Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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, 8th of March 2012
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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.
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Outline 1 2 3
4
5 6
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Measurements Weakly interacting regime : quasi-condensation and sub-poissonian fluctuations Entering the strongly interacting regime Beyond 1D physics Quasi-bec Contribution of excited transverse states in the crossover Third moment of density fluctuations Momentum distributions of a 1D gas focussing technics Classical field analysis Results
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
1D Bose gas with repulsive contact interaction H=−
~2 2m
Z
dzψ +
g ∂2 ψ+ 2 ∂z 2
Z
dzψ + ψ + ψψ,
Exact solution : Lieb-Liniger Thermodynamic : Yang-Yang (60’) n, T Length scale : lg = ~2 /mg, Energy scale Eg = g2 m/2~2 Parameters : t = T/Eg , γ = 1/nlg = mg/~2 n 1e+08
nearly ideal gas
1e+06
ther
10000
t
deg ene
rat
mal
classical
quasi-condensate 100
1
quantum
strongly interacting 0.01
0.0001 0.0001
0.001
0.01
0.1
γ
1
10
100
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Strongly interacting 1D Bose gas • 2 body scattering wave function ψ(z) = cos(kz + ϕ), E = ~2 k2 /m ψ
If E Eg , |ψ(0)| ' 1 If E Eg , |ψ(0)| 1
Eg = mg2 /2~2
ψ(0) z
• Many body system ψ
g(2) (0) ' 0 Fermionization
ψ
1e+08
1e+06
10000
z E = gn Eg ⇒γ 1
z ' 1/n E = ~2 n2 /m Eg ⇒γ 1
Strongly interacting regime : γ 1, t 1
t
100
1
0.01
0.0001 0.0001
0.001
0.01
γ
0.1
1
10
100
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Nearly ideal gas regime : bunching phenomena Two-body correlation function g2 (z) = hψz+ ψ0+ ψ0 ψz i/n2 g2 2
lc = λdB :|µ| T lc ' ~2 n/(mT) : |µ| T
Im ψ
1
Re ψ
0
z0 − z
Bunching effect → density fluctuations. hδn(z) δn(z0 )i = hni2 (g2 (z0 − z) − 1) + hniδ(z − z0 ) Bunching : correlation between particles. Quantum statistic Field theory ψ =
P
ψk eikz , n = |ψ|2 : speckle phenomena
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Transition towards quasi-condensate ρ n
• Repulsive Interactions → Density fluctuations require energy R Hint = g2 dzρ2 ⇒ δHint > 0 Reduction of density fluctuations at low δρ temperature/high density weakly interacting : γ = mg/~2 n 1 z
1 N Hint
∝ gn ' |µ| 2 2√ µ = mT 2 /2~2 n2 , ⇒ Tc.o. ' ~2mn γ Cross-over :
, nc.o. ∝ T 2/3
Transition for a degenerate gas
• For T Tc.o. : quasi-bec regime, g(2) ' 1
g2 2
Im ψ
√ ξ = ~/ mgn
T > gn
1
Reψ
z
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Transition towards quasi-condensate ρ n
• Repulsive Interactions → Density fluctuations require energy R Hint = g2 dzρ2 ⇒ δHint > 0 Reduction of density fluctuations at low δρ temperature/high density weakly interacting : γ = mg/~2 n 1 z
1 N Hint
∝ gn ' |µ| 2 2√ µ = mT 2 /2~2 n2 , ⇒ Tc.o. ' ~2mn γ Cross-over :
, nc.o. ∝ T 2/3
Transition for a degenerate gas
• For T Tc.o. : quasi-bec regime, g(2) ' 1
g2 2
Im ψ
√ ξ = ~/ mgn
T > gn T < gn
1 z
Reψ
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
1D weakly interacting homogeneous Bose gas 1e+08
nco =
t = 2~2 T /(mg2 )
ideal gas
1e+06
1 1/6 λdB t
10000
t
100 1
0.0001 0.0001
0.001
0.01
γ
0.1
quasi-condensate
ideal gas
quasi−condensate
0.01
1
10
100
g2 (0) 2 1
lc ξ = ~/ mgnco
√
λdB
classical field theory 000 111 quantum fluctuations 000 111 g(2) (0) < 1 000 111 000 111 000 111 000 111 000 111 000 111 1/λdB n 000 111 nco 000 111 000 111 Phases fluctuations 000 111 000 111 000 111 Density fluctuations 000 111 000 111 000 111 000 111 000 111 000 111 n 000 regime 111 Quantum decoherent µ 0
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Cross-over towards quasi-bec in a one-dimensionnal gas trapped in a harmonic potential
Local density approach : µ(z) = µ0 −mω 2 z2 /2
V = 1/2mω 2 z 2 z
Local correlations properties : that of a homogeneous gas with µ = µ(z). Validity : lc 1n dn dz . At quasi-condensation transition : lc = ξ 3/2 l⊥ ⇒ T ω⊥ ωω⊥ a Easilly fulfilled experimentally.
