Correlation functions in 1D Bose gases : density fluctuations and momentum distribution

Transcription

1 Correlation functions in 1D Bose gases : density fluctuations and momentum distribution Isabelle Bouchoule, Julien Armijo, Thibaut Jacqmin, Tarik Berrada, Bess Fang, Karen Kheruntsyan (2) and T. Roscilde (3) Institut d Optique, Palaiseau. (2) Brisbane, Australia (3) ENS Lyon Birmingham, 8 th of March 212

2 Correlations in 1D Bose gases Correlation functions : important characterisation of quantum gases Physics in reduced dimension very different from 3D (absence of BEC in 1D ideal Bose gases) Enhanced fluctuations : no TRLO Physics governed by interactions Cold atom experiments : powerful simulators of quantum gases. Reduced dimension achieved by strong transverse confinement. Atom chip experiment : 1D configuration naturally realised.

3 Outline 1 Regimes of 1D Bose gases 2 Experimental apparatus 3 Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime 4 Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover 5 Third moment of density fluctuations 6 Momentum distributions of a 1D gas focussing technics Classical field analysis Results

4 Outline 1 Regimes of 1D Bose gases 2 Experimental apparatus 3 Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime 4 Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover 5 Third moment of density fluctuations 6 Momentum distributions of a 1D gas focussing technics Classical field analysis Results

5 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 (6 ) n, T Length scale : l g = 2 /mg, Energy scale E g = g 2 m/2 2 Parameters : t = T/E g, γ = 1/nl g = mg/ 2 n 1e+8 t 1e thermal nearly ideal gas degenerat quasi-condensate quantum classical strongly interacting γ

6 Strongly interacting 1D Bose gas 2 body scattering wave function ψ(z) = cos(kz + ϕ), E = 2 k 2 /m If E E g, ψ() 1 If E E g, ψ() 1 ψ ψ() E g = mg 2 /2 2 z Many body system ψ ψ g (2) () Fermionization 1e+8 z 1/n E = 2 n 2 /m E g E = gn E g γ 1 γ 1 z t 1e γ Strongly interacting regime : γ 1, t 1

7 Nearly ideal gas regime : bunching phenomena Two-body correlation function g 2 (z) = ψ + z ψ + ψ ψ z /n 2 g l c = λ db : µ T l c 2 n/(mt) : µ T I m ψ R e ψ z z Bunching effect density fluctuations. δn(z) δn(z ) = n 2 (g 2 (z z) 1) + n δ(z z ) Bunching : correlation between particles. Quantum statistic Field theory ψ = ψ k e ikz, n = ψ 2 : speckle phenomena

8 Transition towards quasi-condensate ρ n Repulsive Interactions Density fluctuations require energy H int = g 2 dzρ 2 δh int > δρ Reduction of density fluctuations at low temperature/high density z weakly interacting : γ = mg/ 2 n 1 Cross-over : 1 N H int gn µ µ = mt 2 /2 2 n 2, T c.o. 2 n 2 2m γ, nc.o. T 2/3 Transition for a degenerate gas For T T c.o. : quasi-bec regime, g (2) 1 g 2 2 ξ = / mgn 1 z T > gn I m ψ R e ψ

9 Transition towards quasi-condensate ρ n Repulsive Interactions Density fluctuations require energy H int = g 2 dzρ 2 δh int > δρ Reduction of density fluctuations at low temperature/high density z weakly interacting : γ = mg/ 2 n 1 Cross-over : 1 N H int gn µ µ = mt 2 /2 2 n 2, T c.o. 2 n 2 2m γ, nc.o. T 2/3 Transition for a degenerate gas For T T c.o. : quasi-bec regime, g (2) 1 g 2 2 ξ = / mgn 1 z T > gn T < gn I m ψ R e ψ

