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1 1 1/2 NMR 1.1 ( 10 7 ) 1 H 1/ C 1/ N O 5/ e 1/ NMR rad/ts 1T NMR (MHz) 1

4035 0.4036 3.076 0.001 17 O 5/2-1.894-3.6279 5.

2 S z= γhb S = 1 z B=0 B 1.1 B γhb +> -> 1.2 ± 1/2 0 1/

3 ESR NMR NMR 1 H 1.2 1/2 NMR 1 H 1 ESR 1/2 ±1/2 x ψ(x) dxψ (x)ψ(x) ±1/2 [ ] [ ] 1 0 +,, (1.1) [1 0], [0 1] (1.2) [ 1 0 I = 0 1 ] (1.3) 1/2 2 s, t = c + (t) + + c (t) (1.4) [ ] S z = (1.5) S z + = 1 2 +, S z = 1 (1.6) 2 *1 H + NMR 1 H 1.2 1/2 3

+,, (1.1) 0 1 + [1 0], [0 1] (1.2) [ 1 0 I + + + = 0 1 ] (1.

4 1.3 NMR NMR 1.4 NMR (1) (2) (3) (4) (5) [ 0 1 ] [ 0 0 ] σ + = 0 0, σ = 1 0 (1.7) σ + = +, σ + = (1.8) S z I σ + σ σ + σ [ ] [ ] i σ 1 =, σ 2 = (1.9) 1 0 i 0 4 1

5 hω + σ σ 1.5 σ + = σ 1 + iσ 2,σ = σ 1 iσ 2 (1.10) S z [ ] 1 0 σ 3 = (1.11) 0 1 σ 1 σ 2 σ 3 [σ j,σ k ]=iɛ jkl σ l (j, k, l =1, 2, 3) (1.12) ɛ 123 =1 1 ɛ 132 = ɛ 321 = 1 σ 1 σ 2 σ 3 hs µ µ = γ hs (1.13) γ B E E = µ B (1.14) z B = B z z E = µ z B z = γ hb z S z (1.15) 1.1. z 1.2 1/2 5

6 1.3 x h (t)/2 =γb x (t) hω 0 γ hb z Ĥ = 1 2 hω 0σ h (t)(σ+ + σ ) = 1 2 hω 0σ h (t)σ 1 (1.16). 2 2 [ ] Ĥ = 1 2 h ω0 (t) (1.17) (t) ω 0 [ c+ (t) s, t = c (t) ] (1.18) ī s, t = t (1.19) t hĥ s, [ ] [ ][ ] c+ (t) = i ω0 (t) c+ (t) (1.20) t c (t) 2 (t) hω 0 c (t) (t) t Â A(t) s, t Â s, t (1.21) Z j=± j exp[ βĥ] j (1.22) 6 1

7 1 2 1 i 2 1 i β =1/k B T 1.3 A(t) = s, t j j Â s, t = t s, t j j=± j=± j Â s, = tr{âˆρ(t)} (1.23) ˆρ(t) s, t s, t (1.24) tr ˆB tr{ ˆB} j=± j ˆB j (1.25) tr{â ˆBĈ} = tr{ ˆBĈÂ} = tr{ĉâ ˆB} (1.26) ρ jk (t) = j ˆρ(t) k ˆρ(t) j ρ jk (t) k (1.27) j=± k=± Â A jk j Â k 1.3 7

8 4 Z FU 3 2 S C T Z F E S C T. hω k B. Â j=± j A jk k (1.28) k=± tr{âˆρ(t)} = A jk ρ kj (t) (1.29) j=± k=± A(q) t ρ(t) Â [ ] c+ (t) ˆρ(t) = s, t s, t = [c + c (t) (t),c (t)] [ ] c+ (t)c +(t) c +(t)c (t) = c (t)c +(t) c (t)c +(t) [ ] ρ++ (t) ρ + (t) (1.30) ρ + (t) ρ (t) Ĥ T Z = tr{e βĥ } (1.31) β =1/k B T k B 8 1

