Numerical observation of Hawking radiation from acoustic black holes in atomic Bose-Einstein condensates

Transcription

1 Numerical observation of Hawking radiation from acoustic black holes in atomic Bose-Einstein condensates Iacopo Carusotto BEC CNR-INFM and Università di Trento, Italy Institute of Quantum Electronics, ETH Zürich, Switzerland In collaboration with: Alessio Recati Serena Fagnocchi and Roberto Balbinot Alessandro Fabbri Nicolas Pavloff ( BEC CNR-INFM, Trento, Italy ( Università di Bologna, Italy ( IFIC - Univ. de Valencia and CSIC, Spain ( Université Paris-Sud, Orsay, France

2 The main experimental challenges horizon region Sub-sonic flow c1 > v0 Super-sonic flow v0>c2 v0 σx Numerical simulation of BEC dynamics using Wigner-MC method start from uniform condensate in motion at v0 switch on horizon at a given time t=0 and go to black-hole regime c1 > v0 > c2 + minimize deterministic disturbances, e.g. Landau processes (in super-sonic region and soliton shedding during and after switch-on concentrate on effect of quantum fluctuations + isolate (thermal Hawking emission from background phonons (also thermal

3 (i How to create a clean black hole? From: W. Guerin et al., PRL 97, (2006 Out-coupled atom laser beam: uniform density and velocity v0 Atom-atom interaction constant initially uniform and equal to g1 Within σt around t=0: modulation g1 g2 and V1 V2 in x>0 region only via: Feshbach resonance (g depends on applied B or modify transverse confinement Step in nonlinear coupling constant g step in sound speed. Black-hole formed if c1>v0>c2, thickness σx of crossover region determines surface gravity Chemical potential jump to be compensated by external potential V1+ ng1=v2+ ng2 allows to avoid Cerenkov-Landau phonon emission, soliton shedding

4 (ii How to detect Hawking radiation? Density-density correlation function: ': n0 x1 n 0 x ' 1:( G 0 x, x ' 1= ' n0 x1( ' n0 x ' 1( 021 Prediction of gravitational analogy: entanglement in Hawking pairs gives peculiar Hawking signal in G(2 long-range in/out density correlations G20 x, x ' 1 = 1 $ 16 4 c 1 c 2 [0 c1 c 2 x x' $2 k cosh, 2 c 1 $v v$c 2. n c1$v 10v$c 21 k 2 R. Balbinot, A. Fabbri, S. Fagnocchi, A. Recati, IC, PRA 78, ( ]

5 (2 Experimental measurements of G (x,x' Fully coherent BEC: G(2(x,x' = 1 Atomic HB-T: positive correlation due to thermal Bose atoms (negative for fermions Quantum correlations after collision of two BECs revealed Noise correlations in TOF picture after expansion from lattice Correlations in products of molecular dissociation from molecular BEC Bose Fermi Jeltes et al., Nature 445, 402 (2007 Single-shot image Spatial correlation of noise Rom et al., Nature 444, 733 (2006

6 Atom counting at the single atom level Thermal atom beam Coherent atom laser beam Poissonian statistics A. Öttl, et al. Phys. Rev. Lett. 95, (2005

7 The numerical method: Wigner-Monte Carlo At t=0, homogeneous system: Condensate wavefunction in plane-wave state Quantum + thermal fluctuations in plane wave Bogoliubov modes Gaussian αk, variance < αk 2 > = [2 tanh(ek / 2kBT]-1 ½ for T x,t=01=e i k0 x [. n0,# 0 uk e i k x 2 k,v k e$i k x 2 *k 1 k ] At later times: evolution under GPE!2 2 2 i! "t 50 x1=$ "x 50 x1,v 0 x 150 x1, g 0 x1 %50 x1% 50 x 1 2m Expectation values of observables: Average over noise provides symmetrically-ordered observables 1 / / x ' 1,50 / x'1 5 / 0 x 1( Q '5 * 0 x 1 50 x ' 1(W = ' 5 0 x Equivalent to Bogoliubov, but can explore longer-time dynamics A. Sinatra, C. Lobo, Y. Castin, J. Phys. B 35, 3599 (2002

8 The movie!!!

9 Density plot of : (n ξ1 * [ G(2(x,x' 1 ] Density plot of : (n ξ1 * [ G(2 (x,x' 1 ] (2 Density plot of : (n ξ1 * [ G (x,x' 1 ] IC, S.Fagnocchi, A.Recati, R.Balbinot, A.Fabbri, New J. Phys. 10, (2008

