, 2〉 A P wAHlINbQD] uvG ?BA/DINAHGCL P MXDFECAHG^IKMX GHcIKG GHcDL rsDFE^IKUaU ACD/ P DMbFDwEH?aMWsAGHcD AHG^IKGCDrsDQbFGCLNEwINb^\/GCL@?aGCAACG^IKEHGH?aMW[lmLN?aMG+wUVINb^\IKEHECL `S 0LX P UBL ACLN0DlcINACD] P UtGC?tlUt9?aMW GHcD/IKGHEH?BbQDQA@LKd GHcD/?aMX7?ar?BX P IKU l P UBACDQASECDFrxDIKUBASGHchIKGSGHcDINb/ P ?aECDQXlcINACD?BASIKWxIK?aM −1 # c P A R′
phase
IKUBACL0INbFGCA3Ua?a\xD IAH?aMW]UBD2πn ECLNG^IKGH?BLNM |S, 1〉 7→ −|S, 1〉 IKMX
|D, 2〉 7→ −|D, 2〉
LNM GHcD |S, 0〉 ↔ |D, 1〉 wU P D AH?BXDFwIKMX \_M GHcD |S, 1〉 ↔ |D, 2〉 A P wAHlINbQD ?aGZECDINXA
R′phase = R+(
√2π,
π
2) R+(π, 0) R+(
√2π,
π
2) R+(π, 0).
g ?aW=?'?aUaU P ACGHE^IKGCDQA[GHcDACD/ P DFMbQDQA@?aMGHcD UBLb^cAHlcDFECDl?BbFG P ECD]J@L I P 9?aUa?VIKEHUBDFrxDFU ?BA@ECD/ P ?tE^DFX `3c?Bb^c IK\xDQA0GHcD 0DFGHcLX0LNECD P M?arxDFECA^IKU cDACD/ P DFMbQD G^IK\xDQA I,GCLNG^IKU_LKd 3.4π LKd AH?BXDFwIKMXECLNG^IKGH?BLNMA ) GCDQb^cM?BbIKU X7? b P UaGv LKd3GHcDbQLN lmL]AH?aGCD l P UBACD0GCDQb^cM?* P D bQLNMAH?BACGCA/?aM GHcD bQLNMGHECLNUaUBDQXAH`3?aGCb^c?aMW LKd GHcD UVINACDFEZlcINACDQAACDQD ACDQbFGH?BLNM '7
@
.
~*T/z ~ @ - Z mpzm~ pzh~y[CZC 9S uvM%GHcDwmDFW]?aMM?aMW LKdZGHc?BAACDQbFGH?BLNM GHcD P MXDFEHUa9?aMWlECLNwUBDF LKdSGHcD >[D P GCACb^cnVUsL O A^I,IKUaWsLNEH?aGHc ?BAlECDQACDFMGCDQX 3cD lECLNwUBDF cINA(%ECACGwmDQDFMXDR%MDQXw%>@D P GCA^b^c &% SIKMX?BAGHcDFECDdfLNE^D bQLN 0LNMUaECDd DFEHECDQX GCL0INA v>@D P GCA^b^c AZlECLNwUBDF^ LNMAH?BXDFESIKM P M\ML `3M LLNUBDIKM d P MbFGH?BLNM f(a) `3c?Bb^cINbFbQDFlGCAZLNMD$fbFUVINACAH?BbIKU w?aGZINA ?aMl P GSIKMXlECLX P bQDQA[LNMDfbFUVINACAH?BbIKUZw?aG[INA@L P GHl P GS cDFECDD7?BAHGSdfL P E@A P b^cd P MbFGH?BLNMA IKUaGCLNWsDFGHcDFEm`3c?Bb^cIKECDXDQACbFEH?awmDQX w GHcD GHE P GHcG^IKwUBD
a f1(a) f2(a) f3(a) f4(a) % %% % %
cD%ECAHGZGv`L d P MbFGH?BLNMA f1 IKMX f2 9?BDFUBXIKUa` I7AZGHcD A^IK0D/L P GHl P G?aMXDFlmDFMXDFMG@LKdTGHcD ?aMl P G aIKMX0IKE^D GHcDFECDdfLNECD3wmDZbQIKUaUBDQX^BDC760?R8 47698 R INb^cLNMD3LKdmGHcD3LNGCcDFETGv`L5f3IKMX
f4]cINATwmLNGHc IKMX %ZINAISlmL]A^AH?awUBDL P GHl P GM cDFIKECD bIKUaUaDQX 47G&46B"A d P MbFGH?BLNMA L cDlECLNwUBDF ?BA ML `GCL@XDFGCDFEH ?aMD3`3c?Bb^cbFUVINACA fbQLNMAHGJIKMG@LNESwIKUVIKMbQDQX9_I0W]?arsDFM P M\ML `3Md P MbFGH?BLNMwmDFUBLNMWsA_GCL hcL ` IKM bQLN l P G^IKGH?BLNMIKUAHGCDFlA IKECDMDQDQXDQX J@LNGCDGHcIKG[`D IKECDMLNG INAH\9?aMWYdfLNE[GHcDlIKEHGH?Bb P UVIKE r]IKU P DQA fi(0) IKMX fi(1) w P Gd LNESIW]UBLNwIKU lECLNlmDFEHGv LKd
fi
cD3bFUVINACAC?abIKUrxDFECAH?BLNM0LKdhGHcD lECLNwUBDF(bIKM wmDdfLNEH P UVIKGCDQXINAdfLNUaUBL `ZA [?arsDFMI@LNMDn w?aG bQLN l P GCDFEINA I owUVINb^\wmL `3c?Bb^c bIKUBb P UVIKGCDQA/LNMDLKdGHcDd P MbFGH?BLNMA fi GHcD ACLKdfGv` IKECD_w P G ?aG[?BA MLNG[\9ML `3M`3c?Bb^ck# cDSWsLsIKUh?BA GCLXDFGCDFEH ?aMD9`3c?Bb^c GvlmDSLKdklECLNW]E^IK ? D]`3c?Bb^c GvlmD_LKdd P MbFGH?BLNM 9bQLNMACG^IKMGLNE wIKUVIKMbQDQX9T?BATE P MM?aMWLNMGHcDZbQLN l P GCDFEMT?aGHc I$B3G 4@?D?RO:BD47GbQLN l P GCDFE GHcD d P MbFGH?BLNM P AHG IKUa` I9AwmD0DFrNIKU P IKGCDQX Gv`3?BbQD^GHcD lECLNW]E^IK P ACG wmD E P MGv`3?BbQD[GCL %MX L P G uod dfLNE/DIK lUBD GHcD ?aMl P G% ?BA_b^cL]ACDFM,IKMXGHcDL P GHl P G_G P EHMA@L P GSGCLwmD GHcDFMGHcD d P MbFGH?BLNM,bFL P UaXwmD f1 fbQLNMAHG^IKMG"SLNE f4wIKUVIKMbQDQX9 ) A^DQbQLNMX,DFrNIKU P IKGH?BLNM kGHc?BA[GH?a0D`3?aGHc INA[?aMl P G ?BA@MDQbQDQACA^IKEHGCL XDFGCDFEH ?aMD?td GHcDd P MbFGH?BLNM ?BAZbQLNMAHG^IKMG_LNE3wIKUVIKMbQDQX uvG`3?aUaU3ML ` wmD AHcL `3McL ` `3?aGHc IAC?a lUBDN P IKMG P bQLN l P GCDFE0GHcD AH9 0DGCEH fbQLNMAHG^IKMG0LNEwIKUVIKMbQDQX9 LKd IKM P M\9ML `3M d P MbFGH?BLNM ?BA LNwG^IK?aMDQX ?aM P AHG LNMD,E P Mk cD P IKMG P lECLbQDQX P ECDL P GHUa?aMDQX?aMGHcDdfLNUaUBL `3?aMW ` INA lECLNlmL]ACDQXw UBDFrxD RT\xDFEHG INbQb^c?VIrxDFUaUBLIKMXL]ACbI &%>S d LNE/Ic?BAHGCLNEHLKdGHcD >@D P GCA^b^c7nVUsL O A^I IKUaWsLNEH?aGHc, A^DQDGHcD DFMX,LKdGHc?BA ACDQbFGH?BLNM cDX P IKMG P lECLNwUBDFECD/ P ?aECDQAYINXIKlG^IKGH?BLNMLKd GHcD d P MbFGH?BLNMA c?BA?BAMDQbQDQACA^IKEHINAMLNGYIKUaUZd P MbFGH?BLNMA LKd GHcDACDFG f1 nf4IKECD P M?aG^IKEH `3cDFMGHcDFIKECD ECDINXINA LNlmDFE^IKGH?BLNMA0`3c?Bb^cIKldfECLN LNMD 132 RO:8GCLLNMDN P w?aGuvM4lIKEHGH?Bb P UVIKE_GHcDbQLNMACG^IKMGYd P MbFGH?BLNMA IKECDMLNG cDWsDFMDFEH?BblECLbQDQX P ECD,GCLINXIKlG A P b^cMLNM7nP M?aG^IKEH bFUVINA^AH?BbIKUkLNlmDFE^IKGH?BLNMAZd LNE+ P IKMG P bQLN l P G^IKGH?BLNM?BA3GCL0INXXIKMIKMbF?aUaUVIX P w?aG &% %>o cDbQLN l P G^IKGH?BLNMIKU AHlINbQD/GHc P A@bQLNMAH?BAHGCA@LKd Gv`L P w?aGCAhGHcDF?aE P IKMG P AHG^IKGCD/`3?aUaU wmDXDFMLNG^DFX |a,w〉 uvG@`3?aUaUTwmDAHcL `3M,GCcIKG IYA P ?aG^IKwUBD` IGCL D7GCDFMXGHcDd P MbFGH?BLNMA fi GCLYGHcDGv` LKnV P w?aG AHlINbQD?BA@GCLECDFlUVINbQD GHcDF w 2 60O:8 47E GHE^IKMApd LNEHIKGH?BLNMA
Ui`3c?Bb^cIKECD XDR%MDQX wYGHcD[IKl
Ui : |a,w〉 7→ |a,w ⊕ fi(a)〉,=
`3cDFECD ⊕ ?aMX7?BbIKGCDQA INXX7?aGH?BLNM 0LX P UBLGv`LR P ?ar]IKUBDFMGHUa GCL%GHcDbFUVINACAH?BbIKU[bINACDZGHcD P IKMG P lECLNwUBDF;bIKMML ` wmD D7lECDQACACDQXINAZdfLNUaUBL `ZA @?arxDFMI,wUVINb^\ wmL0 Gv`LKnV P w?aGS P IKMG P bQLN l P GCDFELNM `3c?Bb^cLNMDLKd GHcD olECLNW]E^IK0A Ui
?BA0?a lUBDF0DFMGCDQX IKMX?aG?BAMLNG\9ML `3M `3c?Bb^ck cD WsLsIKU?BAGCLXDFGCDFEH ?aMDY?aMLNMUa LNMDE P MLKdZGHcD P IKMG P bQLN l P GCDFE?tdZGHcDlECLNW]E^IK bQLNEHECDQAHlmLNMXAGCL,I
IH
4 .
⊗
|ψ >0 |ψ >1 |ψ >2 |ψ >3
| >
| >
g ?aW P ECD =?@ P IKMG P bF?aECb P ?aGSECDFlECDQACDFMG^IKGH?BLNMLKd GHcD >[D P GCACb^c7nVUsL O A^IIKUaWsLNEH?aGHc
bQLNMAHG^IKMGLNE IwIKUVIKMbQDQXfi ) \xD%dfDIKG P ECD LKd[GHcD IKUaWsLNEH?aGHc ?BA0GCLDFr]IKU P IKGCD Ui `3?aGHcGHcD Gv`L P w?aGCA ?aMA P lmDFEHlmL]AH?aGH?BLNMA IKGYGHcD?aMl P GZGHc?BA ` I IKUaUBL `3?aMW P IKMG P ?tM9GCDFEpdfDFECDFMbQD,LKd/X7?&Y DFECDFMGbQLN l P G^IKGH?BLNMIKU lIKGHcA &%>o/LNECDlECDQbF?BACDFUa GHcDlE^LxbQDQX P E^D AHG^IKEHGCA@`3?aGHcGHcDACG^IKGCD |a,w〉 = |0, 1〉IKMX GHcDFM GHcD ACD/ P DFMbQD LKdTLNlmDFE^IKGH?BLNMA
UDJ = H1 H2 Ui H2 H1= =
?BAZIKllUa?BDQX W cD%ECAHG3Gv`L [INXIKIKECX WxIKGCDQAZbFECDIKGCD[GHcD A P lmDFEHlmL]AH?aGH?BLNMA7`3c?aUBD[GHcD[Gv`L @INXIKIKECXWxIKGCDQAYIedGCDFEUi
ACDFEHrxD GCLECDQbQLNw?aMDX7?&Y DFECDFMGlIKGHcAYIKMXIK\xD GHcD ?aMG^DEHdfDFECD] cDbQLN lUBDFGCDIKUaWsLNEH?aGHc?BAZXDFl?BbFGCDQX?aMg ?aW0=?@7 jkDFG P A3dfLNUaUBL ` GHcD P IKMG P ACG^IKGCD LKdGHcD AC9AHGCDF GHcECL P W]cGHcDIKUaWsLNEH?aGHc cD ?aMl P G_AHG^IKGCD |ψ0〉 = |0, 1〉 ?BA3GHE^IKMApd LNEH0DQX wYGHcD %ECAHG3G `L [INXIKIKECX WxIKGCDQAZ?tM9GCL
|ψ1〉 =
[ |0〉+ |1〉√2
]⊗[ |0〉 − |1〉√
2
]
=1
2[|0, 0〉 − |0, 1〉+ |1, 0〉 − |1, 1〉] .
J@L `Ui?BA_IKllUa?BDFX ECDQA P UaGH?aMW0?aM GHcD ACG^IKGCD
|ψ2〉 =1
2[|0〉|fi(0)〉 − |0〉|fi(0)⊕ 1〉+ |1〉|fi(1)〉 − |1〉|fi(1)⊕ 1〉]
=1
2
[(−1)fi(0)|0〉(|0〉 − |1〉) + (−1)fi(1)|1〉(|0〉 − |1〉)
]
=
[(−1)fi(0)|0〉+ (−1)fi(1)|1〉√
2
]⊗[ |0〉 − |1〉√
2
]
cD UVINAHG3Gv` L AC?aMW]UBD P w?aGZECLNG^IKGH?BLNMAZ9?BDFUBX GHcD %MIKU ACG^IKGCD[wmDd LNECD GHcD[0DINA P ECDF0DFMG
|ψ3〉 =
[(−1)fi(0)(|0〉+ |1〉) + (−1)fi(1)(|0〉 − |1〉)
2
]⊗ |1〉
=
[(−1)fi(0) + (−1)fi(1)
2|0〉+ (−1)fi(0) − (−1)fi(1)
2|1〉]⊗ |1〉.
(
.
) bFUBL]ACDFE UBLLN\ IKG GHcD %MIKUAHG^IKGCDZLKdGHcD %ECAHG P w?aG ECDFrxDIKUBA GHcIKGS0LX P UBLI@lmL]A^AH?awUBDW]UBLNwIKU d INbFGCLNELKd −1 ?aGSE^DQINXA |0〉 ?td fi(0)⊕ fi(1) = 0`3c?Bb^c?BAZGHcDbINACD ?td IKMXLNMUa ?td"
fibQLNMAHG^IKMGS\_MGHcDLNGHcDFE cIKMX GHcD %ECAHGW P w?aG A AHG^IKGCD3ECDINXA |1〉 ?td fi(0)⊕fi(1) = 1
s`3c?Bb^c0?BATGHcD3bINA^D ?td# IKMX0LNMUa?td"fiwIKUVIKMbQDQX W c?BA?BA GHcD@XDQAH?aECDQX ECDQA P UaGM cD_rNIKU P D@LKdkGHcD+%ECAHG P w?aG: P w?aG a IKGGHcD_DFMXYLKdkGHcDACD/ P DFMbQD?BA |0〉 LNE |1〉 XDFlmDFMX7?aMWLNMGHcD AH 0DFGHECLKd fi ) bQbQLNECX7?aMW]UakGHcD0IKUaWsLNEH?aGHc %M?BAHcDQA`3?aGHcI_0DINA P ECDF0DFMG LKdhGHcDZ%ECACG# P w?aGM cDECDQA P UaG LKdhGHc?BA0DINA P ECDF0DFMG ?aMX7?BbIKGCDQAT`3?aGHc0bQDFEHG^IK?aMGv`DFGHcDFE
fi?BAbQLNMAHGJIKMG LNEwIKUVIKMbQDQX WJ@LNGCD_GHcIKG?aMYGHc?BAlECLNGCLbQLNU GHcDSd P MbFGH?BLNM fi ?BADFr]IKU P IKGCDQX LNMUaLNMbQD P MUa?a\xD_?aMYGHcD@bFUVINACAC?BbQIKU bINACD]TuoM INXX7?aGH?BLNM LNMUaLNMD P w?aG0DINA P ECDF0DFMG3?BA MDQbQDQA^A^IKEH9`3c?aUBDGHcD bFUVINACAH?BbIKUklECLNGCLbQLNU ECD/ P ?aECDQAZGv`L 0DINA P ECDF0DFMGCA
cDSUVINACG [INXIKIKECXYLNlmDFE^IKGH?BLNM LNM GHcD_A^DQbQLNMX P w?aG H2 ?BA MLNG ECD/ P ?aECDQXd LNE GHcDSlmDFEpd LNEHIKMbQDLKd@GHcDIKUaWsLNEH?aGHc w P G ` INA0?aMGHECLX P bQDQXcDFECDYd LNE ECDINACLNMALKd[AH9 0DFGHEHs c?BA `3?aUaUSAH?a lUa?tdf%GHcDGHE^IKMAHUVIKGH?BLNMLKdGHcD IKUaWsLNEH?aGHc?aMGCL UVINACDFE3l P UBA^DQASINbFGH?aMWLNMIGHE^IKllmDQX ?BLNMk cDIKwmL rxDIKUaWsLNEH?aGHc;ECDFlECDQACDFMGCA[IKM?a lECL rxDFXrxDFECAH?BLNM &% %>LKdTGHcD/LNEH?aW]?aMIKUL P IKMG P IKUaWsLKnEH?aGHc lECLNlmL]A^DQX w>@D P GCA^b^c % V9`3c?Bb^c ECDFlECDQACDFMGCA LNMUa I lECLNwIKwUa?BAHGH?Bb[ACLNU P GH?BLNM TuoM GHcD@LNEH?aW]?aMIKUIKUaWsLNEH?aGHcGHcD@d P MbFGH?BLNM fi ?BAZDFr]IKU P IKGCDQX LNMUa LNMbQD[w P GZGHcDFECD/IKECD[GHcECDQD lmL]ACAC?awUBD L P GCbQLN0DQA vbQLNM7nAHG^IKMG D owIKUVIKMbQDQX @IKMXo?aMbQLNMbFU P AH?arxD^TgLNEIKM fi xGHcD_IKUaWsLNEH?aGHc `3?aUaUmL P GHl P G `3?aGHcYI lECLNwIKw?aUa?aGvLKdO' %DF?tGCcDFEvbQLNMAHG^IKMG 0LNEowIKUVIKMbQDQX IKMX,GCcD L P GHl P G/GHcDFM bQLNEHECDQbFGHUaXDQACbFEH?awmDQA fi 0uoMGHcDQACDbINACDQAmGHcDAC 0DFGHEHLKd
fi?BA@LNwG^IK?aMDQX?aMI AH?aMW]UBDDFrNIKU P IKGH?BLNM `3c?Bb^c,bIKM,MLNG@wmDINbQbQLN lUa?BACcDQXw0bFUVINACAH?BbIKUmbQLN l P G^IKGH?BLNMk uoM GHcD_LNGHcDFE ' %*LKd bINACDQA? D]`3cDFMYGHcD_L P GHl P G ?BA o?aMbQLNMbFU P AH?arsDD9ML?aM7d LNEHIKGH?BLNM?BAZLNwG^IK?aMDQX
cDFECD/D7?BAHGCA_IKMD7GCDFMAH?BLNMLKdT>[D P GCACb^c ASlECLNwUBDF,`3c?Bb^c` INA%ECAHGZlECDQA^DFMGCDQX w >[D P GCACb^cIKMXUsL O A^I &%( @?arxDM,I d P MbFGH?BLNM F (a) `3c?Bb^cINbQbQDFlGCA IAHGHEH?aMWYLKd n ?aMl P G@w?aGCA? D]I M P wmDFE@dECLN GCL 2n − 1
IKMX lECLX P bQDQA_LNMD w?aG_INASL P GHl P G LNECDQL rxDFE F ?BAZLKdTLNMD LKdGv`L \9?aMXA DF?aGHcDFEF (a)?BANBDC765?38 47698 dfLNE IKUaU r]IKU P DQA LKd a LNEDFUBA^D F (a) ?BA 47G 476B"A GHcIKG?BATD/ P IKU GCL % d LNEDINbFGHUa%cIKUtdLKd IKUaUlmL]ACAH?awUBD
akIKMX dfLNE[GHcDLNGHcDFE cIKUtdp/uvMGHcDF?aE%ECACG@lE^LNlmLNAJIKUT>[D P GCACb^c IKMXUsL O A^IYlECDQACDFMGI P IKMG P IKUaWsLNEH?aGHc `3c?Bb^cECD/ P ?aECDQAGv`L,DFrNIKU P IKGH?BLNMA LKd F ) UBA^L?aMGHc?BAbINA^D UBDFrsD RT\xDFEHG,INbQb^c?VIrsDFUaUBLIKMXL]ACbI &%>mcIrsD3d L P MX IKM0?a lECL rxDQX P IKMG P IKUaWsLNEH?aGHc W cDZUVIKGHGCDFE ECD/ P ?aECDQALNMUaI ?"OQ67G*A DFr]IKU P IKGH?BLNM LKd F IKMXI ?RO6G A 0DINA P E^D0DFM9G GHc?BA GH?a0D LKd IKM n nV P w?aG ECDFW]?BAHGCDFE"GCLYECDFrxDIKUGHcDW]UBLNwIKU AH9 0DFGHEHlECLNlmDFEHG LKd
F ) X7?aECDQbFG bQLN lIKEH?BACLNM LKd lmDFEpd LNEHIKMbQD`3?aGHcGHcDbFUVINACAH?BbIKU A^LNU P GH?BLNM ?BA cL `DFrxDFE 0LNECD X7? b P UtGGHcIKM d LNE/>[D P GCACb^c ALNEH?aW]?aMIKU lECLNwUBDF UBDIKEHUaTIbFUVINACAH?BbIKU bQLN l P G^IKGH?BLNM `3?aUaUECD/ P ?aECD 2n−1+1
DFr]IKU P IKGH?BLNMAfDINb^c DFr]IKU P IKGH?BLNM ?aMbFU P X7?aMWIAH?aMW]UBDn w?aG0DINA P ECDF0DFMG" GCL,lECL rxD `3?aGHc bQDFEHG^IK?aMGvGHcIKG0GHcD P M\9ML `3Md P MbFGH?BLNM?BA0bQLNMAHG^IKMG @DMbFD GHcDbFUVINACAH?BbIKU A^LNU P GH?BLNMACbIKUBDQA@D7lmLNMDFMGH?VIKUaUa `Z?tGCc n SuoMGHcD/wmDQAHG@bINACDcL `DFrsDFEhGv` L DFrNIKU P IKGH?BLNMA IKECDDFML P W]c ? D]9?td GHcDS%ECAHGGv`LDFr]IKU P IKGH?BLNMA3LKd F `3?aGHc Gv`LE^IKMXLN Ua b^cL]ACDFMa?BDFUBX
0IKMX
1cDFMbQD
F?BA \9ML `3MGCLwmD wIKUVIKMbQDQX9 ) lECLNwIKwIKUa?BAHGH?BbbFUVINACAH?BbIKUSbQLN l P GCDFE`3c?Bb^c DFr]IKU P IKGCDQA F (a) dfLNEI dfDF` E^IKMXLN Uab^cL]ACDFM
akbQL P UBX P ?Bb^\9UaXDFGCDFEH ?aMD `3?aGHc c?aW]clECLNwIKw?aUa?aGv,?td F ?BA bQLNMAHGJIKMG LNEwIKUVIKMbQDQX
) UaGHcL P W]c 9INA lECDQA^DFMGCDQXx?a lECL rxDQX0rxDFECAH?BLNMA LKd GHcDZLNEH?aW]?aMIKUhlECLNlmL]A^IKUBA D7?BAHGIKUaU P IKMG P IKUaWsLKnEH?aGHc0AZX7?BA^b P ACACDQXYcDFECD/IKECD bQLN 0LNMUaYECDd DFEHECDQX GCL0INA v>[D P GCACb^c7nVUxL O A^I IKUaWsLNEH?aGHc0A &% ?aM cLNML P ELKd GHcDbFE P bF?VIKUTX7?BACbQL rxDFEH?BDQA bQLNMG^IK?aMDQX,?aMGHcDLNEH?aW]?aMIKUTlIKlmDFECA_>[D P GCACb^c A IKUaWsLNEH?aGHc ` INA[GHcD%ECACGbQLNMbFECDFGCDXDF0LNMAHGHE^IKGH?BLNMGHcIKG@IX P IKMG P bQLN l P GCDFE@bQL P UBXXL0ACLN0DFGHc?aMWd INAHGCDFEZGHcIKMI bFUVINACAH?BbIKUbQLN l P GCDFE TGHcDD9GCDFMAC?BLNM w>@D P GCA^b^cIKMX UsL O A^I ` INA/GHcDX%ECAHG/DIK lUBDLKdSI ACbIKUVIKwUBD0lECLNwUBDF`3cDFECD/GHcD P IKMG P IKUaWsLNEH?aGHcL P GHlmDFEpdfLNEH0A@GHcDbFUVINACAC?abIKU A^LNU P GH?BLNMECDFWxIKECX7?aMWYGHcDACbIKUa?aMWYLKd GHcDGH?a0D ECD/ P ?aECDQX GCL0ACLNUarsD[GHcD lECLNwUBDF IKGZUBDINAHG3dfLNEZA P ?aG^IKwUa b^cL]ACDFMbFEH?aGCDFEH?VI0LKdbQLN lIKEH?BACLNMF
>
4 .
W ' # 97 !uvM GHc?BA ACDQbFGH?BLNM/GHcD >[D P GCACb^c7nVUxL O A^ISIKUaWsLNEH?aGHc ?BAkGHE^IKMAHUVIKGCDQX/LKd7?aMGCL_I3ACDFEH?BDQA LKd7UVINACDFEkl P UBACDQA INbFGH?aMW_LNMI?RO6G A GHE^IKllmDQX?BLNM N`3c?Bb^c0INbQbQLNECX7?aMW GCL@ACDQbFGH?BLNMS= ECDFlECDQACDFMGCA I_Gv`LKnV P w?aGW P IKMG P -lECLbFDQA^ACLNEuvG cINA wmDQDFM L P GHUa?aMDQX?aM ACDQbFGH?BLNM GHcIKG@INXIKIKECXWxIKGCDQA IKECD MLNGlIKEHGH?Bb P UVIKEHUa`DFUaUA P ?aGCDQXdfLNETIlc7AH?BbIKU?t lUaDF0DFMGJIKGH?BLNMYw P G GHcIKG GHcDF bIKMGv9l?BbIKUaUawmDZGHE^IKMAHUVIKGCDQX?aMGCL/A P ?aG^IKwUBD π/2 ECLNG^IKGH?BLNMA) bQbQLNECX7?aMW]UaT?aMGHcDYbINACD LKdSGHcDY>[D P GCACb^c7nVUsL O A^I,IKUaWsLNEH?aGHc, GHcDYACD/ P DFMbQD LKd_D/Mk= = /bIKMwmDGHE^IKMAHUVIKGCDQX ?aMGCL
U ′DJ = Ry1(π/2) Ry2(π/2) Ui Ry2(π/2) Ry1(π/2).
=?'uvG bIKMYwmD_DINAC?aUa rxDFEH?&%DQX GHcIKGGHc?BAACD/ P DFMbQD `3?aGHcYGHcD@A^IK0D_?aMl P G |ψ0〉 = |0, 1〉 T9?BDFUBXA GHcD@A^IK0D%MIKUkECDQA P UaG_INA3GHcD ACD/ P DFMbQD LKdD/Mk = = J@D9GTGHcD IKGHEH?BbQDQA LKd_GHcD GHE^IKMAHdfLNEHIKGH?BLNMA
UiIKECD bIKUab P UVIKGCDQX INbQbFLNE^X7?aMW,GCLGHcDF?aE XDR%M?aGH?BLNM
= /IKMX GHE^IKMAHUVIKGCDQX ?aMGCLA P ?aG^IKwUBD LNlmDFE^IKGH?BLNMA`3c?Bb^cbIKM wmD?a lUBDF0DFMGCDQX%?aM%L P ED7lmDFEH?a0DFMGU1
?BA3GHcD ?BXDFMGH?aGv
U1 =
1 0 0 00 1 0 00 0 1 00 0 0 1
= I,
? D]U1
bIKM wmD[?a lUBDF0DFMGCDQXw vXLN?aMW0MLNGHc?aMW^
U2 =
0 0 1 00 0 0 11 0 0 00 1 0 0
.
) bFUBL]ACDFE UBLLN\IKG GHc?BA IKGHEC? AHcL `ZA GHcIKGU2
?BA/I LNlmDFE^IKGH?BLNM%LNMGHcD0ACDQbQLNMX P w?aGkcDFMbQDU2 = −iRx2(π)
fbdHA^DQbGC?aLNM &%
U3 =
1 0 0 00 0 0 10 0 1 00 1 0 0
,
U3LNwr9?BL P AHUa ECDFlECDQACDFMGCA[I LNlmDFE^IKGH?BLNM,bQLNMGHECLNUaUBDQXw GHcD%ECACGS P w?aGh`3c?Bb^cINbQbQLNECX7?aMWYGCLACDQbFGH?BLNM bIKM wmD[`3EC?tGCGCDFMINA U3 = Ry2(π/2) Φ Ry2(π/2)
U4 =
0 0 1 00 1 0 01 0 0 00 0 0 1
.
U4G P EHMA[L P G@GCL wmD I LNlmDFE^IKGH?BLNM IKWxIK?aMbQLNMGHECLNUaUBDQX,w GHcD%ECAHG P w?aG[uvG[bIKMGHc P A@wmD`3EH?aGHGCDFMINA
U4 = Rx1(π) Ry2(π/2) Φ Ry2(π/2) Rx1(π)
cD MD9G AHGCDFl/?BA GCLZ?BXDFMGH?tdf GHcD P w?aGCA' cD#%E^AHG' P w?aG a ?BA GHcD LNMD GHcIKG chINA GCLZwmD 0DINA P ECDQX UBDIKEHUa[mGHcD/?aMGCDFEHMIKU# P w?aG_?BA_lE^DJd DFE^IKwUa P ACDQX d LNE_GHc?BA@G^INAH\INA[L P E@DFUBDQbFGHECLNMACcDFUtr9?aMWYGCDQb^cM?* P DlECDQbF?BACDFUaX7?BAHGH?aMW P ?BAHcDQAwmDFGv`DQDFM |S〉 IKMX |D〉 ) X7?aECDQbFGECDINX9noL P G LKdkGHcDSr9?awE^IKGH?BLNMIKU P w?aGLNM GHcD
=
.
| >
| >
|< >|
|ψ >= |0 >0 |ψ >1 |ψ >2 |ψ > = | >3
3
3
⊗
π/2)
| > π/2)
π/2)
π/2)
g ?aW P ECD = H 0 cD bQLN lUBDFGCD/>[D P GCACb^c7nVUsL O A^IIKUaWsLNEH?aGHc;INXIKlGCDQX d LNE3GHcD D9lmDFEH?a0DFMG
LNGHcDFE/cIKMX ?BA MLNG/lmL]ACAH?awUBD0?aMI AH?aMW]UBDAHcLNG/0DQINA P ECDF0DFMGN cD IKUaWsLNEH?aGHc ECD/ P ?aECDQA/GHcD0?aMl P G|0〉 d LNE GHcDS%ECAHG P w?aGSIKMX GHcD[?BLNM MIKG P E^IKUaUaY?BA ?aM GHcD |S〉 ACG^IKGCD IKG3GHcD[wmDFW]?aMM?aMW LKdIACD/ P DFMbQD] cDFECDd LNECD/GHcD/UBDFrsDFUBA@IKECD/?BXDFMGH?&%DQX,INAZd LNUaUBL `ZA |S〉 ≡ |0〉 IKMX |D〉 ≡ |1〉 cD/r?awE^IKGH?BLNMIKUM P w?aGGHcDFM cINA GCL wmD G^IK\xDFM%INA GHcD0ACDQbQLNMX P w?aG w [LKd GHcDIKUaWsLKEC?tGCc w ?BA/A P llmL]ACDQXGCL wmD?aM GHcDUBLNW]?BbYAHG^IKGCD |1〉 IKGGHcDwmDFW]?aMM?aMW,LKdZGHcDYACD/ P DMbFD] cDr9?awE^IKGH?BLNMIKU0LXD IKGGHc?BAlmLN?aMG IedfGCDFEAH?BXDFwIKMXYbQLLNUa?aMW ?BA ?aM GHcD |n = 0〉 AHGJIKGCD]uvG GHcDFECDd LNECD_bQLN0DQAMIKG P E^IKUaUa0GCL/IK\xDSGHcD@INACACLbF?VIKGH?BLNM|n = 1〉 ≡ |0〉 IKMX |n = 0〉 ≡ |1〉 ) ACUa?aW]cG 0LX7?&%bIKGH?BLNM LKdkGHcD[IKUaWsLNEH?aGHc ?BA INXD_w0AClmDQb?tdf9?aMWGHcD_wUVINb^\0wmL INA wmDF?aMW AHUa?aW]cGHUaUVIKEHWsDFEGHcIKM
Ui^ cDMDF`#wUVINb^\ wmL bQLNEHECDQAHlmLNMXAGCLGHcD XINAHcDQX wmL ?aMg ?aW= H A P b^c%GHcIKG?aG?aMbFU P XDQA GHcDGv`L π/2 l P UBACDQA LNMGHcD ACDQbQLNMX P w?aG. c?BA ?BA P AHG I bQLNMbQDFlG P IKU b^cIKMWsD INXDGCLAH?a lUa?tdf GHcDYL rxDFE^IKUaU ACD/ P DFMbQD] c?BAb^cIKMWsD bQLNEHECDQAHlmLNMXAGCL,I Ui `3c?Bb^cLNMGHcDYACDQbQLNMX P w?aGLNlmDFE^IKGCDQAS?aM GHcD(|0〉+ |1〉), (|0〉+ |1〉) wINAH?BA7EJIKGHcDFEZGHcIKM ?aM GHcD |0〉, |1〉 wINAH?BA cD Gv`L
π/2ECLNG^IKGH?BLNMA_LKdGHcD%ECAHGS P w?aGS?tMGHcD/L rsDFE^IKUaU ACD/ P DFMbQD fD/Mk!=?'FR Ry1(π/2)
IKMXRy1(π/2)
NbIKMbQLNMrxDFM?BDFMGHUa wmD ?a lUBDF0DFMGCDQX INA bIKEHEH?BDFE l P UBACDQA `3?aGHcGHcDIKllECLNlEH?VIKGCD UVINACDFE lcINACDQATcIKG ECDFIK?tMAZIKECD@GHcD_d L P E X7?&Y DFECDFMGwUVINb^\wmL9 LNlmDFE^IKGH?BLNMA Ry2(π/2) Ui Ry2(π/2)
9`3c?Bb^c bIKMwmD ?a lUBDF0DFMGCDQXINA3dfLNUaUBL `ZA• i = 1
Ry2(π/2) U1 Ry2(π/2) = Ry2(π/2) I Ry2(π/2) = I
?BA3GHcD[?BXDFM9GH?aGv LNlmDFE^IKGCLNE• i = 2
Ry2(π/2) U2 Ry2(π/2) = Ry2(π/2) Rx2(π) Ry2(π/2)
0LX P UBLIKM?aEHECDFUBDFrNIKMG W]UBLNwIKUlcINACD −i M c?BA?BATI@ACDFEH?BDQATLKdmAH?aMW]UBD P w?aGECLNG^IKGH?BLNMAs`3c?Bb^c IACDQDF DINAH/GCL@?a lUBDF0DFMGT D IKECDcL `DFrxDFEbQLNMbQDFEHMDQX `3?aGHc ECLNG^IKGH?BLNMA_LKd GHcD A^DQbQLNMXGHcIKG3?BA7GHcDXPO REF48:O C7647GM P w?aG cD ` I@GCLZ?a lUBDF0DFMG GHc?BAkchINAkwmDQDFML P GHUa?aMDQX/?aM/ACDQbFGH?BLNMC= kLNlmDFE^IKGH?BLNMA LNM/GHcDTr9?awE^IKGH?BLNMIKU P w?aG3IKECD@INb^c?BDFrsDQX wA^IKMX7`3?Bb^c?aMWGHcD@LNlmDFE^IKGH?BLNM IKllUa?BDQXYGCL/GHcDS?aMGCDFEHMIKU P w?aGwmDFGv`DQDFMGv`L $ LNlmDFE^IKGH?BLNMAS`3?aGHc `DFUaU XDR%MDQX ECDFUVIKGH?arxD lcINACD? D]
Ry2(π/2) U2 Ry2(π/2) = $ (π) Ry1(π/2) Rx1(π) Ry1(π/2)
$ (0). cD bQLNEHECDQAHlmLNMX7?aMW A^DFEH?BDFASLKd UVINA^DFE3l P UBACDQAZECDINXA
Rswap(π) Ry1(π/2) Rx1(π) Ry1(π/2) Rswap(0).
