修士論文 物性研究 電子版 Vol, 2, No. 3, (2013 年 8 月号 ) * Bose-Einstein.

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1 * Bose-Einstein Bogoliubov-de Gennes Bogoliubov-de Gennes Bogoliubov-de Gennes Bogoliubov-de Gennes *1 phyco-sevenface@asagi.waseda.jp

2 Gross-Pitaevskii Bogoliubov-de Gennes ω = ω Bogoliubov-de Gennes GP Bogoliubov-de Gennes Bogoliubov-de Gennes Bogoliubov-de Gennes Y A 31 B 1 BdG 3 B.1 1 (

3 1 194, Bose, Planck. 195, Einstein (, [1]. Bose-Einstein (BEC. Einstein, 4 He [] [3] BEC.,, Einstein., 100% 4 He 10%., BEC, 4 He. BEC. Einstein , BEC [4, 5, 6]. BEC., Doppler,..,, µk BEC. BEC time of flight (TOF [7]. TOF, BEC. 4 He,,.,,, 1., Feshbach [8, 9].,,., [10, 11], [1, 13], [14, 15, 16, 17], - [18], Bloch [19], BEC [0, 1, ],.,,.,,,,,., [3]., BEC.,,,,.,, ( -Goldstone ( -Goldstone [3, 4].., 3

4 .,,....,.., ,., BEC,, Gross-Pitaevslii (TDGP [5], Gross-Pitaevskii (GP [6]. GP Schrödinger,., Bogoliubov-de Gennes (BdG [7, 8]. BdG. 3 BdG, BdG, BdG. BdG, Pauli 3.,., (., BdG. ( BdG,., BdG., BdG BdG [33, 34]., BdG. 4, GP.,.,.,., BdG,.., BdG BEC (, (. 4

5 5, BdG. BdG [9].,. BdG. 6,,,.,,. 5,. BEC.,,.,, BEC., ħ = 1..1 L S L = ψ (x(t K V + µψ(x g ψ (xψ (xψ(xψ(x, (.1 S = d 4 xl. (. x = (x, t, (.3 T = i t, (.4 K = 1 m, (.5 V = V ex (x, (.6, m, µ, V.,., g s a g = 4πa m, (.7. g g > 0, g < 0. g > 0,. ψ(x Π(x Π(x = L ( t ψ = iψ (x. (.8 5

6 ψ(x Π(x,. (.8 [ ˆψ(x, t, ˆΠ(x, t] = iδ(x x, (.9 [ ˆψ(x, t, ˆψ(x, t] = 0, (.10 [ ˆψ(x, t, ˆψ (x, t] = δ(x x, (.11 [ ˆψ(x, t, ˆψ(x, t] = 0, (.1. [ ] Ĥ = dx ˆΠ(x t ˆψ(x L = dx[ ˆψ (x(k + V µ ˆψ(x + g ˆψ (x ˆψ ] (x ˆψ(x ˆψ(x, (.13.. (.13 ˆψ(x e iθ ˆψ(x, (.14 ˆψ (x e iθ ˆψ (x, (.15., Nöther.. c. Nöther N µ N µ =, δψ(x µ N µ (x = 0, (.16 L δψ(x, (.17 ( µ ψ(x δψ(x = iθψ(x, (.18. Nöther 0 N = d 3 xn 0 (x, (.19., N = d 3 xψ (xψ(x, (.0 6

7 ..,. ψ(x,. ˆN = d 3 x ˆψ (x ˆψ(x, (.1.. (.1 (.13.,. (.1 e iθ ˆN iθ ˆψ(xe ˆN = e iθ ˆψ(x, (., ˆN., ˆN e iθ ˆN Ĥe iθ ˆN = Ĥ, (.3..3 Bose-Einstein., (. BEC., BEC,. BEC Ω. e iθ ˆN Ω Ω, (.4 ˆN Ω 0, (.5, 0 Ω ˆψ(x Ω = ξ(x, (.6. ξ(x, ρ c (x, N c ρ c (x = ξ(x, (.7 N c = d 3 xρ c (x = d 3 x ξ(x, (.8. 7

