Development and study of low noise laser diodes emitting at 894 nm for compact cesium atomic clocks

HAL Id: tel-01683150

https://tel.archives-ouvertes.fr/tel-01683150

Submitted on 12 Jan 2018

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Development and study of low noise laser diodes

emitting at 894 nm for compact cesium atomic clocks

Nicolas von Bandel

To cite this version:

Nicolas von Bandel. Development and study of low noise laser diodes emitting at 894 nm for compact cesium atomic clocks. Electronics. Université Montpellier, 2017. English. �NNT : 2017MONTS003�. �tel-01683150�

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h?2 Tm`TQb2 Q7 Kv S?. i?2bBb Bb T`2+Bb2Hv iQ //`2bb i?Bb M22/ M/ iQ /2p2HQT bm+?  bQm`+2 7Q` i?2 TTHB+iBQM- 7`QK i?2 BMBiBH bi`m+im`2 Q7 i?2 Hb2` +pBiv iQ Bib BM/mbi`BH T+F;BM;X "2vQM/ i?i- Kv T`2b2Mi rQ`F HbQ BMp2biB;i2b i?2 i?2K2 Q7 i?2 bT2+i`H Tm`Biv Q7 i?2b2 bQm`+2b- r?B+? Bb HBKBi2/ #v MQBb2 T?2MQK2M QM Bib QmiTmi 7`2[m2M+vX h?Bb iQTB+ Bb i+FH2/ #Qi? QM Bib i?2Q`2iB+H bT2+ib M/- KQ`2 +`m+BHHv- QM i?2 rv iQ +?`+i2`Bx2 2tT2`BK2MiHHv i?Bb T`K2i2`X AMbB;?ib QM i?2 Hb2` /2bB;M M/ ?Qr iQ BKT`Qp2 i?2 bT2+i`H [mHBiv Q7 i?2 bQm`+2 `2 ;Bp2MX 6BMHHv- Bi H2/b K2 iQ T`QTQb2  KQ/2H iQ T`2/B+i i?2 Hb2` 7`2[m2M+v +?`+i2`BbiB+b r?2M Bi Bb +iBp2Hv +QMi`QHH2/ BM i?2 iQKB+ +HQ+F bvbi2KX

h?2 KMmb+`BTi Bb /BpB/2/ BM i?`22 /BbiBM+iBp2 T`ib //`2bbBM; /Bz2`2Mi iQTB+b,

S`i A , Gb2`b 7Q` +2bBmK +HQ+Fb, BM/mbi`BH `2[mB`2K2Mib

h?2 2tT2`BK2Mib /2b+`B#2/ BM i?Bb T`i ?p2 #22M K/2 BM +QHH#Q`iBQM rBi? i?2 iQKB+ *HQ+F a2`pB+2 Q7 h?H2b 1H2+i`QM .2pB+2bX

h?2 }`bi T`i +QKT`Bb2b  mMB[m2 +?Ti2`X *?Ti2` R Bb  b?Q`i BMi`Q/m+iBQM iQ i?2 T?vbB+b Q7

7`2[m2M+v biM/`/b M/ Q7 i?2 +2bBmK iQKB+ +HQ+FX Ai T`2b2Mib i?2 KBM bm#@bvbi2Kb Q7 i?Bb b2imT M/ i?2 bT2+B}+iBQMb bF2/ 7Q` i?2 Hb2` bQm`+2X 6`QK i?i- Bi Bb DmbiB}2/ i?i b2KB+QM/m+iQ` BM@THM2 Hb2`b 2H2+i`B+HHv@TmKT2/ `2 r2HH bmBi#H2 7Q` i?2 TTHB+iBQMX h?2 +m``2Mi bii2@Q7@i?2@`i Bb `2pB2r2/ M/ i?2 BM/mbi`BH T`QD2+i HBMF2/ iQ i?Bb rQ`F Bb T`2b2Mi2/X

k *QMi2Mib

h?2 `2bmHib Q7 i?Bb T`i `2 `2Hi2/ iQ i?2 2M;BM22`BM; /2p2HQTK2Mib Q7 i?2 Hb2` KQ/mH2b /QM2 rBi?BM i?2 BM/mbi`BH H#Q`iQ`v Q7 h?H2b@LQFB AAA@o G#X

h?2 b2+QM/ T`i ?b irQ +?Ti2`bX *?Ti2`k/`rb ;2M2`H +QMbB/2`iBQMb QM Hb2` +pBiB2b- i?2

b2KB+QM/m+iQ` ;BM K2/BmK- M/ ?Qr  bBM;H2@KQ/2 `2;BK2 Bb 2bi#HBb?2/X h?2 KBM /2bB;M T`K2@ i2`b Q7 i?2 bQm`+2 `2 T`2b2Mi2/ M/ i?2B` HBMF rBi? i?2 7mM/K2MiH Hb2` HBM2rB/i? Bb K/2X *?Ti2`

j ?M/H2b HH i?2 T`+iB+H bT2+ib Q7 i?2 b2KB+QM/m+iQ` Hb2` /2bB;M, i?2 2H2+i`B+H M/ QTiB+H +QM@

}M2K2Mi rBi? i?2 p2`iB+H /2bB;M Q7 i?2 b2KB+QM/m+iQ` Hv2`b M/ i?2 Hi2`H M/ HQM;Bim/BMH QTiB+H KQ/2 b2H2+iBQM rBi? i?2 `B/;[email protected]`B#mi2/ 622/#+F bi`m+im`2X AMbB;?i Bb ;Bp2M QM i?2 T`K2i2`b i?i /`Bp2 i?2 Hb2` HBM2rB/i? BM i?2b2 +QKTQM2MibX h?2 i2+?MQHQ;B+H T`Q+2bb Q7 7#`B+iBQM Q7 i?2 /2pB+2b Bb T`2b2Mi2/X h?2M-  7mHH }`bi@Q`/2` 2tT2`BK2MiH bim/v Q7 i?2B` +?`+i2`BbiB+b Bb `2HBx2/X

S`i AAA , Gb2` MQBb2 bim/v, 7`22@`mMMBM; M/ BM@HQQT +QM};m`iBQMb

h?2 rQ`F T`2b2Mi2/ BM i?Bb T`i ?b #22M +QM/m+i2/ mM/2` i?2 i?2Q`2iB+H M/ 2tT2`BK2MiH bmT2`pBbBQM Q7 i?2 qp2b  h2`?2`ix ;`QmT Q7 i?2 AMbiBimi2 7Q` 1H2+i`QMB+b M/ avbi2Kb UlMBp2`bBiv Q7 JQMiT2HHB2`VX

h?2 i?B`/ T`i ?b i?`22 +?Ti2`bX *?Ti2` 9//`2bb2b i?2 bT2+B}+ Bbbm2 Q7 i?2 2H2+i`B+H MQBb2X

lb27mH MQiBQMb mb2/ iQ /2b+`B#2 [mMiBiiBp2Hv i?2 MQBb2 BM i?2 Hb2` T`K2i2`b `2 ;Bp2M }`biX h?2M-i?2 mbmH 2H2+i`B+H MQBb2 bQm`+2b `2 T`2b2Mi2/- b r2HH b h?2M-i?2 2tT2`BK2MiH T`QiQ+QHb iQ K2bm`2 i?2b2 ~m+imiBQMbX J2bm`2K2Mib `2 T2`7Q`K2/ QM i?2 Hb2` KQ/mH2b 7#`B+i2/ M/ `2 +QKT`2/ iQ Qi?2` +QKK2`+BH bQm`+2bX h?2 BKT+i Q7 i?2 BMi`BMbB+ 2H2+i`B+H T2`im`#iBQMb Q7 i?2 /2pB+2b QM i?2

7`2[m2M+v MQBb2 Q7 i?2 QmiTmi #2K `2 /Bb+mbb2/X *?Ti2` 8Bb /2/B+i2/ iQ i?2 QTiB+H MQBb2

bim/v-#Qi? QM  i?2Q`2iB+H M/ 2tT2`BK2MiH TQBMi Q7 pB2rX  M2r K2i?Q/ iQ +?`+i2`Bx2 i?2 7`2[m2M+v MQBb2 BM M #bQHmi2 KMM2`- mMHBF2 i?2 i`/BiBQMH K2i`QHQ;v- Bb #`Qm;?i iQ HB;?iX h?2 HBKBiiBQMb

Q7 i?2 Hb2`b 7#`B+i2/ BM i2`Kb Q7 QTiB+H MQBb2 `2 `2p2H2/X 6BMHHv- *?Ti2` e BMp2biB;i2b i?2

Hb2` MQBb2 T`QT2`iB2b BM  b2`pQ BM Q`/2` iQ bi#BHBx2 i?2 7`2[m2M+v QM  +2bBmK i`MbBiBQM HBM2 BM i?2 iQKB+ +HQ+F X h?2 +T#BHBiB2b Q7 i?2 2H2+i`QMB+ 722/#+F iQ bmTT`2bb i?2 Hb2` 7`2[m2M+v ~m+imiBQMb `2 /Bb+mbb2/X 6BMHHv- i?2 HBMF #2ir22M i?2 Q`B;BMH Hb2` bT2+B}+iBQMb M/ r?i Bb `2HHv M22/2/ BM i2`Kb Q7 bT2+i`H Tm`Biv 7Q` i?2 TTHB+iBQM Bb +H`B}2/X

1TBHQ;m2

h?2 p2`v 2M/ Q7 i?Bb KMmb+`BTi T`2b2Mib i?2 KQbi mT@iQ@/i2 `2bmHib Q7 i?2 iQKB+ +HQ+F T2`7Q`@ KM+2b rBi? i?2 Hb2` /2p2HQT2/ BM i?Bb bim/vX

S`i A

Gb2`b 7Q` +2bBmK +HQ+Fb, BM/mbi`BH

`2[mB`2K2Mib

*?Ti2` R

AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v

TTHB+iBQMb

RXR :2M2`HBiB2b QM iBK2 M/ 7`2[m2M+v K2i`QHQ;v

Hi?Qm;? i?2 +QM+2Ti Q7 iBK2 Bb [mBi2 BMimBiBp2 BM Qm` 2p2`v/v HB72 2tT2`B2M+2- Bi Bb +imHHv [mBi2 ?`/

iQ /2}M2 BM i?2 7`K2 Q7 i?2Q`2iB+H T?vbB+b- #mi Bi Bb HbQ /B{+mHi iQ K2bm`2 T`2+Bb2Hv (.BK`+[ kyR8)X

AM/22/- i?2 722HBM; Q7 iBK2 TbbBM; #v Bb mbmHHv ;Bp2M #v i?2 Q#b2`piBQM Q7 i?2 +iBQM Q7 M 2p2Mi- 7Q` BMbiM+2- i?2 bM/ ~QrBM; BM M ?Qm`;HbbX h?2 i?BM; #Qmi i?2 KQp2K2Mi BM bT+2 Q7 bm+? M 2p2Mi Bb i?i Bi ;2M2`i2b  [mMiBiv Ĝi?2 bM/ BM i?2 HQr2` T`i Q7 i?2 ;HbbĜ i?i ;`Qrb HBM2`Hv- bQ i?i i?2 bbQ+Bi2/ iBK2 K2bm`2K2Mi mMBi Bb /2}M2/ b,

