Topology, Localization, and Quantum Information in Atomic, Molecular and Optical Systems

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1 Topology, Localization, and Quantum Information in Atomic, Molecular and Optical Systems The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Accessed Citable Link Terms of Use Yao, Norman Ying Topology, Localization, and Quantum Information in Atomic, Molecular and Optical Systems. Doctoral dissertation, Harvard University. May 1, :52:49 AM EDT This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at (Article begins on next page)

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8 40 87 G σσ LL (z) Gσσ LR (z) Πσσ LL (z) Πσσ LR (z)

9 ν =1/2

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21 Energy Site H = ij t ij c i c j + h.c. + i µ i n i t ij {µ i } µ i W W =0 t =0 W t t i + j t µ i µ j j

22 e r/ξ ξ H = α c αc α + α,β,γ,δ V α,β,γ,δ c αc β c γc δ.

23 ν =1/2 ν =1/2

24 1/r 2 H sd = 1 2N k,k,α,β J(k k )c k,α σ αβc k,β J(k k ) σ 1 2 J RKKY S 1 S 2 J RKKY r ξ J RKKY

25 J RKKY (r) = 1 8πr 3 J 2 exρ f ν 2 0 cos(2k f r) J ex ρ f ν o k f r ξ J SC RKKY (r) = 1 8πrξ 2 J 2 exρ f ν 2 0 cos(2k f r)e r/ξ. ξ 2 r ξ (ξ/r) 2 r 1 J RKKY H 0 = k,α ɛ k c k,α c k,α + k [c k c k + c k c k ]. Ψ k =(c k,c k,c k,c k )T H 0

26 H 0 = k Ψ k (ɛ kτ 3 + k τ 2 σ 2 )Ψ k. τ 1,2,3 σ 1,2,3 τ {c k,c k } σ G 0 (k, ω) = 1 iω ɛ k τ 3 k τ 2 σ 2. G(k, k,ω)=g 0 (k, ω)δ(k k )+G 0 (k, ω)t (k, k,ω)g 0 (k,ω) J(k k )=J ex T (ω) = 1 N (SJ ex /2) 2 g 0 (ω) 1 (g 0 (ω)sj ex /2) 2 g 0 (ω) =1/N k G 0(k, ω) S k ω + τ 2 σ 2 g 0 (ω) = πρ f 2 ω. 2 E b < ω = E b

27 2 ( 1+π 2 (JS/2) 2 ρ 2 f ) (ω + ωπ2 (JS/2) 2 ρ f ) 2 =0 E b = 1 π2 (JS/2) 2 ρ 2 f. 1+π 2 (JS/2) 2 ρ 2 f J ex 0 E b J ex E b S =1 0 e 0 e 1 e I =1/

28 H e,n = 0 S 2 z + µ e BS z + µ n BI z + AS z I z, 0 =2.87 µ e = 2.8 µ n = 0.43 A =3.0 ẑ

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30 (a) R 1 R 2 R R1 (b) 2R 2,,,,,! 1 "! 2 " = =,,, Pair 1 Pair 2 R2 R R <r<2r t 1/r α α d d d =1

31 α = β β<α β<α d<α d<α d<α d<β d<β d<β+2 d<β/2 d< αβ α+β d<β/2 d<β/2 d<(β +2)/2 d< α(β+2) α+β+4 1/2 S z H = i ɛ i S z i ij t ij r ij α (S+ i S j + h.c.)+ ij V ij r ij β Sz i S z j ɛ i W α β β α ɛ i ɛ j t ij / r ij α R 1 < r ij < 2R 1 N 1 (R 1 ) (ρr d 1) t/rα 1 W ρ N 1 (R 1 ) R 1 d>α d = α δ t/r1 α

32 Dynamic Polarization (D) ν =1/2 α = β =1 α = β =2 (a) (c) α = β =3/2 α = β =3 (b) (d) Disorder Width (W ) d =1 t =1,V =2 α = β =1 α = β =3/2 α = β =2 α = β =3 W c 10 V S z R 2 δ 1,δ 2 >V(R 2 ) δ 1 δ 2 R 2 2R 2 N 2 (R 1,R 2 ) (n 1 (R 1 )R d 2) V/Rβ 2 t/r α 1, n 1 = ρn 1 N 2 R 2 R 1 R 2 d<β R 1 R 2 V/Rβ 2 t/r α 1 [1, V/Rβ 2 ] R t/r1 α 1 R β/α 2 d< αβ α+β

33 N 3 (R 1,R 2,R 3 ) (n 2 (R 1,R 2 )R d 3) V/Rβ 3 V/R β 2 n 2 = n 1 N 2 R 3 R 1,R 2 R 1 R 2 R 3 d<β/2 R 1 R β/α 2 R 2 R 3 t ij,v ij R 1 <R 2 V/R β 2 VR 2 1/R β+2 2 N 2 α = β d<β/2 α<β+4 α>β+4 α d c =1.5 α = β =3 d c =1.5 α =6 β =3 d c 2.3

34 d =1 α = β =1, 3/2, 2, 3 L =14 ν =1/2 W V ij = V =2 t ij = t =1 D ˆF = j Sz j e i2πj/l η k Dη k =1 k ˆF k k ˆF k k ˆF ˆF. k D Dη k L D 1 D 0 D α =2, 3 α =1 α =3 α =3/2 d =1 1 <α c < 3 α c =2

35 (a) (b) (c) z y x E (d) 1, 1 (e) 1, 1 1, 1 1, 1 1, 1 2B 0, 0 i t ij 1 r 6 ij ŷ ẑ ˆx = 1, 1, = 0, 0 α = β =3 α =6 j ŷ ẑ t ˆx α E d V d 2 /R 3 β =3 B W

36 V d 2 E 0 V 0 ν =1/2, 1/3, 1/4 L =16, 18, 20 H m = BJ 2 d z E = J =0,m j =0 = J =1,m j =1 H dd = 1 d i (1 3ˆr ij ˆr ij )d j 2 i j d rij 3 H dd {, } ɛ i = j i d s d a α = β =3 d rij 3 s,a = 1 dz 1 ± 0 d z 0 2 ɛ i W d sd a a ν(1 ν) 3 a0 0 d =1 d c =3/2 α = β =3

37 Dynamic Polarization (D) (a) β =3 V/t = ν =1/ Disorder Width (W ) Disorder Width (W ) (b) V/t =1 V/t =2 V/t =4 MBL Ergodic /4 1/3 1/2 Filling Fraction (ν) α β =3 W W c 1.4t V/t =1, 2, 4 d c =3 d =2 B = J =1,m j = 1 = J =1,m j =1 J z H = t 2 [ ij r (d i ij 6 + ) 2 (d j ) 2 +(d i ) 2 (d j +) 2] α =6 β =3 d =2 d c 2.3 d c =2.5 H NV = D 0 S 2 z + µ e BS z D 0

38 e Dk2t D D =0 T 1 k T 2 D T2 a 2 0/T 2 T 2 T 1 D T2 D e a 2 0/T a0 T a0 l (l 2 + Dt) d/2 e t/t T a0 10µ T T 1 25

39 2 3 T a0 1µ T 1,T 2 10

40 π/2 τ z i ±1

41 Ĥ = i h i τ z i + ij J ij τ z i τ z j + ijk J ijk τ z i τ z j τ z k +... J ij, J ijk,... ξ τi z τ z i Ĥ... + =( + )/ 2 h ( ) = h + j J jτ z j + j,k J jkτ z j τ z k +... π t/2 h ( ) t/2 t/2 π h ( )

42 spin echo DEER F(t) D(t) W L J z = J F (t) d ξ t/2 π π/2 τ z

43 ψ(t) = R π/2 e iĥ t 2 R π R π/2 e iĥ t 2 R π/2 ψ(0), Rr π/2 = j r (ˆ1 iˆσ y j )/ 2 Rr π =(Rr π/2 ) 2 D(t) ψ(t) ˆτ z ψ(t) = ( 1+e 2iJ Ij τ j ) t 2 j II N τ j j D(t) J Ij j I N J Ij t 1 J Ij t 1 1 D(t) D(t) D osc (t) D(t) = D(t)+D osc (t), D(t) =1/2 N (t), 1/2

44 J Ij exp( j I /ξ) j I ξ log(t) t t 0 /J Ik k = I + d N fast =0 t 0 t t 0 e N/ξ N (t) ξ log(t/t 0 ) D(t) t ξ ln 2 t t 0 e N/ξ D( ) 2 N (1 + t D(t) 2 /t 2 0) α/2 t t 0 e N/ξ =, 2 N t t 0 e N/ξ α = ξ ln 2 D osc (t) D osc (t) D osc (t)

45 1 0.8 fit J z =0.1J J z = J W =6J =3 d =3 N =7 Ĥ = J 2 (Ŝ+ i ij Ŝ j + Ŝ+ j Ŝ i )+J z Ŝi z Ŝz j + i ij h i Ŝ z i Ŝa j a {x, y, z} ±1/2 Ŝ± j = Ŝx j ± iŝy j h i [ W ; W ] J z =0 Ĥ W>0 J z 0 W/J S τ

46 fit J z =0.1J J z = J d =3 d =7 W =8J N =3 α W d = 7 N = 3 α = c 1 / ln(c 2 W ) ξ 1/ ln(w ) D( ) N J z D( ) J z N c/1.8 N L =12 S z ψ(0) D(0) = ψ(0) ˆσ z ψ(0) > 0 D(t) π/2 t =0

47 D(t) =A/(1 + t 2 /t 2 0) α/2 A D osc (t) D(t) D(t) d t 0 [ exp(d/ξ)] d α α α = c 1 / ln(c 2 W ) α = ξ ln 2 ξ 1/ ln(w ) N f(k) =c/1.8 N 1/2 N

48 J,J z T 1 1 J ξ ln(j 1 ) 6ξ 10 J /J J (1 10) 5µ 1 100µ 1 /J 1 (0.5 5) 10 3 J /J

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50 1/2 H = ij JS + i S j + ij J σ + i σ j + i J z S z i σ z i + h.c S σ XY J J J z J 0 σ S {σ i } ±J z

51 J!" J #" J! J J J J J J z S J σ J N =20 Stot z = σtot z =0 σ J =1.0 J z 10.0 J J S ent = ρ A log ρ A = ρ B log ρ B A B D

52 N =8 Sent J =1.0 J =0.01 J z = 10.0 J =1.0 J =0.001 J z = 10.0 J =1.0 J = J z = 10.0 Fractional Polarization k =1 k =2 Time (1/J) Time (1/J) Time (1/J) Time (1/J) N = 16 Fractional Polarization Sent k =1 J =1.0 J =0.01 J z = 10.0 S-chain!!-chain" J =1.0 J =0.001 J z = 10.0 J =1.0 J = J z = 10.0 Time (1/J) Time (1/J) Time (1/J) N = 8 S z tot = σ z tot =0 k =1 S σ k =2 t int t d N = 16 k =1

53 ˆF = j Sz j e i2πj/l Sj z σj z k D k =1 k ˆF k k ˆF k k ˆF ˆF, k D D k N =8 J =10 2, 10 3, 10 4 J S ent t 1/J t 1/J J t 1/J J t int t>t int t>t d t int t d J S ent

