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1 REPORT DOCUMENTATION PAGE Form Approved OMB NO The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggesstions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA, Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any oenalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE Ph.D. Dissertation 4. TITLE AND SUBTITLE Topology, Localization, and Quantum Information in Atomic, Molecular and Optical Systems 3. DATES COVERED (From - To) - 5a. CONTRACT NUMBER W911NF b. GRANT NUMBER 6. AUTHORS Norman Ying Yao 5c. PROGRAM ELEMENT NUMBER d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAMES AND ADDRESSES Massachusetts Institute of Technology (MIT 77 Massachusetts Ave 8. PERFORMING ORGANIZATION REPORT NUMBER Cambridge, MA SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS (ES) U.S. Army Research Office P.O. Box Research Triangle Park, NC DISTRIBUTION AVAILIBILITY STATEMENT Approved for public release; distribution is unlimited. 10. SPONSOR/MONITOR'S ACRONYM(S) ARO 11. SPONSOR/MONITOR'S REPORT NUMBER(S) PH-MUR SUPPLEMENTARY NOTES The views, opinions and/or findings contained in this report are those of the author(s) and should not contrued as an official Department of the Army position, policy or decision, unless so designated by other documentation. 14. ABSTRACT The scientific interface between atomic, molecular and optical (AMO) physics, condensed matter, and quantum information science has recently led to the development of new insights and tools that bridge the gap between macroscopic quantum behavior and detailed microscopic intuition. While the dialogue between these fields has sharpened our understanding of quantum theory, it has also raised a bevy of new questions regarding the out-ofequilibrium dynamics and control of many-body systems. This thesis is motivated by experimental advances that make it possible to produce and probe isolated, strongly interacting ensembles of disordered particles, as found in 15. SUBJECT TERMS QuISM 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF a. REPORT b. ABSTRACT c. THIS PAGE ABSTRACT UU UU UU UU 15. NUMBER OF PAGES 19a. NAME OF RESPONSIBLE PERSON Paola Cappellaro 19b. TELEPHONE NUMBER Standard Form 298 (Rev 8/98) Prescribed by ANSI Std. Z39.18

2 Report Title Topology, Localization, and Quantum Information in Atomic, Molecular and Optical Systems ABSTRACT The scientific interface between atomic, molecular and optical (AMO) physics, condensed matter, and quantum information science has recently led to the development of new insights and tools that bridge the gap between macroscopic quantum behavior and detailed microscopic intuition. While the dialogue between these fields has sharpened our understanding of quantum theory, it has also raised a bevy of new questions regarding the out-ofequilibrium dynamics and control of many-body systems. This thesis is motivated by experimental advances that make it possible to produce and probe isolated, strongly interacting ensembles of disordered particles, as found in systems ranging from trapped ions and Rydberg atoms to ultracold polar molecules and spin defects in the solid state. The presence of strong interactions in these systems underlies their potential for exploring correlated many-body physics and this thesis presents recent results on realizing fractionalization and localization. From a complementary perspective, the controlled manipulation of individual quanta can also enable the bottom-up construction of quantum devices. To this end, this thesis also describes blueprints for a room-temperature quantum computer, quantum credit cards and nanoscale quantum thermometry.

3 4 HARVARD UNIVERSITY Graduate School of Arts and Sciences DISSERTATION ACCEPTANCE CERTIFICATE The undersigned, appointed by the Department of Physics have examined a dissertation entitled Topology, Localization, and Quantum Information in Atomic, Molecular and Optical Systems presented by Norman Ying Yao candidate for the degree of Doctor of Philosophy and hereby certify that it is worthy of acceptance. Signature /?4 :::_~ ::,, ~= Typed name: ~ikhail Lill<in, Chair Szgnature -----""-= :y,r~,::~_e_:_p~~~o~ft-es_s_o_r_e_u_g~_n_e_d-em-le_r Typed name: Professor Subir Sachdev Date: May 7, 2014

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9 10 W

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11 G σσ LL (z) Gσσ LR (z) Πσσ LL (z) Πσσ LR (z)

12 13 ν =1/2

13 14 W

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21 22 ß

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24 25 Energy Site H = t ij c i c j + h.c. + ij 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

25 26 e r/ξ ξ H = α c αc α + α,β,γ,δ V α,β,γ,δ c αc β c γc δ.

26 27 ν =1/2 ν =1/2

27 28 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

28 29 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

29 30 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

30 31 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/

31 32 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

32 33

33 34 (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

34 35 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

35 36 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

36 37 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

37 38 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

38 39 (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

39 40 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

40 41 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

41 42 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

42 T a0 1µ T 1,T 2 10

43 44 π/2 τ z i ±1

44 45 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 ( )

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

46 47 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

47 48 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)

48 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

49 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

50 51 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

51 52 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

52 53

53 54 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

54 55 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

55 56 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

56 57 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

57 58 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

58 59 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

59 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

60 61 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

61 62 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

62 63

63 64 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

64 65 ±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

65 66 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

66 67 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 ),

67 68 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

68 69 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

69 70 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

70 71 (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

71 72 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

72 73

73 74 E H m = BJ 2 d z E E 0, 0 J =1 1, 1 1, 0 1, 1 J ẑ J, m E

74 75 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

75 76 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

76 77 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

77 78 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

78 79 σ xy ν =1/2 ν =1/2

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

80 81 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

81 82 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

82 83

83 84 (a) (b) E b " η η " BCS ψ BCS E b r < ξ η η r ξ r r<ξ

84 85 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

85 86 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

86 87 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)

87 88 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

88 89 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 )

89 90 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

90 91 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.

