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I(t) C V C L V L C L Q(t) Φ(t) (L, C) Q(t) V C + V L =0 Q(t) C + Q(t) Ld2 =0 dt 2 d 2 Q(t) + Q(t) dt 2 LC =0 d 2 Q(t) + ω dt 0Q(t) 2 =0 2 Q(t) ω0 2 = 1 LC

V L + V C =0 d 2 Φ(t) + Φ(t) dt 2 LC =0 d 2 Φ(t) + ω dt 0Φ(t) 2 =0 2 Φ(t) ω 2 0 = 1 LC

1) 2) I(t) C C V C L Z 0 = 50 Ω L R V L V R C L Z 0 = 50 Ω R Q(t) Ω R V R = dq(t) R + V dt N(t) V N (t)

d 2 Q(t) dt 2 d 2 Q(t) dt 2 + dq(t) R dt L + Q(t) LC =0 +2α dq(t) dt + ω 2 0Q(t) =0 α = R 2L ω2 0 = 1 ζ LC ζ = α ω 0 = R 2 L C = R 2ω 0 C = ω 0 = 1

P d P d = V 2 2R E c E c = 1 4 V 2 C E i V 2 E i = 1 4 ω0l 2 = ω 0 1 4 V 2 C + 1 4 V 2 /(ω 2 0L) V 2 2R = ω 0 RC = ω 0 τ τ

= ω 0 ω 0 = f 0 f 0 Z(ω) = Im[Z(ω)] Re[Z(ω)] Y (ω) = Im[Y (ω)] Re[Y (ω)]

C c Z 0 C L C c Z 0 Z 0 Ω C c Z 0 Ω Y s Y s = jωc c 1 jq 1+q 2 q = ωz 0 C c Re[Y s ] q2 Z 0

q 1 Q = ω 0C Re[Y s ] Q = ω 0CZ 0 q 2 ω 0 Z c ω 0 = Z c = 1 LC L C C = 1 ω 0 Z c Q = 1 q 2 Z 0 Z c Q 1 q 2 C c = 1 ω 0 Z 0 Q

Ω 10 10 ϵ = ϵ r + iϵ i i = 1

A d! =! r + i! i C r G shunt C r G shunt Y c = jωc C ω j j = i C pp = ϵ A d ϵ A d

Y cpp = jωϵ A d Y cpp = jω(ϵ r + iϵ i ) A d A Y cpp = jωϵ r d + jiωϵ A i d Y cpp = jωϵ r A d + ωϵ i A d Y cpp = jωc r + G E D G shunt cpp = Im[Y cpp] Re[Y cpp ] cpp = ϵ r = 1 ϵ i tan δ

A 1 A 2 C 1 C 2 d! 1 =! 1r + i! 1i! 2 =! 2r + i! 2i G 1 G 2 E D ϵ 1,2 C 1,2 G 1,2 tan δ E D ϵ 1 ϵ 2 E D

A 1,A 2 C 1 = ϵ 1r A 1 d G 1 = tanδ 1 C 1 ω C 2 = ϵ 2r A 2 d G 2 = tanδ 2 C 2 ω C tot G tot A 1 C tot = C 1 + C 2 = ϵ 1r d + ϵ A 2 2r d G tot = G 1 + G 2 =tanδ 1 C 1 ω +tanδ 2 C 2 ω Z cp = G tot + jωc tot

Z cp cp = Im[Z cp] Re[Z cp ] cp = C tot tan δ 1 C 1 +tanδ 2 C 2 1 = C 1 tan δ 1 + C 2 tan δ 2 cp C tot C tot 1 = C 1 1 + C 2 1 cp C tot Q 1 C tot Q 2 1 = p 1 + p 2 cp Q 1 Q 2 p i = C i C tot 10 7 10 5

1 µ µ ϵ 1 ϵ 2 E D Y cs1 = jωc 1 + ωc 1 tan δ 1 = G 1 + jωc 1 Y cs2 = jωc 2 + ωc 2 tan δ 2 = G 2 + jωc 2

A C 1 G 1 d 1! 1 =! 1r + i! 1i d 2! 2 =! 2r + i! 2i C 2 G 2 E D ϵ 1,2 C 1,2 G 1,2