=%
4 .
| >
| >
π/2,π/2
π/2,−π/2
| >
| >
π/2,π/2
π
π/2,π/2 π,π
π/2,−π/2 π/2,−π/2
| >
| >
π/2,π/2
π/2,−π/2
| >
| >
π/2,π/2
Φ
π,π π/2,−π/2
π,π
Φ
| >
| >
| >
| >
g ?tW P ECD =?( LN lUBDFGCD/ACD/ P DFMbQD@d LNESIKUaU dfL P ESbINACDQA i = 1 . . . 4LKd GHcD >[D P GCACb^c7nVUsL O AJI0IKUaWsLNEH?aGHc
• i = 3Ry2(π/2) U3 Ry2(π/2) = Ry2(π/2) Ry2(π/2) Φ Ry2(π/2) Ry2(π/2) = Φ
-9L GHcDwUVINb^\YwmL bQLNEHECDQAHlmLNMXASGCL0IlcINACD WxIKGCD]• i = 4
Ry2(π/2) U4 Ry2(π/2) = Ry2(π/2)Rx1(π)Ry2(π/2) ΦRy2(π/2)Rx1(π)Ry2(π/2)
= Rx1(π) Φ Rx1(π)NJ_LNGCDGHcIKGAH?aMW]UBD P w?aGECLNG^IKGH?BLNMAINbFGH?aMWLNM%X7?&Y DFECDFMG P w?aGCAbQLNn P GCD]M c P As?aM0bINACD = `D WsDFG I@lcINA^D WxIKGCDxA P EHECL P MXDQX0wGv`L π n ECLNG^IKGH?BLNMA LKdmGHcD ?aMGCDFEHMIKU P w?aG
-9DQbFGH?BLNM = X7?BACb P ACA^DQX GHcIKG dfLNE GHcD ?a lUBDF0DFMG^IKGH?BLNM LKdhIZlcINACD WxIKGCD Φ `D cIrxD GHcD b^cLN?BbQD wmDFGv`DQDFMGHcECDQD X7?&Y DFECDFMG0DFGHcLXA, L?a lUBDF0DFMGGHcD >[D P GCACb^c7nVUsL O A^IIKUaWsLNEH?aGHc`D XDQbF?BXDQX GCL P ACD0GHcDbQLN lmL]AH?aGCD l P UBACDYGCDQb^cM?* P DYd LNE Φ %SDF0DFwmDFE GHcIKGd LNEGHcDYr9?awE^IKGH?BLNMIKU+ P w?aGcDFECD GHcDN P w?aG?BXDFMGH?&%bIKGH?BLNM |n = 1〉 ≡ |0〉 IKMX |n = 0〉 ≡ |1〉 ` INA/b^cL]ACDFMk0uoMGHc?BA wINAH?BA LKd UBLNW]?Bb0AHG^IKGCDQAkGHcDbQLN lmL]AH?aGCDlcINA^D0WxIKGCD ACD/ P DFMbQD LKdZD/MkZ= [lECDQbF?BA^DFUa?a lUBDF0DFMGCAGHcD XDQAH?aECDQX lcINACD0WxIKGCD ΦfD/Mk0 ?(FAC?aMbQD ?aG IKllUa?BDQAI@lcINACD d INbFGCLNE LKd −1 GCL[GHcD3UBLNW]?BbZAHG^IKGCD |1, 1〉 = |D, 0〉 L cD $LNlmDFE^IKGH?BLNMLKdTD/xMk!=&%RcL `DFrxDFE P AHGZ0LX7?&%DQXX P D GCLGHc?BAZlIKEHGH?Bb P UVIKES P w?aG3?BXDFMGH?&%bIKGH?BLNM )bFUBL]ACDFE IKMIKUa7AH?BA@ECDFrxDIKUBA@GHcIKG[?aG A P bQDQA[GCL P A^D RGH25A0?aMAHGCDINX,LKd E^DFX,AH?BXDFwIKMXl P UBACDQA ACL GHcIKG[GHcDACD/ P DFMbQD GCLwmD P ACDQX cDFECD ECDINXA
Rswap(φ0) = R+(π√2, φ0) R
+(2π√2, φ0 + φswap) R
+(π√2, φ0).
=
/ 4
) P IKMG P bF?aECb P ?aGECDFlECDQACDFMG^IKGH?BLNMLKdZGHcD bQLN lUBDFGCD ACD/ P DFMbQDQA?BAAHcL `3M%?aMg ?aW=?( uvM GHcDINbFG P IKUmD9lmDFEH?a0DFMGGHcDZwUVINb^\ wmL wmL P MXIKEH ?BA LKdkbQL P ECACD_I[d LNEHIKUhbQLNMAHGHE P bFG INA GHcDSD7lmDFEH?a0DFMGCDFE\9ML `ZA`3c?Bb^c4\9?aMX LKd[d P MbFGH?BLNM4?BAwmDF?aMW%lmDFEpd LNEH0DQX uvM lEH?aMbF?alUBD3cL `DFrxDFEZI GHc?aECX lIKEHGv?BA?aMr9?aGCDQX,?aMGCL GHcD UVIKwmLNE^IKGCLNEHGCL A^DFG[GHcD owUVINb^\wmL Nfb^cLL]ACDGHcD0bFLN l P GCDFE lECLNW]E^IK @`3c?tUBD GHcDD7lmDFEH?a0DFMGCDFE ?BA0MLNGlECDQACDFMGZIKMX4IedfGCDFE0GHc?BAGHcDD9lmDFEH?a0DFMGCDFEYE P MAGHcD `3cLNUBDIKUaWsLNEH?aGHcwINXX7?aMWGHcD[G `Lπ/2
l P UBACDQA3GCL GHcD ACD/ P DFMbQD/IKMX ECDINX7?aMW L P GZGHcD[?aMGCDFEHMhIKU! P w?aG Z_zh S;| zh3zhCZ pzm~ 4 Z 9ZuvM IAHGCEH?aMWLKdGHE^IKllmDQX ?BLNMA7OQE A B/8 AHl?aM7noAHl?aM GvlmDbQL P lUa?aMW wmDFGv`DQDFMKMDF?aW]cwmLNEH?aMW ?aMGCDFEHMIKU P w?aGCA?BAMDFW]Ua?aW]?awUBD INAGHcDY?BLNMA0IKECD ACDFlIKE^IKGCDQXw%ACDFrxDFE^IKU ?BbFECLN0DFGCDFECA `3c?aUBD GHcDF?aE?aMX7?ar?BX P IKU` IrxDd P MbFGH?BLNMA D9GCDFMXL rxDFEYACDFrxDFE^IKU_MIKMLN0DFGCDFECA LNMUa fbdp A^DQbFGH?BLNM uoMLNECXDFE GCL lmDFEpd LNEHI P IKMG P WxIKGCDwmDFGv`DQDFM4Gv`L ?aMGCDFEHMIKU P w?aGCA ?BLNM %IKMX?BLNM GHcD lECLNlmL]A^IKU_LKd ?aE^INbIKMX]mLNUaUBDFE = P ACDQAZLNMD LKdGHcD r?awE^IKGH?BLNMIKU 0LXDQAZGCL 0DQX7?VIKGCD GHcD bQL P lUa?aMW c?BAZr9?awE^IKGH?BLNMIKU 0LXD?BAZ?BXDFMGH?&%DQXINA@IX P w?aG P AHG_INAZ?aMGHcD lECDFr9?BL P ASA^DQbFGH?BLNMA@IKMX cINAZGCL0wmD lECDFlIKECDQX?aMGHcD |n = 0〉AHG^IKGCD]W cD D9lmDFEH?a0DFMGJIKU lECLNGCLbQLNU GHcDFM ECDINXA• -GCDFlD%T,IKl GHcD$: P IKMG P ?aM7dfLNEHIKGH?BLNMLKd?BLNMD%[GCL GHcD[r9?awE^IKGH?BLNMIKU' P w?aG• -GCDFl TDFEpdfLNEHGHcD XDQAH?aECDQX WxIKGCD wmDFGv`DQDFMGHcD r?awE^IKGH?BLNMIKU' P w?aGSIKMX ?BLNM • -GCDFl T,IKl whINb^\5GHcD ?aM7dfLNEHIKGH?BLNM dfECLN GHcD[r9?awE^IKGH?BLNMIKU' P w?aGZGCL ?BLNMT%]
gLNE/GHcDIKll?aMW,LKd P IKMG P ?aM7d LNEHIKGH?BLNM%dfECLNIKM%?BLNM GCLGHcDr?awE^IKGH?BLNMIKU0LXD IKMX whINb^\5RIAH?aMW]UBDπn l P UBACDLNMGHcD/ECDQXAH?BXDFwIKMXLNlGH?BbIKUaUaINXX7ECDQA^ACDQXLNM?BLNM %h?BA[IKllUa?BDQX =GHcDbQLNEHECDnAHlmLNMX7?aMW LNlmDFE^IKGH?BLNM?BA3?aUaU P AHGHE^IKGCDQX ?aM g ?aW0=?>0IKMX `3?aUaUkwmD XDFMLNGCDQX $
$ (φ0) = R−(π, φ0).
) A GHcD0LXD?BA/?aM?aGH?VIKUaUa?aM%GHcD |n = 0〉 AHG^IKGCD GHcD |D, 1〉 AHG^IKGCD0MDFrxDFE/wmDQbQLN0DQAlmLNl P UVIKGCDQX%?aMGHcDL rsDFE^IKUaU lECLbQDQX P ECD IKMXACL?aGCA@bQL P lUa?aMWYGCL GHcD |S, 2〉 AHG^IKGCDL P GCAC?BXD/GHcDbQLN l P G^IKGH?BLNMIKU AHlINbQDr9?VI GHcD_E^DFX AH?BXDFwIKMX XLDQAMLNG bQLN0D@?aMGCLlUVIx ) UaUmD7lmDFEH?a0DFMG^IKU GCDQb^cM?* P DQA ECD/ P ?aECDQXd LNEACGCDFl cIrxD@wmDQDFMX7?BACb P ACACDQX ?aM ACDQbFGH?BLNM = WJ_LNGCD[GHcIKG3?aM bQLNMGHE^INAHGZGCLA^DQbFGH?BLNM = `3cDFECD[GHcD@r9?awE^IKGH?BLNMIKU P w?aG?BAd P UaUa P ACDQXINA0IKM?aMXDFlmDFMXDFMG P w?aG cDFECD ?tG0LNMUaA^DFEHrsDQA0INAI8:EF4760? ARE P w?aGIKMX?BAGHcDFECDd LNECDLKdfGCDFMECDd DFEHECDQXGCLYINA_GHcD132547698:2 ; R2 ? c?BA@r?awE^IKGH?BLNMIKU# P w?aG_ECDFlmDIKGCDQX7UaAHGCLNECDQA[IKMXlINACACDQA/LNM P IKMG P ?aM7dfLNEHIKGH?BLNM w P GbQLN0DQAwINb^\,GCL GHcDAHG^IKGCD |n = 0〉 IedfGCDFEbQLN lUBDFGH?BLNMLKdSIGv`LKn ?BLNMWxIKGCD]uvMX7?ar?BX P IKUE^LNG^IKGH?BLNMA LKdGHcD3r9?twE^IKGH?BLNMIKU9 P w?aG IKECD3MLNG MDQDQXDQX ) wmDI P GH?td P Ud DIKG P ECDLKdmGHcD ?aE^INbnI]mLNUaUBDFE lECLNG^LxbQLNU?BATGHcIKGTGv`LKnV P w?aGTWxIKGCDQA bIKM wmD lmDFEpdfLNEH0DQX0X7?aECDQbFGHUawmDFGv`DQDFM IKMGv`L?BLNMAZ?aMIAHGHEH?aMWLKdF7?aM lEH?aMbF?alUBDhIKEHw?aGHEJIKEH?aUt IKMY?BLNMA WU P AHG_INAZ`3cDFM GHcD $ (φ0) A^D3 P DFMbQD` INA P ACDQX dfLNE@AH?aMW]UBD/?BLNM P IKM9G P UBLNW]?Bb P M` IKMGCDQXlcINACD d INbFGCLNECA@bIKMwmDIrxLN?BXDQXcDFECD/w P AH?aMWX7?&Y DFECDFMG_UVINACDFESlcINACDQA
φ0 = 0IKMX
φ0 = πd LNESGHcD Gv`L A P EHECL P MX7?aMW $ LNlmDFE^IKGH?BLNMAZgLNE_AHGCDFl GHcD UVINACDFEScINAZGCL0wmDINXX7ECDQACACDQXLNM?BLNM 2`3c?aUBD/GHcD UVINACDFE3d Lb P AZ P ACGZwmD/AH`3?aGCb^cDQXGCL ?BLNM 1
d LNE 20+#/ 0C *+3< "*)(+<+#$&+ 7@*+<*38+8$C,. %+8 ( *+"/ 0$&+8$
φD*+38 0;&;(+<; "*@)$&+<$
R(θ, φ) ;+ 5 +<;+<7?F
*($D($D'+83? 0@*$&+ + *04 0(%Ry
.$D3E%$&+8 .;π/2
;.4 .*$ 0;.@**5+x :($D+9BD)(734$&"*+<;+%F
*($-380;&;+8$&"%*5$C! z ;04 0(% '7=
π/2 + BD)(734 $&"*+<;+ *+#(7+<;/ 0) 7@*'(2;%@,%%@ *+ *.)+
.)(,%0; (3E 3E* .,%+8$*0*(*,#% $%+<;4 .)()"+<; 0; .38+%F
=
4 .
>||
>|| >|||
>||
>||
>||
π
>||
>|| >|||
>||
>||
>||
π
g ?aW P ECD =?> -GCDFlE%0IKMXAHGCDFl LKdGHcD ?aE^INbnI]mLNUaUBDFE/lECLNGCLbQLNU LNGHc%W P ECDQA/X7?BAHlUVIGHcD?aMGCDFEHMIKU P w?tG@AHG^IKGCDQA@LKdO C76bQLNw?aMDQX`3?aGHcGHcD/ACG^IKGCDQA_LKdTGHcD/r9?awE^IKGH?BLNMIKU 0LXD] IS-GCDFl% cD0LXDcINA@wmDQDFM,lECDFlIKECDQX,?aM |n = 0〉 `3c?aUBDGHcD?aMGCDFEHMIKUW P w?aG@?aM,WsDFMDFE^IKU?BA?aM ACLN0D_A P lmDFEHlmL]AH?aGH?BLNMk ) bQbQLNECX7?aMW]Ua0GHcD_bFLNw?aMDQX AC9AHGCDF(?BA ?aMYI A P lmDFEHlmL]AC?tGC?tGC?aLNM LKd|S, 0〉 IKMX |D, 0〉 `3c?Bb^c ?BA ?aMX7?BbIKGCDQX wI/wUVINb^\YIKMX I W]ECDF bF?aECbFUBD]# cD_IKll?tMWECDQXAH?BXDFwIKMX
πn l P UBACD0bQLN lUBDFGCDFUaGCE^IKMApdfDFECA GHcD |D, 0〉 lmLNl P UVIKGH?BLNM GCL |S, 1〉 W]ECDF,bF?aECbFUBD`3?aGHcYXINAHcDQX wmL P MXIKEH5 ) dfGCDFE GHc?BALNlmDFE^IKGH?BLNM 9GHcDS?aMGCDFEHMIKU P w?aG ?BA ?aM GHcD |S〉 AHG^IKGCD`3c?aUBDZGHcDZA P lmDFEHlmL]AH?aGH?BLNM?BA ML ` DFMbQLXDQX0?aM0GHcD30LNGH?BLNMIKUmAHG^IKGCDQA |n = 0〉 IKMX |n = 1〉 wL-GCDFl ?aMGHcD 0DIKMGH?a0DfAHGCDFl IS P IKMG P -WxIKGCD wmDFGv` DQDM GHcD r9?awE^IKGH?BLNMIKU0 P w?aGIKMX?BLNM cINA wmDQDFMlmDFEpd LNEH0DQX]`3c?Bb^c?aMWsDFMDFE^IKUb^cIKMWsDQAGHcDAHG^IKGCD IK lUa?aG P XDQATLKdGHcDr?awE^IKGC?aLNMhIKUM P w?aGh`3c?aUBDGHcD?aMGCDFEHMIKUTAHG^IKGCDLKd ?BLNM %ECDFIK?aMA P MI Y DQbFGCDQXm? D]h?BA |S〉 c P AGCcD@bQLNw?aMDQX AH7AHGCDF#?BA?aM I/MDF` A P lmDFEHlmL]AH?aGH?BLNM LKd |S, 0〉 IKMX |S, 1〉 ?aMX7?BbIKGCDQXw bF?aECbFUBDQA@LKd X7?&YDFECDFMG@W]ECDFxnoACbIKUBD] cD/ECDQXAH?BXDFwIKMX
πn l P UBACD d LNESGHcD/wINb^\n IKll?aMWGHE^IKMApd DFECA GHcD |S, 1〉 lmLNl P UVIKGH?BLNM bFLN lUaDFGCDFUaGCL |D, 0〉 fbF?aECbFUBDS`3?aGHcYXINAHcDQXwmL P MXIKEH0uoM GHc?BA3` IYGHcD P IKMG P ?tMdfLNEHIKGH?BLNM`3c?Bb^c` INAZDFMbQLXDQX ?aM GHcD r9?awE^IKGH?BLNMIKU' P w?aG?BAGHE^IKMApd DE^DFXwINb^\GCL@GHcD ?aMGCDFEHMIKU0 P w?aGTLKdh?BLNM %N`3c?aUBD GHcD r?awE^IKGH?BLNMIKU0LXD ECDFG P EHMAGCL GHcD |n = 0〉 ACG^IKGCD]
GHcD A P EHECL P MX7?aMW $ WxIKGCDQAJ@LNGCD@GHcIKGZGHc?BA lECLbQDQX P ECDIKUaUBL `ZA3GHcD[?a lUBDF0DFMG^IKGC?BLNMLKd476 P IKMG P WxIKGCD[wmDFGv`DQDFM GHcD[Gv`L?aMGCDFEHMIKU' P w?aGCA uod GHc?BA P IKMG P WxIKGCD ?BA9d LNESDIK lUBDhI WxIKGCDGHcD AHG^IKMXIKECX 0DFGHcLXLKd?a lUBDF0DFMGH?aMW AHGCDFl wmDFGv`DQDFMr9?awE^IKGH?BLNMIKUIKMX ?aMGCDFEHMIKUL P w?aG" ?BASGCLA^IKMX7`3?Bb^cI lcINACDWxIKGCDΦwmDFGv`DQDFMr9?awE^IKGH?BLNMIKU3IKMX%?aMGCDFEHMIKU P w?aG" wmDFGv`DQDFMGv`LAH?aMW]UBD P w?aG π/2 n ECLNG^IKGH?BLNMALNM?BLNM fACDQD ACDQbFGH?BLNM W c?BAZ?BDFUBXA_I bQLNMGHECLNUaUBDQX wYGHcD P IKMG P w P A
→2 = Ry2(π/2) Φ
↔2 Ry2(π/2).
c P A GCcDSL rxDFE^IKUaUhA^D/ P DFMbQDZdfLNEI wmDFGv`DQDFM GCcDZGv`L/?BLNMAx`3cDFECDS?BLNMU%Z?BA GHcDSbQLNMGHECLNU P w?aGECDINXA 1→2 = R−
1 (π, π) Ry2(π/2) Φ
↔2 Ry2(π/2) R−1 (π, 0).
) A@?aMGHcD AC?aMW]UBD?BLNM bINACD,fACDQbFGH?BLNM3= RkGHcDECDFIK?aM?aMW dECDQDQXLN Ua?BDQA[?aMGHcD0DFGHcLX IKllUa?BDQXGCL?a lUBDF0DFMGΦ
↔24uoMGHcDN%ECAHGD7lmDFEH?a0DFMGCA LNM GHcD Gv`LKn ?BLNM LTGHcD bQLN lmL]AC?aGCD l P UBACDGCDQb^cM?* P D` INAYDF lUBL xDQX gLNE GHcDbQLN lmL]AH?aGCDlcINACDWxIKGCD Φ′ LKd/D3Mk= RZbQL P lUa?aMWLNM4GHcD
==
/ 4
wU P DAH?BXDFwIKMX?BA P ACDQX SgECLN IKMD9lmDFEH?a0DFMG^IKUlmLN?aMG[LKd r?BDF` ?aG_?BA@XDQAH?aE^IKwUBDGCL P ACDI0 ?aM?a P M P wmDFE@LKdTX7?&Y DFECDFMGSUVINACDFEZdECD/ P DFMbF?BDQA ) bQbQLNECX7?aMW]UaGHcD `3cLNUBD/ACD/ P DFMbQD ?BAZECDIKEHE^IKMWsDQX?aMLNECXDFEGCL bQLN lEH?BACDLNMUawU P DAC?BXDFwIKMXIKMX bIKEHEH?BDFE l P UBACDQA \_MDbIKM d LNEDIK lUBD ?a lUBDF0DFMGYGHcD $ LNlmDFE^IKGH?BLNMA3w π n l P UBACDQA3LNM GHcD RG 20A/AH?BXDFwIKMX R+(π, φ0) ?aMAHGCDINX LKd GHcD@ECDQX AH?BXDFwIKMX IKMXbQLN lmDFMA^IKGCD d LNEZGHc?BA_b^cIKMWsD w IKllUa9?tMWYI →2 ?aMAHGCDINXLKdTGHcD →2 WxIKGCD] cD UVIKGHGCDFESbIKM wmD ECDIKUa? O DQXINAfACDQD A^DQbFGH?BLNMA IKMX =
→2 = Ry2(π/2) Φ′
↔2 Ry2(π/2)
cD bQLN lUBDFGCD/ACD/ P DFMbQD[dfLNE3GHc?BAZGv`LKn ?BLNM L7D7lECDQACACDQX ?aM UVINACDFEZl P UBACDQAGHc P A3ECDINXA 1→2 = R+1 (π, π) Ry2(π/2) Φ
′2 Ry2(π/2) R
+1 (π, 0),
=?@`3?aGHc
Φ′2 = R+2 (π,
π
2) R+2 (
π√2, 0) R+2 (π,
π
2) R+2 (
π√2, 0).
P IKMG P bQLN l P G^IKGH?BLNMD9lmDFEH?a0DFMGCA@`3?aGHcGv`L fLNE_0LNECDZ?BLNMA[IKECDh?aMWsDFMDFE^IKU m0LNECDX7? b P UaGGHcIKM`3?aGHc I_AH?aMW]UBD ?BLNMkL cDI pLNE E^DQINA^LNMd LNE GHc?BA Ua?BDQA ?aMGHcD ECDQX P bQDQX. P IKMG P -WxIKGCDAHlmDQDQX`3c?Bb^c?BA D7lUVIK?aMDQX ?aM GHcD_dfLNUaUBL `3?aMW uoG?BA3L P GHUa?aMDQX ?aM ) llmDFMX7?t ) GHcIKG dfLNE Gv`LKn ?BLNM D9lmDFEH?a0DFMGCA3GHcDj IKw7no>@?Bb^\xD0d INbFGCLNEη?aMDFr9?aG^IKwUa?BA/AHIKUaUBDFE/GHcIKM ?aM GHcDAH?aMW]UBDn ?BLNM%D7lmDFEH?a0DFMGCA ? D]
η = 1.8%fbQLN lIKECDQXGCLη = 6.8%
?aMAH?aMW]UBDn ?BLNM,bINACD c?BA_?BA P UaGH?aIKGCDFUaX P D/GCLGHcD/MDQDQX d LNES?aMX7?ar9?BX P IKUINXX7ECDQACAH?aMW ?aM,GHcD P UaGH?tn ?BLNM bINACD] ) bQbQLNECX7?aMW GCLY?aMD/ P IKUa?tG ] IYAHIKUaUBDFE η bQLNEHECDQAHlmLNMXA GCL IAHIKUaUBDFEΩS7cDFMbQDIECDQX P bQDQX WxIKGCD AHlmDQDQX
J@LNGCD GHcIKGS`3?aGHcbQLN lmL]AH?aGCD l P UBACDQAS`D cIrxD/IGCDQb^cM?* P DIKGZcIKMX `3c?Bb^cIKUaUBL `ZASGHcD ECDIKUa? O IKGH?BLNMLKdTIECDIKU $ LNlmDFE^IKGH?BLNMk cDFECDd LNECD`D[bIKM ?aM lEH?aMbF?alUBD P ACD@GHcD[0LXD INAZINB C; < G A38VA( P w?aGDFrxDFM?aM,GCcD P UaGH?tn ?BLNM bINACD]. c?BAIKWxIK?aM0DIKMA GHcIKG `3?aGHc Gv`L GHE^IKllmDQX?BLNMA `D bQL P UBXlmDFEpd LNEHIKEHw?aGHE^IKEH GHcECDQD+ P w?aG P IKM9G P (bQLN l P G^IKGH?BLNMA \_MDSbFL P UaXDFrxDFMbQLNMbQDF?arsDSLKd P AH?aMWX?ARP@AREF4G0LXDQALKd I P UaGH?tn ?BLNM AHGHEH?aMW0INA3INXX7?aGH?BLNMIKU P w?aGCA ) GlECDQACDFMG7`D@bIKM MLNGd P UaUalECL@%GdfECLN GHc?BA3LNlGH?BLNMwmDQbI P ACD3`3?aGHc Gv`L LNET0LNECD ?BLNMAT`DZIKECDd LNECbQDQXGCL LNlmDFE^IKGCDSIKGd IK?aEHUaUBL `AH?BXDFwIKMX0_IKw?7dECD/ P DFMbF?BDQAd LNE3ECDINACLNMAZW]?arsDFMIKwmL rxD]# cDE^DJd LNECDhIKG30L]ACG3Gv`L $ LNlmDFE^IKGH?BLNMA@IKECD lmL]ACAH?awUBD/IKG3lECDQACDFMGuod IGv`LKn ?BLNM P IKMG P WxIKGCD cINA0wmDQDFM ?a lUBDF0DFMGCDQXSI,MIKG P E^IKUSWsLsIKUS?BA0GCLlECLNwmD GHcD lcINACDbQLNcDFECDFMbQD LKd GHcDWxIKGCD] c?BA[bIKMwmDXLNMDw ECDQbQLNECX7?aMWN P IKMG P L]ACbF?aUaUVIKGH?BLNMA =6> ' ( %o c?BAGv9l?BbIKUaUa ?aMrxLNUarsDQA lECLNw?aMWGHcDS P IKMG P #WxIKGCDS`3?aGHc P w?aGA P lmDFEHlmL]AH?aGH?BLNMA3INA ?aMl P GCA lECDFlIKECDQX w_IK0A^DFxn Gv9lmDπ/2
l P UBACDQA"[IKMX,ECDINX7?aMWL P G GHcD?aMGCDFEpdfDFECDFMbQD0lIKGHGCDFEHM `3?aGHc INXX7?aGH?BLNMIKU _IK0ACDFnGv9lmDπ/2
l P UBACDQATINXXDQX IedGCDFE GHcDWxIKGCDLNlmDFE^IKGH?BLNM,GHcDlcINACDLKd`3c?Bb^c?BA ACbIKMMDQX9FLJ@LNGCD GHcIKGA P b^cIKM?aMGCDFEpdfDFECLN0DFGHEH?Bb 0DFGHcLX &%> ?BAbQLNMbQDFl P G^IKUaUaD/ P ?ar]IKUBDFMGGCLGHcD ` I GHcDYLNlmDFE^IKGH?BLNMA Ui IKECDlECLNwmDQX?aMGHcD>[D P ACb^c7nVUsL O A^I IKUaWsLNEH?aGHc _ GHcDA^IK0D/Gv9lmD/LKd lECLbQDQX P ECDGHcD( P IKM9G P lcINACDQALKdIKM?a lUBDF0DFMGCDQX,WxIKGCDLNlmDFE^IKGH?BLNMbIKMwmD0DINA P ECDQX?aMGHc?BA_` I lmDFEpdfLNEH ?aMW I P IKMG P ;WxIKGCDGCLN0LNW]E^IKlc?( (.=ogLNE I CNOT WxIKGCDI[lIKEHGH?Bb P UVIKEHUa ?aMGCDFECDQACGH?aMWbINACDSIKEH?BACDQA ?tdGHcDBDC7698:EFC7GN?BLNM?BAT?aMI[A P lmDFEHlmL]AH?aGH?BLNM IKGTGHcD3?aMl P GgLNETGHc?BAT\9?tMX LKdm?tMl P GTGHcD+ P IKMG P *WxIKGCD3lECLX P bQDQA DFMG^IKMW]UBDQXAHG^IKGCDQAh? D]SA3GG 8 48VA"?3 INA@IKM,L P GHl P G_gLNE@DIK lUBDIKM?aMl P G[AHG^IKGCD (|S〉 + |D〉)|S〉 f`3cDFECD/GCcD%ECAHG3?BLNM ECDFlECDQACDFMGCAZGHcD bQLNMGHECLNU! P w?aG" ?BA3GHE^IKMApd LNEH0DQX ?aMGCLGCcD DFUaUkAHG^IKGCD |S, S〉+ |D,D〉 w IKM?BXDIKUCNOT
LNlmDFE^IKGH?BLNMk
=6'
4 .
