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1 In the format provided by the authors and unedited. NATURE PHYSICS 1

2 285 x Ω/µ 3 x C c = µ 2 ε r = µ NATURE PHYSICS 2

3 a V SQUID A R 1 A R 2 V s2 V g V s1 V V T room T MC Cu-P RC Cu -P RC Cu -P RC Cu -P RC Cu -P RC Cu -P RC Cu -P RC Cu -P RC Cu-P RC Cu-P RC Cu-P RC 3μm b 12kΩ 4fF 1kΩ 12kΩ 12kΩ Φ 4μm 5nm 5nm 12kΩ 12kΩ 4fF 12kΩ 1kΩ 12 NATURE PHYSICS 3

4 [Z(ω)] [Z(x)] = Z Q 1+ Q2 (1 x x 2 ), 2 2 x = ω/ω ω =(LC) 1/2 C Q = R Z L = L/C I(V ) E J L J NATURE PHYSICS 4

5 a 4 measurement fit b 4 4 mt 3mT I spec (na) I spec (na) I spec (na) 1 2 measurement fit I(V ) 3 4 I spec (na) ev 2 = 482 µ NATURE PHYSICS 5

6 R J Ω J µ E J = hi 2e µ I = J 2eR J L J = Φ 2I R µ I R Ω C Ω = 2eR E c = 2e2 C µ f p = 1 2 L J C L Z = J C Q = R C L J Ω NATURE PHYSICS 6

7 I(V ) [Z(ω)] d 2 I/dV 2 (V ) I(V ) [Z(ω)] 1.5 µ d 2 I/dV 2 (V ) > I(V ) d 2 I/dV 2 (V ) 4 I spec (na) d 2 I/dV 2 (S/V) I(V ) d 2 I/dV 2 (V ) d 2 I/dV 2 (V )= V g = 141 ϕ = NATURE PHYSICS 7

8 ϕ = ϕ = 15 a I sub (na) 1 b I spec (na) I sub (na) V g =-141 mv device φ I (ϕ) = I (ϕ) I (ϕ = ) φ = I (ϕ = ) I (ϕ = ) V g = 141 NATURE PHYSICS 8

9 V = I(V ) V g ev = 2 J µ > 2 G V = 4 φ = I SQUID (na) -4 V g = -1.6V V g = -1.62V V g = -1.65V V g = -1.75V -4 4 V SQUID I(V ) V g = 1.75 I(V ) x V NATURE PHYSICS 9

10 a -d 2 I/dV 2 (S/V) -4 8 b 1 Ψ(δ,σ) 2 Excitation 2 - δ 5 Ψ(δ,σ) 2 Excitation 1 - δ σ = e σ = g φ = device 1 Ψ(δ,σ) 2 GS - δ 2 φ V g = = 97.5±1.7 µ T =.9±.1 ϕ = V g ϕ = NATURE PHYSICS 1

11 -d 2 I/dV 2 (S/V) -4 8 B = -4mT B = mt B = 4mT 1 5 φ device 1 device 1 device φ φ ϕ = B = ±4 V 1 µ V g = 77 NATURE PHYSICS 11

12 ±E (ϕ) hω p = 2E J E C Ĥ = hω p (â â )+E ˆσ 3 H g (ϕ) =g(ϕ) z (â +â)ˆσ 1. z = E C /2E J g(ϕ) g(ϕ) = T 1 T sin 2 (ϕ/2) E (ϕ). ϕ = ε = T z z z.52 Ω x (ϕ) zg(ϕ) NATURE PHYSICS 12

13 Ĥ = E C ˆN 2 + E J (1 cos ˆδ)+Ĥ (ϕ ˆδ), [ˆδ, ˆN] =i Ĥ (φ) = Û(φ) [ cos(φ/2) ˆσ T sin(φ/2) ˆσ 2 ] Û (φ), Û(φ) = exp ( i 1 T ˆσ 1 φ/4) ˆσ 2 ˆσ 3 Ĥ (φ) =V 2 (φ)ˆσ 2 + V 3 (φ)ˆσ 3 V 2 V 3 V 2 (φ) = ( 1 ) ( 1 ) 1 T sin (φ/2) cos T φ/2 cos (φ/2) sin T φ/2, V 3 (φ) = ( 1 ) ( 1 ) 1 T sin (φ/2) sin T φ/2 + cos (φ/2) cos T φ/2, g e Ĥ (φ) g = E (φ) g, Ĥ (φ) e =+E (φ) e, E (φ) ± ˆσ 3 ˆσ 3 ± = ± ± g = c g+ (φ) + + c g (φ), e = c e+ (φ) + + c e (φ), c g+ (φ) =i c e+ (φ) = i E A (φ) V 3 (φ) 2EA (φ)[e A (φ) V 3 (φ)], c g (φ) = E A (φ)+v 3 (φ) 2EA (φ)[e A (φ) V 3 (φ)], c e (φ) = V 2 (φ) 2EA (φ)[e A (φ) V 3 (φ)], V 2 (φ) 2EA (φ)[e A (φ)+v 3 (φ)]. NATURE PHYSICS 13