54 t int 1/J t int 1/J 2 J, J J z t int J z t int J J S t d J z /J 2 J 2 /J z S t d D k=1 k =2 t e L k =1/L e Dk2t t d L 2 /D D J 2 /J z N =16 e 8 J t int t d

55 Sent J int J SPL quasi MBL thermal t L 2 /D t e L Short chain thermal t L 2 /D J d J 2 /J z SPL t e L Long chain 1/J 1/J int t (1/J) 1/J d 1/J 1/J int 1/J d t (1/J) ρ N = 8 N = 12 N = 16 d ρ/dw W/J S ent 1/J t>t int =1/J int J t>t d =1/J d D J 2 /J z S ent e L L 2 N

56 2.0 Sent J independent J 2 Time (1/J) t int J J 2 J L 2 t d 1/Dk 2 L 2 J =10 2 t int t d J J =10 5 S ent t d σ σ J

57 H W = i b z i S z i + i b z i σ z i b, b W H W ρ ψ = 1 N N i ψ Si+1 z Si z ψ H W ( ) N d ρ dw c. W/J d ρ dw N H T = H + H W J =1.0 J =0.01 J z = <W < N =8, N =16 ρ ψ J/3 ρ H W

58 H h = ij JS i S j + ij J σ i σ j + i J z S z i σ z i. J 0 t int 1/J J S J 2 J S ent J S σ JJ /J z 1/t d J 2 /J z

59

60 ẑ Θ 0, Φ 0 {X, Y, Z} ij {X, Y } H dd = 1 κ 2 i j R 3 ij [ ] d i d j 3(d i ˆR ij )(d j ˆR ij ), κ 1/4πɛ 0 µ 0 /4π R ij d i d j 0 ±1 ẑ

61 ±1 0 Ω + Ω Ω +, Ω 0 B = α( 1 + β 1 ) D = α ( β ) α =Ω + / Ω αβ =Ω / Ω Ω = Ω 2 + Ω + 2 E 0 = Ω 2 / E B = + Ω 2 / E D = d R 0 κd 2 /R0 3 Ω 2 / D 0 B 0 D B 0 a i = B 0 i n i = a i a i H B = ij t ij a i a j + 1 V ij n i n j, 2 i j t ij = B i 0 j H dd 0 i B j t ii = j i ( 0 i0 j H dd 0 i 0 j B i 0 j H dd B i 0 j ) V ij = B i B j H dd B i B j + 0 i 0 j H dd 0 i 0 j B i 0 j H dd B i 0 j 0 i B j H dd 0 i B j N i = i a i a i κd 2 /R0 3 H dd H B

62 J =1 Ω (r) Ω + (r) Ω ± δ J = 1 l p l t l t l W (p l) β π/n κ =1,i j t ij = d2 01 R 3 [ χ i (q 0 + [q 2 ]σ x + [q 2 ]σ y )χ j ], t ii = j i 2 q 0 R 3 (d0 d B i (d 0 ) 2 ), V ij =2 q 0 R 3 [ d B i d B j d 0 d B i d 0 d B j +(d 0 ) 2], d 0 d B ẑ 0 B d χ i = α i (1,β i ) T i q 0 = 1(1 2 3cos2 (Φ Φ 0 )sin 2 (Θ 0 )) q 2 = 3[cos(Φ Φ 2 0)cosΘ 0 i sin(φ Φ 0 )] 2 σ (R, Φ) R ij ij R Φ q 0 q 2 d B i

63 π/n (Θ 0, Φ 0 )=(sin 1 ( 2/3),π/4) q 0 =0 ˆX Ŷ H dd d + i d+ j d i d j d ± = (d x ± id y )/ 2 t ˆX ij = d2 01 χ R0 3 i tŷij = d2 01 χ R0 3 i [ 1 2 σx [ 1 2 σx σy 3 2 σy ] ] χ j, χ j. β =Ω /Ω + β β B 1 1 β Φ=π/4 W (p) = p t ij l Ψ l =arg[w (p l )] = arg[t 2 l t 2 l ] t l t l θ l =arg(t l ) θ l =arg(t l )= θ l+1 Ψ l =2θ l 2θ l+1 π/n θ l+1 = η l π η 2N β θ l β l+1 β l = sin( π 3 η + l π 2N ) sin( π 3 + η l π 2N ),

64 4N β 1 π/n t ij H B β = β 1,β 2 β 1 β 2 β 1 β 2 d B i ±1

65 β = β 1,β 2 Ψ, Ψ g 1 g 2 /R 3 β 2 β 1 c = 1 4π dkx dk y ( kx ˆd ky ˆd) ˆd H(k) = d(k) σ + f(k) ẑ H m = BJ 2 d z E + H D, B J d z ẑ E H D J =1 J, M E M J 0 ±1 ±1

66 d 1 d B i = d 1 V ij = 2 q 0 R 3 (d0 d 1 ) 2. V ij /t ij (d 0 d 1 ) 2 /d Ω H B N ν

67 (Θ 0, Φ 0 )=(0.46, 0.42) β 1 =3.6e 2.69i β 2 =5.8e 5.63i 1/27R 0 > 10 S(R, 0) = n(r)n(0) ν = 1/2 (Θ 0, Φ 0 )=(0.66,π/4) β 1 = 2.82e iφ 1 β 2 = 4.84e iφ 2 (d 0 d 1 ) 2 /d ĝ 1 ĝ 2 N s = 24 2π θ 1 4π θ 2

68 c = 1 ν =1/2 (d 0 d 1 ) 2 /d S(R, 0) = n(r)n(0) N s =32 ĝ 1 ĝ 2 φ 1 = φ 2 =0.1 k 2 =0,k 1 =0, 2π/3, 4π/3 ĝ 2 N s =24 ĝ 2

69

70 E H m = BJ 2 d z E E 0, 0 J =1 1, 1 1, 0 1, 1 J ẑ J, m E

71 φ {X, Y } ẑ Θ 0 Φ 0 {x, y, z} {X, Y, Z} J =0, 1 0, 0 M J = N s = 24 Θ 0 E =0 J =1

72 J =1 e 1 e 2 M H r = [ e 1 (Ω 1 1, 1 +Ω 2 1, 0 )+ e 2 (Ω 3 1, 0 +Ω 4 1, 1 )+ ] Ω i = 1 Ω(Ω 2 Ω 4 1, 1 Ω 1 Ω 4 1, 0 +Ω 1 Ω 3 1, 1 ) Ω H dd = 1 κ 2 i j R 3 ij [ ] d i d j 3(d i ˆR ij )(d j ˆR ij ), κ =1/(4πɛ 0 ) R ij i j d i d j d R 0 d κd 2 /R0 3 Ω i {, } 2B a i = i j i t ij = i j H dd i j

73 V ij = i j H dd i j + i j H dd i j i j H dd i j i j H dd i j H B = ij t ij a i a j + 1 V ij n i n j, 2 i j N = i a i a i t ii {a, b, A, B} J =1 a i t ij V ij g 1 g 2 C = 1 f 11.5 J =1 ν ν =1/2 N s =44 Θ 0 =cos 1 (1/ 3) ν =1/2

74 a b A B J =1 φ g 1 g 2 f 11.5 C = 1 {Θ 0, Φ 0 } = {0.68, 5.83} σ xy = 1 2π F (θx,θ y )dθ x dθ y = 0.5 {θ x,θ y } F (θ x,θ y )= ( Ψ θ y Ψ θ x Ψ θ x Ψ θ y ) ν =1/2 Q torus = ( N uc +1 N b N uc+1 2N b ) ( Nuc 1 N b N uc+1 2N b ) Nuc = N s /2 N b ν =1/2 Θ 0

75 σ xy ν =1/2 ν =1/2

76 d µ e 1 e 2 J,m = 2, ±2 v =41 (3) 1 Σ +

77 ν = 1/2 ν =1/2 N s = 24 N b =6 1/(3R 0 ) 3 (k x,k y )=(0, 0) (k x,k y )=( π, 0) ν =1/2 k x,k y N s = 16 N s = 44 N b =5 36 Q torus µ = E Nb +1 E Nb E Nb N b ν =1/2 ν =1/2 de/dθ 0 Θ 0 de/dθ 0 E =0.4 8

78 H lattice H hf Ω i H lattice H hf H hf 1 =160 E = B/d 0.5 Ω i E 1,0 E 1,1 M H lattice H hf Ω i

79

80 (a) (b) E b " η η " BCS ψ BCS E b r < ξ η η r ξ r r<ξ

81 r ξ H 0 =,σ ɛ c,σ c,σ + [c c + c c ]. ψ BCS J E b = 1 (πjsn 0/2) 2 β2 = 1 1+(πJSN 0 /2) 2 1+β 2 N 0 tan(δ) β = πjsn 0 /2 E b = cos(2δ) E b δ

82 T K exp ( 1/JN 0 ) T K β 1/ ln( /T K ) 1 φ sh ( ) 1 e /ξ sin(2δ) r ξ r<ξ η η η η r L r R ẑ H int = J σ d σ[s L f( L )c σ( )c σ ( )+S R f( R )c σ( )c σ ( )], S L(R) f( ) H int = J σ d d σ[s L e i( )r L f, + S R e i( )r R f, ]c σ, c σ, f

83 Ψ =(c,,c, ) H 0 = d Ψ [ɛ τ z + τ x ]Ψ τ H int = J d d Ψ [S Le i( )r L f, + S R e i( )r R f, ]Ψ + E 0 E 0 = J d f, [S L + S R ] H T = H 0 + H int d n = d (u n, ψ, + v n, ψ, ) H T = ε n d nd n 1 ε n = 2 n n n E tot = 1 ε n = E V n ε n (d nd n 1 2 ). dɛ ɛ δρ(ɛ) E V δρ(ɛ) I(r) I(r) =E, tot E, tot = 1 2 dɛ ɛ [δρ, (ɛ) δρ, (ɛ)]. δρ(ɛ) = 1 { [G π, (z) G(0) (z)]} z = ɛ + i0 + G (0) (z) =[z (ɛ τ z + τ x )] 1 G, (z)

84 T G, (z) =G (0) (z)+g(0) (z)t, G(0) (z), T, T T δρ(ɛ) = 1 π { [G(0) (z)t, G(0) (z)]} = 1 π { [JSΠ(1 JSG) 1 ]} Π G S 4 4 Π ll (z) = G ll (z) = d G (0) (z)g(0) (z)ei ( l l ) d G (0) (z)ei ( l l ) S ll = S l δ ll τ 0. τ 0 l l {L, R} J 1 [JSΠ(1 JSG) 1 ] [J 2 SΠSG] I( ) = E fβ 2 π(k f r) cos(2k fr)e 2r 3 ξ F1 [ 2r ξ ] + β2 (k f r) 2 sin2 (k f r)e 2r ξ F2 [ 2r ξ ]. k f r = L R

85 Interaction (khz) 400! 0! -400! 100! 150! R (nm) J YSR! RKKY E b ! E f = 11.7 k f = N 0 = 35 3 ξ =1.6µ J YSR E b 10 2 J YSR r ξ F 1 [α] =α dxe α( x ) F 0 2 [α] = 2 π x ) dx e α( 0 (x 2 +1) 1/r 2 /E f r E f /( k f ) ξ I( ) e 2r ξ O(J 2 )

86 J E b 0 η ɛ ɛ 0 [G, (z)] E b F (E b ) [1 SG(E b )] = 0. k f r 1 η η/e b F (E b )+η F (E b )=0

87 1 β η = 1 cos 2 (k f r) 1 β 2(k f e 2r r) 2 ξ. F (E b )+η F (E b )+ 1 2 η2 F (E b )+ 1 6 η3 F (E b )=0 η = 2 cos(k f r) (k f r) 2 e 2r ξ β 1 I(r) J YSR = η η E b 0 β 1 J YSR η J YSR = 1 cos 2 (k f r) 2r 1 β 2(k f r) 2 e ξ, 1 1 β U(r)c L, c R, U(r) =cos(k fr)/(k f r) 2E b β E b J YSR J YSR k f r> 1 1 β > ξ r.

