Numerical observation of Hawking radiation from acoustic black holes in atomic Bose-Einstein condensates Iacopo Carusotto BEC CNR-INFM and Università di Trento, Italy In collaboration with: Alessio Recati (BEC CNR-INFM, Trento, Italy) Serena Fagnocchi and Roberto Balbinot (Università di Bologna, Italy) Alessandro Fabbri (Universidad de Valencia, Spain) Nicolas Pavloff (Université Paris-Sud, Orsay, France)
Outline What is an acoustic black hole? What link to general relativity and quantum field theory in curved space-time? The numerical calculation: Wigner-Monte Carlo Signatures of Hawking radiation of phonons Quantitative comparison with QFT prediction Experimental perspectives
The acoustic black hole horizon region v 1 v 2 Sub-sonic flow c 1 > v 1 Super-sonic flow v 2 > c 2 Steady but non-uniform flow in the atomic condensate Horizon separates region of sub-sonic and super-sonic flow No sonic perturbation can propagate back from super-sonic region
Bogoliubov modes are dragged by flow horizon region Sub-sonic flow c 1 > v 0 Super-sonic flow v 0 >c 2 v 0 v 0 Sub-sonic region Super-sonic region v g = v 0 - c 1 < 0 v g = v 0 + c 1 > 0 v g = v 0 - c 2 > 0 Sound can not propagate back! v g = v 0 + c 2 > 0 Only single particle excitations can emerge from black hole
The analogy with QFT in curved space-time Low-k, hydrodynamic region: linear phonon dispersion ω = (c s +v) k Mathematical analogy with light propagation in curved metric ds=g dx dx = n x mc s x [ c s x 2 dt 2 d x v x dt d x v x dt ] Wave equation for BEC phase 1 G [ G G ] x,t =0 Once quantized quantum field theory in a curved space time
Hawking radiation in black-hole geometries Astrophysical black-holes: emit Hawking radiation at solar mass BH: T H = 0.4 µk to be compared to cosmological background at 2.73 K ħ c 3 T H = 8 G M k B Hawking mechanism in acoustic black holes parametric emission of quantum correlated phonon pairs out phonon exits from BH, in phonon falls into it tracing out the in phonon, one obtains thermal state for out radiation To observe it: low temperature, steep velocity gradients ħ [ 4 k B c s d H ] T H = in nk range for µm-sized ultracold atomic BECs d x c 2 s v 2 (not so bad...) W. G. Unruh, PRL 46, 1351 (1981); Artificial Black Holes, eds. Novello, Visser, Volovik (2002)
How to create a clean acoustic black-hole? horizon region Sub-sonic flow c 1 > v 0 Super-sonic flow v 0 >c 2 v 0 σ x One-dimensional geometry: atom laser beam in atomic waveguide Initially: uniform density n, flowing at v 0, interaction constant g 1 Within σ t around t=0: modulation g 1 g 2 and V 1 V 2 in x>0 region only via: Feshbach resonance, modify transverse confinement Black-hole formation if c 1 >v 0 >c 2, horizon thickness σ x Other possibility: Laval nozzle, i.e. nonlinear shock wave at defect
Atom laser beam From: W. Guerin et al., PRL 97, 200402 (2006) Out-coupled beam: uniform density and velocity Horizon: step in nonlinear coupling constant g step in sound speed (depends on B field via Feshbach effect and on transverse confinement) Chemical potential jump to be compensated by external potential V 1 +ng 1 =V 2 +ng 2 to avoid Cerenkov-Landau phonon emission, soliton shedding, etc.
How to detect Hawking radiation? Density-density correlation function: G 2 x, x ' = : n x n x ' : n x n x ' Prediction of gravitational analogy: entanglement in Hawking pairs results into peculiar Hawking signal in G (2) long-range in/out density correlations G 2 x, x ' = 1 1 2 16 c 1 c 2 k 2 n 2 1 2 c 1 c 2 c 1 v v c 2 cosh 2 [ k 2 x c 1 v x ' v c 2 ] R. Balbinot, A. Fabbri, S. Fagnocchi, A. Recati, IC, preprint arxiv:0711.4520.