91 92 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

92 93 2 T K I I T K

93 94 ɛ = E b E b E b

94 95 ( /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

95 96 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

96 97 (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

97 98 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

98 99 /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

99 100 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

100 101 (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

101 102 T 0 D + D T 0 S 0 hν = E (D +,S 2 ) (D +,T 0 ) S 2 S 0

102 103

103 104 g κ

104 105 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

105 106 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

106 107 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 )]

107 108 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

108 109 =E k t k E =0 g L g R g L g R τ N/g

109 110 N =7 g/κ ɛ 0 = 10 3 τ 1/κ

110 111 g

111 112 T 1

112 113 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

113 114 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).

114 115 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

115 116 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 =

116 117 a b = a b = a b a b 0

117 118 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

118 119 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

119 120 [ 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) ],

120 121 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

121 122 {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

122 123 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

123 124 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)

124 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

125 126 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

126 127 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

127 128 {J i } J κ =1 g 0.7 T 1 N =11 N =51 1 ɛ 200 N =51 T 1

128 129 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

129 130 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

130 131 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

131 132 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

132 133 XX N =12 90% N =10 98%

133 134 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

134 135 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

135 136 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

136 137 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,

137 138 NV 1 NV 2 N b Q M ( H i CP i ) n+1

138 139!"#$%&'()'*+#,-+%%&.'/-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

139 140

140 141 1/r 2 A B a R

141 142 %! $ " # A B a R λ Ω

142 143 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

143 144 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

144 145 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

145 146 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

146 147 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

147 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

148 149 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

149 150 γ 0

150 151 r a S =1 W

151 152 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

152 153 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.),

153 154 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/

154 155 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

155 156 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

156 157 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

157 158 R = 100 nm W T 2 T 1 1 T 1

158 159 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

159 160

160 161

161 162

162 163 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

163 164 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

164 165 0 e I z B z, µ 10µ

165 166 B z (y) = db z dy y + B z,0

166 167 6B;m`2 93, h?2 `+?Bi2+im`2 7Q` im`2 i2 [m MimK +QKTmi2`X U V irq@ /BK2MbBQM H?B2` `+?B+ H H iib+2 HHQrBM; 7Q` H2M;i?@b+ H2 # b2/ +QMi`QH- r?b+? 2M #H2b 7mHHv T ` H@ H2H QT2` ibqmbx i i?2 HQr2bi H2p2H- BM/BpB/m H TH [m2ii2b `2 QmiHBM2/ BM ;`2v M/ 2 +? +QMi BMb bbm;h2 +QKTmi ibqm H Lo `2;Bbi2`X i i?2 b2+qm/ H2p2H Q7?B2` `+?v- bmt2`@th [m2ii2- QmiHBM2/ BM r?bi2-2m+qkt bb2b H iib+2 Q7 TH [m2ii2bc 2 +? bmt2`@th [m2ii2 Bb b2t ` i2hv K MBTmH i2/ #v KB+`Q@bQH2MQB/ +QM}M2/ KB+`Qr p2 }2H/bX AM Q`/2` iq HHQr 7Q` [m MimK BM7Q`K ibqm i` Mb@ 72` +`Qbb #QmM/ `B2b Q7 bmt2`@th [m2ii2b- i?2`2 2tBbib /m H bmt2`@th [m2ii2 H iib+2 QmiHBM2/ BM `2/X U#V h?2 b+?2k ib+ Lo `2;Bbi2` BKTH Mi ibqm rbi?bm bmt2`@th [m2ii2x hrq `Qrb Q7 BM/B@ pb/m H TH [m2ii2b rbi?bm bmt2`@th [m2ii2 `2 b?qrmx Lo `2;Bbi2`b- +QMbBbiBM; Q7 M 2H2+i`QMB+ U;`22MV M/ Mm+H2 ` Uv2HHQrV btbm `2 /2TB+i2/ rbi?bm bi ;;2`2/ mt@bhqtbm; `` v r?b+? Bb `Qr@ `2T2iBiBp2X AM/BpB/m H `Qrb rbi?bm bbm;h2 TH [m2ii2 `2 bt2+b}2/ #v M BMi2;2` n rbi? n = 1 #2BM; i?2 #QiiQK `Qr M/ n = M #2BM; i?2 iqt `QrX hq +?B2p2 bi ;;2`2/ bi`m+im`2- r2 bt2+@ B7v mmb[m2 BKTH Mi ibqm `Qr rbi?bm 2 +? TH [m2ii2 r?2`2bm bbm;h2 BKTm`BiB2b `2 BKTH Mi2/ M/ bm#b2[m2mihv MM2 H2/X 6Q` ;Bp2M `Qr Q7 TH [m2ii2b- i?2 BKTH Mi ibqm `Qr +Q``2bTQM/BM; iq i?2 H27i@KQbi TH [m2ii2 Bb M 4 R- r?bh2 i?2 TH [m2ii2 iq i?2 BKK2/B i2 `B;?i? b BKTH Mi ibqm `Qr M 4 kc i?bb T ii2`m +QMiBMm2b mmibh i?2 }M H TH [m2ii2 BM ;Bp2M `Qr- r?b+? #v +QMbi`m+iBQM? b i?2?b;?2bi BKTH Mi ibqm `Qr MmK#2`X h?2 BKTH Mi ibqm T`Q+2bb Bb `2T2 i2/ 7Q` 2 +? `Qr Q7 TH [m2ii2b rbi?bm i?2 bmt2`@th [m2ii2 M/ +`2 i2b M `` v Q7 Lo `2;Bbi2`b- r?b+? 2 +? Q++mTv mmb[m2 `Qr BM i?2 bmt2`@th [m2ii2x abm+2 2 +? Lo `2;Bbi2` Q++mTB2b mmb[m2 `Qr rbi?bm i?2 bmt2`@th [m2ii2- i?2 K ;M2iB+ }2H/ ;` /B2Mi BM i?2 y /B`2+iBQM HHQrb 7Q` BM/BpB/m H bt2+i`qb+qtb+ //`2bbBM; Q7 bbm;h2 `2;Bbi2`bX *Q?2`2Mi +QmTHBM; Q7 bt ib HHv b2t ` i2/ Lo `2;Bbi2`b BM /D +2Mi TH [m2ii2b Bb K2/B i2/ #v / `F btbm +? BM / i #mb U.a*"V M/ Bb b+?2k ib+ HHv `2T`2b2Mi2/ #v i?2 +m`p2/ HBM2 +QMM2+iBM; BM/BpB/m H `2;Bbi2`bX h?2 b2+qm/ BKTH Mi ibqm bi2t +Q``2bTQM/b iq i?2 +`2 ibqm Q7 i?2b2?q`bxqmi H M/ p2`ib+ H / `F btbm +? BMbX R98