9 S S S S 1.8 S ˆρ eq = e βĥ Z (1.32). ˆρ eq = 1 j j e βĥ k k Z = j=± j=± k=± k=± j ρ eq jk k (1.33) ρ eq jk 1 Z j e βĥ k (1.34) β =1/k B T A < Â> eq tr{âˆρ eq } = j=± k=± A jk ρ eq kj (1.35) + ρ eq jk (1.16) (t) = 1.3 9

10 [ 1 2 hω 0 s h h 1 2 hω 0 s ] = 0 (1.36) s 2 ( h ) 2 ( 1 2 hω 0) 2 = 0 (1.37) s = ± h ( ) 2 +( 1 2 ω 0) 2 (1.38) E = ± 1 2 h 2 + ω0 2 ± hω /2 1 = [ ] 1 ω0, 2 + ω0 2 (1.39) 2 = [ ] ω0 2 ω 0 (1.40) Z Z = ( ) β hω n e βĥ n = 2cosh 2 n (1.41) ˆρ eq jk 1 ) ( 1 e Z j j β hω / e β hω /2 2 k k (1.42) Z [ ( )] β hω F = k B T ln Z = k B T ln 2 cosh 2 ( = β hω + k B T ln 1+e β hω ) (1.43) 2 E = ( ) β hω ln Z = β hω tanh β 2 2 S = F [ ( )] T = β hω β hω + k B ln 2 cosh 2 2 C = E T = 10 1 k B ( ) β hω 2 2 cosh 2 (β hω over2) (1.44) (1.45) (1.46)

41) ˆρ eq jk 1 ) ( 1 e Z j j β hω /2 1 + 2 e β hω /2 2 k k (1.

11 (a) S (b) S S S S Ω 1.9 Ω S S ω 0 γb S 3 Ω + Ω S ˆρ(t) 1.19 t ˆρ(t) = ī h [Ĥ, ˆρ(t)] (1.47) quantal Liouville equation [ ] ([ ][ ] ρ++ (t) ρ + (t) = i ω0 (t) ρ++ (t) ρ + (t) t ρ + (t) ρ (t) 2 (t) ω 0 ρ + (t) ρ (t) [ ][ ]) ρ++ (t) ρ + (t) ω0 (t) ρ + (t) ρ (t) (t) ω 0 (1.48). ρ ++ (t) 0 0 (t) (t) ρ ++ (t) ρ (t) = i 0 0 (t) (t) ρ (t) t ρ + (t) 2 (t) (t) ω 0 0 ρ + (t) ρ + (t) (t) (t) 0 ω 0 ρ + (t) (1.49)

12 (t). ρ = ρ + ρ ++ ρ ρ ++ +ρ =1 (positivity condition) ρ ++ > 0 ρ > 0 (1.16) ρ(t) =S 0 I + S 1 (t)σ 1 + S 2 (t)σ 2 + S 3 (t)σ 3 (1.50) S1 2 + S2 2 + S3 2 =1 (1.16) σ i Ṡ 0 = 0 (1.51) Ṡ 1 = ω 0 S 2 (1.52) Ṡ 2 = ω 0 S 1 + (t)s 3 (1.53) Ṡ 3 = (t)s 2 (1.54) (1.51) S 0 ρ ++ + ρ = S 0 S 1 0 ω 0 (t) S 1 d dt S 2 = ω 0 0 (t) S 2 (1.55) (t) (t) 0 S 3 S (S 1,S 2,S 3 ) (1.52)-(1.54) ds dt =ΩF S (1.56) 12 1 Ω F =( (t), 0,ω 0 ) (1.57) M B dm = γb M (1.58) dt 1/2 i =1, 2, 3 S 3

13 w ' u ' Ω v S π 3π 4π γβ t 1.11 S 3 = 0, 0.5, 1. x y z 1.5 (t) =2γB 1 cos ωt (1.59) Ω F Ω F =Ω + (t)+ω (t)+ω 0 (1.60) Ω 0 =(0, 0,ω 0 )

14 w u Ω v 1.12 π Ω + =( γb 1 cos ωt, γb 1 sin ωt, 0) (1.61) Ω =( γb 1 cos ωt, γb 1 sin ωt, 0) ω 0 γb Ω F z S z 1.9(a) S Ω + Ω F Ω F =Ω + (t)+ω 0 (1.62) (1.52)-(1.54) d dt S 1 S 2 S 3 = 0 ω 0 γb 1 sin ωt ω 0 0 γb 1 cos ωt γb 1 sin ωt γb 1 cos ωt 0 S 1 S 2 S 3 (1.63) z 1.9(a) u v w 14 1

