Bose-Einstein Hawking Hawking Hawking Hawking nk Hawking Bose-Einstein Hawking 1 Bekenstein[1] Hawking 1974 [2,

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1 Bose-Einstein Hawking Hawking Hawking Hawking nk Hawking Bose-Einstein Hawking 1 Bekenstein[1] Hawking 1974 [2, 3] Hawking Hawking 6nK Hawking Hawking 3K Hawking Hawking Unuh 1981 [4, 5] [6] Hawking Bose-Einstein 3 Hawking 4 Bose-Einstein 5 2 Bose-Einstein Bose-Einstein ε, ε 1, ε 2,... N Pauli Bose-Einstein : n ε = exp 1 ε µ k B T, 1 1 1

2 µ k B Boltzmann T ON Bose-Einstein µ = ε 1 Bose-Einstein Bose Einstein Bose-Einstein [7] Bose-Einstein Bose-Einstein Bose-Einstein 21 Nobel : V = 1 2 mω2 ho 2, m 3 Hawking Hawking Einstein R µν 1 2 g µνr = 8πG c 4 T µν. 1 µ = ε N ε µ = O1/N 24 He Bose-Einstein 2

3 R µν Ricci g µν R G T µν Einstein Schwazschild Schwazschild ds 2 = 1 g c 2 dt 2 d2 1 g 2 dθ 2 2 sin 2 θdϕ 2 g g = 2GM/c 2 g = 3 km g = 1 cm g Planck v = c g / t cd t = cdt v/c 1 v 2 /c 2 d ds 2 = c 2 d t 2 d vd t 2 2 dθ 2 2 sin 2 θdϕ 2 2 ds = ct = ± + g ln 1 g Schwazschild g µν 1 gg µν ϕ g x µ x ν = 3 g = detg µν s Minkowskii δ = 2 g + 2 g log min g min k B T H = c 3 /8πGM Planck Planck 3

4 4 Bose-Einstein Bose-Einstein Bose-Einstein Goss-Pitaevskii Bogoliubov-de Genne Bose-Einstein 4.1 Bose-Einstein 87 Rb 2 Bose-Einstein N N = ON Φ a a a Φ = N Φ, a Φ = N + 1 Φ. N = ON N + 1 N Bose [8] φ = φ, t S = dt d 3 [ i φ t φ φ 2 2m mω2 ho 2 φ 12 ] Uφφφφ U s φ = Φ Goss- Pitaevskii i t Φ = 2 2m mω2 ho 2 Φ + UΦ ΦΦ. 4 Goss-Pitaevskii [9] l = / mω ho u = NU/ ω ho l 3 l Oµm ω ho = OnK u = 5 u = 5 1 4

5 .6.5 Φ hamonic osci. stong coupling.6.5 Φ hamonic osci. stong coupling.4.4 Φ.3 Φ a b 1: a u = 5 bu = 5 ϕ Φ ϕ i t ϕ = 2 2m mω UΦ Φ ϕ + UΦ 2 ϕ. 5 Φ ϕ [ ϕ, t, ϕ, t ] = δ 5 Bogoliubov ϕ, t = [ ] A α, t b α + Bα, t b α 6 α A α, t B α, t A α, t K UΦ 2 i t = B α, t UΦ 2 K A α, t B α, t, 7 K = 2 2m mω UΦ Φ Bogoliubov-de Gennes 7 7 A α, t = Φ, t, B α, t = Φ, t Goss-Pitaevskii ϕ E i t A α, t B α, t = K E UΦ 2 UΦ 2 K E A α, t B α, t 5

6 Goss-Pitaevskii Φ Φ Φ = ρ 1/2 exp iθ Goss-Pitaevskii 4 1/ ul Φ 2 t θ 1 2 θ gρ t ρ + ρ θ = θ t + v t + v θ c 2 s θ v = θ c s = UΦ Φ/m g µν 3 c 2 s v 2 v x v y v z v x 1 g µν = c s v y 1 v z 1 2 v c s θ θ = 1 Φ 2iρ ϕ Φ ϕ ρ = Φ ϕ + Φ ϕ Bogoliubov b α,b α θ = f α b α + fαb α α µν ρ = α g α b α + gαb α p, q = 4πi u f α, f β = δ αβ d 2 [p t + v q [ t + v p ] q ] Klein-Godon [1] 6

7 5 Bose-Einstein Hawking Bose-Einstein Bose-Einstein Hawking v > c s Laval [11] [12] [13] Feshbach [14] u = 5 Goss-Pitaevskii ω ho ω ho /2 2a t =.4 c s v ω 1 ho > h h 3.67 v > c s > h < h c s v H T H a t b 2: at =.4 c s v b H k B T H = 2π v c =H Hawking 2b backeaction backeaction?? WKB Hawking ω ho OnK Hawking Hawking Goss- Pitaevskii Bogoliubov-de Gennes t Φ, t Φ, t Φ Bβα = f 2 β, f α 1 7

8 .1 8e-5 α=: T H =.61 α=1: T H = 1.32 α=2: T H = e-5 4e-5 2e ω 3: t = 1.44 Planck t = Planck Planck Hawking backeaction quantum depletion Goss-Pitaevskii Bose-Einstein quantum depletion, Bα, t B α, t α quantum depletion 4a 4b quantum depletion Josephson disentangle 8

9 .25 t =.1 t = 83.4 Quantum Depletion Quantum Depletion a b 4: Quantum depletion a t = bt = Planck Bose-Einstein Hawking Hawking Hawking OnK quantum depletion [1] J. D. Bekenstein: Phys. Rev. D [2] S. Hawking: Natue [3] S. W. Hawking: Commun. Math. Phys [4] W. G. Unuh: Phys. Rev. Lett

10 [5] : [6] M. Novello, M. Visse, and G. Volovik: Atificial black holes Wold Scientific, Singapoe, 22. [7] C. Pethick and H. Smith: Bose-Einstein Condensation in Dilute Gases Cambidge Univesity Pess, Cambidge, 21. [8] J. W. Negele and H. Oland, Quantum Many-Paticle Systems Westview Pess, Boulde, [9] F. Dalfovo, S. Giogini, L. P. Pitaevskii, and S. Stingai: Rev. Mod. Phys [1] N. Biell and P.C.W. Davies, Quantum Fields in Cuved Space Cambidge Univesity Pess, Cambidge, [11] M. Sakagami and A. Ohashi: Pog. Theo. Phys [12] L. J. Gaay, J. R. Anglin, J. I. Ciac, and P. Zolle: Phys. Rev. Lett [13] Y. Kuita, M. Kobayashi, T. Moinai, M. Tsubota, and H. Ishihaa: Phys. Rev. A [14] Y. Kuita and T. Moinai: Phys. Rev. A

1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1)

1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) 1 9 v..1 c (216/1/7) Minoru Suzuki 1 1 9.1 9.1.1 T µ 1 (7.18) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) E E µ = E f(e ) E µ (9.1) µ (9.2) µ 1 e β(e µ) 1 f(e )