88 r ξ J YSR r λ f ξ ξ / k f ξ J YSR = η η 1 r 2 β J YSR J YSR J YSR >J RKKY r J YSR

89 2 T K I I T K

90 ɛ = E b E b E b

91 ( /T K ) c S =1/2 /T K > ( /T K ) c S G =1/2 /T K < ( /T K ) c S G =0 H BCS = dk [ ξ (2π) 3 k c kσ c kσ + ( c k c k + h.c.)] σ 1/2 S 1 S 2 H int = J 2 S 1ψ 1σ ψ 1 + J 2 S 2ψ 2σ ψ 2. ψ 1 ψ 2 H T = H BCS + H int T K

92 H T P S 2 =0 SU c (2) Q x =(Q + + Q )/2 Q y =(Q + Q )/2i Q z = 1 dk 2 σ (c (2π) 3 kσ c kσ 1) 2 Q+ = dk c (2π) 3 k c π k Q = ( Q +) 0 U c (1) Q P S Q S 0 S =0 P = Q =0 T 0 S =1 P =+ Q =0 D + D S =1/2 P = ± Q =1 S 2 Q =2 P = π

93 (S, Q, P ) S 0 (0, 0, ) T 0 (1, 0, +) D ± ( 1, 1, ±) 2 S 2 (0, 2, ) /T K /T K 1 S S = sin(k F R) k F R R = R 1 R 2 k F S D ± I I J I/T K

94 S = 0.1 I/T K /T K S =1 S =1/2 S =0 S 2 S 0 /T K T 0 I<0 S 0 I>0 I T K, =0 S 2 I, T K Q =0 Q =2 S 0 S 2 =0 =0

95 /T K I/T K = 0.58 S = 0.1 S 2 D + T 0 S 0 T 0 I eff = E S0 E T0 I T K T K ω T K S =0 S =1/2 T K I 0 D + T K S =0 S =0 D ± /T K D ± D + S =1 1/2 /T K

96 S 2 φ ( ) 1 e /ζ sin(2δ) E = 1 β2 1+β 2 ζ β tan(δ) =JSN 0 π/2 N 0 T k F R β E sh =0 I <0 I >0 F 1 S 1 σψ 1 T 0 D + D 4

97 (a) 1.0 Esh/ Esh/ (b) I<0 I>0 Molecular Doublet Mol. Triplet Molecular Doublet Triplet Kondo Mol. Singlet β k F R 4.1 Kondo Singlet k F R 2.6 k F R 4.1 β = JNSπ/2 β 0.86 β 1.3 k F R Pb =1.55 T 20 mk T 0 S 0 E = hν S 0 S 0 D ± E S 0 di/dv S 0 T 0

98 T 0 D + D T 0 S 0 hν = E (D +,S 2 ) (D +,T 0 ) S 2 S 0

99

100 g κ

101 1/2 H = H 0 + H H 0 = N 1 i=1 κ(s+ i S i+1 + S i S+ i+1 ) H = g(s 0 + S1 + S + N+1 S N + ) S ± = S x ± is y S = σ / 2 σ =1 g κ H c i = e iπ i 1 0 S + j S j S i H 0 H 0 = N 1 i=1 κ(c i c i+1 + c i c i+1 ) f k = 1 N jkπ A j=1 sin N+1 c j k =1,,N A =(N+1 2 )1/2 H 0 = N k=1 E kf k f k E k =2κ cos kπ N+1 t k = g A H = N t k (c 0f k +( 1) k 1 c N+1 f k + ), k=1 kπ sin N N+1 k = z (N +1)/2 H t z g/a

102 g κ κ a b k = z g κ/ N E z E z±1 κ/n ( 1) z 1 c N+1 H eff = t z (c 0f z + c N+1 f z + ) H eff τ = π 2tz U eff = e iτh eff =( 1) f z f z (1 (c 0 + c N+1 )(c 0 + c N+1 )) {(1,c 0,c N+1,c 0c N+1 ) 00 0,N+1} U fermi eff =( 1) n 0+n N+1 +n z ( 1) n 0n N+1 0,N+1, n θ = f θ f θ H eff

103 0,N+1 =( 1) n 0n N+1 Φ i =(α + β ) 0 (α + β ) N+1 Ψ M,nz Ψ M = N j=1 S+ j S j n z N Φ f =( 0,j N+1,j ) 0,N+1 0,N+1 Φ i j=1 a a = a a = a a a b a b b b b F = i=1,2,3 [σi E(σ i )]

104 E e ikτ K N N H = i,j K i,js + i S j ɛ =1 F g/κ g κ ɛ ( ) 2 5 t k k z 3 E k [1 + ( 1) k+z cos(e k τ)] z = N+1 2 ( ) 10 t 2 k k z 3 E k N κ ɛ 0 g τ τ N t z t z κ/n τ 1/t z N/κ N N N E =0 H = (S0 z + SN+1 z ) N

105 =E k t k E =0 g L g R g L g R τ N/g

106 N =7 g/κ ɛ 0 = 10 3 τ 1/κ

107 g

108 T 1

109 L =12 N 1 H = κσi x σi+1 x + i=1 N i=1 Bσ z i κ B σi x σ ± i =(σi x ± iσ y i )/2 c i = σ+ i e iπ i 1 j=1 σ+ j σ j N 1 H JW = κ(c i c i+1 + c i c i+1 c ic i+1 c ic i+1 ) i=1 N B(c i c i c i c i ) i=1

110 H JW φ A φ φ =(c 1,c 2,...,c N,c 1,c 2,...,c N )T A ɛ ɛ Λ= 0 0 ɛ ɛ 2 ψ ψaψ T =Λ ±ɛ k d k = ψ 2k 1,j φ j d k = ψ 2k,jφ j k =1,,N H JW = N ɛ k (d k d k d k d k ), k=1 d ɛ k κ 2 + B 2 2Bκcos q k q k = kπ/(n +1) 0 N +1 g B H = g(σ x 0σ x 1 + σ x Nσ x N+1)+B (σ z 0 + σ z N+1).

111 H JW = g(c 0c 1 + c 0c 1 + c 1c 0 c 0 c 1 ) g(c N c N+1 + c N c N+1 + c N+1 c N c N c N+1 ) + B (c 0c 0 c 0 c 0 + c N+1 c N+1 c N+1 c N+1 ). B = ɛ z d z c i = N k=1 (ψt ) i,2k 1 d k + N k=1 (ψt ) i,2k d k c 1 c N d gψ 2z 1,1 = gψ 2z 1,N B, ɛ z ɛ z±1 H eff ɛ z (d zd z d z d z)+ɛ z (c 0c 0 c 0 c 0)+ɛ z (c N+1 c N+1 c N+1 c N+1 ) gψ 2z 1,1 (c 0d z + d zc 0 ) gψ 2z 1,N (c N+1 d z + d zc N+1 ). B<κ

112 g t "E g t "E >> gt (1) (2) g τ = π 2gψ2z 1,1 U eff = e iτh eff =( 1) n z ( 1) (c 0 +c N+1 )(c 0+c N+1 )/2 = ( 1) nz (1 (c 0 + c N+1 )(c 0 + c N+1 )), n z = d zd z U eff Ψ={ Ω,c 0 Ω,c N+1 Ω,c 0c N+1 Ω } Ω c 0 c N U eff Ψ=( 1) n z Ψ e iπ i 1 j=1 σ+ j σ j 2 =

113 a b = a b = a b a b 0

114 0 (QR) 1 2 N-1 N N+1 (QR) 0 N +1 X F = [ σ i E(σ i ) ], i=x,y,z E H = g(σ + 0 σ 1 + σ + N σ N+1 + )+ N 1 i=1 κ(σ+ i σ i+1 + ) U H t =2τ ρ DS ch

115 F DS = = = i=x,y,z i=x,y,z i=x,y,z [ σ i 0U(σ i 0 ρ DS ch )U ] [ U σ i 0U(σ i 0 ρ DS ch ) ] [ σ i 0(t)(σ i 0 ρ DS ch ) ], σ i 0(t) M = e ikt K (N +2) (N +2) H = N+1 i,j=0 K ijc i c j c m = i n K mnc n c m (t) = n M mnc n σ + 0 (t) =U σ + 0 U = U c 0U = i M 0ic i = i M 0iσ + i e iπσ+ l σ l, l<i σ0(t) z = 2c 0(t)c 0 (t) 1= 1+2 M0iM 0j c i c j ij = 1+2 M0iM 0j σ + i σ j e iπσ+ l σ l, ij i<l<j c 0 F DS σ ± =(σ x ± iσ y )/2 [σ0(t)(σ x 0 x ρ ch )] = [ (σ 0 + (t)+σ0 (t))((σ σ0 ) ρ ch ) ] σ 0 + (t)(σ0 ρ ch ) σ0 (t)(σ 0 + ρ ch ) i =0

116 [ σ + 0 (t)(σ0 ρ ch ) ] [ = ( ] M0iσ + i e iπσ+ l σ l )(σ 0 ρ ch ) i l<i = [ ] M00σ 0 + σ0 ρ ch = M 00. [ σ 0 (t)(σ + 0 ρ ch ) ] = M 00 σ z [σ0(t)(σ z 0 z ρ ch )] = [ σ0 z ρ ch ] [ + (2 ] M0iM 0j σ + i σ j e iπσ+ l σ l )(σ z 0 ρ ch ) ij i<l<j = [ 2M 00M 00 σ + 0 σ 0 σ z 0 ρ ch ] =2 M00 2, i = j i = j =0 [σ z 0]=0 F DS = (M 00 + M 00 + M 00 2 ). N +1 0 F SS = i=x,y,z [ σ i 0(t)(ρ SS ch σ i N+1) ],

117 ρ SS ch {0,,N} F SS σ x 0(t) = c 0(t)+c 0 (t) = i M 0ic i + M 0ic i = i i 1 [{ (M 0i )σi x + (M 0i )σ y i } ( σl z )]. l=0 i N +1 [σ x 0(t)(ρ ch σ x N+1)] = 2 (M 0,N+1 ) [ρ SS ch N ( σl z )]. l=0 σ y σ z [σ z 0(t)(ρ SS ch σz N+1 )] = 2 M 0,N+1 2 F SS = [2 (M 0,N+1) [ρ SS ch N ( σl z )] + M 0,N+1 2 ). l=0 F SS =1 M 0,N+1 =1 [ρ SS N ch l=0 ( σz l )] =1 P = N l=0 ( σz l ) = a b = a b {0 a, 0 b, 1,,N,(N +1) b, (N +1) a } U b {0 b, 1,,N,(N +1) b } U a