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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
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 − 10 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)
0.69 −0.077
T ' 400 − 15nK ' 3.0 − 0.1~ω⊥ N ' 5000 − 1000.
y B
t ' 80 − 1000 Weakly interacting gases
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 − 150 kHz
I = 1A
ωm = 200kHz
AlN
Diverse longitudinal potentials Dynamical change of g
15µm
• Approaching strong interactions ω⊥ /(2π) = 19 kHz, ωz = 7.5 Hz 45 40
Yang-Yang solution t = 4.3
35 30 25
hN i
Experimental sequence : ? Cooling at moderate ω⊥ ? Increase ω⊥ Problem : exess of heating (Entropy not preserved)
20 15 10 5 0 0
10
20
30
40
z/∆
50
60
70
80
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Density fluctuation measurements • Statistical analysis over hundreds of images Contribution of atomic fluctuations
100 N
lc , ∆ L → local density approximation δN binned according to hNi hδN 2 i versus n = hNi/∆ Optical shot noise substracted
Mean curve Optical shot noise
0 100 150 200 z/∆
hδN 2 i : Two-body correlation function integral Z Z hδN 2 i = hNi + n2 dz(g2 (z − z0 ) − 1) lc ∆ ⇒ hδN 2 i = hNi + hNin ⇒ Thermodynamic quantity
R
dz(g2 (z − z0 ) − 1) ∂n hδN 2 i = kB T∆ ∂µ
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Effect of finite spatial resolution • Absorption of an atom spreads on several pixels → blurring of the image δ
Decrease of fluctuations : hδN 2 i = κ2 hδN 2 itrue
∆
→ Correlation between pixels 0.8
Ci,i+j
0.6
δ deduced from measured correlations between adjacent pixels κ2 deduced
j=1
0.4 0.2
j=2
0 0
20
40
60
80 N
100 120
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Expected behavior in asymptotic regimes hδN 2 i
• Ideal gas regime
Z = hNi + hNin |
g2 2 1
0
lc = λdB :|µ| T lc λdB : |µ| T z0 − z
dz(g2 (z) − 1) {z } lc
. Non degenerate gas : nlc 1 hδN 2 i ' hNi . Degenerate gas : nlc 1 hδN 2 i ' hNinlc = hNi2 lc /∆ hδN 2 i
• Quasi-bec regime Thermodynamic : µ ' gn ⇒ hδN 2 i = ∆T/g hNi
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Experimental results 1e+08
nearly ideal gas
1e+06
deg
T = 15 nK' ~ω⊥ /10 t ' 64 µ ' 30 nK' 0.2 ~ω⊥
ene ra
therm
classical
t
al
10000
quasi-condensate
t
100
1
quantum
strongly interacting 0.01
0.0001 0.0001
0.001
0.01
0.1
1
10
100
γ
γ
50
0.2
0.05
0.02
0.01
Poissonian level Ideal Bose gas
hδN 2 i
40 30
Exact Yang−Yang thermodynamics Quasi−cond (beyond 1D)
20 10 0
0
20
40
60
80
hNi
100 120 140
Strong bunching effect in the transition region Quasi-bec both in the thermal and quantum regime
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Quasi-condensate in the quantum regime We measure : hδN 2 i < hNi R hδN 2 i = hNi + hNin dz(g2 (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
hδn2k i
Non trivial quantum fluctuations dominate
Quantum fluctuations Thermal fluctuations T/µξ
1/ξ
k
For our datas : lT = (T/µξ)−1 < 450 nm lT ∆, δ
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
From weakly to strongly interacting gases 1e+08
nearly ideal gas
1e+06
de
gen e ther mal rate
10000
classical
quasi-condensate
hN i
100 1
quantum
strongly interacting 0.01 0.0001 0.0001
t 1
0.001
0.01
0.1
1
10
hδ N 2 i
t
hδ N 2 i
hδ N 2 imax /hN i ∝ t1/3
t 1
100
γ
hN i
smaller t : smaller bunching hδN 2 imax /hNi ∝ t1/3 t 1 : Fermi behavior → from poissonian to sub-poissonian. No bunching anymore t small, γ large ⇒ g large ⇒ large ω⊥
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Density fluctuations close to the strongly interacting regime
hδ N 2i
Moderate compression : ω⊥ /2π = 18.8 kHz T = 40 nK' ~ω⊥ /20 t'5 µ/T ' 1.9
