Conference on Research Frontiers in Ultra-Cold Atoms. 4-8 May Bose gas in atom-chip experiment: from ideal gas to quasi-condensate

2030-25 Conference on Research Frontiers in Ultra-Cold Atoms 4-8 May 2009 Bose gas in atom-chip experiment: from ideal gas to quasi-condensate BOUCHOULE Isabelle Chargee de Recherche au CNRS Laboratoire Charles Fabry de l'institut d'optique Groupe d'optique Atomique, Campus Polytechnique RD 128, 91127 Palaiseau Cedex FRANCE

Density fluctuations in a very elongated Bose gas : from ideal gaz to quasi-condensate Isabelle Bouchoule Institut d Optique, Palaiseau. Trieste, May 2009

Physics in 1D gases 1D physics very different from 3D : no bec phenomena (no sturation of excited state population) Enhanced effect of interactions Special case of quantum correlated systems : exact solutions exist Chip experiment well suited to study 1D physics

Outline 1 Theoretical results on weakly interacting 1D gases Homogeneous gases harmonically trapped gas 2 Experimental study of the cross-over towards quasi-bec Observation of bunching effect Inhibition of bunching in the quasi-bec regime Failure of Hartree-Fock theory to explain the transition towards a quasi-bec 3 Other results 4 Conclusion

1D Bose gas with repulsive contact interaction H = 2 2m dzψ + 2 z 2 ψ + g 2 dzψ + ψ + ψψ, Exact solution : Lieb-Liniger Thermodynamic : Yang-Yang (60 ) n, T Length scale : l g = 2 /mg, Energy scale E g = 2 /2mlg 2 Parameters : t = T/E g, γ = 1/nl g = mg/ 2 n 1e+08 t 1e+06 10000 100 1 0.01 thermal nearly ideal gas degenerat quasi-condensate quantum classical strongly interacting 0.0001 0.0001 0.001 0.01 0.1 1 10 100 γ

Nearly ideal gas regime : bunching phenomena For each quantum state, Boltzmann distribution particle number fluctuations : n 2 k nk 2 = n k }{{} Two-body correlation function g 2 (z) g 2 2 l c = λ db : µ T shot noise l c 2 n/(mt) : µ T + n k 2 }{{} bunching 1 z z Bunching effect density fluctuations. n(z) n(z ) = n 2 g 2 (z z)+ n δ(z z ) Bunching : correlation between particles. Quantum statistic Classical field ψ = a k e ikz : speckle phenomena n k = a k 2 n 2 k nk 2 = n k 2

Crossover towards quasi-condensate ρ n Repulsive Interactions Density fluctuations require energy δρ dzδρ = 0 H int = g 2 dzρ 2 = g 2 dzδρ 2 > 0 z Reduction of density fluctuations at low temperature/high density Cross-over temperature : 1 N H int gn µ T c.o. 2 n 2 2m γ For T T c.o. : quasi-bec regime bunching effect killed g 2 2 ξ = / mgn 1 z T > gn

1D weakly interacting homogeneous Bose gas t = 2 T/(mg 2 ) n co = 1 λ db t 1/6 g 2 (0) 2 1 l c ξ = / mgn co ideal gas 1/λ db quasi-condensat classical field theory n co quantum fluctuations n λ db Quantum decoherent regime n µ<0 µ gn > 0

One-dimensional gas trapped in a harmonic potential Competition : Interaction-induced cross-over / finite size BEC Cross over towards quasi-bec ω small Local density approach L V = 1/2mω Cross-over : n(z = 0) =n 2 z 2 c.o. N c.o. k BT ω ln (t 1/3) z Validity of LDA : l c L At cross-over ω (mg 2 T 2 / 2 ) 1/3 Finite size condensation phenomena at large ω N c = k BT ω ln (2T/ ω) LDA condition in 3D world : g = 2 ω a for a l = /mω ω ω (T/ ω ) 2/3 (a/l ) 2/3 Easily satisfied Bouhoule et al., PRA 75 031606 (2007)

Realisation of very anisotropic traps on an atom chip Use of a H shape trap I 1 L=2.8 mm I 1 + I 2 z I 2 B ext Transverse confinement : I 1 + I 2 = 3A and B ext = 30G : ω = 2π 2800Hz Longitudinal confinement : ω z (I 1 I 2 ) 2 : 6Hz 20Hz 87 Rb atoms MOT transfered into the magnetic trap (3 10 6 atoms) Radio-frequency evaporative cooling T 200nK 1.5 ω, N 5000.

Absorption images Imaging geometry = 6 µm Atom chip z y x 2 pictures taken : * With atoms and trap still on * Without atoms (delay of 200 ms) CCD camera Typical image y Imaging beam Atom per pixel : n at = 2 σ ln( I 2 I1 ). Error if variation of density on a scale smaller than pixel size and high optical density Transverse integration : N at = y n at N at 0.36 mm z 600 µm z

Noise measurement N at Inspired by Föling et al. Nature 434, 481, M. Greiner et al., PRL 94, 110401 Statistical analysis over about 300 images taken in the same condition. Presence of real atomic fluctuations Reference curve : average over 20 images (running average) and normalised to the same N tot A typical curve Contribution of photon shot noise substracted L = 600 µm z Pixel size : l c L LDA valid N 2 N 2 = ( n(z)n(z ) n 2) = N + n 2 dz(g 2 (z) 1) z l c z

Atom-number fluctuations in an ideal Bose gas For each eigenstate : n 2 n 2 = n + n 2 Many occupied eigenstates N 2 t Nt 2 = N t + n i 2 For G equally occupied states, N 2 t Nt 2 = N t + 1 G N t 2 i To observe bunching : G should not be too high Ratio bunching/shot noise : N t /G = psd ( ) T 2 G ω Ω for non degenerate gas. Ω= mk B T G when N for a degenerate gas.