=6@
- , "*
( Z zh~ CT 3| cD%ECAHG LNMDZIKMX0I@cIKUtdmxDIKECATLKdmGHc?BA Tc>`LNEH\ P ACDQX0I_Ua?aMDIKE ?BLNM GHE^IKl `3c?Bb^c0cINXwmDQDFM0XDQAC?aW]MDQXIKMXlE^DlhIKECDQX w P G/MLNG xDFG P A^DQX9@w [IKMMApn cEH?BAHGCLNlc J[iKWsDFEHU ?aM c?BA/Tc> ?' %IKMX ?aG `3?aUaU wmDECDd DFEHECDQX GCL?aG3INA GCcD vLNUBX GHE^IKl ^#3cD_`LNEH\ IKMXYGHcD_ECDQA P UaGCA LNwG^IK?aMDQXY`3?aGHc GHc?BA GCE^IKl IKECD_ECDFlmLNEHGCDQX?aMGHcD Tc>GHcDQAH?BALKd [IKE^IKUBX_LNcXD ?' o ) A^b^cDFIKGH?Bb l?BbFG P ECDLKd@GHcD LNUBXGHE^IKl?BA0ACcL `3M?aMg ?aW'7&%] @DFECD GHcDDFMX9nobIKlA/`DFECD0INXD INA EH?aMWsAIKECL P MX GHcDSg n ECLXA cDDFMX9nobIKl rxLNUaG^IKWsD` INA_Ua?a ?aGCDQXw GHcDAHlmDQbF?&%bIKGH?BLNMA[LKd GHcDbQLNEHECDQAClmLNMX7?aMW d DQDQX9n GHcECL P W]cA[LKd GHcD/r]INb PP ;rsDQACA^DU GCLVcap ≤ 2000
) GVcap = 2000
#IKM Ie9?VIKU dfECD/ P DFMbF ωz = 2π · 700 \ O `3INA@ECDINb^cDQX gECLNGHc?BA D9lmDFEH?a0DFMG^IKU_ECDQA P UaG IWsDQLN0DFGHEHd INbFGCLNE α = 0.81bIKM wmD XDQX P bQDQX fD/xM ?@F@INX7?VIKUdfECD/ P DFMbF?BDQA ωr LKd P lGCL 2π · 1.8 # O `DFECDLNwG^IK?aMDQX`3?aGHc,Sg noX7EH?arxD lmL `DFECA@LKd P lGCL 5
T IKG[IdfECD/ P DFMbF Ω = 2π · 16 O ) MDF` GHE^IKlXDQAH?aW]M TbIKUaUBDQX%GHcD oMDF` GHE^IKl D` INADFMr9?BA^IKWsDQX ?aM%GHcD sDIKE ]]] GCL?a lECL rxD LNMACDFrxDFE^IKUlECLNwUBDF0A DFMbQL P MGCDFECDQXY`3?aGHc GHcD_LNUBXGHE^IKlk# cDQACDSlECLNwUBDF0A3IKMXGHcD_XDQAH?aW]M lIKGHcYGCL ` IKECXAGHcD MDF` GHE^IKl `3?aUaUkwmD DFUVIKwmLNE^IKGCDQX?aM GHcD[d LNUaUBL `3?aMWTcDFMUBLsINX7?aMWI GHE^IKl w?BLNM? O ?aMW IenvIKGCLN0A dfECLNIKM%IKGCLN ?Bb0LersDFM r?VI DFUBDQbFGHECLNM ?a lINbFG/?BLNM7n? O IKGH?BLNM IKM AHGCE^IDFUBDQbFGHECLNMA IKECD XDFlmL]AH?aGCDQX%LNMDFUBDQbFGHEH?BbIKUaUa ?aMA P UVIKGH?aMWA P Epd INbQDQA d LNEDIK lUBDLNM%bQDFE^IK ?Bb A P llmLNEHGAHGCE P bFG P ECDQALKd3GHcDGHE^IKl%bQLNMAHGHE P bFGH?BLNMk^-lmDQbF?VIKU D"Y LNEHG/GCL d Lb P A/GHcD0?BLNM? O ?aMWDFUBDQbFGHECLNM wmDIK `DFUaUT?aMGCL GHcDGHE^IKl bQDFMGCDFE ?BA@d P GH?aUBDwmDQbI P ACDGHcDlECDQACDFMbQD LKdGHcDGHE^IKll?aMW E^INX7?BLKndfECD/ P DFMbF.%DFUBXA bI P A^DQAI UVIKEHWsD_XDdfLb P ACAC?aMWLKd GHcDSwmDIK T>[DFlmL]AH?aGCDQX b^cIKEHWsDQAbFECDIKGCD_> %DFUBXAIKGGHcD_GHE^IKl bQDFMGCDFE `3c?Bb^cY P ACGwmD@bQLN lmDFMA^IKGCDQXYdfLNE# c?BA?BA INb^c?BDFrxDQXYw INXX7?aGH?BLNMIKU bQLN lmDFMAJIKGH?BLNM + +*+ *+< &;4 ." .$#5+<$&,.*+?5 &;4 ." )% %5(,6 .$ $ ()() 3? ;&;+85 %@: '7= +8)(+<3<&;% "* .3< (% 0(%F *($ +E* 5 .$ ) 0+<;;+8") 038+?5 '7="*..(%* (% /$&+<+-$&+<3<(%:F
z
L
g ?aW P ECDC'7&%>[DQAH?aW]M,LKd GHcDLNUBXGHE^IKlk L ?BAS% IKMXr0n GHcDACcLNEHGCDQAHG[X7?BAHG^IKMbQDdfECLN;GHcDGHE^IKlIe7?BA GCL/GHcD@_g n ECLX A P Epd INbQDQA n?BA 1.2 # cD@Sg n ECLXA cIrxD[I/X7?VIK0DFGCDFE3LKd 0.6
= H
4
DFUBDQbFGHECLXDQA IKECL P MXGHcDGHE^IKlGCL `3c?Bb^c IKllECLNlEH?VIKGCD> rxLNUaG^IKWsDQA IKECDIKllUa?BDQX [g ?aMX7?aMW GHcDEH?aW]cGbQLN lmDFMA^IKGH?BLNMrxLNUaG^IKWsDQAT?BA I P ?aGCD GCDQX7?BL P AIKMXGH?a0DZbQLNMA P ?aMW/lECLbQDQX P ECD(fACDQDZIKUBACL b^cIKlGCDFE @7 cD LNUBXGHE^IKl` INA[cDFUBXG^LNWsDGCcDFE wECDFUVIKGH?arxDFUaUVIKEHWsD P MbQL rxDFECDQXbQDFE^IK ?BblUVIKGCDQA `3c?Bb^c bQLNUaUBDQbFGCDQXUVIKEHWsD/IK0L P MGCA_LKdTDFUBDQbFGHECLNMA gLNEZGHcD MDF` GHE^IKlXDQAH?aW]MbQDEJIK ?ab lIKECGCAZ`DFECD GCL wmD[ ?aM?a ? O DQXIKMXbQL rxDFECDQX w 0DFG^IKU `3cDFECDFrsDFESlmL]ACAH?awUBD]uvG ` INA dfL P MXYD9lmDFEH?a0DFMG^IKUaUa `3?aGHc GHcD@LNUBXGHE^IKl!GHcIKGGHcDSE^INX7?VIKUm0LXD[IKG 2π · 1.8 O INA ` DUaUINAGHcD Ie9?VIKU0LXD IKG
2π · 700 \ O IedGCDFE3>[LNllUBDFEZbQLLNUt?aMW ` DFECD@UBDdG `3?aGHc 0DIKM lcLNMLNM M P wmDFECAnLKd IKwmL P G 45 ?' `3c?Bb^c?BAZAHGH?aUaUkECDFUVIKGH?arxDFUt c?aW]ckZ cD DFMr?BAJIKWsDQX D9lmDFEH?a0DFMGCAcL `DFrsDFEECD/ P ?aECDWsLLX bQLLNUt?aMW?aMGCDFEH0A LKd I UBL `
ndfLNE47GGr9?awE^IKGH?BLNMIKUh0LXDQA bJdHxACDQbFGH?BLNM H ) UBL ` n LKd I[0LXD?BA 0L]AHG DINAH?aUaECDINb^cDQX0w I[c?aW]c0LXD dfECD/ P DMbxgLNE GHcD3E^INX7?VIKUdECD/ P DFMbF?BDQA GHcDFECDZ?BA DQA^ACDFMGH?VIKUaUaMLUa?a ?aG[IKMX?tM,IACDFMACD oGHcD/c?aW]cDFE@GHcD/wmDFGHGCDFE ^S cDIe9?VIKU dfECD/ P DFMbFcL ` DFrsDFEmbIKMMLNG@wmD/GCLLc?aW]cINASLNlGH?BbIKU AH?aMW]UBD ?BLNMINXX7ECDQACAH?aMW?BAZECD/ P ?aECDQX TcDFM GHcD MDF` GHE^IKl ` INASXDQAC?aW]MDQXGHcD ` IK?BAHGLKdmGCcDZINXX7ECDQA^AH?aMW wmDIK#IKG HN >[M ` INA 0DINA P ECDQX0GCL[wmDSIKwmL P G H µ ?' oM c?BA Ua?a ?aGCA ?aMGCDFEpn ?BLNMACDFlIKE^IKGH?BLNMAGCL,I ?tM?t P LKd_IKwmL P G H µ `3c?Bb^cIKWxIK?aMUa?a ?aGCA Ie9?VIKU dfECD/ P DFMbF?aDQAGCLMLNG0LNECDGHcIKM HN] \ O w b^cIKMbQD[GHcD@Ua?a ?aG3LKd GHcD[LNUBXYGCE^IKl# cD[MDF` GHE^IKl ` INA ?aMGCDFMXDQX GCL9?BDFUBX Ie9?VIKUdfECD/ P DFMbF?BDQA LKd P lGCL %]?' # O wmDQbI P ACDlmL]ACAH?awUBDd P G P ECD?a lECL rsDF0DFMGCA LKd GHcD INXX7ECDQACAH?aMW LNlGH?BbQA`L P UBX GHcDFMIKUaUBL ` GHcD LNlmDFE^IKGH?BLNMLKdGHcD[GHE^IKl `3?aGHcIAHGHECLNMWsDFE_Ie7?VIKUkbQLNM %MDF0DFMGZ c?BA3?BASINX7r]IKM7nG^IKWsDQL P A3wmDQbI P ACD[?aGd INbF?aUa?aG^IKGCDQA3W]ECL P MX AHG^IKGCD[bQLLNUa?aMW0LKd GHcD Ie7?VIKU0LXDfA" IKMX IKUBA^L?aMbFECDINA^DQA GHcDAHlmDQDQX LKdM P IKMG P WxIKGCDQA(fACDQD D/xMk! = F
TcDFM LNlGH?a ? O ?aMW GHcD,GHE^IKl XDQAH?aW]M dfLNE c?aW]cDFE E^INX7?VIKU dfECD/ P DFMbF?BDQAACDFrsDFE^IKU/INAClmDQbFGCA P AHG wmDG^IK\xDFM?aMGCLINbQbQL P MGg ?aECAHGHUakGHcD0AHG^IKw?aUa?aGvlIKE^IK0DFGCDFE q AHcL P UBXwmD0bFUBDIKEHUawmDFUBL ` ?>70uvM LNECXDFEGCLIrxLN?BXlmLNGCDFMGH?VIKU MLNMUa?aMDIKE/ECDQACLNMhIKMbQDFA fA^DQD0ACDQbFGH?BLNM &%[?aG ?BA/XDQAH?aE^IKwUBD0GCL \xDQDFl q wmDFUBL ` ?'7gECLN D/xMk ?(/?aG?BAd L P MXTdfLNEDIK lUBD GHcIKG`3?aGHcGHcD LNUBXGHE^IKlGHcDYGv9l?BbIKU `LNEH\?aMWlmLN?aMGωr0 = 2π · 1.8 O Ω = 2π · 16 # O ` INA q = 0.32
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(@ M W c?BAZIKUaUBL `ZA3A P lmDFEH?a lmL]AC?aMW UVINACDFE wmDIK0ASIKGO(@@M IKMX ('.= M LNMGCL I > H M wmDIK ) AH?aMW]UBD ' ?' wmDIK AClUa?aGHGCDFE ?BA P ACDQX,GCL AHlUa?aG GHcDUVINACDFE wmDIK0AIKGC(@@YMIKMXD('.= MIKMX IKGGHcDYA^IK0D GH?a0DYL rxDFEHUVIKlGHcD ECDQA P UaGH?aMW,wmDIK0A - P wA^D3 P DFMGHUa%DINb^cLKdSGHcDQACDYXL P wUBD ` IrxDFUBDFMW]GHc?aM7dfE^IKECDQXwmDIK0A ?aA L rxDFEHUVIKllmDQX `3?aGHc I > H M(wmDIK # cDSECDQA P UaGH?aMWGHEH?alUBDS` IrxDFUBDFMW]GHc wmDIK0A IKECDbIKUaUBDQX >[LNlG%/IKMX >[LNl uvGZbIKMIKUBA^LwmD ACDQDFM GCcIKGSDINb^c UVINACDFE3wmDIKbIKM wmD AC`3?aGCb^cDQX LNMIKMX L@Y`3?aGHcDF?aGHcDFE@IAC?tMW]UaD LNEIXL P wUBDlINACA@INbQL P ACGCLKnoLNlGH?BbIKU0LX P UVIKGCLNEX ) \@ UVIKwmDFUaUBDQX ) \ % . . . ) \ @7 ) \ ) \ ) \ = IKMX) \ ' IKECD X7EH?arxDFM%IKG %DQXdfECD/ P DFMbF?BDQA fDF?aGHcDFE ( LNE IHN # O [IKMX IKECD P A^DQXdfLNE AC`3?aGCb^c?aMWIKMXUa?aW]cG ?aMGCDFMAH?aGv bQLNMGHECLNU TgECD/ P DFMbFG P M?aMWLKdmGHcD3bQLNEHECDQAHlmLNMX?tMW/UVINACDFECA IKG > H M,IKMX8('.= M*?BAINb^c?BDFrxDQXw A^bIKMM?aMWGHcDSUBDFMW]GHcYLKdkGHcDF?aE ECDdfDFECDFMbQD@bIr?aGH?BDQA P AH?aMWl?BD O L]A# cDSECDdfDFECDFMbQD@bIr?aGH?BDQAd LNEO(@@MIKMX HN >M XLMLNG dfDIKG P ECD[l?BD O L]A n IKMX bQLNMACD/ P DFMGHUa IKECD[MLNG3A^bIKMMIKwUBD@n ?aM LNECXDFEGCLYINb^c?BDFrsD ?aM?a P ;Ua?aMDF`3?BX7GHc IKMX dECD/ P DFMbF X7EH?tdfGh?tMlIKEHGH?Bb P UVIKE@dfLNE_GHcD HN >0MUa?aW]cGS cDF?aEdfECD/ P DFMbF?BDQAIKECDYG P MDQX`3?aGHcGHcD cDFUalLKd_GHcD XL P wUBD lINACA ACDFG P lAw%bQLNM9GHECLNUaUa?tMW GHcD _g X7EH?arsDdfECD/ P DFMbF?BDQAZLKd ) \ % IKMX ) \ @7TgLNE3GHcD LNlGH?BbIKUkb^cIKMMDFU bIKUaUBDQX 397σ
bF?aECb P UVIKEHUaYlmLNUVIKEH? O DQX Ua?aW]cG3?BAlECLX P bQDQX IKMX,GHcDFMAHcLNMD?aM,lIKE^IKUaUBDFUTGCLYGHcDIKW]MDFGH?Bb.%DFUBXrxDQbFGCLNE IKG[GHcDlmL]AH?aGH?BLNM LKd GHcDGHE^IKlbQDFMGCDFE3?aM LNECXDFE3GCLlECL r?BXDσ−
lmLNUVIKEH? O DQX > H M Ua?aW]cG ) \ ECDFlECDQACDFMGCASIKM INXX7?aGH?BLNMIKUkAH`3?aGCb^c GCLlECL r9?BXDLNlGH?a P bIKMbQDFUaUa?aMW LKd > H M;Ua?aW]cG X P EH?aMW lmDFEH?BLXA LKd bQLNcDFECDFMG[IKM?al P UBIKGC?aL]M LNMGHcDHN >0M;GHE^IKMAH?aGH?BLNMk@SDQAH?BX P IKU > H MUa?aW]cG@`L P UBXLNGHcDFEH`3?BACDwECLsINXDFM,GHcD S1/2 W]ECL P MXAHG^IKGCDfA"AH?aW]M?&%bIKMGHUa IKMX GHc P ASXDQAHGHECL bQLNcDFECDFMbQD/LNM GHcD P w?aG3GHE^IKMAH?aGH?BLNMIKG HN >M uvM0INXX7?aGH?BLNM0GCL[`3cIKGT?BATACcL `3M ?aM g ?aWI'7?' sACIKUaU7dE^INbFGH?BLNMAdfECLN(IKUaU` IrsDFUBDFMW]GHcACL P ECbQDQA IKECD3A^DM9GGCLcLN0D w P ?aUaGZ` IrxDF0DFGCDFECAIKUaUBL `3?aMW d LNEZbQLsIKECACD ` IrsDFUBDFMW]GHc0DINA P ECDF0DFMGCAS`3?aGHcIKMINbQb P E^INbF LKdIKwmL P G 10−6 cD Ua?aW]cGIKG HN >3M ?BA DF?aGHcDFE ACDFMG GCLZGHcD HN > ) LNE GHcD HN >7Jb^cIKMMDFU K`3c?aUBD GHcD ECDQAClmDFbFGH?arxD LNGHcDFEb^cIKMMDFU?BA@GHcDFMwUBLb^\xDQX cD HN > ) Ua?aW]cG[lINACACDQA@GHcECL P W]c I AHlmDQbF?VIKULNlGH?BbQA[lECLX P bF?aMW IGH?aW]cGHUad Lxb P A^ACDQXwmDIKIKG[GHcDGHE^IKl bQDFMGCDFE XDQAH?aW]MDQXGCL INXX7ECDQA^A[?aMX7?ar?BX P IKUT?BLNMA LKd IYb^cIK?aM fACDQDACDQbFGH?BLNMH = W cD HN >7J Ua?aW]cGZ?BA dfLb P A^ACDQX LNMGCLGHcD ?BLNMA3wYAH?aMW]UBD UBDFMA3lECLX P bF?aMWYIbQLN lIKE^IKGH?arxDFUa UVIKEHWsDUVINACDFE ` IK?BAHGLKd IKwmL P G 20 µ j INA^DE^A ?aM GHcD_b^cIKMMDFUBA >[LNlG%9>[LNl IKMX 397σ
IKECD@IKUBACL d Lb P ACACDQXw P AHGI AH?aMW]UBDSUBDFMA M cD_bQLNEHECDQAHlmLNMX7?aMWUVINACDFE ` IK?BAHGCA IKG GHcDSGHE^IKl bQDFMGCDFE IKECD_LNM GHcD_LNECXDFELKd 50 µ d LNESIKUaU > H M wmDIK0A_IKMXIKwmL P G 300 µ d LNE3GHcDC('.=0MIKMX (@@M wmDIK0A
@T S ~ Z@z mS z cD P IKMG P bQLN l P G^IKGH?BLNM lECLNGCLbQLNUBA3lECDQACDFMGCDQX ?aM b^cIKlGCDFECA IKMX=ECD/ P ?aECD@MLNG3LNMUa IlECDQbF?BACDUVINACDFE_lmL `DFE IKMX dfECD/ P DFMbF w P G[IKUBACL AClmDQbF?BIKU lcINACDbQLNMGHECLNU m?aMlIKEHGH?Bb P UVIKE@dfLNE_GHcDIKllUa?BbIKGH?BLNMLKdbQLN lmL]AH?aGCDl P UBACD0LNlmDFE^IKGH?BLNMAfbdH= ) bQLN lUBDFGCD P IKMG P WxIKGCD0lECLNGCLbQLNU ?aM%L P E/D7lmDFEH?a0DFMGGv9l?BbIKUaUaECD/ P ?aECDQA/IKG UBDINACG %rsD X7?&Y DE^DM9G UVINACDFE dECD/ P DFMbF?BDQA/IKG HN >YMdfLNE W]ECL P MX AHG^IKGCD bQLLNUa?aMW
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φ
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g ?aW P ECDC'7?@ J@DFGv`LNEH\ dfLNE@UVINACDFE[lmL `DFEmdfECD/ P DFMbFIKMXlcINACDbQLNMGHECLNU IKG HN > M cDDFUBDF0DFMGCAg %nog 'IKECDAH?aW]MIKU WsDFMDFE^IKGCLNECA@E P MM?aMW IKG_bQLNMAHG^IKMGSdfECD/ P DFMbFIKMXlmL `DFE@X P EH?aMWYLNMDXIKG^I[INb/ P ?BAH?VIKGH?BLNM0ACbIKM xA%noA% XDFMLNGCD 3j bQLNMGHECLNUaUBDQX0SgAH`3?aGCb^cDQAsb%nob*= IKE^DlmL `DFEbQLNw?aMDFECAIKMX, ?BA/I ?tDFE. cDUBLbIKU L]A^bF?aUaUVIKGCLNE/j'\#?BA LNlmDFE^IKGCDQX IKG ∼ 600# OIKMXGHcD,ACL P ECbQDQA g %nog 'IKGIKECL P MX 370
# O cDL P GHl P G dECD/ P DFMbF4?BAYGHc P A ∼230
# O GHcDA P dfECD/ P DFMbF LKd ∼ 1+ O ?BA_AHGHECLNMW]Ua A P llECDQA^ACDQX w GHcD d LNUaUBL `3?aMWSgIK lUa?&%DFECA@IKMX GCcD HN >M ) \@
IKMX bQLNcDFECDFMG IKM?al P UVIKGH?BLNMA LN lmL]AH?aGCDl P UBACDLNlmDFE^IKGH?BLNMA bIKM wmDAH?a lUa?&%DQXw P G DFrxDFMGHcDFMECD/ P ?aECD/ACDFrxDFE^IKU `DFUaU XDR%MDQXAH`3?aGCb^cIKwUaDUVINACDFESlcINACDQA@IKG_bQDFEHG^IK?aMdfECD/ P DFMbF?BDQA cD IKM?al P UVIKGH?BLNMLKd HN >,M Ua?aW]cG0?tMGHcD D7lmDFEH?a0DFMG0?BAINb^c?BDFrsDQXwIKllUa9?aMW A P ?aG^IKwUBD l P UBACD ACD/ P DFMbQDQA ?aMGHcDSgE^IKMWsDIKECL P MX ] ] O GCL GHcD bQLNEHECDQAHlmLNMX7?aMWXL P wUBD lINACA ) \@ ) \ %?aM g ?aW'7?'X cDP UaGH?aIKGCDGCLLNU_dfLNE GHcDECD/ P ?aECDQX4GH?a ?aMW03lmL `DFE IKMXdfECD/ P DFMbFbQLNMGHECLNU@`L P UBX4wmDI AH?aMW]UBD_gl P UBACD ACL P ECbQD `3c?Bb^ccINA P AHGLNMD L P GHl P Gb^cIKMMDFU IKMX%bIKM%wmD bQLN l P GCDFEbQLNMGHECLNUaUBDQX - P b^cIKM?aMAHGHE P 0DFMG@`L P UBXMDQDQXGCLwmDAHlmDQbF?VIKUaUaXDQAH?aW]MDQXIKMXlECLX P bQDQXdfLNE_GHcD/lIKEHGH?Bb P UVIKE IKllUa?BbIKGH?BLNM MDFrxDFEHGHcDFUBDQACAZ?aM GHcD[UBLNMW0GCDFEH GCcD D7lmDFEH?a0DFMGS`3?aUaUkECD/ P ?aECD A P b^cIXDr9?BbQD]gLNE[GHcDD7lmDFEH?a0DFMGCA lECDQACDFMGCDQX,cDFECD`DcIrxDw P ?aUaG/IYE^INX7?BLKn dECD/ P DFMbFMDFGv`LNEH\`3c?Bb^c A^LNUarsDQAGHcDQACDZlECLNwUBDF0A d LNE GHcD3GH?a0DZwmDF?aMWTg ?aW#'7?@ AHcL `ZA GHcD3MDFGv`LNEH\IKMXAHcL P UBX0wmDZACDQDFMYINA I [AR6 ARE"O BECDFlECDQACDFMG^IKGH?BLNMk cD@ECDIKU ACDFG P l ?BA A P w HDQbFGGCLbQLNMGH?aM P IKU 0LX7?&%bIKGH?BLNMA3GCL0DQDFG GHcD@ECD/ P ?aECDF0DFMGCALKd DINb^c?aMX7?ar9?BX P IKUD7lmDFEH?a0DM9G cD/ECDQA^LNMIKMbQDQA@?aMGHcD S1/2 ↔ D5/2AClmDQbFGHE P IKECD/rxDFEH AHG^IKwUBD`3?aGHcYECDQAHlmDQbFGGCLDINb^c LNGHcDFE9Gv9l?BbIKUaUa0wmDFGHGCDFEGHcIKM %S\ O M cD@LNMUab^cIKMWsD[IKEH?BACDQA dfECLN#GHcD@X7EH?tdGLKdSGHcD HN >MECDd DFECDFMbQD bIr9?aGvx cDFECDd LNECDY`D P ACDdfECD/ P DFMbF ?t7?aMW,GCLlECLX P bQD GHcDYL P GHl P GAH?aW]MIKU cD INX7rNIKMG^IKWsD ?BAGHcIKG GHcD dfECD/ P DFMbF%A^L P ECbQDQA g %nog 'bIKMwmD %7DQX `3c?aUBD GHcD L rxDFE^IKUaUX7EH?tdfGS?BA_bQLN lmDFMA^IKGCDQXd LNESw GHcD/UBLbIKU L]ACbF?aUaUVIKGCLNE.fjL\SIKUBLNMD]Zg=YIKMXg ' IKECD P ACDQX dfLNE_AC?BXDFwIKMX
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bQLLNUa?aMWj INACDFE lcINACDQA0IKECD bQLNMGHECLNUaUBDQXwAH`3?aGCb^c?aMWwmDFGv`DQDFM X7?&Y DFECDFMG0AH?aW]MIKU3lIKGHcA P AC?aMW,GHcDAH`3?aGCb^cDQA/A H noA >7. cDUBDFMW]GHcA/LKdGHcD0bIKwUBDQA XDFMLNGCDQX φ0 n φ3 IKECD0bIKECDd P UaUaG^IK?aUBLNECDQXGCL ?a lUBDF0DFMGGHcD XDQAC?tE^DFX%lcINACDQA`3c?Bb^cXDFlmDFMXLNM%GHcD D7lmDFEH?a0DFMGGCLwmDlmDFEpd LNEH0DQX @DFMbQD0d LNEIbQDFEHG^IK?aMD7lmDEC?t0DFM9G^IKUZlECLNGCLbQLNU3lECDFlIKE^IKGH?BLNM4LKd_GHcD IKllECLNlEH?VIKGCD ACDFG LKd@bIKwUBDQAφ0nφ3
?BAECD/ P ?aECDQX c?BA0DFGHcLX cINA ACDFEHrxDQX INAZIKMY?aMGCDFEH?a ACLNU P GH?BLNM LKd GHcD@lcINACD[AH`3?aGCb^c?aMW lECLNwUBDF d LNEGHcD[D9lmDFEH?a0DFMGCAlECDQACDFMGCDQX?aMGHc?BA GHcDQAC?BA P AHG^LNn INXDlcINA^DAH`3?aGCb^c?aMW XDFr9?BbFDQA dfLNE GHcD ECD/ P ?aECDQXE^INX7?BLKdECD/ P DFMbFE^IKMWsD3IKECDbQLN 0DFECbF?VIKUaUa IrNIK?aUVIKwUBD]w P G dfDIKG P ECDUBLNMW[XDFUa?arxDFEH/GH?a0DQAL cD MDFG `LNEH\LKdlcINACDnobIKwUBDQAIKMX AH`3?aGCb^cDQA(wmDFGv`DQDFMA @ IKMX b*= ?BAZb P EHECDFMGHUa wmDF?aMW0ECDFlUVINbQDQX w A P b^cINfAH?aMW]UBD XDFr9?BbQD T Z |/zh Z cD X7?&Y DFECDFMG3UVINACDFE3wmDIK0AZX7?BACb P ACACDQX ?aM GHcD[lECDFr9?BL P AZb^cIKlGCDFE_IKECD[X7?aECDQbFGCDQX GCLGHcD[GHE^IKl bQDFMGCDFESIKGr]IKEH?BL P AZIKMW]UBDQA `3?aGHc ECDQAClmDQbFG GCLGHcD@GHE^IKlIe7DQAZIKMXYGHcD@IKW]MDFGH?Bb %DFUBX Tg ?aW0'7 H ACcL `ZAGHcD[WsDFLNnDFGHEHLKdkGHcD_rNINb PP b^cIKwmDFEW cD_Ua?aMDIKEGHE^IKlY?BA IKG GHcD@bFDFM9GCDFE LKdkGHcD_rsDQACA^DFUhbQLNMAHGHE P bFGH?BLNM 7? D]LKdGHcD
x′ny′nz′bQLLNECX7?aMIKGCDAH7AHGCDF,mIKMX?BA_?aMX7?BbIKGCDQXw Gv`L ACLNUa?BXwIKECA +J@LNGCD/GHcIKG_GCc?BA_bQLLNECX7?aMIKG^DAH7AHGCDF(X7?&Y DFECAwI ECLNG^IKGH?BLNMTdECLN(GCcD
xnynzbQLLNECX7?aMIKGCD@AC9AHGCDF#DF lUBL xDQXGCL/XDQACbFEH?awmDSGHcDSGHE^IKl?aGCACDFUtd fbdp ACDQbFGH?BLNM &% cD GHE^IKlIe7?BA z ?BAcLNEH? O LNMG^IKU ? D] ?tGUa?BDQA?aMGHcD x′ n y′ lUVIKMD IKMX%?BAECDFlECDQACDFMGCDQX%w I XINAHc7noXLNGHGCDQX%Ua?aMD]N cD_g DFUBDQbFGHECLXDwUVINXDQAIKECDYIKGC=6' `3?aGHc%ECDQAHlmDQbFGGCL GHcD
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x′nvIe9?BAZw IKMIKMW]UBD
α = 22.5 cD ECDINA^LNMd LNE GHc?BA ?BA GHcIKG ?aM%L P E lECDQA^DM9G/ACb^cDF0D0GHcD HN > ) wmDIK P AHGINXX7ECDQACA?aMX7?ar9?BX P IKU ?BLNMAw P G0IKGGHcD A^IK0D GH?a0DYcIrxDYACLN0D lECL HDQbFGH?BLNMIKUBLNMW,GHcDYGHE^IKlIe9?BA?aMLNECXDFEGCLbQL P lUBD GCL0AH?BXDFwIKMXASLKd Ie7?VIKU 0LNGH?BLNMfACDQD IKUBA^L ) llmDFMX7?t ) gLNEGHcD D7lmDFEH?a0DFMGCA `DYMDQDQXI`DFUaUtnoXDR%MDQXIKW]MDFGH?Bb%DFUBXGCL,AClUa?tG GHcD,]mDQDFIKM7noA P wUBDFrxDFUBALKd GHcD
S1/2 ↔ D5/2GHE^IKMAH?aGH?BLNMk cD0bF?aECb P UVIKEHUalmLNUVIKEH? O DQX 397σ
Ua?tW]c9G/X7EH?arxDQAσGHEJIKMAH?aGH?BLNMAIKMXGHcDFECDFwlECL r9?BXDQA LNlGH?BbIKUTl P l?aMW ?aMGCL LNMDIKW]MDFGH?Bb A P wUBDFrxDFUTLKd GHcD S1/2 ACG^IKGCD] gLNE[GHc?BA[GHcDUVINACDFESwmDIK P AHGSlmLN?aMG@IKUBLNMWGHcD IKW]MDFGH?Bb%DFUBXIe7?BA uvM GHcD D7lmDFEH?a0DFMGCA_lECDQACDFMGCDQXcDFECDGv`LX7?&Y DFECDFMG[IKW]MDFGH?Bb.%DFUBXbQLNM %W P E^IKGH?BLNMA cIrxDwmDQDM P ACDQX GHcD0LNGH?ar]IKGH?BLNM,dfLNE[GHc?BA[?BA XDFG^IK?aUBDQX,?aM
) llmDFMX7?t ) uvMGHcD %ECAHGbQLNM %W P E^IKGH?BLNM `3c?Bb^c ` INA P ACDQXd LNE0AH?aMW]UBD ?BLNM D7lmDEC?t0DFM9GCA GHcDIKW]MDFGH?Bb%DFUBX lmLN?aMGCA_IKUBLNMW0GHcDx′nvIe7?BAf %3IKMX ?BA3lECLX P bQDQX w GCcD bQLN?aU # cD LNGHcDFESbQLN?aUBAIKECD P ACDQXGCLM P UaUIKUaUD9GCDFEHMIKU IKW]MDFGH?Bb%DFUBXA_lmDFEHlmDFMX7?Bb P UVINE[GCL x′ ) bQbQLNECX7?aMW]Ua[hGHcD 397σ Ua?aW]cG` INA_X7?aECDQbFGCDQX,IKUBLNMW GHcD
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λdF
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λ = 729eFSgFWYPiUXW
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Vcomp = Voffset + cVend−cap PRGE;GbYDFcCW;bnUXE;G
Voffset = −44.5
c = 0.0667UXeRf
Voffset = −7.7\
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866 nm
729 nm
854 nm
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397 nm
866 nm
854 nm
397 nm σ-
729 nm
counter
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WYEQUXeRbYACWYAZ[%eF,%eWYPRG8RE;bYWOmU%b;GWYPRGAZ[%eAZb^fRUXEYnWQ[WYPRGcU%b;GTEEQU%fAUXWYAZ[%eUXeRfPRGTeROG\[%eFc0.gbYWYEQU2.wcCACBP<WxUXWxIeFS AZbO[%cCcCGOTW;Gf[%eWYPRG 9CBPkNp3ACWYPO[%DFe<WhEQUXW;Gb[X]UXoFoFE;[mACSVUXW;GTc0. xh ,!%] PR[pGT_7GTEm7W;PRGrAZ[%eO?.OTcZGb[%eVWYPRG
S1/2 ↔ P1/2WYEQUXeRb;ACWYAZ[%eUxzi[%DFE;Gb;OGTeROG
EQUXW;G[X]3HW;[0 Lhg¡ o GTEkAZ[%eN¢hAZbk[%~FWQUXACeRGf5, PFAZb.LAZGTcZfFbU_7GTE1.B[<[<ffAZbQOTEYACSACeiUXWYAZ[%e ¡> 99
N¢~ GTW pGGTe , AZ[%ezRDR[%E;Gb;OGb2- ¡⇔ S1/2
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@ACB8,FI , F¡U¢bYPR[prb3Unb;ACeFBcZGhAZ[%e|PFAZbYW;[%BEQUXS [X]soFPR[%W;[%SnDFcCWYACoFcZAZGTErO[%DFeLW;bO[%cCcZGOTW;Gfw[_7GTESVUXe/.Gqo GTE uACSGTe<WQUXcbQG7DRGTeROGb2,+% eWYPFAZbkOmU%b;G WYPRG\AZ[%epU%bkGqFOTACW;GfE;GbQ[%eiUXe<WYc0.UXWnIJeFS fDFEYACeFBwWYPRGoFE;[%~ Go GTEYAZ[<f5, " WYPFE;GbYPR[%cZfPgAZbrb;GTW3~ GTW pGGTegzRDR[%E;Gb;OGTeROG 1 # UXeRf|zRDR[%E;Gb;OGTeROG 1 @@ ,G\WK.<oFAZOmUXcCc0.o GTE ][%EYS £b;DROQPGqLo GTEYACSGTe<W;b`p3ACWYPWYPRGb UXSG\oiUXEQUXSGTW;GTE;b`W;[E;GO[%E;f[%eRGfRUXWQU
o [%ACe<W2,+PRGcUXWYW;GTExAZbkOmUXcZOTDFcUXW;GfU%bxWYPRGn]tEQU%OTWYAZ[%e[X]zRDR[%E;Gb;OGTeROG 1 @@GT_7GTe<W;b2,&% WE;GToFE;Gb;GTe<W;bkWYPRGD5/2
bYWQUXW;GoFEQ[%~iUX~FACcCAW . UXeRfp3ACcCc~ GfFGTeR[%W;GfPD
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UX~ [%DFWrIJ`Sb7ACeROTcCDRfACeFBWYPRGhbQGTWYWYACeFB\[X]s_%UXEYAZ[%DRbfFGT_LAZOGbUXeRfWYPRGTACEO[%SSnDFeFAZOmUXWYAZ[%ewp3ACWYPWYPRGO[%euWYE;[%csO[%SoFDFW;GTE2,)% W3AZb3DRb;GfU%b3cZ[%eFBU%b3S[%WYAZ[%eiUXcsBE;[%DFeRfbYWQUXW;G`O[<[%cCACeFBAZbreR[%W3EQGT<DFACE;Gf5,)B[%E;G[_7GTEmU%bnACW\AZbneR[%Wb1.LeROQPFE;[%eFACmGfW;[WYPRGo [pGTE\cCACeRGV]tE;G7DRGTeRO?.[X]HlWYPRGG GOTWYAC_7GSVUXBeRGTWYAZO iGTcZfzRDROTWYDiUXWYAZ[%eRbV¡tfDFEYACeFB|WYPRG\oFE;[%~ G\o GqE;AC[<fFbq¢UXE;G\b;[%SGTp3PiUXWxcUXEYBGTExA]cCAeiG\b1.LeROQPFE;[%eR[%DRbzRDROTWYDiUXWYAZ[%eRb
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S1/2(m = −1/2) ↔ D5/2(m′)
UXE;G fFGTeR[%W;Gf ~/. m′\p3PRGTE;G
m′ =−5/2,−3/2,−1/2,+1/2,+3/2 ,!PRG?.nUXE;GbYoFcCACW~/.nUlSVUXBeRGTWYAZO NGqcZf[X], H(QkUXDRbQb2," LAUXcLbYAZfFGT~iUXeRfFb^UXE;GcUX~ GTcZGfn~/. " EQU%fAUXcbYAZfFGT~iUXeRfFb~/.H- bYACBeRbdACeRfAZOmUXW;G~FcCDRGbYAZfFGqu~iUXeRfFbp3PFACcZG − bYACBeRbE;Gq]tGTEW;[nE;Gf|bYAZfFGT~iUXeRfFb2,@i[%EWYPFAZb3b;OmUXeLWYPRGxU:AUXcN]tE;G7DiGqeiO7.VpU%bWYDFeRGfW;[
ωz = 2π · 1.715 Bln~/.UXoFoFc0.<ACeFBwUXeGTeRfLuOmUXo_[%cCWQUXBG\[X]Vcap = 2000
I,1 e WYPRG|bYWYE;[%eFBGb;WOmUXEYEYAZGTE
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b;GGTeF, PRGwE U%fAUXcrWYEQUXo ] E;G<DRGTeRO?. AZbωr ' 2π · 4.9 Bl , PRGwWYEQUXeRbYACWYAZ[%eRb|
−3/2UXeRf +1/2
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UXE;GoFE;Gb;GTe<W¡tb;GGbQGOTWYAZ[%e I , ¢7, " cCACeRG|b1.LeROQPFE;[%eR[%DRboFDFcZb;Gf b;o GTOTWYEQ[b;O[%o/.E;G7DFACE;GbUSACeFACSnDFScZGTeFBWYP[X]CSbmFb;GG`b;GOTWYAZ[%eI ,/,/\][%ElU\BGTeRGTEYAZO`oFDFcZb;G`b;G7DRGTeROGC,
PRGbYACSoFcZGbYWkWK.Lo G[X]3U *+ b;G7DRGTeROGAZbU !0' %#" " ! UXWnIJeFSg p3PFAZOQP¤AZbkOQPiUXEQU%OTW;GTEYACmGf~/.ACW;bfDFEQUXWYAZ[%e
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D5/2(m′ = −5/2) WYEQUXeRbYACWYAZ[%eF, - UXeRf " fFGTeR[%WQGU:LAUXclUXeif EQU%fAUXchUXeRf U:LAUXclbYAZfFGqu
~iUXeRfFbmEQGTb;o GOTWYAC_GTc0. ,CPRG`U:AUXc WYEQUXo|]tE;G7DRGTeRO?.AZbωz = 2π · 960 Lh ,#h[%W;GxWYPiUXW3WYPiG
EQU%fAUXcLbYAZfFGT~iUXeRfFb^UXWωr ' 2π·4.95 BhUXE;GW p[Xu]t[%cZfp3ACWYPUlbYoFcCACWYWYACeFB[X] ' 2π·70 Lh ,PFAZbAZbnUwE;GbYDFcCWn[X]lUwbYcCACBP<WBG[%SGTWYEYAZOACSo GTE ]tGOTWYAZ[%e [X]3WYPRGWYEQUXocCA] WYACeFBfFGTBGTeRGTEQU%O?.