C 1,2 = ϵ 1,2r A d 1,2 Z tot Z tot = 1 Y cs1 + 1 Y cs2 Z tot C 2 tan δ 1 + C 1 tan δ 2 + j(c 1 + C 2 ) ωc 1 C 2 C tot = C 1C 2 C 1 +C 2 1 = Re[Z tot] tots Im[Z tot ] 1 = p 1 tan δ 1 + p 2 tan δ 2 tots p i = Ctot C i p 1 = C tot C 1 = C 2 C 1 + C 2 = ( ϵ 2r A/d 2 = 1+ ϵ ) 1 1r d 2 ϵ 1r A/d 1 + ϵ 2r A/d 2 ϵ 2r d 1

A L' l µ = µ r + iµ i R l A µ n L R E D B H n

µ A l L = µn 2 Al Z solenoid = jωl = jωµn 2 Al = jω(µ r +iµ i )n 2 Al = ωµ i n 2 Al+jωµ r n 2 Al = R+jωL solenoid = Im[Z solenoid] Re[Z solenoid ] solenoid = µ r = 1 µ i tan δ l tan δ l = µ i µ r B H

µ 1 = µ 1r + iµ 1i A L 1 l 1 R 1 l 2 L 2 µ 2 = µ 2r + iµ 2i R 2 µ 1,2 B H L 1,2 R 1,2 L 1,2 = µ r 1,2 n 2 Al 1,2 R 1,2 = ω tan δ l 1,2 L 1,2 L tot = n 2 A(µ r 1 l 1 + µ r 2 l 2 ) R tot = ωn 2 A(tan δ l 1 µ r 1 l 1 +tanδ l 2 µ r 2 l 2 )

µ 1 = µ 1r + iµ 1i A 1 l L 1 R 1 L 2 R 2 A µ 2 = µ 2r + iµ 2 2i µ 1,2 B H L 1,2 R 1,2 1 = Re[Z tot] tots Im[Z tot ] 1 = p 1 tan δ l 1 + p 2 tan δ l 2 tots B H p i = µ r i l i /(Σµ r j l j )

B H L 1,2 = µ r 1,2 n 2 A 1,2 l R 1,2 = ω tan δ 1,2 L 1,2 Z 1,2 = R 1,2 + jωl 1,2 Z net = Z 1 Z 2 Z 1 + Z 2 Z net ω2 L 2 1R 2 + ω 2 L 2 2R 1 + jω 3 L 1 L 2 (L 1 + L 2 ) ω 2 (L 1 + L 2 ) 2 1 totp = Re[Z tot] Im[Z tot ] 1 totp = p 2 tan δ l 2 + p 1 tan δ l 1 L tot = L 1L 2 L 1 +L 2 p i = L tot L i

p 1 = L tot L 1 p 1 = ( ) 1 µr 1 A 1 +1 µ r 2 A 2

H ϵ r W t Z c Z c = 60 ( ) 4H ln ϵr.67πw(.8+ t ) W

t W H 2! r H 2 H ϵ r W t t W Z c = 60 ( ) 4H ln ϵr.54πw Ω ϵ r =10

f c,mn a b f c,mn = c (mπ ) 2 ( nπ 2π + µ r ϵ r a b ) 2 c µ r ϵ r m, n

a b c ˆb, ĉ f c, = c 2a µ r ϵ r

a b c f mnl = c (mπ ) 2 ( nπ 2π + µ r ϵ r a b ) 2 + ( lπ c ) 2 c µ r ϵ r m, n l 101 c>a>b 101 c a ϵ r =10

a 1.2mm Al b 25mm Al Vacuum λ/2 Al/Nb 1mm Al/Nb Al/Nb Al/Nb 1mm Sapphire Sapphire c 25mm Al SMA Sapphire Wire bonds Al Wire bonds SMA