10 1D weakly interacting homogeneous Bose gas t 1e+8 1e ideal gas quasi condensate γ ξ = / mgn co λ db g 2 () l c t = 2 2 T/(mg 2 ) ideal gas 1/λ db classical field theory n co Quantum decoherent regime n co = 1 λ db t 1/6 quasi-condensate quantum fluctuations g (2) () < 1 n Phases fluctuations Density fluctuations n µ < µ co µ gn >

11 1D weakly interacting homogeneous Bose gas t 1e+8 1e ideal gas quasi condensate γ large transition quantum decoherent regime barely exists for t < ξ = / mgn co λ db t = 2 2 T/(mg 2 ) g 2 () l c ideal gas classical field theory n co Quantum decoherent regime n co = 1 λ db t 1/6 quasi-condensate quantum fluctuations n n µ < µ gn >

12 Cross-over towards quasi-bec in a one-dimensionnal gas trapped in a harmonic potential Local density approach : µ(z)=µ mω 2 z 2 /2 Local correlations properties : that of a homogeneous gas with µ = µ(z). Validity : l c 1 dn n dz. At quasi-condensation transition : l c = ξ T ω ( ω ω ) 3/2 l a Easilly fulfilled experimentally. V = 1/2mω 2 z 2 z

13 Outline 1 Regimes of 1D Bose gases 2 Experimental apparatus 3 Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime 4 Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover 5 Third moment of density fluctuations 6 Momentum distributions of a 1D gas focussing technics Classical field analysis Results

14 Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum Realisation of very anisotropic traps on an atom chip Magnetic confinement of 87 Rb by micro-wires H-shape trap IH ω /2π = 3 4 khz ωz /2π = 5 1 Hz IH IZ 3 mm z 1D : T, µ ~ω g = 2~ω a chip mount trapping wire z CCD camera In-situ images absolute calibration (a) T ' 4 15nK ' 3..1~ω N ' 5 1. y B t ' 8 1 Weakly interacting gases

15 Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum Reaching strongly interacting gases on an atom-chip A new atom-guide 3 wire modulated guide Roughness free (modulation) Quadrupolar field 1.4 mm Strong confinement : 1 15 khz I = 1A ωm = 2kHz AlN Diverse longitudinal potentials Dynamical change of g 15µm Approaching strong interactions ω /(2π) = 19 khz, ωz = 7.5 Hz 45 4 Yang-Yang solution t = hn i Experimental sequence :? Cooling at moderate ω? Increase ω Problem : exess of heating (Entropy not preserved) z/

16 Outline 1 Regimes of 1D Bose gases 2 Experimental apparatus 3 Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime 4 Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover 5 Third moment of density fluctuations 6 Momentum distributions of a 1D gas focussing technics Classical field analysis Results

17 Density fluctuation measurements Statistical analysis over hundreds of images l c, L local density approximation δn binned according to N δn 2 versus n = N / Optical shot noise substracted N z/ 2 δn 2 : Two-body correlation function integral δn 2 = N + n 2 dz(g 2 (z z ) 1) l c δn 2 = N + N n dz(g 2 (z z ) 1) Thermodynamic quantity δn 2 = k B T n µ Contribution of atomic fluctuations Mean curve Optical shot noise

18 Effect of finite spatial resolution Absorption of an atom spreads on several pixels blurring of the image Decrease of fluctuations : δn 2 = κ 2 δn 2 true δ Correlation between pixels i,i+j C j=1 j= N δ deduced from measured correlations between adjacent pixels κ 2 deduced

19 Expected behavior in asymptotic regimes g 2 Ideal gas regime 2 l c = λ db : µ T 1 l c λ db : µ T z z δn 2 = N + N n dz(g 2 (z) 1) } {{ } l c Non degenerate gas : nl c 1 δn 2 N Degenerate gas : nl c 1 δn 2 N nl c = N 2 l c / δn 2 Quasi-bec regime Thermodynamic : µ gn δn 2 = T/g N