∆Q = k× ∆t URXRV

r?2`2 Q Bb i?2 KQmMi Q7 i?2 [mMiBiv Q#b2`p2/ Ĝ?2`2 i?2 Kbb Q7 bM/Ĝ M/ k  +Q2{+B2Mi i?i +QMp2`ib i?2 Kbb iQ  iBK2 K2bm`2K2Mi- i?2 KQbi +QKKQM QM2 #2BM; i?2 a2+QM/X h?2 2[mBpH2Mi +QM+2Ti 7Q` /BbiM+2 K2bm`2K2Mi rQmH/ #2  `mH2`- rBi? ;`/miBQMb 2[mHHv bT+2/ /2}MBM;  /BbiM+2@b+H2X

>Qr2p2`- bm+?  /2}MBiBQM Bb MQi p2`v T`+iB+H B7 mb2/ rBi? Ki2`BH Q#D2+ib b Bi rQmH/ `2[mB`2 iQ Q#b2`p2  ;`QrBM; Ki2`BH [mMiBiv iQ BM}MBivX  KQ`2 mb27mH iBK2 K2bm`2K2Mi i2+?MB[m2 +M +2`iBMHv #2 ;`QmM/2/ QM i?2 Q#b2`piBQM Q7  T2`BQ/B+H T?2MQK2MX  p2`v bBKTH2 2tKTH2 Q7 i?i Bb  T2M/mHmK- Q` i?2 Hi2`MiBM; #2ir22M /v M/ MB;?iX AM i?i +b2- i?2 iBK2@b+H2 rBHH #2 i?2 /m`iBQM #2ir22M i?2 irQ KQK2Mib r?2M i?2 bvbi2K Bb BM i?2 bK2 bii2X PM2 +M i?2M /2}M2 

T2`BQ/ T M/  7`2[m2M+v f- HBMF2/ #v i?2 `2HiBQM,

f = 1/T URXkV

A7 T Q` f M2p2` p`B2b b i?2 bvbi2K 2pQHp2b- QM2 +M bv i?i Bi Bb  T2`72+i ǶiBK2@`mH2`Ƕ Q7 iBK2@ ;`/miBQM T X  b?Q`i2` T - Q` 2[mBpH2MiHv-  ?B;?2` f rBHH T`QpB/2  KQ`2 T`2+Bb2 iBK2@+QmMi2` U};m`2

URXRVVX h?Qb2 p2`v bBKTH2 +QMbB/2`iBQMb H2/ mb iQ i?2 +QM+HmbBQM i?i  Ƕ;QQ/ +HQ+FǶ b?QmH/ ?p2 i?2

7QHHQrBM; [mHBiB2b U};m`2 URXkVV,

Ç ++m`+v U2t+iBim/2V, Bi Bb i?2 #BHBiv Q7 i?2 +HQ+F iQ ;Bp2  T2`BQ/ Q` 7`2[m2M+v 2[mH iQ i?2 QM2 +?Qb2M b iBK2@/2}MBiBQM rBi?Qmi Mv bvbi2KiB+ #Bb

Ç ai#BHBiv Ubi#BHBiûV, Bi Bb i?2 7+i i?i i?2 T2`BQ/ Q` i?2 7`2[m2M+v Q7 i?2 +HQ+F F22Tb i?2 bK2 pHm2 Qp2`  ;Bp2M Q#b2`piBQM iBK2

Ç _2T2i#BHBiv U`2T`Q/m+iB#BHBiûV, r?2M i?2 T2`BQ/ Q` 7`2[m2M+v Q7 i?2 +HQ+F Bb +QMbBbi2Mi Qp2` `2T2i2/ 2tT2`BK2Mib Ui?i Bb- BM/2T2M/2Mi Q7 i?2 QT2`iQ`V

e *?Ti2` RX AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v TTHB+iBQMb !"#$ % "& ' ( ) _* + *_,

6B;m`2 RXR, _2T`2b2MiiBQM Q7 irQ T2`BQ/B+ bB;MHb rBi? T1= 4T2X aB;MH BM `2/ Bb  #2ii2` iBK2@`mH2`

#2+mb2 Bi ?b  ?B;?2` 7`2[m2M+v +QKT`2/ iQ i?2 #Hm2 QM2X ! "!#$%& $'! ()! #**'+#!& , , -! .**'+#!& #(/ 0!#$%& , , -! 1)! 0!#$%& #(/ ()! #**'+#!& , , -! .**'+#!& 2)( #3&+#4&5 $'! ()! 0!#$%& , ,

-6B;m`2 RXk, .Bz2`2M+2 #2ir22M ++m`+v M/ ai#BHBivX 7i2` M BHHmbi`iBQM Q7 CQ?M oB; `2T`Q/m+2/

BM (.BK`+[ kyR8)

6`QK M ?BbiQ`B+H TQBMi Q7 pB2r- i?2 }`bi BMbi`mK2Mib mb2/ iQ K2bm`2 iBK2 r?2`2 #b2/ QM K2+?MB+bX h?2 QrM `QiiBQM Q7 2`i? rBi? `2HiBQM iQ i?2 bmM /2}M2b i?2 bQH` /v- r?B+? Bb T`2+Bb2Hv i?2 /m`iBQM #2ir22M i?2 irQ KQK2Mib i?2 bmM BHHmKBMi2b i?2 bK2 TQBMi QM 2`i? BM i?2 bK2 /B`2+iBQM Ui?2 K2`B/BMVX am+? iBK2@b+H2 Bb MQi p2`v T`+iB+H b ?mKM +iBpBiB2b iF2 TH+2 BM Km+? bKHH2` iBK2@`M;2bX A7 QM2 /v Bb /BpB/2/ BM k9 ?Qm`b- r?B+? BM im`M ++QmMib 7Q` 3e9yy b2+QM/b- QM2 KB;?i ?p2 7QmM/  /2}MBiBQM Q7 i?2 Hi2` mMBi b+H2- T`QpB/2/ i?i i?2 bQH` /v Bb `2T2i#H2 M/ bi#H2X

h?Bb Bb +imHHv MQi i?2 +b2X AM/22/- #2+mb2 i?2 2`i? `Qii2b `QmM/ i?2 bmM BM M 2HHBTiB+H Q`#Bi- i?2 /m`iBQM Q7 i?2 /v Bb MQi i?2 bK2 i?`Qm;?Qmi i?2 v2`X HbQ- i?2 tBb Q7 `QiiBQM ?b  bKHH iBHi- b r2HH ?b  T`2+2bbBQM KQp2K2Mi UT`2+2bbBQM Q7 i?2 2[mBMQt2bV r?B+? KF2b i?2 /v H2M;i? ~m+imi2 bHQrHv U72r ms T2` v2`VX h?2 bQHmiBQM Bb i?2M iQ /2}M2 i?2 /v b M KQmMi Q7 iQiH M;H2 Q7 i?2 2`i? `QiiBQM QM Bib2H7X "mi i?Bb [mMiBiv Bb MQi HBM2`X 1`i?[mF2b- iB/2b M/ KMv Qi?2` T?2MQK2M KQ/B7v i?2 M;mH` bT22/ bQ i?i Bi HbQ p`B2b bHQrHv BM iBK2X PM2 rQmH/ i?2M +QMi2KTHi2 iQ ;`QmM/ i?2 /2}MBiBQM Q7 i?2 b2+QM/ QM M p2`;2 /v iBK2 Q#b2`p2/ Qp2`  v2`X h?Bb Bb +imHHv ?Qr Bi rb /2}M2/ iBHH RN8eX >Qr2p2`- i?Bb rb MQi +QKTH2i2Hv biBb7+iQ`v b H2T b2+QM/b ?p2 `2;mH`Hv iQ #2 //2/ iQ i?2 mMBp2`bH iBK2X q2 b?QmH/ HbQ F22T BM KBM/ i?i i?MFb iQ i?2 _2HiBpBiv Q7 1BMbi2BM- r2 FMQr i?i i?2 iBK2b+H2 Bb `2HiBp2 iQ i?2 Q#b2`p2`- bQ 7i2` HH- i?2 /2}MBiBQM Q7  iBK2b+H2 Bib2H7 Bb `#Bi``vX h?Bb /Q2b MQi bQHp2 i?2 Bbbm2 Q7 i?2 bi#BHBiv Q7 i?Bb

RXRX :2M2`HBiB2b QM iBK2 M/ 7`2[m2M+v K2i`QHQ;v d

/2}MBiBQM Qp2` iBK2 BM  ;Bp2M T?vbB+H 7`K2X

h?2 }`bi bi2T iQ +?B2p2 #2ii2` iBK2 K2bm`2K2Mi Bb iQ mb2  7bi2` T2`BQ/B+H T?2MQK2MQM- rBi?  ?B;?2` 7`2[m2M+vX h?2 bvbi2Kb i?i ;Bp2b ivTB+HHv bm+? T`K2i2`b `2 Qb+BHHiQ`b #b2/ QM +`vbiH pB#`iBQMb UbQmM/ rp2bV Q` Mv Qi?2` 2H2+i`Q@K;M2iB+ bvbi2K 7`QK `/BQ@7`2[m2M+B2b U_6V iQ Hb2`

HB;?i U};m`2 URXjVVX ! !"# <sub>!"</sub>$ <sub>!"</sub>% <sub>!"</sub>!& <sub>!"</sub>!' ( )*+,-./+-0 12+/00-3142 56*.7808)9 :*0*23/-. ;17/*2<<<= *0*+341>)*+,-./+-0 12+/00-3142 5?@?A9 B8-43C<<<= *0*+341./+ 12+/00-3142 5?/+41D-E* F*.*4-31429 GH +/4+8/32<<<= I,131./+ 12+/00-3142 5J-2*42= 6B;m`2 RXj, 6`2[m2M+v@b+H2b ;Bp2M #v b2p2`H ivT2b Q7 Qb+BHHiQ`b

lM7Q`imMi2Hv- bm+? Qb+BHHiQ`b `2 mbmHHv ?B;?Hv b2MbBiBp2 iQ i?2 T`K2i2`b i?2v `2 #mBHi QM b r2HH b QM 2ti2`MH T2`im`#iBQMb UH2M;i? Q` Kbb ~m+imiBQMb- i?2`KH 2MpB`QMK2Mi- ;BM;XXXVX >2`2 +QK2b i?2 T`BM+BTH2 Q7 i?2 iQKB+ +HQ+F- r?B+? Bb iQ mb2 #bQHmi2 iQKB+ `272`2M+2b QM r?B+? i?2 bvbi2K Bb 7Q`+2/ iQ Qb+BHHi2- bQ i?i Bib 7`2[m2M+v rBHH- BM i?2Q`v- M2p2` p`vX >pBM;  /22T2` HQQF

!"#$%%&'!( )*+,-./0 1,23-4445 6137.-89:72 &'!;$#<$!=$# (6>6(6=#6 6_? 6_@ 𝜈 𝜈₀ 78--37.:89 𝜈 𝜈 ℎ𝜈0 ,.8A:7 2:B9,1 >-3*+397C 8+.D+. 𝜈 𝜀 ∼ 𝜈 − 𝜈₀ ℐ

6B;m`2 RX9, S`BM+BTH2 Q7 i?2 iQKB+ +HQ+F,  HQ+H Qb+BHHiQ` Bb 7Q`+2/ iQ Qb+BHHi2 QM M #bQHmi2 `272`2M+2- r?B+? Bb i?2 BMi2`@bT+2 2M2`;v Q7 irQ iQKB+ Q` BQMB+ H2p2Hb

BMiQ i?2 iQKB+ +HQ+F K+?BM2`v U};m`2 URX9VV- QM2 }M/b M 2KBiiBM; Qb+BHHiQ` Q7 7`2[m2M+v ν- mbmHHv