15 u v w = cos ωt sin ωt 0 sin ωt cos ωt S 1 S 2 S 3 (1.64) u 0 (ω 0 ω) 0 d dt v = (ω 0 ω) 0 γb 1 w 0 γb 1 0 u v w (1.65) ρ (u, v, w) T Ω ( γb 1, 0,ω 0 ω) T d dt ρ = Ω ρ (1.66) v tan χ = ω 0 ω γb 1 (1.67) u v w = cos χ 0 sin χ sin χ 0 cos χ u v w (1.68) (u,v,w ) u u d dt v = 0 0 Ω( ) v (1.69) 0 Ω( ) 0 w Ω(δ) = δ 2 +(γb 1 ) 2 δ = ω 0 ω Ω 1.10 u u 0 v = 0 cos Ωt sin Ωt v 0 (1.70) 0 sin Ωt cos Ωt w (u 0,v 0,w 0) T (u 0,v 0,w 0 ) T u 0 cos χ 0 sin χ u 0 v 0 = v 0 (1.71) sin χ 0 cos χ w 0 z 0 z 0 w

67) u v w = cos χ 0 sin χ 0 1 0 sin χ 0 cos χ u v w (1.68) (u,v,w ) u 0 0 0 u d dt v = 0 0 Ω( ) v (1.

16 (u, v, w) T u cos χ 0 sin χ cos χ 0 sin χ v = cos Ωt sin Ωt w sin χ 0 cos χ 0 sin Ωt cos Ωt sin χ 0 cos χ (γb 1) 2 +δ 2 cos Ωt δ δγb1 sin Ωt (1 cos Ωt) u Ω 2 Ω Ω 2 0 = δ Ω sin Ωt cos Ωt γb 1 sin Ωt Ω v 0 δγb1 Ω (1 cos Ωt) γb1 2 Ω sin Ωt δ 2 +(γb 1) 2 cos Ωt Ω w 2 0 w δγb 1 w = u 0 δ 2 +(γb 1 ) (1 cos Ωt) v γb sin Ωt δ2 +(γb 1 ) 2 + w 0 δ 2 +(γb 1 ) 2 cos Ωt δ 2 +(γb 1 ) 2 (1.73) u 0 v 0 w 0 (1.72) u 0 = v 0 =0,w 0 = 1 w = 1+ 2(γB 1) 2 ( ) Ωt δ 2 +(γb 1 ) 2 sin2 2 (1.74) S 3 = w 1.11 σ ± H rot = hω 0 2 σ 3 + hγb ( e iωt σ + + e iωt ) σ (1.75) e iωt σ + e iωt σ e iωt σ + e iωt σ Π Π/2 δ =0 π/2γb 1 w w 0 = 1 π/2 w =0 (u, v, w) u v 1.13 π/γb 1 π w =

+(γb 1 ) 2 + w 0 δ 2 +(γb 1 ) 2 cos Ωt δ 2 +(γb 1 ) 2 (1.73) u 0 v 0 w 0 (1.72) u 0 = v 0 =0,w 0 = 1 w = 1+ 2(γB 1) 2 ( ) Ωt δ 2 +(γb 1 ) 2 sin2 2 (1.74) S 3 = w 1.

17 w u- v u 1.13 v u- v S 3 π/2 u v H = 1 2 hω 0σ h σ 1 (1.76) H = 1 2 hω 0σ h σ 2 (1.77)

19 2 1/ (1.56) ds dt = Ω S (2.1)?? z z T 1 ds 3 dt = Seq z S 3 T 1 (2.2) z?? x Bloch ds 1 dt ds 2 dt ds 3 dt =(S Ω) 1 S 1 T 2 =(S Ω) 2 S 2 T 2 (2.3) =(S Ω) 3 + Seq z S 3 T 1 19