8 ˆψ(x ˆψ(x = ξ(x + ˆφ(x. (.9 ˆφ(x (.11, (.1,. [ ˆφ(x, t, ˆφ (x, t] = δ(x x, (.30 [ ˆφ(x, t, ˆφ(x, t] = 0, (.31 Ω ˆφ(x Ω = 0, (.3 (.9 Ĥ, ˆφ(x Ĥ0 ĤI Ĥ = Ĥ0 + ĤI. (.33 Ĥ0, ĤI, [ Ĥ 0 = d 3 x ˆφ (x(k + V µ + g ξ(x ˆφ(x + g ] (ξ (x ˆφ(x ˆφ(x + ξ (x ˆφ (x ˆφ (x, (.34 [ Ĥ I = d 3 x ξ (x(k + V µ + g ξ(x ˆφ(x + ˆφ (x(k + V µ + g ξ(x ξ(x + gξ (x ˆφ (x ˆφ(x ˆφ(x + gξ(x ˆφ (x ˆφ (x ˆφ(x + g ] ˆφ (x ˆφ (x ˆφ(x ˆφ(x, ( Ĥ ˆφ(x ˆφ(x e iθ ˆφ(x, (.36 ˆφ (x e iθ ˆφ (x, (.37. ˆN q = d 3 x ˆφ (x ˆφ(x, (.38., BEC N q. 3 Bogoliubov-de Gennes, Bogoliubov-de Gennes (BdG., BdG,, BdG. 8

9 3.1 Bogoliubov-de Gennes, Gross-Pitaevskii (GP. Heisenberg ˆψ(x i t ˆψ(x = [ ˆψ(x, Ĥ] = (K + V µ + g ˆψ (x ˆψ(x ˆψ(x, (3.1., ˆψ(x c ψ(x i t ψ(x = (K + V µ + g ψ(x ψ(x. (3. Gross-Pitaevskii (TDGP., ψ(x ξ(x, GP. (K + V µ + g ξ(x ξ(x = 0, (3.3 TDGP (3.4, BdG., TDGP ψ(x ψ(x = ξ(x + δψ(x. δψ(x, δψ(x.,, δψ(x, (3.4 i t δψ(x = (K + V µ + g ξ(x δψ(x + gξ (xδψ (x, (3.4 L = K + V µ + g ξ(x, (3.5 M = gξ (x, (3.6 δψ(x = u n (xe iω nt + v n(xe iω n t, (3.7 T y n (x = ω n y n (x, (3.8., T, y n ( L M T = M L ( un (x v n (x y n (x =, (3.9, (3.10. BdG., T, ω n. BdG ω n, δψ., ω n,. 9

10 3. Bogoliubov-de Gennes BdG T., σ 1, σ 3 Pauli. ( ( σ 1 =, σ = 0 1 σ 1 T σ 1 = T, (3.11 σ 3 T σ 3 = T, (3.1, (3.13 T y n (x. T (s, t d 3 xs (xσ 3 t(x, (3.14. Pauli 3. T (s, T t = (T s, t, (3.15., T (3.1 Pauli σ i. s = (s, s = = 1 (i = 1,, 3 d 3 xs (xσ 3 s(x, (3.16. σ 3 s.,. BdG.,,,,. 3.. ω R. y n (x, y m (x ω n, ω m. (y n, T y m = ω m (y n, y m = (T y n, y m = ω n (y n, y m, (3.17 (ω n ω m (y n, y m = 0, (

11 , ω n ω m y n (x, y m (x. ω n = ω m Gram-Schmidt, y n (x, y m (x. (3.11, ω n y n (x ω n z n (x σ 1 y n(x, (3.19 T z n (x = T σ 1 y n(x = σ 1 T y n(x = ω n σ 1 y n(x = ω n z n (x. (3.0 z n (x z n = d 3 x(σ 1 y n σ 3 (σ 1 y n = d 3 x(y n σ 1 σ 3 σ 1 y n = d 3 x(y n σ 3 y n = y n, (3.1, y n (x.,., y n (x, z n (x. y n (x, z n (x (y n, y m = δ nm, (3. (z n, z m = δ nm, (3.3 (y n, z m = 0, ( , BdG T y 0 (x = 0, (3.5..,. BdG. GP ξ ξ + iδξ., δξ 1 ( δξ(x T δξ = 0, (3.6 (x., GP. 11