118 {0 a, 1,,N,(N +1) a } U = U b U a F enc = i=x,y,z [ σ i N+1(t)(σ i 0 ρ PP ch ρ N+1 ) ]. ρ PP ch {1,,N} σi 0 0 ρ N+1 (N +1) 0a,N+1 a 0b,N+1 b = P 2 = H Ua U a H Ua H Ua = g(c 0 a e iπn 0 b c 1 + c N eiπn (N+1) b c (N+1)a + ) F enc = 1 6 (2 M 0,N+1 2 [ M 2 0,N+1 M 0,0 M N+1,N+1 ] + M 0,N i M N+1,i M i,0 2 ) M

119 1/r 3 H B = N ωa i a i + i=1 N 1 i=1 κ(a i a i+1 + a i+1 a i). b k = 1 A A = (N +1)/2 k =1,,N H = k (ω + ɛ k)b k b k j sin jkπ N+1 a j ɛ k =2κcos( kπ ) N+1 H B = g(a 0a 1 + a N a N+1 + )+ω (a 0a 0 + a N+1 a N+1) g ω a 1 a N b k H B + H B = N t k (a 0b k +( 1) k 1 a N+1 b k + ) k=1 + ω (a 0a 0 + a N+1 a N+1)+ N (ω + ɛ k )b k b k, t k =(g/a)sin[kπ/(n +1)] b z ω = ω + ɛ z t z ɛ z ɛ z±1 H B eff = 2t z (η 0b z + b zη 0 ) η 0 =1/ 2(a 0 + a N+1 ) k=1

120 ξ ± =1/ 2(η 0 ± b z ) H B eff = 2t z (ξ +ξ + + ξ ξ ). H B eff τ B = π/( 2t z ) U B eff = e ihb eff τ B =( 1) ξ + ξ + ( 1) ξ ξ (U B eff ) ξ ± (U B eff )= ξ ± a 0 a N+1 a 0 (τ) (U B eff) a 0 (U B eff) = a N+1, a N+1 (τ) (U B eff) a N+1 (U B eff) = a 0, 0 1 a N+1 (τ) =M N+1,0 a 0 + ɛa ɛ ɛ =1 M N+1,0 2 g 2 a ɛ a i i =1,...,N +1 N +1 n N+1 (τ) =(1 ɛ) n 0 + ɛ n ɛ n i = a i a i n ɛ kt/ω > 1 g g ω/(kt)

121 40 Number of States T1 NV (units of ms) T1 NV (units of s) Participation Ratio Number of States Participation Ratio Number of States Participation Ratio Number of States Participation Ratio N = 11 T 1 d = 10 κ = 50 σ d σ d g L g R N = 51 N = 51 κ = 50

122 g g T 1 ɛ = L k z(g 2 ψ k,l 2 + g 2 R 2 k ψ k,r 2 2 k )+N t T 1, g L(R) ψ k,l(r) k z k N t T 1 N

123 g L g R t z = g L ψ z,l = g R ψ z,r t = π/ 2t z ɛ = k z ( ψk,l 2 gl 2 2 k + ψ ) z,l 2 ψ k,r 2 Nπ + ψ z,r 2 2 k 2T1 g L ψ z,l, ( g L = Nπ 3 2 2T 1 ψ z,l k z ψ k,l 2 2 k ) + ψ 1 z,l 2 ψ k,r 2. ψ z,r 2 2 k 1/ T 1 10

124 {J i } J κ =1 g 0.7 T 1 N =11 N =51 1 ɛ 200 N =51 T 1

125 N PR = 1 N i=1 ψ i 4 N PR O(N) N PR N N PR σ κ σ κ 0.5κ < 2/3 T 1 5 gψ κ/n g = g M (N) κ N +2

126 g/κ N 1/6 N > 90% N = 100 J i = 1 2 (i +1)(N +1 i) H = N i=0 N+1 J i (σ + i σ i+1 + h.c.)+ i=0 h 2 σz i, h H = ij K ijc i c j K ij = J i δ j,i+1 + J j δ i,j+1 + hδ i,j H = N+1 k=0 ω kf k f k ω k = k + h N+1 2 c i (t) = j M ij(t)c i (0) h = N+1 t =2π M(2π) = 2 c i (2π) =c i (0) {J i } J i = J N i ψ H ψ ik =( 1) N+1+k ψ N+1 i,k

127 h = 3 (N +1) t = π 2 M ij = k ψ N+1 i,k ψ jk = δ N+1 i,j. {0, 1,...N} [ρ SS ch P ]=1 M 0,N+1 =1 F SS =1 h = 3(N +1) U 2 P = ( i) N+1 UP 2 = ( 1) N+1 F SS = [2 M 0,N+1 + M 0,N+1 2 ), F enc = [2 M 0,N+1 2 M 2 0,N+1 M 0,0 M N+1,N+1 + M 0,N i M N+1,i M i,0 2 ]. M 0,N+1 1 J i = 2 1 (i +1)(N +1 i) g κ J 0 = J N = g J 1 = J 2 =... = J N 1 = κ g/κ

128 1/T 1 M 0,N+1 1 N =2, 3 J i = 1 2 (i +1)(N +1 i) N>3 g = g M (N) ω k k N+1 2 h =0 F enc g M N 1/6 τ N 1/r 3 e iπ i 1 j=1 σ+ j σ j

129 XX N =12 90% N =10 98%

130 1/2 H N = γ e B S γn B I + A S z I z + A (S x I x + S y I y ), S 1/2 I A = A = N 1 H N = κ Si z Si+1 z + i=1 N (ω 0 + δ i )Si z, i=1 κ ω 0 δ i a 1 H eff = κsi z Si+1 z + JSNV z (Sa z + Sb z )+ i=1 N 1 i=b κs z i S z i+1, J a b

131 a)! 5-1%3-"617'01-"8234"53'. "!!!"#$%&'( "!!!!!!!!!!!! /&0010"(234 "!!!!)*"+,+-.( " b) J 9" 3 " % " * "!" c) - - N (1,N) (2,N 1) N +1 Q = H CP H CP Q M Q L Q M Q L

132 U eff = e ih eff T /2 S x NV e ih eff T /2 S x NV = e iκ S z i Sz i+1 T U local = e ih eff T e iκ S z i Sz i+1 T = e ijsz NV (Sz a+s z b )T κ(t + T )=2πm m U local N N th (N 1) st N +1 Q n+1 =( H i CP i ) n+1 U local Q M

133 Magnetic field gradient : Nitrogen impurity : Two-qubit NV Register Q L Q M Q L U directed = e ijsz NV Sz N T b H med = J(S z NV 1 + S z NV 2 )S z N b,

134 NV 1 NV 2 N b Q M ( H i CP i ) n+1

135 !"#$%&'()'*+#,-+%%&.'/-0",-1-2'!3/4'51,& '! "! "! "! #! "! $!!!! " "!!!! # #!!!! = %!! " 678'929%&: '! Q k m U p (k )! Q k #! 678'929%&: '! $ A B X U p = X 1 x π CP X 1 CP CP Q k =( H CP) k U (k) p = Q k U pq k n n +1 k = n 1 U swap = HU (k) p H XU (k) p Z HU (k) p H, X x Z z π

136

137 1/r 2 A B a R

138 %! $ " # A B a R λ Ω

139 1/r 6 1/2 H = 2 σi z + Ω 2 i σi x + i i<j C p r i r j p P i P j. Ω C p P i = =(1+σ z i )/2 m s =0 m s =1 H AB = i C p r A r i P p A P C p i + r B r i P p B P i. σa z σz B a R =[ζ(p)(p +1)C p / ] 1/p Ω=0

140 R max p r (r) p r (r) =n exp( nr) n R max = n 1 log N N = C p (a R R max ) p, L L A B E int = E E E + E, E αβ α A β B E int b 2 /L b

141 ε ε ε 1 + ε 2 ε 1 ε 2 ε 1 =exp[ 2 G t g/( λ)] G t g λ ε 1 =exp( c G t g / ) c G 1/L α = c G / = α 0 /L ε 2 = ε 2 (γt g ) γ t g ε 2 =1 exp[ (γt g ) δ ] (γt g ) δ δ

142 γ L γ = γ 0 L/L 0 γ 0 L 0 L 0 a R ε =exp( α 0 /Lt g )+(γ 0 L/L 0 t g ) δ. t opt α γ t opt = δl log[l 0 α 0 /(L 2 γ 0 )]/α 0 ε = ( δ L2 γ 0 log L ) δ 0α 0. L 0 α 0 L 2 γ 0 1/L 2 N a b ψ A,B =( A,B + A,B )/ 2

143 ψ SC = i i Ω(t) = ( ) Ω 0 sin 2 8t/t0 1+16t 2 /t 2 0 (t) = 0 [1 5exp( 4t/t 0 )], t 0 t = t 0 t π = π /E int π H H t g =2t 0 + t π F = ρ 2 AB ρ AB A B G t 0 F =1 cexp( d G t 0 ) c d ρ AB

144 F G t 0 G t 0 N = 34 t 0 Ω 0 p =3 C 3 = 100Ω 0 a 3 0 =2.3Ω 0 b =3a t 0 L/ a R F = 1 2 [1 c exp( d Gt 0 )] { 1+exp [ (γ 0 L/ a R t g ) 3]} δ =3 β t 0 G E int t 0 A B

145 S p =6 p =3 n =43 γ 0 = 10 KHz F =0.95 Ω 0 =2π 3.2 MHz 0 =2π 7 MHz C 3 n 3 =2π 320 MHz a =1µm γ 0 = 100 Hz Ω 0 =2π 80 KHz 0 =2π 170 KHz a =2nm T 2 m s =+1 m s = 1 T 1

146 γ 0

147 r a S =1 W

148 W 2.87 m s = 1 m s =1 m Ω m 0 1 H r = NΩ( 0 W +h.c.) 0 m s =0 W W = i.... N i NΩ/ H r = J W W J = NΩ 2 / m J J V dd W

149 m m>1 0 W Ω ext V dd J V dd Ω ext N =100 r =20nm m s =0, 1 σ α V ij = ( ) 1 3cos 2 µ 2 ϑ ij r i r j 3 { 1 4 [ 1+σ (i) z ][ 1+σ (j) z ] (i) σ + σ (j) } σ (i) σ (j) +, r i µ ϑ ij r i r j H = /2 i σ(i) z +Ω i σ(i) x + i<j V ij 0 m s =1 W H eff = N c µ 2 R 3 ( 1 q, 0 0 q,w +h.c.),

150 N c W 1 q, 0 m s =1 0 R =100nm m 2 m =3 φ ( N 2 N c = i...0 N φ ), i N c N Ω h 100 MHz N c 70 N Ω N c > 50 N c N c Ω NΩ/

151 N c Ω=0 Ω=h 110 MHz N c 70 = h 4 GHz) N c Ω 1 q, 0 m s =1 p q π t π R H eff R r V c t π 1/R 3

152 m s = 1 p q t π 600 µs V c 1/R 3 N N W N T 2 i W [ p W = p T2 1 W σ (i) z W 2] = 4 ( N p T ), N p T2 N