 Zm`ix rQ`FBM; BM i?2 _6 /QKBM Q`- TQbbB#Hv BM i?2 7mim`2-  Hb2` 2KBiiBM; BM i?2 h>x bT2+i`mKX PM i?2 Qi?2` bB/2- i?2 2M2`;v bT+BM; #2ir22M irQ bii2b Q7 i?2 iQKB+ Q` BQMB+ bT2+B2b i?i Bb mb2/

b `272`2M+2 Bb HBMF2/ iQ M [mMiBx2/ 2H2+i`QK;M2iB+ 2M2`;v Q7 7`2[m2M+v ν0- ++Q`/BM; iQ [mMimK

K2+?MB+b (hMMQm/DB RNdj),

hν0 = E2− E1 URXjV

r?2`2 h Bb i?2 SHM+F +QMbiMi- Ei i?2 2M2`;v H2p2H M/ |ii i?2 +Q``2bTQM/BM; [mMimK bii2X

h?2M-i?2 iQKB+ bvbi2K Bb 2t+Bi2/ r?2M ν ≈ ν0 M/ ?b  `2bQMMi `2bTQMb2 rBi? i?2 BMTmi 7`2[m2M+v

νX h?Bb T`BM+BTH2 Bb mb2/ BM bT2+i`Qb+QTv iQ K2bm`2 2M2`;v H2p2Hb BM iQKB+ bT2+B2b M/ ;Bp2b i?2

r2HH@FMQrM ~mQ`2b+2M+2 2KBbbBQM Q` #bQ`TiBQM HBM2bX b M iQKB+ 2t+Bi2/ bii2 /2+vb- T?QiQMb +Q``2bTQM/BM; iQ i?2 /Bz2`2M+2 BM 2M2`;v H2p2Hb rBi? i?2 ;`QmM/ bii2 `2 2KBii2/ M/ /2i2+i2/X q?i2p2` i?2 Mim`2 Q7 i?2 [mMiBiv i?i T2Fb rBi? i?2 BMTmi 7`2[m2M+v ν- Bi HHQrb iQ TQBMi Qmi

3 *?Ti2` RX AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v TTHB+iBQMb M #bQHmi2 +QMbiMi ν0- FMQrM iQ M2p2` ~m+imi2 Qp2` iBK2 UmMH2bb i?2 7mM/K2MiH +QMbiMib

Q7 i?2 lMBp2`b2 `2 MQi `2HHv +QMbiMi- r?B+? ?b M2p2` #22M T`Qp2/ iQ i?Bb /i2VX 6QHHQrBM; i?i-M /2[mi2 2H2+i`QMB+b rBHH HHQr iQ ;2M2`i2 i?i-M 2``Q` bB;MH ε 7`QK i?2 T`2pBQmb 7mM+iBQM i?i

+Q``2+ib i?2 Qb+BHHiQ` 7`2[m2M+v 7`QK /2pBiBQM iQ i?2 iQKB+ `272`2M+2- bQ i?i ν ≈ ν0X *HQbBM; i?2

+QMi`QH@HQQT rBHH bi#BHBx2 T2`KM2MiHv i?2 Qb+BHHiQ` iQ i?2 iQKB+ `272`2M+2 r?B+? rBHH T`QpB/2 i?2 Q#b2`p2`  p2`v bi#H2 M/ ++m`i2 bB;MH Q7 7`2[m2M+v νX

6`QK i?i TQBMi Q7 pB2r- i?2 iQKB+ +HQ+F ?b #22M  /Bb`mTiBp2 iBK2@+QmMiBM; i2+?MQHQ;v +QK@

T`2/ iQ i?2 7Q`K2` K2+?MB+H #b2/ +HQ+Fb U};m`2 URX8VVX AM/22/- i?2 }`bi BbQ+?`QMQmb Qb+BHHiQ`b

U>mv;2Mb T2M/mHmK BM Reey- +HQ+F Q7 a?Q`ii BM RNky- Zm`ix Qb+BHHiQ`bXXXV K/2 iBK2 2``Q`b BM i?2 `M;2 Q7 QM2 b2+QM/ T2` /v iQ QM2 b2+QM/ T2` v2`fi2Mb Q7 v2`bX h?2 }`bi ++m`i2 iQKB+

+HQ+F-r?B+? rb #b2/ QM i?2 +2bBmK Rjj `272`2M+2- rb #mBHi BM RN89 (1bb2M RN88)X aBM+2 i?i iBK2- /`Ki@

B+H BM+`2b2 BM T`2+BbBQM M/ bi#BHBiv ?b #22M +?B2p2/- 7`QK i?2 +QKKQM mb2 Q7 +QKT+i BM/mbi`BH +2bBmK +HQ+Fb- iQ H#Q`iQ`v b2imTb i?i iF2 #2M2}i 7`QK [mMimK 2z2+ib BM +QH/ iQK 7QmMiBMb M/ BM /B`2+i QTiB+H BMi2``Q;iBQM Q7 BQMb BM QTiB+H +HQ+FbX hQ/v- i?2 KQbi bi#H2 M/ ++m`i2 +HQ+Fb

KF2 M 2``Q` Q7 QM2 b2+QM/ QM i?2 ?Q`BxQM Q7 i?2 lMBp2`b2Ƕb ;2 (>mMi2KMM kyRe)X

!"# $% &'# ()*+#,-# ! " # $ % & ' () * # # + + % + , -(# + .#,+ !"#_$%&'( !")_$%&'( !"*_$%&'( ! $%&' ! +&$ ! ,-.' !)"" !/"" !0"" !#"" 1"""

!-&,$)$.*/012 .#/'0)*/01 3 #1#/&,*/01 /1$/4 #,0- !&$.*/

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6B;m`2 RX8, aBM+2 i?2 RN8yb-  ?m;2 T`Q;`2bb ?b #22M K/2 QM i?2 T2`7Q`KM+2b Q7 i?2 +HQ+Fb +QKT`2/

iQ i?2 T`2pBQmb +2Mim`B2bX 7i2` (.BK`+[ kyR8)

"27Q`2 ?pBM;  HQQF i i?2 TTHB+iBQMb Q7 i?2 iQKB+ +HQ+Fb iQ/v- r2 b?HH }`bi b22 ?Qr i?2 bi#BHBiv M/ ++m`+v +M #2 /2}M2/ BM  `B;Q`Qmb rv #2+mb2 i?2b2 irQ T`K2i2`b `2 +`m+BH 7Q` i?2 T2`7Q`KM+2b `2[mB`2/ BM KMv bvbi2KbX A7  iBK2@/2T2M/2Mi bB;MH s /2HBp2`2/ #v  7`2[m2M+v biM/`/ ?b i?2 7QHHQrBM; 7Q`K,

s(t) = s0sin (2πν(t)t) URX9V

rBi? M KTHBim/2 s0 M/  iBK2@/2T2M/2Mi 7`2[m2M+v ν(t)- i?2M i?2 Hii2` T`K2i2` +M #2 r`Bii2M

b, ν(t) = ν0(1 + ε + y(t)) URX8V rBi?, y(t) = δν(t) ν0 = 1 2πν0 dϕ(t) dt URXeV

RXRX :2M2`HBiB2b QM iBK2 M/ 7`2[m2M+v K2i`QHQ;v N

ν0 Bb i?2 2t+i 7`2[m2M+v- ε Bb i?2 bvbi2KiB+ U+QMbiMiV 7`2[m2M+v #Bb- ϕ(t) i?2 bB;MH T?b2 M/

y(t) i?2 MQ`KHBx2/ 7`2[m2M+v ~m+imiBQMb BM iBK2X  7`2[m2M+v biM/`/ ++m`+v Bb HBMF2/ iQ

i?2 mM+2`iBMiv QM i?2 +Q``2+iBQM Q7 εX  7`2[m2M+v biM/`/ Bb mMbi#H2 B7 y(t) Bb MQM@x2`QX

 rv iQ K2bm`2 i?2 7`2[m2M+v /2pBiBQM Qp2` iBK2 Bb iQ mb2  biiBbiB+H 2biBKiQ` Q7 i?2b2 7`2[m2M+v ~m+imiBQMbX AM/22/- Bi Bb MQi `2H2pMi iQ [mHB7v  `M/QK bB;MH #v Bib KTHBim/2 QMHv, i?2

`QQi K2M b[m`2 Q` p`BM+2 pHm2 Bb `i?2` mb2/X AM i?Bb T2`bT2+iBp2-  7KQmb +`Bi2`BQM 7Q` HQM;@i2`K

bi#BHBiv 2pHmiBQM U& 1sV Bb i?2 HHM o`BM+2 (HHM RNee)- /2}M2/ ?2`2 7Q`  /Bb+`2i2 iBK2@b2`B2b,

AV AR(τ ) = σ_y2(τ ) = 1

2h(¯yn+1− ¯yn)

2_{i URXdV}

r?2`2 ¯yn Bb i?2 nth K2M 7`2[m2M+v ~m+imiBQM Qp2` i?2 Q#b2`piBQM iBK2 τ M/ i?2 #`+F2ib /2MQi2

i?2 MmK#2` p2`;2 Qp2` HH i?2 yi bKTH2b Q7 i?2 iBK2 b2`B2bX h?Bb [mMiBiv ?b i?2 /pMi;2

iQ +QMp2`;2 7Q` KMv ivT2b Q7 `M/QK MQBb2b z2+iBM; i?2 yiiBK2 b2`B2b- mMHBF2 i?2 i`/BiBQMH p`BM+2X

h?2 b[m`2@`QQi Q7 i?2 o_- +HH2/ i?2 HHM .2pBiBQM U.1oV- ?b  /B`2+i T?vbB+H BMi2`@ T`2iiBQMX h?2 [mMiBiv,

∆t = σy(τ )× τ URX3V

Bb i?2 Q`/2` Q7 K;MBim/2 Q7 i?2 iBK2 /2pBiBQM- BM b2+QM/- Q7 i?2 Qb+BHHiQ` 7`2[m2M+v Q7 .1o σy(τ )

/m`BM; M Q#b2`piBQM iBK2 τX 6Q` BMbiM+2-  +HQ+F Q7 σy(τ = 1 day) = 10−14?b  `2HiBp2 mM+2`iBMiv

Q7 ∼ 1 ns Qp2` QM2 /vX JQbi Q7 i?2 iBK2- σ2

y Bb  +?M;BM; 7mM+iBQM Q7 τ- #2+mb2 i?2 +HQ+F Bb T2`im`#2/

#v p`BQmb BMi2`MH 7+iQ`bX h?2 bHQT2 Q7 i?2 o_ `2~2+ib- BM i?i +b2- i?2 Mim`2 Q7 i?2 QM@;QBM;

7`2[m2M+v T2`im`#iBQMX b M 2tKTH2-  bHQT2 Q7 −1 BM  HQ;@HQ; (τ, σ2

y(τ )) /B;`K b?Qrb i?i i?2

Qb+BHHiQ` Bb QMHv z2+i2/ #v r?Bi2 7`2[m2M+v MQBb2 Ui?i Bb- i?2 TQr2` Q7 i?2 /2pBiBQMb #Qmi i?2 K2M pHm2 Bb +QMbiMi r?i2p2` i?2 7`2[m2M+v Q7 MHvbBbVX

*QMb2[m2MiHv- i?2 bi#BHBiv Ui?2 o_V /2+`2b2b UBM+`2b2bV bi2/BHv b i?2 Q#b2`piBQM iBK2