12 GP.,. η, ξ(x ξ(x + iηξ(x, (3.7., (3.6 ( ξ(x ξ, (3.8 (x. BEC,., [33, 34]. GP.,.. d, x., (3.6. ξ ξ + d ξ, (3.9 x ( i ξ x (x i ξ x (x, (3.30,.,., y 0,θ = ( ξ(x ξ (x x ( i ξ y 0,x = x (x i ξ x (x., (3.31, (3.3 y 0,i (x(i. (3.6 y 0,i = d 3 x( δξ(x δξ(x = 0, (3.33.,. y 0,i (x y 0,i (x (y 0,i, y n = 0, (3.34 (y 0,i, z n = 0, (3.35 (y 0,i, y 0,i = 0, (3.36 1

13 ,.,.,., y 0,i (x σ 1 y 0,i(x = y 0,i (x, (3.37, y 0,i (x z 0,i (x y 0 (x, BdG {y n (x, z n (x, y 0,i (x}., BdG, BdG y 0,i (x y n (x, z n (x,. [33, 34] T y 1,i (x = I i y 0,i (x. (3.38 y 1,i (x (adjoint, BdG T y 0,i h i (x y 1,i (x = ( hi (x h i (x, (3.39., I i, y 1,i (x y 0,i (x 1. y 1,i (x BdG, T. (3.38 (y 1,i, y n = 0, (3.40 (y 1,i, z n = 0, (3.41 (y 1,i, y 0,i = 1, (3.4 (y 1,i, y 1,i = 0, (3.43 (3.44, (y 1,i, y 0,j = 0, (i j (3.45 (y 1,i, y 1,j = 0, (i j (3.46 ( A., y 1,i (x BdG.. (3.31. y 1,θ (x = ( ξ(x N ξ (x, I θ = µ N (3.47 BdG N [3].,. 13

14 , (3.38.,.,,..,, ( , y 1 (x σ 1 y 1,i(x = y 1,i (x, (3.48, y 1,i (x z 1,i (x {y n (x, z n (x, y 0,i (x, y 1,i (x} n (xy n=1{y n(x z n (xz n(x } + {y 0,i (xy 1,i (x + y 1,i (xy 0,i (x } = σ 3 δ(x x, i=1 (3.49. s(x s(x = {a n y n (x b n z n (x} + i y 0,i (x + d i y 1,i (x}, (3.50 n=1 i=1{c., y n (x, z n (x, y 0,i (x, y 1,i (x a n = (y n, s, (3.51 b n = (z n, s, (3.5 c i = (y 1,i, s, (3.53 d i = (y 0,i, s, (3.54., BdG BdG. 3.3 ( , BdG. ω µ y µ (x. y µ (x (y µ, T y µ = ω µ (y µ, y µ = (T y µ, y µ = ωµ(y µ, y µ, (3.55 (ω µ ωµ y µ = 0, (

15 ω µ Im ω µ 0..,. y µ = 0, ( (y ν, T y µ = ω µ (y ν, y µ = (T y ν, y µ = ω ν(y ν, y µ, (3.58 (ω µ ων(y ν, y µ = 0, (3.59, ω µ ων.,, (y n, y µ = 0, (3.60 (y 0, y µ = 0, (3.61. ω µ y µ (x ω ν = ωµ y ν(x., y µ (x. ω µ y µ (x ωµ, (3.1., ω µ y µ (x Det T ω µ = 0, (3.6., 0 = Det T ω µ = Det T ωµ = Det σ 3 T σ 3 ωµ = Det T ωµ, (3.63, T y µ(x = ωµy µ(x. ωµ y µ(x. y µ (x, y µ (x (y µ, y ν = δ µν, (3.64., z µ σ 1 y µ, (3.65 ωµ z µ = 0, (3.66 (z µ, z ν = δ µν, (3.67 (z µ, y ν = 0, (3.68 (z µ, y ν = 0, (