ACe|EQU%fAUXcfACE;GOTWYAZ[%eF,)PRGkfFGq]t[<OTDRb;bYACeFBG GOqWr[X]dWYPRGxEQU%fAUXc] E;G<DRGTeRO?.fFDRGW;[\WYPRG`U:LAUXcO[%e<ReRGTSGTe<W|¡tb;GGG<eF,¡,C£¢Y¢xAZb`EQUXWYPRGTEnbYSVUXcCcACe¤WYPFAZbOmU%b;GVU%b
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WYPiUXegWYPRG , oFDFE;GL-xEQU%fAUXcsWYEQUXow] E;G<DRGTeRO?.ωr0 ' 2π · 5.05 Bl ,
S1/2,m = −1/2 bYWQUXW;G7GTcCACSACeiUXWYACeFB\Hxo [b;bYAC~FcZG3WYEQUXeRbYACWYAZ[%eRb2,S% eVU%fFfACWYAZ[%e7PRGTE;G3WYPRGφ = 45, γ = 0O[%e<RBACEQUXWYAZ[%e ¡tbQGGkbQGOTWYAZ[%e, , H¢pU%blOQPR[b;GTeg]t[%ErWYPRGkBG[%SGTWYE1.w~ GTW pGGTeIJeFS cU%b;GTEhUXeRfgSVUXB%u
eRGTWYAZOMiGTcZf p3PFAZOQPUXcCcZ[prbr[%eFc0. 4m = 0UXeRf 4m = ±2 WYEQUXeRbYACWYAZ[%eRb2,C#[%W;GhWYPiUXW3WYPRG`IJneFS cU%b;GTE
fFGTWYDFeFACeFBAZb\OmUXcZOTDFcUXW;Gf U%bnW p3AZOGWYPRGfEYAC_GV] EQGT<DRGTeRO?.[X]lWYPRGO[%EYE;GbYo [%eRfACeFB¤fF[%DF~FcZGquvoiU%b;b " 1 B¡tb;GGb;GOTWYAZ[%eH, ¢p3PFAZOQP UXWRRE;bYWm8.LACGTcZfFbr]tE;G7DRGTeROTAZGbh[%eWYPRG[%E;fFGTEh[X]
2π · 460 Bh ,@i[%ExIJeFScU%b;GTEr]tE;G<DRGTeRO?.wb;OmUXeRb\¡tbYDROQPU%bhGqOTACWQUXWYAZ[%eb;o GOTWYEQU¢rfAZbYoFcU2.GfACeWYPFAZblWYPRGbYAZbRWYPRGk]tE;G<DRGTeRO?.gU:AZbAZbE;GOmUXcCAC~FEQUXW;GfbYDROQPVWYPiUXWACW;b[%EYACBACe¡ mGTE;[nfFGTWYDFeFACeFB<¢O[%ACeROTACfRGTbp3ACWYP[%eRGn¡ WK.LoFAZOmUXcCc0.WYPRGlb;WYE;[%eFBGbYW ¢OmUXEYEYAZGTErE;GbQ[%eiUXeROGC,@ACB8,I , HbYPR[prbxU~ GTWYW;GTEE;Gb;[%cC_7Gfb;OmUXe ¡ WQUXGTep3ACWYPpGmUXGTExGqOTACWQUXWYAZ[%eO[%SoiUXE;GfWQ[@ACB8,I , 0<¢
UXE;[%DFeRf [%eRG[X]rWYPRGOmUXEYEYAZGTE;bmE;GT_7GmUXcCACeFBfFGTWQUXACcZb\[X]rWYPRGVS[%WYAZ[%eiUXcbYAZfFGT~iUXeRf bYWYEYDROTWYDFE;GC,PRGbQUXSGWK.Lo Gk[X]b;OmUXewp3ACWYPgW p[AZ[%eRblUXeRfgbYcCACBP<WYc0.fFA GTE;GTe<WrWYEQUXoFoFACeFBoiUXEQUXSGTW;GTE;blAZbrbYPR[p3ewACeg@ACB8,iI ,/7,C% eBGTeRGTEQUXcpGVGT_UXcCDiUXW;GVW p[XuvAZ[%ezRDR[%E;Gb;OGTeROGfRUXWQUg~/.b;GTWYWYACeFBW p[gWYPFE;GbYPR[%cZfFb][%EoFPR[%W;[%SnDFcCWYACoFcCAZGTEO[%DFe<W;bmb;GG@AB8,I , , PRG
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′ = −5/2) WYEQUXeRb;ACWYAZ[%eF,PRG_%UXEYAZ[%DRbdbYAZfFGT~iUXeRfFbO[%EYE;GbYo [%eRfW;[rfA GTE;GTe<Ws_LAC~FEQUXWYAZ[%eiUXc<S[<fFGb¡tb;GGb;GOTWYAZ[%e\,/,/¢qK" .xU:LAUXc 1 B ! . !E;GmUXWYPFACeFBR-l.hE;[<OQLACeFBR5-.hEQU%fAUXc 1 B , PRGU:AUXcoiUXEQUXSGTW;GTE;bUXE;GUXB7UXACe
ωz = 2π · 960 LhVcap = 600
p3PFACcZGWYPRGEQU%fAUXc^]tE;G7DRGTeRO?.pU%bωr '
2π · 3.0 Bl ,~PRGVEQU%fAUXcbYoFcCACWYWYACeFBAZbPiUXE;fc0.¤_<AZb;A~RcCGV~ GOmUXDRb;GVWYPRGbYW;GTo bYACmGV[X]rWYPRGb;OmUXegAZb3W;[<[O[UXE;b;GC,
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%] dAe O[%e<WYEQU%bYWnW;[bYDROQPGqFOTAUXWYAZ[%e bYo GOTWYEQUpGVo GTE ]t[%EYS " ! " %-#! !" ! " '!'!"" "$#' !UXE;Gx[%~RbQGTEY_Gf5,@R[%E3WYPRGcUXWYW;GTE3pGWK.LoFAZOmUXcCc0.b;GTW
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fEYAC_LACeFBhACenWYPFAZbdpU2.`WYPRGO[%EYE;GbYo [%eRfACeFBxWYEQUXeRbYACWYAZ[%e\EQGTbQ[%eiUXe<WYc0. ,!% enWYPRGGqLo GTEYACSGTe<W;boRE;Gb;GTe<W;GfnPRGTE;GpGUm_7[%AZf[ NuvE;GbQ[%eiUXe<W`GqFOTACWQUXWYAZ[%eRb¡tOq] ,b;GOTWYAZ[%e, ,/7¢x[X][%WYPRGTEkWYEQUXeRb;ACWYAZ[%eRbU%bkSnDROQPU%bko [b;bYAC~FcZG~/.7GGToFACeFB\WYPRGhO[%DFoFcCACeFBg¡ A+, GC,<WYPRGlcCACBP<Wo [pGTE ¢bYD<KVOTAZGTe<WYc0.bYSVUXcCc+, 4 RUXSoFcZGb[X]-hUX~FA [b;OTACcCcUXWYAZ[%eRbUXE;G`bYPR[p3egACewWYPRGx][%cCcZ[p3ACeFBVbQGOqW;AC[%e,
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Excitation time t in µs0 20 40 603010
Excitation time t in µs
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[X]3¡U¢3UXeRf¡ ~N¢qiU\W;PRGTEYSVUXcsb;WQUXW;Gxp3ACWYPn ≈ 20
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0 20 40 60 80 100 120 140 160 1800.0
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Excitation time t in µsExcitation time t in µs
Excitation time t in µs
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bms¡ ~N¢w = 72.2 µ
bms¡tO¢w = 58.5 µ
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S1/2(m = −1/2)↔ D5/2(m′ = +3/2)WYEQUXeRbYACWYAZ[%ePiU%blWYPRGnbYWYE;[%eFBGbYWhSVUXBeRGTWYAZO&iGTcZffFGTo GTeRfFGTeROG eiUXSGTc0.UE;GTcUXWYAC_7G
gu]U%OTW;[%Ex[X]),/5NbQGG
@ACB8, ,JILUXeRfAZblWYPRGTE;Gq][%E;G\bYWYE;[%eFBc0.g~FE;[U%fFGTeRGfA]SVUXBeRGTWYAZO+iGTcZfgzRDROTWYDiUXWYAZ[%eRb`UXE;GnoFE;Gb;GTe<W2,xM<DROQPSGmU%bYDFE;GTSGTe<W;bnpGTE;GVOmUXEYEYAZGf[%DFW[%e_%UXEYAZ[%DRbOmUXEYE;ACGTE;b\[X]WYPRG ( GGTSVUXeSVUXeFA]t[%cZfUXeRfUXWfA GTE;GTe<WfFGTcU2.b[X]xWYPRGwoFE;[%~ GwoFDFcZbQG|p3ACWYP EQGTb;o GOTWW;[WYPRGwo [pGTEcCACeRGwoFPiU%b;GC, PRG
S1/2(m = −1/2) ↔D5/2(m
′ = −1/2) WYEQUXeRbYACWYAZ[%ePiU%brWYPRGkpGmUX7GbYWrSVUXBeRGTWYAZO iGTcZfgfFGTo GTeRfFGTeROGNeiUXSGTc0.gUE;GTcUXWYAC_Ggu
]vU%OqWQ[%El[X]C, 0RRp3PFAZOQPAZblb;GT_7GTegWYACSGblbYSVUXcCcZGTElWYPiUXeg]t[%ErWYPRGS1/2(m = −1/2)↔ D5/2(m
′ = +3/2)
)(UOx O_V+* T0Oc`,*<X/r2k T mA\Fs2r2k/k/X/\(T 0| twrJ(o
5
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@ACBDFE;GnI ,C£[0RK ,ACeRGhOGTe<W;GTE[X]WYPRGS1/2(m = −1/2)↔ D5/2(m
′ = −5/2) E;Gb;[%eiUXeROG¡tfRUXWQU`o [%ACe<W;b ¢UXeRf O[%EYE;GbYo [%eRfACeFBcCACeRGTp3AZfWYPRbg¡ ~iUXE;b ¢nUXWfAGTE;GTe<WfFGTcU2.Lb[X]rWYPiGGqFOTACWYACeFB¤IJeFSoFE;[%~ GoFDFcZb;G¡tfDFEQUXWYAZ[%e £VSb ¢`p3ACWYP E;GbYo GOTW\W;[WYPRGo [pGTE\cCACeRGoFPiU%b;GC, PRG[%EYACBACe[X]WYPRG
yu U:LAZbkE;GToFE;Gb;GTe<W;b`WYPRGUm_7GTEQUXBG[X]3UXcCcfRUXWQUo [%ACe<W;b2, #h[%W;GWYPiUXWGTEYE;[%Ek~iUXE;b`[%e
WYPRGgcCACeRGOGTe<W;GTE;bUXE;GbYSVUXcCcZGTEVWYPiUXe WYPRGwcCACeRGTp3AZfWYPRbeiUXSGTc0. ~ . U]U%OTW;[%E[X]UX~ [%DFWWYPFE;GGC, PRG
S1/2(m = −1/2) ↔ D5/2(m′ = −5/2) WYEQUXeRbYACWYAZ[%epU%bkOQPR[b;GTePRGTE;GU%b
ACW;brbYo GOTWYEQUXco [bYACWYAZ[%efFGTo GTeRfFblbYWYE;[%eFBc0.|[%ewWYPRGkSVUXBeRGTWYAZO(iGTcZf¡ E;GTcUXWYAC_Ggu]vU%OTW;[%E'¢7,
WYEQUXeRbYACWYAZ[%eF,j%]rcU%b;GTEnzRDROTWYDiUXWYAZ[%eRbpGTE;GeRGTBcCACBA~RcCGPRGTeROGA]lWYPRGcCACeRGb\pGTE;G~FE;[U%fFGTeRGf DFeFAZ<DRGTc0.~/.gSVUXBeRGTWYAZO+iGTcZfzRDROTWYDiUXWYAZ[%eRbmsUVbQGT_GTeWYACSGbeiUXEYE;[pGTExcCACeRGw¡ cCAeiGqprACfFWYP
2π · 170 l:¢rp[%DFcZf~ GGqo GOTW;Gf ]t[%EWYPRG
S1/2(m = −1/2) ↔ D5/2(m′ = −1/2) E;GbQ[%eiUXeROGC, Gw[%~RbQGTEY_GPR[pGq_7GTEm
UcCACeRGTp3AZfWYP [X]`UX~ [%DFW2π · 400 h¡ ][%EWYPRGgbQUXSGwcCACeRGgWYEYACBBGTEfFGTcZU.F¢7, ¤GgO[%eROTcCDRfFGgWYPiUXWVWYPRG
S1/2(m = −1/2) ↔ D5/2(m′ = −1/2) E;Gb;[%eiUXeROG|AZb~FE;[U%fFGTeRGf ~/.SVUXBeRGTWYAZOiGTcCf UXeRf cU%b;GTE
zRDROTWYDiUXWYAZ[%eRbrW;[UXcCS[bYWrG<DiUXc oiUXEYW;bmp3PFACcZG][%ErUXcCc WYEQUXeRbYACWYAZ[%eRb3prAW;PUnPFACBPRGTErE;GTcUXWYAC_Ggu]vU%OTW;[%E3WYPRG
cCACeRGk~FE;[U%fFGTeFACeFBAZbrfF[%SACeiUXW;Gfg~ .SVUXBeRGTWYAZO iGTcZf|zRDROTWYDiUXWYAZ[%eRb2, PRGkcCACeRGTp3AZfWYP[X]U\BAC_7GTegWYEQUXeRbYAuWYAZ[%egfFGTo GTeRfFbrbYWYE;[%eFBc0.[%e|WYPRGcCACeRGxWYEYACBBGTErfFGTcU2.7FU%b3bYPR[p3ewACew@AB8,iI ,C£[08,C%W3OmUXe|~ Gxb;GGTewWYPiUXWWYPRGcCACeRGAZboiUXEYWYAZOTDFcUXEYc0.¤eiUXEYE;[p UXWnOGTEYWQUXACe fFGTcU2.b2, % W`AZbWYPRGTE;Gq][%E;GU%f_UXe<WQUXBG[%DRbnW;[bQGTW`WYPRGWYEYACBBGTE[X][%DFE$SbhoFDFcZbQG\b;G<DRGTeROGTbkbYDROQPWYPiUXWWYPRG\oFE;[%~ Gno GTEYAZ[<f ¡ prPRGTE;GO[%PRGTE;GTe<WxSVUXeFACoFDFcUXWYAZ[%eRbUXE;Go GTE ][%EYSGfi¢`]vUXcCcZb\W;[bYDROQP UfFGTcU2. ,¤le][%EYWYDFeiUXW;GTc0.WYPRGoiUXWYW;GTEYe [X]@ACB8,^I ,C£[0AZb\eR[%WbYWQUX~FcZG|[_7GTEcZ[%eFBVWYACSGnb;OmUXcZGbU%blbYPR[pre~/.wbYACSACcUXEhGqo GTEYACSGTe<W;bhp3PFAZOQPpG`PiUm_7GOmUXEYEYAZGf[%DFWUXWlACe<W;GTEY_%UXcZb[X]U]GTp S[%e<WYPRb2,)#GT_7GqE;WYPRGTcZGTbQbmFWYPRGb;G`GqLo GTEYACSGTe<W;blE;GT_7GmUXcWYPiUXWrWYPRGTE;GnUXE;GkcCACeRGqub1.LeROQPFE;[%eR[%DRbmRo GTEYAZ[<fAZOzRDROTWYDiUXWYAZ[%eRbloFE;Gb;GTe<WlAe[%DFElb1.bYW;GTS@,
#[%ecCACeRGqub1.LeROQPFE;[%eR[%DRbzRDROTWYDiUXWYAZ[%eRbkOmUXe~ GnoFE;[%~ Gf~/. -hUXSb;G?.bYo GOTWYE;[b;O[%o/.U%bGqoFcUXACeRGfACe@ACB8,I ,C£mH,%WnOmUXe~ Gb;GGTeWYPiUXW][%E
τ = 100 µbnWYPRG[%~Rb;GTEY_7GTf -UXSb;G?.]tEYACeFBGbnPiUm_7G_7GTE1.PFACBP
¡ON¢O[%e<WYEQU%bYW2,SPRG3b UXSGWK.<o G3[X]b;OmUXeRb^E;GO[%E;fFGf][%E^fA GTE;GTe<W^pUXACWYACeFB`WYACSGbτE;GT_7GmUXcFWYPRGoFPiU%b;G
fFGO[%PRGTE;GTeROGp3PFAZOQPcZGmU%fFbkW;[wUcZ[b;bk[X]O[%e<WYEQU%bYWkp3ACWYPACeROTE;GmU%b;ACeFBτ b;GG@ACB8,I ,C£7,n@i[%Ek<DiUXe<WYDFS
O[%SoFDFWQUXWYAZ[%e GqLo GTEYACSGTe<W;bVUSACeFACSnDFS O[%e<WYEQU%bYW[X]xUX~ [%DFW*5ON AZb\E;G<DFACE;Gf5, 1 DFEGqLo GTEYACSGTe<W;bbYPR[p WYPiUXWWYPRGO[%e<WYEQU%bYWAZb`E;GfDROGf¤W;[5ON U:] W;GTE
τ = 450µbmsp3PFAZOQPU%OO[%E;fACeFBc0.¤bQGTW;bkWYPRGWYACSG
5
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τ,SPRGrcCACBP<Wo [pGTEAZbU%fF; DRbYW;GfbYDROQPVWYPiUXW
GmU%OQPVoFDFcZb;G3E;GToFE;Gb;GTe<W;bUπ/2
E;[%WQUXWYAZ[%eVA]WYPRG3cU%bQGqE] E;G<DRGTeRO?.\AZbE;Gb;[%eiUXe<WW;[UkOGTEYWQUXACeOmUXEYEYAZGTEWYEQUXeRbYACWYAZ[%eF,@i[%EU -hUXSbQG?. bYo GOTWYEYDFSgWYPRGgcZU%bQGTE]tE;G7DRGTeRO?.
ωlAZbb;OmUXeFeRGf
UXE;[%DFeRfnWYPFAZbOmUXEYEYAZGTEp3PFACcZGUXcCcL[%WYPiGqEoiUXEQUXSGTW;GTE;b^UXEQGCFGf5,!% fFGmUXcCc0.7XA+, GC,p3ACWYPR[%DFWoFPiU%b;GfFGO[%PRGTE;GTeROG
PDbYPR[%DFcZfGqLPFAC~FACWkUVS[<fDFcUXWYAZ[%e~ GTW pGGTemGTE;[|UXeRf[%eRG\UXEQ[%DFeRfWYPRG
E;Gb;[%eiUXeROGC,g¡ ~N¢4-hUXSb;G?.bYo GOTWYEYDFS [%eWYPiGS1/2(m = −1/2) ↔ D5/2(m
′ = −1/2)OmUXEYEYAZGTE2,'PRG`cU%bQGTEhfFGTWYDFeFACeFBAZblBAC_7GTep3ACWYPE;GbYo GOTWhW;[VWYPFAZbhOmUXEYEYAZGTE2,'PRG`Wp[oFDFcZb;Gb[X]cZGTeFBWYP
9.5 µb3pGTE;G`b;GToiUXEQUXWQGTfg~/.
τ = 100µb2,
6 78696 :9686 ;9686 <=686 >%69686 >=786966796:96;96<=6>=686
?A@BCBDEFCBGIHJKBDMLON
@ACBDFE;GnI ,C£7K [%e<WYEQU%b;WV[X]xWYPRGS1/2(m = −1/2) ↔ D5/2(m
′ = −1/2) -hUXSbQG?.oiUXWYW;GTEYeU%bVU]tDFeROTWYAZ[%e [X]
τ, ML[%cCAZf OTACE;OTcZGbmKwfRUXWQUWQUX7GTe p3ACWYPR[%DFWSVUXBeRGTWYAZO~iGTcCf O[%So GTeRbQUXWYAZ[%eF,
1 o GTeOTACE;OTcZGbmKkfRUXW;GWQUX7GTep3ACWYP¤O[%So GTeRbQUXWYAZ[%e~FDFWkp3ACWYPU|bYACeFBcZG?NGqcZf¤b;GTeRb;[%E`[%eFc0.¡tb;GGW;GqW ¢7,aML[%cCAZf cCACeRGKIQkUXDRb;bYAUXe RWgp3ACWYP
e−(τ/τ0)2 .LAZGTcZfACeFB
τ0 = 0.94(5)Sb2,
jkU%bYPRGfwcCACeRGKGqo [%eRGTe<WYAUXc RW3p3ACWYPe−τ/τ0
.LAZGTcZfACeFBτ0 = 1.4(2)
Sb2,
5F0
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e−τ/τ0^p3ACWYP
τ0 = 1/4ω ,GPFAZbhS[<fFGTcoFE;GfAZOTW;b`U ,[%E;GTe<WYAUXeE;Gb;[%eiUXeROGncCACeRG\[X]p3AZfWYP 4ω ,@dACB8, I ,C£7bYPR[prbmPR[pGT_7GTEmWYPiUXWUXe GqLo [%eRGTe<WYAUXcRRWW;[[%DFEfRUXWQUAZb_GTE1.o [<[%Ep3PFACcZbYWVU6QkUXDRb;b;AUXe>RWAZbACe B[<[<fUXBE;GGTSGTe<W`p3ACWYPWYPRGE;GbYDFcCW;b,n@RE;[%S WYPFAZbxpGO[%eROTcCDRfFGWYPiUXWUXeGqFOGb;bk[X]zRDROTWYDiUXWYAZ[%eRbO[%SoiUXE;GfW;[VU\p3PFACW;GkeR[%AZbQGxS[<fFGTcsACbroFE;Gb;GTe<WrACeg[%DRErb .LbYW;GTS ][%E] E;G<DRGTeROTAZGb
> 1.5h ,
" bSGTe<WYAZ[%eRGf 3UXe U%OTWYAC_7GwOmUXeROGTcCcUXWYAZ[%e b .LbYW;GTS ]t[%ESVUXBeRGTWYAZO~iGTcZf zRDROTWYDiUXWYAZ[%eRbVPiU%bE;GOGTe<WYc0.~ GGTeACeRbYW UXccZGf|[%eWYPRGGqLo GTEYACSGTe<W2,@RE;[%S fACE;GOTW3OmUXEYEYAZGTEbQOmUXeRbpG'ReRfWYPiUXWp3ACWYPWYPRGxU%OTWYAC_7GO[%Suo GTeRbQUXWYAZ[%eACeDRb;G WYPRGnE;Gb;[%eiUXeROGn]tE;G7DiGqeiOqAZGbk[X]fAGTE;GTe<WkOmUXEYEYAZGTE;b`UXE;G\Gb;b;GTe<WYAUXcCc0.O[%eRbYWQUXe<Wxp3ACWYPE;GbYo GOTWW;[nWYPRGho [pGTEcACeRGxfFGTcU2. ,PFAZbPiU%bWYPRGxU%f_UXe<WQUXBGWYPiUXWWYEYACBBGTEYACeFB[X]sWYPRGhoFDFcZb;Gb;G7DRGTeROGb[%eWYPRGo [pGTEncCACeRGAZbeR[%WOTEYACWYAZOmUXc3UXe/.S[%E;G|UXeRfW;PiUXWnpGU%OQPFAZGT_7GVACe UHgSbnoFDFcZb;GbQG7DRGTeROG|UbYACSACcUXE`] E;G<DRGTeRO?.b;WQUX~FACcCACWK.¤U%bkACeU|cCACeRGWYEYACBBGTE;Gf wSbkoFDFcZbQGb;G<DRGTeROGC,V[pGT_GTEm " ' ' -#! -!pGTE;G[%eFc0.bYcCACBP<WYc0.E;GfDROGf¤~/.WYPRGOmUXeROGTcCcUXWYAZ[%eb .LbYWQGqS@,*B[%EQGT[:_GTEmd@ACB8,sI ,C£7wE;GT_GmUXcZb`WYPiUXW`WYPRGO[%e<WYEQU%bYW[X] -UXSb;G?.¤bYo GOTWYEQUU%bUw]tDFeROTWYAZ[%e [X]rWYPRGVpUXACWYACeFBWYACSG
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" cCcGqLo GTEYACSGTe<W;bxoFE;Gb;GTe<W;Gfb;[]UXEhpGTE;G\OmUXEYEYAZGf[%DFWhp3ACWYPR[%DRWSVUXBeRGTWYAZO+iGTcZfO[%So GTeRbQUXWYAZ[%e[%Ep3ACWYPUVbYACeFBcZGSVUXBeRGTWYAZO+iGTcZfbQGTeRb;[%Er][%EhWYPRGnO[%So GTeRbQUXWYAZ[%eb .LbYW;GTS@,ljDRGW;[VWYPRGnO[%eRb;WYEYDROTWYAZ[%e[X]WYPRGx_U%OTDFDFS _7Gb;b;GTcUXeRf|WYPRGkbYACmGk[X]WYPRGkbYPFAZGTcZfFGfgb;GTeRb;[%E~ [m¤¡UX~ [%DFW`£OTS¢qFWYPRGkb;GTeRb;[%ErOmUXeg[%eFc0.~ GloFcU%OGf|UXWU`fAZbYW UXeROGh[X]dUX~ [%DFWx£mH`OTS ] E;[%S WYPRGlWYEQUXoOGTe<W;GTEmLPRGTeROGfF[<GbeR[%WoFAZOQDFoSVUXBeRGTWYAZOiGTcZfzRDROTWYDiUXWYAZ[%e GqRU%OTWYc0.p3PRGTE;GVWYPRG?.UXE;GW;[w~ GVOmUXeROGTccZGf5,% e¤[%E;fFGTEW;[gACSoFE;[_GV[%eWYPFAZbbYPR[%EYW uO[%SACeFBUwb;GO[%eRfb;GTeRb;[%E`PiU%bk~ GGTeACeRbYWQUXcCcZGf[%e¤WYPRG[%oFo [b;ACW;Go [b;AW;AC[%e[X]WYPRGWYE UXo¤WYEQUXoOGTe<W;GTE2," egU_GTEQUXBGfgbYACBeiUXc ]tE;[%S WYPRGxWp[b;GTeRb;[%E;b3AZb3WYPRGTew]Gf|W;[WYPRG`OUXeiOTGTcCcUXWYAZ[%eb1.bYW;GTS@,PFAZblUm_7GTEQUXBGAZbrGqo GOTW;GfwW;[E;GToFE;Gb;GTe<WUb;[%SGTp3PNUXWrACSoFE;[_7GfwSGmU%bYDFEQG`[X]W;PRG`U%OTWYDiUXczRDROTWYDiUXWYAZ[%eRbUXWrWYPRGkWYEQUXoOGTe<W;GTE2,% e U%fFfACWYAZ[%eSGmU%bYDFE;GbpGTE;GWQUX7GTe W;[E;GfDROG|GTcZGOTWYE;[%eFAZOweR[%AZb;G|[%e WYPRGcU%b;GTE\cZ[<OQ ,GPiUm_7G[%~Rb;GTEY_7GfWYPiUXWp3ACWYP|WYPRGb;GxOQPiUXeFBGbmWYPRG-hUXSb;G?.VO[%e<W;EQU%bYWPiU%bACeRfRGTGf|ACSoFE;[_7Gf ~ GTACeFBeR[p5ON U:]tW;GTEUX~ [%DFW
700 µb2,h[pGT_7GTEmib1.bYW;GTSVUXWYAZObYWYDRfAZGbh[%eWYPRG`o GTE ][%EYSVUXeROGn[X]^WYPRGOmUXeROGTcCcUXWYAZ[%e
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B[bYWk[X]WYPRG\E;Gb;DFcCW;bxoFE;Gb;GTe<WQGTfACeWYPFAZbkb;GOTWYAZ[%eUXEQGfFGTWQUXACcZGfACe 9 :,?PFAZbkE;Gq]tGTE;GTeROGVU%fFfACWYAZ[%euUXcCc0.fAZb;OTDRbQb;GboFE;GTcCACSACeiUXE1.SGmU%bYDFE;GTSGTe<W;bkpG\PiUm_7GOmUXEYEYAZGf[%DFWkGTSoFcZ[.<ACeFBU -hUXSVUXeWYEQUXeRbYACWYAZ[%eW;GOQPFeFAZ<DRGC, % e WYPRGbQGwGqo GTEYACSGTe<W;bm -UXSVUXe WYEQUXeRb;ACWYAZ[%eRbUXE;GwfE;A_7GTe ~ GTW pGGTe WYPiG|W p[SVUXBeRGTWYAZObYDF~FcZGT_7GTcZb¡
m = ±1/2 ¢h[X]WYPRG S1/2 BE;[%DReRfbYWQUXW;GC,+PFAZbkW;GOQPFeFAZ<DRGUXcCcZ[prbkWYPRGfFGTW;GTEYSACeiUXW;AZ[%e[X]SVUXBeRGTWYAZO$iGTcZf|zRDROTWYDiUXWYAZ[%eRbUXcZ[%eRGnUXeRfwE;GTSVUXACeRbrACSSnDFeRG`W;[cU%b;GTE3zRDROTWYDiUXWYAZ[%eRb2,
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WYPRGhU:LAUXcRWYEQUXo]tE;G7DRGTeRO?.pU%bωz = 2π ·940 h%]tE;[%S p3PFAZOQPUXeAZ[%eVfAZb;WQUXeROG 4z =
6.3 µS AZbrOmUXcZOTDFcUXW;Gf RWYPFAZblO[%EYE;GbYo [%eRfFblW;[UoFE;[F; GOTW;GffAZbYWQUXeROG 4zp = 5.8 µ
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AZbxSGmU%bYDFE;Gf p3PFAZOQP.LAZGqcZfFbUwoFE;[FRcZG[X]rWYPRGVcU%b;GTE`][<OTDRb2, !Gq]t[%E;GVWYPiUXWmWYPRGcU%bQGTE`][<OTDRbAZb\U%fF;DRb;W;Gf[%e<W;[WYPRGVAZ[%eRb~/.WYDFeFACeFBVWYPRGnU%fFfE;Gb;bYACeFB[%oFWYAZObr]t[%ErSVU:ACSnDFS GqFOqACWQUXW;AC[%e, % eUb;GO[%eRfbYW;GTo8-UX~FAs[b;OTACcCcUXWYAZ[%eRbUXE;GE;GO[%E;fFGf5,@i[%E`WYPRGb;GGqo GTEYACSGTe<W;bmdjx[%oFoFcZGTEnO[<[%cCACeFBAZbbYD<KVOTAZGTe<W\UXeRf¤b;[UWK.LoFAZOmUXcoFDFcZb;GcZGTeFBWYPb;OmUXecZ[7[%bcCAC7G\@ACB8,I ,JI¡U¢7,`@i[%cCcZ[p3ACeFB|eRGqWmpG\b;GTcZGOTW`UVoFDFcZbQG\fDFEQUXWYAZ[%e
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t0 = 4 µb2,6#[p U
!; # *!" AZbo GTE ]t[%EYSGf K ωl UXeRf t0 UXE;G3GToFWFFGfp3PFACcZGrWYPRG3_[%cCWQUXBGr[%eWYPRG3~ GmUXS fFGqziGOTW;[%EAZbb;OmUXeFeRGf5, 4 U%OQPfFGqziGOTW;[%E3o [b;ACWYAZ[%egO[%EYE;GbYo [%eRfFb3W;[U\OGTEYWQUXACewcCACBP<W3ACe<W;GTeRbYACW .
I[%e|WYPRGxAZ[%es¡tb ¢7,CPRG
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]tE;[%S mGTE;[W;[w[%eRGw¡tb;GG@ACB8,sI ,JI¡U¢Y¢hAZbBAC_7GTeV~/.
PD = sin2(Ωt/2)W;[\Uk_GTE1.B[<[<fUXoFoFE;[mACSVUXWYAZ[%eF,C% W^][%cCcZ[prbWYPiUXW
PD = sin2(a√It0/2)
,bYACeFBWYPFAZb]t[%EYSnDFcUFWYPRGxGqFOTACWQUXWYAZ[%ewoFE;[%~iUX~FACcCACWK.oFEQ[FRcZGkOmUXe|~ GxE;Gb;OmUXcZGf|ACe<W;[UXe|ACe<W;GTeRbYACWK.oFE;[FRcZGC,4 RUXSoFcZGbl[X]fFGqziGOTW;[%Ehb;OmUXeRbrUXE;G`bYPR[p3ewACeg@ACB8,iI ,£I ,C%]dpGxo GTE ][%EYS b;DROQPgUb;OmUXewp3ACWYPU\W p[XuvAZ[%eOTE1.bYWQUXc¡t@dACB8,I ,C£I¡ ~N¢Y¢kpGVOmUXeOmUXcCAC~FEQUXW;GWYPRGVfFGqziGOTW;[%E\_[%cCWQUXBGbACe<W;[cUXW;GTEQUXcWYEQUXeRbYcUXWYAZ[%eRb\[X]3WYPRGcU%b;GTE]t[<OTDRb2,d[fF[b;[FLWYPRGxU:AUXcWYEQUXo] E;G<DRGTeRO?.
ωzAZbfFGTW;GTEYSACeRGf|] E;[%S UXewGqOTACWQUXWYAZ[%egbYo GOTWYEYDFS
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H, 0RdWYPRGWYEQUXo U:AZb`AZbeR[%Wo GTEYo GTeRfAZOTDFcUXE\W;[gWYPRGU%fFfE;Gb;bYACeFB~ GmUXS OQPiUXeReRGTcIJ " U%b`WYPRGW;EQUXoAZbWYACcCW;Gfw~/.~, H ,[%eib;G7DRGTe<WYc0.7FWYPRG`G GOTWYAC_GV¡ oFE;[F;YGOTW;Gfi¢AZ[%efAZbYWQUXeROG`AZb
4zp = 4z · cos 22.5 .¡vI ,C£¢
@RE;[%S bYACeFBcZGAZ[%e fFGqziGOTW;[%Eb;OmUXeRbU%b\ACe @ACB8,I ,C£I¡U¢qU QkUXDRbQbYAUXe]t[<OTDRbpUXAZbYW[X]2.5 µ
S AZbfFGquW;GTEYSACeRGf DRbYACeFBWYPFAZbOmUXcCAC~FEQUXWYAZ[%eF,>PRG|_%UXcCDRGw[X]
2.5 µS E;GToFE;Gb;GTe<W;bWYPRG|[%oFWYACSnDFS U%OQPFAZGT_7Gf ~/.