L c V U tot L = 1 2 cv 2 t L U surf = 1 2 Lt ϵ E surf 2 dw E surf

σ E surf = σ ϵ 0 ϵ r E surf Q L W E surf = σ ϵ 0 ϵ r = Q LW ϵ 0 ϵ r Q L = cv E surf = cv Wϵ 0 ϵ r U surf = 1 2 Lt ϵ E surf 2 dw U surf = 1 ( cv 2 LtW ϵ 0ϵ r Wϵ 0 ϵ r U surf = 1 t (cv ) 2 L 2 Wϵ 0 ϵ r ) 2

p surf = p surf = p surf = ( )( ) 1 Usurf Utot L L ( )( 1 t 1 (cv ) 2 2 Wϵ 0 ϵ r 2 cv 2 tc Wϵ 0 ϵ r ) 1 t w ϵ r c l c Z c = l c v = 1 lc c = 1 Z c v

p surf = t Wϵ 0 ϵ r Z c v c ϵ r v = c ϵr p surf = t Wϵ 0 ϵr Z c c µ Z c =50Ω 10 5

P surface [10-5 ] 4 3 2 1 Theory Simulation t=1nm 0 0 100 200 300 400 500 Center Trace Width [µm] 600 µ µ

U bulk = 1 2 ϵ 0 ϵ r E 2 dv U vac = 1 2 ϵ 0 E 2 dv p E p E = p E = p E = U bulk U vac + U bulk 1 ϵ0 ϵ 2 r E 2 dv 1 ϵ0 E 2 2 dv + 1 ϵ0 ϵ 2 r E 2 dv ϵ r 1+ϵ r ϵ r =10 p E =.91 p E =.92 α X k = ωµ 0 λ 0 = ωl k

ω µ 0 λ 0 µ 0 λ 0 L k w l k = L k w = µ 0λ 0 w l k l g α = l k l g l g 1 v = lg c g Z 0 = l g c g l g = Z 0 ϵr c α sl = l k l g = µ 0λ 0 c wz 0 ϵr

Kinetic Inductance Ratio [10-3 ] 1.6 1.2 0.8 0.4 0.0 100 200 300 400 Theory Simulation λ 0 =50nm 500 600 Center Trace Width [µm] λ 0 =50 ϵ r =6 Z 0

( πx ) ( πy ) E x,t EM (x, y, z) = E 0x cos sin e j(βz ωt) ( a b πx ) ( πy ) E y,tem (x, y, z) = E 0y sin cos e j(βz ωt) ( a b πx ) ( πy ) H x,t EM (x, y, z) = H 0x sin cos e j(βz ωt) ( a b πx ) ( πy ) H y,tem (x, y, z) = H 0y cos sin e j(βz ωt) a b β β = k = 2π λ mn mn mn E x,t E (x, y, z) = jωµnπ A kcb 2 mn cos E y,te (x, y, z) = jωµnπ kca A 2 mn sin H x,t E (x, y, z) = jβmπ kca A 2 mn sin H y,te (x, y, z) = jβmπ k 2 cb A mn cos ( mπx a ( mπx a ( mπx a ( mπx a ) sin ) cos ( nπy b ( nπy b ( nπy ) cos ) sin b ( nπy b ) e j(βz ωt) ) e j(βz ωt) ) e j(βz ωt) ) e j(βz ωt)

mn E x,t M (x, y, z) = jβmπ k 2 ca B mn cos E y,tm (x, y, z) = jβnπ k 2 cb B mn sin H x,t M (x, y, z) = jωϵmπ B kcb 2 mn sin H y,tm (x, y, z) = jωϵmπ kca B 2 mn cos ( mπx a ( mπx a ( mπx a ( mπx a ) sin ) cos ) cos ( nπy b ( nπy b ( nπy b ( nπy b ) sin ) e j(βz ωt) ) e j(βz ωt) ) e j(βz ωt) ) e j(βz ωt) k k c β = k 2 k 2 c k<k c O = a b 0 0 E TEM E TE/TM dxdy

11 11 11 E x,hw (x, y, z) = 0 E y,hw (x, y, z) = E 0y Θ(y b ( ) (2m +1)π 2 )cos z e jωt L H x,hw (x, y, z) = H y,hw (x, y, z) =0 Θ(y b ) 2 t =0 11 Q c Q c e 2βz z Q c

a Q couple 10 8 10 6 10 4 10 2 10 0 0 20 Room Temperature Data Simulation Evanescent Coupling TM 11 40 Distance [mils] 60 80 b Q couple 10 9 10 7 10 5 10 3 10 1 400 Evanescent coupling TM 11 Evanescent coupling TE 01 Simulation First generation stripline 800 1200 Gap [microns] 1600 S 21 2 β β 2β =5.2mm 1 2β =2.7mm 1 2β =8.2mm 1 2β =8.3mm 1