16 ,,. Reω µ 0 ω µ, ωµ, ωµ, ω µ y µ (x, z µ (x, y µ (x, z µ (x 4., n (xy n=1{y n(x z n (xz n(x } + {y 0,i (xy 1,i (x + y 1,i (xy 0,i (x } i=1 + {y µ (xy µ(x + y µ (xy µ(x z µ (xz µ(x z µ (xz µ(x } µ = σ 3 δ(x x, (3.70. s(x s(x = n=1{a n y n (x + a nz n (x} + j = 1{ iq j y 0,j (x + P j y 1,j (x} ( µ {A µ y µ (x + B µ y µ (x + A µz µ (x + B µz µ (x}, (3.7., a n = (y n, s, a n = (z n, s, (3.73 Q j = i(y 1,j, s, P j = (y 0,j, s, (3.74 A µ = (y µ, s, B µ = (y µ, s, (3.75 A µ = (z µ, s, B µ = (z µ, s, ( BdG (.34.,.,., ˆΦ(x ˆΦ(x ( ˆφ(x ˆφ (x, (3.77. Ĥ 0 = 1 d 3 x ( ˆφ (x ˆφ(x ( L M M L = 1 d 3 xˆφ (xσ 3 T ˆΦ(x ( ˆφ(x ˆφ (x = 1 (ˆΦ, T ˆΦ, (3.78, ˆΦ(x BdG ˆΦ(x = n=1{â n y n (x + â nz n (x} i ˆQy 0 (x + ˆP y 1 (x. (

17 , â n = (y n, ˆΦ, (3.80 â n = (z n, ˆΦ, (3.81 ˆQ = i(y 1, ˆΦ, (3.8 ˆP = (y 0, ˆΦ, (3.83., ˆQ, ˆP, ˆQ, ˆP, ân, â n (3.79. Ĥ 0 = 1 (ˆΦ, T ˆΦ [ ˆQ, ˆP ] = i, (3.84 [â n, â m] = δ nm, (3.85 others = 0. (3.86 = ˆP I + ω n â nâ n + (c number, (3.87 n=1 (3.87 1,., BdG.,, Fock.,, ω 0 = 0,. 3.5 Bogoliubov-de Gennes,., ω n > 0,,....,,., BdG [35].. Landau. Landau. 3,.,,.., 3. 17

18 .,, Landau..,,. 4, 1 GP,., BdG.,., x Gross-Pitaevskii 1 GP { 1 d } m dx µ + gξ(x ξ(x = 0, (4.1, ξ(x 0 = 0, ξ( = n c. ξ(x = n c tanh{α(x x 0 }, (4. µ = gn c, (4.3., α = mµ = mgn c., x 0.,.,,.,, x 0 = Bogoliubov-de Gennes BdG ( ( ( L M u(x u(x M = ω L v(x v(x,, (4.4 L = 1 d m dx gn c + gn c tanh (αx, M = gn c tanh (αx, (4.5., u(x, v(x tanh N A u(x = A n tanh n (αxe ipx, (4.6 n=0 N B v(x = B n tanh n (αxe ipx. (4.7 n=0 18

19 , N A, N B. BdG tanh { } NA (N A µ A NA tanh NA+ (αx + µb NB tanh NB+ (αx = 0, (4.8 { } NB (N B µ B NB tanh NB+ (αx + µa NA tanh NA+ (αx = 0, (4.9. A NA, B NB 0, (4.8, (4.9 N A = N B (= N., tanh { } N(N + 1 A N + B N = 0, (4.10 { } N(N + 1 B N + A N = 0, (4.11. N 1., N A = N B = 1 A 1 = B 1, N A = N B = A = B., M = M., BdG ( 0 L M L + M 0., ( u(x + v(x u(x v(x ( u(x + v(x = ω u(x v(x, (4.1 u(x + v(x = ( A + ib tanh(αx + C tanh (αx e ipx, (4.13 u(x v(x = (A + ib tanh(αx e ipx, (4.14,., tanh 1 i. (4.13,(4.14 BdG ( ϵ p µ αp ϵ p µ αp ϵ p µ αp µ ϵ p αp 0 0 A B C A B = ω A B C A B, (4.15., ϵ p = p m., (4.15, BdG.,. 3µA αpb + (ϵ p + 3µC = 0, (4.16 µb + αpc = 0, (4.17 ω = 0, ±ϵ p, ± ϵ p (ϵ p + µ, (