167 168 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

168 169 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 )

169 170 Ω i Ω j g κ g Ω

170 171 H FFST g B z (y)

171 172 N = T1 NV N = T1 NV T N 1 T N T NV 1

172 173 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

173 174 T NV T 1 T 1 10 T 1 50 ɛ 1.4%

174 175

175 176 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

176 177 κ + κ 2 / 2 0 (α + β ) (α + β D ) π D D D Ω D

177 178

178 179

179 180 1/2

180 181 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

181 182 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

182 183 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

183 184 Ûi,j γ 0 i π π/2 U ij w(p) = 1

184 185 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

185 186 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

186 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

187 188 T N v n p e v/t n p π N v π ξ a a π

188 189 v H p H T H p H e = v dp 2π pc pc p = v dx γ(x)(i )γ(x),

189 190 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)

190 191 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

191 192 e S/k B T S l

192 193 L R R L

193 194

194 195

195 196 N ρ = i ρ i ρ i Q = { +,, + i, i, 0, 1 } ρ F tol ρ i M i

196 197 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

197 198 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

198 199 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

199 200 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

200 201, 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

201 202 c

202 203 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

203 204 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

204 205 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

205 206 (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) ],

206 207 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

207 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

208 209 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

209 210 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

210 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

211 212 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

212 213 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

213 214 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. )

214 215 4

215 216 σ y Q = ν/ ν ν ν ν

216 217 Excitation Laser Si Photodiode Diamond Film Bragg Mirror Microwave Line µ > 0 ±

217 218 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

218 219 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

219 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

220 221 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

221 222 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

222 223 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

223 224 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

224 225 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

225 226 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

226 227 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

227 228 (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

228 229 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

229 230 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

230 231 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

231 232 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

232 233 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

233 234 σ y = τ 1/2 9 /

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

235 236 " #!!"#$%&'( )%*%&+( 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

236 237! 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 /

237 238 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

238 239 N 500 = nm 2.5

239 240 " # 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

240 ± 0.1 µ 0.8 µ δt =(44± 10) T (r) = Q 4πκr Q κ r 72 ± µ 1 T =( 20 ± 50)

241 µ 0.5 ± µ 3.9 ± µ /

242 243 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) /

243 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

244 245 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)

245 246 9 / 250 µ 13 T 1 / µ / n 200 µ B µ 50 µ

246 247 2

247 248 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

248 249 (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

249 250 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

250 251 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

251 252 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

252 253 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

253 254 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

254 255 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

255 256 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.

256 257 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

257 258 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

258 259 ( 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

259 260 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

260 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

261 262 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

262 263 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

263 264 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

264 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