20 z y B 0 x 2.1 T 2 T Bloch x, y, z Bloch (2.3) (A) (B = B 0 ẑ) ω 0 = γb 0 Bloch ds 1 dt ds 2 dt ds 3 dt = ω 0 S 2 S 1 T 2 = ω 0 S 1 S 2 T 2 (2.4) =0 20 2

21 2.2 S 1 = me t/τ cos φt S 2 = me t/τ sin φt (2.4) φ sin φt 1 τ cos φt = ω 0 sin φt 1 cos φt (2.5) T 2 φ = ω 0 τ = T 2 S 1 = me t/t2 cos ω 0 t (2.6) S 2 = me t/t2 sin ω 0 t (2.7) (B) B =(B 1 cos ωt, B 1 sin ωt, B z ) ω 0 = γb 0 Bloch ds 1 dt ds 2 dt = 1 T 2 S 1 ω 0 S 2 γb 1 sin ωts 3 = ω 0 S 1 1 T 2 S 2 + γb 1 cos ωts 3 (2.8) 2.2 Bloch 21

22 ( ) ( ) ds 3 dt = γb 1 sin ωts 1 γb 1 cos ωts 2 + Seq z S 3 T 1 S 1 = u(t) cos ωt v(t) sin ωt S 2 = u(t) sin ωt + v(t) cos ωt (2.8) u + a 2 u + bv =0 v + a 2 v bu + S 3 = 0 (2.9) S 3 + a 1 S 3 v = a 1 Sz eq a 1 = 1/γB 1 T 1 a 2 = 1/γB 1 T 2 b = (ω 0 ω)/γb 1

23 u = v = S 3 =0 (2.9) γb 1 T 2 δt 2 u = 1+(δT 2 ) 2 +(γb 1 ) 2 Sz eq T 1 T 2 γb 1 T 2 v = 1+(δT 2 ) 2 +(γb 1 ) 2 Sz eq (2.10) T 1 T 2 S 3 = 1+(δT 2 ) 2 S 1+(δT 2 ) 2 +(γb 1 ) 2 z eq T 1 T 2 δ = ω 0 ω χ S = χb (S 1 + is 2 )=(χ iχ )(B x + ib y ) (2.11) u = χ B 1 v = χ B 1 χ γt 2 (δt 2 )χ 0 B 0 = 1+(T 2 δ) 2 +(γb 1 ) 2 T 1 T 2 (2.12) χ γt 2 χ 0 B 0 = 1+(T 2 δ) 2 +(γb 1 ) 2 T 1 T 2 (2.13) χ 0 = Sz eq /B 0 γb 1 χ χ γb 1 χ χ Bloch 23

25 3 3.1 T 1 T 2 ρ = i/ h[h, ρ]+γρ ψ A+B > ψ A > A B ρ A+B (t) ρ A+B (t) = ψ A+B (t) ψ A+B (t) (3.1) X A (t) <X A (t) >= tr A tr B {X A ρ A+B (t)} (3.2) tr B {ρ A+B (t)} = ρ A (t) (3.3) B ρ A <X A (t) >= tr A {X A ρ A (t)} (3.4) ρ A 25

26 3.1 A < ˆµ ( t) > K H A H I H B H H = H A + H I + H B (3.5) H A (t) = hω 0 2 σ 3 + hkɛ(e iωt σ + + e iωt σ ) (3.6) H B [b q,b q ]=δ qq H B = q ω q b qb q (3.7) H I H I = i q (g q σ + b q g qσ b q) (3.8) H I b q σ

27 σ x H = H + H + H A I B 3.2 A B σ b q b q σ + σ b q d dt ψ A+B >= ī h H ψ A+B > (3.9) ρ A+B (t) d dt ρ A+B(t) = ī h [H, ρ A+B(t)] ī h Lρ A+B(t) (3.10) H I H I H 0 (t) H I H(t) =H 0 (t)+h I ρ(t) = ī h [(H 0(t)+H I ),ρ] ī L(t)ρ(t) (3.11) h U 0 + (t) 1 *1 U + 0 (t) = i/ hh 0(t)U + 0 (t)