20 4..1 ω = 0 ω = 0 (4.15 (4.17 ( u(x v(x = C ( 1 tanh (αx 1 tanh (αx + i B ( tanh(αx tanh(αx = c 1 ( dξ dx dξ dx + c ( ξ ξ, ( (4.19.,. 4.. ω 0, ( ω (ϵ p µωa + αpeb = ω A, (4.0 ϵ p ωb = ω B, (4.1 µωa αpωb = ω C, (4. ( {(ϵ p µ ω }A + αp(ϵ p µb µ(5ϵ p µc = 0, (4.3 (ϵ p ω B αpϵ p C = 0, (4.4 µ(ϵ p µa + αp(µ ϵ p B + (4µϵ p µ ω C = 0, (4.5. ω = ϵ p, (4.3, (4.4, (4.5 ϵ p = 0 (p = 0 or C = 0, (4.6. p = 0 ω = 0. C = 0 A B.,. ω = ± ϵ p (ϵ p + µ p = 0 ω = 0. p 0 A = ϵ p + µ µ C, B = p α C, A = ω µ C, B = ω C, (4.7 αp 0

21 , { u(x + v(x = C ϵ p + µ µ u(x v(x = C i p } α tanh(αx + tanh (αx } { ω µ iω αp tanh(αx, (4.8, ( Bogoliubov-de Gennes, 1. BdG { u(x + v(x = C ϵ p + µ µ u(x v(x = C, p 0 ω = ± ϵ p (ϵ p + µ, (4.30 i p } α tanh(αx + tanh (αx } { ω µ iω αp tanh(αx ( u(x v(x, (4.31, (4.3 = c 1 y 0,θ + c y 0,x, (4.33., ,., T y 1,θ (x = I θ y 0,θ (x, (4.34 T y 1,x (x = I x y 0,x (x. (4.35, y 1,θ, y 1,x,., ( hi (x y 1,i (x = h i (x, (4.36., i = θ, x. 1

22 4.3.1.,, 1. (4.34, h θ (x = 1 ( tanh αx + αx(1 tanh αx, (4.37 I θ = α, (4.38., , 1 (4.35 h x (x = i, (4.39 I x = 1, (4.40.,. 5,.,. 5.1 GP GP. ε δv ε = εδv (x. GP { 1 } m + εδv (x µ ε + g ξ ε (x ξ ε (x = 0, (5.1 ε ξ ε (x = ξ (0 (x + εξ (1 (x +, (5. µ ε = µ (0 + εµ (1 +. (5.3 ε 1 { 1 } m µ (0 + g ξ (0 (x ξ (1 (x + gξ (0 (xξ (1 (x = (µ (1 δv (xξ (0 (x, (5.4.

23 (5.4., N = = dx ξ ε { } dx ξ (0 + ε(ξ (0 ξ (1 + ξ (0 ξ (1 + O(ε dxξ (0 ξ (1 + O(ε, (5.5 = N + εre, Re dxξ (0 ξ (1 = 0, ( Bogoliubov-de Gennes BdG GP [9].,., GP..,,., BdG.,.., 0 BdG BdG.,.., BdG Bogoliubov-de Gennes BdG [9]. δv ε = εδv (x BdG T ε y ε n(x = ω ε ny ε n(x, (5.7 T ε = T (0 + εt (1 +, y ε n(x = y (0 n (x + εy (1 n (x +. (5.8 BdG 1,., T (0, T (1 (T (0 ω (0 n y (1 n (x + (T (1 ω (1 n y (0 n (x = 0, (5.9 L ε = 1 m µ ε + g ξ ε (x, M = gξ ε (x, (5.10 3

24 L (0 = 1 m µ (0 + g ξ (0 (x, (5.11 L (1 = µ (1 + V (x + g(ξ (0 (xξ (1 (x + ξ (0 (xξ (1 (x, (5.1 M (0 = gξ (0 (x, (5.13 M (1 = gξ (0 (xξ (1 (x, (5.14 ( T (0 L (0 M = (0 M (0 L (0 (, T (1 L (1 M = (1 M (1 L (1, ( Bogoliubov-de Gennes BdG T ε y ε 0(x = δω0y ε ε 0(x, (5.16. ε 0 δω0 ε 0., BdG T ε ε., y ε 0(x = i α i y 0,i (x + δy ε 0(x, (5.17, δy ε 0(x 0 BdG δy ε 0(x = i { C ε i y 0,i (x + D ε i y 1,i (x } + n=1 {A ε ny n (x + B ε nz n (x}. (5.18, y ε 0(x y ε 0(x = i { (αi + C ε i y 0,i (x + D ε i y 1,i (x } + n=1 {A ε ny n (x + B ε nz n (x}, (5.19. ε BdG, {y 0,i, y 1,i, y n, z n },., δt ε = T ε T (0. (5., (5.3, (y 0,i, δt ε y ε 0 = δω ε 0D ε i, (5.0 I i D ε i + (y 1,i, δt ε y ε 0 = δω ε 0(α i + C ε i, (5.1 ω n A ε n + (y n, δt ε y ε 0 = δω ε 0A ε n, (5. ω n B ε n + (z n, δt ε y ε 0 = δω ε 0B ε n, (5.3 A ε n = (y n, δt ε y ε 0 δω ε 0 ω n, Bn ε = (z n, δt ε y ε 0 δω0 ε + ω, (5.4 n 4

