le LPTMS en Bretagne... photo extraite du site

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1 le LPTMS en Bretagne... 1 photo extraite du site

2 le LPTMS en Bretagne... 1

3 2 Quantum signature of analog Hawking radiation in momentum space Nicolas Pavloff LPTMS, CNRS, Univ. Paris Sud, Université Paris-Saclay D. Boiron S. Fabbri P.-É. Larré C. Westbrook P. Zin

4 quasi-1d Bose-Einstein condensates 3 quasi-1d condensate longitudinal size 1 2 µm transverse size 1µm radial confinement pulsation ω x y x (a) (b) z BEC magnetic trap rf knife 1.6 mm optical guide g harmonic radial confinement : V ( r ) = 1 2 m ω2 r 2. 1D model : ψ(x, t) Guerin et al., Phys. Rev. Lett. (26)

5 4 Analogous Hawking radiation Unruh, Phys. Rev. Lett. (1981) amont : subsonique aval : supersonique sens de l écoulement horizon

6 4 Analogous Hawking radiation Unruh, Phys. Rev. Lett. (1981) U(x) = Λδ(x) 1 n(x) n u amont : subsonique aval : supersonique sens de l écoulement V u < c u Delta peak V d > c d x ξ u horizon U(x) U Waterfall V u < c u = Θ(x) 1 1 n(x) n u V d > c d x ξ u

7 4 Analogous Hawking radiation Unruh, Phys. Rev. Lett. (1981) amont : subsonique horizon aval : supersonique sens de l écoulement gravitational black hole horizon Hawking radiation 75

8 4 Analogous Hawking radiation Unruh, Phys. Rev. Lett. (1981) amont : subsonique aval : supersonique sens de l écoulement V u < c u V d > c d horizon amont : subsonique vitesse amont Hawking (V u c u) horizon aval : supersonique vitesse aval partenaire (V d ± c d )

9 Quantum correlations Balbinot, Carusotto, Fabbri, Fagnocchi, Recati, Phys. Rev. A & New J. Phys. (28) upstream region downstream region ω [a.u.] u out u in d1 out d1 in Ω Ω d2 out d2 in example of induced correlation: u out d1 out in q [a.u.] out out q [a.u.] in x = (v d + c d )t x = (v u c u)t correlates with Hawking radiation in the u out channel. Equivalent to a black body radiation of temperature T H 1% µ affects the density correlation pattern : n(x)n(x ) : 2 x, / ξ u 1 d1-d2 u-d1-1 u-d2 Larré et al., Phys. Rev. A (212) x / ξ u

10 One body momentum distribution in the presence of a horizon 6 T =, adiabatic opening of the trap Boiron et al. PRL (215) ˆn(p) [arb. units] Hawking u out channel S ud2 2 Partner d2 out channel S d2u 2 + S d2d1 2 d1 out channel S d1d2 2 P u P d p 1 P u P d p ξ u

11 Two body momentum distribution in the presence of a horizon 7 p, q : absolute momenta in units of ξ 1 u T = adiabatic opening Boiron et al. PRL (215) right plot: g 2(p, q) where g 2(p, q) = : ˆn(p)ˆn(q): ˆn(p) ˆn(q) 2 u out - d1 out d1 out - d2 out d1 out - d1 out ω [a.u.] upstream region u out u in Ω downstream region Ω d1 out d2 in d1 in d2 out q 1-1 u out - d2 out u out - u out d2 out - d2 out k [a.u.] k [a.u.] P u P d p k : momentum relative to the condensate p = k + P (u/d) where P (u/d) = mv (u/d) without horizon: g 2 1

12 Violation of Cauchy-Schwarz inequality (T ) 8 C.-S. violation : g 2 (p, q) u out d2 out > g 2 (p, p) g 2 (q, q) 2 u out d2 out Boiron et al. PRL (215) g 2 (p,q) uout - d2 out T=1.2 T=.8 T= T=1.6 T= p P u T in units of µ T H =.13 V u/c u =.5 V d /c d = 4 V d /V u = 4 n u/n d = 4

13 Cauchy-Schwarz : a wrong theorem?... cf. sub-poissonian fluctuations... def = Tr(ρ...) ˆn 2 = â ââ â = â â â â + â 1 â = â â â â + ˆn δn 2 def = ˆn 2 ˆn 2 = â â â â ˆn 2 + ˆn }{{} sign?

