REPORT DOCUMENTATION PAGE

REPORT DOCUMENTATION PAGE Form Approved OMB NO. 0704-0188 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, 22202-4302. 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 30-12-2014 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-11-1-0400 5b. GRANT NUMBER 6. AUTHORS Norman Ying Yao 5c. PROGRAM ELEMENT NUMBER 611103 5d. 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 02139-4307 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS (ES) U.S. Army Research Office P.O. Box 12211 Research Triangle Park, NC 27709-2211 12. DISTRIBUTION AVAILIBILITY STATEMENT Approved for public release; distribution is unlimited. 10. SPONSOR/MONITOR'S ACRONYM(S) ARO 11. SPONSOR/MONITOR'S REPORT NUMBER(S) 59745-PH-MUR.85 13. 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 617-253-8137 Standard Form 298 (Rev 8/98) Prescribed by ANSI Std. Z39.18

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.

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

12 40 87 G σσ LL (z) Gσσ LR (z) Πσσ LL (z) Πσσ LR (z)

13 ν =1/2

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

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.

27 ν =1/2 ν =1/2

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

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 100 1 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

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

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 =1/2 15 13

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

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

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

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

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

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

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

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

41 Dynamic Polarization (D) (a) β =3 V/t =2 1.0 0.8 ν =1/2 0.6 0.4 0.2 0.0 0.1 1.0 10.0 Disorder Width (W ) Disorder Width (W ) (b) 2 2.0 1.5 1.0 1.5 1 0.5 V/t =1 V/t =2 V/t =4 MBL Ergodic 1 1 1 4 3 2 1/4 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

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 1 40 87 3 532 100 T a0 10µ T 2 100 T 1 25

43 2 3 T a0 1µ T 1,T 2 10

44 π/2 τ z i ±1

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 ( )

46 0.8 1 0.6 0.4 0.2 spin echo DEER 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 0 4 6 8 10 F(t) D(t) W L J z = J F (t) d ξ t/2 π π/2 τ z

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

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)

49 1 0.8 fit 0.6 0.4 0.2 0 0 2 4 6 8 10 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

50 10 0 10-1 10 0 10 4 10 8 10 12 0.26 0.22 0.18 0.14 3 4 5 6 7 8 0.4 0.3 0.2 0.1 fit 0 0 2 4 6 8 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

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

52 J,J z T 1 1 J 10 1 10 500 ξ ln(j 1 ) 6ξ 10 J 100 1 10 1 /J 1 5 10 3 J (1 10) 5µ 1 100µ 1 /J 1 (0.5 5) 10 3 J 50 1 25 1 /J 1 8 10 3

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

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 30 100 D

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 =0.0001 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 =0.0001 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

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

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

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

60 2.0 Sent 1.5 1.0 J independent 0.5 0.0 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

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 =10.0 10 6 <W <10 4 10 3 N =8, 12 10 2 N =16 ρ ψ J/3 ρ H W

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

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

65 ±1 0 Ω + Ω Ω +, Ω 0 B = α( 1 + β 1 ) D = α ( β 1 + 1 ) 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

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 01 1 0 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

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 + 3 2 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 ),

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

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

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 2 01 100 Ω H B N ν

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 2 01 2.8 ĝ 1 ĝ 2 N s = 24 2π θ 1 4π θ 2

72 c = 1 ν =1/2 (d 0 d 1 ) 2 /d 2 01 6 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

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

75 φ {X, Y } ẑ Θ 0 Φ 0 {x, y, z} {X, Y, Z} J =0, 1 0, 0 M J =1 40 87 N s = 24 Θ 0 E =0 J =1

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

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

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

79 σ xy ν =1/2 ν =1/2

80 40 87 7 133 41 87 87 133 d 3 532 1µ 100 340 395 40 87 e 1 e 2 J,m = 2, ±2 v =41 (3) 1 Σ +

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

82 H lattice H hf Ω i H lattice H hf 40 87 H hf 1 =160 E = B/d 0.5 Ω i E 1,0 E 1,1 M H lattice H hf Ω i

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

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

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

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 1 2 2 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)

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

89 Interaction (khz) 400! 0! -400! 100! 150! R (nm) J YSR! RKKY E b 10 2 200! E f = 11.7 k f = 20.1 1 N 0 = 35 3 ξ =1.6µ J YSR E b 10 2 J YSR r ξ F 1 [α] =α dxe α( x 2 +1 1) F 0 2 [α] = 2 π x 2 +1 1) 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 )

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

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.

