Breathing modes of a 1D Bose gas - Seras Club Nautique

Experimental tools. 2. Hydrodynamic breathing mode. Theory. Measurement. 3. Beyond .... Images taken at focus (tv = 15ms) for different initial temperatures ..... Fit using YY of the wings. 100 ... Change the 1D-ness (difficult) or add a lattice.
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Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Breathing modes of a 1D Bose gas Bess Fang , Isabelle Bouchoule, Thibaut Jacqmin, Tarik Berrada, Aisling Johnson Institut d’Optique, Palaiseau.

Institut d’Optique, 30 Mai 2013

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Outline 1

Experimental techniques and physical background Apparatus Regimes of 1D Bose gases Experimental tools

2

Hydrodynamic breathing mode Theory Measurement

3

Beyond self similar solution Observations Origin ?

4

frequency and lifetime

5

Long-lived out-of-equilibrium stationnary states

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Outline 1

Experimental techniques and physical background Apparatus Regimes of 1D Bose gases Experimental tools

2

Hydrodynamic breathing mode Theory Measurement

3

Beyond self similar solution Observations Origin ?

4

frequency and lifetime

5

Long-lived out-of-equilibrium stationnary states

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Realisation of very anisotropic traps on an atom chip Magnetic confinement of 87 Rb by micro-wires A 3-wire guide 

z

ω⊥ /2π = 2 − 40 kH ωz /2π = 5 − 10 Hz

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

y B

T ' 400 − 15nK ' 3.0 − 0.1~ω⊥ N ' 5000 − 1000.

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

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

deg

ther

10000

t

mal

ene

rat

classical

Bosonic bunching g(2) (0) ' 2 µ ~/ξ























Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Outline 1

Experimental techniques and physical background Apparatus Regimes of 1D Bose gases Experimental tools

2

Hydrodynamic breathing mode Theory Measurement

3

Beyond self similar solution Observations Origin ?

4

frequency and lifetime

5

Long-lived out-of-equilibrium stationnary states

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Hydrodynamic breathing mode Long wave length perturbation : hydrodynamic equations  ∂t ρ + ∂z(ρv) = 0 mρ∂t v + ρ∂z (V + mv2 /2) + ∂z p = 0 Quasi-condensate equation of state : p = −∂E/∂V = gn2 /2 √ V = 0, small perturbations : phonons ω = ck, c = ρg V harmonic and ρ(t = 0) inverted parabola : self similar solution ρ

z

ρ(t, z) = ρ(t = 0, z/η)/η

v

z

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Experimental observation V

Quench of the longitudinal confinement z Mom ent um profile

21

6.0

20

5.5

19

5.0

18

rm s widt h

rm s widt h

Insit u profile

17 16

4.5 4.0 3.5

15

3.0

14

2.5

13

0

100

200

300

400 t

500

600

700

2.0

0

100

200

300

400

500

600

700

t √ Insitu profile : ωB = 0.94 3ωD Momentum profile : ωBp = 2ωB Very large amplitude in momentum space : P ' (~/ξ)∆σ/σ First observation in momentum space.

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Breathing mode excited by strong evaporation Cutting into the Thomas-Fermi profile by RF knife Cut into quasi−condensate

5 × 104 Nat

U µ

µU

0 −1200 −800 −400 t

Mom ent um profile 3.2

√ ωBp = 0.98 × 2 3ω

3.1

rm s widt h

3.0 2.9 2.8 2.7 2.6 2.5 2.4 2.3

50

100

150 t

200

250

0

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Outline 1

Experimental techniques and physical background Apparatus Regimes of 1D Bose gases Experimental tools

2

Hydrodynamic breathing mode Theory Measurement

3

Beyond self similar solution Observations Origin ?

4

frequency and lifetime

5

Long-lived out-of-equilibrium stationnary states

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Ab-normal breathing mode resulting from evaporation Insitu profile

center (pixel)

310

rms width (pixel)

Insitu profile

308 306

0

200 400 600 t (ms)

16 14

√ ωB = 1.01 × 3ωD

12

800

0

200 400 600 800 t (ms)

rms width (pixel)

Momentum distribution 5.0 4.5 4.0 3.5

ωBp = 1.00 ×

3.0 2.5 2.0 0 100 200 300 400 500 600 700 800 9001000

t (ms)



3ωD

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

rms width (pixels)

Co-existence of both modes

3.5

3

2.5 0

250 t (ms) √

√ 3ω

500

2 3ω

S(ω)

40 30 20 105 0 0

5

10

15

20

ω/(2π)

25

30

35

40

750

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

g

rms width (pixels)

Effect of a quench of g

5 ms g ∝ ω⊥

3g1

Insitu profile √ ω = 3ωD

18 16 14

g1 0

t

250 500 t (ms)

750

√ 2 3ωD

14 12

3.2

10

S(ω)

rms width (pixel)

Momentum distribution

2.8

8 6 4

2.4

2

0

200

400 t (ms)

600

Both frequency visible.