The measurement of G (2) (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 Single-shot image Fermi Spatial correlation of noise Jeltes et al., Nature 445, 402 (2007) Rom et al., Nature 444, 733 (2006)
Atom counting at the single atom level Thermal atom beam Coherent atom laser beam Poissonian statistics A. Öttl, et al. Phys. Rev. Lett. 95, 090404 (2005)
Numerical calculations: Wigner-Monte Carlo At t=0, homogeneous system: Condensate wavefunction in plane-wave state Quantum + thermal fluctuations in plane wave Bogoliubov modes Gaussian α k, variance < α k 2 > = [2 tanh(e k /2k B T)] -1 ½ for T 0. x,t=0 =e i k 0 x [ n 0 k At later times: evolution under GPE i ħ t x = ħ 2 2m 2 x x V x x g x x 2 x Expectation values of observables: u k e i k x k v k e i k x k ] Average over noise provides symmetrically-ordered observables x x ' W = 1 2 x x ' x ' x Q Equivalent to Bogoliubov, but can explore longer-time dynamics A. Sinatra, C. Lobo, Y. Castin, J. Phys. B 35, 3599 (2002)
A movie!!!
Numerical observations Dynamical Casimir emission Density plot of : (n ξ 1 ) * G (2) (x,x') - 1 Many-body antibunching Hawking in / out Hawking in / in
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)
Dynamical Casimir emission of phonons In x>0 region g 1 g 2 within σ t Non-adiabatic time modulation of Bogoliubov vacuum Weaker effect for slower switch-on Not depending on flow pattern, also in homogeneous system 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 c s See also: M. Kramer et al. PRA 71, 061602(R) (2005); K. Staliunas el al., PRL 89, 210406 (2002).
The Hawking signal (I) Negative correlation tongue extending from the horizon x=x'=0 long-range in/out density correlation which disappears if both c 1,2 <v 0 length grows linearly in t peak height, FWHM constant in t v 0 c 2 v 0 c 1 slope agrees with prediction pairs emitted at all t from horizon, propagate at sound speed length cuts of G (2) (x,x') - 1 peak FWHM growing time t
The Hawking signal (II) 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 v 0 c 2 1 v 0 c 1 Slope of tongues, v 0 c 2 v 0 c 2 1 5
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 ξ 1 / σ x «1
Effect of an initial T>0 Hawking signal remains visible also for initial T comparable to T H Stimulated Hawking emission Extra tongues due to partial scattering of thermal phonons on horizon distinguishable from Hawking emission by the slope with Black Hole without Black Hole
A simple physical interpretation Reflected Incident Transmitted Anomalous ransmitted Incident plane wave reflected, transmitted and anomalous transmitted Anomalous transmitted: sits on Bogoliubov negative branch a an.tr. =u a inc zero-point fluctuations in incident beam becomes real particles ω conserved: only possible in low-energy region in the presence of horizon also called: superradiance, stimulated parametric emission...
Wavepacket dynamics Signature of Hawking processes: appearance of anomalous transmitted wavepacket total phonon number is not conserved, but N inc = N refl + N tr - N an.tr
Outlook Signature of Hawking radiation of phonons from acoustic black-hole observed in density correlation function: 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 is atomic shot noise calculations based on microscopic theory: role of trans-planckian, high-k modes in horizon region under full control For more details: arxiv/0803.0507, arxiv/0711.4520
Perspectives * Perform Wigner-MC evolution for longer times: back-action of fluctuations on condensate black-hole evaporation by Hawking emission * Study other geometries, e.g. condensate expansion: models for universe expansion and inflation * More refined detection schemes to overcome shot-noise: light scattering on flowing condensate sensitive to density (phase contrast imaging) or current (slow-light imaging) look for signature in phase and/or intensity correlations of scattered light * Other systems: moving refractive index jumps in nonlinear optical waveguides (Philbin et al. Science, 2008)