ϵ r =10 2β 8.2 1 01 ϵ r 10 01 2β =5.2mm 1

2β meas = 2.7mm 1 1 Q tot = k p k Q k Ω

Total Quality Factor [10 5 ] 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 Temperature [K] 0.8 1.0 1200 µ 265, 000 500, 000

Q i Q i 1 Q tot = 2 Q c + 1 Q i Q i = ( 2 1 ) 1 Q c Q tot 2 Q c 1 Q tot

R s X s 1 Q tot +2j df f = α ωµλ 0 (R s + jδx s ) α ω λ 0 δ α T c

a [ppm] df f 0 90 180 270 360 450 0 200 400 600 800 Temperature [mk] 1000 b [ppm] df f 0 20 40 60 80 100 120 0 200 400 600 800 Temperature [mk] 1000 α total = 1.06 10 3 α outer =2.4 10 4 α center =.82 10 4 µ.8 10 3.6 10 3 µ α c.8 10 3

50mm 6B;m`2 jxry, a2+qm/ ;2M2` ibqm bi`bthbm2 /2bB;MX AM i?2 b2+qm/ ;2M2` ibqm Q7 i?2 bi`bthbm2?qh/2`b i?2 K DQ` Bbbm2b iq //`2bb r2`2 i?2 +QmTHBM; iq i?2 `2bQM iq`b b r2hh b i?2 7`2[m2M+v bi #BHBiv Q7 i?2 `2bQM iq`bx hq //`2bb i?2 +QmTHBM; BM@ /BmK ;`QQp2b r2`2 //2/ iq BKT`Qp2 +QMi +i #2ir22M i?2 irq? Hp2bX HbQ- i?2 +mi r b MQr TH +2 i i?2 iqt Q7 i?2 `2+i M;mH ` bi`m+im`2 ` i?2` i? M i i?2 KB//H2X hq //`2bb 7`2[m2M+v bi #BHBiv i?2 KB+`Qr p2 H mm+?2`b r2`2 /2bB;M2/ iq #2 Qp2` i?2 +?BT bq i? i BM/BmK b r2hh b TQ;Q TBMb U#2`vHHBmK@+QTT2` +2M@ i2`v +QmH/ +H KT i?2 2/;2b Q7 i?2 +?BTX lm7q`imm i2hv- HH Q7 i?2b2 +? M;2b /B/ MQi BKT`Qp2 i?2 +QmTHBM; MQ` i?2 7`2[m2M+v bi #BHBivX 6`QK i?bb r2 +QM+Hm/2 i? i? pbm; i`m2 _6 b?q`i Bb M2+2bb `v 7Q`?B;? [m HBiv 7 +iq` `2bQM iq`b BM i?bb /2bB;M b r2hh b Qi?2` /2bB;Mb i? i /`Bp2 +m``2mi +`Qbb M BMi2`7 +2 bbkb@ H `Hv iq r? i Bb 7QmM/ #v Qi?2`b (dj)x ek

H JC = ω c a a + ω a σ z 2 + g(aσ + + a σ ) ω c a, a ω a σ z σ + σ g, e σ z = e e g g σ + = e g σ = g e

g κ γ g κ, γ 2g n

I J V J I J = I 0 sin δ(t) V J = 2e δ(t)

a b c V J s I s! l! r I J I J V J δ = δ l δ r L J

I 0 δ(t) = δ l δ r e δ(t) δ(t) I J = I 0 cos δ(t) δ(t) δ(t) = I J I 0 cos δ(t) δ(t) V J = 2e δ(t) = 2eI 0 cos δ(t) I J V J I J L J L J = 2eI 0 cos δ(t)

q φ H LC = q2 2C + φ2 2L ω Z ω = 1 LC Z = L C q φ a a H LC = ω(a a + 1 /2) a a [a, a ]=1 φ =Φ ZPF (a + a ) q = iq ZPF (a a )

Z Φ ZPF = 2 Q ZPF = 2 Z ( ) H T = ω a (a a + 1 /2) E J cos(φ)+ φ2 2 E J = I 0Φ 0 2π I 0 Φ 0 = h 2e cos φ H T = ω a a a E J 24 φ4 + O(φ 6 ) φ L L J H T ω a a a E J 24 Φ4 ZPF(a + a ) 4

a) b) L J! q C l! Mg E J L J C φ 1 H T ω a a a α 2 a 2 a 2 ω a = ω a α α = E J 12 Φ4 ZPF α α = E 0 E 1