25 α i + C ε i, Dε i (ij., Y n,m = (y n,i, δt ε y m,j { } (α j + Cj ε Y (ji 0,0 + Dε jy (ji 0, 1 + O(ε, (5.5 (y 0,i, δt ε y ε 0 = j (y 1,i, δt ε y ε 0 = j { } (α j + Cj ε Y (ji 1,0 + Dε jy (ji 1, 1 + O(ε, (5.6., α i + Ci ε, Dε i { } (α j + Cj ε Y (ji 0,0 + Dε jy (ji 0, 1 = δω0d ε i ε, (5.7. j j { (α j + C ε j Y (ji 1,0 + Dε j(i j δ ji + Y (ji 1, 1 } = δω ε 0(α i + C ε i, ( Y BdG Y (ij n,m. BdG (3.11, (3.1. Y (ij n,m σ 1 δt ε σ 1 = δt ε, (5.9 σ 3 δt ε σ 3 = δt ε, (5.30., Y (ij 0,0 = (y 0,i, δt ε y 0,j = (y 0,i, σ 1 δt ε σ 1 y 0,j = (σ 1 y 0,i, δt ε σ 1 y 0,j = (y 0,i, δt ε y 0,j = Y (ij 0,0 = Y (ji 0,0, (5.31 Y (ij (ij (ji 1,0 = Y 1,0 = Y 0, 1, (5.3 Y (ij 1, 1 = Y (ij 1, 1 = Y (ji 1, 1, (5.33., Y (ij 0,0, Y (ij (ij 1, 1, Y 1,0., GP, 1 Y (θi 0,0. 1 GP L (0 ξ (1 (x + M (0 ξ (1 (x = (µ (1 V (xξ (0 (x, (5.34 BdG ( T (1 = ( µ (0 ξ + δv σ 3 + g (0 ξ (1 + ξ (0 ξ (1 ξ (0 ξ (1 ξ (0 ξ (1 (ξ (0 ξ (1 + ξ (0 ξ (1, (5.35 5

26 . Y (θi 0,0 0,0 = ε dx ( ξ (0 ξ ( (0 σ 3 T (1 fi fi [ ] = ε dx ( µ (0 + δv (ξ (0 f i + ξ (0 fi + g(ξ (0 ξ (1 f i + ξ (0 ξ (1 fi = ε dx [f i ( µ (0 + δv + gξ (0 ξ (1 ξ (0 + fi ( µ (0 + δv + gξ (0 ξ (1 ξ (0] [ = ε dx f i ( L (0 ξ (1 M (0 ξ (1 (x + M (0 ξ (1 ] + fi ( L (0 ξ (1 M (0 ξ (1 (x + M (0 ξ (1 [{ = ε dx f i L (0 fi M (0} ξ (1 + {f i L (0 fi M (0 } ξ (1] [ { } { }] = ε ( ε dx ξ (1 L (0 f i M (0 fi + ξ (1 L (0 fi M (0 fi Y (θi = ε (, (5.36., Y (θi 0,0. 0, Y (θi 0,0 0. Y (θi 0, , 1. (5.7, (5.8 Y (θθ 1,0 δωε 0 I θ + Y (θθ 1, 1 Y (θi 1,0 Y (θi 1, 1 Y (θθ 0,0 Y (θθ 0, 1 δωε 0 Y (θi 0,0 Y (θi 0, 1 Y (iθ 1,0 Y (iθ 1, 1 Y (ii 1,0 δωε 0 I i + Y (ii 1, 1 Y (iθ 0,0 Y (iθ 0, 1 Y (ii 0,0 Y (ii 0, 1 δωε 0. δω ε 0 I θ Y (θθ δω0 ε 0,0 + I i Y (ii 0,0 = ± ± (I θ Y (θθ 0,0 + I i Y (ii 0,0 4I θ I i (Y (θθ., Y. 0,0 Y (ii α θ + C θ D θ α i + C i D i 0,0 Y (θi = 0, (5.37 0,0 Y (iθ 0,0 ( + O ε 1 4, (5.38,.,., BdG [9].,..,. 6