153 N T 2 W T eff 2 T 1 m s =1 m s =0 W m s =1 m s =0 p 1 0 T 1 p 0 1 T 1 p 1 0 T 1 0 m s =0 W 0 p W 0 = p T σ (i) 1 W 2 = p T1 1 0 N, N T W m s =1 T p 0 1 T 1 N p 0 1 T 1 T 1 T 2 T 1 /N T 2

154 R = 100 nm W T 2 T 1 1 T 1

155 t g t π T 1 /N T 2 ε =1 exp[ (4t π /T eff 2 ) 3 ] ε =10 2 T eff 2 =11ms 0 1 W W W 100 khz m s =1 A 2.14 MHz 14 N N N t π =70µs ε =10 2 T eff 2 =700µs ε =10 4 T eff 2 =3ms W

156

157

158

159 I =1/2 S =1 B Ω MW 0 e 1 e Ω RF 1 e t e n π 1 e n e τ π/2 n e e n n e S =1 0 e

160 0 e 1 e I =1/2 15 H e,n = 0 S 2 z + µ e BS z + µ n BI z + AS z I z, 0 =2.87 µ e = 2.8 µ n = 0.43 A =3.0 ẑ n e

161 0 e I z B z, µ 10µ

162 B z (y) = db z dy y + B z,0

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164 1/2 H int =4κSzS 1 z 2 + (ω 0 + δ i )Sz, i κ ω 0 δ i H drive = i=1,2 2Ω isx i cos[(ω 0 + δ i )t] (x, y, z) (z, y, x) i=1,2 H int = κ(s + 1 S 2 + S 1 S + 2 )+Ω 1 S 1 z +Ω 2 S 2 z. H int Ω 1 Ω 2 κ Ω 1 Ω 2 κ Ω 1 Ω 2 H int = i κ(s+ i S i+1 + S i S+ i+1 )+ i Ω isz i

165 κ 1/2 κ i,i+1 Ω i Ω i+1 H FFST = g(s + NV 1 S 1 N 1 + S + NV 2 S N + )+ i=1 κ(s + i S i+1 + S i S+ i+1 )

166 Ω i Ω j g κ g Ω

167 H FFST g B z (y)

168 N = T1 NV N = T1 NV T N 1 T N T NV 1

169 p SS err N(p SS off + p adia + p dip + p SS T 1 + p SS T 2). ( ) p SS off Ω 2 i g Ω i g p adia ( ) κ 2 p dip Ω i p SS T 1 T 1 p SS T 2 N N N p FFST err p FFST off + p fermi + p g + p FFST T 1 + p FFST T 2. p SS err p FFST err p fermi g/ N κ/n ( g/ N κ/n )2 p g g p SS err t FFST

170 T NV T 1 T 1 10 T 1 50 ɛ 1.4%

171

172 g H = ( ) Ω( ) Ω N S N x +4κS NV z S N z, Ω Ω N B = D = B + 2Ω 0 0 2Ω B Ω H = D D ( + 2Ω2 ) Ω2 1 2 Ω N( + N + N ) +2κ( B D + D B )( + N + N + ), ± N = N ± 2 N S N x 2κ { ( + D + D + ) 2Ω ( D + D ) } ( + N N + N + N ). { D, } g κ Ω

173 κ + κ 2 / 2 0 (α + β ) (α + β D ) π D D D Ω D

174

175

176 1/2

177 6B;m`2 8R, a+?2k ib+ `2T`2b2Mi ibqm Q7 iqtqhq;b+ HHv T`Qi2+i2/ bi i2 i` Mb72` h?2 ;`2v /`QTH2i `2T`2b2Mib k. `` v Q7 BMi2` +ibm; btbmb imm2/ BMiQ i?2 *ag" T? b2x Zm MimK btbm@`2;bbi2`b +QKTQb2/ Q7 i` Mb72` [m#bi U;`22MV M/ K2KQ`v [m#bi U;QH/V `2 `` M;2/ `QmM/ i?2 2/;2 Q7 i?2 k. /`QTH2i M/ +QmTHBM; #2ir22M i?2k Q++m`b i?`qm;? i?2 +?B` H 2/;2 KQ/2X URV "v K T@ TBM; i?2 [m MimK BM7Q`K ibqm QMiQ 72`KBQMB+ r p2@t +F2i U#Hm2V i` p2hbm; HQM; i?2 2/;2- i?2 [m MimK bi i2 + M #2 i` Mb72``2/ iq `2KQi2 `2;Bbi2`X h?2 r p2t +F2i i` p2hb QMHv BM i?2 /B`2+@ ibqm Q7 i?2 #Hm2 ``Qrc i?bb +?B` HBiv T`2p2Mib KQ/2 HQ+ HBx ibqm M/ /2bi`m+iBp2 # +Fb+ ii2`bm;x i bt2+b}2/ ibk2 i i?2 `2KQi2 `2;Bbi2` HQ+ ibqm- i?2 +QmTHBM; Bb im`m2/ QM M/ i?2 r p2t +F2i Bb + Tim`2/ UkVX :Bp2M M M+BHH `v K2KQ`v [m#bi M/ HQ+ H `2;Bbi2` K MBTmH ibqmb- irq@[m#bi ; i2 UjV + M #2 T2`7Q`K2/ #27Q`2 i?2 [m MimK bi i2 Bb i` Mb72``2/ # +F iq i?2 Q`B;BM H `2;Bbi2` M/ biq`2/ U9@8VX h?bb HHQrb 7Q` mmbp2`b H +QKTmi ibqm #2ir22M i?2 K2KQ`v [m#bib Q7 bt ib HHv b2t ` i2/ `2;Bbi2`bX?QM2v+QK# H iib+2 b /2TB+i2/ BM 6B;X 8k (jkd)x h?2 bbq+b i2/ > KBHiQMB M M im` HHv ;2M2` HBx2b i?2 EBi 2p KQ/2H (jrn) M/ 72 im`2b +?B` H btbm HB[mB/ ;`QmM/ bi i2 U*aG" T? b2vh0 = 1! x x 1! y y 1! z z κσi σj + κσi σj + κσi σj, 2 x,x! 2 y,y! 2 z,z! links links UdXRV links r?2`2 1σ `2 S mhb btbm QT2` iq`b U! = 1VX h?2 KQ/2H K v #2 bqhp2/ #v BMi`Q/m+BM; 7Qm` J DQ` M QT2` iq`b- {γ 0, γ 1, γ 2, γ 3 } 7Q` 2 +? btbm- b b?qrm b+?2k ib+ HHv BM 6B;X 8k M/ #v `2T`2b2MiBM; i?2 btbm H;2#` b, σ x = iγ 1 γ 0 - σ y = iγ 2 γ 0 - σ z = iγ 3 γ 0 (jk8- jkd)x h?2 J DQ` M QT2` iq`b `2 >2`KBiB M M/ b ibb7v i?2 bi M/ `/ MiB+QKKmi ibqm `2H ibqm {γ l, γ m } = 2δlm X h?2 >BH#2`i bt +2 bbq+b i2/ rbi? i?2 T?vbB+ H btbm Bb irq@/bk2mbbqm H bm#bt +2 Q7 i?2 2ti2M/2/ 7Qm`@/BK2MbBQM H J DQ` M >BH#2`i bt +2c i?mb- r2 Kmbi BKTQb2 i?2 ; m;2 T`QD2+iBQM- P = 1+D 2 r?2`2 D = γ 1 γ 2 γ 3 γ 0 (jk8)x R8N

178 H γ = i 4 κ i,j Û i,j γ 0 i γ 0 j, Ûi,j = iγ α i γ α j α ij ij Ûi,j Û i,j Ûl,m {U i,j = ±1} {U i,j } γ 0 U i,j 2 w(p) = ij p U i,j p p ij U i,j =+1 γ 0 π/2 w(p) =+1 π π/2 π/2 w(p) = 1 v Q κ i,j Q k,i(iu i,j )Q k,j = δ kk ɛ k H γ = 1 N/2 2 k= N/2 ɛ kc k c k c k = 1 2 j Q k,jγj 0 N k

179 ɛ k = ɛ k c k = c k k>0 H γ = k>0 ɛ k (c k c k 1 2 ), c k c k b U i,j H T = H 0 + H int H int L R H int = S 2 (σz L + σ z R)+g L σ β L σβ a + g R σ η R ση b. S β,η g L g R

180 Ûi,j γ 0 i π π/2 U ij w(p) = 1

181 a b H int = S 2 (iγl 3γ0 L + iγ3 R γ0 R )+g LγL 1γ1 aγl 0γ0 a + g R γr 1 γ1 b γ0 R γ0 b σ x σ x U i,j U L,a U R,b H T = H + H int = k>0 ɛ k (c k c k 1 2 ) + S (c L c L 1 2 )+ S(c R c R 1 2 ) g L U L,a (c L + c L ) i 2 ( k Q k,ac k + k Q k,a c k ) g R U R,b (c R + c R ) i 2 ( k Q k,bc k + k Q k,b c k ), c L,R =1/2(γL,R 0 iγ L,R 3 ) c L,R =1/2(γ L,R 0 + iγ L,R 3 ) σ z

182 L R S > 0 g L,g R < S c k S k ɛ k H eff = i g L Q k,a c L 2 c k i g R Q k,b c R 2 c k + h.c. τ ɛ κ/l l τ l/κ

183 61 40 y k y a = π b =0.46κ 0.14κ 0.17κ S g L (t) g L (t) = vf(t), dt t f(t ) 2 f(t) v g R (t) S l

184 T N v n p e v/t n p π N v π ξ a a π

185 v H p H T H p H e = v dp 2π pc pc p = v dx γ(x)(i )γ(x),

186 c p = c p {c k } p {γ(x),γ(y)} = δ(x y) c p H e = λ dx γ(x)(i )γ(x)(i ) 2 γ(x)(i ) 3 γ(x), λ Γ int p ɛ p k B T Γ int p Γ int p λ2 p 13 + λ2 p 11 T 2 + O(T 4 ). v v κ λ κ( κ κ )2 a 7 Γ int p κ2 S ( κ κ )4 (ap) 14 S = vp p<1/a Γ dec p σ α i Γ dec p S(ω)

187 e d/ξ d i S(ω 0 ) e ω 0/k B T ω 0 =2 v S l Γ dec p e S/k B T + le V /k B T. S(ω) 1 ω 2 +1/t 2 c t c Γ dec p 1/ 2 v v 1/t c

188 e S/k B T S l

189 L R R L

190

191

192 N ρ = i ρ i ρ i Q = { +,, + i, i, 0, 1 } ρ F tol ρ i M i

193 F i =1/ Q ρ i Tr[ρ i M i (ρ i )] p h = 1 Q Tr[P acc M(ρ)] 1 e ND(F exp F tol ), ρ Q Q = Q N P acc M = i M i F exp =1/N i F i D F tol F tol N F tol N F tol N (2F tol 1)N 2F tol 1 p d = 1 Q ρ Q Tr [ P 2 acct (ρ) ] e ND(2F tol 1 2/3), T 2F tol 1 > 2/3 5/6

194 a b original qticket:... cloned qticket:... F tol N " 1 " 2 " 3 minimum overlap " N #2 F tol N " 1 " 2 " 3 " N #1 " N 1" " N #2 " N #1 " N "# +" 0" " Z challenge questions " X F tol F tol N (2F tol 1)N Ftol cv = 1+1/ 2 2