Observation of atomic shot noise N 2 at Nat 2 80 60 40 20 0 0 100 200 300 400 N at High temperature cloud Slope smaller than 1 : due to finite optical resolution δ δ Contribution of each atom on the absorption in a given pixel reduced by a factor δ slope of shot noise reduced For Gaussian resolution of rms width δ, reduction by κ = Measured reduction κ = 0.17 δ = 10µm. Measured resolution : 8µm 2 πδ.

Observation of atomic bunching and evidence for quantum decoherent regime 80 60 Temperature deduced from longitudinal profile. < d N 2 > 40 20 0 0 100 N T~10 hw T=2.1hw 200 Dotted : non-degenerate gas approximation Dashed-dotted : Exact formula

Expected atom number fluctuations in a one dimensional system in quasibec regime Pixel size much larger than healing length ξ = / mgn Relevant excitations are phonons Energy of a phonon of wave vector k : ( g H k = 2 δρ2 k + n 2 k 2 ) 2m θ2 k Thermodynamic equilibrium : ψ = n + δρ e iθ ρ c n δρ k B T 2 = g 2 δρ2 k (k B T gn) Atom number fluctuations : VarN = δρ(z)δρ(z ) = k δρ2 k eik(z z ) = 2 δρ 2 0 VarN = k BT g z

Expected atom number fluctuations in a nearly one dimensional system in quasibec regime n 1/a : Purely one dimensional case recovered. (g = 2 ω a) n of the order or larger than 1/a : Transverse breathing associated with a longitudinal phonon has to be taken into account. Thermodynamic argument : Var(N) =k B T N ) µ Analytic expression : µ hω 1 + 4na 1 + 4na Var(N) =k B T 2 ω a T In good agreement with a 3D Bogoliubov calculation of Var(N).

Experimental results in the quasibec regime < δn 2 > 400 300 200 100 T~10 hω T=1.4 hω 200 N T=1.4 hω 0-400 z (µm) 400 0 0 100 200 N 300 400 Temperature fitted from the wings of the profile Good agreement with theory for low temperature Estève et al. PRL 96, 130403

Conclusion on density fluctuations measurement Most features of weakly interacting 1D Bose gases observed t 1e+08 1e+06 10000 100 1 nearly ideal gas thermal degenerat classical quasi-condensate quantum strongly interacting 0.01 0.0001 0.0001 0.001 0.01 0.1 1 10 100 γ <dn 2 > 400 300 200 100 0 0 T~10 hw T=1.4 hw 100 200 N 200 N 0 400 z(!m) 400 300 T=1.4 hw 400 quasi condensate quantum decoherent classical decoherent

Density profile through the transition towards quasibec Nature of the transition towards quasi-bec : driven by interactions In situ density profile by absorption imaging Graph 600 k B T = 5.7 ω k B T = 2.75 ω k B T = 2.3 ω µ = 3.6 ω µ = 1.6 ω µ = 1.4 ω Graph Graph 400 600 600 N 200 N 400 200 N 400 200 0 0 50 100 150 200 z/delta Good agreement with ideal Bose gas 0 40 60 80 100 120 140 z (µ m) Appearance of quasi-bec at the center 0 40 60 80 100 120 140 z (µ m) Good agreement with quasibec shape Wings : ideal Bose gas equation of state Quasi-bec shape : µ loc = µ mω 2 z 2 /2 = ω 1 + 4na

Success of Hartree-Fock theory in 3D Bose gases 3D ideal Bose gases : BEC for ρ c = 2.612.../λ 3 db For weak interactions (ρa 3 1), Mean-field theories accurate. For ρ<ρ c : Hartree-Fock theory variational method : non interacting Bosons that experienced V eff. ρ c unchanged V eff (r) =2gρ(r), g = 4π 2 a/m for a gas trapped in harmonic potential, small shift of N c (Gerbier et al., Phys. Rev. Lett. 92, 030405 (2004)) Beyond mean-field Validity of Mean-Field (Landau-Ginzburg criteria) : T T c /T c > aρ 1/3. Beyond mean-field effects : small shift of T c, change of critical exponent (Donner et al., Sience 315 :1556 (2007)

Failure of Hartree-Fock theory : a quasibec without condensation Graph N 600 400 200 Hartree-Fock calculation *Population in the ground state N 0 /N tot 3 10 3 1. 0 40 60 80 100 120 140 z (µ m) The appearance of quasi-bec is not explained by Hartree-Fock theory. First failure of mean field theory in a weakly interacting regime. J.-B. Trebbia et al. PRL 97, 250403 (2006)

Other experimental results Phase fluctuations measurement in weakly interacting gases. ψ + (z)ψ(0) = ne mtz/2n 2 Dettmer et al. PRL 87, 160406 (2001), Richard et al. PRL 91, 010405 (2003) Quantum phase fluctuations in weakly interacting 1D gas S. Hofferberth et eal., Nature Physics 4, 489 (2008) Strongly interacting 1D gases : fermionization Kinoshita et al. PRL 95, 190406 (2005)

Conclusion Prospects Study of 1D gases in the strong interaction regime. expected : ω = 40 khz, n = 1 at/µm Better imaging system Higher order of the distribution g 3 Reaching quantum fluctuations Study of correlation length of density fluctuations in 1D gases. Correlation between pixels Tomography method Study of density fluctuations in 2D gases. Use of rf dressed potentials

Collaborators Members of the chip experiment Theoreticians collaborators Karen Kheruntsyan Chris Westbrook Gora Shlyapnikov Jérôme Estève Micro-fabrication Thorsten Schumm LPN laboratory Jean-Baptiste Trebbia Dominique Mailly Carlos Garrido-Alzar Julien Armijo