BM+`2b2b U/2+`2b2bV U};m`2 URXeVVX PM i?2 +QMi``v- Qi?2` 2z2+ib UHBF2 i?2`KH QM2bV +?`+i2`Bx2/ #v

`M/QK rHF MQBb2 BKTHB2b  +1 bHQT2 BM i?2 /B;`K- /2;`/BM; i?2 bi#BHBiv Q7 i?2 +HQ+F Qp2` iBK2X HbQ- Bi Bb +`m+BH iQ K2MiBQM i?i bm+?  K2bm`2K2Mi- /QM2 #v bKTHBM; Qp2` iBK2 Q7 i?2 +HQ+F bB;MH-?b iQ #2 K/2 rBi?  KQ`2 bi#H2 Qb+BHHiQ` i?M i?2 QM2 r?B+? Bb T`Q#2/X AM/22/- i?2 +QKTmi2/ o_ Bb Hrvb i?2 QM2 Q7 i?2 H2bb bi#H2 Qb+BHHiQ`X h?2 pBH#BHBiv Q7 p2`v bi#H2 +HQ+Fb 2M#H2b-QM i?i TQBMi Q7 pB2r- iQ i2bi MmK2`Qmb 7`2[m2M+v biM/`/bX h?2 o_ ?b #22M T`2b2Mi2/ ?2`2 7Q` T2/;Q;B+H Tm`TQb2b- #mi KMv Qi?2` `2}M2/ b+?2K2b 2tBbi iQ #2ii2` bb2bb i?2 bi#BHBiv i`2M/b UJQ/B}2/ HHM o`BM+2- >/K`/ o`BM+2XXXVX

Ai Kv #2 ?`/ i }`bi ;HM+2 iQ b22 i?2 TQBMi BM K2bm`BM; iBK2 rBi? H2bb i?M  ns Q7 2``Q`X +imHHv- Bi Bb +`m+BH 7Q` KMv i2+?MB+H TTHB+iBQMb Q7 iQ/vǶb KQ/2`M rQ`H/X h?2 i?`22 KBM `2b r?2`2 iQKB+ +HQ+Fb `2 BMi2MbBp2Hv mb2/ `2 i?2 7QHHQrBM;,

Ç 6mM/K2MiH 2tT2`BK2Mib, 7mM/K2MiH +QMbiMib i2biBM;- 7bi T?2MQK2MQMb i`+FBM; UG`;2 >/`QM *QHHB/2` Q` Qi?2` T`iB+H2b ++2H2`iQ`b 7Q` BMbiM+2V

Ç qQ`H/rB/2 i2H2+QKKmMB+iBQMb, M2irQ`Fb bvM+?`QMBxiBQM- `/BQ M/ ho #`Q/+biBM;- MpB;@ iBQM M/ TQbBiBQMBM; U:Sa- :HBH2QXXX r?B+? T`2+BbBQM Q7 HQ+HBxiBQM Bb /B`2+iHv HBMF2/ iQ i?2 o_ pHm2V- *QQ`/BMi2/ lMBp2`bH hBK2 Ulh*V- AMi2`MiBQMH iQKB+ hBK2 UhAV

Ry *?Ti2` RX AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v TTHB+iBQMb log 𝜎 𝑦(𝜏 ) log 𝜏 ! "" # $ % # &' # $ ( ) *+',) -&)./)$(0 -"'(1)& -&)./)$(0 2$,)3&#,'4$ ,'5) &#$645 *#"1 -&)./)$(0 -&)./)$(0 6&'-, 78 9 :8 :;

6B;m`2 RXe, h?2 bHQT2 Q7 i?2 o_ BM HQ;@HQ; THQi pbX τ Bb HBMF2/ iQ  bT2+B}+ ivT2 Q7 MQBb2X h?2 Q#b2`piBQM iBK2 b?HH #2 K2bm`2/ #v  KQ`2 bi#H2 +HQ+F i?M i?2 QM2 i2bi2/X

hrQ TTHB+iBQMb `2 rQ`i? iQ K2MiBQM ?2`2X h?2 }`bi QM2 Bb i?2 /2}MBiBQM Q7 i?2 iBK2b+H2 hA- M/ Bib r2HH@FMQrM /2`BpiBQMb lh* M/ :Sa hBK2 U:SahVX h?2 hA Bb 2biBKi2/ 2p2`v KQMi? #v i?2

"m`2m AMi2`MiBQMH /2b SQB/b 2i J2bm`2b U"ASJV BM aĕp`2b #v p2`;BM; i?2 iQKB+ iBK2 Q7 M2`Hv

98y b2+QM/`v 7`2[m2M+v biM/`/b M/ Rj S`BK`v 6`2[m2M+v aiM/`/b Ur?B+? `2 ?B;? ++m`+v iQKB+ +HQ+Fb HQ+i2/ BM 72r BMbiBimi2b Q7 K2i`QHQ;v BM i?2 rQ`H/VX h?Bb mMBi Q7 K2bm`2- iF2M b M

mHiBKi2 iBK2@`272`2M+2 7Q` b+B2MiB}+ 2tT2`BK2Mib- Bb b ++m`i2 b 2 × 10−16 _{UBM kyR8V M/ Bb HBMF2/}

iQ i?2 lh* #v //BM; Q` bm#i`+iBM; i?2 ǶH2T b2+QM/bǶ- bQ b iQ `2+Qp2`  iBK2@b+H2 +HQb2 iQ i?2 QM2 ;Bp2M #v i?2 2`i? `QiiBQM QM Bib2H7X :Sa iBK2 /2}MBiBQM Bb HbQ ;`QmM/2/ QM hA, i?2v /Bz2` QMHv 7`QK M BMi2;2` MmK#2` Q7 b2+QM/bX

MQi?2` TTHB+iBQM Bb i?2 ;2QHQ+HBxiBQM T`BM+BTH2 #v i?2 mb2 Q7 QM@#Q`/ +QKT+i iQKB+ +HQ+Fb BM i?2 :HQ#H LpB;iBQM ai2HHBi2 avbi2K U:LaaVX .Bz2`2Mi :Laa +QMbi2HHiBQMb `2 pBH#H2- i?2 KQbi 7KQmb QM2 #2BM; i?2 :HQ#H SQbBiBQMBM; avbi2K U:SaVX am+? bvbi2Kb 2M#H2b  :Laa `2+2Bp2` U +2HH@T?QM2 QM 2`i? 7Q` BMbiM+2V iQ HQ+i2 Bib2H7 QM i?2 ;2QB/ #v `2+2BpBM; i?2 bB;MHb 7`QK Ui H2biV 7Qm` bi2HHBi2b BM i?2 bFvX h?2 }`bi i?`22 bi2HHBi2b b2M/ i?2B` +QQ`/BMi2b M/ i?2B` HQ+H iBK2 U;Bp2M #v i?2 HQ+H iQKB+ +HQ+F- r?B+? Kmbi #2 +Q``2+i2/ 7`QK _2HiBpBbiB+ 2z2+ib #27Q`2V M/ i?2 7Qm`i? bi2HHBi2 mbmHHv ;Bp2 i?2 iBK2 /Bz2`2M+2 #2ir22M i?2 :Sa iBK2 M/ i?2 HQ+H iBK2 Q7 i?2 `2+2Bp2`X  7Qm` 2[miBQMb bvbi2K Q7 7Qm` mMFMQrMb i?2M 2M#H2b i?2 `2+2Bp2` iQ `2+Qp2` Bib HQ+HBxiBQM QM 2`i?X HbQ- i?2 :Laa bvbi2Kb `2 M BKTQ`iMi K2M iQ /Bbb2KBMi2 i?2 mMBp2`bH iBK2 Qp2` i?2 MmK2`Qmb i2H2+QKKmMB+iBQM +2Mi2`bX AM bQK2 +b2b-  HQ+H iQKB+ +HQ+F Bb M22/2/- r?2M  M2irQ`F Q`  bvbi2K +MMQi ?p2 T?vbB+H ++2bb iQ i?2 ;HQ#H iBK2 Ubm#K`BM2b- BbQHi2/ i2H2+QKKmMB+iBQM +2Mi2`bXXXVX AM kyRe- `QmM/ 8yy iQKB+ 7`2[m2M+v biM/`/b r2`2 BM@Q`#Bi `QmM/ i?2 1`i?X

hQ/v- BM+`2bBM; ii2MiBQM Bb /2pQi2/ iQ i?2 /2p2HQTK2Mi Q7 +QKT+i M/ p2`v bKHH iQKB+ +HQ+FbX h?2 p2`v }`bi +?BT@b+H2 +HQ+F rb #b2/ QM +2bBmK Rjj BMi2``Q;iBQM M/ b?Qr2/ p2`v ;QQ/

b?Q`i@i2`K bi#BHBiv 7Q` Bib `2/m+2/ bBx2/ U};m`2 URXdVV (EMTT2 kyy9)X am+? /2pB+2b `2 THMM2/ iQ

rQ`F b KQ`2 bi#H2 HQ+H Qb+BHHiQ`b BM BbQHi2/ 2[mBTK2Mib i?i +MMQi bvM+?`QMBx2 `2;mH`Hv iQ i?2 :LaaX

RXkX h?2 PTiB+H *2bBmK 6`2[m2M+v aiM/`/ UP*6aV RR

6B;m`2 RXd, h?2 *?BT@a+H2 iQKB+ *HQ+F U*a*V Q7 i?2 LiBQMH AMbiBimi2 Q7 aiM/`/b M/ h2+?@ MQHQ;v i`;2ib b?Q`i@?mH MpB;iBQM M/ HQ+H +QKKmMB+iBQMbX

RXk h?2 PTiB+H *2bBmK 6`2[m2M+v aiM/`/ UP*6aV

RXkXR #Qmi *2bBmK 6`2[m2M+v aiM/`/b U*6aV

h?2 +2bBmK +HQ+F rb i?2 }`bi ivT2 Q7 bi#H2 iQKB+ +HQ+F 2tT2`BK2MiHHv `2HBx2/ BM RN88 (1bb2M RN88)X

h?2 +2bBmK Bb M HFHB Q7 i?2 +QHmKM A Q7 i?2 +HbbB}+iBQM r?B+? ?b  mMB[m2 pH2M+2 2H2+i`QM

Ubi`m+im`2 Bb (s2)RbR_{VX h?2`27Q`2- Bib iQKB+ 2t+Bi2/ bi`m+im`2 U1aV +M #2 `i?2` bBKTHv /2b+`B#2/X}

>BbiQ`B+HHv- i?Bb 2H2K2Mi rb +?Qb2M 7Q` i?2 bi#BHBiv Q7 Bib ;`QmM/ bii2 U:aV BM i?2 2tT2`BK2MiH +QM/BiBQMb Ui?2 iQKB+ Kbb Bb ?B;? 2MQm;? 7Q` i?2 2H2K2Mi iQ #2 mb2/ BM #2Kb rBi? 72r T2`im`#iBQM 2z2+ib /m2 iQ +QHHBbBQMbVX >pBM;  /22T2` HQQF BMiQ Bi- QM2 }M/b  }M2 bi`m+im`2 `BbBM; 7`QK i?2 bTBM@ Q`#Bi 2M2`;v i2`K- r?B+? bTHBib BMiQ i?2 ?vT2`}M2 bi`m+im`2 r?2M i?2 Mm+H2mb KQK2Mi Bb iF2M BMiQ ++QmMiX lM/2` 2ti2`MH K;M2iB+ }2H/- i?2b2 H2p2Hb bTHBi ;BM BMiQ i?2 w22KM bi`m+im`2- ++Q`/BM;

iQ i?2 [mMimK K;M2iB+ MmK#2` pHm2 m (hMMQm/DB RNdj)(ai2+F kyy3)U};m`2 URX3VVX

h?2 +2bBmK Bib2H7 ?b  mb#H2 bi#H2 BbQiQT2 Ui?i Bb MQi `/BQ+iBp2V r?B+? Bb i?2Rjj<sub>*bX ++Q`/@</sub>