φ q g, e H Tr = ω a e e H T

2C c L J C! C L 2C c L J C Σ C C H LC H O H tot = H T + H LC + H O H T H T ω T a a α 2 a 2 a 2 H LC H LC = ω r (b b + 1 /2)

q T q LC H O = q T q LC C net C net C net = CC Σ + C Σ C C + C C C C C H O = ( iq T,ZPF (a a ) )( iq r,zp F (b b ) ) 1 C net H O H O,rwa = Q T,ZPFQ r,zp F C net (a b + ab (ab + a b )) = Q T,ZPFQ r,zp F C net (a b + ab ) H O,rwa = g(a b + ab ) g g = 1 2 Z T Z LC C net

g ω g ω C c 2 C Σ C C =320 C Σ =70 C c = 5 H tot ω r (b b + 1 /2)+ ω T a a α 2 a 2 a 2 + g(a b + ab ) g g γ T,κ r γ T κ r g 100 γ T 3 κ r 24

a b c d e 6B;m`2 9Xd, ak HH CmM+iBQM UEJ/EC 1V amt2`+qm/m+ibm; Zm#BibX V h?2 Q`B;BM H bk HH DmM+iBQM bmt2`+qm/m+ibm; [m#bi, i?2 *QQT2` T B` #QtX #V h?2 QzbT`BM; Q7 i?2 *QQT2` T B` #Qt `2 i?2 + T +BiBp2Hv b?mmi2/ CQb2T?bQM DmM+iBQM- i` MbKQM M/ +V i?2 BM/m+iBp2Hv b?mmi2/ CQb2T?bQM DmM+iBQM- ~mtqmbmkx /V M/ 2V `2 i?2 j. p2`bbqmb Q7 i?2 i` MbKQM M/ ~mtqmbmk r?b+? /m2 iq i?2b` r2hh /2}M2/ 2H2+@ i`qk ;M2iB+ 2MpB`QMK2Mi? p2- i i?2 r`bibm; Q7 i?bb i?2bbb- i?2 #2bi +Q?2`2M+2 M/ 2M2`;v `2H t ibqm ibk2b Q7 Mv bmt2`+qm/m+ibm; [m#bix 3j

= ω r ω T g H tot ω T (A A+ 1 /2)+ ω r(b B+ 1 /2) α T 2 A 2 A 2 α r 2 B 2 B 2 χa AB B χ 2g2,α T E c,α r 0 φ H 4 = i (ω i a a α i 2 a 2 a 2 ) χ ij a ab b i,j i ω i α i χ ij i j

Y (ω) Y (ω i )=0 L Z c,i = i C i Y (ω) =jωc i + 1 jωl i + 1 R i Y (ω) =j(c i + 1 ω 2 L i )

ω 2 i = 1 L i C i L i = 1 ω 2 i C i Y (ω) =2jC Z c,i = 1 ω i C i C i C = Im[Y (ω)] 2 Z c,i = 2 ω i Im[Y (ω i )] Y (ω) L J C j R j

a b c ω)] (ms) 1 2 3 L j

φ H m = ω m a a α m a 2 a 2 ω m α m ω m = ω 0 α m α m = e2 Z 2 c L j

χ ij 2 α i α j

ϵ r, 9.3 ϵ r, 11.5 µ µ µ

A bridge B substrate substrate C D substrate substrate

0 10 8

< n > ω κ P =< n> ωκ

300K 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 A B 4K 313 LNF 20dB 20dB 359D CalTech 20dB 20dB 20dB 20dB 20dB 20dB 20dB 20dB 20dB 20dB 20dB 20dB 20mK LP 10GHz Ecco 14_1 30dB 30dB 30dB 30dB 30dB 30dB LP 12GHz 10dB 10dB 10dB LP 10GHz LP 10GHz Ecco 14_2 LP 10GHz LP 10GHz LP 10GHz LP 12GHz 10dB 20dB 10dB 20dB 10dB 10dB Ecco 14_4 Ecco 14_2 10dB Ecco 14_7 10dB Ecco 8 Ecco 7 Ecco 4 Ecco 5 180-H Ecco C 10dB 180-H S S D_1_re FS_2 S S JPC 180-H JPSX 43 I I