30 25 20 15 10 5 0 0
γ = 0.5
10
20
30
40
hN i
No bunching seen anymore, at a level of 20%. Behavoir close to that of a Fermi gas
50
60
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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) = kB T ∂N ∂µ T
T = 96 nK ' 0.5~ω⊥
Heuristic √ equation : µ = ~ω⊥ 1 + 4na
160 140 120
hδN 2 i
100
Quasi-bec prediction
80 60 40
Modified Yang-Yang prediction
20
Efficient thermometry
0 0
50
100
hNi
150
200
250
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Modified Yang-Yang model If µco ~ω⊥ , transversally excited states behave as ideal Bose gases. V
n
~ω⊥ ~ω⊥
0
ρ
0 µco
~ω⊥
µ
2/3 1/3 2/3 1/3 a µco ' Eg T 2/3 = mg2 /~2 T ⇒ µco /(~ω⊥ ) ' ~ωT⊥ l⊥ For our parameters : a/l⊥ ' 0.025 Modified Yang-Yang model : Transverse ground state : Yang-Yang thermodynamic Excited transverse states : ideal 1D Bose gases First introduced by Van Druten and co-workers : Phys. Rev. Lett. 100, 090402 (2008)
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Modified Yang-Yang model
600
T = 150 nK ' 1.0~ω⊥
T = 490 nK ' 3.2~ω⊥ 1600 1200
hδN 2 i
hδN 2 i
400
200
800 400 0
0 0
100
hNi
Very good agreement
200
300
0
100 200 300 400 500 600 700 800
hNi
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Third moment of density fluctuations
800 700
number of counts
Non gaussian fluctuations We look at hδN 3 i • Motivation Learn about 3 body correlations • A thermodynamic quantity :
600 500 400 300 200 100
3
hδN i = ∆T
2∂
2n
0
−100
∂µ2
• Effect of finite spatial resolution : hδN 3 i = κ3 hδN 3 itrue , κ3 depends on resolution. κ3 infered from neighbour pixels correlation.
−50
0
N − Nmedian
50
100
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Third moment of density fluctuations T = 380 nK
T = 96 nK 1200
2000
800
200 0
200 hδN 2 im
400
hδN 3 im
hδN 2 im
hδN 3 im
3000
hNi 300
1000
100
0 hNi
400
200
0 0
−400 0
100 150 200 250 300 350 hNi
0
50
100 hNi
150
200
0.8 0.4 0
γm
γm
0.4 0.2 00
50
100
200 hNi
300
0
100 hNi
Measured 3rd moment compatible with MYY. Skweness vanishes in quasi-bec regime (as expected) γm = hδN 3 i/hδN 2 i3/2
200
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Three-body correlations hδN 3 i function of g(2) and g(3) . g(3) function contains g(2) We measure : H = hδN 3 i + 2hNi − 3hδN 2 i = n3 R R R h(z1 , z2 , z3 ), h(z , z , z ) = g(3) (z , z , z )−g(2) (z , z )−g(2) (z , z )−g(2) (z , z )+2 1 2 3 1 2 3 1 2 1 3 2 3 8000 6000 4000 2000 0
H
800 0 −800 −1600 0
100
200 hNi
300
0
50 100 150 200 hNi
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
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
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Momentum distribution measurements : motivation
Momentum distribution : TF of first order correlation function Z nk = n dug(1) (u)eiku 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
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Focussing techniques Momentum distribution measurement p
• Short kick of a strong harmonic potential ⇒ δp = −Az
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 z5 )
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Focussing
Images taken at focus (tv = 15ms) for different initial temperatures (initial RF knife position) 250
500 450 400 350 300 250 200 150 100 50 0 −50
200
N
150 100 50 0 −50 0
50
100
150
z/∆
200
250
1200
1000 900 800 700 600 500 400 300 200 100 0 −100 0
50
100
z/∆
150
200
250
1000 800 600 400 200 0 −200 0
50
100
150
z/∆
200
250
0
50
100
150
z/∆
200
250
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Expected results in limiting cases Momentum distribution not knowned exactly.