BM; iQ i?2 [mMimK K2+?MB+b- B7 QM2 2t+Bi2b i?2 bT2+B2b i?MFb iQ i?2 M2`@BM7``2/ `/BiBQMb D1

Urp2H2M;i? BM B` λ1 = 894.4 nm Q` ν1 = 335.1 T HzV M/ D2 Uλ2 = 852.1 nm Q` ν2 = 351.7 T

HzV-i?2 2H2+i`QM rBHH DmKT iQ  ?B;?2` 2M2`;v H2p2H M/ HzV-i?2 ;HQ#H bii2 Q7 HzV-i?2 iQK rBHH +?M;2X HbQ- HzV-i?2 TQbbB#H2 bm#@i`MbBiBQMb BM i?2b2 HBM2b +Q``2bTQM/ iQ /2}M2/ bii2 Q7 TQH`BxiBQM Q7 i?2 HB;?i Uπ Q` σVX 7i2` i?2 p2`;2 HB72iBK2 Q7 i?Bb bii2- i?2 2H2+i`QM rBHH /2+v iQ HQr2` 2M2`;v H2p2Hb- r?BH2 2KBiiBM;

~mQ`2b+2M+2 T?QiQMbX Ai b?QmH/ #2 MQiB+2/ i?i #2+mb2 Q7 i?2 b2H2+iBQM `mH2b- QMHv bQK2 i`MbBiBQMb

#2ir22M i?2 ;`QmM/ bii2b M/ i?2 2t+Bi2/ bii2b `2 HHQr2/ bQ i?i  b2H2+iBp2 ~mQ`2b+2M+2 /B;`K

Bb Q#iBM2/ B7 HH i?2 i`MbBiBQMb `2 T`Q#2/ U};m`2 URXNVVX

PM i?2 Qi?2` bB/2- `2;`/BM; i?2 iQKB+ `2bQMiQ`- Bi Bb #b2/ QM i?2 /Bz2`2M+2 BM 2M2`;v Q`

7`2[m2M+v Q7 i?2 irQ ?vT2`}M2 H2p2Hb F = 3 M/ F = 4 Q7 i?2 ;`QmM/ bii2 U:aV 62_S

1/2 (ai2+F kyy3)

U};XRX3V r?B+? Bb,

ν0 = 9.192631770 GHz URXNV

am+? i`MbBiBQM HB2b BM i?2 _6 /QKBM bQ Bi +M #2 /B`2+iHv //`2bb2/ mbBM; KB+`Qrp2 ;2M2`iQ`bX

++Q`/BM; iQ i?2 T`BM+BTH2 Q7 i?2 iQKB+ +HQ+F /2b+`B#2/ 2`HB2` U};m`2 URX9VV- bQK2 iQKB+ T2F bB;MH

?b iQ #2 `2+Qp2`2/ 7`QK i?i 2t+BiiBQM- ++Q`/BM; iQ i?2 /2imMBM; Q7 ν rBi? `2HiBQM iQ ν0 U};m`2

URXRyVVX A7 r2 +QMbB/2`  H`;2 TQTmHiBQM Q7 iQKb- bm+? bB;MH 7mM/K2MiHHv `2~2+ib i?2 TQTmHiBQM

/Bz2`2M+2 Q7 i?2 iQKb BM :a UF = 3V M/ i?Qb2 BM :a UF = 4VX h?2 T`Q#H2K Bb i?2M ?Qr iQ Ƕ+QmMiǶ i?2 MmK#2` Q7 iQKb i?i `2 QM 2+? H2p2H- r?BH2 i?2 ν 7`2[m2M+v Bb +?M;2/X

Rk *?Ti2` RX AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v TTHB+iBQMb !"#! $%& !"#! $%& !_" #$"%& !_' #()%& '() *+,-./0 12(/3.1.4/ '5) 6789./0 12(/3.1.4/ :;<= >?_* @A? >?_* #A? B_#C@D=E B_?CDD=! B_?CD<= >?_F #A? :;D= :;D= :;D :;@= :;@= :;@ :;?= ##>G H%& #<# H%& ?I# H%& ?<# H%& ## ! J ! J ! J < :./K F12+81+2K %7-K2L./K F12+81+2K MKK,(/

6B;m`2 RX3, >vT2`}M2 bi`m+im`2 Q7 i?2 +2bBmK RjjX PMHv  72r HHQr2/ i`MbBiBQMb `2 `2T`2b2Mi2/X h?2`2 `2 Q7 irQ ivT2b, UV h?2 SmKTBM; i`MbBiBQMb, i?2 1a +M /2+v BMiQ i?2 irQ ?vT2`}M2 :a bm#@H2p2HbX U#V h?2 *v+HBM; i`MbBiBQMb, i?2 1a +M QMHv /2+v BM QM2 Q7 i?2 ?vT2`}M2 :a

!"# !$# %!&' %!%' %!(' (!)' %!(' %!%' (!%' (!(' (!(' (!%' 𝞶 𝞶 * ) + , -. / * ) & , -. / * , & , -. / * ) + , -. / *,,01 -./ *,,01 -./

6B;m`2 RXN, UV h?2 bBt ~mQ`2b+2M+2 bT2+i`mK Q7 i?2 D2 HBM2X U#V h?2 7Qm` ~mQ`2b+2M+2 bT2+i`mK Q7

i?2 D1 HBM2X 7i2` (.BK`+[ RNNR)X

M 2bb2MiBH TQBMi iQ F22T BM KBM/ Bb i?i bm+?  bB;MH +MMQi #2 `2+Q`/2/ B7 i?2 TQTmHiBQM Q7 *b iQKb Bb i i?2`KQ/vMKB+ 2[mBHB#`BmK i `QQK i2KT2`im`2X AM/22/- #2+mb2 i?2 2M2`;v /Bz2`2M+2

#2ir22M i?2 irQ ;`QmM/ bii2b Bb Q7 i?2 Q`/2` Q7 40 µeV - p2`bmb i?2 i?2`KH 2t+BiiBQM kBT = 25 meV

i `QQK i2KT2`im`2- QM2 rBHH }M/ `Qm;?Hv ?H7 Q7 i?2 iQKb BM i?2 :a UF = 3V M/ i?2 Qi?2` ?H7 BM i?2 :a UF = 4V- bQ i?i MQ `2bQMM+2 +M #2 `2i`B2p2/ #2+mb2 i?i T`iBiBQM rBHH #2 i?2 bK2 7i2`

i?2 ν02t+BiiBQM- 7Q` bvKK2i`v Tm`TQb2bX h?i K2Mb QM2 ?b iQ +?B2p2 TQTmHiBQM BMp2`bBQM bQ i?i

HH i?2 iQKb `2 BM i?2 bK2 [mMimK bii2 |F = 3i Q` |F = 4i #27Q`2 i?2v +M #2 2t+Bi2/ iQ i?2 Qi?2` H2p2H M/ +QmMi2/X h?2`2 `2 #bB+HHv irQ rvb Q7 /QBM; i?i,

Ç iQKB+ b2H2+iBQM, i?2 iQKb Q7 QM2 Q7 i?2 irQ bii2b `2 /Bb+`/2/ Ç iQKB+ TmKTBM;, i?2 iQKb `2 HH Tmi BM i?2 bK2 bii2

h?2 }`bi TT`Q+? +M #2 `2HBx2/ i?`Qm;? K;M2iB+ b2H2+iBQM- 7QHHQrBM; i?2 7mM/K2MiH B/2

RXkX h?2 PTiB+H *2bBmK 6`2[m2M+v aiM/`/ UP*6aV Rj

i?2`KH iQKB+ #2K Q7 bT22/ v +M #2 /2pBi2/ 7`QK i?2B` i`D2+iQ`v mbBM; K;M2ib r?B+? +`2i2  BM?QKQ;2M2Qmb K;M2iB+ }2H/ BM i?2 i`Mbp2`b2 /B`2+iBQM iQ i?2 #2KX h?2 /2pBiBQM Bb imM2/ ++Q`/BM; iQ i?2 iQiH bTBM Q7 i?2 iQK r?B+? +?M;2b /2T2M/BM; QM r?B+? bii2 i?2 iQK Bb BMBiBHHvX am+? T`+iB+H K2M iQ b2H2+i i?2 BMBiBH ;`QmM/ bii2 Q7 i?2 +2bBmK Bb i i?2 #bBb Q7 i?2 J;M2iB+

*2bBmK 6`2[m2M+v aiM/`/ UJ*6aV- bBM+2 i?2 }`bi T`QQ7 Q7 +QM+2Ti #v 1bb2M (1bb2M RN88)X

PM i?2 Qi?2` ?M/- iQKB+ TQTmHiBQM BMp2`bBQM +M #2 +?B2p2/ mbBM; i?2 +QM+2Ti Q7 TmKTBM;

(EbiH2` RN8y) r?2`2 i?2 iQKB+ bii2b `2 KMBTmHi2/ mbBM; Hb2` `/BiBQMbX h?2 7mM/K2MiH

/Bz2`2M+2 ?2`2 Bb i?i- i i?2 2M/ Q7 i?2 TQTmHiBQM BMp2`bBQM T`Q+2bb- MQ iQKb `2 T?vbB+HHv HQbi QM i?2 +QMi``v iQ i?2 K;M2iB+ b2H2+iBQM TT`Q+? r?2`2 i H2bi ?H7 Q7 i?2 T`iB+H2b `2 i?`QrM rvX h?Bb K2Mb i?i  ;`2i2` MmK#2` Q7 iQKb BM :a +M #2 BMi2``Q;i2/ mbBM; i?2 _6 2t+BiiBQM- bQ i?i i?2 `2bQMM+2 bB;MH Q7 i?2 +HQ+F `2+Q`/2/ Bb ?B;?2` 7Q`  ;Bp2M iQKB+ ~mtX h?Bb B/2 ;`QmM/b i?2 T`BM+BTH2 Q7 i?2 *2bBmK 6`2[m2M+v aiM/`/ PTiB+HHv TmKT2/- r?B+? rBHH #2 +HH2/ ?2`2 ǶP*6aǶX _2;`/BM; i?2 +QmMiBM; T`Q+2bb- BM i?2 J*6a- i?2 iQKb i?i `2KBM 7i2` K;M2iB+ b2H2+iBQM

Ui?mb BM i?2 bK2 bii2V `2 BQMBx2/ M/ /2i2+i2/ mbBM; M 2H2+i`QM KmHiBTHB2` (*miH2` kyy8)X AM i?2

P*6a- i?2 iQKb `2 BM/B`2+iHv +QmMi2/ mbBM;  +Q?2`2Mi HB;?i bQm`+2 BX2X,  Hb2` `/BiBQM i QM2 Q7 i?2 A_ i`MbBiBQMb K2MiBQM2/ #27Q`2, r?2M i?2 iQK bii2 /2+vb 7i2` i?2 2t+BiiBQM- Bi 2KBib T?QiQMb i?i +M #2 bBKTHv /2i2+i2/ #v  T?QiQ/BQ/2 M/ +QMp2`i2/ iQ M 2H2+i`B+H bB;MHX HH BM HH- i?2 QTiB+H i2+?MQHQ;v QmiT2`7Q`Kb i?2 K;M2iB+ QM2 #2+mb2 Bi Bb i2+?MB+HHv bBKTH2` iQ QT2`i2 M/ Bi T`QpB/2b ?B;?2` iQKB+ bB;MH- M/ i?mb  #2ii2` bi#BHBiv Ub22 7m`i?2`VX