14 c g+ (φ) 2 + c g (φ) 2 = c e+ (φ) 2 + c e (φ) 2 =1. Ĥ Ψ = E Ψ δ, ± δ ± ˆδ ˆσ 3 ˆδ ˆσ3 δ, ± =(ˆδ δ ) (ˆσ 3 ± ) =± δ δ, ± [, ) M 2/M 2M δ k ± δ k =2k/M k =, ±1, ±2,...,±(M 1)/2 ± ˆσ 3 2M 2M M ω n = E n E Ψ { g, e } φ = ϕ Ψ = δ σ=g,e Ψ(δ, σ) δ, σ, Ψ(δ, σ) = δ, σ Ψ, δ, σ = δ (c σ+ (ϕ) + + c σ (ϕ) ). Ψ(δ, σ) 2 Ψ σ = e φ GS Ĵ(ϕ) Ψ GS Ψ Ĥ Ĵ(ϕ) =E J sin(ˆδ)+ H (ϕ ˆδ) ˆδ. NATURE PHYSICS 14

15 .1 numerical analytical, Eq. (S17) δ.1 2 φ δ = 122 µ T =.57 E J = 165 µ φ φ ϕ φ = ϕ δ δ GS ˆδ GS δ E C = δ E J sin(δ)+ T 4 sin(δ ϕ) 1 T sin 2 [(ϕ δ)/2] =. E J T/4 NATURE PHYSICS 15

16 δ T 4E J ( T/E J ) sin(ϕ) 1 T sin 2 (ϕ/2). φ = ϕ δ δ.12 ϕ /2 φ ϕ ϕ = n n ϕ B s H ( ) 2 = x 2m E F τ z iα x s z τ z + E Z s x + e iφ θ(x) τz τ x + Vδ(x) τ z. m =.23m θ τ x,y,z s x,y,z α E = mα 2 /2 E Z = 1gµ 2 BB E F h =1 x< x> φ x = H Ψ(x) =E Ψ(x), E < E F E,E Z, E F = NATURE PHYSICS 16

17 V =,φ = E = E F H (a) H (b) H (a) = iv x τ z σ z vq τ z ρ z + αk F vq H (b) = iα x τ z σ z + τ x E Z σ z (1 ρ z ). τ x σ z + E Z vq τ y ρ y, τ x,y,z ρ x,y,z s x,y,z σ x,y,z σ k F = 2mE F v = k F /m vq = α 2 k 2 F + E2 Z E F = E,E Z T = E = E F T E Z = E 1 F E Z = T =1/(1 + V 2 /α 2 ) T T =4k 2 F /(4k2 F + V 2 ) T x = E det [ 1 G(E) τ z σ z ( e iφ τ z/2 T 1 )] =, G(E) G(E) =v dq 2i e iq [H (q) E] 1, NATURE PHYSICS 17

18 H (q) v α G(E) α/e Z G(E) G(E) (B) (B) E G(E) (B) (B) () (B) B 2 B () (B) B (B) E F = (B) NATURE PHYSICS 18

19 () (B) B E F 1..8 Δ(B)/Δ E SO = E SO E F /Δ=.1 E SO E F /Δ= ½gμ B B/Δ 1. (B) E EF E Z = 1 2 gµ BB (B) E,± (φ, B) = (1 B 2 /B ) 2 1 T sin 2 (φ/2) ± (1/2)gµ B B. B Φ /A A Φ = h/2e δ(x) NATURE PHYSICS 19

20 (1 B 2 /B 2 ) B = 4 ± 2 g g E,+ + E, B <B E, (φ, B )= B B g φ = g = 1 T µ B B ( 1 B 2 /B 2 ) 3 B > 3 g g < 5.8 g =5 gµ B B (B) = (1 B 2 /B ) 2 1gµ 2 BB (B) E,± (B) g =5 B 15 B E (B ) = (B ) B = 4 B>B E E (B) (B) 15 <B<3 NATURE PHYSICS 2

21 E (B) B = 5, 75, 1, 15, 2 3 B ϕ = NATURE PHYSICS 21

22 a b c E (μev) d E+ ABS () E- ABS () E F E SO -k SO 2E Z kso E tot ()/2 2 4 B (mt) 15 B = 1mT Δ(B) E (μev) e 15 1 E+ ABS () E- ABS () E F = E SO -k SO 2E Z k SO 5 E tot ()/2 Δ(B) 2 4 B (mt) 15 B = 1mT E (μev) f Δ(B) E- ABS () E + () ABS 2 B (mt) B = 1mT E tot ()/ g 15 B = 3mT device 2 h 15 B = 3mT device 2 i 15 B = 3mT device φ device 2 2 φ device 2 2 φ device 2 2 E ± (B) E (B) =E + (B)+E (B) ϕ = (B) B E E ϕ B = 1 B = 3 g = 14.7 ±.6 E EF / =.32 ±.2 g = 11.2 ±.1 B = 4 ± 2 g =5 B > 3 E () = NATURE PHYSICS 22

23 NATURE PHYSICS 23

24 NATURE PHYSICS 24

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