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

93 2 T K I I T K

94 ɛ = E b E b E b

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

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

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

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

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

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

101 (a) 1.0 Esh/ Esh/ 0.5 0.0 (b) 1.0 0.5 0.0 I<0 I>0 Molecular Doublet 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 Mol. Triplet Molecular Doublet Triplet Kondo 0.6 0.8 1.0 1.2 1.4 1.6 Mol. Singlet 0.0 0.5 1.0 1.5 2.0 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 2.6 500 0.14 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

102 T 0 D + D T 0 S 0 hν = E (D +,S 2 ) (D +,T 0 ) 2 1 3 2 S 2 S 0

104 g κ

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

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

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 = 1 + 1 2 12 i=1,2,3 [σi E(σ i )]

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

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

110 N =7 g/κ ɛ 0 = 10 3 τ 1/κ

112 T 1

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

114 H JW φ A φ φ =(c 1,c 2,...,c N,c 1,c 2,...,c N )T A ɛ 1 0 0 0 0 ɛ 1 0 0 Λ= 0 0 ɛ 2 0 0 0 0 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).

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

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+1 1 0 0 0 U eff Ψ=( 1) n z 0 0 1 0 Ψ. 0 1 0 0 0 0 0 1 e iπ i 1 j=1 σ+ j σ j 2 =

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

118 0 (QR) 1 2 N-1 N N+1 (QR) 0 N +1 X F = 1 2 + 1 12 [ 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

119 F DS = 1 2 + 1 12 = 1 2 + 1 12 = 1 2 + 1 12 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 0 + + σ0 ) ρ ch ) ] σ 0 + (t)(σ0 ρ ch ) σ0 (t)(σ 0 + ρ ch ) i =0

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 = 1 2 + 1 6 (M 00 + M 00 + M 00 2 ). N +1 0 F SS = 1 2 + 1 12 i=x,y,z [ σ i 0(t)(ρ SS ch σ i N+1) ],

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 = 1 2 + 1 6 [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

122 {0 a, 1,,N,(N +1) a } U = U b U a F enc = 1 2 + 1 12 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+1 2 + i M N+1,i M i,0 2 )+ 1 2. M

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

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)

125 40 Number of States 30 20 10 0 T1 NV (units of ms) T1 NV (units of s) 26 28 30 32 34 Participation Ratio Number of States 20 15 10 5 0 10 15 20 25 30 Participation Ratio Number of States 8 6 4 2 0 5 10 15 20 Participation Ratio Number of States 10 8 6 4 2 0 3 4 5 6 Participation Ratio N = 11 T 1 d = 10 κ = 50 σ d σ d g L g R N = 51 N = 51 κ = 50

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

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 1/2 10 50 T 1 10

128 {J i } J 0 0 1 κ =1 g 0.7 T 1 N =11 N =51 1 ɛ 200 N =51 T 1

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

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

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 = 1 0 0 0 ( i) N+1 UP 2 = 1 0 0 ( 1) N+1 F SS = 1 2 + 1 6 [2 M 0,N+1 + M 0,N+1 2 ), F enc = 1 2 + 1 6 [2 M 0,N+1 2 M 2 0,N+1 M 0,0 M N+1,N+1 + M 0,N+1 2 + 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

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

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

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 = 159.7 A = 113.8 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

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

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

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,

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

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

141 1/r 2 A B a R

142 %! $ " # A B a R λ Ω

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

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

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

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

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

148 1 0.9 0.8 F 0.7 0.6 0.5 0 2 4 6 8 10 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

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

150 γ 0

151 r a S =1 W

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

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.),

154 N c W 1 q, 0 m s =1 0 R =100nm m 2 m =3 φ ( N 2 N c = 0 1...1 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/

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

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 2 1 1 ), N p T2 N

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

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

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

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

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

165 0 e I z B z,0 1 100 500 10µ 10µ

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

167 6B;m`2 93, h?2 `+?Bi2+im`2 7Q` `QQK@i2KT2` im`2 bqhb/@bi 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

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

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 )

170 Ω i Ω j g κ g Ω

171 H FFST g B z (y)

172 N = 18 8.71 18.1 T1 NV N =7 12.6 16 T1 NV T N 1 T N 1 50 1 T NV 1

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 5 100 N 20 500 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