20 Experimental results 1e+8 1 T = 15 nk ω /1 t 1 t 64 1 µ 3 nk.2 ω.1.1 1e+6 thermal nearly ideal gas degenerat quasi-condensate quantum classical strongly interacting γ δn 2 γ N Poissonian level Ideal Bose gas Exact Yang Yang thermodynamics Quasi cond (beyond 1D) Strong bunching effect in the transition region Quasi-bec both in the thermal and quantum regime

21 Quasi-condensate in the quantum regime We measure : δn 2 < N δn 2 = N + N n dz(g 2 (z) 1) g (2) < 1 : Quantum regime However we still measure thermal excitations Shot noise term, removed from the g (2) fonction, IS quantum. Low momentum phonons (k T/µξ) : high occupation number δn 2 k Non trivial quantum fluctuations dominate Quantum fluctuations Thermal fluctuations For our datas : l T = (T/µξ) 1 < 45 nm l T, δ T/µξ 1/ξ k

22 From weakly to strongly interacting gases t 1e+8 1e thermal nearly ideal gas degenerate quasi-condensate classical δn 2 δn 2 max / N t 1/3 N t 1 1 quantum.1 strongly interacting γ δn 2 N t 1 smaller t : smaller bunching δn 2 max / N t 1/3 t 1 : Fermi behavior from poissonian to sub-poissonian. No bunching anymore t small, γ large g large large ω

23 Density fluctuations close to the strongly interacting regime Moderate compression : ω /2π = 18.8 khz T = 4 nk ω /2 t 5 µ/t 1.9 δn γ = N No bunching seen anymore, at a level of 2%. Behavoir close to that of a Fermi gas

24 Outline 1 Regimes of 1D Bose gases 2 Experimental apparatus 3 Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime 4 Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover 5 Third moment of density fluctuations 6 Momentum distributions of a 1D gas focussing technics Classical field analysis Results

25 1D-3D Crossover in the quasi-bec regime µ ω (na 1) Pure 1D quasi-bec µ ω : 1D 3D behavior Transverse breathing associated with a longitudinal phonon has to be taken into account. Thermodynamic argument : Var(N) = k B T N µ ) T Heuristic equation : µ = ω 1 + 4na T = 96 nk.5 ω δn Quasi-bec prediction Modified Yang-Yang prediction Efficient thermometry N

26 Modified Yang-Yang model If µ co ω, transversally excited states behave as ideal Bose gases. n V ω µ co µ co Eg 1/3 T 2/3 = ( mg 2 / 2) 1/3 T 2/3 µ co /( ω ) For our parameters : a/l.25 Modified Yang-Yang model : µ ( T ω Transverse ground state : Yang-Yang thermodynamic Excited transverse states : ideal 1D Bose gases ) 2/3 a l First introduced by Van Druten and co-workers : Phys. Rev. Lett. 1, 942 (28) ω ω ρ

27 Modified Yang-Yang model δn T = 15 nk 1. ω N Very good agreement δn T = 49 nk 3.2 ω N

28 Outline 1 Regimes of 1D Bose gases 2 Experimental apparatus 3 Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime 4 Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover 5 Third moment of density fluctuations 6 Momentum distributions of a 1D gas focussing technics Classical field analysis Results

29 Third moment of density fluctuations Non gaussian fluctuations We look at δn 3 Motivation Learn about 3 body correlations A thermodynamic quantity : number of counts δn 3 = T 2 2 n µ 2 N N median Effect of finite spatial resolution : δn 3 = κ 3 δn 3 true, κ 3 depends on resolution. κ 3 infered from neighbour pixels correlation. 1

30 Third moment of density fluctuations δn 3 m γm δn 2 m 4 2 T = 38 nk 3 N N N δn 3 m T = 96 nk δn 2 m 2 1 N N γm N Measured 3 rd moment compatible with MYY. Skweness vanishes in quasi-bec regime (as expected) γ m = δn 3 / δn 2 3/2