195 X Z { 0, +, 0,, 1, +, 1,, +, 0,, 0, +, 1,, 1 }. n r n r 2 X Z Ftol cv r n F exp >Ftol cv n

196 r p cv h ( rd(fexp F cv 1 e )) n tol. F cv tol > 1+1/ / Ftol cv p cv d ( ) 2 v ( 2 1/2+e rd(f tol 1+1/ 2 2 )) n, v F dishonest <F tol <F exp

197 , 0 0, + 1, + 0, + 0, + +, 1, 0 1, + Z X 0, 0 0, 1 1, 1 0, 1, + +,, + +, n = 4 r =2 8 F tol =3/4 F tol =3/4 F tol > 1/2+1/ 8

198 c

199 c (c+1)(c+2) X 1,...,X n {0, 1} δ i S {1,...,n} Pr [ i S X ] i i S δ i Pr[ n X i γn] e nd(γ δ) i=1 δ := n 1 N i=1 δ i γ δ γ 1 D(p q) =p ln p q +(1 p)ln 1 p 1 q (X =1)=p (X =1)=q

200 P ρ acc N F tol ρ = N i=1 ρ i 0 F tol 1 P ρ acc = N ( bi ρ i + b i ρ i ). b: b 1 F tol N i=1 b {0, 1} N N b 1 = N i=1 b i b i =1 b i ρ i = ρ i b 1 F tol N 1 b M i F i F exp =1/N i F i F exp >F tol p v 1 e ND(F tol F exp). X =(X 1,...,X N ) Pr[ X = b]= 1 Q = N i=1 [ Tr M(ρ) ρ Q N ( bi ρ i + b ) ] i ρ i i=1 1 Tr [ M i (ρ i )(b i ρ i + 6 b i ρ i ) ] ρ i Q

201 X i Pr[X i ]=F i 1 Q ρ Q Tr[P accm(ρ)] ρ = Pr[ N i=1 X i F tol N] L 1 µ

202 (a) (c) (b) (d) ±1 { +1, 1 } S =1 m s =0, ±1 E B ( =1) H NV =(D 0 + d E z )S 2 z + µ B g s S B d [ Ex (S x S y + S y S x )+E y (S 2 x S 2 y) ],

203 D 0 /2π 2.88 g s 2 µ B d d ±1 B = g s µ B B z / E ±1 E = E 0 (a + a ) a ω m E 0 ±1 0 = B ω m D 0 ±1 0 H i = g ( σ + i a + a σ i ) σ ± i = ±1 i 1 i g J z = 1 2 i 1 i 1 1 i 1 J ± = J x ± ij y = i σ± i H = ω m a a + B J z + g ( a J + aj + ), g

204 L w, h ( ) 1/2 g 2π 180 L 3 w GHz, ρe ρ E (L, w, h) =(1, 0.1, 0.1) µ ω m /2π 1 g/2π 1 g e /2π 10 T 2 η = g2 T 2 γ n th γ = ω m /Q n th =(e ω m/k B T 1) 1 T Q =10 6 T 2 =10 T =4 η 0.8 g = B ω m H ( ) e R He R R = g a J aj + (g/ ) 2 H eff = ω m a a + ( B + λa a ) J z + λ 2 J +J, λ =2g 2 / J + J = J 2 J 2 z + J z J

205 J 2 z ψ 0 x J x ψ 0 = J ψ 0 J 2 y = J 2 z = J/2 J 2 z z J z J 2 min = 1 2 ( V + V 2 + V 2 yz ξ 2 = 2J J min 2 J x 2, ) V ± = J 2 y ± J 2 z Vyz = J y J z + J z J y /2 ξ 2 < 1 H eff [ ρ = i λ 2 J z 2 + ( B + λa a ) ] J z,ρ + 1 D[σ 2T z]ρ i 2 +Γ γ ( n th +1)D[J ]+Γ γ n th D[J + ], i D[c]ρ = cρc 1 2 ( c cρ + ρc c ) T 1 2 λ/2 B T 1

206 Γ γ = γg 2 / 2 n = a a n ψ 0 N =100 n th T 1 2 Γ γ ξ 2 Γ γ T 2 N J 1 n th 1 ξ 2 4Γ γ n th Jλ 2 t + t T 2. t ξ 2 opt 2 Jη, t opt = T 2 / Jη J ξopt 2 J 2/3 n = a a J z

207 1 0.5 ξ (a) 10 n th = J λt ξ 2 opt 0.1 (b) N N N N = 100 T 2 = 10 n th n th =1 T 2 =1 ω m /2π =1 g/2π =1 Q = 10 6 T 2 H int (t) =λj z f(t)δn(t). f(t) J z J z π δn(t) =n(t) n n

208 n S n (ω) = dte iωt δn(t)δn(0) J + (t) = e χ e iµ(jz 1/2) J + (0), J 2 +(t) J + (t)j z (t) χ µ dω χ = λ 2 F (ωτ) 2π µ = λ 2 dω 2π ω 2 Sn (ω), K(ωτ) A ω 2 n (ω), S n (ω) =(S n (ω)+s n ( ω)) /2 A n (ω) =(S n (ω) S n ( ω)) /2 F (ωτ) = ω2 dte iωt f(t) 2 τ 2 π K(ωτ) µ F K F M =4 S n (ω) =2γ n th ( n th +1)/(ω 2 + γ 2 ) ω =0 χ 0 (t) = 1λ2 n 2 2 th t2 n th 1 T2 2/λ n th t = t opt n th > J M π

209 t χ th λ 2 γ n 2 th t3 /M 2 Γ γ T 1 2 M n th γt2 ω dr = ω m + δ S n (ω) ω = ±δ n dr n th n dr ( n th +1) n dr t/m =2π/δ π χ dr ( λ δ ) 2 ndr n th γt µ λ2 δ n drt n dr n th χ th Γ γ n dr M =4 g J eff <J J 1 g i i g i/ i g2 i i g2 i / i g4 i g i

210 Sn(ω) f (t) t/τ F (ωτ) ξ (a) 0.1 (b) K (ωτ) δ 0 δ ω J λt ξ 2 min (c) F K M =4 f(t) M =4 n th = 10 n dr = 10 3, , 10 6 n th = 50, 10 M =4 g/2π =1 T 2 = 10 N = 100 ω m /2π =1 Q = 10 6 n dr (1, 0.1, 0.1) µ N N eff 100 η 1 H int = λ ( σ 1 + σ2 +h.c. )

211 4

212 σ y Q = ν/ ν ν ν ν

213 Excitation Laser Si Photodiode Diamond Film Bragg Mirror Microwave Line µ > 0 ±

214 H gs =(D gs + d σ z )S 2 z + gµ b S B + d σ x (S x S y + S y S x )+d σ y (S 2 x S 2 y) d, C 3v D gs µ b S k k = {x, y, z} σ D gs S z ω 0 ±1 ω D gs ω D gs B 1 =2b 1 cos(2πωt)ˆx H gs V = e 2πiωtS2 z H gs =(D gs + d σ z ω)s 2 z + gµ b B z S z + gµ b b 1 S x

215 ρ = 1 i [H gs,ρ]+ k L k ρl k 1 2 L k L kρ 1 2 ρl k L k ρ L k r k I Ω=gµ b b 1 =D gs ω F (I,Ω, ) = γρ ss 22 + γ2 κ+γ ρss 33 ρ ss 22 ρ ss 33 F σ y (τ) = πQ (S/N) τ τ S/N C 1

216 A V 3 E 3 A I D es γ orbital states electron spin sublevels D gs κ λ 1 E 1 A1 S B 9.5 x 105 I 0 9 F(!) (photons/sec) C ν Detuning from resonance,! (Hz) 3 E 3 A 1 E 1 A I γ κ λ S z 0 ±1 C 1 T2 =88 σ y (τ) = τ 1/2 Q S/N

217 D gs T δω δωt T 1 T T c T c T c T 2 m f =0 π T T 2 π ) 2 x ) τ π x τ π ) 2 x θ ) φ θ φ t p θ = gµ b b 1 t p Ω 1 gµ b b 1 (D gs Ω),gµ b B z

218 U echo = e i π 2 Sx e i(δωs2 z +bs z )τ e i π 2 Sx e i π 2 Sx e i π 2 Sx e i(δωs2 z +bs z )τ e i π 2 Sx = WW W W = e i π 2 Sx e i(δωs2 z +bs z)τ e i π 2 Sx = e i(δω ˆX bs y )τ, W W = e i π 2 S x e i(δωs2 z +bs z )τ e i π 2 S x = e i(δω ˆX+bS y)τ, ˆX ˆX = [ ˆX,S y ]=0 W W U echo = e 2i(D gs Ω)τ ˆX ˆX 0 π 4 ˆx U 45 = e i π 4 S y U 45 0 = 1 2 ( 0 i + + ±1 S z U echo U 45 0 = 1 ( ) ie iδωτ + + e iδωτ 0. 2 Sz 2 δν S z B A S z I z π/2 π/4 S z

219 Optical Pumping Z π π π 4 T T 4 TF } S 2 z X Y S z S z S 2 z S z S 2 z Sz 2 2T S z ψ 0 = m s =0 U echo ψ f = U echo U 45 ψ 0 = 1 2 sin(φ) cos(φ)( ) φ =(D gs ω)t = δωt ˆM ˆM = a b ( ) a b ˆM

220 ψ f δω ω 0 = 1 ω 0 ˆM ˆM / ω ˆM 2 = ˆM 2 ˆM 2 2a 3b ˆM M τ = M T δω ω 0 M = ξ D gs Tτ ξ 5 b 0 a λ/γ T = T 2 1 D gs =2870 δω/ω 0 = / τ 0.2% N 1/ N N µ 3 3 1/ N / τ / τ 100µ 1 2 σ pulsed y / τ

221 T 1/2 2T e (2T/T 2) n n 3 T 2 T e n =1 2 S/N T e D gs 1µ

222 T 2, 10 20µ D gs dd gs /dt = 74.2(7) dd gs /dt 100 dd gs /dt =100 D gs dd gs /dt 75 dd gs /dt

223 δ D gs V ij S i S j = κ 2 3ˆri ˆr j δ ij r 3 S i S j S i κ a r (1) = aŷ r (2) = 3a ˆx a 2 2ŷ r(3) = 3a a ˆx 2 2ŷ H = V ij S i S j = 3κ 4a 3 (S2 z 2/3). ɛ kl H = S i S j V ij = κ 2 ( ) 3ˆr i ˆr j δ ij ɛ r k r 3 kl ˆr l S i S j = 3κ 2a 3 [ 3 4 (ɛ xx + ɛ yy )(S 2 z 2/3) (ɛ xx ɛ yy )(S 2 x S 2 y) (ɛ xy + ɛ yx )(S x S y + S y S x ) 1 2 ɛ zx(s x S z + S z S x ) 1 2 ɛ zy(s y S z + S z S y )]. α = 3 4 (ɛ xx + ɛ yy ) 3κ 2a 3 3κ 4a 3 a =2.38 α = 4.32(ɛ xx + ɛ yy ) =2.88