28 U + 0 (t) = exp ī h t 0 [ ī h t 0 ] dτ H 0 (τ) dτ H 0 (τ) 1 h 2 t 0 dτ τ 0 dτ H 0 (τ)h 0 (τ )+ (3.12) exp τ >τ >τ > U 0 + (t) ρ I (t) U 0 (t)ρ(t)u + 0 (t) (3.13) H I (t) U 0 + (t)h IU0 (t) (3.14) U 0 + (t) U 0 (t) ρ I (t) = ī h [H I(t),ρ I (t)] ī h L I(t)ρ I (t) (3.15) ρ I (t) H I (t) ) ρ I (t) =ρ I (t 0 ) ī h t t 0 dt 1 L I (t 1 )ρ I (t 1 ) (3.16) ρ I (t 1 ) t ρ I (t) =ρ I (t 0 ) ī dt 1 L I (t 1 )ρ I (t 0 ) h t 0 1 t t h 2 dt 1 dt 2 L I (t 1 )L I (t 2 )ρ 2 (t 0 )+ (3.17) t 0 t 0 L I (t 1 )L I (t 2 )ρ I (t 0 )=[H I (t 1 ), [H I (t 2 ),ρ I (t 0 )]] (3.18) H 0 = H A + H B ρ I (t) =ρ I (t + t) ρ I (t) (3.19) t 0 t t t + t ρ I (t) = ī h 1 h 2 t+ t t t dt 1 L I (t 1 )ρ I (t) t+ dt 1 dt 2 L I (t 1 ){L I (t 2 )ρ I (t)} + (3.20) t

29 H I ρ I (t) ρ I (t) =ρ I A(t) 0 BB 0 (3.21) 0 BB 0 ρ I A (t) ρ I A (t) =tr B{ρ I (t)} (3.22) ρ I (t) tr B { ρ I A (t) = ī t+ t h tr B dt 1 L I (t 1 ) ( ρ I A (t) 0 BB 0 )} t { 1 t+ t1 h 2 tr B dt 1 dt 2 L I (t 1 ) ( L I (t 2 )ρ I A(t) 0 BB 0 )} t t + (3.23) H I (t) = i ( ) g q e ī h dth A(t) σ + e ī h dth A(t) e ī h HBt b q e ī h HBt q +c.c. (3.24) {[ tr B σ I + (t)b I q(t),ρ I A(t) 0 BB 0 ]} { = σ+(t)ρ I I A(t)tr B b I q (t) 0 BB 0 } ρ I A (t)σi + (t)tr { B 0 BB 0 b I q (t)} (3.25) 0 tr B {[H I (t 1 ), [H I (t 2 ),ρ I A (t) 0 BB 0 ]]} H I (t) tr B { bq (t 1 ) 0 BB 0 b + q (t 2) } 0 (3.26) tr B { bq (t 1 )b + q (t 2 ) 0 BB 0 } e iωq(t1 t2) (3.27) tr B { b + q (t 1 ) 0 BB 0 b q (t 2 ) } e iωq(t1 t2) (3.28) tr B { b + q (t 1 )b q (t 2 ) 0 BB 0 } 0 (3.29) ρ I A(t) = 1 t+ t t1 h 2 gqg q dt 1 dt 2 q t t [ e iωq(t1 t2) σ (t I 2 )ρ I A(t)σ+(t I 1 )

30 +e iωq(t1 t2) σ I (t 1 )ρ I A(t)σ I +(t 2 ) e iωq(t1 t2) σ+ I (t 1)σ I (t 2)ρ I A (t) ] e iωq(t1 t2) ρ I A(t)σ+(t I 2 )σ (t I 1 ) (3.30) σ + (t 1 )=e iω0t σ + σ (t 2 )=e iω0t σ t 1 t 2 e ±i(ωq ω0)(t1 t2) ( ) 3 L gq g q = dω dωω 2 gq 2πc g q = dωn (ω)g 2 (ω) (3.31) q ω2 N(ω) = 2πc 3 (3.32) g 2 (ω) = L3 4π 2 dωg q gq (3.33) eq.(3.30) t+ t1 dt 1 dt 2 dωn (ω)g 2 (ω)e i(ω ω0)(t1 t2) (3.34) t t N(ω)g 2 (ω) ω t 1 t 2 t+ t1 dt 1 dt 2 dωn (ω)g 2 (ω)e i(ω ω0)(t1 t2) t t dτ dωn (ω)g 2 (ω)e i(ω ω0)τ 0 0 dτ e i(ω ω0)τ = πδ(ω ω 0 ) P =(K + iδ) τ (3.35) 1 ω ω 0 (3.36) K = dωn (ω)g 2 (ω)πδ(ω ω 0 )=πn(ω 0 )g 2 (ω 0 ) (3.37) δ = P dω N(ω)g2 (ω) (3.38) ω ω 0 ρ I A (t) t =2Kσ ρ I A σ + (K + iδ)σ + σ ρ I A (t) (K iδ)ρ I A(t)σ + σ (3.39) 0 ρ I A (t)/ t ρi A (t) 30 3