174 T NV 1 100 10 2 T 1 T 1 10 T 1 50 ɛ 1.4%

176 g H = ( 1 1 + 1 1 ) Ω( 0 1 + 0 1 + ) Ω N S N x +4κS NV z S N z, Ω Ω N B = 1 + 1 2 D = 1 1 2 + B + 2Ω 0 0 2Ω B Ω H = D D ( + 2Ω2 ) + + + 2Ω2 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

177 κ + κ 2 / 2 0 (α + β ) (α + β D ) π 0 1 0 1 0 D D D Ω D

180 1/2

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

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

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

184 Ûi,j γ 0 i π π/2 U ij w(p) = 1

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

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

187 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

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

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

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)

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

192 e S/k B T S l

193 L R R L

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

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

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

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

200 r p cv h ( rd(fexp F cv 1 e )) n tol. F cv tol > 1+1/ 2 2 1+1/ 2 1.707 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

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

203 c +1 1 1 (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

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

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

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) ],

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

208 0.03 1 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

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

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

211 1 0.5 ξ 2 100 0.1 (a) 10 n th =0 0 0.2 0.4 0.6 J λt 1 0.5 ξ 2 opt 0.1 (b) 0. 50 N 0. 67 N 10 50 100 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

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

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

214 Sn(ω) f (t) 1 0 1 0 2 4 t/τ F (ωτ) ξ 2 1 0.5 (a) 0.1 (b) K (ωτ) δ 0 δ ω 0 5 10 J λt ξ 2 min 1 0.5 0.1 (c) 10 3 10 4 10 5 10 6 F K M =4 f(t) M =4 n th = 10 n dr = 10 3, 5 10 4, 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 200 2 N eff 100 η 1 H int = λ ( σ 1 + σ2 +h.c. )

216 σ y 2 10 12 3 Q = ν/ ν ν ν ν

217 Excitation Laser Si Photodiode Diamond Film Bragg Mirror Microwave Line µ > 0 ±1 0 0 1

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

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) = 1 1 1 2πQ (S/N) τ τ S/N C 1

7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 220 A V 3 E 3 A I D es γ orbital states electron spin sublevels 3 2 1 0 D gs κ λ 1 E 1 A1 S B 9.5 x 105 I 0 9 F(!) (photons/sec) 8.5 8 C ν 7.5 500 0 500 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) =8.124 10 5 τ 1/2 Q S/N

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

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 = 1 2 0 1 2 0 1 0 1 2 0 1 2. [ 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

223 Optical Pumping Z 1 2 3 4 5 6 π π π 4 T T 4 TF } S 2 z X Y S z 1 2 3 4 5 6 0 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) 0 1 2 cos(φ)( +1 + 1 ) φ =(D gs ω)t = δωt ˆM ˆM = a 0 0 + b ( +1 +1 + 1 1 ) a b ˆM

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 0.031 λ/γ T = T 2 1 D gs =2870 δω/ω 0 =8.8 10 9 / τ 0.2% N 1/ N N 1.74 10 23 3 10 11 µ 3 3 1/ N 2 10 13 / τ 2 10 9 / τ 100µ 1 2 σ pulsed y 6.7 10 13 / τ

225 T 1/2 2T e (2T/T 2) n n 3 T 2 T e n =1 2 S/N T e 10 17 3 10 13 50 100 D gs 1µ

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

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) + 1 8 (ɛ xx ɛ yy )(S 2 x S 2 y) + 1 8 (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

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 ( ) 1.6 10 6 =( 4.32 2) 15, 1K 5 δ 3 ddgs dt 742 2.58 10 7

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 0 10 2 10 3

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 β1 1+2. ω T =T 0 β 1,2

231 D ZFS (khz) (a) (b) (c) 0 1 2 3 Brass Tungsten 2.443 khz, 6.92 mk 4 0 2 4 6 8 10 Temperature (mk) D ZFS (khz) 2 2.2 2.4 2.6 2.8 T=5 mk T=10 mk 3 0.2 0.1 0 0.1 0.2 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

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

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 10 15 10 10 10 5 Allan Deviation after 1s averaging Single NV CW / f 40 1000 N 2

234 σ y =2 10 12 τ 1/2 9 /

235 200 m s =0 m s = ±1 d /dt = (2π)77 / N

236 " #!!"#$%&'( )%*%&+( Temperatureaccuracy Kelvin 10. 1. 0.1 0.01 CdSe QD SThM Nano Diamond Raman Liquid crystal Infrared Green FluorescentProtein Seebeck! "!! "!,-./0'( )%*%&+( #! " " 0.001 ProjectedNano Diamond Bulk Diamond 0.01 0.1 1. 10. Sensor size um ±1 0 (T ) δ

