Bose-Einstein Hawking E-mail: moinai@yukawa.kyoto-u.ac.jp 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
µ k B Boltzmann T ON Bose-Einstein µ = ε 1 Bose-Einstein Bose Einstein 1995 2 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
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 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 2 + 1 2 mω2 ho 2 φ 12 ] Uφφφφ U s φ = Φ Goss- Pitaevskii i t Φ = 2 2m 2 + 1 2 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
.6.5 Φ hamonic osci. stong coupling.6.5 Φ hamonic osci. stong coupling.4.4 Φ.3 Φ.3.2.2.1.1 1 2 3 4 5 1 2 3 4 5 a b 1: a u = 5 bu = 5 ϕ Φ ϕ i t ϕ = 2 2m 2 + 1 2 mω2 2 + 2UΦ 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 2 + 1 2 mω2 2 + 2UΦ 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
Goss-Pitaevskii Φ Φ Φ = ρ 1/2 exp iθ Goss-Pitaevskii 4 1/ ul Φ 2 t θ 1 2 θ 2 1 2 2 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
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 2.5 2 c s v 6 5.6.5 1.5 H 4 3.4.3 T H 1 2.2.5 1.1 1 2 3 4 5 6 7 a 2 4 6 8 1 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
.1 8e-5 α=: T H =.61 α=1: T H = 1.32 α=2: T H = 1.219 6e-5 4e-5 2e-5 1 2 3 4 5 ω 3: t = 1.44 Planck t = 1.44 3 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
.25 t =.1 t = 83.4 Quantum Depletion.2.15.1.5 Quantum Depletion.8.6.4.2 1 2 3 4 5 6 1 2 3 4 5 6 a b 4: Quantum depletion a t = bt = 83.4 6 Planck Bose-Einstein Hawking Hawking Hawking OnK quantum depletion [1] J. D. Bekenstein: Phys. Rev. D 7 1973 2333. [2] S. Hawking: Natue 248 1974 3. [3] S. W. Hawking: Commun. Math. Phys. 43 1975 199. [4] W. G. Unuh: Phys. Rev. Lett. 46 1981 1351. 9
[5] : 76 21 328. [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, 1998. [9] F. Dalfovo, S. Giogini, L. P. Pitaevskii, and S. Stingai: Rev. Mod. Phys. 71 1999 463. [1] N. Biell and P.C.W. Davies, Quantum Fields in Cuved Space Cambidge Univesity Pess, Cambidge, 1982. [11] M. Sakagami and A. Ohashi: Pog. Theo. Phys. 17 22 1267. [12] L. J. Gaay, J. R. Anglin, J. I. Ciac, and P. Zolle: Phys. Rev. Lett. 85 2 4643. [13] Y. Kuita, M. Kobayashi, T. Moinai, M. Tsubota, and H. Ishihaa: Phys. Rev. A 79 29 43616. [14] Y. Kuita and T. Moinai: Phys. Rev. A 76 27 5363. 1