31 ρ A (t) = ī h [H A(t),ρ A (t)] + 2Kσ ρ A (t)σ + Kσ + σ ρ A (t) Kρ A (t)σ + σ iδ [σ + σ,ρ A ] = ī h [H A(t) hσ σ +,ρ A ] Γρ A (t) (3.40) Γρ A 2Kσ ρ A σ + Kσ + σ ρ A ρ A σ + σ (3.41) [ hδσ + σ,ρ A ] ω 0 +δ ω 0 ω 0 ρ A (t) = ī h [H A(t),ρ A (t)] Γρ A (t) (3.42) Γρ A Γρ A 1/T 1 =2K 1/T 2 = K 0K 1/T 1 =2/T (0K) 1/T 1 =2/T 2 1/T 1 < 2/T 2 1/T 1 < 2/T 2 H A (t) H B H I H = H A (t)+h I + H B (3.43) H I = σ + σ c j (b j + b j ) (3.44) j H B = h ω j b j b j (3.45) j V = σ + σ H I B 0K H I

32 ρ A+B (t) ρ A (t)e βh B (3.46) β = 1/k B T tr B {b j (t 1)e βh B } tr B {b j (t 1 )e βh B } ( n n 0) tr B {b j (t 1)b j (t 2 )e βh B } X j b j + b j { tr B e βh B X j (t 2 )X j (t 1 ) } = n j e njβωn j nj n j X j n j +1 n j +1 X j n j e ωn j (t1 t2) n j + n j n j +1 e (nj+1)βωn j nj +1 n j +1 X j n j n j X j n j e iωn j (t1 t2) = 1 1 e βωn j 1 e njβωnj = n j 1 e βωn j [ e iωn j (t1 t2) + e βωn j e iω nj (t 1 t 2) ] (3.47) (3.48) Γρ = j { c 2 j 1 h 2 t+ t t1 dt 1 dt 2 e iωn j (t1 t2) t t 2 [ V I (t 1 ),V I (t 2 )ρ ] 2e βωn j 1 e βωn j 1 e βωn j [ V I (t 1 ),ρv I (t 2 ) ]} (3.49) V I (t) exp [ ī h t 0 ] [ ī dth A (t) (σ + σ )exp h t 0 ] dth A (t) (3.50) J(ω) = j c j δ(ω ω nj ) (3.51) 1 h 2 t+ t t dt 1 t1 t dt 2 dωj(ω) [ ( ) β hω coth cos (ω(t 1 t 2 )) [ V I (t 1 ), [ V I (t 2 ),ρ ]] 2 i sin ω(t 1 t 2 ) [ V I (t 1 ), {V I (t 2 ),ρ} ]] (3.52) 32 3

33 {, } [ ( ) ] β hω dωj(ω) coth cos ω(t 1 t 2 ) γδ(t 1 t 2 ) (3.53) 2 dωj(ω) sin ω(t 1 t 2 ) δ(t 1 t 2 ) (3.54) V σ + σ σ + σ = σ + σ Γ ρ = t [2γσ + σ ρσ + σ γ(σ + σ ρ ρσ + σ ) +iδ(σ + σ ρ + ρσ + σ )] (3.55) β h γ >>iδ iδ ρ = ρ z σ z + ρ I σ I + ρ + ρ + + ρ σ (3.56) Γ ρ = γ(σ + σ ρ ρσ + σ ) (3.57) B B ρ A = ī h [H A(t),ρ A (t)] Γρ A Γ ρ A (3.58) Γ tot =Γ+Γ Γ tot ρ =2Kσ + ρσ K (σ + σ ρ ρσ + σ ) (3.59) K = K + γ K <2K 1/T 1 < 2/T A ω j j a j â j H A (t) = j hω j 2 â jâj + he(t) jk µ jk (â jâk +â kâj) (3.60) H I H I = i g q A jk (b q + b q)(â jâk +â kâj) (3.61) q jk