U%fF; DRbYWYACeFBVWYPRGU%fFfE;GbQbYACeFB[%oFWYAZOb2, PFAZbh[%oFWYACSACUXWYAZ[%eAZblfF[%eRG`~/.o GTE ][%EYSACeFBb;ACeFBcZGkAZ[%efFGqziGOTW;[%Eb;OmUXeRbkUXWxfA GTE;GTe<WxcZ[%eFBACWYDRfACeiUXc^o [bYACWYAZ[%eRbk[X]cZGTeRb,_[X]WYPiGU%fFfE;Gb;bYACeFBw[%oFWYAZOb¡tb;GG\b;GOTWYAZ[%e¤H, H¢qp3PFAZOQPO[%EYE;Gb;o [%eRfFbrW;[bYPFA] W;AeRBWYPRGkcU%b;GTE3]t[<OTDRbhUXW3WYPRGkWYEQUXoOGTe<W;GTErACegcZ[%eFBACWYDRfACeiUXcfACE;GOTWYAZ[%eF,
" eR[%WYPRGTESGTWYPR[<f W;[OQPiUXEQU%OTW;GTEYACmGwWYPRGwU%fFfE;Gb;bYACeFB¤O[%eRbYAZbYW;b[X]fEYAC_<ACeFBOmUXEYE;ACGTEA-UX~FA[b;OTACcCcUXWYAZ[%eRb
57
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0.7
0 20 40 60 80 100 120 1400.0
0.1
0.2
0.3
0.4
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Excitation time t in µsExcitation time t in µs
@ACBDFE;GnI ,C£5K " fFfE;Gb;b;GfOmUXEYEYAZGTE -UX~FA[b;OTACcCcUXWYAZ[%eRbno GTE ]t[%EYSGf[%e U|W p[XuvAZ[%e OTE1.Lb;WQUXcACe U wSboFDFcZb;Gb;G<DRGTeROGACeROTcCDRfACeFBhBE;[%DFeRfnbYWQUXW;GbYAZfFGT~iUXeRfO[<[%cCACeFB[X]F~ [%WYPWYPRGU:LAUXc 1 B UXeRfWYPRG~FE;GmUXWYPFACeFBS[<fFGC,R% e ¡U¢WYPRGcU%b;GTEhpU%bU%fFfE;Gb;bQGf[%eLW;[V[%eRGn[X]WYPRG`W p[AZ[%eRbmiACe¡ ~N¢^[%e<W;[kW;PRGr[%WYPRGTE2,%WOmUXe~ Glb;GGTeWYPiUXW]t[%EWYPRGrBAC_GTeU%fF;DRb;WYSGTe<W[X] WYPRGr[%oFWYAZObm7WYPRGU%fFfE;Gb;b;ACeFBV<DiUXcCACWYAZGblfA GTElbYACBeFA iOmUXe<WYc0.gfFGTo GTeRfACeFB[%egp3PFAZOQPAZ[%egAZbhU%fFfE;Gb;b;Gf Kh¡U¢Ω1 = 2π · 35.5(1) Lh Ω2 = 2π · 2.46(7) LhRU%fFfE;Gb;bYACeFB7DiUXcCACWK. AQ = 6.9(1)
NEQUXWYAZ[[X]cCACBP<WrACe<W;GTeRbYACWYAZGb£K/£,h¡ ~N¢
Ω1 = 2π · 39.7(2) Lh Ω2 = 2π · 1.16(5) lU%fFfE;Gb;b;ACeFBV7DiUXcCACWK.AQ = 2.9(1)
NFEQUXWYAZ[[X]cCACBP<WlAeLW;GTeRbYACWYAZGb£KC£,
ACeU\W p[XuvAZ[%eOTE1.LbYWQUXcp3PRGTE;GkWYPRGnIJeFS cZU%bQGTE3AZblU%fFfE;GbQb;Gfg[%e<W;[[%eRG`[X]WYPRGkAZ[%eRb2,CPRGU%fFfE;Gb;b;GfAZ[%eVp3ACcCciDFeRfFGTEYB[n[b;OTACcCcUXWYAZ[%eRbp3ACWYPU6-hUX~FAF]tE;G<DRGTeRO?.
Ω1p3PFACcZGlWYPRGl[%WYPRGTEAZ[%e[b;OTACcCcUXW;GbUXWUkSnDROQP
bYcZ[pGTEx]tE;G<DRGTeRO?.Ω2
fDRGW;[|E;GbYAZfDiUXccCACBP<W`ACe<W;GTeRbYACWK.UXWkACW;bxo [b;ACWYAZ[%eF, % eWYPiGGqLo GTEYACSGTe<WmsoFDFcZb;GcZGTeFBWYPgb;OmUXeRbrUXE;Go GTE ]t[%E;SGTfUXeRfWYPRGW p[XuvAZ[%egGqFOTACWQUXWYAZ[%ewoFE;[%~iUX~FACcCACWK.
PDAZb3fFGTW;GTEYSACeRGf|] E;[%S WYPRG
oFPR[%W;[%SnDFcCWYACoFcCAZGE`O[%DFe<W;bkU%bxGqoFcUXACeRGfACeb;GOTWYAZ[%e¤I ,C£C, 4 FGTSoFcUXE1.b;OmUXeRbkUXE;G\bYPi[p3eACe@ACB8, I ,C£5,PRG|fRUXWQUUXE;G RWYW;Gf p3ACWYP UbYDFS [X]hW p[
sin2] DFeROTWYAZ[%eRb¡UXSoFcCACWYDRfFGb , H¢q^WYPRGo GTEYAZ[<fAZOTACWYAZGb[X]
p3PFAZOQP .LACGTcZfWYPRG -hUX~FAi]tE;G7DiGqeiOqAZGbΩ1
UXeRfΩ2
,^@RE;[%S WYPRGb;GRRWE;GbYDFcCW;bWYPRGU%fFfEQGTbQbYACeFB7DiUXcCACWK.AQAZbOmUXcZOTDFcUXW;Gf p3PFAZOQPpGnfFGLReRGU%b
AQ = Ω2/Ω1,G#h[%W;GnWYPiUXW-hUX~FAs]tE;G<DRGTeROTACGbkUXE;GnoFE;[%o [%EYWYAZ[%eiUXc
W;[WYPRGGTcZGOTWYEYAZO iGTcZf UXSoFcCACWYDRfFG^PRGTeROGoFE;[%o [%EYWYAZ[%eiUXc3W;[WYPRGb;<DiUXE;GVE;[<[%W[X]rWYPRGVcCACBP<W\ACe<W;GTeRb;AW . ,[%eRb;G<DRGTe<WYc0.7FWYPRGxE UXWYAZ[[X]cCACBP<WrACe<W;GTeRb;ACWYAZGbl[%egWYPRGxW p[AZ[%eRbrAZb3BAC_7GTew~/.
AQ2,
" ~ GTEYEQUXWYAZ[%eRb[X]kWYPRGU%fFfE;Gb;bYACeFB [%oFWYAZObVWK.LoFAZOmUXcCc0. fF[¤eR[%Wb;[¤SnDROQP ~FE;[U%fFGTe W;PRGw]t[<OTDRbVpUXAZbYWmEQUXWYPRGTEcZGmU%f WQ[U~iU%OQ<BE;[%DFeRf OmUXEYo GTW[X]lcCACBP<WUXE;[%DFeRf WYPRG QkUXDRb;b;AZUXe oFE;[FRcZGC,%e [%DFEOmU%b;G^WYPFAZb~iU%OQLBE;[%DFeRffF[%SACeiUXW;Gb`[_GTExWYPRG*QkUXDRb;bYAUXe~ GTPiUm_LAZ[%DFE`UXWkfAZbYWQUXeROGb
> 5 µS ]tE;[%S WYPRGOGTe<W;GTEk[X]
WYPRG][<OTDRb2, " bUO[%eRbQGT<DRGTeROGpGw[%~Rb;GTEY_G|WYPiUXWWYPRGgU%fFfE;Gb;b;ACeFB¤7DiUXcCACWK.AQ
fF[<GbeR[%WbYACBeFA FuOmUXe<WYc0.VACSoFE;[_7Gh]t[%EAZ[%ewfAZbYWQUXeROGb D 4zp > 5 µ
S@,PFAZbo [b;GbO[%eRbYAZfFGTEQUX~FcZGxoFE;[%~FcZGTSbACew7DiUXe<WYDRSO[%SoFDFWQUXWYAZ[%eiUXcoFE;[%W;[<O[%cZbmKhWYPRG\~ GbYWkU%fFfE;GbQbYACeFBw7DiUXcCACWYAZGbxpG\PiUm_7GU%OQPFAZGT_7GfUXE;G[%eWYPRG[%E;fFGTEk[X]AQ = 2.9(1)
N U%bACe@ACB8,<I ,C£5F¡ ~N¢7,#h[%W;G3WYPiUXWWYPFAZbO[%EYE;GbYo [%eRfFbW;[k_GTE1.\B[<[<fVU%fFfE;Gb;bYACeFBACeW;GTEYSb[X]hE;GbYAZfDiUXc3ACe<W;GTeRbYACWK. [%e WYPRGeR[%eu U%fFfE;Gb;b;Gf AZ[%ep3PFAZOQP AZbbYSVUXcCcZGTE~/. U]U%OTW;[%E[X]\£ ¡J¢ U%bO[%SoiUXE;GfW;[WYPRGU%fFfE;Gb;b;GfAZ[%eF,+PRGoFE;[%o GTEYWK.[X] -UX~FA] E;G<DRGTeROTAZGbxWQ[~ G\oFE;[%o [%EYWYAZ[%eiUXcW;[WYPRGL*FZAu r uv knmAu+T]r.X/ZL\ Xn| tmA\Fs2rW|
4xp > 15 µJ(
AQR ZLyk+uwZLFm]Fk r r2\ twyJmAknk X6J.uwZ r \FZAtwX/sm]k/[o 0yFs&knmAu+T]r
X/| tm]\s2r2|2 FZ1Rr r u]s2ZAuvuwr2|vZL\ twZ rWuvSk/Z1R mpXnmAk tvumA uwr yFr2\s2X/r2|ωz < 2π·200
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0.2
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TUXeRf|SGmU%bYDFEYACeFBWYPRGkGqOTACWQUXWYAZ[%egoFE;[%~iUX~FACcCACWK.¡ |D〉 bYWQUXW;GoRE;[%~iUX~FACcCACWK. PD ¢UXWGmU%OQP
Tuv_UXcCDRGC,'PRGE;GbYDFcCWhAZbxUXeGqOTACWQUXWYAZ[%eoFE;[%~iUX~FACcCACWK._%UXE .<ACeFBbYACe<DRb;[%AZfRUXcCc0.U%bU]tDFeROTWYAZ[%e[X]
Tp3ACWYPU\]tE;G7DiGqeiO7.
δAC,@dACB8, I ,/VbYPR[prbhUXeGqRUXSoFcZGn[X]^bYDiOQPUb;OmUXeF,gBUXe/.wb;OmUXeRbr]t[%El_%UXEYAZ[%DRb
, bYPFA]tW;GTE2-l]tE;G<DRGTeROTAZGTbωs
.LAZGTcZfwUXe " uM<WQUXEYbYPFA]tWb;o GOTWYEYDFSδAC(ω
s)LWYP<DRb3SVUXoFoFACeFBG<eF, ¡, 7¢7,
% W3AZbhU%b;bYDRSGTfgPRGTE;GRWYPiUXWlWYPRG`bQUXSGGqLo GTEYACSGTe<WQUXcO[%e<RBDFEQUXWYAZ[%esiACeoiUXEYWYAZOTDFcUXElACeW;GTEYSbl[X]^~ GmUXSBG[%SGTWYE1.7AZbkOQPR[b;GTe ][%Ep3PFAZOQP¤G7eF,^¡, 7¢lPNU%b~ GGTefFGTEYAC_7Gf eiUXSGTc0.WYPRG ! ' % " ') b;WQUXeRfRUXE;fO[%e<RBDFEQUXWYAZ[%e¡ " oFo GTeRfA " , ¢7,!%v]RpGfFGLReRG
∆U%bdWYPRGfFGTWYDFeFACeFBk[X]RWYPRGbYPFA] WQGqEd] E;[%S WYPiG |S〉 ↔ |D〉E;Gb;[%eiUXeROGReiUXSGTc0.gU%b
∆ = ωs − ω−1WYPRGkcUXWYW;GTElG<DiUXWYAZ[%egWYEQUXeRb ][%EYSbrW;[
δAC(∆) =Ω2−14
(d− a−5
∆+∆mag− 2
∆− a+3
∆−∆mag
),
¡vI ,/¢
p3PRGTE;G∆mag = 2π · 2.4µBB
~= 2π · 3.36B ! #"$
$Jw C[[<J wW
"! $#% '&
()
*
+
, -. , -.
@ACBDFE;GnI ,/K-,Gq]tWmK 9^E;[%~ GgoFDRcCbQGgb;G<DRGTeROG|][%EM<WQUXE; M<PFA]tWSGmU%bYDFE;GTSGTe<W;b,Isp[ -hUXSbQG7.π/2
uoFDFcZb;Gb¡ p3PFACW;G¢UXE;GhUXoFoFcCAZGf[%eVWYPiG |S〉 ↔ |D〉 WYEQUXeRbYACWYAZ[%e¡tfDFEQUXWYAZ[%e
8 µbm7bQGToiUXEQUXWYAZ[%e
τ = 200 µb ¢7, !GTW pGGTe WYPRGgW p[¤oFDFcZbQGbm[ uvE;Gb;[%eiUXe<W , bYPFA] W;GTE2-gcCABPLW¡ BE;G?.R¢UXWU
fFGTWYDFeFACeFB∆
AZbUXoFoFcCAZGf ]t[%EUWYACSGT
, -rACBP<WmK 4 FUXSoFcZGg[X]xUb;OmUXe E;GO[%E;fFGf UXWUfFGTWYDFeFACeFB
∆ = −1.20 Bl lWYPRG~RW .LAZGTcZfFbUo GTEYAZ[<f [X]£7, F¡ £¢µbWYP<DRb
δAC =2π · 61.5(5) l ,SPRGE;GfDROGfwO[%e<WYEQU%bYW`¡ ≈ 90
N¢AZbfDRGW;[fFGOT[%PiGqEQGqeiOTG`OmUXDRb;Gf~/.cU%b;GTE\U%b`pGTcCcU%b`SVUXBeRGTWYAZO*iGTcZfzRDROTWYDiUXWYAZ[%eRb[_GTEWYPRGpUXACWYACeFBWYACSG
τdb;GGVb;GOTWYAZ[%e
I , ,
AZbhWYPRG( GGTSVUXeb;oFcCAW;WYACeFB~ GTW pGGTeWYPRG\bYWQUXW;GbD1/2(m
′ = −1/2) UXeRf D1/2(m′ = −5/2) ¡ E;GbYo GOquWYAC_7GTc0. ~ GTW pGGTeD1/2(m
′ = +3/2)UXeRf
D1/2(m′ = −1/2) ¢qrOq]1,@ACB8,,JI , " bVSGTe<WYAZ[%eRGf
∆¡ E;GbYo GOTWYAC_7GTc0.ωs¢^bYPR[%DFcZfUXcCpU2.b~ G3]vUXEGTeR[%DFBPV]tE;[%S UXe/.OmUXEYEYAZGTE[%Eb;ACfRGq~NUXeRfE;GbQ[%eiUXeROGlACe[%E;fFGTE
W;[Um_7[%AZfo [%oFDFcUXWYAZ[%eWYEQUXeRb ]tGTE2,@i[%EWYPRG3cU%b;GTEACe<W;GTeRbYACWYAZGbDRb;GfACeVWYPFAZbGqo GTEYACSGTe<Wm<ACWpU%bbYD<KVOTAZGTe<WW;[gUm_7[%AZf¤UEQUXeFBG[X] ±2π · 400 hUXE;[%DFeRf¤GmU%OQPOmUXEYEYAZGTEnUXeRfUEQUXeFBG[X] ±2π · 50 hUXE;[%DFeRfGmU%OQPbYAZfFGT~iUXeRfgACeg[%E;fFGTErWQ[\]tDFc RcCc WYPFAZbl[ uvE;Gb;[%eiUXeROG`OTEYACW;GTEYAZ[%eF,
/10%2 43 6587 95 PRGnGqo GTEYACSGTe<W;bpGTE;Go GTE ]t[%EYSGfACeWYPRG !0' %#" &') bYWQUXeRfRUXE;fO[%euRBDFEQUXWYAZ[%e¡UXoFo GTeRfFA " , ¢7,SPRG
π/2OmUXEYEYAZGTEoFDFcZb;Gb[%eVWYPRG |S〉 ↔ |D〉 WYEQUXeRb;ACWYAZ[%ePNU%fU`fDFEQUXWYAZ[%e[X]85
µb,(PRG\WYACSG\ACe<W;GTEY_UXc
τ~ GTW pGGTeWYPiGqS pU%bxb;GTWW;[~
µb2,kjEYA] W;bx[X][%DFEE;Gq]GTE;GTeROGOmUm_LACWK.
fDFEYACeFBWYPRG`U%O7DFAZb;ACWYAZ[%ewWYACSGE;Gb;DFcCWACegU\b;cCACBP<WYc0.]vU%bYW;GTE3[%ErbYcZ[pGTE3[b;OTACcCcUXWYAZ[%eRfFGTo GTeRfACeFBV[%ewWYPRGTACEbYACBeF,+PiG7.OmUXe~ GO[%So GTeRbQUXW;Gf][%ExA]WYPRGfEYA]tWxAZbxDFeFA][%EYS [_7GTExWYPRGWYACSGb;OmUXcZG[X]3U%O7DFAZb;ACWYAZ[%e¡UX~ [%DFWW p[\SACe<DFW;Gb ¢uprPFAZOQP|W;[\[%DFE<eR[p3cZGfBGxAZb3UR;DRb;WYA iGf|U%b;bYDFSoFWYAZ[%eu^~/.VfF[%ACeFBGT_7GTE1.Vb;OmUXeW p3AZOGK^[%eROGl][%EYpUXE;f b;OmUXeFeFACeFB
T]tE;[%S mGqEQ[nW;[\b;[%SGbYDFACWQUX~FcZG
TendUXeRfWYPRGTewUXeR[%WYPRGTE3b;OmUXe] E;[%S
TendfF[p3eVW;[`mGqEQ[ ,SPRGhUm_GTEQUXBGl[X] WYPRGrW p[`[bQOqACcCcUXWYAZ[%eV]tE;G7DRGTeROTAZGbWYPRGTeVE;GToFE;Gb;GTe<W;bUXeVACSoFE;[_Gf
_UXcDiG][%EδAC
,% WAZbACSo [%EYWQUXe<W3]t[%EWYPRGbQGhSGmU%b;DFE;GTSGTe<W;b3WYPiUXWWYPRGcU%b;GTEo [pGTE3AZbO[%eRbYWQUXe<Wr[_GTErUXcCc b;OmUXeRb[%ErUXW
cZGmU%bYWWYPiUXW3[%eRGGGToRbWYEQU%OQ[X]fAGTE;GTeLWo [pGTE;b2,C%e|[%DFE3Gqo GTEYACSGTe<WmWYPRG`IJneFS cU%b;GTEAZb3U%OTWYAC_7GTc0.ACe<W;GTeRbYACWK.bYWQUX~FACcCACmGfW;[p3ACWYPFACe~ GTWYW;GTEkWYPiUXe £N ,&PRGGLKVOTAZGTeRO?.[X]WYPRGfF[%DF~FcZGoiU%b;b " 1 B b;GTWYDFop3PFAZOQPAZblDRbQGfgW;[bYp3ACW;OQPUXeRfg] E;G<DRGTeRO?.7uvWYDReRGWYPRGcCACBP<WmNPR[pGT_GTEmNPiU%bUVOGTEYWQUXACe]tE;G<DRGTeRO?.gOQPiUXE uU%OTW;GTEYAZbYWYAZOC,@i[%EWYPRG4-hUXSb;G?.oFDFcZbQGbWYPFAZbAZbeR[%W3U`oFE;[%~FcZGTS U%bWYPRG?.E;GTSVUXAegUXWWYPRGbQUXSGl] E;G<DRGTeRO?.p3PFACcZG\WYPRGb;PFA] W;GTExp3PFAZOQPAZbkb;OmUXeFeRGf[_7GTEkU%bxSnDROQP¤U%b
2π · 70 BhOQPiUXeFBGbxBE;GmUXWYc0.ACeACe<W;GTeRbYACWK.
F0
w T W 6L J[ &-]
200 210 220 230 240 250 260
3,0
3,5
4,0
4,5
5,0
5,5
6,0
6,5
Ligh
t pow
er in
mW
AOM frequency in MHz
@ACBDFE;GnI ,/F0RKsEQUXeRbYSACWYW;GfcCACBP<Wro [pGTElWYPFE;[%DFBPWYPRGfF[%DF~FcZG`oiU%b;b " 1 B b;GTWYDFogDRbQGf|W;[WYDReRGUXeRfbYp3ACW;OQPcU%b;GTEcCACBP<WdUXWIJeFS@,_PRG
S1/2(m = −1/2)↔ D5/2(m′ = −1/2) E;Gb;[%eiUXeROGO[%EYE;GbYo [%eRfFblW;[UXe " 1 B ]tE;G<DRGTeRO?.[X]C Bl ,#! .|Ub;ACSoFcZGxE;GOmUXcCAC~FEQUXWYAZ[%e[X]WYPRG
]tE;G7DRGTeRO?. U:AZbm^WYPRGcU%b;GTEo [pGTEP (∆)
U%bU] DFeROTWYAZ[%e [X]hfFGTWYDFeFACeFB∆
AZb[%~FWQUXACeRGf]tE;[%S WYPFAZbroFcZ[%W2,
[_7GTEhWYPFAZbb;o GOTWYEYDFS@,G% e[%E;fFGTEW;[O[%EYE;GOTW]t[%EWYPFAZbG GOTWmNWYPRG]tE;G7DRGTeRO?.fFGTo GTeRfFGTeROG[X]WYPRGncCACBP<Wo [pGTEhWYPFE;[%DFBPWYPiGfF[%DF~FcZGoiU%b;b " 1 B pU%blE;GO[%E;fFGf5,RPFAZblp3U%bhOmUXEYEYAZGf[%DFWr][%ElWYPRGGqRU%OTWhbQUXSG-l@ fEYAC_7Gno [pGTEB[%ACeFB|W;[WYPRG " 1 B U%bhACeWYPRG\Gqo GTEYACSGTe<W2,$PRGncCACBP<Wxo [pGTEpU%bSGmU%bYDFE;GffAuE;GOTWYc0.Ae]tE;[%e<W3[X]sWYPRGAZ[%e|WYEQUXowW;[nSVUX7GxbYDFE;GhWYPiUXWrG GOTW;b3[X]fAGTE;GTe<WrO[%DFoFcCACeFB[%DFW3[X]sWYPRGxfF[%DR~FcZGoiU%b;b " 1 B ACe<W;[nWYPRG'R~FE;GhWYEQUXeRbYSACWYWYACeFBW;PRGlcCACBP<WW;[WYPRGGqo GTEYACSGTe<WrUXE;GxUXcZb;[nWQUXGTeACe<W;[U%OO[%DFe<W2,PRG`E;GO[%E;fFGfOTDFE;_G`AZblbYPR[p3egACe@ACB8,NI ,/F08,8PRG |S〉 ↔ |D〉 E;GbQ[%eiUXeROGO[%EYE;GbYo [%eRfFbhW;[VUXe " 1 B]tE;G7DRGTeRO?.|[X]C Bh ,
" W;[%WQUXc [X]!75n~iU%OQUXeRf]t[%EYWYPwb;OmUXeRbpGTE;GE;GO[%E;fFGf|E;Gb;DFcCWYACeFBACewF0δAC
_UXcCDRGb][%EfAGTE;GTe<W∆
," fFfACWYAZ[%eiUXcCc0.C-UX~FA<[b;OTACcCcUXWYAZ[%eRb[%enWYPiG |S〉 ↔ |D〉 WYEQUXeRbYACWYAZ[%e\pGTE;GE;GO[%E;fFGf\ACe\[%E;fFGTEdW;[hfFGTW;GTEYSACeRGWYPRG3] E;G<DRGTeRO?.
Ω−1[X] G<eF,i¡vI ,/¢q<WYPRG3E;Gb;DFcCW~ GTACeFB
Ω−1 = 2π ·376(7) l , " cCci[%~FWQUXACeRGfδAC
_UXcuDRGb^pGTE;G3E;GTeR[%EYSVUXcCACmGfVW;[xWYPRGrbQUXSGcU%b;GTEo [pGTE~/.\fFA_LAZfACeFB`WYPRGTS ~/.
P (0)/P (∆)¡tOq]1,@dACB8,<I ,/F0<¢7,
PRGVO[%SoFcZGTW;GfRUXWQUgb;GTWAZbbYPR[p3eACe@ACB8,I ,/HGqOGToFW`][%EW p[fRUXWQUwo [%ACe<W;bUXWcZUXE;BGVfFGTWYDFeFACeFBbmp3PFAZOQPp3PRGTE;Gn[%SACWYW;GfACe[%E;fFGTElW;[7GGTogWYPRG`oFcZ[%WhO[%SoiU%OTW2,gPRG?.|E;GmU%f
δAC = 2π · 5.88 hUXeRf2π · 8.49 lx][%ErfFGTWYDFeFACeFBb ∆ [X]
2π · 40.0 BlUXeRf2π · 60.0 BlE;GbYo GOTWYAC_7GTc0. ,
#[%W;GWYPiUXWkWYPFAZbkSGmU%bYDFE;GTSGTe<W`oFE;[<OGfDFE;GV[%eFc0. .<AZGTcZfFbkWYPRGS[<fDFcCDRbk~FDFWkeR[%WkWYPRGb;ABe¤[X]δAC
,PRGVbYACBeRb`pGTE;GUXWYWYEYAC~FDFW;GfW;[wWYPiGSGmU%bYDFE;GTSGTe<WnE;Gb;DFcCW;bnU%OO[%E;fACeFBW;[wWYPRGWYPRG[%E;GTWYAZOmUXcOTDFEY_7GW;[~ G\Gqo GOTW;Gf WYPiUXWxAZbxG7eF,^¡vI ,/¢7,$PRGbYo GOTWYEQUXco [bYACWYAZ[%eRbk[X]WYPRG\WYPFE;GGOmUXEYEYAZGTExE;Gb;[%eNUXeROGTbkp3PRGTE;GfFGTW;GTEYSACeRGffACE;GOTWYc0.]tE;[%S UXe¤GqOTACWQUXWYAZ[%ebYo GOTWYEYDFS ¡tOq] , @AB8,sI , 0<¢7,@RE;[%S WYPFAZbm
∆magpU%b`fFGTW;GTE u
SACeRGfW;[k~ G2π ·8.40 Bh3p3PFAZOQPO[%EYE;GbYo [%eRfFbW;[nUSVUXBeiGqW;ACORiGTcZfV[X]
B = 2.50Q:,OPRGrGqo GOqWQGTf
_UXcCDRGb][%E3WYPRGkE;GTcUXWYAC_GkcCACeRG`b;WYE;GTeFBWYPRbn¡tOq] ,Rb;GOTWYAZ[%e, ,/7¢3UXE;Ga−5 = 0.278
UXeRfa+3 = 0.0556
,PRG[%eRc.O[%SoFcZGTW;GDFeFLeR[p3e ]t[%E\WYPRG RW[X]lWYPRGfRUXWQUp3ACWYP WYPFAZbG<DiUXWYAZ[%e pU%b
d, PRGTE;Gq]t[%E;G
ACeWYPRG&RWYWYACeFBoFE;[<OGfDFE;GsUXW'RE;b;W[%eFc0.dpU%b_UXEYAZGTfUXeifW;PRGTebYDF~Rb;G<DRGTe<WYc0.UXcZb;[
Ω−1a−5
UXeRfa+3
,r@ACB8, I ,/HbYPR[prbUXeGqFOGTcCcZGTe<WxUXBE;GGTSGTe<Wk[X]^W;PRG`SGmU%b;DFE;Gf_UXcCDRGbhp3ACWYPWYPRG ReiUXcERWmNp3PFAZOQP.LAZGTcZfFbWYPRG][%cCcZ[p3ACeFB_UXcCDRGbmK
Ω−1 = 2π · 357(3) l a−5 = 0.318(15)a+3 = 0.051(17)
UXeRf
H
$Jw C[[<J wW
-10 -5 0 5 10 15
-100
-50
0
50
100
8.4-8.4
! #"%$ ,'&(*),+.-0/1 325436!879-δAC
/4:/;79!<>=?-3A@<B@17C4#6!87D-#"? :E!"?-3!<!<!(∆)
,GFH/1-0/JI@<.-#4LK94#M.N/1 #"4POQ/1 #"=R/1SA=?!S/1-#"ET7D #@U
T V 4#=R/1<>4WKX/4Y<Z8,>$ ,'&[OY/1<>E]\ .J #"?<>@ 3U^/1S_<>J-36>"H@\!-0/1<>"E δAC` /1S>"4a-#@L@<>": #"b7X"? #"?<>="cS/4#"? YI@edY"? 2,gfG6>"ihN-a>4#<!L"M.<F,jKk$ ,'&O:K94#@SAElS<>"eOmA4a< ` "b .@[email protected]/1 #""?UJ"?<.-:do-365-36>"4#"WE>/1-0/,
d = 0.224(10)/2πK9pi_qO −1 ,srt-c=R/1<u\"L4#""?<v-36N/1-H-36>";436!87D-HE*>"L-#@]E*I@SA"L-3 0/1<>43-3A@<>4Rwj<x@-36>"?
dY@ #E!4dwyA4s #"?S/1-3 ` "?S0.u4#U^/1SS+,JfG6>";h>-3-#"E Ω−1
E*8z"? #4H79 #@U|-36>" ` /1S>"UJ"R/43! #"E~\/.uE* #"=?-H-c/1\!@4#=?SS/1-3A@<>4o\/.](do6!A=06TA4GS/1 3"? G-36N/1<Z-36>"H"? 3 #@ G\N/1 #4o@17"R/=06 ` /1S>"wdo6!A=06n/1 #"H@<T-36>"H@ #E!"? o@17 V & ,gfG6!A4aE!" ` /1-3A@<]=@!SAE^\"i"qI!S/1<>"El\/.43SA@ed=06N/1<!"4a<^-36>"QS/40"? mI@edY"? Y@ ` "? m-36>":=@! #4#"i@17-36>"o"q*I"b #8UJ"?<-2,C+.>=06JI@edY"? CE* 3879-#4g=R/1<\"o=R/1>40"?Eyw7X@ g"q>/1UI!SA"w87,-36>"o@I!-3A=R/1S!h>\"? C-3 0/1<>43U-3-3<!-36>"sS6.-:/1-c$&L<!U43SA@edQS.] #@-0/1-#"4i-36>"HI@S/1 3_e/1-3A@<NWE*N"H-#@-36>"? 3U^/1S"bzj"=?-#4,go-o-36>"sI@S/1 3_<!\"R/1U43I!S-3-#"? ;/q79-#"? %-36>"JhN\"b ewg/5I@4#4#\!SA"^I@S/1 3_e/1-3A@< #@-0/1-3A@<A4LE* #"=?-3S0.~-3 0/1<>4379@ 3UJ"E<.-#@v/=06N/1<!"W<5-#6>"s-3 0/1<>43U-3-#"En<.-#"?<>43-K. ,afG6>"Hh>-i #"43!S-o7X@
a−5S"4c/1\@ ` "W-36>"s-36>"@ #"?-3A=R/1S ` /1S>"s\/.
/1\@!- R do6!A=06=R/1<\"J"q*I!S/1<>"E\/.~/ZE!" ` /1-3A@<7D #@U-36>"JAE!"R/1Sa"?@UJ"?-3 .~@<-36>"^@ #E!"? %@17R <v\"R/1UE* #"=?-3A@<x@ cI@S/1 3_e/1-3A@<F, ! .Z-36>"@ 1.n-36>"% 0/1-3A@l\"?-tdY""?<
a−5/1<>E
a+3A4c=@<>4#-0/1<.-
!<>E!"? "#"" =@<>E*-3A@<>42,;fG6.>4Rw87 a−5 SA"4W/1\@ ` "L-36>";-36>"@ #"?-3A=R/1S ` /1S>"w-36>"40/1UJ";4#6>@!SAEv\";-3 3>"7X@ a+3
w.do6!A=06TA4Y<>@-Y-36>"c=R/4#"C,^c@edY" ` "? Rw.-36>"H"qI"? 3UJ"?<.-0/1S ` /1S>":79@ a+3 =@<.-0/1<>4Q/S/1 3"H"? 3 #@ /4s-36>"=@!I!S<!n-#@T-36>"J=@ 3 #"43I@<>E*<!n-3 0/1<>4#-3A@<~A4 ` "? 1.BdY"R/12x/1<>E~@<!S0.B79"?dE>/1-0/TI@<.-#4W6N/ ` "\""?<5 #"=@ #E!"E5<5-36>" ` A=?<!-K.T@17-36>" −1/2↔ +3/2
-3 P/1<>43-3A@<F,rt<5@ #E!"? i-#@J=? #@404 V =06>"=02T-36>"W@\!-0/1<>"E ` /1S>"4Q79@ a−5 /1<>E a+3 w8-c/1\!@4#=?SS/1-3A@<>4:@<5-36>"W-36! #""=R/1 3 3A"? #4dY"? #"a #"=@ #E!"EL!<>E!"? -36>"Y40/1UJ"Y=@<>E*-3A@<>4QKD\"R/1U"@UJ"?-3 1.j43<!SA"aA@<yw43AE!"?\N/1<>EL=@.@SA"ENO
<L/Q4#"?IN/1 0/1-#"m"q*I"? 3UJ"?<.-2,6.-I@edY"? #4/1-$&o<>UdY"? #"CUJ"R/43! #"EW7X@ y-36>"C-36! 0"?"aE*8z"? #"?<.-YK9:cpxO79 #"M>"?<>=?A"42,GfG6>"@\!-0/1<>"E ` /1S>"4idY"? #"W>4#"E5-#@J<>@ 3U^/1S_"%-36>"@\!-0/1<>"E -:/1\!y7D #"M.>"?<>=?. ` /1S>"4q> e¡k¢'£q¤¥X¦1§1Ry¥>¨k¦©#¨N¥X1¤ª¦ak§P¨ª©0¨k¢'§P£1¥,§¤¤qN«?£Y¢¬®¯k¥>¥X¡¤¢'®13°G©0¥!±D¡§P²©#k¢ 3©0¨k¢'§b£o³Q©P¢'£¨ª©P¢'£q¢´£TPµ¶©0kj1¥X3°·
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! "# "%$&(')*+ ",-.//')"
0 20 40 60 80 1000,0
0,2
0,4
0,6
0,8
1,0-1/2 +3/2
Exc
itatio
n P
roba
bilit
y
Time T in µs0 10 20 30 40 50 60
0,0
0,2
0,4
0,6
0,8
1,0 -1/2 -5/2
Exc
itatio
n P
roba
bilit
y
Time T in µs0 10 20 30 40
0,0
0,2
0,4
0,6
0,8
1,0-1/2 -1/2
Exc
itatio
n P
roba
bilit
y
Time T in µs
02143 0253 076-3
! #"%$ ,'&¸*)8-c/1\!c@40=?SS/1-3A@<>4Z@< -36>"u[I@4043\!SA"v=R/1 3 3A"? T-3 0/1<>4#-3A@<>4Tdo6>"?< 4#-0/1 3-3<!79 #@U -36>"S1/2(m = −1/2) 4#-0/1-#"o<-36>"Q\"R/1U"@UJ"?-3 1.