31 Three-body correlations δn 3 function of g (2) and g (3). g (3) function contains g (2) We measure : H = δn N 3 δn 2 = n 3 h(z 1, z 2, z 3 ), h(z 1, z 2, z 3 )=g (3) (z 1, z 2, z 3 ) g (2) (z 1, z 2 ) g (2) (z 1, z 3 ) g (2) (z 2, z 3 )+2 H N N

32 Outline 1 Regimes of 1D Bose gases 2 Experimental apparatus 3 Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime 4 Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover 5 Third moment of density fluctuations 6 Momentum distributions of a 1D gas focussing technics Classical field analysis Results

33 Momentum distribution measurements : motivation Momentum distribution : TF of first order correlation function n k = n dug (1) (u)e iku Cannot be derived from the equation of state. Its measurement opens perspectives (study of dynamics for example) Problem : global measurement averaged over many different situations in presence of a smooth longitudinal potential

34 Focussing techniques Momentum distribution measurement Short kick of a strong harmonic potential δp = Az p z p Free fligh until focuss z Final spatial distribution : initial momentum distribution, averaged over the initial position Experiment well adapted : longitudinal confinement undependent on transverse one Purely harmonic potential (up to order z 5 )

35 Focussing N Images taken at focus (t v = 15ms) for different initial temperatures (initial RF knife position) z/ z/ z/ z/

36 Expected results in limiting cases Momentum distribution not knowned exactly. ideal gas quasi-condensate classical degenerate thermal quantum classical field theory valid Fixed T 1/λ db n co n gaussian n n p n p p mk B T/ n mk B T/2 n p Lorentzian p Lorentzian p Beyond this approach, some knowed results : 1/p 4 tail Mean kinetic energy (second moment of n p ) via Yang-Yang

37 1D classical field calculation ψ(z) : complex field of energy functionnal [ 2 ] E[{Ψ}] = dz dψ 2 2m dz + g 2 Ψ 4 µ Ψ 2 No high energy divergence (as in higher dimension). Mapped to a Schrödinger equation, evolving in imaginary time Ĥ = ˆp2 2M + [ g k B T 2 (ˆx2 + ŷ 2 ) 2 µ(ˆx 2 + ŷ )] 2, M = 3 mk B T For L, only ground state contribute to g (1) : g (1) (z) = φ (ˆx iŷ)e Ĥz/ (ˆx + iŷ) φ Very fast calculation (as opposed to Monte Carlo sampling).

38 Comparison with exact calculations ideal gas quasi condensate classical degenerate classical field theory valid Fixed T T µ ideal gas classical field interpolation Comparison with quantum Monte Carlo calculation nk 2 t = 6, γ =.27 µ >, n > n co p(mt/(n )) nk t = 6, γ =.56 µ <, n < n co p(mt/(n )) Gaussian or 1/p 4 tails Not reproduced by CF

39 Result : degenerate ideal Bose gas regime nk(µm) Classical Field Fit : T=88. nk k(µm 1 ) Classical field Quasi condensate nk(µm) /p k(µm 1 ) VarN Nat Modified YY In agreement with insitu density fluctuations Good agreement with CF Lorentzian behavior : 1/p 2 tail

40 Result : quasi-condensate regime nk(µm) 9 8 Bogoliubov 7 Classical field k(µm 1 ) nk(µm) Fit : T=55 nk 1/p k(µm 1 ) VarN Nat In agreement with insitu density fluctuations Good agreement with CF Lorentzian behavior : 1/p 2 tail not in agreement with Bogoliubov

41 Conclusion and prospetcs Conclusion Prospects Precise density fluctuation measurement Good thermometry Investigation of the quasi-condensation transition Strong anti-bunching Higher order correlation functions Dimensional crossover Momentum distribution measurement Investigation of out-of-equilibrium situations : dynamic following a quench, relaxation towards non thermal states Using tomography to gain in spatial resolution and investigate g (2) (r) Investigating the physics of the Mott transition in 1D using the probes we developped : pinning transition.