34 (3.46) (3.52) [ V I (t) exp ī t ] dth A (t) h 0 jk â jâk [ ī exp h t 0 ] dth A (t) (3.62) (3.54) (3.55) (3.61) a 2â1ρa 1â2 J(ω) A (3.49) [ V I (t 1 ),V I (t 2 )ρ ] [ V I (t 1 ),ρv I (t 2 ) ] ( he(t) jk µ jk) H A (t) HA 0 = hω j j 2 â jâj A ρ lm l m l m j Γˆρlk Γ Γ jklm j [ V I (t 1 ),V I (t 2 ) l m ] lk j V I (t 1 )V I (t 2 ) l m k = α A jα A αl e iωjαt1 e iωαlt2 δ mk (3.63) j V I (t 1 ) l m V I (t 2 ) k = A jl A mk e iωjlt1 e iωmkt2 (3.64) j l m V I (t 2 )V I (t 1 ) l j V I (t 2 )l m V I (t 1 ) l (3.65)

35 Ω(t) t P(Ω;0) P(Ω;t 1) P(Ω;t ) 2 P(Ω;t) 3.3 Ω(t) P (Ω; t) Ω(t) H(t) =H + H I (Ω(t)) (3.66) H I (Ω(t)) z H I (Ω(t)) = Ω(t)σ Z (3.67) Ω(t) Ω(t)

36 Ω(t) Ω(t) (??) (Ω 1, Ω 2, Ω 3, ) Ω(t) Ω(t) Ω P (Ω,t) Ω(t) P (Ω,t) Ω t P (Ω,t)=ˆΓP (Ω,t) (3.68) ˆΓ Ω t P (Ω,t) P (Ω,t) t P (Ω,t)= Γ ΩP (Ω,t) (3.69) Γ Ω P (Ω,t) Ω Two-state jump model Ω(t) ± { t P (+,t)= γp(+,t)+γp(,t) t P (,t)=γp(+,t) γp(,t) (3.70) + (1, 0) T (0, 1) T P = ΓP (3.71) t ( ) P (+,t) P P (,t) ( ) 1 1 Γ γ 1 1 (3.72) (3.73) 36 3

37 Ω) t P (Ω,t)= (Ω ˆΓ Ω )P (Ω,t) (3.74) Ω Ω P (Ω,t)= γ(ω)p (Ω,t)+ t dω (Ω ˆΓ Ω )P (Ω,t) (3.75) dωp (Ω,t) = 1 (3.76) γ(ω) = dω (Ω Γ Ω ) (3.77) r = Ω Ω w(ω,r) = (Ω ˆΓ Ω ) Chapman- Kormogrov t P (Ω,t)= drw(ω,r)p (Ω,t)+ drw(ω r, r)p (Ω r, t)(3.78) e γ ( γ) n q f(q) = n! n=0 ( ) n f(q) =f(q r) (3.79) q Chapman-Kormogrov t P (Ω,t)= ( 1) n ( ) n drw(ω,r)p (Ω,t)+ dr r n w(ω,r)p (Ω,t) n! Ω n=0 ( 1) n ( ) n = drr n w(ω,r)p (Ω,t) (3.80) n! Ω n=1 n α n (Ω) α n (Ω) = drr n w(ω,r) (3.81) eq.(3.80) ( t P (Ω,t)= Ω α 1(Ω) ) 2 Ω 2 α 2(Ω) P (Ω,t) (3.82) P (Ω,t Ω 0,t 0 ) eq.(3.82) t = t + t dωωp (Ω,t) dωω 2 P (Ω,t) < Ω > t = α 1 (Ω 0 ) < ( Ω) 2 > t = α 2 (Ω 0 ) (3.83)