φ = 45, γ = 0,fG6>"Q=@ 3 #"4#I@<>E*<!
$&<!U S6.-ZI@edY"? #4n/1<>E -36>" -c/1\!c7D #"M.>"?<>=?A"4]79 #@U -36>"vh>-#45/1 #" KX/O158 µ 8
2π · 32.15(12) 2i_JKD\,O 137 µ 8 2π · 50.20(10) 2L:_JK9=eO 128 µ 8 2π · 10.71(6) 2:_ ,
-#@]-36>";4P/1UJ"LS86-H<.-#"?<>43-K. ,fG6>";-36! #"";40=R/1<>4s/1 #";4#6>@edo<u<x8,y$ ,'&¸,fG6>"; #"?S/1-3 ` "4#M.N/1 #"4H@17-36>"W<>@ 3U^/1S_R"E -:/1\!y7D #"M.>"?<>=?A"4 .A"?SE5-36>" ` /1S>"4 a−5 = 0.357(17)
/1<>Ea+3 = 0.049(5)
w>do6!A=06/1 #"s<n/1 #""?UJ"?<.-:do-365-36>" ` /1S>"4i@\!-0/1<>"EZ7D #@U -36>"Hh>-o-#@-36>"W:9 V +.-0/1 32Z436!879-oE>/1-P/,
;=< ?> @ > '32A < 2 3 fG6>"LUI!SA"?UJ"?<.-0/1-3@<@17YM.N/1<.-3!U|/1S@ 3-36!UJ4W<x@! s"q VI"? 3UJ"?<.-c"?<.-0/1SA4:40"M>"?<>="4:@17C=R/1 3 3A"? c/1<>EB43AE!"?\N/1<>E5I!!SA40"?4,B9Y@!I!S<!l-#@l/=R/1 3 3A"? :-3 0/1<>43-3A@<BA4 #"?S/1-3 ` "?S0.^43-3 #@<!>wdo6!A=06]IN/1 0/E!@m*A=R/1SS0.]/1UJ@!<.-#4G-#@;+.-0/1 32l436!87D-#4a<>E*>="E]\/.^=/1 # 3A"b GI!!SA4#"4Y\"?<!<>"?S\!SA"%=@UIN/1 #"En-0@-36>""bz"=?-i@17C43AE!"?\N/1<>EnI!!SA4#"42,YfG6>"W #"R/4#@<579@ i-36!A4QA4Q-36N/1-H/; #"4#@<N/1<.-3S0.E* 3 ` "?< -3 0/1<>43-3A@< E!@."4DCFEHGL436!87D-8-04#"?S87 , 8 6>"?< E* 3 ` <!/v43AE!"?\N/1<>E -3 0/1<>43-3A@< #"4#@<N/1<.-3S0.wm-36>"<>"R/1 #"43-s@-36>"? c-3 0/1<>43-3A@<uA4c-K..I!A=R/1SS0.v/l=R/1 3 3A"? Rwjdo6!A=06vI! #@.E*>="4s43-3 #@<!5+-0/1 32n436!879-#42, 8 6>"?<ywj@<-36>"J@-36>"? W6N/1<>Ew/T=R/1 3 3A"? A4WE* 3 ` "?<x-#6>"?<~-36>";<>"?6.\@ #8<>n-3 0/1<>43-3A@<>4L/1 #";-K.I!A=R/1SS0.u4#AE!"?\N/1<>E!42,QE!E*-3A@<N/1SS0.w/l43AE!"?\N/1<>Ev-3 0/1<>43-3A@<u<x@ #E!"? H-#@l"?-Rw40/2.wj/J79!SS
π V I!!SA4#"%U%>4#-c\"LE* 3 ` "?<vSA@<!"? \/.x/l7k/=?-#@ 1/η
87G-36>"J40/1UJ";S6.-I@edY"? WA4/1I!I!SA"E5,^N@ WM.N/1<.-3!U /1S@ 3-36!UJ4L/4s-36>"?.u=R/1<~\" #"R/1S_R"Exdo-36@! W"q*I"? 3UJ"?<.-0/1SmIN/1 0/1UJ"?-#"? #4Rw-36!A4sUJ"R/1<>4s-36N/1-=R/1 3 3A"? W<>E*>="E~I9 V +.-0/1 32v4#6!87D-#4/1 #"s<>"?S\!SA"Wdo6!SA"W43AE!"?\N/1<NEZ<>E*>="En436!87D-#4oU%>43-:/1\>4#@S!-#"?S0.T\"s=@UI"?<>4P/1-#"EZ79@ ,:I!I!S0..<!T/S/4#"? c@<B-36>"W\!S>"L/:*/1S4#E>"b\,/1<>EB@17g-36>" |S〉 ↔ |D〉 -3 0/1<>43-3A@<KKJª/1-#"%S/4#"? MLOi/1-c/E!"?-3!<!<!]@17
∆/2π N ,´$p:_s #"4#!S-#4Q<B/;<>"?/1-3 ` ":9 V +.-0/1 32Z4#6!87D- δAC ,Y+.6!<!<!l<B/4#"=@<>ES6.-shN"?SAEK9=R/1SS"E&Jt=@UI"?<>40/1-0@ MLk<-36>";79@SSA@edo<!.OW/1-%/]S/4#"? E!"?-3!<!<!ndo6!A=06 ..A"?SAE!4%/TI@43-3 ` "I9 V +.-0/1 32n436!87D-c=R/1<B<.!SS-36>"%@ ` "? 0/1SS4#6!87D-2, 8 "W-K.I!A=R/1SS0.n4#"?-:-36>"%=@UI"?<>40/1-#@ c79 #"M>"?<>=?.5-#@^-36>" #"E%43AE!"a@17>-36>"
S1/2(m = −1/2)↔ D5/2(m = −5/2) #"40@<N/1<>="a\/. 2π·800 2i_ ,o--36N/1-I@8<--I! #@.E*>="4Y #"R/4#@<N/1\!S0.JS/1 3"c436!879-#4Y/1<>E]/1-a-36>"c40/1UJ"i-3UJ"iA4a-#@.@W7k/1 C79 #@U /1</. #"4#@<N/1<>=":-#@<>E!>="@1z V #"4#@<N/1<.-i"q!=?-0/1-3A@<>42,Y:! QUJ"?-36>@.E5-#@JE!"?-#"? 3U<>"W-36>"W@I!-3U%!U 40"b-#-3<!^@17-36>"W=@UI"?<>4P/1-3A@<S/4#"? =@<>43A43-#4W@17G-36>";79@SSA@edo<!B43-#"?I>4R)s #43-WdY"E!"?-3!<>"J-36>"/1-#"SA/40"? s\/. ' 2π · 80 2:_L79 #@U-36>"43AE!"?\N/1<>En #"4#@<N/1<>="-#@^/ ` @AEn"q*=?-0/1-3A@<B<.-#@J-36>" |D〉 43-0/1-#"^KD-#4:I9 V +.-0/1 32Z"bz"=?-i6>@edY" ` "? :A443-3SSg/1SUJ@43-cAE!"?<.-3A=R/1S-#@l-36N/1-H@17a/^S/4#"? :hN"?SAEB #"4#@<N/1<.-Hdo-36v-36>"L43AE!"?\N/1<>ENO7,HfG6>"?<vdY"%U<!U_R"-36>"L-#@-0/1SgI9 V +.-0/1 32B"bz"=?-H\/.v/EO>43-3<!T-36>"L<.-#"?<>43- .v/1<>EuE!"?-3!<!<!5@17a-36>";=?@UI"?<>4P/1-3A@<xS/4#"? hN"?SAEu43>=06v-36N/1-H-36>"L@4#=?SS/1-3A@<>4s<x/
T V 4#=R/1<uE*A40/1I!I"R/1 2, [email protected]"?SAE!4W/1 #"%"?<>"? 0/1-#"Eu\/.B/1I VI!S0.<!;-tdY@:-: V 7D #"M.>"?<>=?A"4o43U%!S-0/1<>"@>43S0.]-#@L-36>"W$&%<>U E!@!\!SA" V IN/4#4Q:cp K9i <58,'(,'(O7,+.<>="]\@-36S6.-LhN"?SAE!4J/1 #"]E!"? 3 ` "E7D #@U-36>"]4P/1UJ"lS/4#"? Rwg<.-#"?<>43-K.P>>=?-3,/1-3A@<>4JE!@B<>@-J/qzj"=?-
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T1 = 0wdo6!-#";4#MN/1 #"4e)QE>/1-0/
I@<.-#4Qdo-36T2 = 200µ
46.
-36>"L=@UI"?<>40/1-3A@<9.sfo6>";/==?! 0/=5n-#@ldo6!A=06v-36>"LI9 V +.-0/1 32B"bzj"=?-H=R/1<v\"%<.!SSA"EvA4:I! #@I@ 3-3A@<N/1S-#@(2 TR)
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200 µ4bO:/1<NE
S/N-36>"Z43<N/1So-#@u<>@A4#"T 0/1-3A@x@17:-#6>"T4#-0/1-#"]UJ"R/4#! #"?UJ"?<.-6.rt<.-#"? 0/1-3<!~-36!A4;UJ"R/4#! #"?UJ"?<.-7X@ ;SA@<!-3UJ"4Q-#@UI! #@ ` " S/N A4oSU-#"E5\5]-36>"s7D #"M.>"?<>=5TE* 3879-i@17-36>"sS/4#"? i4#@! #="s<>"R/1 H$&;<!U KD-5I VA=R/1SS;5 ≤ :_<14PObw43<>="^/TE* 3879-W@17G-36>"; 0"bS/1-3 ` "I!6N/40"@17G-36>"=3c/1UJ4#"5BI!!SA4#"4WUUA=4%/] #"4#AE*N/1SI9 V +.-0/1 32"bz"=?-6.uf@v@ ` "? #=@UJ"]-36!A4LI! #@\!SA"?UdY"]@I!-3U_R"T-36>"l<.-#"?<>43-5@17i-36>"]=@UI"?<>40/1-3A@<S/4#"? c\5n/1S-#"? 3<N/1-3<!3:/1UJ4#"55"q*I"? 3UJ"?<.-#4cdo-36
T1 = 0/1<>E
T2 ' 200µ46.:fG6.>4Rwj/l43SA@edE* 3879-:A4
E*A4#=? 3U<N/1-#"E/1/1<>43-W/^ #"43AE*,/1SI!6N/4#";436!87D-HE!>"%-#@]UI"? 7X"=?-s=@UI"?<>40/1-3A@<9.%U-#"Eu\5n-36>"436>@-Q<>@A40"W@17
PD(T )wN/1<5ZI9 V +.-0/1 32Z"bz"=?-i=R/1<n\"W=R/1<>="?SSA"En-#@Jdo-36!< ' 2π · 30 i_W<v¸ 46.arª<
I! 0/=?-3A="wdY"Gh> #4#-g>4#"QM!-#"Q4#6>@ 3-m/1S-#"? 3<N/1-3A@<>4ew7X@ C"q>/1UI!SA"o\"?-tdY""?<T1 = 0
/1<>ET2 = 20 µ
4a/1<>E-3 %5]-#@U<!U_R"-36>"WE*8z"? #"?<>="W\"?-tdY""?<v/1S-#"? 3<N/1-3<!]E>/1-0/;I@<.-#4i\5Z/EO>4#-3<!J-36>"sI@edY"? :@17-36>"=@UI"?<>40/1-#@ 6.WfG6>"?<udY"L43-#"?Iv!I
T2/1<>EBh><>"%-3!<>"LI@edY"? W/1<>EvE!"?-3!<!<!5@17m-36>"L=@UI"?<>4P/1-#@ W/1-
"R/=06l43-#"?I9.C+.-0/1 3-3<!LE* #"=?-3S;5do-36]/s #"?S/1-3 ` "?S;5SA@<! T2 @17 200 µ 4m=R/1 3 3A"4a-36>"Q 3A432;@17/EO>43-3<!-36>"/==?!U%!S/1-#"E5I!6N/4#"H-#@4#@UJ"HU%!S-3I!SA"W@172π
0/1-36>"? Q-36N/1< .C1.N$/.'&$L436>@edQ4Q=@UI"?<>40/1-3A@<BE>/1-0/@ ` "? G-36>"s=@! #4#"H@17-3UJ"w!"R/=065E>/1-0/LI@<.-Q=@ 3 #"43I@<NE*<!J-#@ R #"?I"?-3-3A@<>4i@17-36>"s"q*I"? 3UJ"?<.-0/1S4#"M.>"?<>=";do-36/1</=M!A4#8-#@<-3UJ"@17Q&T46.F:! #8<>5-36>"J/=M.!A43-3A@<@17Y-36N"?40"E>/1-0/T<>@5/4#I"=?-W@17-36>"W"qI"? 3UJ"?<.-idG/4i=06N/1<!"E5<5@ #E!"? Q-#@J=06>"=02]-36>"W43-0/1\!S-5Z@17-36>"W=@UI"?<>40/1-3A@<9.
>@?%> ACBD1EGF*HJIKEGLJB −1 M LONPDQBSR"TUNWV8FYXOB[Z:N"HJX2π
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4/479@!<>E79 #@U-36>"mh>-6.GKD\,O 3:/1UJ40"5π<&iI!!SA4#"4@<L-36>" |S〉 ↔ |D〉=R/1 3 3A"? %-3 0/1<>43-3A@<"b<N=bSA@40"J-36>" 3:/1\!+PN@I!I!<!v@<-36>"^43AE!"?\N/1<>E^.TfG6>"JI!6N/4#"J@17o-36>"
|S〉 4#-0/1-#"6N/4s/J79!SS=5*=?SA"%I"? [email protected]@17a&($*KX&O µ 46.;K9=eOc+*/1UJ"LI! #@.="E*! #"/4c< KD\,OQ\!!-do-365-36>"WI9 V +.-0/1 32T436!879-Q=@UI"?<>40/1-3A@<n-3> 3<>"E5@1z.
fG6>"2π
#@-0/1-3A@<yw><>"R/1 [ µ4Q8<-#"? 0/=?-3A@<B-3UJ"w>4#6>@edQ4o>IB/4i/ −1 I!6N/40"W436!879-K D V 4#-0/1-#"sI! #@\N/ V\!S-5 ' 1
Obw.do6!SA"H/q79-#"? G/1\@!-o&¸ µ4Y/W7D!SS,I"? [email protected]*=@ 3 #"43I@<NE*<!;-#@L/43AE!"?\N/1<NE
4π #@-0/1-3A@<ywA4
=@UI!SA"?-#"E^.cfG6N"Wh>-:<v-36!A4c=R/4#" 5A"?SAE!4257(2) µ
46.ifG6>"% 0/1-3A@]@17gh>-3-#"EBI"? [email protected]!4cA4 .'¸!KX[OQdo6!A=06/1 #""4QdY"?SSydo-365-36>"W"q*I"?=?-#"E ` /1S>"W@17C&2.rt< -36!A4J"q>/1UI!SA"wa-36>"5=@UI"?<>40/1-3A@<S/4#"? J6N/E-#@~=@ 3 #"=?-J79@ ^/1< I9 V +.-0/1 32436!879-J@17 δAC ≈
2π · 3.5 2:_ . fG6!A4^436!87D-l/1SA@<>"ZdY@!SAE 6N/ ` "T #"4#!S-#"E </1</E!E*-3A@<N/1S:I!6N/4#"KD\"?-tdY""?< |S, n〉/1<>E |D,n〉 <>E!"?I"?<>E!"?<.-H@17 n wN=b7 .N4#"=?-3A@< i2. Oo@17 ≈ 0.82π< [
µ46.Gf@^436>@ed6>@edE!"?-3 3UJ"?<.-0/1S
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T V 4#=R/1<l-36.>4@\!-0/1<>"EuA4s436>@edo<u<x1.y$/.'&!K9=eO .fG6>"Lh>-c7D!<>=?-3A@<xA4s/ldY"?6.-#"Ex43>U@17a-tdY@cos2
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n · δac
@1z4#"?-G4#6!87D-G=@UUJ@<5-#@ " SA" ` "?SA4G<Z1.i2.'&2.m+.>=06n/1<Z@ ` "? 0/1SSy436!87D-Rw6>@edY" ` "? Rw!E!@."4G<>@-G #"43!S-G</W #"?S/1-3 ` "iI!6,/4#"i\"?-tdY""?<]S/4#"? Y/1<>ElA@<yw6>"?<>=":A4m<>@-YE!"?-#"=?-0/1\!SA"w>/1<>E^-36>"? #"b79@ #":-36!A4a@ ` "? 0/1SS436!879-doSS>\"G<>@ #"EJ<J-36>"Y7X@SSA@edo<!1.fo6>"o"q*I"? 3UJ"?<.-0/1SNI! #@.="E*! #"Q<>@ed >4#"4C@<!S;5L-36>"Q="?<.-3 0/1S>I,/1 3-C@17-36>"iI! #@\"c40"?M.>"?<>="LK940@SAE^S<>"4PObw.do6>"? #":-36>"cE*! 0/1-3A@<
τ@17y-36>":I9 I"? [email protected]=R/1<!<>"E^.0/>43-o/4m<
4#"=?-3A@<J$/.'¸*w1-36>"m7D #"M.>"?<>=5@17>-36>"m #"4#!S-3<!H@4#=?SS/1-3A@<>4A4"q!/=?-3S;5W-36>"m #"?S/1-3 ` "aSA" ` "?S*436!87D-GKD\"?-tdY""?<|S, 0〉 /1<>E |D, 0〉 OY<>E*>="En\5Z436!87D-#"? c/1<>En=@UI"?<>40/1-#@ H/1I!I!SA"EB43U%!S-0/1<>"@>43S;5 .YfG6>"W@/1S-36>"?<A4%-#@nh><>E-36>"]=@UI"?<>40/1-#@ <.-#"?<>43-5~do6>"? #"]-36!A4%7D #"M.>"?<>=5~\"=@UJ"4L_R"? #@!wg ."_.do6>"? #"l-36N"b 0"lA4<>@LUJ@ #"s@4#=?SS/1-3A@<9.CN@ G-36!A4GI! #@.="4#4GdY"HE!"?-3!<>"H-36>"83c/1UJ4#"5 V I!!SA4#"4G4#S6.-3S;5J79 #@U #"4#@<N/1<>="s4#@-36N/1-mdo-36
τ = 0dY"i"?<>E!I^<J-36>"oUAE!E*SA":@17y/ 3c/1UJ4#"5 V 79 3<!"w./1- PD ≈ 50%
.fG6!A4m #"?UJ"QA4C-36>"UJ@43-m4#"?<>43-3 ` "G-#@HI!6N/4#"4m/=M.! #"E<-36>"o-3UJ"o\"?-tdY""?< 3 /1<>E 3:&2.:<J"q>/1UI!SA"Q4#=R/1<do6>"? #"o-36>"=@UI"?<>40/1-#@ QI@edY"? QA4GdY"?SSy4#"?-GA4o436>@edo<Z<Z1.1i2.'[;/1-o-36>"HS/1\"?SJ
n = 0LY.3:"43AE*N/1SP>>=?-3,/1-3A@<>4
=R/1<]\"c40""?<9.fo6>"4#"H/1 #"cE*>"cIN/1 3-3/1SS;5^-#@% 0/1<>E!@U U^/1<>"?-3A=chN"?SAE!4o/4YdY"?SS/4YS/4#"? a79 #"M.>"?<>=5l/1<>EI@edY"? P>>=?-3N/1-3A@<>46. *c@-#"s6>@edY" ` "? Rw>-36N/1-i\5]U^/12<! R "bSA"=?-3 #@<v436>"?S ` <!^"q*I"? 3UJ"?<.-#4i79@ i@<>"E>/1-0/ZI@<.-dY"l/1SdG/65*4s7k/=" #"4#E!N/1Sa43-0/1-3A43-3A=R/1ScK94#6>@-W<>@A4#"JSU-#"ENO P>>=?-3N/1-3A@<>4L@17 R .]fG6>"
R (
*+ ! " 4) "% :.4I "%$= ') /
/ ` "? 0/1"c"q*=?-0/1-3A@<TI! #@\N/1\!S-5w*6>@edY" ` "? Rw*@<!S;5^=06N/1<!"4Gdo-36Z/ 0/1UIT@17 −6(2) · 10−4µ 4 −1 KDS<>"R/1 h>-POcdo6!A=06~ ` "4/1<"43-3U^/1-#"J@17 δres ≤ 2π · 180 i_;7X@ s-36>" #"43AE*N/1SC #"?S/1-3 ` "SA" ` "?SC4#6!87D-W@17G-36!A44#=R/1<9.c4<;-36>"o"q*I"? 3UJ"?<.-#4C@17j4#"=?-3A@<l$/.'¸*wE* 387D-#4g@17-36>"G #"b79"? #"?<>="Q=R/ ` -a5%@17-36>"i$&c<!U S/40"b m/1 #"=@UI"?<>40/1-#"E;79@ \5WI"? 7X@ 3U<!W" ` "? %540=R/1<L-tdoA="w@<>"a79@ 3dG/1 #EJ/1<>EL-36>"?<J/1<>@-36>"? C4#=R/1<;\N/=02.do/1 #E79 #@U
τ = 250 µ4WE!@edo<~-#@
τ = 0 µ4 .;> #@U@! WE>/1-0/TdY"J=R/1SA=?!S/1-#"J-36N/1-Wdo-36~-36!A4sUJ"?-36>@.ExdG"
=R/1<<.!SSY-36>"^ #"?S/1-3 ` "lSA" ` "?SY436!879--#@Bdo-36!< 2π · 300 :_ .nfG6!A4L@I!-3U%!U /EO>43-3UJ"?<.-;@17i436!879-#"? /1<>E=@UI"?<>4P/1-#@ ;I@edY"? ;dG/4L2"?I!-%7X@ ;/1SSY79@SSA@edo<!x"q*I"? 3UJ"?<.-#46.xfG6>"]<>E* ` AE*N/1SoSA" ` "?SG4#6!87D-#4!<>E!"? Y-36>"4#":=@<>E*-3A@<>4o/1 #":<>E*A=R/1-#"El<]1./i2.'&Wdo-36]S/1\"?SA4 E 3 s"R/=06]SA" ` "?Sk)-36>"c=@UI"b<N40/1-#@ "bz"=b-# ` "?S;5] #"E*>="4i-36>"s4#6!87D-Q@17C/1SS |S, n〉 /1<>E |D,n〉 SA" ` "?SA4o\5 δS + δ
/1<>EδDw* #"43I"=?-3 ` "?S;5 .
rt<Z-36>"W<>"q*-i43-#"?In@17g-36>"W"q*I"? 3UJ"?<.-i-36>"WA@<nA4o-3 0/1<N479"? 3 #"E5<.-#@J-36>" |S, 1〉 43-0/1-#"W<5-36>" g #"?IN/ V 0/1-3A@<ZI"? [email protected]*do-36Z-36>"c #"4#-G@17-36>"cI! #@.="E*! #"s #"?U^/1<!<!-36N"H40/1UJ"_.gfG6!A4GI! 0"bI,/1 0/1-3A@<5A4G=R/1 3 3A"E@!-i\55/1I!I!S;5.<!T/; 0"?40@<N/1<.-i\!S>"4#AE!"?\N/1<>E
π V I!!SA4#"s7X@SSA@edY"En\5Z/ #"4#@<N/1<.-c=R/1 3 3A"? π V I!!SA40"w>-36>"-3 0/1<>47X"? "^=?A"?<>=?A"4^@17cdo6!A=06 /1 #"5_i/1<>E ]wm #"43I"=?-3 ` "?S;5 .fG6>"Z40/1UJ"T-5.I"Z@17 τ V 4#=R/1<>4^/4/1\@ ` "W/1 #"HI"? 7X@ 3UJ"E^. 8 "H<>@ed"qI"=?-Q-#@J/=?-3N/1SS;5Z4#""W/1<5@4#=?SS/1-3A@<5dQ8-#6n/%79 #"M>"?<>=5δac = 2δdo6>"?<4#=R/1<!<!<!
τwo/4JdY"Z<>@edI! #@\"5-36>"5 #"?S/1-3 ` "5I!6N/40"Z\"?-tdY""?< |S, 1〉 /1<>E |D, 1〉 .>@ ^-36!A4
UJ"R/43! #"?UJ"?<.-Rw,-36N/1-iA4o7X@ Q-36>"s79 #"MN"b<N= 55@17-36>"@4#=?SS/1-3A@<yw,-QA4i<>@-:"4#4#"?<.-3/1Sy7X@ Q-36>"WI!!SA4#"43 /1<>E 3:&v-#@u\"Z"q>/=?-3S;5@< #"4#@<N/1<>="_. 9Y@<>4#"M.>"?<.-3S;5wa4#S6.-JE!"?-3!<!<!4l@< -36>"5@ #E!"? ^@17s/B7X"?d6.!<>E* #"E:_WdY"? #"%/=="?I!-#"Ew/4iI"b #U^/1<>"?<.-c=@<.-3 #@S@17-36>"W #"4#@<N/1<>="%=@<>E*-3A@<BA4Q-3UJ"=@<>4#!U V<!1.ofG6N"b 0"P7X@ #"-36>"W7X@SSA@edo<!]4#=R/1<>4:E!@^<>@-i<>"="4040/1 3S;5543-0/1 3-c/1-
PD = 1.G:<B"q!"?UI!S/1 %554#=R/1<BA4
436>@edo<^<]1./i2.'[^Kn = 1
O .[745;-36>":40/1UJ"QUJ"?-36>@.Ew.-36>" |S, 2〉 /1<>EJ-36>" |S, 3〉 43-0/1-#"4m6N/ ` ":/1SA4#@W\""?<I! #"?IN/1 #"E^\5J/1I!I! #@I! 3/1-#":=@U%\!<N/1-#@<N4Y@17 #"4#@<N/1<.-m\!S>"i4#AE!"?\N/1<>El/1<>E^=R/1 3 3A"? aI!!SA4#"4Rw."qI"=?-3<!79 #"M>"?<>=?A"4l@17
2δac/1<>E
3δac< @!
τ V 4#=R/1<>46. 1. i2.'[/1SA4#@~436>@edQ4^"q>/1UI!SA"4J79@ J-36>"4#"5=R/4#"4 .
0 1 2 30
2
4
6
8
10
!#" $
! #" i2. c>)
-#@-0/1SW@17L[(\N/=02 /1<>E 7X@ 3-36 4#=R/1<>4]dY"? #"v #"=@ #E!"E7X@
n = 0, 1, 2@ ;[n/4<!I!>-L43-0/1-#"46.51.i2. cv436>@edQ4
-36>"" ` /1SN/1-3A@<v@17-36>"Wdo6>@SA"E>/1-0/J40"?-6.arª-i=R/1<n\"W4#""?<-36N/1-;-36>"]UJ"R/43! #"E I9 V +.-0/1 32"bz"=?-; #"R/1SS;5A4JtM.N/1< V-3_R"E.L0wC/4"qI"=?-#"E^. S<>"R/1 ;h>-@17:-36!A4;I!SA@-=5"?SAE!4δac = 2π · 2.71(6) 2L:_ .fG6>"v7X@SSA@edo<! doSSWE!"4#=? 3\"x6>@ed -36!A45"q*I"? 3UJ"?<.-0/1S4#=06>"?UJ"i=R/1<^\"o>40"E-#@sUI!SA"?UJ"?<.-G/WMN/1<.-3!U/1-#"Q<-36>"s-tdY@ V M.!\!-:=@UI!!-0/1-3A@<N/1S43IN/="W=@UI@4#"En@17-36>"43-0/1-#"4&% |S, 0〉 w |D, 0〉 w |S, 1〉 w |D, 1〉 ' .fG6>"v/1 3 0/1<!" VUJ"?<.-W@17a@I"? 0/1-0@ sU^/1-3 3A="4s #"b79"? #4H-#@]-36>"LS/1-3-#"? W@ #E!"? @17\N/4#A4Q4#-0/1-#"46.my"?-
Φ(τ)\"W-36>"s!<!-0/1 %5ZU^/1-3 3Z #"?I V #"4#"?<.-3<!L-36>"c/=?-3A@<]@17O>4#-C-36>":436!879-#"? )(c=@UI"?<>40/1-#@
I!!SA4#"Q@17jSA"?<!-36τ.i79-#"? m/H43!-0/1\!S;5=06>@4#"?<
τ0w<N/1UJ"?S;5
43>=06 -36N/1-δac · τ0 = π
wm-36>"Z4#-0/1-#"4 |S, 1〉 /1<>E |D, 1〉6N/ ` "W/=M.! #"E5I!6N/4#"4i@17 e±iπ/2 = ±i w! #"43I"=?-3 ` "?S;5w>do6!SA43- |S, 0〉 /1<>E |D, 0〉 E!@<>@-:/=M.! #"/1<5I!6N/4#"^K9=b7 .>1.1i2.'&Obw!-36>4
Φ(τ0) =
1 0 0 00 1 0 00 0 i 00 0 0 −i
.
pv@ #"@ ` "? Rwy-36>"43! 3 #@!<>E*<! 3c/1UJ4#"5nI!!SA40"4HdY"? #"L79 #@U <>@ed@<~=R/1 #"b7D!SS;5v2"?I!-s@<x #"40@<N/1<>=";-#@
R ¸
/$ "%$"%$ ?$D I *+ +F0 .
do-36!< ±100 :_ .rt<]/Wh> #43-Y"qI"? 3UJ"?<.-Rw*-36>"iS/40"? mhN"?SAE]@17yI!!SA4#"@3:&WdG/4Y4#"?-a-#@6N/ ` "c/WI!6N/40":@17 πdo-365 #"4#I"=b-i-#@ 3 .CfG6>"5T=R/1<5-36>"? #"b7X@ #"s\"W4#""?<B/4Rx1(π/2)
/1<>ERx1(π/2)
I!!SA4#"46. *c@-#"s-36N/1--36>"SA@edY"? 8<NE!"q 4W>4#"Ex-#@T<>E!=R/1-#"J-36N"8<-#"? 3<N/1SmM.!\!-Rw4#""4#"=?-3A@< c1.'&2.JfG6>"J=@ 3 #"4#I@<>E*<!U^/1-3 3A="4i #"R/E
1√2
1 ±i 0 0±i 1 0 00 0 1 ±i0 0 ±i 1
,
do6>"? #"n-36>"+4#<>4J 0"P7X"? ^-#@ 3
= Rx1(π/2)/1<>E -36>" − 4#<>4^-#@ 3:&
= Rx1(π/2). fG6>"5U^/1<
4#"M.>"?<>="s-36>"?<5 #"R/E!4
C = Rx1(π/2) · Φ(τ0) ·Rx1(π/2) =
1 0 0 00 1 0 00 0 0 −10 0 1 0
.
0!="?I!-L79@ L-36>"T@<>"lU<.>443<ywgdo6!A=06doSSY\"]E!40=?>4#4#"E\"?SA@edWwC #"?I! #"4#"?<.-#4J/#DFEG H /1-#"
do6>"? #"5-36>" ` \! 0/1-3A@<N/1S:M.!\!-^=@<.-3 #@SA4^-36>"5<.-#"? 3<N/1S:M.!\!-Rwado-36 -36>"ZUJ@ #"5<.-3!-3 ` "B/1SSA@.=R/1-3A@<|0〉 = |n = 0〉 /1<>E |1〉 = |n = 1〉 .rk7^3 /1<>E 3c&c6N/ ` "o-36>"i40/1UJ"oS/4#"? mI!6N/4#"K 3 N 3:& N Rx1(π/2)
Obw-36>"?<5-36>"W@I"? 0/1-3A@<n #"R/E!4
C ′ = Rx1(π/2) · Φ(τ0) ·Rx1(π/2) =
0 1 0 01 0 0 00 0 −1 00 0 0 1
.
fG6!A4A4YKX/1/1<%"q!="?I!-y7X@ y-#6>"CU<.>443<,Oy/ DFEG H~/1-#"Cdo-36-36>"m< ` "? #4#"a/1SSA@.=R/1-3A@< |0〉 = |n = 1〉/1<>E |1〉 = |n = 0〉 79@ G-36>"H=@<-3 #@SyM!\!-6.Crª-o=R/1<yw!@17=@! #4#"w>/1SA4#@L\"s4#""?<5/4Q/4DFEGIH V S2":/1-#"do-365-36>"W<.-3!-3 ` "%/1SSA@.=R/1-3A@<579 #@U /1\@ ` "_.mfG6>"WI! #@I"? 3-3A"4:@17-36>"W@I"? 0/1-3A@< C <5IN/1 3-3A=?!S/1 i-36>"7k/=?-s-36N/1-W-sA4/]!<! ` "? #40/1Sg-dY@ V M!\!-M.N/1<.-3!U|/1-#"w6N/ ` "\""?<E*A4#=?>4#4#"Eu<4#"=?-3A@<&2.'&2.fG6>"40/1UJ"s #"R/4#@<!<!^6>@SAE!4G7X@
C ′ .fG6>"gh>-@17!1. i2. cQ ` "4 τ0 = (2·2.71 2i_ )−1 = 185 µ
46.Fc! 3<!i-36>"g 3!<!<!<!:"q*I"? 3UJ"?<.-τ0do/4
E!"?-#"? 3U<>"E^79 #@U /s #"E*>="El4#"?-a@17yE>/1-0/*w.-36>"Q #"4#!S-m\"?<!;& µ46.>@ a-36>"o7X@SSA@edo<!L"qI"? 3UJ"?<.-#4
τdG/4Y-36.>4 %l-#@
200 µ46. 8 "i6N/ ` ":-36>"?<TI! #@\"E]-36>":I! #@\N/1\!S-5l-3 3!-36T-0/1\!SA"H@17-36>":/1-#" C \5
43!\>4#"M.>"?<.-3S;5uI! #"?IN/1 3<!B-36>" c5I@4#43\!SA"J<!I!!-%43-0/1-#"4lKX/1/1<do-36/1I!I! #@I! 3/1-#"l=@U%\!<N/1-3A@<>4L@17 #"4#@<N/1<.-m=R/1 3 3A"? a/1<>EJ43AE!"?\N/1<>E
π V I!!SA4#"4C<-36>" C #"?IN/1 0/1-3A@<^I"? [email protected]/1<>E\5%I"? 7X@ 3U<! ECHC G3:/1\!@4#=?SS/1-3A@<>4:@<5-36>"s\!S>"W43AE!"?\N/1<>E5<5-36>" J3:/1\! LHI"? [email protected]/4i/43-0/1-#"s #"R/E!@!-6.mfG6>"s #"4#!S-3<!4#=R/1<>4o@17-#6>":J3:/1\!-3UJ"L
T/1 #"H436N@edo<]<Z1.i2.'(2.gfG6>"HE>/1-0/LdY"? #"s/1<N/1S;5*4#"E]\5Jh> #43-GI"b 379@ 3U<!