237! 1.1 " 1.05 1!"#$!" "!" 1 0.8 Normalized fluorescence 0.95 0.9 0.85 population 0.6 0.4 0.2 0.8 0 0.75 0 100 200 300 400 500 2τ (us) 24.2 24.22 24.24 24.26 24.28 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 0.03 1 / 200 600

238 ω 1 2 ( 0 + B ) B = 1 2 ( +1 + 1 ) τ 2π +1 1 τ ±1 99.99 % 12 13 0.5 2τ η =(9± 1.8) / 2τ =250µ δt =1.8 ± 0.3 2τ (2d /dt 2τ) 1 2τ < 2τ 1 µ

239 N 500 =2.87 100 100 nm 2.5

240 " # 40 1 42 7000 Normalized fluorescence 0.98 0.96 0.94 0.92 0.9 data max. slope meas. freq. y ( µm) 44 46 48 50 52 54 56 58! $ " #!"#$ % & 6000 5000 4000 3000 2000 1000 0.88 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.9 ω (GHz) 60 0 50 55 60 65 70 x ( µm)! 4 $ 15 3.5 3 off AU Fit on AU Fit " theory NV data 2.5 10 T (K) 2 1.5 1 0.5 0 T (K) 5! # $ % & 0.5 0 50 100 150 200 250 300 350 laser power ( µw) 0 0 1 2 3 4 5 6 7 8 9 distance ( µm) 500 0.8 µ

241 0.8 ± 0.1 µ 0.8 µ δt =(44± 10) T (r) = Q 4πκr Q κ r 72 ± 6 1 2 1 2 5 7 µ 1 T =( 20 ± 50)

242 12 µ 0.5 ± 0.2 10 120µ 3.9 ± 0.1 80 80 µ /

243 y ( µm) " 10 5 0 5 10 15 5 0 5 10 15 x ( µm) # $ +$'()*!"#$$$$$$$!"%$$$$$$$$!#$$$$$$$$%$$$$$$$$$#$$$$$$$$"%$$$$$$$"#!" &'()#$!"$ &'()#$ &'()#%!"%!"#$$$$$$$!"%$$$$$$$$!#$$$$$$$$%$$$$$$$$$#$$$$$$$$"%$$$$$$$"# & '()* T K 2.5 2.0 1.5 1.0 0.5 0.0 Pos 1 Pos 2 y ( µm) 250 200 150 100 50 30 35 40 45 &'()! T (K) 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0!"#$ 0 20 40 60 80 100 Au Fluorescence (kcps) *#+#$%-. /#+#,15#3,1%#4 y ( µm) 35 40 45 50 55!"% *#+#$%,-. /#+# 60 /#+#012#3,1$#4 50 55 60 65 70 75 50 60 70 80 x ( µm) x ( µm) 3500 3000 2500 2000 1500 1000 500 532 638 1 2 1 2 494/528 515 617 532 630

244 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

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)

246 9 / 250 µ 13 T 1 /2 3 1000 80 µ / n 200 µ B 10 10 6 5 20 µ 50 µ

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

249 (a), (b) r 1 r 2 1 2 =, =, 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

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

251 V (r) 1/r β V ab = V 13 V 14 V 23 + V 24 ( 1 = 1 ) ( 1 + 1 ) R β 13 R β 14 R β 24 R β 23 ( ) ( ) 1 = R 2 + r 4 r 1 1 1 β R + 2 + 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 2 2 2 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

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

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

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

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

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 3 0 1 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 3 0 1 l 6.

257 V (d a ) 2 ( d s 1 R 2 a 0 (ν ν d a ν 2 ) ) 1 2d 3 1. 2 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

258 H B = ij t ija i a j + 1 2 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

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

260 ij d i j d z d z + 1 2 (d +d + d d + ) i j = d 2 0, i j d z d z + 1 2 (d +d + d d + ) i j = d i d 0, i j d z d z + 1 2 (d +d + d d + ) i j = d 0 d j, i j d z d z + 1 2 (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

261 40 87 40 87 I 1 =4 I 2 =3/2 H Q 1 40 87 1, ±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

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

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

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

265 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