38 t < Ω(t) >= γ <Ω(t) > t < Ω2 (t) >= α<ω(t) > α 1 (Ω) = γω α 2 (Ω) = α γ 2 (3.84) eq.(3.82) t P (Ω,t)= γ ( 2 ) Ω Ω +Ω P (Ω,t) Γ Ω P (Ω,t) (3.85) t P (Ω,t)=0 Γ ΩP (Ω,t)=0 P eq (Ω) = 1 e Ω2 2 2 (3.86) 2π Ṡ = il(ω(t))s (3.87) P (S,t) P (S,t) P (S,t+ δt) =P (S il(ω(t))δt,t) (3.88) t P (S,t)= S ) (ṠP (S,t) = ( il(ω(t))p) (3.89) S Ω(t) P (S, Ω,t) Γ Ω P (S, Ω,t)=i t S ( il(ω)p (S, Ω,t)) Γ ΩP (S, Ω,t) (3.90) S(Ω,t)= dssp (S, Ω,t) (3.91) eq.(3.90) stochastic Liouville t S(Ω,t)= L(Ω)S(Ω,t)+Γ ΩS(Ω,t) (3.92) ω

39 H(t) =H A (t)+ hω(t)σ z (3.93) t ρ(t) = ī h [H A(t),ρ] iω(t)[σ z,ρ] = ī h L A(t)ρ iω(t)σz ρ (3.94) A ρ Aρ ρa P(t) = ilp(t) iω(t)vp (3.95) t Ω(t) Ω P(t) P(Ω,t) P(Ω,t)= i (L +ΩV) P(Ω,t)+ΓP(Ω,t) (3.96) t Γ= γ ( 2 ) Ω Ω +Ω (3.97) P(Ω,t 0 )=P(t 0 ) 0 0 = e Ω2 2 2 / 2π P (Ω,t)=e Ω2 4 2 P(Ω,t) b b t P(Ω,t)= i ( L +(b + b )V ) P(Ω,t) γb bp (Ω,t) (3.98) b b n > P(Ω,t)= P n (t) n > (3.99) eq.(3.98) Ṗ n (t) = i (L + nγ) P n (t) i 2 VP n+1 (t) i 2nVP n 1 (t)(3.100) P n (t) P 0 (t) < Ω(t)Ω(t ) >= 2 e γ(t t ) eq.(3.100) P n (t) P n (s) eq.(3.100) s + il i 2 V 0 0 P 0 (s) P 1 (s) i 2 V s + γ + il i 2 V 0 = P 2 (s) 0 i 2 2 V s +2γ + il i 2 V 0 0 i 3 2 V s +3γ + il.. P 0 (s) P 0 (0) P 1 (0) P 2 (0). (3.101)

40 P 0 (s) = s + il + V 2 s+γ+il+v s+2γ+il+v s+3γ+ 3 2 V V V P 0(0) 1 s + il +Γ(s) P 0(0) (3.102) il K 2K 0 0 il = 0 0 K iω K + iω 0 (3.103) (s + il +Γ(s)) 1 = 1/s /(s +2K) G + (s) G (s) (3.104) G ± (s) = 1 s + K ± iω s+γ±iω s+2γ±iω 0 + (3.105) [ 2 (s + K ± G ± (s) = exp iω0 ) 2 ] [ ] (s + K ± iω0 ) 2 2 Erfc 2 π [ (s + K ± iω0 ) 2 ] exp (3.106) (s ± iω 0 )/ 2 << 1 Erfc[(s + K ± iω 0 )/ 2 ] π/2 eq.(3.106) γ = 2 /γ γ >>ω 0 G ± (s) = 1 s + K + γ ± iω 0 (3.107) 3.4 I(ν) = G (iν) 2 ω

41 [h] ???? 2-2?? 2-3?? 2-4??

43 , 10, 10, 12, 6,9, 35, 7, 12, 5, 35, 35, 7, 5, 10, 6,8, 10, 35, 7, 11 29

３

- { } / f ( ) e nπ + f( ) = Cne n= nπ / Eucld r e (= N) j = j e e = δj, δj = 0 j r e ( =, < N) r r r { } ε ε = r r r = Ce = r r r e ε = = C = r C r e + CC e j e j e = = ε = r ( r e ) + r e C C 0 r e =