RXkXk S`BM+BTH2b #2?BM/ i?2 P*6a

++Q`/BM; iQ };m`2 URXRyV- i?2 *6a ?b 7Qm` KBM T`ib,

RX h?2 HQ+H 2H2+i`QMB+ Qb+BHHiQ`

kX h?2 KB+`Qrp2 2H2+i`QMB+b 7Q` _6 KM;2K2Mi jX h?2 iQKB+ `2bQMiQ`

9X h?2 2H2+i`QMB+b Q7 `2;mHiBQM i?i im`Mb i?2 `2bQMM+2 bB;MH BMiQ M 2``Q` bB;MH

!"#$%& '()*(+ ,%-./0 123-4#%-$ 𝜈0= 9.192631770 GHz 5620%$-4/0 127"6#%/-4 8/0$-9#:2 5620%$-4/03 𝜈′_{= 10 𝑀 𝐻 𝑧 ';32$3+} $23-4#%-$ $23<-432 2$$-$ 3/74#6 ∼ 𝜈 − 𝜈₀ !"#$%&$ !'&(')* +,*&-./0%&$ !'&(')* '=+ '>+ '?+ '@+ 𝜈 = 9.19 𝐺𝐻 𝑧

6B;m`2 RXRy, AM i?2 +2bBmK 7`2[m2M+v biM/`/- i?2 BM@HQQT mb2` 7`2[m2M+v ν′i 10 MHz Bb mT@+QMp2`i2/

iQ 9.19 GHz iQ ;2M2`i2 i?2 iQKB+ `2bQMM+2 `2bTQMb2

h?2 T`ib URV- UkV M/ U9V `2 BM+Hm/2/ BM i?2 2H2+i`QMB+b T+F;2 r?BH2 T`i UjV Bb +HH2/ i?2

T?vbB+b T+F;2X "`B2~v- i?2 Qb+BHHiQ` Q7 URV Bb M Pp2M@*QMi`QHH2/ *`vbiH Pb+BHHiQ` UP*sPV Q7 p2`v

;QQ/ b?Q`i@i2`K bi#BHBiv Uσy = 10−13 i 1sV i?i Bb imM2/ i ν′ = 10 M Hz r?B+? Bb HbQ i?2 }MH

R9 *?Ti2` RX AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v TTHB+iBQMb

2H2+i`QMB+b- r?B+? Bb im`M2/ BMiQ  `2bQMM+2 `2bTQMb2 i ν0 BM UjVX h?2M-  b2+QM/ 2H2+i`QMB+b U9V

;2M2`i2b  bB;MH 2``Q` 7`QK i?2 iQKB+ `2bQMM+2 r?B+? +M #2 mb2/ b  722/#+F BM i?2 b2`pQ@HQQT iQ bi#BHBx2 URV BM i?2 HQM;@i2`KX 6`QK MQr QM- r2 rBHH 7Q+mb QM i?2 T?vbB+b T+F;2 #27Q`2 //BM; bQK2 72r rQ`/b #Qmi i?2 2H2+i`QMB+b i i?2 2M/ Q7 i?Bb b2+iBQMX

RXkXkXR S?vbB+b T+F;2

h?2 iQKB+ `2bQMiQ` +QMbBbib BM  i?2`KH +2bBmK #2K r?B+? ?b M BMi2MbBiv Q7 ~mt Q7 I ∼

1010atoms.s−1QM  72r mm2+B`+mH` bm`7+2X h?2 iQKB+ bT22/b `2 ;Bp2M #v i?2 Jtr2HH@"QHixKMM

/Bbi`B#miBQMX i 100◦<sub>C- i?2 KQbi T`Q##H2 bT22/ Bb v</sub>

p = 216 m/sX h?2 ~mt BMi2`+ib b2p2`H iBK2b

rBi?  Hb2` #2K i 852 nm Q` 894 nm T2`T2M/B+mH` iQ i?i iQKB+ #2K i?`Qm;? M QTiB+H #2M+?

TH+2/ HQM; i?2 +2bBmK im#2 U};m`2 URXRRVVX

!"#$% &'()$ *"+#$, -".'/, ! 01/'2# 3$425 -# 6$"+ -# (.$4 78 9:; <#'=4"> (? @#$%A BC7B D:; E F@+1'4= )$/$2/(% )$/$2/(% EG 3 <6A *H %$=@>"/'(4 <"A !"#$% )'()$ %$=@>"/'(4 I0 JK

6B;m`2 RXRR, a+?2KiB+ Q7 i?2 iQKB+ `2bQMiQ` Q7 i?2 P*6aX h?2 i?2`KH *b #2K BMi2`+ib b2p2`H

iBK2b rBi? i?2 Hb2` i D1 Q` D2 i`MbBiBQM HBM2b

i i?i TQBMi- Bi b?QmH/ #2 K2MiBQM2/ i?i i?2 Hb2` mb2/ BM i?Bb bvbi2K Bb mbmHHv- 7Q` BM/mbi`BH TTHB+iBQM-  Hb2` /BQ/2- r?B+? /2p2HQTK2Mi Bb T`2+Bb2Hv i?2 Tm`TQb2 Q7 i?Bb KMmb+`BTiX h?2 iQTB+ rBHH #2 //`2bb2/ 2ti2MbBp2Hv BM i?2 +QKBM; +?Ti2`b- #mi 7Q` i?2 iBK2 #2BM;- Bi +M #2 bb2`i2/ i?i i?2b2 /2pB+2b `2 bmBiBM; i?2 M22/ Q7  bQm`+2 i?i +M #2 imM2/ 2bBHv BM i?2 `M;2b Q7 i?2 +2bBmK i`M@

bBiBQMb `QmM/ i?2 D1HBM2 Q` i?2 D2HBM2- i?MFb iQ i2KT2`im`2 Q` +m``2Mi imMBM; Q7 i?2 /2pB+2X

JQ`2 T`2+Bb2Hv- i?`22 KBM xQM2b Q7 *b@HB;?i BMi2`+iBQM +M #2 B/2MiB}2/X h?2v `2 +HH2/ xQM2 -xQM2 Ƕ M/ -xQM2 "X HbQ-  7Qm`i? -xQM2 HHQrb i?2 *b #2K iQ BMi2`+i rBi? i?2 KB+`Qrp2 `/BiBQM

M/ Bb +HH2/ i?2 _Kb2v +pBivX PM  7mM+iBQMH TQBMi Q7 pB2r- T`i UV BM };m`2 URXRRV `2;mHi2b

i?2 Hb2` /BQ/2 7`2[m2M+v 2KBbbBQM- r?BH2 T`i U#V #2HQM;BM; iQ i?2 2H2+i`QMB+ T+F;2 ?M/H2b i?2 _6 `2;mHiBQM M/ i?2 +HQ+F@bB;MH Bib2H7X q2 MQr ?p2  /22T2` HQQF i i?2 T?vbB+H K2+?MBbKb mM/2`HvBM; i?2 QT2`iBQM Q7 i?2 +HQ+FX hQ /Q i?i- K2bm`2K2Mib QM  `2H T`QiQivT2 Q7 P*6a Uh?H2b 1H2+i`QM .2pB+2bV ?p2 #22M T2`7Q`K2/X

Ç wQM2 , Gb2` /BQ/2 `2;mHiBQM

h?2 QMHv 7mM+iBQM Q7 xQM2  Bb iQ bi#BHBx2 i?2 /BQ/2 Hb2` rp2H2M;i? QM i?2 /2bB`2/ *b i`MbBiBQM

Ub22 7m`i?2` b2+iBQM (RXkXkXk)VX AM/22/- i?2 Hb2` 7`2[m2M+v +M /`B7i [mB+FHv rBi? i?2 2MpB`QMK2Mi

RXkX h?2 PTiB+H *2bBmK 6`2[m2M+v aiM/`/ UP*6aV R8

i?i ?p2 p`BQmb T?vbB+H Q`B;BMb UBMbB/2 Q` QmibB/2 i?2 Hb2` +pBivV T2`im`# i?2 QTiB+H 7`2[m2M+vX

h?2b2 mMrMi2/ b?B7ib rBi? `2HiBQM iQ ν0 +M #2 +QmMi2`+i2/ UiQ bQK2 2ti2M/ QMHvV- rBi?  +`27mH

/`BpBM; Q7 i?2 i2KT2`im`2 M/ +m``2Mi Q7 i?2 bQm`+2X A/2HHv- i?2b2 T`K2i2`b Kmbi #2 QTiBKBx2/ BM `2H@iBK2 rBi?  T2`KM2Mi i`+FBM; M/ +Q``2+iBQM Q7 i?2 7`2[m2M+v, M 2H2+i`QMB+ b2`pQ HQQT Bb mb2/ iQ +?B2p2 i?iX h?Bb T`Q+2/m`2 rBHH #2 /2b+`B#2/ BM i?2 2H2+i`QMB+ T+F;2X 6Q` i?2 KQK2Mi- r2 rBHH Dmbi bim/v i?2 T?vbB+H HB;?i@iQK BMi2`+iBQM M/ i?2 bB;MH r2 +M `2i`B2p2 7`QK BiX

hQ ?B;?HB;?i i?2 ~mQ`2b+2M+2 T`QT2`iB2b Q7 *b- QM2 }t2b i?2 +QMbiMi BMTmi +m``2Mi I0 i?i ;Bp2b

i?2 `2[mB`2/ QTiB+H TQr2` i i?2 Hb2` QmiTmi- i?2M i?2 i2KT2`im`2 Q7 i?2 /BQ/2 Bb bi#BHBx2/ i 

}t2/ pHm2 T0 bQ i?i i?2 7`2[m2M+v Q7 i?2 Hb2` #2K 7HHb `Qm;?Hv i i?2 D1 Q` D2 HBM2b- M/ 7i2`

i?i- i?2 +m``2Mi Q7 i?2 Hb2` Bb bHB;?iHv KQ/mHi2/ rBi? KQ/mHiBQM /2Ti? ∆I i  7`2[m2M+v fI bQ

b iQ b+M i?2 b2p2`H i`MbBiBQMb Q7 2+? HBM2X h?Bb Bb b?QrM BM };m`2b URXRkV M/ URXRjVX 6Q` i?2

irQ i`MbBiBQMb HBM2b- i?2 +m``2Mi Q7 i?2 Hb2` /BQ/2 Bb KQ/mHi2/ ?2`2 i HQr 7`2[m2M+v UfI ∼ 40 HzV

M/  [mBi2 ?B;? /2Ti? U∆I ∼ 3 mAV rBi?  briQQi? T`Q}H2X h?2 T?QiQMb 2KBii2/ #+F #v i?2 /2@2t+BiiBQM T`Q+2bb `2 i?2M +QHH2+i2/ #v  H`;2 `2 T?QiQ/BQ/2 i?i im`Mb i?2 QTiB+H TQr2` BMiQ  pQHi;2 T`Q}H2 `2/ QM M Qb+BHHQb+QT2X h?2 imMBM; 2{+B2M+v Q7 i?2 7`2[m2M+v Urp2H2M;i?V #2BM; M2;iBp2 UTQbBiBp2V rBi? `2HiBQM iQ i?2 +m``2Mi U`QmM/ −1 GHz/mA 7Q` i?2 Hb2` mb2/ ?2`2V- i?2 b+MMBM; Bb 7`QK ?B;?2` 2M2`;v iQ HQr2` 2M2`;v rBi?  TQbBiBp2 bHQT2 Q7 i?2 +m``2Mi `KT- bQ i?i r2