quasi-condensate ideal gas degenerate classical thermal quantum classical field theory valid Fixed T 1/λdB
np mkB T/~n
np
gaussian
p
n
nco
Lorentzian p
np mkB T/2~n
Lorentzian p
Beyond this approach, some knowed results : ? 1/p4 tail ? Mean kinetic energy (second moment of np ) via Yang-Yang
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
1D classical field calculation ψ(z) : complex field of energy functionnal " # Z ~2 dΨ 2 g 4 2 + |Ψ| − µ |Ψ| E[{Ψ}] = dz 2m dz 2 No high energy divergence (as in higher dimension). Mapped to a Schr¨odinger equation, evolving in imaginary time h i 2 ˆ = pˆ + ~ g (ˆx2 + ˆy2 )2 − µ(ˆx2 + ˆy2 ) , H 2M kB T 2
M=
~3 mkB T
For L → ∞, only ground state contribute to g(1) : ˆ
g(1) (z) = hφ0 |(ˆx − iˆy)e−Hz/~ (ˆx + iˆy)|φ0 i Very fast calculation (as opposed to Monte Carlo sampling).
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Comparison with exact calculations ideal gas quasi−condensate degenerate classical classical field theory valid Fixed T −T ideal gas
µ
0
classical field
interpolation
Comparison with quantum Monte Carlo calculation t = 60, γ = 0.027 µ > 0, n > nco
t = 60, γ = 0.056 µ < 0, n < nco
2
2
10
10
1
1
10
10 0
nk
nk
0
10 −1
10
Gaussian or 1/p4 tails Not reproduced by CF
10 −1
10
−2
−2
10
10
−3
10−40−30−20−100 10 20 30 40
p(mT/(n~))
−3
10−20−15−10−5 0 5 10 15 20
p(mT/(n~))
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Result : degenerate ideal Bose gas regime Classical Field Fit : T=88.0 nK
3
1/p2
10 600
400 nk (µm)
300
2
nk (µm)
Classical field Quasi−condensate
500
10 1
10
200 100
1
0 −10 −5 0 5 k(µm−1)
10
70 60 VarN
50
Modified YY
15
−10−8−6−4−2 0 2 4 6 8 10 k(µm−1)
In agreement with insitu density fluctuations
40
Good agreement with CF
30 20
Lorentzian behavior : 1/p2 tail
10 0 0
10 20 30 40 50 60 Nat
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Result : quasi-condensate regime Fit : T=55 nK 3
nk (µm)
10
1/p2
2
10 nk (µm)
900 800 Bogoliubov 700 Classical field 600 500 400 300 200 100 0 −15−10−5 0 5 10 15 20
1
10
0
10−15 −10 −5 0 5 10 15 k(µm−1)
k(µm−1) 25
VarN
20 15
In agreement with insitu density fluctuations
10 5 0 0 5 101520253035404550 Nat
Good agreement with CF Lorentzian behavior : 1/p2 tail not in agreement with Bogoliubov
Regimes of 1D Bose gases Experimental apparatus Density fluctuations Beyond 1D physics Third moment of density fluctuations Momentum
Conclusion and prospetcs Conclusion Precise density fluctuation measurement Good thermometry Investigation of the quasi-condensation transition Strong anti-bunching Higher order correlation functions Dimensional crossover
Momentum distribution measurement Prospects 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.