27 , y ε 0(x = 1 z ε 0(x = 1 δω ε 0 y Y (xx 0,x (x + 0,0 δω ε 0 y Y (xx 0,x (x + 0,0 Y (xx 0,0 δω0 ε Y (xx 0,0 δω ε 0 y 1,x (x, (5.39 y 1,x (x, (5.40 (, O ε 1 4., , ,.. GP, 0 (5.37, ( 1 d m dx + εδ(x µ ε + g ξ ε (x ξ ε (x = 0, (6.1 ξ ε (x = n c tanh (αx, (6. µ ε = gn c = α m. (6.3 δω0 ε = 0, ± I x Y (xx 0,0, (6.4., I x, Y (xx 0,0, ± I x Y (xx 0,0.,., y ε 0(x = 1 z ε 0(x = 1 δω ε 0 y Y (xx 0,x (x + 0,0 δω ε 0 y Y (xx 0,x (x + 0,0 Y (xx 0,0 δω0 ε Y (xx 0,0 δω ε 0 y 1,x (x, (6.5 y 1,x (x, (6.6. δω0 (ε ε O 1, Y (xx 0,0 O ( ε 1., 0,. 7

28 6. k, δv (x = gn c sin(αkx tanh(αx. Fig. 4.1., 1 GP (5.4. Y (θθ 0,0, Y (θx 0,0, Y (xx 0,0 ( 3 tanh (αx + 1 k, (6.7 ξ (1 (x = n c sin(αkx, (6.8 µ (1 = 0, (6.9 Y (θθ 0,0 = 0, Y (xx 0,0 = ε gn c Y (θx 0,0 = 0, dx(1 tanh (αx sin(αkx tanh(αx (6.10 } {3 tanh (αx + 1 k, (6.11 (6.1.,.., (6.11, Y (xx 0,0 k = k c (= 1.73., I x, k > k c,. Fig Fig

29 ., k,,.,,.,.,.,. ω µ, ωµ, ωµ, ω µ {y µ, y µ, z µ, z µ } 4., {y 0,x, y 1,x }., ω µ = ωµ, ωµ = ω µ., 1.,,.,. 7,.., BEC BEC., TDGP, GP, BdG. 3 BdG, 1 BdG. BdG.. BdG., BdG.,. 4, 1 GP,., BdG,.,,. 5 BdG. BdG., ε 0. 0 BdG,..,,. 6 BEC,,., 9

30 ., 1 1 BEC,..,,,.,,,.,.,.,., snake instability [10, 11]. 30

31 A. T ỹ 0,i = 0, (A.1 (, 1. Gram-Schmidt (., (ỹ 0,i ỹ 1,j, ỹ 1,i ỹ 0,j, ỹ 1,i ỹ 1,j., Gram-Schmidt,. 1 ( i y 0,i = ỹ 0,i, y 1,i = ỹ 1,i, (A.. j y 0,j = ỹ 0,j (y 1,i, ỹ 0,j y 0,i, y 1,j = ỹ 1,j (y 0,i, ỹ 1,j y 1,i (y 1,i, ỹ 1,j y 0,i, (A.3 (A

32 B 1 BdG 1, BdG.., A. B.1 1 ( GP (4.1 ξ(x = ξ ( x., ξ(χ = [ n c 1 κ tanh ] 1 κ χ iκ e iκχ, µ = gn c (1 + κ, (B.1 κ = k gnc, χ = gn c x, (B.. BdG., ( ξ(x ỹ 0,θ (x = ξ (x ỹ 1,θ (x = ( fx (x, ỹ 0,x (x = f x(x ( hθ (x h θ (x, ỹ 1,x (x = ( hx (x h x(x (B.3, (B.4, ( f x (x = i κ + iκ 1 κ + (1 κ (1 tanh 1 κ χ e iκχ, ( h θ (x = 1 tanh 1 κ χ + χ(1 tanh 1 κ χ e iκχ, 1 κ { h x (x = i ( 1 + i 3 κ tanh 1 κ χ + χ(1 tanh } 1 κ χ e iκχ, 1 κ (B.5 (B.6 (B.7., (ỹ 0,θ, ỹ 1,x = κ, (ỹ 1,θ, ỹ 0,x = κ, (ỹ 1,θ, ỹ 1,x = 0, (B.8., A., y 0,θ = ỹ 0,θ, y 1,θ = ỹ 1,θ, (B.9. 3