#"E*>="Ex=@<.-3 0/43-s43<.>40@AE>/1Sh>-#4H-#@l-36>";=R/4#"4HLw 9 /1<>EuF<u@ #E!"? s-#@]E!"?-#"? 3U<>";-36>"LI! 3<>=?IN/1S79 #"M>"?<>=?A"4@17c@4#=?SS/1-3A@<9.fG6>"] #"4#!S-#4/1 #"
ωA = (2π)11.9(1)wωC = (2π)12.0(2)
/1<>EωD =
(2π)16.95(5)2i_ .cfG6>"L 0/1-3A@4
ωD/ωA/1<>E
ωD/ωC/1 #"L"q*I"=?-#"EB-#@l\" √
2w,do6!A=06uA4 ` "? %55dY"?SS
=@<*h> 3UJ"E\5x-36>"Jh>-L #"43!S-#4R)(ωD/ωA)
2 = 2.02(4)w(ωD/ωC)
2 = 1.99(7).niSSm7X@! LE>/1-0/n4#"?-#4
dY"? #"ih>-3-#"EZ/1/1<lh>-3-#"El-0/12<!L-36>"4#"i7D #"M.>"?<>=?A"4o/4mhF!"E^.gfG6!A4Y-3UJ"H/43!I"? 3I@4#-3A@<T@17/1SS,79@> 43-0/1-#"4QdG/4i/1SSA@edY"EB/4i/%h><,/1S43-0/1-#"s@17-36>"s/1-#"_.CfG6>"/1I!I!SA"EZh>-G79!<>=?-3A@<5 #"R/E!4
PD(T ) = 1− 1
2(a2 + a3 + a4 − (a2 − a3) cos(ω1 · T ) + a4 cos(ω2 · T )),
R $
*+ ! " 4) "% :.4I "%$= ') /
0,0
0,5
1,0
0,0
0,5
1,0
0,0
0,5
1,0
0 20 40 60 80 1000,0
0,5
1,0
!"# >|!$"# >|
% "# >|% "&# >|
!"' >|% "' >|
% "(' >|!"' >|
)*
) *
) *
) *
+-,. /0 123.
4 52/6. /7
8 3.23.
9 :;=<
9 :;>
9 :; 9
9 :;>?
! #" i2.'(*)g #@\!<!^-36>"s!<! ` "? #40/1SMN/1<.-3!U/1-#"C.
do-36ω1 = (2π)12.0
2>@:_/1<>Eω2 = (2π)16.95
2>@:_ .CfG6>"Hh>-QIN/1 0/1UJ"?-#"? #4ai/1 0"H-36N"HI> #@\N/1\!S-3A"4
@17G-36>"79@! 4#-0/1-#"46.^fG6>"7D!<N=b-#@<=@<.-0/1<>4L@<!S;5v-36>"JIN/1 0/1UJ"?-#"? #4a2
K9=@ 3 #"43I@<>E*<!B-#@ |D, 1〉 Obwa3
K |S, 0〉 OQ/1<>E a4 K |S, 1〉 O .YfG6N"WI! #@\N/1\!S-5 a1 @17 |D, 0〉 A4iE!"?-#"? 3U<>"En79 #@U -#6>"W<>@ 3U^/1S_e/1-3A@<=@<>E*-3A@<
a1 + a2 + a3 + a4 = 1.fG6>"Y4#@SAE%S<>"4<;1.i2.'(:=@ 3 #"43I@<>EL-0@i-36>"4#"mh>-#46.fG6>"m7X@SSA@edo<!c-0/1\!SA"YSA43-#4-36>"Y@\!-0/1<>"E
ai/1<>E
=R/1<^\"o #"?/1 #E!"E]/4C-36>"o-3 3!-36^-0/1\!SA"i@17j-36N"oUI!SA"?UJ"?<.-#"E]MN/1<.-3!U/1-#"Q<J-#"? 3UJ4a@17jI! #@\N/1\!S-3A"4e)
↓ <!I!!- |S, 0〉 |D, 0〉 |S, 1〉 |D, 1〉|S, 0〉 .' K O . ¸!K O . * KX&O . [!K O|D, 0〉 . !K O . i!K O . K O . &!K O|S, 1〉 . K O . [!K O . ¸!KX&O . i KX&O|D, 1〉 . $*K O . K O . idcNKX&O . !KX&O
rt<;7X/=?-Rw\/5%>4#<!W-36!A4gh>-g7D!<>=?-3A@<^-36>"i/4#43>UI!-3A@<JA4CU^/E!"o-36N/1-m-36>" JªSA"R/12/1"i@17jI! #@\N/1\!S-5Lo<.-#@
R i
/$ "%$"%$ ?$D I *+ +F0 .
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aiE!"?-#"? 3U<>"EB/1\@ ` "s/1 0"c<T7k/=?-oI"?4043UA43-3A=s"43-3U^/1-#"4i@17-36>"H/1-#"sI"? 79@ 3U^/1<>="W\"=R/1>4#"
-36>"5x/1SA4#@T<>=?S>E!"J-36>"UI"? 79"=?-3A@<>4%@17G-36>";<!I!!-4#-0/1-#";I! #"?IN/1 0/1-3A@<9.lrk7G@<>"^/4#43!UJ"4/T=R/1 3 3A"? π V I!!SA4#":hNE!"?S-5l@17 99% /1<>E
98%79@ Y-36>"c43AE!"?\N/1<>E
π V I!!SA4#"4G@17-36>" C #"?IN/1 0/1-3A@<ZI"? [email protected]!@<>":"?-#4-36>"W7X@SSA@edo<!T=@ 3 #"=?-#"EvI! #@\N/1\!S-5n-3 3!-36v-0/1\!SA"_.:rt-:E!"4#=? 3\"4c-36>"WhNE>"bS-5B@17C-36>"%M.N/1<.-3!U /1-#"-#4#"?S87%.
↓ <!I!!- |S, 0〉 |D, 0〉 |S, 1〉 |D, 1〉|S, 0〉 .' K O . ¸!K O . * KX&O . [!K O|D, 0〉 . i!K O .' K O . K O . &!K O|S, 1〉 . K O . * K O . (!KX&O . idcNKX&O|D, 1〉 . (!K O . K O . i¸!KX&O . !Kk&O
rt<~/T4#"=@<>Ex4#"? 3A"4W@17G"q*I"? 3UJ"?<.-#4Rwy-36>"C ′ /1-#";6N/4s\""?<xUI!SA"?UJ"?<.-#"E~\5B>4#<!5"MN/1SgI!6N/4#"4
7X@ W-36>" 3:/1UJ4#"5uI!!SA4#"4%/1<>E/1/1<do-36τ = 200 µ
4shF!"E^.]fG6>"^40/1UJ"JI! #@.="E*! #"^dG/4L/1I!I!SA"Ew<>=?S>E*<!C3:/1\!@4#=?SS/1-3A@<>4s/1<>E5-36>"? ih>-#4c/4c/^43-0/1-#"W #"R/E V @!-6.QfG6>" #"43!S-3<!lI> #@\N/1\!S-55-3 3!-36-0/1\!SA"w CEHGY=@ 3 #"=?-#"EZ7X@ oUI"? 79"=?-i<!I!!-i43-P/1-#"HI! #"?IN/1 0/1-3A@<ywN #"R/E!4
↓ <!I!!- |S, 0〉 |D, 0〉 |S, 1〉 |D, 1〉|S, 0〉 . R K O . i$*K O . * K O . &!K O|D, 0〉 . i_i!K O . i!K O . K O . cNK O|S, 1〉 . $*K O . K O . i(!KX&O . i!KX&O|D, 1〉 . &!KX&O . cNKX&O . cNKX&O .i Kk&O
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R
% ')- P ') /7 ? A LOB = B 9 EGD : L@R DQN*I"R ; F E Ly"?->4h> #4#-do 3-#"Y@!--36>"a=@UI!SA"?-#"YI!!SA40"a4#"M.>"?<>="4
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7X@ o-36>"H79@! i=R/4#"4Rw*do6!A=06ndY"? #"WE!" ` "?SA@I"E5<540"=?-3A@<c1.'[2.'&2.• P1 = R(π/2,−π/2) R(π/2, π/2)
• P2 =R(π/2,−π/2) | Rswap(π) R(π/2,−π/2) R(π, π) R(π/2, π/2) Rswap(0) | R(π/2, π/2)
• P3 = R(π/2,−π/2) | Rphase | R(π/2, π/2)
• P4 = R(π/2,−π/2) | R(π, π) Rphase R(π, π) | R(π/2, π/2)m"? 3-3A=R/1SjE>/436>"4 | U^/1 32J-36>":\@!<>E>/1 %5l@17y-36>":\!S/=02J\@m^@I"? 0/1-3A@<yw*do6!A=06]<D9G/40" =@<>43A43-#4 O>43-@17G/]I!! #";dG/1-3<!Z-3UJ"_.LfG6>";-tdY@ V M!\!-@I"? 0/1-3A@<>4Rwydo6!A=06/1 #"J/1SSUI!SA"?UJ"?<.-#"E~\5n>43<!Z\!S>"43AE!"?\N/1<>E5I!!SA4#"4Rw* #"R/E
Rphase = R+(π,π
2) R+(
π√2, 0) R+(π,
π
2) R+(
π√2, 0)
/1<>ERswap(φ0) = R+(
π√2, φ0) R
+(2π√2, φ0 + φswap) R
+(π√2, φ0).
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π V -3UJ"?4;@<-36>"]@ #E!"? ;@17 R µ 4POdo6!A=06v40@.@<B=@< P>A=?-#4cdo-36v@! c=@6>"? #"?<>="%-3UJ"4c@<B-36>"%@ #E!"? H@17Pc µ46. 8 -36
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UJ"?-36>@.Ev7X@ ` "? 38hN=R/1-3A@<@17a-36>";=@ 3 #"=?-H79!<>=?-3A@<!<!5@17G/]4#"M.>"?<>="LA4H-#@&G&( -36>"=@UI!SA"?-#"@I V"? 0/1-3A@<<-#"? 3UJ4L@17D5/2 V
43-0/1-#"^I! #@\N/1\!S-5PD
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t/1<>E~ #" ` "R/1S PD
"M.N/1Sm-#@ 〈1|a(t)〉 w4#""^1.9c1.´$/.Zrª<-36>"^"q*I"? 3UJ"?<.-Rw-36>"^S/4#"? I@edY"? #4G79@ o43AE!"?\N/1<>E5/1<>EZ=R/1 3 3A"? Q"q*=?-0/1-3A@<5U%>43-G\"s/EO>43-#"El7X@ Y-36>"sE!"43 #"E 3:/1\!7D #"M.>"?<>=?A"46.pv@ #"@ ` "? RwI9 V +.-0/1 32B436!879-c=@UI"?<>4P/1-3A@<x6N/4c-#@]\";/1 3 0/1<!"Ev7X@ c-36>"%\!S>";43AE!"?\N/1<>Ex"q*=?-0/1-3A@</1<>E>40"E/1SA@<!vdo-36 " ` "? %5~I!!SA4#"_.u1.[i2.´$B436>@edQ4;"qI"? 3UJ"?<.-0/1So-3 0/="4@17i-36>" Φ
/1-#"Z/1<>E@17-36>"
(φ0 = 0)@I"? 0/1-3A@<x/1SA@<>"_. *c@-#"-36N/1-c<B-36>";/1\>4#"?<>="L@17a@-36>"? cI!!SA4#"4Rw
φ0A4:<>@-cdY"?SS
E!"bh><>"Ew\!!-;=R/1</1SdG/65*4%\"l4#"?--#@ \5~E!"bh><!-3A@<9.vfG6>"lE>/1-0/v/1 #""ldY"?SSado-36-36>"]=R/1SA=?!S/1-#"EAE!"R/1S" ` @S!-3A@<9.afG6>"sS/1-3-0"b :=@<.-0/1<>4i<>@;h>-QIN/1 0/1UJ"?-#"? #4H/1<>En/4#43!UJ"4Q<>@J"? 3 #@ #4LKX/4QE!"=@6>"? #"?<>="@ i@1z V #"4#@<N/1<-i"q*=?-0/1-3A@<>4bO .+.!\>4#"M.>"?<.-3S;5]-36>"H=@UI!SA"?-#"W4#"M.>"?<>="4o@17g/1SSj79@! o=R/40"4GdY"? #"HUI!SA"?UJ"?<.-#"En/1<>ET-3 0/="EZ\5^-36>"40/1UJ"JUJ"?-36>@.Ewg4#""^1.i2. i2.Trt</ Jt43-0/1<>E>/1 #E LJ 3!<@17o-36>"^FH"?!-#40=06 V /@_R40/B/1S@ 3-36!U5w PD dY@!SAE@<!S;5J\"i #"R/E]@!-o/1-a-36>"c"?<>E]@17y-36>"c4#"M>"?<>="H/1<>ElAE!"R/1SS;5JdY@!SAEl\" <]=R/40"4G[%/1<>E cTKD\N/1S/1<>="E !j¦1¢'¥j£Rq³Y¯t,¤§0Xk¥X¡§b£°e¥¨k§g¨k¦1y¨k¢'³o#"y¦1£ "¢´³o¡q²'³o£¨k3°a¨k¦q%$q¨k¥X¤ª¦'&)(P§+*¥k©m©P²-,P§Pk¢¨k¦1³H·/.R§§b£:©#¬ ¨k0"j©0ª°e¥1"y¢¨k¦%¨k¦1G¦q²'¡L§0¬y©P£L©P¤¨k¢'«?G¤§P³o¡£1¥k©#¨k¢´§P£L¥32e¥9¨k³ ¬8§0m³Q©4,b£q¨k¢'¤Y®1²´°65q¤¨k1©0¨k¢'§b£q¥17"j8"jkQ©0¯1²'G¨k§c¡ek§b²'§b£9,¨k¦1¤§P¦1k£1¤¨k¢'³o¨k§;:+<+<
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π/
√2, 0
√2π, 0.303π π/
√2, 0
π/√2
, 0
π, π/2 π/
√2, 0
π, π/2
! #" i2.´$)10*I"? 3UJ"?<.-0/1SNUI!SA"?UJ"?<.-0/1-3A@<N4a@17-tdY@HU^/ O3@ g\!!SAE*<!W\!SA@.=02*47X@ -36>"QFc"?!-#40=06 V /@_R40//1S@ 3-36!U.Z 0Cfs) (0)
@I"? 0/1-3A@<yC-36>"JS/4#"? I@edY"? %dG/4L/EO>4#-#"Ex79@ L/Z\!S>"43AE!"?\N/1<>E
2π3:/1\!iI"? [email protected]
T ! 2π @1757.7 µ
46. 3:r ! @:fs) g6N/4#"n/1-#"ΦdQ8-#6
T ! 2π =94.9 µ
4 .fG6>":E>/436N"?E^S<>"4aU^/1 32-36N"i4#do-#=06!<!%I@<.-#4Y@17y-36>"iS/4#"? mI!6N/40"_.C+@SAEl #"5S<>"4i/1 #"c<.!UJ"? 3A=R/1S4#U%!S/1-3A@<>4Q@17-36>"cAE!"R/1Sy" ` @S!-3A@<Zdo6!A=06n=@<.-0/1<Z<>@J/EO>43-0/1\!SA"IN/1 0/1UJ"?-#"? #46.
79!<>=?-3A@<>4POi/1<>E <v=R/4#"4 /1<>Ev&BK9=@<>43-0/1<.-:79!<>=?-3A@<>4PO .ifG6>"-3 0/=?<!]dG/4c"q-#"?<>E!"Ev4#@UJ"?do6N/1-\"5@<>EZ-36>"W"?<>E5@17-36>"/=?-3N/1S4#"M.>"?<>="4%K94#""W1.Qi2. iOG<5@ 0E!"? Q-#@^/=M! #"%/;@[email protected]/1-3A43-3A=4i@<-36>"J@!-3I!!-6.fG6>"UJ"R/1< ` /1S>"@17Y-36>"40"Lh><N/1SCI@<.-#4=@ 3 #"43I@<>E!4W-#@Z-36>"; #"R/E!@!- |〈1|a〉|2 @17Y-36>"/1S@ 3-36!U.mfG6>"s #"43!S-#4:/1 #"sSA43-#"EZ<Z-36>"H7X@SSA@edo<! 3:"R/E!@!-if/1\!SA" )
"q*I"=?-#"E UJ"R/43! #"E "qI"=?-#"E UJ"R/43! #"E↓ =R/40" |〈1|a〉|2 |〈1|a〉|2 |〈1|w〉|2 |〈1|w〉|2f1
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,=LOW65 O4S WXNNM=HEFJ G,@ WXE:Em[ JT;Y8MJTG4WXG,=LJ @XP GTSM= EF@>A=H<QE;5'7:N9 JTG4WXG,= )_.Q0 SM==O2+!= ,?7/(=H<,=HN'G
3 × 3G,=HNMJ,@"<,J
c(q)WX<,= E:7FJTG,=-, 7:N <,=\P%. b g eh.<0 SM= G,=H<TB OL@"N'G4WX7:N7:N9 JT7 N';B 8 =H<,J7:N <,@";N2,K8<4W"O!=HG,J7FJ NM@"G W B WXGT<T7
8;GZW J4@X[ OWXE:EF=-, 7:9*NM=H<$[ J%5'B 8 @"E 7 <,=HC<,=LJ4=\N GT7:N9 WQN';B 8 =H<6.HPG 7FJ <,=HE WXG,=-,VG,@ GTSM= EF=?8MJ,O4S?[ !U@"< ,MWXNOL@'= OH7F=HN'G,J 85 GTSM=I=LK!; WXGT7F@"N b)Ba ] e
〈j1; j2;m1;m2‖j3;m3〉 = (−1)j1−j2+m3(2j3 + 1)1/2(j1 j2 j3m1 m2 −m3
).
PN @";<OW"J,= 7 JTG4WXG,= ) WXN2,bL WX<,=B WX9*N =\G,7:OIJT;Y8EF=?+ =HEFJ@XP G,SM=S1/2
9*<,@";N2, JTG4WXG,=UWXN2, GTSM=QBV=HG4W"JTG4WA8EF=D5/2
JTG4WXG,= 7 <,=LJTC =LOHGT7+ =HE;5 7 7:N @"GTSM=H< >A@"< ,Jj = 1/2
WXN2,j′ = 5/2
. 7:GTS GTS7FJ 7 WXN2,b85 <,=HCE W"OH7:N9 7:N=LK'N9. 3 .)5 GTSM= 7:9*NM=H< [ZJ%5 B 8 @"EFJ 85 EF=?8MJ4O4S?[ ! @"< ,MWXNOL@'= OH7F=HN'G,J 7 G4W 7:N9 7:N G,@ W"OLOL@";N'GGTS WXG1W±1 J,7:9*N 7:N @";<1OW"J4= B W!=LJNM@ ,?7/(=H<,=HNMOL= 8 =LOWX;MJ,=U7:NGTSM=I=HN2, GTSM= BV@J,?;E:;MJ 7FJG4W!=HN 7 >A=U9 =HG >
Ω =eE02~
√15A
cαk3
2∑
q=−2
∣∣∣∣⟨1
2; 2;−m; q
∥∥∥52;−m′
⟩∣∣∣∣ c(q)ij εinj .
3 . L65
C<,@"C =H<TG5 @XPIGTS = EF=?8MJ4O4S?[ ! @"< ,MWXN OL@'= OH7F=HN'G,J 7FJ GTS WXG GTSM=5f,?7/ =H< P<,@"B =H<,@ @"NE;5 7 Pm3 =
m1 +m2. 0 S7FJ OL@"N2,?7:GT7F@"N 7 <,=W@,?7:N9
q = ∆m := m −m′ 7:N @";< OW"J,= 7 B W!=LJGTSM= JT;B @+!=H<q7:N
=LK'N9. 3 . L65A@@8MJ4@"EF=\G4= 7 W"JGTSM=I@"NE;5 NM@"N?[ =H<,@VG,=H<TB 7:NGTSM=UJ,;B 7FJ >
Λ(m,m′)c(∆m)ij εinj>7:GTSGTSM=",JTC =LOH7O O- EF=?8MJ,O4S?[ ! @"< ,MWXNYOL@'= OH7F=HN'G,J
Λ(m,m′) :=
⟨1
2; 2;−m;∆m
∥∥∥52;−m′
⟩.
0 SM=Λ(m,m′)
@XP87:N'G,=H<,=LJTG SM=H<,= WX<,= 3$J4=L=QPR@"<1=qWXBDCEF=IG4WA8EF=LJ17:N <,=\P%.^b)Ba*ge5 >
)BL )
m) <@L ) <@L ) <@L ) <@L ) <@L
m′ ] <@L <@L ) <@L [ ) <@L [ <@L
Λ(m,m′)) √
4/5√
3/5√
2/5√
1/5
NE;5 G,SM= OW"J,=LJ >7:GTSm = 1/2
WX<,= E:7FJTG,=-, W"J GTSM=Λ(m,m′)
WX<,= 7:NJ+"WX<T7 WXN'GD;N2,=H<DGTSM= 7:NJ+!=H<,JT7F@"N(m→ −m; m′ → −m′) 7 J,@ GTS WXG1GTSM= m = −1/2 J,=HG OWXNb8 =I=W"JT7:E;5K,=H<T7+!=-,b85 J*5'BDBV=HGT<%5 .
0 SM=U9 =L@"BV=HGT<%5'[$,=HC =HN2,=HN G C WX<TG17FJ1OL@"N G4WX7:NM=-,7:N GTSM=IK'; W@,?<4WXGT7FO PR@"<TB
g(q) :=∣∣∣c(q)ij εinj
∣∣∣ .0 SM= C @"E WX<T7 WXGT7F@"N WXN2,YGTSM= > WB+ = +!=LOHG,@"< OWXN 8 = C WX<4WXBV=HGT<T7FJ4=-, 85GTSM= WXN9*EF=
φ8 =HGe>A=L=HN GTSM= E W"J,=H<
8 =WXB WXN2, GTSM= B WX9*NM=HGT7FO O =HE , WXN2,f85 GTSM= +*WX<T7 WA8EF=γ
,=LJ,OH<T787:N9 GTSM= WXN9*EF=M8 =HGe>A=L=HN GTSM= C @X[E WX<T7 WXGT7F@"N WXN2, GTSM= B WX9*NM=HGT7FOEO =HE , +!=LOHG,@"<C<,@ /=LOHG,=-, 7:N'G,@ GTSM= CE WXNM=%@XP 7:NMOH7 ,=HNMOL=_. SM@'@*JT7:N9B = B0(0, 0, 1)
@"NM= @@8G4WX7:NMJk = k(sinφ, 0, cosφ)
VWXN2,ε = (cos γ cosφ, sin γ,− cos γ sinφ)
.0 SM= P$;NMOHGT7F@"NMJ
g(q),=HC =HN2, @"NE;5 @"NGTSM= BV@J,?;E:;MJ@XP
qWXN2, WX<,=U9*7+!=HNb85 >
g(0) = 12 |cos γ sin(2φ)|
g(±1) = 1√6|cos γ cos(2φ) + i sin γ cosφ|
g(±2) = 1√6
∣∣12 cos γ sin(2φ) + i sin γ sinφ
∣∣ .08W 7:N9 WXE:E GTSM=LJ,=U<,=LJT;E:G,JWXN2, NM@"G4WXGT7F@"NMJ17:N'G,@ W"OLOL@";N G 7 =LK'N9. 3 . L65 OWXNR8 = ><T7:GTG,=HN W"J
Ω =eE02~
√15A
cαk3Λ(m,m′)g(∆m)(φ, γ) .
? N"XJF*R8 T ; B 9OBH:WE ; N<? X; FBA"B0 SM=.,?<T7+ = +!@"E:G4WX9 = P$@"<GTSM= WX;EGT<4WXC 7FJ9 =HNM=H<4WXG,=-, 85 OL@"NNM=LOHGT7:N9 @"NM=J3 6 [ =HEF=LOHGT<,@J,= C WX7:<QG,@ GTSM=@"C =HN=HN2, @XPW SM=HE:7FOWXE <,=LJ4@"N WXG,@"< b)Ba e 3$K'; WXE:7:G5 PW"OHG,@"< ≈ 200
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f85 W@, /;MJTGT7:N9 G,SM=U3 6 C @>A=H<6. 0 SM=YSM=HE:7FOWXE<,=LJ,@"N WXG,@"< 7FJBV@";N'G4=?, @"N G,@"C @XP GTSM= +"W"OH;;B O4S WXB 8 =H< 7:NMJT7 ,= W BV=HG4WXE JTS7F=HE ,YGTS WXGQC<,=?+!=HN'G,JQGTSM==HEF=LOHGT<,@"N7FOLJ 7 PR@"<8=qWXBDCEF=ZGTSM= ,?7F@J,=ZE W"J,=H<,J 7 P<,@"BC7FO 7:N9;C GTSM= ,?<T7+ = P$<,=LK!;M=HNMO5 . E:E @"GTSM=H<8=HEF=LOHGT<T7FOOL@"NNM=LOHGT7F@"NMJ =q 7:GT7:N9 GTSM= +"W"OH;;B J%5?JTG,=HBWX<,=IEF@> [C W"J4J OME:G,=H<,=-,PR@"<1GTSM= J WXBV=I<,=W"J,@"N9.C0 SM=IC @>A=H<9 @"7:N9Q7:N'G,@GTSM=Z<,=LJ,@"N WXG,@"< W"JX> =HE EW"J GTS = <,= =LOHG,=-, C @>A=H< WX<,=ZBV=W"JT;<,=-, .X0 SM=Z<,=LJ,@"N WXNMOL=ZP<,=LK';M=HNMO57FJ W@, /;MJTG,=-,b85 BD7:N7:BD7L7:N9 GTSM=U<,= =LO\G4=?, C @>A=H<6.
0 SM= ,=LJT7:9*N%@XPAW SM=HE:7FOWXE <,=LJ,@"N WXG,@"< P$@"< GTSM= NM=?> GT<4WXC C @*J,=-, OL@"NMJ,7 ,=H<4WA8EF= C<,@@8EF=HBVJIOL@"BDC WX<,=-,G,@D@"GTSM=H<1GT<4WXCMJ1GTS WXGS WB+ = 8 =L=HN;MJ,=-, 7:N @";<9*<,@";CGTS<,@";9*S GTSM= E W"J,G\5 =WX<,J6.H0SM= ;MJT; WXEC<,@'OL=-,?;<,=> W"JAG,@.8;7:E , W=8 WX<,=Q<,=LJ,@"N WXG,@"< >7:GTSW <,=LJ,@"N WXNMOL=QP<,=LK';M=HNMO5 JTE:7:9*S'GTE;5 S7:9*SM=H< GTS WXN GTSM= ,=LJT7:<,=-, @"NM= ! 2'&
k! #" $%& #'(" * *)',+.- /'( 2.0213+.)'12! 04%32. &)0265
-78 )'194<;:!<=>5?!7#121?.@1A)'-B1A/CDFE &':5G78 H.12(-BI " *)KJ <(L 102 "M+B12%N')&'13O-BP'94<QLR5ST " *)U ' )#$ 1" 1.@V%&WB: <7X5
)BL*L
!
" $#%'&)(
"! ! *+"!
,$% # -/.1023
4
5 6
7 8
687:9*;<,= .) > @Q=HE:7FOWXE <,=LJ4@"N WXG,@"<6.H0 SM=OL@"CC =H< O5 E:7:N2,=H< WXN2, GTSM=OL@"CC =H<ASM=HE:7VP$@"<TB W K!; WX<TG,=H< > WB+ =<,=LJ,@"N WXG,@"<6.
3G5 C7FOWXE:E;5 ) ] p @5 . @"NNM=LOHGT7:N9 7:GQG,@ GTSM= GT<4WXC JTS7 PG,=-, GTSM= <,=LJ,@"N WXN OH=1,@>NF85 +"WXE:;M=LJ @XPA@"NE;5L]p @ @"<UEF=LJ,J6. 6 @"< GTSM= NM=?> GT< WXC 7 W J,@"BV=?>S WXGIS7:9*SM=H< ,?<T7+!= P$<,=LK!;M=HNMO5 @XPWA8 @";G=LdcZp @.>W"JWX7:BV=-, PR@"<K3$O\P . J4=LOHGT7F@"N ] .)5 . 6 @"E:EF@>7:N9 GTSM= WA8 @+!= C<,@'OL=-,?;<,= 7 WSM=HE:7FOWXEA<,=LJ4@XN WXG,@"<.>7:GTS WR8 WX<,=<,=LJ,@"N WXNMOL=ZP<4=HK';M=HNMO5 @XP L_i:p @ > W"JX8;M7 E:G6.@Q@>A=?+ =H< 7 >SM=HN OL@"NNM=LOHG,=-, G,@GTSM=ZGT<4WXC 7 GTSM=Z<,=LJ,@"N WXNMOL=@XPGTSM=Q@+!=\< WXE E(J%5?JTG,=HB > W"J W"JZEF@> W"J )6i%p @ . J 8;7:E ,?7:N9DWISM=HE:7FOWXE(<,=LJ,@"N WXG,@"< OWXN 8 =K';7:G,= GT7:BV=OL@"NMJT;BD7:N9 7 GTS7FJ@@8 J,=H< +"WXGT7F@"NVJT;9*9 =LJTG,=-, G,@QOM<,JTGH,=?+ =HEF@"CVWBV@J,=HE PR@"<8GTSM= OL@"B 87:NM=-,DJ*5 JTG4=\B WXN2, G,@,=LJT7:9*N GTSM=<,=LJ,@"N WXG4@"<AW"OLOL@"< ,?7:N9 G,@IGTS7FJ BV@J,=HE .HP NVGTSM= PR@"E:EF@>7:N9 7 JT;MO4S OL@"NMJT7 ,=H<4WXGT7F@"NMJAOL@"NMOL=H<TN7:N9SM=HE:7FOWXE <,=LJ,@"N WXG,@"< ,=LJT7:9*NYWX<,= ,?7FJ,OH;MJ4J,=-,^.
JUWXN G,@K,=LJ,OH<T78 = GTSM= ! " ' '"J ! &' & ' %YGTSM= P$@"E:EF@>7:N9 W"J,J,;BDCGT7F@"NMJIWX<,=B W@,= b)Ba e >• 0 SM=QJ4=\G,;C 8 =HS WB+ =LJ E:7 = W OL@ W:?7 WXE(OL@"N2,?;MOHG,@"< >7:GTS GTSM=QOL@"7:EW"JZGTS =QOL@"<,= WXN;, GTSM=QO5 E:7:N2,=H<1W"JGTSM=IJTS7F=HE ,^.
• 0 SM= <,=LJ,@"N WXNMOL= P<,=LK';M=HNMO5 7FJ ,=HG,=H<TBD7:NM=-, 8/5%Wλ/4
[ OL@"N2,?7:GT7F@"N /;MJTG E:7 = 7:N WOL@"NJ+!=HN'GT7F@"N WXEOL@ W: 7 WXE K'; WX<TG,=H< > WB+!= <,=LJ,@"N WXG,@"< .