AWG5014C D 1 I 1 Q 1 I 2 Q 2 D 2 D 3 RF S S Quantum DUT Reference Signal LO AlazarTech

21 20 10 3 S 21 2 = P 2 P 1

S 21 2 I = VG tot P 1 = I2 4G 1 P 2 = I2 G G 2 2 tot S 21 2 = I 2 4G 1 I 2 G 2 tot G 2 S 21 2 = G2 tot 4G 1 G 2 S 21 2 = Q 1Q 2 4Q 2 tot S 21 2 = Q2 C 4Q 2 tot Q C = 2Q tot S 21

g < 1 χ a, b a, b H disp = χa ab b

Amplitude a Readout spectrum f C e χ f C g b Frequency Dispersive readout qubit spectrum

Signal (mv) 100 80 60 40 20 Qubit excited state Qubit ground state 0-4 -2 0 2 4 Readout power (dbm) 6 8

Z 0 cold bath 1%

ω ds Ω f 0 1 ω ds ω dc Ω

a Storage cavity Cooling cavity 1 f c f s,1 2 f c 0 f s,0 1 As κ s χ sc κ c b Ω S Ω C 1,0 Ω C!! sc... (n!1)!! sc n!! sc (n +1)!! sc 1,1 Ω S 0,1... 0,n-1 0,n 0,n+1 0,0 A s χ sc Ω Ω 0, 0

ω dc = ω 0 c χ sc Ω Ω κ s κ

α α α κ κ κ κ κ Ω κ κ κ n 1 κ κ = κ = κ κ Ω κ κ Ω κ

! BC 1,0 1,α! AB!! 0,α 0,0! DA κ = ( 1 κ + 1 κ + 1 ) 1 Ω Ω κ κ κ 3 κ κ κ κ n

κ P (1) = κ P (0) P (1) P (1) + P (0) = 1 P (1) = ( 1+ κ ) 1 κ P (1) ( P (1) = 1+ κ ) n 1 κ n =3 κ κ P (1) 0.9 H/ = ω s b b + ω c c c A s 2 b 2 b 2 A c 2 c 2 c 2 χ sc b bc c

b, c b, c χ sc >κ c κ s A s κ s Ω (b +b) Ω (c + c) Ω Ω U s = (ib bω ds t) U s = (ic cω dc t) s = ω s ω ds c = ω c ω dc ω ds ω dc H/ = s b b + c c c A s 2 b 2 b 2 A c 2 c 2 c 2 χ sc b bc c +Ω (b + b)+ω (c + c) κ s κ c ρ = i[h,ρ]+κ s D[b]ρ + κ c D[c]ρ

κ κ/9 ω d =(ωc g +ωc)/2 e I I m σ I m σ = 1 Tm ηκt m α g (t) α e (t) T 2 dt 2ηκT m n sin m 0 ( ) θ 2 T m η κ α e,g (t) n

α g,α e θ 2σ

( Im σ ) 2 ( ) θ 2ηκT m n sin 2 Γ φ 2 T A = exp(γ φ T ) ( ( θ A = exp 2ηκT m n sin 2 2 A = exp(γt )exp ) ) T ( 2κT m n sin 2 ( θ 2 )) χ κ sin 2 ( θ 2 ) χ2 χ 2 + κ 2 Ae Bn B = 2κT m sin 2 ( θ 2) P d P d De CP d B = C ) χ n = (2κT 2 1 m CP χ 2 + κ 2 d

C π 2 ω d π 2 C ω d =(ωc g + ωc)/2 e

a b Readout Signal (µv) R ˆx (! 2 ) 300 250 200 150 0 Cavity Drive 3! 2 2µs 500 ns R ˆx (!) 5! 7! 9! 2 2 2 Rotation Angle (radians) Measurement 11! 2 π /2 (ωc g + ωc)/2 e κ 1 c 100 π /2

AC Stark Cal. Amplitude (uv) 70 60 50 40 30 20 10 0 10-7 10-6 10-5 10-4 Power (Watts) C

A s χ sc ω i A i χ ij H/ = ω q a a + ω s b b + ω c c c A q 2 a 2 a 2 A s 2 b 2 b 2 A c 2 c 2 c 2 χ qs a ab b χ qc a ac c χ sc b bc c