224 (a) "(T) (b) area A!d, Ed Diamond 1 Diamond 2 "0 clock 1 clock 2 (c)!1, E1 clamp1 Diamond clamp2 z T0 T!2, E2 Pz=Fz/A=Y! #T dd gs /dt η 1,2 E 1,2 ɛ xx = ɛ yy = ɛ zz dd gs dt ( ) =( ) 15, 1K 5 δ 3 ddgs dt

225 E 1 η c1 E 2 η c2 <η c1 T ɛ 1,2 η d (1 + η c1,2 E c1,2 /E d ) T T T 0 ω 0 T T

226 ω 1 (T ) = ω 0 + β 1 T ω 2 (T ) = ω 0 + β 2 T β 1,2 = dɛ dd gs dt dɛ = η d (1 + η c1,2 E c1,2 /E d )(dd gs /dɛ) τ =0 T 0 T 0 t φ 1,2 (t) =ω 0 t + t 0 β 1,2 T (t )dt ± φ 0, φ 0 = ξ t/ T 2,N φ(t) =φ 2 (t) φ 1 (t) = t 0 β 1,2 T (t )dt ± 2 φ 0 β 1,2 β 2 β 1 φ 1 (t) φ 2 (t) t φ 1(t) = φ 1 (t) β 1 T (t )dt 0 ( t ) = φ 1 (t) β 1,2 T (t )dt 0 = φ 1 (t) β 1 β 1,2 ( φ(t) 2 φ 0 ). β 1 β 1,2 t t ω 1 ω 1 = = ( ξ β1 (1+2 T2 NtD gs β 1,2 ) 2 ) 1/2 ( ) ( ( ) ) 2 1/2 ω β ω T =T 0 β 1,2

227 D ZFS (khz) (a) (b) (c) Brass Tungsten khz, 6.92 mk Temperature (mk) D ZFS (khz) T=5 mk T=10 mk Position (mm) diamond d clamp D gs β 1,2 β 1 ν beat 10 Q 10 6 D gs β 1,2 η d (1 + η c1,2 E c1,2 /E d )/( 75 T 0.01 dd gs /dt

228 D gs σ ɛ F ijkl S i S j ɛ kl F Sz 2 (D gs + A 1 (ɛ xx + ɛ yy )+A 2 ɛ zz )(S 2 z 2/3) A 1 A 2 ɛ xx = ɛ yy = ɛ zz dd gs /dt =2A 1 + A 2 T T 0 L d L = L d η d T P = E d L d /L = E d ε d = E d Tη d L T 0.01K

229 Al + ion clock Ensemble NV Echo (0.01ppb x 1mm 3 ) TXCO Commercial Rb Single NV Echo Rb Chip Clock Ensemble NV CW SAW Oscillator Allan Deviation after 1s averaging Single NV CW / f N 2

230 σ y = τ 1/2 9 /

231 200 m s =0 m s = ±1 d /dt = (2π)77 / N

232 " #!!"#$%&'( )%*%&+( Temperatureaccuracy Kelvin CdSe QD SThM Nano Diamond Raman Liquid crystal Infrared Green FluorescentProtein Seebeck! "!! "!,-./0'( )%*%&+( #! " " ProjectedNano Diamond Bulk Diamond Sensor size um ±1 0 (T ) δ

233 ! 1.1 " !"#$!" "!" Normalized fluorescence population τ (us) T ( o C) 2τ 50 2π 2τ = 250 µ 2τ = 50 µ 2τ 2τ m s =0 m s = 1 η = C d /dt 1 T Nt, T t C T C /

234 ω 1 2 ( 0 + B ) B = 1 2 ( ) τ 2π +1 1 τ ± % τ η =(9± 1.8) / 2τ =250µ δt =1.8 ± 0.3 2τ (2d /dt 2τ) 1 2τ < 2τ 1 µ

235 N 500 = nm 2.5

236 " # Normalized fluorescence data max. slope meas. freq. y ( µm) ! $ " #!"#$ % & ω (GHz) x ( µm)! 4 $ off AU Fit on AU Fit " theory NV data T (K) T (K) 5! # $ % & laser power ( µw) distance ( µm) µ

237 0.8 ± 0.1 µ 0.8 µ δt =(44± 10) T (r) = Q 4πκr Q κ r 72 ± µ 1 T =( 20 ± 50)

238 12 µ 0.5 ± µ 3.9 ± µ /

239 y ( µm) " x ( µm) # $ +$'()*!"#$$$$$$$!"%$$$$$$$$!#$$$$$$$$%$$$$$$$$$#$$$$$$$$"%$$$$$$$"#!" &'()#$!"$ &'()#$ &'()#%!"%!"#$$$$$$$!"%$$$$$$$$!#$$$$$$$$%$$$$$$$$$#$$$$$$$$"%$$$$$$$"# & '()* T K Pos 1 Pos 2 y ( µm) &'()! T (K) !"#$ Au Fluorescence (kcps) *#+#$%-. /#+#,15#3,1%#4 y ( µm) !"% *#+#$%,-. /#+# 60 /#+#012#3,1$# x ( µm) x ( µm) /

240 50 µ 5 =2.87 f 1,2 f (ω )+ f ω ω f 3,4 f (ω + )+ f ω ω+ ( ) δω + δb + δt d dt ( ) δω δb + δt d dt δt = δω (f 1 + f 2 ) (f 3 + f 4 ) d /dt (f 1 f 2 )+(f 3 f 4 ), ω ± δω δb

241 C 0.03 δt σ δt = σ/(c d dt 2τ) c 2τ t<30 s t η = δt t m 1 δt = N 1 ΣN i=1 (T i mp i ) 2 T i P i σ (δt) =δt 1 2 Γ 2 (n/2) Γ( ) N 1 Γ 2 ((n 1)/2)

242 9 / 250 µ 13 T 1 / µ / n 200 µ B µ 50 µ

243 2

244 S 1,S 2 R H 12 = ɛ 1 S z 1 + ɛ 2 S z 2 + t 12 (S + 1 S 2 + S 1 S + 2 )+V 12 S z 1S z 2 ɛ i W t 12,V 12 R R t 12,V 12 ɛ i W t 12 = = S z =0 δ a = ɛ 1 ɛ 2 δ a t a = t 12 τ α a, S z =0

245 (a), (b) r 1 r =, =, t R 2 = (c) a δ a + δ b + V ab,,, δ b δ a V δ ab a δ b V ab, = 3 r 3 r 4 4 b, (δ a + δ b )+V ab R 1 S 1 S 2 a S 3 S 4 b t a,t b H ab H a = δ a τ z a + t a τ x a τ x a V 12 S z 1S z 2 S z =0 a =12 b =34 R 1 R 2 t(r 2 ) ɛ i W a b S z a =0 S z b =0 H ab = δ a τ z a + t a τ x a + δ b τ z b + t b τ x b + V ab τ z a τ z b V ab = V 13 V 14 V 23 + V 24 τa z = S1 z = S2 z H ab H ab

246 R 1,R 2 V ab 0 t a/b 0 τ z δ a/b t a/b t 2 a + δ 2 a, t 2 b + δ2 b V ab t a t b µ α µ z H ab τ µ O(1) τ x τ z H = H ab + H cd + H int = ab µ z ab + cd µ z cd + V αβ µ α abµ β cd, α,β {x,z} H int = V ac τ z a τ z c + V ad τ z a τ z d + V bcτ z b τ z c + V bd τ z b τ z d µ V (r) 1/r 1/r 3

247 V (r) 1/r β V ab = V 13 V 14 V 23 + V 24 ( 1 = 1 ) ( ) R β 13 R β 14 R β 24 R β 23 ( ) ( ) 1 = R 2 + r 4 r β R r 3 r 1 β R 2 + r 3 r 2 1 β R 2 + r 4 r 2 β 1 ( ) β 2r3 r 1 +2r 4 r 2 2r 4 r 1 2r 3 r 2 R2 1. R β 2 R R β+2 2 R 1 <R 2 /2 R 2 1 R ij V ab = V 13 V 14 V 23 + V 24 a b V ab R 2 1/R 5 2 V ab N 2 (R 1,R 2 ) V/R β 2 VR 2 1/R β+2 2

248 R 1 R 2 N 2 (R 1,R 2 ) (n 1 (R 1 )R2) d VR2 1/R β+2 2 t/r1 α R d (β+2) 2, n 1 = ρn 1 n 1 (R 1 )R d 2 R 2 2R 2 VR2 1 /Rβ+2 2 t/r1 α R 1 R 2 d>β+2 R 2 V (R 2 ) t(r 1 ) V (R 2 ) t(r 1 ) R2 1 /Rβ+2 2 1/R1 α 1/10 R 2 R α+2 β+2 1 N 2 (R 1,R 2 ) R1 d+2 (R α+2 β+2 1 ) d (β+2) α+2 d α+d β+2 = R1. d> α(β+2) α+β+4

249 N 3 (R 1,R 2,R 3 ) (n 2 (R 1,R 2 )R d 3) Ṽ/Rβ 3 Ṽ/R β 2 =(n 2 (R 1,R 2 )R3) d ṼR2 1/R β+2 3 ṼR1/R 2 β+2 2 = R 2d α+2 1 R d 2 R d β 2 3 n 2 = n 1 N 2 R 3 R 1,R 2 R 1 R 2 R 3 d>(β +2)/2 R 1 R (β+2)/(α+2) 2 R 2 R 3 N 3 V σ z α>β+4 R 1 R 2 R 3 α>β R 2 R 1 R β+2/α+2 2 R 1 R β/α 2 R <R 2 R 2 O(1) R 1 R <R 2

250 R N 2 (R 1, R) ρn 1 (R 1 ) R d R d α 1 R d. O(1) R R α/d 1 1 R 2 V (R 2 ) t(r 1 ) R >R 2 R = R 2 R 1 R = R 2 V R <R 2 R 2 R R 2 R α/d 1 1 R α β 1 d c = αβ α+β d< αβ α+β R 2 O(1) N 2 (R 2 ) 1 R 2

251 R 2 N 2 (R 1,R 2 ) R 2 α = β R 1 R 2 R 2 N 2 (R 2 ) 1 1 N 2 (R 2 )=ρ 2 R 2d β 2 V W = R2d β2 = 1 W ρ 2 V = 1 1 ρ 2 a β 0 W V/a β 0 D = W V/a β 0 ρ 1/a d 0 R 2 a 0 D 1/(2d β), d = β =3 R 2 a 0 3 D d =2 R 2 a 0 D α = β =3 d c =1.5 D>1 d c α = β =3 d z i d z j

252 i<j d z i d z j r 3 ij = i<j (d s i + d a i σ z i )(d s j + d a j σ z j ) r 3 ij = i<j (d s ) 2 r 3 ij + i<j d a d s (σ z rij 3 i + σj z )+ i<j (d a ) 2 σ z rij 3 i σj z, d z = d s + d a d z = d a d s ɛ i = i j d a d s i ɛ rij 3 i = j i d a d s Q rij 3 j = ν da d s a 3 0 l lat 1 ν l 3 Q j =1 0 δɛ 2 i = ɛ2 i ɛ i 2 ɛ 2 i = ( )( d a d s d j i k i Q a d s rij 3 j Q rik 3 k ) Q j Q k = νδ jk + ν 2 (1 δ jk ) ( ) 2 [ ɛ 2 i = (ν ν 2 ) 1 l lat + ( ν ) 1 2 l 6 l lat ] l 3 d a d s a 3 0 W = ɛ ɛ 2i = 2i ɛ i 2 = da d s (ν ν a 3 2 ) 1 0 l, 6 l lat ν 1 W ν da d s ρ ν/a 3 0 W = da d s (ν ν a 2 ) 1 3 l lat 0 l 6 a N 2 (R 2 )=ρ 2 R 2d β 2 V W = R2d 32 = 1 W ρ 2 V = 1 1 ρ 2 a 3 0 d a d s (ν ν a 2 ) 3 l lat 0 V/a l 6.