33 y 0,x = ỹ 0,x (y 1,θ, ỹ 0,x y 0,θ, y 1,x = ỹ 1,x (y 0,θ, ỹ 1,x y 1,θ (y 1,θ, ỹ 1,x y 0,θ, (B.10 (B.11. y 0,x (x = ( fx (x f x(x ( hx (x, y 1,x (x = h x(x, (B.1 f x (x = i(1 κ (1 tanh 1 κ χe iκχ, ( { tanh 1 κ h x (x = i 1 + iκ χ + χ(1 tanh } 1 κ χ e iκχ, 1 κ (B.13 (B.14 T y 1θ (x = gn c y 0,θ (x, T y 1,x (x = (gn c κ y 0,x(x, (B.15 (B.16. (B.13 (B.5.. f x (x ( x ik ξ(x, (B.17 33

34 ,,.,.... D1.,.. M1,.., B4,.. 34

35 [1] A. Einstein, Sitzber. Kgl. Preuss. Akad. Wiss. 61 (194; Sitzber. Kgl. Preuss. Akad. Wiss. 3 (195. [] F. London, Nature (London 141, 643 (1938. [3] J. L. Lin and J. P. Wolfe, Phys. Rev. Lett. 71, 1 (1993. [4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 69, 198 (1995. [5] K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995. [6] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995. [7] F. Dalfovo, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999. [8] H. Feshbach, Ann. Phys. (N.Y. 5, 357 (1958. [9] H. Feshbach, Ann. Phys. (N.Y. 19, 87 (196. [10] S. Burger et al., Phys. Rev. Lett. 83, 5198 (1999. [11] A. E. Muryshev, H. B. van Linden, van den Heuvel, and G. V. Shlyapnikov, Phys. Rev. A 60, R665 (1999. [1] L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 96, 190 (00. [13] K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature 417, 150 (00. [14] M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 83, 498 (1999. [15] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (000. [16] E. Hodby, G. Hechenblailkner, S. A. Hopkins, O. M. Margao, and C. J. Foot, Phys. Rev. Lett. 88, (00. [17] P. C. Haljan, I. Coddington, P. Engels, and E. A. Coenell, Phys. Rev. Lett. 87, (001. [18] M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, Nature (London 415, 39 (00. [19] M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Phys. Rev. Lett. 76, 4511 (1996. [0] M. R. Andrews, C. G. Townsend, H. -J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 75, 637 (1997. [1] A. Röhrl, M. Naraschewski, A. Schenzle, and H. Wallis, Phys. Rev. Lett. 78, 4143 (1997. [] D. S. Hall, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett. 81, 1543 (1998. [3] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 1, 345 (1961. [4] J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 17, 965 (196. [5] F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Rev. Mod. Phys. 71, 463 (

36 [6] E. P. Gross, Nuovo Cimento 0, 454 (1961; J. Math. Phys. 4, 195 (1963. L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. [Soc. Phys. JETP] 40, 546 (1961; Sov. Phys. JETP 13, 451 (1961. [7] N. N. Bogoliubov, J. Phys. (Moscow 11, 3 (1947. [8] P. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, [9] Y. Nakamura, M. Mine, M. Okumura, and Y. Yamanaka, Phys. Rev. A 77, (008. [30] M. Okumura and Y. Yamanaka, Phys. Rev. A 68, (003. [31] M. Mine, T. Koide, M. Okumura, and Y. Yamanaka, Prog. Theor. Phys. 115, 683 (006. [3] K. Kobayashi, M. Mine, M. Okumura, and Y. Yamanaka, Ann. Phys. 33, 147 (008. [33] M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996. [34] H. Matsumoto and S. Sakamoto, Prog. Theor. Phys. 107, 679 (00. [35] L. Pitaevskii, and S. Stringari, Bose-Einstein Condensation, (Oxford University Press, New York,

25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3

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