14 Cauchy-Schwarz : a wrong theorem?... cf. sub-poissonian fluctuations... def = Tr(ρ...) ˆn 2 = â ââ â = â â â â + â 1 â = â â â â + ˆn δn 2 def = ˆn 2 ˆn 2 = â â â â ˆn 2 + ˆn }{{} sign? Cauchy-Schwarz: Â 2 Â Â Hence ââ 2 â â â â But â â 2 â â â â

15 Cauchy-Schwarz : a wrong theorem?... cf. sub-poissonian fluctuations... def = Tr(ρ...) ˆn 2 = â ââ â = â â â â + â 1 â = â â â â + ˆn stupid theoretical example average over a number state: ρ n n ââ 2 = â â â â = n(n 1) â â 2 = n 2 â â â â a number state is clearly sub-poissonian! δn 2 def = ˆn 2 ˆn 2 = â â â â ˆn 2 + ˆn }{{} sign? Cauchy-Schwarz: Â 2 Â Â Hence ââ 2 â â â â But â â 2 â â â â

16 Cauchy-Schwarz : a wrong theorem?... cf. sub-poissonian fluctuations... def = Tr(ρ...) ˆn 2 = â ââ â = â â â â + â 1 â = â â â â + ˆn stupid theoretical example average over a number state: ρ n n ââ 2 = â â â â = n(n 1) â â 2 = n 2 â â â â a number state is clearly sub-poissonian! δn 2 def = ˆn 2 ˆn 2 = â â â â ˆn 2 + ˆn }{{} sign? experimental results Cauchy-Schwarz: Â 2 Â Â Hence ââ 2 â â â â But â â 2 â â â â Poissonian limit : δn 2 =.34 N Ideal Bose gas Yang-Yang Quasi-cond. Jacqmin et al., PRL (211)

17 Quantum effects within 1D mean field? Popov, Teor. Mat. Fiz. (1971) The NLS Gross-Pitaevskii eq. is a nonlinear quantum field eq. : 2 2m 2 x ˆψ + g ˆψ ˆψ ˆψ = i t ˆψ, with [ ˆψ(x, t), ˆψ (y, t) ] = δ(x y). BEC : macroscopic occupation of the lowest quantum state: ˆψ(x, t) = ψ () (x, t) + ˆφ(x, t) (Bogoliubov 1947) ψ () ˆφ : solution of the (classical) NLS : solution of a linearized (quantum) eq. makes it possible to consider vacuum fluctuations. In particular : Hawking radiation in a stationary, non uniform setting. q!"$ " #"$! "$$ & #"!" %!" #" " #"!" q Mathey, Vishwanath, Altman, PRA (29) Bouchoule, Arzamasovs, Kheruntsyan, Gangardt, PRA (212)

18 truly 1D? No: transverse excitations when ω µ : ω nk [ω ρ ] (a) η= k [a ρ ] modified dispersion relation : ω(q) 2 = c1d 2 q ( (qr 48 ) ) new channels : ω 2 n 1(q) = 2n(n + 1) ω (qr ω ) these new channels will be populated at T = Zaremba, PRA (1998) Stringari, PRA (1998) Fedichev & Shlyapnikov, PRA (21) Tozzo & Dalfovo, PRA (22) mass term Klein-Gordon new in modes

19 Conclusion BECs offer interesting prospects to observe Hawking radiation [Steinhauer, Nat. Phys. (214)] general perspective : quantum effects with nonlinear matter waves One- and two-body momentum distributions accessible by present day experimental techniques provide clear direct evidences of the occurrence of a sonic horizon. of the associated acoustic Hawking radiation. of the quantum nature of the Hawking process. The signature of the quantum behavior persists even at temperatures larger than the chemical potential.