0 0.0 0.5

3.0 √ ω/( 3ωD )

1.0 1.5 2.0 2.5

3.5 4.0

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Physical origin ? Adiabatic temperature change to the compression of the gas ? Not the good order of mangnitude Gross-Pitaevski calculations. Initial thermal state : use Local density approximation h(θ(z) − θ(z0 ))2 i = mT/(~2 ρ)|z − z0 | ⇒ Random walk p 0 hδρ(z)δρ(z0 )i = mT/(~2 ) mg/(~2 ρ)e−2|z−z |/ξ ⇒ Ornstein-Uhlenbeck 1.8

1.4

0.5

1.2

ρ

1.6

1

θ

1.5

0

1

0.8 −0.5

0.6

−1

0.4 0.2

−1.5 2000

2500

3000

3500

z

4000

4500

0 1000

1500

2000

2500

3000

z

3500

4000

4500

5000

5500

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Gross-Pitaevski calculations

momentu width

Time evolution according to Gross-Pitaevski after a quench of ωz Result of GP fitted by a prabola

1.0 0.9 0.8

Self-similar solution

0.7 0.6 0.5 0.4 0.3

Effect of compression

0.2 0.1 0.0 0

1

2

3

4

5

6

7



Do not account for experimental observation

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Gross-Pitaevski calculations : quench of g Time evolution according to Gross-Pitaevski after the quench of g. Experimental parameters used

Do not account for experimental observation

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Outline 1

Experimental techniques and physical background Apparatus Regimes of 1D Bose gases Experimental tools

2

Hydrodynamic breathing mode Theory Measurement

3

Beyond self similar solution Observations Origin ?

4

frequency and lifetime

5

Long-lived out-of-equilibrium stationnary states

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Breathing mode frequency Quasi-condensation cross-over : ω :



3ω → 2ω

Dimensionnal pcrossover (within quasi-bec) : √ ω : 3ω → 5/2ω Insit u profile Insit u profile 27

15

26 25 rm s widt h

rm s widt h

14 13 12

ωB = 1.89ωD T from profile : 80 nK, T from fluctuations : 20 nK

23 22 21

11 10 -800 -700 -600 -500 -400 -300 -200 -100 t (m s)

24

20 -0

19 -800

-700

-600 -500 t (m s)

-400

T ' 180nK = 1.9~ω⊥ µ/(~ω⊥ ) ' 0.1

-300

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Lifetime of the breathing mode 3D case : lifetime limited by phonons/phonons coupling at non zero temperature. 1D case ? Insit u profile Insit u profile

18

26

17

25

16

rm s widt h

rm s widt h

27

24 23 22 21

14 13

20 19 -800

15

-700

-600 -500 t (m s)

-400

T ' 180nK = 1.9~ω⊥

-300

12 -800 -700 -600 -500 -400 -300 -200 -100 t (m s)

T ' 30 nK τ = 850 ms

-0

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Outline 1

Experimental techniques and physical background Apparatus Regimes of 1D Bose gases Experimental tools

2

Hydrodynamic breathing mode Theory Measurement

3

Beyond self similar solution Observations Origin ?

4

frequency and lifetime

5

Long-lived out-of-equilibrium stationnary states

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Long-lived out-of-equilibrium stationnary states Predictions for Tprofile

Qusi-condensate (T=)

60

hδN 2 i

Prediction for T=) 30

0 0

40

80

hNi Experimental data

Relax time of 950 ms !

80

N

Modified Yang-Yang profile : T= 140 nK Contribution of transverse excited states

40

0 100

z (pixels)

200

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Long lived states with strange profile

Experimental data

80

Best YY fit

N

Thomas−Fermi parabola Fit using YY of the wings

40

0 100 z (pixel)

200

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Satbility of the non-equilibrium state

T from fluctuation T from profile

T

120

70

20 450

1050 t(ms)

1350

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Outline 1

Experimental techniques and physical background Apparatus Regimes of 1D Bose gases Experimental tools

2

Hydrodynamic breathing mode Theory Measurement

3

Beyond self similar solution Observations Origin ?

4

frequency and lifetime

5

Long-lived out-of-equilibrium stationnary states

Experimental techniques and physical background Hydrodynamic breathing mode Beyond self similar solution frequency and lifetime Long-li

Conclusion and prospetcs Conclusion First observation of the breathing mode in momentum space A strange behavior of the momentum distribution associated with breathing mode at finite temperature Very long lived out of thermal equilibrium states Prospects Investigating the breathing mode frequency and lifetime across the quasi-condensation. Investigate the breathing mode frequency and lifetime across the dimensional crossover Relation of the long lived out of thermal equilibrium states with intergrability ? Change the 1D-ness (difficult) or add a lattice. Investigating effect of quenches of g. Going towards strongly interacting gases