0 SM= 3CS W"J,=5X+ =HEF@'OH7:G5 WXEF@"N9UGTSM=OL@"7:E?7FJ9*7+!=HN 85υ =√LC 7 >SM=H<,= L WXN2, C WX<,= 7:G,J7:N2,?;MOHG4WXNMOL= WXN2,
OWXC W"OH7:G4WXNMOL=AC =H< ;N7:G EF=HN9*GTS 7 <,=LJTC =LOHGT7+!=HE;5 .X0 SM=λ/4
<,=LJ,@"N WXNMOL=AOL@"N2,?7:GT7F@"N <,=W@,Jb = λ/4 = υ/4f0 7
)BL
687:9*;<,= L > W65 '7:BDCEF=VOH7:<,OH;7:G=,?7 WX9*<4WXB PR@"< GTSM= ,=q?G,=HNMJT7F@"N#- OL@"BDC<T7FJT7:N9GTSM= GT<4WXC%7:G,J <4W@,?7F@ P$<,=\[K';M=HNMO"! OL@"NNM=LOHGT7F@"N PGUJ,SM@>1JUGTSM=DOWXC W"OH7:G4WXNMOL=J @XPAGTSM= P$=L=-, [GT<,@";9*S 3
Cf5 7 GTSM=1>7:<,=
3Cw
5ZWXN2, GTSM=QGT<4WXCE3Ct
5ZW"J >A=HE:EW"JAGTSM=7:N2,?;MOHG4WXNMOL=U@XP GTSM=Q>7:<,=13Lw
5 P1
WXN2,P2
WX<,=GTSM= CM7 N J @XP GTSM= =HE:=LOHGT<T7FOWXEPR=L=-, [GTS<,@";9*S G,@ GTSM=.+"W"OH;;B +!=LJ,J,=HE 0 SM= 9*<,@";N;, E:7:NM= 7:N<,=WXE:7:G! OL@"NMJT7FJTG,J@XP GTSM= >S @"E:= +*W"OH;;B + =LJ,J4=HE?W"JX>A=HE:EMW"J8=?+!=H<#! GTS7:N9 >S7FO4S 7FJ89*<,@";N2,=-,7:NMJT7 ,= 895%$@ W@,=-,V<,=LJ,@"N WXG,@"< P G,J <,=LJ,@"N WXNMOL= P$<,=LK';M=HNMO"!
f7FJ<,=OMNM=-,18&!
l = λ/4 0SM=
9*<,="! J,OWXEF=LJ1<,=HC<,=LJ,=HN'G ,?7/ =H<,=HN'G1CS W"J,=G+!=HEF@'OH7:GT7F=LJ
>SM=H<,=f07FJGTSM= <,=LJ,@"N WXNMOL= P$<,=LK!;M=HNMO"! 6 @"<TB ;E W"= P$@"< L WXN2, C WX<,=.,=H<T7+ =-,Y7:N <,=\P (' )Ba ) PNYGTS WXG
C WXC =H< 7 W4,?7:BV=HNMJT7F@"N WXE8<4WXGT7F@d/D = 0.55
7FJJT;M9*9 =HJ,G,=-,P$@"< WXNY@"CG,7 B ;B 3S7:9*S95 PRW"OHG,@"< 08W'7:N9GTS7FJ17 N G,@VW"OLOL@";N GWXN2, C;GTGT7:N9 WXE:E P$@"<TB ;E W"=IG,@"9 =HGTSM=H< 7 @"NM= O N2,JGTSM=IJT7:BDCEF=U<,=HE WXGT7F@"N
f0 =2670
N · d MHz · cm, 3 65
>SM=H<,=N7FJ8GTSM=AG4@"G4WXE?N';B 8 =H< @XP9>7:N2,?7:N9 J @XP GTSM=ASM=HE:7 +* @"G,= 7 GTS WXG NM@>
D ≈ d/0.55 7FJ =LJ,J,=HN'GT7 WXE:E,!OF=-, >7:GTSGTSM= O4S @"7:OL=I@XPd 0 SM= @XP8GTSM= <,=LJ,@"N WXG,@"<17FJ NM@> ,=HG,=H<TBD7:NM=-,R8&! WXE:E <,=LJT7FJTGT7+!=QEF@*J,J,=LJ 7:N
GTSM=IJ-! JTG,=HB 0 SM=I;MJ,=I@XPS7:9*S OL@"N2,;MOHGT7+'7:G! B WXG,=H<T7 WXEFJ 7 7:N C WX<TGT7FOH;E WX<1PR@"<1GTSM=USM=HE:7 7 ! 7F=HE ,J1S7:9*S +*WXE ; =HJ 6 @"<AGTSM=QJ4WXBV=<,=W"J,@"N 7 J,OWXE:7:N9 ;C GTSM=QJ4=HGT;C =\NMS WXNMOL=LJ W"JA7 GA<,=-,?;MOL=LJ GTSM=G,@"G4WXEH3$J,;</PRW"OL=5<,=LJT7FJTG4WXNMOL= P N @"< ,=H<G,@!=L=HC GTSM= >SM@"EF= J,=HGT;C OL@"BDC W"OHG 7 WOL@"BDC<,@"BD7FJ,=C> W"JO4SM@*J,=HN ;MJT7:N9UWOL@"CC =H<O"! E 7:N2,M=\<.>7:GTS
D = 10OHB P$@"< GTSM= @";G,=H< OL@"N2,?;MOHG,@"< @"NMJ,=LK';M=HN'GTE,! 7 GTSM= OL@"7:E > W"J >A@";N2, >7:GTS
WK,?7 WXBV=HG,=H<d ' 5.5
OHB PN <,=\P .' )Ba ) 7 WXN%@"CGT7:B ;B <4WXN9 =1 < b/d < 4
P$@"<UGTS = <4WXGT7F@b8 =HGe>A=L=HNSM=H7:9*S'GIWXN2, ,?7 WXBV=HG,=H<I@XP GTSM= OL@"7:E87FJQ9*7+ =HN 0 S';MJ 7 W OL@"7:E8SM=H7:9*S'G b ' 2d = 11
OHB >W"JQ=HNJ+ 7FJ4WX9 =-, !Q7+!=HN GTSM=LJ4= ,?7:BV=HNMJ,7F@"NMJ 7 7:N C WX<TGT7FOH;E WX< d 7 =LK'N 3 65 ,=HG,=H<TBD7:NM=LJGTSM=QC WX<4WXBV=HG,=H<
NP$@"<1W.,=LJT7:<,=-,
f0 S WXGI7FJUNM@> GTSM=0/12'43 1 5/-1 "671 5
f' 81 '%:9";241-<=265>2?1% 7 7 = GTSM=DSM=HE:7FOWXE <,=LJ,@"N WXG,@"<>7:GTS%GTSM= GT<4WXC W"J W EF@ W@,A@ SM=HN GTS = GT<4WXC%7FJIOL@"NNM=LOHG,=-, G4@ GTSM=D@"C =HN =HN2, 7 7:G JT7:BDCE,!Y<,=HC<4=HJ4=HN'G,JWXN =q G,=HNMJT7F@"N @XPUGTSM=<,=LJ,@"N WXG,@"< 0 SM= =HN'GT7:<,=EF=W@, P$<,@"B GTSM= =HN2, @XP GTS = SM=HE:7 G,@ GTSM=GT<4WXC OWXN
8 = + 7F=?>A=-, W"J W GT<4WXNMJ,BD7FJ,JT7F@"N E:7:NM= WXN2, OWXN 8 =V<,=HC<,=LJ,=HN'G4=?, 7:N W E:;BDC =-, OH7:<,OH;7:G.,?7 WX9*<4WXB W"JI7:N687:9 B L*W65 P G 7FJ W"J,JT;BV=-, GTS WXGGTSM==> WB+!=HEF=HN9*GTS @XP GTSM=BC 6 JT7:9*N WXE GT<4WB+!=HE:7 NM9 WXEF@"N9 GTS WXGE:7:NM= 3 PR@"<GTSM=DP$<,=LK!;M=HNMO"! <4WXN9 = @XP7:N'G,=H<,=LJ,G5 7FJ B ;MO4S E WX<T9 =H< GTS WXN WXE EZ7:N2,?7+'7 ,?; WXEZC7F=LOL=LJ @XPGTSM= OH7:<,OH;7:G P G,J
)BL>D
=HEF=HBV=HN'G,J1OWXN GTSM=H<,=\PR@"<,= 8 =Q<,=H9!WX< ,=-,W"JOL@"N'GT7:N';M@";MJTE,!K,?7FJTGT<T78;G,=-, P N @"GTSM=H<C> @"< ,J GTSM=QE:7:NM= S W"J1WEF=HN9*GTS
l2WXN2, WXN 7:N2,?;MOHG4WXNMOL=DWXN2, OLWXC W"O\7:G4WXN OH= C =H<Q;N7:G EF=HN9*GTS @XP L2 = L2/l2
WXN2, C2 = C2/l2 7<,=LJTC =LOHGT7+!=HE,! L2
WXN2,C2
WX<,= GTSM=VJT;BVJI@XPWXE:E 7:N2,?7+'7 ,?; WXE 7:N2,?;MOHG4WXNMOL=LJ WXN2,%OWXC W"OH7:G4WXNMOL=LJ 7:N GTS7FJJ#!?JTG,=HB 0 S7FJ 7:NMOHE:;2,=LJ E:;BDC =-, =HEF=HBV=HN'G,J 7 PR@"< =qMWXBDCEF= GTSM=YOWXC W"OH7:G4WXNMOL= @XPUGTSM= GT<4WXC 7 W"J4>A=HE:EW"J BV@"<,= OL@"N'GT7:N';M@";MJTE,! ,7:J,GT<T78;G,=-, =HEF=HBV=HN'G,J J,;MO4S W"J GTSM= 7:N2,?;MOHG4WXNMOL= @XP1GTSM= EF=W@,J GTS<,@";9*S GTSM=+*W"OH;;B +!=LJ,J,=HE G,@ GTSM=QGT<4WXC =HEF=LOHGT<,@J,=LJ 0 SM=U=q G,=HNMJT7F@"Nb8&! 7 G4J,=HE P8OLWXN GTS';MJ 8 =QGT<,=WXG,=-, 7:N GTSM= J4WXBV=> W !YW"JQGTSM= SM=HE:7FOWXE<,=LJ,@"N WXG,@"<IWA8 @+ = WXN2, 7:G,JQ@>N
λ/4<,=LJ4@"N WXNMOL= OL@";E , 8 = OWXEFOH;E WXG,=-,%W"OHOL@"< ,7 NM9
G,@f2 =
√L2C2/4l2 7 7 P L2
WXN2, C2> =H<,= =qMW"OHGTE,! NM@>N 0 SM= +"WXE:;M= >A@";E ,M8 =.,=HG,=H<TBD7:NM=-, 8&!
=HEF=LOHGT<T7FOWXE(<,=LJ,7:J,G4WXNMOL=LJAG,@ 9*<,@";N2, @"<WXEF@"N9 GTSM=E:7:NM= 3>S7FO4S WX<,=Q@"BD7:GTG4=?, 7:N 687:9 L*W65 PR@"<ZGTSM=QJ4W!=@XP JT7:BDCE:7FOH7:G!25 687:9 LA895<,=HC<,=LJ,=HN'G,JUW J,7 BDCME 7O =-, C7FOHGT;<,= @XP GTSM= EF@ W@,=-,YSM=HE:7FOWXE<,=LJ,@"N WXG,@"< 6M@"<GTSM=
λ/4<,=LJ,@"N WXNMOL=I@XPGTSM=I@+ =H<4WXE:E J#!?JTG,=HB @"NM= OMN2,J >
1
f=
1
f0+
1
f2.
0 S7FJ P$@"<TB ;E W OWXNE8 = 7:N'G,=H<TC<,=HG,=-, 7:N%@";<UOL@"N'G,=q?G > JQEF@"N9 W"Jf27FJ B ;MO4S S7:9*SM=H<IGTS WXN
f0 7OL@"N?[
NM=LOHGT7:N9 GTSM=GT<4WXC G,@UGTSM=SM=HE:7FOWXE <,=LJ,@"N WXG,@"< >7:E:EMNM@"GZO4S WXN9 =G,SM=<,=LJ,@"N WXNMOL=P<,=LK';M=HNMO"! OL@"NMJ,7 ,=H<4WA8E,! 7W"JS W"J 8 =L=HN GTSM= OW"J4= 7:NYPR@"<TBV=H< J,=HGT;CMJQ;MJ,=-,Y7:N @";<Q9*<,@";C 6M@"<QGTSM= NM=?> GT<4WXC 7 Ge>A@ GTS7:N9 J >A=H<,=,?7/ =H<,=HN'G > GTSM=K,=LJT7:<,=-,dOMN WXEAP$<,=LK';M=HNMO"! > W"J S7:9*SM=H<DGTS WXN 8 =\PR@"<,= WXN2, WXG GTSM= J4WXBV= GT7:BV=
f2>W"J
WXCC WX<,=HN'GTE,! EF@>A=H< 7 C<,@@8 WA8E,! ,?;M= G,@ GTSM= S7:9*SM=H<VOWXC W"OH7:G4WXNMOL=Ct
@XPGTSM= GT<4WXC 6M<,@"B GTSM=WA8 @+ =BV=HN'GT7F@"NM=-, @@8MJ,=H< +*WXGT7F@"NE3GTSM=QEF@ W@,=-, <,=LJ,@"N WXNMOL=
f> W"J )%p @ P$@"<
f0 = 28p @5 WXN
f2@XP8WA8 @";G
] a5p @ > W"J OWXEFOH;E WXG,=-, JIGTSM=4,M=HJ,7:<,=-,f
>W"J.L>DBp @ 7 GTSM=VNM=?> SM=HE:7FOWXEZ<,=LJ4@"N WXG,@"< > W"J NM@>,=LJT7:9*NM=-, P$@"<
f0 ≈ 46p @
OLOL@"< ,?7:N9 G,@ =LK'N 3 65 7 GTS7FJ <,=LK';7:<,=LJ WIOL@"7:E9>7:GTSN ' 11
>7:N2,?7:N9 J C =\P 7' )Ba ) JT;9*9 =LJTG,JZGTS WXGGTSM=UOHEF=WX<T7:N9 8 =HGe>A=L=HN G!>A@ OL@"7:E >7:N2,?7:N9 J JTSM@";E ,K8 = <,@";9*SE,! =LK'; WXE GTSM= ,?7 WXBV=HG,=H< @XP8GTSM=UOL@"7:E >7:<,= P P @"NE,! )*)=>7:N2,?7:N9 JUWX<,= G,@48 = ,?7FJTGT<T78;G,=-, @+!=H<
b = 11OHB 7 GTS7FJ<,=LK';7:<,=LJ W >7:<,= ,?7 WXBV=HG,=H< @XPZW"J
B ;MO4S%W"J ] BDB 7 >S7FO4S 7:N PW"O\G > W"JQ;MJ4=-, PR@"<QGTSM= NM=?> SM=HE:7FOWXE<,=LJ4@"N WXG,@"< 0 SM= E WXGTG,=H<UGT;<TNM=-, @";GG,@ S WB+!= W18 WX<,= <,=LJ,@"N WXNMOL= @XP
f0@XP D2)%p @ SM=HN OL@"NNM=LOHG,=-, G,@ GTSM= GT<4WXC J4=HGT;C 7 7:G <,=LJ,;E:G,=-, 7:N
W OMN WXE P$<,=LK!;M=HNMO"!f@XP
23.5p @ 3>S7FO4S 7FJU7:N%<4WXGTSM=H< 9 @'@J, WX9*<,=L=HBV=HN'G >7:GTS%GTSM=VOH7:<,OH;7:G BV@J,=HE
B W@,=5
? <; 9I; "!$# ?% @; & '
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e>A@ <,@'O'7:N9 BV@J,=LJ( JT7:N9 W <4W@,?7 WXE BV@J,= >A@";E , GTSM=H<,=\PR@"<,= OL@"<T<,=LJTC @"N2, G,@ W BV@"GT7F@"N WXE1K!;Y87:G,?7FJTGT<T78;G,=-, @+ =H<$ J 'K'; W"JT7 [Ge>A@X[EF=?+!=HE J#!?JTG,=HBVJ7:N W<4WXGTS =\<8;NMOL@"N'GT<,@"E:EF=-, > W ! 7 >S7FO4S 7FJ ;N2,=LJT7:<4WA8EF= @"NMJ,=LK';M=HN'GTE,! 7 >A= S WB+!=UG,@ ;MJ,=43$@"NM=I@XP5ZGTSM= W:?7 WXE BV@J,=63$J5
0 SM=H<,=Q7FJW E WX<T9 =QN';B 8 =H<1@XP J,=HGT;C C WX<4WXBV=HG,=H<,J >S7FO4S OWXN 8 = +"WX<T7F=-, 7:N GTSM= =q?C =H<T7:BV=HN'G 0 SM=LJ,=7:NMOHE:;2,= 7 P$@"< =qWXBDCEF=
• GTSM=UB WX9*N7:GT;2,= @XP8GTS = B WX9*NM=HGT7FOGO =HE , 3)+* ' )' ! 0&)$ 'B12%+B , B )$ 2 13%@021A ' +-, %& U%& *'13 =;52:/.1" 12$71;B " 10 ) 1;$F , % %&%& '13 0202,'B12 , "-B' %& KB%&13+B )$ '13%U %' K%&021A&'12 , ' /)$ +B13 0 )' 12;-B 2 13%306 *)02 %&, 020,
≈ 1 4 .*5
)BL ]
• GTSM=<,=HE WXGT7+!= ,?7:<,=LOHGT7F@"NMJZ@XP B WX9*NM=HGT7FO O =HE , 7 @L $ NB 8 =LWXB WXN2,V7:G,J C @"E WX<T7 WXGT7F@"NE3RWXN9*EF=LJφWXN2,
γ 7 J4=L=UJ,=LOHGT7F@"N ] 53
• GTSM= G,<4WXCC WX<4WXBV=HG,=H<,J 7 N WXBV=HE,! C6d,?<T7+!= C @> =\<Prf
WXN2,=HN2, [ OWXC + @"E:G4WX9 =Vcap
3
• GTSM=IWXNM9*E:= β 8 =HGe>A=L=HN GTSM= @L $ NB 8 =WXBWXN2, GTSM=UGT<4WXCYW: 7FJ +!=H< GTSM=1OL@";<,J,=1@XP(GTS7FJ S >A@"< 7 B WXN&! ,?7/(=H<,=HN G OL@"B 87:N WXGT7F@"NMJZ@XP(GTSM=LJ,=C WX<4WXBV=HG,=H<,J S WB+ = 8 =L=HN;MJ,=-, WXN2,%G,=LJTG,=-, J W B WXGTG,=H< @XP PRW"OHG 7 SM@>A=?+!=H< 7 @"NE,! Ge>A@ ,?7/ =H<,=HN'G OL@"N OM9*;< WXGT7F@"NMJ S WB+ = 8 =L=HN=HBDCEF@!!=-, G,@%W"O4S7F=?+ = GTSM= B W0T@"<D<,=LJT;ME G4JV@XPQGTS7FJDGTSM=LJT7FJ [ @"NM= PR@"<VJT7:N9*EF= 7F@"N WXN2, GTSM= @"GTSM=H<DPR@"<Ge>A@X[7F@"NY=q C =H<T7:BV=HN'G,J
PN 8 @"GTS OW"J,=LJ >A=1P$@";N2, GTSM= OL@"B 87:N WXGT7F@"N %φ = 45, γ = 0 ' BV@*JTG ;MJ4=\P;E 7 8 =LOWX;MJ,= GTSM= GT<4WXN?[
JT7:GT7F@"N >7:GTS GTSM=4>A=W!=LJTG B WX9*NM=HGT7FO4O =HE , ,=HC =HN2,=HNMOL= 7 N WXBV=HE,! S1/2(m = −1/2) ↔ D5/2(m′ =
−1/2) 7FJQB W:?7:B WXE:E,!YJTGT<,@"NM9 0 SM= E WXGTG4=\<U7FJGTSM=HN ;MJ,=-, G,@ =HNMOL@J,= GTSM= K';Y87:G GQGTSM= J4WXBV= GT7:BV= 7GTSM=S1/2(m = −1/2)↔ D5/2(m
′ = −5/2) GT<4WXNMJ,7:GT7F@"N 7:N GTS7FJZJ,=HGTGT7:N9 7FJZJTGT7:E:E JTGT<,@"N9 =HNM@";M9*S G,@ 8 =;MJ,=-, P$@"<1J,7 ,=?8 WXN2,OL@'@"E:7:N9 0 SM= ,?7/(=H<,=HNMOL=LJ@XP8GTSM=UGe>A@VJ,=HGTGT7:N9 JWX<,= ,?7FJ,OH;MJ,J4=?, 7:N GTSM= PR@"E:EF@>7:N9 > !"#$% @'@"E:7:N9D7:N G,=H<TBVJ1@XP GTSM=QBV=WXN @'OLOH;C WXGT7F@"NN';B 8 =H<1@XPW + 7 [
8<4WXGT7F@"N WXEMBV@J,=C8 =LOL@"BV=LJ =W"J,7F=H< GTSM=AS7:9*SM=H< GTSM=AP<,=LK';M=HNMO"! @XP GTSM= BV@J,= PNVW@,Y,?7:GT7F@"N 7 WQS7:9*SW: 7 WXEGT<4WXCYP$<,=LK';M=HNMO"!7FJQ<,=LOL@"BDBV=HN2,=-, PR@"<PRW"JTGQ9!WXG,= @"C =H<4WXGT7F@"NMJ 3$=LK'N 3 >DJ5 5 0 SM=H<,=\P$@"<,= 77:NW JT7:N9*EF=\[7F@"N=q?C =H<T7:BV=HN'G >SM=H<,= 7:N2,?7+ 7 ,?; WXE W@,Y,?<,=LJ4JT7:N9 7FJ NM@"G WXN 7FJ,JT;M= 7 >A= GT;<TN GTSM= GT<4WXCC WX<4WXBV=HG,=H<,JAW"J S7:9*S W"J C @*J,JT78EF= 0 S7FJ 7FJ 7:N @";< OW"J4='&
Prf = 13−15 WXN2,
Vcap = 2000 7<,=LJT;E:GT7:N9V7:N<4W@,?7 WXEGT<4WXC P$<,=LK!;M=HNMOH7F=LJ
ωr@XP
2π · 4.4− 4.7p @ WXN2, WXN W:?7 WXE(P$<,=LK';M=HNMO"!
ωz = 2π · 1.715 p @ .pY@"<,=L@+!=H< 7 >A= > WXN'G GTSM= 8 =LJTG C<,@/=LOHGT7F@"N C @*J,J,78EF= @XPGTSM= @L $ NB
k[+ =LOHG,@"< @"N'G,@G,SM=VGT<4WXC
W: 7FJI7:N @"< ,=H< G,@ S WB+ =DB W:?7:B ;B OL@";CE:7:N9 3η
5QG,@ GTSM= W: 7 WXE BV@J,= 7 >S7FO4S 7FJI;MJ,=-, P$@"<IGTSM=BV@"GT7F@"N WXE K!;Y87:G 0 SM=DJTB WXE:EF=LJTG WXN9*EF=
β8 =HGe>A=L=HN @L $ NB 8 =WXB WXN2, GTSM= GT<4WXC W:?7FJ 7:N%@";<
J,=HGT;C 7FJ1W"O4S7F=?+ =-,K8&! ;MJT7:N9 GTSM=U@"CGT7FOWXEO4S WXNNM=HE@L $ * 3$O\P J,=LOHGT7F@"N ] DJ5 7 N WXBV=HE,! β = 22.5
0 S7FJ ! 7F=HE ,J W $ WXB 8?[Q7FO =IC WX<4WXBV=HG,=H<
η = 6.8%
P$@"<1GTSM= W: 7 WXEBV@J,= 0 SM= WXN9*EF=φ = 45
8 =HGe>A=L=HN @L $ NB 8 =WXBWXN2, GTSM=UB WX9*NM=HGT7FO O =HE , 7FJGTSM=HN W"O4S7F=?+!=-, 8&!DWXCCE,! 7:N9 GTSM=B WX9*NM=HGT7FO O =HE ,V7:N
x[$,?7:<,=LOHGT7F@"NF3)( )1@XP 687:9 ] *5 B WX9*NM=HGT7FO
O =HE , @XPGL ] a ! 7FJ WXCCE:7F=-, 0 SM= @L $ NMB C @"E WX<T7 WXGT7F@"N 7FJ <,@"G4WXG4=?, 7:N'G,@ GTSM=x[yCE WXNM= 7:N
@"< ,=H<VG,@ C<,@+ 7 ,=γ = 0
E:E =q C =H<T7:BV=HN'G,Jc>A=H<,= C =H</PR@"<TBV=-, 7:N W L*a BVJDE:7:NM=GT<T7:9*9 =H<,=-,C;EFJ,=1J4O4SM=HBV= 7 7:NMOHE:;2,?7:N9 JT7 ,=?8 WXN2, OL@'@"E:7:N9 @XP(GTSM=W: 7 WXE BV@J,=1G,@U9*<,@";N2, JTG4WXG,=CM<,@@8 WA87:E:7:GT7F=LJ9*<,=WXG,=H<1GTS WXN $ 4N
*,+ %- !"#. PNIGTSM= B ;E:GT7 [7F@"N OW"J4=H>A= @@8J+ 7F@";MJTE,! NM=L=-,IG,@ O4SM@'@*J,=ZW ,?7:<,=LOHGT7F@"N@XPGTSM= @L $ NB E W"J,=H< >7:GTS <,=LJTC =LOHGAG,@ GTSM=GT<4WXC W: 7FJ >S7FO4S WXE:EF@>1JZP$@"<Z7:N2,?7+ 7 ,?; WXE W@,Y,?<,=LJ,J,7:N9 0 SM=@"CGT7:B ;B PR@"<DG,S7FJV7FJ @XPUOL@";<,J,=C =H<TC =HN2,?7FOH;E WX< G,@%GTSM= 7F@"N JTGT<T7:N9]3
β = 905 0 S7FJ 7SM@>A=?+ =H< 7 OL@"<T<,=LJTC @"N2,J G,@
η = 0PR@"<VGTSM=YW: 7 WXE BV@J,=LJ >S7FO4S]>A@";E , B W =7:GV7:BDC @*J,JT78EF=
/*L >-B03+112O.)'1212B03 41 ")0" 1A'O%&031-,;'0281-, ) 0 02- %1"-.'B12%%"@-B0 + 06 )02P%& +B ' )'12)$ ' >' &)$ B139,%N$ "B13021A 25
)BL*g
G,@R,?<T7+!=VJT7 ,=?8 WXN2,%GT<4WXNMJ,7 G,7:@"N J WXN2,%=HNMOL@J,= Wb+'78<4WXGT7F@"N WXEAK!;Y87:G 7:N'G,@@"NM= @XP GTSM=LJ,=VBV@J,=LJ @"NMJ,=LK';M=HN'GTE,! 7 WUOL@"BDC<,@"BD7FJ,=1B ; JTG<8 =B W@,= PNV@";< C<,=LJ,=HN'G J4=HGT;C 7 GTSM= WXN9*EF= 8 =HGe>A=L=HN GTSM=W@,Y,?<,=LJ,JT7:N9 @L $ NB O4S WXNNM=HE 3 @L $ 5QWXN2,YGTSM= GT<4WXC%W: 7FJQ7FJ
β = 67.5 7 >S7FO4S 7FJ ,=OMNM=-, 8&!GTSM= OH@"N JTGT<T;MOHGT7F@"N%@XP GTS = +*W"OH;;B +!=LJ,J,=HE P N @"< ,=H<UG,@ OL@"BDCE,!M>7:GTS
φ = 45>A=DWXCCE,! GTSM=
B WX9*NM=HGT7FO O =HE ,b,?7 WX9 @"N WXE:E,! >7:GTS GTSM=QSM=HE:C@XP8OL@"7:E ,E3)( L @XP6 7:9 ] *5 /*;MJTG1W"J 7:N G,SM= JT7:N9*EF=7F@"N%JTG4WXN2,MWX< ,%OL@"N OM9*;<4WXGT7F@"N 7 GTSM= @L $ NB C @"E WX<T7 WXGT7F@"N E:7F=LJU7:N GTSM=
x[yCE WXNM= 3
γ = 05
B WX9*NM=HGT7FO1O =HE , @XP a ! 7FJ WXCCE:7F=-, J ,?7FJ,OH;MJ,J,=-, 7:N J,=LOHGT7F@"Ndg ) 7 >7:GTS W Ge>A@X[7F@"N OH<#! J,G4WXEGTSM= C6 GT<4WXC ,?<T7+!= C @>A=H< S W@, G,@M8 = <,=-,?;MOL=-, G,@
Prf = 12 PR@"< JTG4WA8EF= GT<4WXCC7:N9 6 @"<
GTSM= J4WXBV= <,=W"J,@"N 7:G >W"JUW@, +*WXN'G4WX9 =L@";MJUG,@ <,=-,?;MOL= GTSM= =HN2, [ OWXCE+!@"E:G4WX9 = G,@Vcap = 1000
0 SM=LJ,= +"WXE:;M=LJ ! 7F=HE , WXN W:?7 WXE p P$<,=LK!;M=HNMO"!
ωz@XP
2π · 1.215 p @13RW"J=q C =LOHG,=-, P$<,@"B GTSM=√Vcap
J,OWXE:7:N9J5 WXN2, W <4W@,?7 WXE p P<,=LK';M=HNMO"!ωr
@XP2π · 4.35 p @ .* @"G,=QGTS WXG 7:N @";< GT<4WXCGTSM==,=H9 =HNM=H<4W"O"!R8 =HGe>A=L=HNYGTSM=IG!>A@V<4W@,?7 WXE p BV@J,=LJ7FJ1E:7 P$G,=-, 0 SM=
ωr+"WXE:;M=LJ 9*7+!=HN SM=H<,=
<,=HC<,=LJ,=HN'G WB+ =H<4WX9 = +*WXE:;M=LJ G WXN&! <4WXG,= 7 GTSM=VC<,=LOH7FJ,= C @*JT7:GT7F@"N @XPGTSM=V<4W@,?7 WXEZBV@J,=LJ 7FJINM@"G<,=HEF=?+"WXN'G1P$@"<1@";< =q?C =H<T7:BV=HN'G,J JU@";GTE:7:NM=-, 7:N%J,=LOHGT7F@"N g 7 7:G 7FJIW@, +*WXN'G4WX9 =L@";MJIG,@ ;MJ,= GTSM= + 78<4WXGT7F@"N WXE BV@J,=D@XPAS7:9*SM=LJTGC @*J,JT78EF=IP$<,=LK!;M=HNMO"!YW"JGTSM= K'; WXN'GT;B 8; J 7 J,=L= =LK'N 3 >DJ5 GGTSM= J4WXBV= GT7:BV= 7 7:GS W"JQ8 =L=HNWX<T9*;M=-,YWA8 @+!=UGTS WXG @"NE,!W:?7 WXE BV@J,=LJQWX<,= WXE:EF@>A=-, @"NMJ,=LK';M=HN'GTE,! 7 GTSM=US7:9*SM=LJ,GQW: 7 WXE BV@J,=>7:E:ET8 =IO4SM@*J,=HNPR@"<1GTSM=IK'; WXN'GT;B 8;MJ 7 >S7FO4S 7:NGTS =UGe>A@X[7F@"N OW"J,=U7FJ1GTSM= 95/-1-3 ; % '< 11WXGW P$<,=LK!;M=HNMO"!
ωb =√3ωz 7 SM=HNMOL=
ωb = 2π · 2.104 p @ . OLOL@"< ,?7:N9 G,@ =LK'N 3 *a65 GTSM= $ WXB 8?[Q7FO!=PRW"OHG,@"< @XP GTSM=Q8<,=WXGTS7:N9 BV@J,=P$@"<WXN =q?OH7:G4WXGT7F@"NGTS<,@";9*S GTSM=IW@,Y,?<4=HJ4JT7:N9 O4S WXNNM=HE 7FJGTSM=HN
η = 1.8% .
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β
%=IS WB+!=UC<4=+ 7F@";MJTE,!@@8MJ,=H< +!=-,GTS WXGJT7 ,=?8 WXN;, OL@'@"E:7:N9 @XP W1+ 78<4WXGT7F@"N WXE BV@J,= SM=WXG,J ;C GTSM=@"GTSM=H< BV@J,=LJ 8 ! <,@";9*SME ! @"NM= CSM@"NM@"N =W"O4S ' ] L ) 6M@"< @";< K'; WXN'GT;B OL@"BDC;G WXGT7F@"N =q?C =H</[7:BV=HN'G,J 7 @"CGT7:B ;B 9*<,@";N2, JTG4WXG,= OL@'@"E:7 NM9 3$NM=WX<b)Ba*a6N15 7:J <,=LK!;7:<,=-, @"NE,!%PR@"< GTSM= K'; WXN'GT;B8;MJ 7 SM=HNMOL=1P$@"< GTSM= 8<,=WXGTS7:N9 BV@[,M= 6M@"< WXE E @"GTSM=H<ZBV@J,=LJ 7 BV=WXN CSM@"NM@"N N';B 8 =H<,JA@XP ;C G,@n = 10
WX<,= G,@"EF=H<4WA8EF=G3RO P J4=LOHGT7F@"N L L65 0 SM=H<,=\PR@"<,= 7 7:N GTSM=ZC;EFJ,=AJ,=LK!;M=HNMOL=AJ,7,M=8 WXN2, OL@'@"E:7:N9> W"J OM<,J,G1C =H</PR@"<TBV=-, @"N GTSM= W: 7 WXE p BV@J,=IP$@"< D BVJ ! 7F=HE ,?7:N9 WDBV@"GT7F@"N WXE 9*<,@";N2, JTG4WXG,=C<,@@8 WA87:E:7:G! @XPWA8 @";G$*a6N * =q?G 7 JT7 ,=?8 WXN;, OL@'@"E:7:N9 @"N GTSM= 8<,=WXGTS7:N9BV@J,=1> W"JIWXCCE:7F=-,P$@"< DBVJ1<,=LJ,;E:GT7:N9 7:N WD9*<,@";N2,YJTG4WXG,=IC<,@@8 WA87:E:7:G!9*<,=WXG,=H<GTS WXN $ cN J1BV=HN'GT7F@"NM=-, 7 GTS7FJSM=WXG,J GTSM=W: 7 WXE p BV@J,= G,@
n ≈ 1+* @"G,=1GTS WXG GTSM=1<4W@,?7 WXE BV@J,=LJ 3<4W@,?7 WXE p 7 <,@'O'7:N9J5
WX<,= WXGn ≈ 5
,?;M= G,@ @"CCEF=H< OL@'@"E:7:N9
)BL
)BL
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: 1B!0'6& ;' =H<,J,=H;MJ ( @'@! J 7 C =W@,?7:N9 7 p W"J,J 3 )%$ $ 65 ' ) C 6M="! NB WXN A; % 6,3 ; >2; 52C.;* '% 46&?15/2 PN'G / 0 SM=L@"< S&! J ED,F 7 D[g c3 )%$ *L65 ' $ ) C 6 ="!'NB WXN ) 6 3 6&% % 1 37; 3 '% 6#?15/2 6M@";N2,MWXG,7:@"N J @XP S&!?JT7FOLJ FHG 7 ] a 3 )%$ *g65 ' )Ba ) =H;G,J,O4S ) 6831 6 % 81 '/"JI 81 # 6 / B8(6 /; 4/; "; - 1:3 < 81 6&7;%!>15/2"3 "6 3 6&%
'% 6#?15/ <,@'O C @'O $ @"N2,@"N K+-L-L 7 $*gR3 )%$ ] 5 ' )*)5) =H;G,J,O4S ) 6 3 6&% '% 46& 3 ;' 3 1M '/ 2 <,@'O C @'O $@"N2,@"N N+,DPO 7 3 )%$ $65 ' )BL ) @ (A=HNNM=HGTG WXN2,Q0 (A<4W"J,J4WX< , 679"; 1B < ; 2 /; 956# ;*' 3 < '; '>22; <,@'O P RR
P N'G,=H< @"N?P @"N @"BDC;G,=H<,J 7 &!?JTG,=HBVJ WXN2, '7:9*N WXE <,@'OL=HJ4JT7:N9VC WX9 =HJ=) ]/S )!$ 3 )%$ >DJ5 ' ) )Lp * 7F=HEFJ,=HNWXN2, P $ S'; WXN9 ) 6831 6 % # '% 6& 3 ;*' 3 <T) 6 3 6&% '/%:3 ;'
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O,F 7 )Ba ) ] 3 )%$ $ ] 5
)BL $
' ) ) =H;G,J,O4S%WXN2, C @*J W : 3 ; < 2'6# ;*' ' / ' 9" 1% 2 9" "6 3 6 % '% 6& 3 ;*' <,@'O C @'O $@"N2,@"N K+ 7 ]*] b3 )%$ $*L65
' )%$ )BC EF=?+ = 7 B !=H<TG 7 p W"OLOH7 WB+ =HE:EF@ 7 WXN2,Jp p @*J,OW ) 6 3 6&% 3 '/;* % 2 /-1"!; 25; ?1-< <,@'O C @'O $@"N2,@"N K+,O+ 7 $b3 )%$ $ 65
' L*a )B$ 0Q<,@+!=H< "3>2 "6 3 6&% % 1 3A; 3 3 ' /;* % ' / < 3 3 9-3>2 1 2 1-3/ <,@'O L XGTS NN'; WXE p &!'BDC @*JT7:;B@"N G,SM=G0 SM=L@"<#! @XP @"BDCM;GT7:N9 7 C WX9 ==L )BLR3 )%$ $*g65
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' L*L )BC $ C 7+ =LJTG 7 B 'S WXBD7:< 7 WXN2, $ p ,?EF=HB WXN % 1 &'<+' '&9 3; 7;2 <; ; 3.2; 3 6&/-1523 < 679"; 1B "/" '>2" 2?1% 2 @"BDB p D F 7 )BL*aR3 )%$ 65
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' L $ ) = * W*WXB ;<4W 7 = W"JTS '7:N 7 WXN2, 7 0 J4WX7 # ' 815/1 ' /(' ' %:3 "/('2 ' ; "6 3 6&%2 3?12 ;2 3:2; 1 # ' ' 715/ 3; / 9 ' * WXG,;<,= PV 7 *gR3 )%$ $ $65
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' )5) p E WXGLB WXN WXN2,]p P ! B WXN ) 6 3 6&% '% 6# ;21.;* 1" 1 /(' 2 '3 ; ' ;"6&; < 1";6&% OH7F=\N OH= DPV+ 7 )%$*g b3 )%$ $ $65
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UE+ 7 D[a $ )c3 )%$ $ ] 5
) *a
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' *g ).P $ S'; WXN9 7 *B 0 =H<,J,SM=HN O =HE , 7 WXN2,p ;Y87:NM=LO ; 715/5;2% 1 3 ; % ,1% 11 3 ;*' ' 3>2"683 6 % 2 1-3/ ;2 S&! J C =?+ $=HGTG PV-L 7 >D[a R3 )%$ $ 65
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) )
' ] a ) ?W"O =HGTG 7 7F=HE:C7:NMJ 7 7 ( 7:N9 7 $ WXN9 =H< 7 p ="!!=H< 7 7 p !!WXGTG 7 p C @>A= 7 B0 ;<,O4SM=HGTG,= 7 p PG4WXNM@ 7 7 7:NM=HE WXN2, 7 WXN2, p @"N<,@'= ; 715/5; %:1 3%1 3 1% 11 ' )'6 / 3/ ; ,152 * WXGT;<,= +-L+ 7 L ] gR3 L*a*a*a65
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' ] ) =H;G,J,O4S 7 (WX<,=HNMOL@ 7 WXN2, E =H<TG 7; !15/2"3; M ; "6 3 6&% '% 6# 3 ;*' <,@'O C ?@!O $@"N2, +P+ 7 )%$ b3 )%$ $ ] 5
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