`2+Qp2` i?2 HBM2b BM i?2 BMp2`b2 Q`/2` Q7 };m`2 URXNVX !"#$ !$#% $#$ $#% "#$ !$#$&$ !$#$'% !$#$'$ !$#$"% $#$ $#% "#$ "#% '#$ '#% ()**+,- ./)*0+ 1/2)34-5/, ! "# $ % & '( ) 657+ 89: .5;,43 /< 2+-+0-/* $#$ $#% "#$ "#% '#$ !$#$&$ !$#$'% !$#$'$ !$#$"% $#$ $#% "#$ "#% '#$ '#% ()**+,- ./)*0+ 1/2)34-5/, ! "# $ % & '( ) 657+ 89: .5;,43 /< 2+-+0-/* = %> = => = &> & => & &> & '>

6B;m`2 RXRk, PM i?2 H27i, mTT2` ;`T? BM ;`22M, aB;MH +Q``2bTQM/BM; iQ i?2 /BQ/2 Hb2` +m``2Mi KQ/mHiBQM BM iBK2X GQr2` ;`T? BM TBMF, iQKB+ ~mQ`2b+2M+2 bB;MH Q7 i?2 ?vT2`}M2 i`MbBiBQMb iQ

F = 4Q7 i?2 D2 HBM2 `2+Q`/2/ b  7mM+iBQM Q7 iBK2 BM xQM2  /2i2+iQ`X PM i?2 `B;?i, bK2 b QM i?2

H27i #mi 7Q` i?2 ?vT2`}M2 i`MbBiBQMb iQ F = 3 Q7 i?2 D2 HBM2

Gi2` BM i?Bb b2+iBQM- i?2 /2b+`BTiBQM rBHH #2 ?QH/ rBi? i?2 D1 HBM2 i 894 nmX LQr i?i i?2

?vT2`}M2 i`MbBiBQMb `2 /2i2+i2/-  b2`pQ HQQT +M #2 BKTH2K2Mi2/ QM QM2 Q7 i?2b2 i`MbBiBQMbX 6Q`

BMbiM+2- i?2 U43′_{σV 2t+BiiBQM b+?2K2 Ur?B+? Bb i?2 MQiiBQM 7Q`  i`MbBiBQM ;QBM; 7`QK |F = 4i iQ}

Re *?Ti2` RX AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v TTHB+iBQMb ! "# " !# ! !# " "# $%&' $%&%( %&%%

$%&%)( $%&%)% $%&%'( $%&%' %&% %&) %&" %&* %&+ ,-../01 23-.4/ 536-781930 :9;087 3< 6/1/413. !"#$ %&' ( ") * + , %-'

6B;m`2 RXRj, lTT2` ;`T? BM ;`22M, aB;MH +Q``2bTQM/BM; iQ i?2 /BQ/2 Hb2` +m``2Mi KQ/mHiBQM BM iBK2X GQr2` ;`T? BM TBMF, iQKB+ ~mQ`2b+2M+2 bB;MH Q7 i?2 ?vT2`}M2 i`MbBiBQMb iQ F = 3 M/ F = 4

Q7 i?2 D1 HBM2 `2+Q`/2/ b  7mM+iBQM Q7 iBK2 BM xQM2  /2i2+iQ`

K2MiBQM2/ i?i i?2 Q#b2`p2/ ~mQ`2b+2M+2 HBM2b ?p2  }MBi2 HBM2rB/i?X h?Bb 7+i `Bb2b 7Q` irQ KBM `2bQMb,

Ç h?2 lM+2`iBMiv S`BM+BTH2, [mMimK K2+?MB+b bii2b i?i i?2 HQM;2` i?2 /2+vBM; iBK2 Q7 M iQKB+ 2t+Bi2/ bii2- i?2 b?Q`i2` i?2 2M2`;vf7`2[m2M+v #`Q/2MBM;- ++Q`/BM; iQ i?2 `2HiBQM ∆ν∆t & h

Ç .QTTH2` #`Q/2MBM;, BM  ;b- i?2 T`iB+H2b ?p2  /Bbi`B#miBQM Q7 bT22/b i?i #`Q/2Mb i?2 7`2[m2M+v Q7 #bQ`TiBQM Q7 i?i T`iB+H2b U7Q` BMbiM+2- rBi? BM+`2bBM; i2KT2`im`2V

AM  i?2`KH Cs #2K- QM2 +M 2tT2+i iQ ?p2 M2;HB;B#H2 .QTTH2` #`Q/2MBM;- r?B+? Bb i?2 +b2

?2`2-bQ i?i i?2 ~mQ`2b+2M+2 HBM2rB/i? Bb i?2 Mim`H QM2X AM i?2 +b2 Q7 i?2 D1 HBM2b- i?2 i`MbBiBQM iBK2

∆tdec M/ i?2 bbQ+Bi2/ HBM2rB/i? ∆νCs `2 (ai2+F kyy3),

∆tdec= 35 ns URXRyV

∆νCs= 4.6 M Hz URXRRV

6Q` +2`iBM ivT2b Q7 iQKB+ +HQ+Fb r?B+? rQ`F rBi? ;b +2HHb- .QTTH2` #`Q/2MBM; ?b iQ #2 iF2M BMiQ ++QmMiX h?2 #Qp2 pHm2 Bb p2`v BKTQ`iMi #2+mb2 i?2 ~mQ`2b+2M+2 bB;MH /2T2M/b QM ?Qr KMv

iQKb mM/2`;Q i?2 U43′<sub>σV i`MbBiBQM, B7 i?2 Hb2` HBM2rB/i? ∆ν</sub>

l Ui?i Bb- i?2 2M2`;v #`Q/2MBM; Q7 i?2

`/BiBQMV Bb rv H`;2` i?M BM URXRRV- i?2 bB;MH BMi2MbBiv rBHH #2 H2bb i?M i?2 KtBKmK TQbbB#H2

pHm2 M/ Kv p`v Qp2` iBK2- +`2iBM; /2i2+iBQM MQBb2X h?2 [m2biBQM Q7 i?2 Hb2` bT2+i`H Tm`Biv

RXkX h?2 PTiB+H *2bBmK 6`2[m2M+v aiM/`/ UP*6aV Rd

Ç wQM2 Ƕ, SmKTBM; Q7 i?2 Cs #2K

q?2M i?2 Hb2` Bb bi#BHBx2/ BM 7`2[m2M+v QM QM2 Q7 i?2 i`MbBiBQM Q7 i?2 D1 HBM2 U43′σV- i?2 M2ti

bi2T Bb iQ +?B2p2 TQTmHiBQM BMp2`bBQM `2;`/BM; iQ r?i ?b #22M bB/ #27Q`2X h?Bb +M #2 /QM2

mbBM; QTiB+H TmKTBM; U};m`2 URXR9VVX b i?2 iQKb `2 2t+Bi2/ i?`Qm;? U43′_{σV- i?2 iQKB+ TQTmHiBQM}

!"# !"# !" #$%&' $("$ )*+ $%_& '(% )*+, )*+ )*-, )*-$%_. '(% /01234205 67849 97670 :;<<;=>

6B;m`2 RXR9, S`BM+BTH2 Q7 QTiB+H TmKTBM; BM i?2 F = 3 bii2X b i?2 Hb2` `/BiBQM +ib- i?2 H2p2H Bb T`Q;`2bbBp2Hv }HH2/ mT rBi? i?2 7mHH iQKB+ TQTmHiBQMX

brBi+?2b iQ i?2 bii2 |3′_{i M/ /2+v 2Bi?2` BM |3i Q` |4iX "mi b i?2 2t+BiiBQM ;Q2b QM- i?2 7`+iBQM Q7}

iQKb BM |4i mM/2`;Q i?2 bK2 +v+H2 b #27Q`2- r?BH2 i?2 7`+iBQM BM |3i bivb BM Bib bii2X A7 i?2 Hb2` TQr2` Bb bm{+B2Mi `2;`/BM; i?2 iQKB+ ~mt- 7i2`  +2`iBM iBK2- HH i?2 iQKb rBHH TQTmHi2 bii2 |3iX P7 +Qm`b2- r2 +QmH/ HbQ +?QQb2 iQ TQTmHi2 i?2 Qi?2` bii2 |4i rBi? i?2 TT`QT`Bi2 i`MbBiBQMX

hQ ;Bp2 bQK2 Q`/2` Q7 K;MBim/2- i?2 TQr2` M22/2/ 7Q` +QKTH2i2 TmKTBM; Q7 i?2 Cs #2K Q7 h?H2b 1H2+i`QM .2pB+2b P*6a Bb #Qmi 2 mW M/ i?2 iBK2 M22/2/ iQ /Q i?i Bb `QmM/  72r µsX 1tT2`BK2MiHHv- i?2 TmKTBM; Bb `2HBx2/ BM i?2 xQM2 Ƕ Q7 i?2 `2bQMiQ`X h?2M- i?2 ~mQ`2b+2M+2 bB;MH Bb /2i2+i2/ BM xQM2 "- r?2`2  i?B`/ Hb2` BMi2`+iBQM iF2b TH+2- rBi? M BMi2``Q;iBM; TQr2` Q7 ∼ 150 µW Dmbi b BM xQM2 X Ai Bb BMi2`2biBM; iQ HQQF i i?2 QTiB+H bB;MH /2i2+i2/ i?2`2 b  7mM+iBQM

!"#$ !"%! !"%$ !"&! !"&$ !"# $"! $"% $"& $"' $"# %"! ! "# $ % & ' ( )* +* , +' -./ 0 1"2* .30 ()*+,-./-0/- 10 2+0- 3 40/,-5.106 +781/5) 7*97106 7+:-, & ;<

6B;m`2 RXR8, .2i2+i2/ ~mQ`2b+2M+2 bB;MH BM xQM2 "- mbBM; i?2 U43′_σV i`MbBiBQMX b Hb2` TQr2` Bb

BM+`2b2/ BM xQM2 Ƕ- KQ`2 M/ KQ`2 iQKb `2 Tmi QM F = 3 H2p2H- bQ i?i i?2 ~mQ`2b+2M+2 bB;MH

/2+`2b2b BM xQM2 "- mMiBH i?2 #+F;`QmM/ MQBb2 Q7 /2i2+iBQM Bb `2+?2/

R3 *?Ti2` RX AMi`Q/m+iBQM iQ iBK2 M/ 7`2[m2M+v TTHB+iBQMb

Bb ?B;?X b i?2 TmKT bi`ib iQ +i- i?2 KtBKmK K;MBim/2 /2+`2b2b iBHH i?2 T2F pHm2 `2+?2b

i?2 #+F;`QmM/ MQBb2 ~QQ`X h?Bb #2?pBQ` +M #2 mM/2`biQQ/ i?MFb iQ i?2 2M2`;v /B;`K URXReVX

AM/22/- b TQTmHiBQM BM bii2 |3i BM+`2b2b iQ i?2 /2i`BK2Mi Q7 |4i- H2bb M/ H2bb iQKb BMi2`+i rBi?