10 The numerical observations (ii Dynamical Casimir emission Density plot of : (n ξ1 * [ G(2(x,x' 1 ] (i Many-body antibunching (iii Hawking in / out (iv Hawking in / in IC, S.Fagnocchi, A.Recati, R.Balbinot, A.Fabbri, New J. Phys. 10, (2008

11 Feature (i : Many-body antibunching present at all times due to repulsive interactions almost unaffected by flow See e.g.: M. Naraschewski and R. J. Glauber, PRA 59, 4595 (1999

12 Feature (ii: Dynamical Casimir emission of phonons Fringes parallel to main diagonal intensity depends on speed of switch-on only in x>0 region, move away in time do not depend on flow pattern, also present in homogeneous system Physical explanation: in x>0 region g1 g2 within short time σt : non-adiabatic time modulation of Bogoliubov vacuum phonon pair emission at t=0, from all points x>0 fringes depend on x-x' : quantum correlations in counter-propagating pairs correlations propagate away at speed 2cs See also: M. Kramer et al. PRA 71, (R (2005; K. Staliunas el al., PRL 89, (2002.

13 Feature (iii : The Hawking signal Negative correlation tongue extending from the horizon x=x'=0 long-range in/out density correlation which disappears if both c1,2<v0 length grows linearly in t peak height, FWHM constant in t slope v 0 $c 2 v 0$c1 agrees with theory + pairs emitted at all t from horizon + propagate at sound speed length cuts of G(2(x,x' - 1 peak FWHM growing time t

14 Quantitative analysis Prediction of QFT in curved space-time Proportional to emission temperature Proportional to surface gravity Analog model prediction quantitatively correct in hydrodynamic limit ξ1 / σx «1

15 Features (iii,iv : More on the Hawking signal Two parametric Hawking processes: in/out: vacuum α + β (feature iii in/in: vacuum β + γ (feature iv Energy conserved only if sub/super-sonic Momentum provided by horizon Slope of tongues v 0 $c 2 &$1 v 0 $c 1, v 0$c 2 1 & v 0,c 2 5

16 Effect of an initial T>0 Hawking signal remains visible also for initial T comparable to TH Stimulated Hawking emission Extra tongues (v due to partial scattering of thermal phonons on horizon distinguishable from Hawking emission by different slope with Black Hole 0v 0 $c1 1 0v 0,c2 1 without Black Hole

17 How I physically understand Hawking radiation Bogoliubov dispersion on sub- and super-sonic sides Incident Reflected Transmitted Anomalous ransmitted Incident plane wave reflected, transmitted and anomalous transmitted Anomalous transmitted wavepacket only exists if black hole c1 > v0 > c2 Similar phenomenology to classical hydrodynamics experiments

18 (Classical wavepacket dynamics

19 Hawking radiation : parametric emission of phonon pairs Inc Inc2 Refl Input-output formalism of quantum optics An. inc. Tr An. tr a refl ainc atr =M ainc2 aan.tr aan.inc. aan.inc., aan.tr. creation operators for Bogoliubov ghost branch zero-point fluctuations in incident beam becomes real transmitted particles ' a an.tr. aan.tr. (=%M 3,3%2 ' aan.inc. a an.inc. (,- energy conserved thanks to super-sonic flow; momentum provided by horizon See also: Leonhardt & Philbin, cond-mat/arxiv: and Macher-Parentani's talks.

20 Why correlations? Quantum correlations in emitted pairs: 'a refl. aan.tr. (= M 1,3 M *3,3 'a an.inc. a an.inc. ( 'a tr. a an.tr. (=M 2,3 M 3,3 ' a an.inc. a an.inc. ( * two-mode squeezing, thermal statistics when looking at one component simultaneous emission at all times t at horizon position propagate from the horizon with group velocity x x' visible in density correlation function as signal peaked on lines = v g1 v g2 slopes determined by c1, v0, c2: + in-out: vg1=v0-c1, vg2=v0-c2 + in-in: vg2=v0+c2, vg2=v0-c2 v 0 $c 2 &$1 v 0 $c 1 v 0 $c 2 1 & v 0,c 2 5

21 Outlook Analog Hawking radiation of phonons from acoustic black-hole numerically observed via density correlation function microscopic simulations of condensate dynamics parametric emission of entangled phonon pairs from the horizon propagating phonons responsible for correlated density fluctuations Hawking signal easily distinguished from other processes (e.g. Landau-Cerenkov, background thermal phonons appreciable signal intensity for realistic parameters, worst enemy: atomic shot noise trans-planckian, high-k modes in horizon region under control R. Balbinot, A. Fabbri, S. Fagnocchi, A. Recati, IC, PRA 78, (2008 IC, S. Fagnocchi, A. Recati, R. Balbinot, A. Fabbri, New J. Phys. 10, (2008