Readout (µv) 400 380 360 340 320 300 280 8.480 f s,1 2 f s,0 2 2 f s,0 1 8.485 8.490 8.495 Frequency (GHz) 8.500 Data Model f 0 2 2 f 1 2 f 0 1 A s κ s f 0 1 f i = ω i/2π

Readout (µv) 6 4 2 f c 3 f c 2 χ sc f c 1 f c 0 Data Fit 0 9.312 9.316 9.320 Frequency (GHz) n 1.5 χ sc (2π) 1 =2.59 ±.06 f 0 2 2 f 0 1 i = A i/2π s =4.0 c =300 q =26.1 Ω

µ n 1.5 χ sc/2π =2.59 ±.06 χ qs µ χ qc π/2 µ χ ij = χ ij/2π i = A i/2π

f q 7249.46 ±.01 f s 8493.73 ±.02 f c 9320.11 ±.02 q 26.1 ±.3 s 4.0 ±.1 c 300 ± 80 χ qs 21.1 ±.1 χ qc 4.9 ±.1 χ sc 2.59 ±.06 κ s 65 ± 5 κ c 1.7 ±.1 µm

π χ sc >κ c κ s A s κ s π fq n fq 0 nχ qs π π π

n 2 n 2 π π S π S = (1+ π 8 exp ( (χ qs /σ w ) 2)) 1 σ w χ qs χ qs π χ qs =21.1 σ w =4 π π N =0 N =1

Readout (mv) 16 12 8 4 N0 N1 N2 N3 N4 0 0 2000 4000 6000 Amplitude (DAC) 8000 π π

Ω κ f s,0 1 Ω κ κ c

b c Probability P drive (db) 5 0-5 -10 1.0 0.5 0.0 a Stabilization drives ON Stabilization time (Ts) -1 0 1 2 3 Drive frequency ( sc ) 300 ns 1.0 0.8 0.6 0.4 0.2 0.0 Probability -1.5dB -8.5dB P drive (db) detect N in storage N R ˆx,! Probability 5 0-5 -10 1.0 0.5 0.0 Measurement N=0 N=1-1 0 1 2 3 Drive frequency ( sc ) -1.5dB -8.5dB T s π =ω 0 c ω χ /χ 1

π σ t =40 f,0 1 κ c =ω 0 ω χ sc = χ sc 1.5χ sc

0.993 ± 0.005 0.96 ± 0.03 κ c /κ (q,s) A (q,s)/a c χ (q,s)c/κ c N P (N) N =0 N =3 P (0) = 0.96 ± 0.03 P (1) = 0.63 ± 0.02

Q N (α) = 1 π N D α Ψ 2 D α Ψ Ψ N = 3 W (α) = 2 π ( 1) n Q n (α) n=0

χ qs π π σ t =5 P (0) = 0.37 ± 0.03 P (1) = 0.63 ± 0.02 N = 2, 3

p p = P (0) P (1) P (0)+P (1) B T

Wigner Tomography a Stabilization drives ON D!! N R ˆx,! Measurement Stabilization time (Ts) 300 ns b 0.3 α 0.0 Parity -0.3 c α α Parity α α N =1 N =0 (α) (α) N =1

T = hf 0 1 2k tanh 1 (p) h k 0.77 ± 0.06 Ω p =0.95 ±.04 p = 0.26 ± 0.04 P (1) > 0.99

detect N in storage a Stabilization drives ON Stabilization time (Ts) b P drive (db) 5 0-5 300 ns N R ˆx,! Measurement 1.0 0.0-1.0 Polarization c d Polarization Polarization -10 1.0 0.0-1.0-1 1.0 0.0-1.0 0.0 0 1 2 3 Normalized detuning ( sc ) region of T<0 0.5 1.0 40 80 120 160 200 Stabilization time (µs) -1.5 db -8.5 db T s π =ωc 0 ω dc χ /χ sc 1

κ c/κ s 25 26 κ g 2 g 2 κ 2 2 χ κ 2α P (1) >.87 p<.74

P (1) >.99

ω N C = ω0 C Nχ SC =1, =0 N > 1

ω 00 c ω NN c

680µ 2 220µ 2 2 2