253 V (d a ) 2 ( d s 1 R 2 a 0 (ν ν d a ν 2 ) ) 1 2d l 6 l lat ν 1 ( ) 1/3 d d =3 R 2 s a0 d a / ν d =2 R 2 ds d a a 0 /ν 3/2

254 H B = ij t ija i a j i j V ijn i n j X Y (Θ 0, Φ 0 ) = s 1, 1 + v 1, 1 + w 1, 0 s =Ω 2 Ω 4 / Ω v =Ω 1 Ω 3 / Ω w = Ω 1 Ω 4 / Ω i j R =(R, θ, φ) {x, y, z} H = 1 6 4πɛ 0 R 3 2 ( 1) q C q(θ, 2 φ)tq 2 (d (i), d (j) ), q= 2 C k q (θ, φ) k z q T 2 2 ( ) T±2(d 2 (i), d (j) )=d (i) ± d (j) ± T±1(d 2 (i), d (j) )= d (i) z d (j) ± + d (i) ± d (j) z / 2

255 ( T0 2 (d (i), d (j) )= d (i) d (j) + +2d (i) z d (j) z ) + d (i) + d (j) / 6 d ± = (d x ± id y )/ 2 1, 0 1, ±1 T 2 ±1 t ij i j T 2 0 i j = 2 3 [d2 00w i w j 1 2 d2 01(v i v j + s i s j )], i j T 2 +2 i j = d 2 01(v i s j ), i j T 2 2 i j = d 2 01(s i v j ), d 00 = 1, 0 d z 0, 0 d 01 = 1, ±1 d ± 0, 0 t ij V ij V ij = i j H dd i j + i j H dd i j i j H dd i j i j H dd i j i d i = d 1 ( s i 2 + v i 2 )+µ 0 w i 2 d 1 = 1, ±1 d z 1, ±1 µ 0 = 1, 0 d z 1, 0 1, 0 V ij

256 ij d i j d z d z (d +d + d d + ) i j = d 2 0, i j d z d z (d +d + d d + ) i j = d i d 0, i j d z d z (d +d + d d + ) i j = d 0 d j, i j d z d z (d +d + d d + ) i j = d i d j 1 2 µ2 01(s i wi w j s j + w i vi v j wj + ) i j d + d + i j = µ 2 01(s i wi w j vj + w i vi s j wj ), i j d d i j = µ 2 01(w i s i v j wj + v i wi w j s j), d 0 = 0, 0 d z 0, 0 µ 01 = 1, ±1 d ± 1, 0 1, 0 1, ±1 H dd t ii = j i ( i j H dd i j i j H dd i j ) t ii 2B {a, b, A, B} a A b B t ij V ij g 1, g 2 w t ij V ij a A b B g 2 w a/b = w A/B w a/b = w A/B

257 I 1 =4 I 2 =3/2 H Q , ±1 T±2 2 H dd Ω M H hf {a, b, A, B} A H hf A = a H hf a = B H hf B = b H hf b 10 3

258 A B t ii E(R, t) =E(R)e iωt + ˆX H lattice = E(R) α(ω)e(r) E(R) = E(R) p β p(r)e p e p α(ω) H lattice = E 2 (R) [ 2α α 3 +(α α ) p C 2 pγ p ] α α γ 0 = β 0 2 1/3 γ ±1 =1/ 3(β0β ± β β 0 ) γ ±2 = 2/3β β ± [ ] H lattice = E 2 2α α (R) +(α α ) 0, 0 C0 2 0, 0 γ 0 3 H lattice = E 2 (R)[ 2α α 3 +(α α ){γ 0 ( s 2 1, 1 C 2 0 1, 1 + v 2 1, 1 C 2 0 1, 1 + w 2 1, 0 C 2 0 1, 0 )+γ 2 sv 1, 1 C 2 2 1, 1 + γ 2 s v 1, 1 C 2 2 1, 1 }]. δe = H lattice H lattice A B {s, v, w} t ii

259 σ + π σ γ ±2 γ ±1 =E 1,0 E 1,1 H lattice ˆx ŷ ẑ λ 0 {a, b, A, B} λ L = R 0 λ 0 λ L λ L k ˆX Ŷ λ L k ( ˆX ± Ŷ ) 2λ L ˆk ẑ Ω 2 Ω 3 Ω 1 =Ω 4 =0 Ω 1 =Ω 4 Ω 2 =Ω 3 =0 M Θ 0 =0.68, Φ 0 =5.83

260 k x k y Θ 0 =0.05 E 32 Θ 0 =1.05 E 28 Θ 0 =0.68 E 36 Θ 0 =0.25 E 40 {θ a,θ b,φ a,φ b,α a,α b,γ a,γ b } = {0.53, 0.97, 1.36, 3.49, 2.84, 2.03, 4.26, 3.84} s i =sin(α i )sin(θ i ) v i =sin(α i )cos(θ i )e iφ i w i =cos(α i )e iγ i ν =1/2 N s =24 {Θ 0, Φ 0,θ a,θ b,φ a,φ b,α a,α b,γ a,γ b } = {0.65, 3.68, 2.4, 2.97, 6.06, 4.1, 0.97, 2.74, 3.44, 1.74} f 7 A B a b γ A = π + γ a γ B = π + γ b w a/b = w A/B E 8 ν =1/2 Θ 0

261 40 87 Θ 0 A B d 00 = d 01 1, ±1 s i s i d 00 /d 01 v i v i d 00 /d 01 (0, 0) ( π, 0) E < 4 A B

262

263 I(r) = β2 4π 0 [G 0 (r; z)g 0 (r; z)]dx G 0 (r; z) = d 3 k z+ɛτ z+ τ x z 2 ɛ 2 2 G 0 (r; z = ix) = 2πρ 0 k f r e 2 +x 2 /v f r (cos(k f r) ) 2 + x 2 τ z +sin(k f r)[ix + τ x ]. 2 + x 2

264 [G 0 (r; z)g 0 (r; z)] I(r) = = k f r 0 x dx e 0 β 2 π(k f r 0 ) 2 E f β 2 [ 2 + x 2 cos(2k f r 0 )] π(k f r 0 ) 2 (x ) [ cos(2k f r 0 ) 0 dx e k f r 0 x E f ] +2 2 sin 2 (k f r) k f r 0 x dx e E f 0 (x ) x x I(r) = β 2 cos(2k π(k f r 0 ) 2 f r 0 ) 0 dx [ e k f r 0 x 2 +1 E f ] +2sin 2 (k f r) 0 k f r 0 x 2 +1 E f dx e (x 2 +1) dx [e k f r 0 ] x E f = E k f r 0 f 1 k f r 0 e E f 0 dx [ e k f r 0 x 2 +1 ] E f = E f k f r 0 e k f r 0 E f [ ] kf r 0 F 1 E f F 1 F 1 [α] =α 0 dxe α( x ). k f r 0 x 2 +1 dx e E f 0 (x 2 +1) = π k f r 0 [ ] 2 e E kf f r 0 F 2 E f

265 F 2 F 2 [α] = 2 π 0 x dx e α( ). (x 2 +1) I(r) = E fβ 2 π(k f r 0 ) cos(2k fr 3 0 )e k f r 0 [ ] E kf f r 0 F 1 + β2 = E [ fβ 2 π(k f r 0 ) cos(2k fr 3 0 )e 2r 0 2r0 ξ F1 ξ k f r 0 E f (k f r 0 ) 2 sin2 (k f r)e E f ] [ + β2 (k f r 0 ) 2 sin2 (k f r)e 2r 0 2r0 ξ F2 ξ [ ] kf r 0 F 2 E f ] F 1 F 2 F 1 [α] =α 0 dxe α( x ) 1.25(α +0.65) 1/2 F 2 [α] = 2 π 0 x dx e α( ) (x 2 +1) 0.8. (α +0.65) 1/2 [G, (z) G (0) (z)] = [SΠ+SΠSG + SΠSGSG +...]= [SΠ(z)(1 SG(0) (z)) 1 ]. S S

266 SC Interaction (khz) Interaction Strength Hz 6! ! Full SC Correction Perturbative SC Correction β =0.01 0! 3! 5! 7! Distance between spins nm R (nm) β =0.01 =0 S [SΠ(z)(1 SG (0) (z)) 1 ] [SΠ(z)SG (0) (z)]. Π(z) G(z) [SΠ(z)SG (0) (z)] = n,m Ψ n S Ψ m Ψ m S Ψ n. (z ε m ) 2 (z ε n ) E tot +iλ 0 iλ = = iλ iλ iλ dɛ 4π 2Re[ iɛ (iɛ ε m ) 2 (iɛ ε n ) ] ɛdɛ ɛ 4πi (iɛ ε m ) 2 (iɛ ε n ) dɛ 4π ɛ (ɛ + iε m ) 2 (ɛ + iε n ). ±i iε m,n iλ

267 iλ iε n iλ dɛ ɛ iλ 4π (ɛ + iε m ) 2 (ɛ + iε n ) = 1 ε n (ε n Λ). 2 (ε n ε m ) 2 m iλ dɛ ɛ iλ 4π (ɛ + iε m ) 2 (ɛ + iε n ) = 1 ε n (ε n Λ) + ε m (ε m Λ) 4 (ε n ε m ) 2 = 1 (ε n Λ) 4 (ε n ε m ) = 1 4 iλ iλ dɛ 1 2π (ɛ + iε m )(ɛ + iε n ). 4δE tot =2 +iλ 0 0 dɛ 4π 2 [i [SG(0) (iɛ)sg (0) (iɛ)]] dɛ 4π 2 [i [SG(0) (iɛ)sg (0) (iɛ)]] iλ 0 dɛ 4π 2 [i [SG(0) (iɛ)sg (0) (iɛ)]], G σσ LL (z) Gσσ LR (z) Πσσ LL (z) Πσσ LR (z) G σσ LL (z) Gσσ LR (z) Πσσ LL (z) Πσσ LR (z) L R G σσ LL (z) =Gσσ RR (z)

268 G σσ LR (z) =Gσσ RL (z) G σσ LL (z) = d 3 1 k z 2 2 ɛ 2 z ɛ k k z + ɛ k = ρ 0 2 z 2 z z G σσ LR (z) G σσ LR(z) = d 3 1 k z 2 2 ɛ 2 z ɛ k k z + ɛ k e ik r 0, r 0 = r L r R G σσ LR (z) = 2 z 2 r v 0 f ρ 0 e k f r 0 2 z 2 z sin(k fr 0 )+ 2 z 2 cos(k f r 0 ) sin(k f r 0 ) sin(k f r 0 ) z sin(k f r 0 ) 2 z 2 cos(k f r 0 ). Π σσ LL (z) Π σσ LL (z) = ρ 0 ( 2 z 2 ) 3/2 z z. k f r 1