!"#"$ %&''$( !" #$%&' $("$ )*+ )*₊ ,-* ./01 ./0 ./21 ./2 )*₃ ,-*

6B;m`2 RXRe, q?2M +QKTH2i2 TmKTBM; Q7 H2p2H 4 Bb +?B2p2/- i?2 iQKB+ TQTmHiBQM Bb i`MbT`2Mi iQ

i?2 U43′_{σV `/BiBQM}

i?2 Hb2` #2K M/ i?2 MmK#2` Q7 T?QiQMb /2i2+i2/ BM " /2+`2b2b T`QTQ`iBQMHHvX h?mb- i?2 2z2+iBp2 iQiH TmKTBM; Q7 i?2 *b #2K Bb +?B2p2/ r?2M i?2 T2F bB;MH pMBb?2b iQiHHvX >Qr2p2`- b Bi +M #2 MQiB+2/-  HBiiH2 Q7 i?2 bB;MH `2KBMb BM i?2 BbH2b- r?B+? Bb /m2 iQ i?2 7+i i?i  .QTTH2` bT`2/BM; biBHH z2+ib i?2 HQr2` T`i Q7 i?2 HBM2, bQK2 iQKb Q7 p2`v /Bz2`2Mi bT22/b `2 MQi BM~m2M+2/ #v i?2 Hb2` 7`2[m2M+vX

hrQ //BiBQMH TQBMib `2 rQ`i? iQ K2MiBQMX h?2 }`bi QM2 Bb i?i- BM T`+iB+2- i?2 HQ+FBM; Q7 i?2 Hb2` QM i?2 i`MbBiBQM HBM2 BM xQM2  H`2/v TmKTb i?2 iQKB+ #2K- Hi?Qm;? Bi Bb MQi +QKTH2i2 7i2` i?2 iQKb ?p2 +`Qbb2/ i?i xQM2X Ai +QmH/ #2 TQbbB#H2- BM i?2Q`v- iQ +?B2p2 #Qi? HQ+FBM; M/ TmKTBM;

i i?2 bK2 iBK2X >Qr2p2`- i?Bb Bb TQbbB#H2 QMHv rBi? i?2 D1 HBM2 #2+mb2 i?2 TmKTBM; i`MbBiBQMb Q7

i?2 D2 `2 Qp2`b?/Qr2/ #v i?2 p2`v #`B;?i +v+HBM; QM2 UD2 : 45′σ- i?i T`Q/m+2b i H2bi i2M iBK2b

KQ`2 T?QiQMb i?M i?2 TmKTBM; i`MbBiBQMbV r?B+? T`2p2Mib 2{+B2Mi ~mQ`2b+2M+2 bB;MH i`+FBM;X

h?2 Qi?2` TQBMi Bb i?i- +QMi``v iQ i?2 bBKTH2 +b2 ;Bp2M BM (RXkXR)- i?2 2{+B2M+v Q7 *b TmKTBM;

M/ ~mQ`2b+2M+2 /2i2+iBQM Bb MQi 2[mBpH2Mi 7Q` HH i`MbBiBQMbX 6B`bi- QM2 ?b +imHHv iQ iF2 BMiQ ++QmMi i?2 7+i i?i i?2 ?vT2`}M2 H2p2Hb `2 /2;2M2`i2/ BMiQ i?2 2F + 1 w22KM bm#@H2p2Hb- r?B+? bTHBi r?2M  r2F UM/ Hrvb T`2b2MiV K;M2iB+ }2H/ Bb TTHB2/X AM/22/- r?2M TmKTBM; Bb

+QMbB/2`2/-Bi Bb +imHHv BMiQ  T`iB+mH` |F, mFi bii2X "2+mb2 Q7 i?2 p2`v bKHH 2M2`;v /Bz2`2M+2 #2ir22M i?2b2

bii2b- i?2 /2+vBM; Q7 i?2 2t+Bi2/ iQKb mbmHHv bT`2/ Qp2` HH i?2 mF MmK#2`b rBi?  T`QTQ`iBQM

;Bp2M #v i?2 *H2#b+?@:Q`/M +Q2{+B2Mib Q7 i?2 i`MbBiBQM (oMB2` RN3N)X 6Q` i?2 QT2`iBQM Q7 i?2

+HQ+F-i?2 TmKTBM; Q7 +HQ+F-i?2 H2p2H |F, mF = 0i Bb +?Qb2M, Bi Bb BM/22/ MQi b2MbBiBp2 iQ i?2 }`bi@Q`/2` p`BiBQMb

Q7 i?2 K;M2iB+ }2H/ Ui?mb KFBM;  KQ`2 bi#H2 +HQ+F rBi? `2HiBQM iQ 2MpB`QMK2MiH T2`im`#iBQMbVX a2+QM/- QM2 b?QmH/ MQi2 i?i i?2 KtBKmK TQTmHiBQM +?B2p#H2 ∆n /2T2M/b QM i?2 i`MbBiBQM mb2/

(.BK`+[ RNNR)X 6Q` BMbiM+2- i?2 U43′_{σV `2+Q`/2/ #Qp2 ?b  p2`v ;QQ/ TQTmHiBQM BMp2`bBQM `iBQ}

∆n = 16.5% BM |3, mF = 0iX HbQ- 7Q` i?Bb FBM/ Q7 TmKTBM; i`MbBiBQM- i?2 MmK#2` Q7 ~mQ`2b+2M+2

T?QiQMb 2KBii2/ T2` iQK Bb +QMbiMi QM p2`;2- r?i2p2` i?2 /2i2+iBQM +QM/BiBQMb, BM i?2 T`2pBQmb +b2- β = 4 /atomX  `mH2 Q7 i?mK# Bb i?2M iQ `MF i?2 ;HQ#H 2{+B2M+v Q7 i`MbBiBQMb #v i?2B` T`Q/m+i

β_{× ∆nX}

Ç _Kb2v *pBiv M/ wQM2 ", AMi2``Q;iBQM M/ .2i2+iBQM Q7 i?2 *b #2K

PM+2 i?2 #2K Bb TmKT2/ BMiQ |F = 3i- i?2 iQKb +`Qbb M 2H2+i`QK;M2iB+ +pBiv- +HH2/ i?2

RXkX h?2 PTiB+H *2bBmK 6`2[m2M+v aiM/`/ UP*6aV RN

HQM; i?2 #2K /B`2+iBQM- bQ i?i HH i?2 w22KM bm#@H2p2Hb `2 r2HH b2T`i2/X AM i?2 K2MiBK2- i?2

*b mM/2`;Q2b i?2 |3i → |4i mM/2` _6 TQr2` TTHB2/- rBi?  7`2[m2M+v νHF ;2M2`i2/ #v i?2 HQ+H

Qb+BHHiQ`X h?Bb Bb /QM2 #v mbBM; i?2 Ƕ_Kb2v b+?2K2ǶX JQ`2 T`2+Bb2Hv- irQ >6 }2H/ 2t+BiiBQMb BHF i

7`2[m2M+v νHF = νR≈ 9.19 GHz Q`i?Q;QMH iQ i?2 #2K /B`2+iBQM- QM2 i i?2 2Mi`v Q7 i?2

`2bQMiQ`-i?2 Q`2bQMiQ`-i?2` i `2bQMiQ`-i?2 2tBi UbK2 BMi2`+iBQM iBK2 τV +?M;2 `2bQMiQ`-i?2 [mMimK bii2 Q7 `2bQMiQ`-i?2 T`iB+H2- r?BH2 Bi Bb

7`22@2pQHpBM; #2ir22M i?2b2 irQ TQBMib Ui?2 7`22@2pQHmiBQM iBK2 Bb T VX A7 i?2 7`2[m2M+v νR = ν0 Q7

URXNV- i?2M i?2 r?QH2 iQKB+ TQTmHiBQM rBHH ~BT BMiQ |4i i i?2 2M/ Q7 i?2 `2bQMiQ`X A7 i?2`2 Bb 

HBiiH2 /2imMBM; 7`QK i?i 7`2[m2M+v- bQK2 Q7 i?2 iQKb rBHH #2 BM |4i- r?BH2 bQK2 Q7 i?2K rBHH `2KBM BM |3i UBM 7+i- 7`QK  [mMimK K2+?MB+H TQBMi Q7 pB2r- bm+? BMi2`K2/Bi2 *b #2K bii2 Bb  KBt2/

bmT2`TQbBiBQM Q7 i?2 irQ ?vT2`}M2 bii2bVX A7 i?2`2 Bb  i?B`/ U43′_{σV Hb2` BMi2`+iBQM BM i?2 xQM2}

"-Bi rBHH #2 TQbbB#H2 iQ `2/ Qmi i?2 MmK#2` Q7 iQKb BM bii2 |4i- 7Q` i?2 `2bQM K2MiBQM2/ #Qp2 UB7 QM2 `2+Q`/b  ~mQ`2b+2M+2 bB;MH- i?i K2Mb i?2 iQK rb BM bii2 |4i rBi?  T`Q##BHBiv ;Bp2M #v [mMimK +Q2{+B2MibVX !"#$ !"#% "#" "#% !"# $"! $"% $"& &'()*+,-+.-+ /. 0).+ 1 2/3+ 4,5 6 /7 . 8 ' 49 5 ! "# $ % & ' ( ) * +* , +' -./ 0 1"2* .30 &'()*+,-+.-+ /. 0).+ 1 3:" ;<"=>0 3:< 3:% 3:? 3:!< 3:!% 3:!? 1:" _1@"

6B;m`2 RXRd, JBM :`T?, lM/2` KQ/mHi2/ 7`2[m2M+v νHF Q7 i?2 _6 TQr2` BHF M/ biiB+ K;M2iB+

}2H/ B0- i?2 BMi2`72`2M+2 Tii2`M Q7 i?2 _Kb2v *pBiv Bb /2i2+i2/ BM xQM2 "X h?2 b2p2M ;`QmTb Q7

7`BM;2b +Q``2bTQM/ iQ i?2 b2p2M w22KM bm#@H2p2Hb r?B+? `2 2M2`;2iB+HHv bT+2/ Q7 bQK2 10 kHz BM i?2 2tT2`BK2MiH +QM/BiBQMbX "Qt, i x2`Q B@}2H/b M/ MQ _6 TQr2`- i?2 ~i xQM2 #2ir22M i?2 BbH2b Q7 i?2 .QTTH2` #`Q/2MBM; ?b 7mHH TmKTBM; BMiQ |F = 3i bii2

LQr- B7 i?2 2t+BiBM; νR Bb br2Ti `QmM/ ν0- i?2 ~mQ`2b+2M+2 BM xQM2 " rBHH b?Qr i?2 BMi2`72`2M+2

Tii2`M Qp2` iBK2 Q7 i?2 _Kb2v *pBiv rBi? i?Bb 7`2[m2M+v /2imMBM;X h?Bb Bb r?i Bb ?B;?HB;?i2/ BM

};m`2 URXRdVX

h?2 bTHBiiBM; BM 2M2`;v BMiQ i?2 w22KM H2p2Hb Bb +H2`Hv MQiB+2#H2 b b2p2M ;`QmTb Q7 7`BM;2b `2 T`2b2MiX h?2 m = 0 Bb i?2 QM2 Q7 BMi2`2bi 7Q` i?2 +HQ+F QT2`iBQMX AM i?i ;`QmT Q7 Qb+BHHiBQMb- r2 `2

7m`i?2` BMi2`2bi2/ BM i?2 +2Mi`H 7`BM;2 Q7 ?B;?2bi KTHBim/2 U};m`2 URXR3VVX h?2 +?`+i2`BbiB+b Q7 i?i