Dynamic structure factor and drag force in a strongly interacting 1D - - PowerPoint PPT Presentation

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Dynamic structure factor and drag force in a strongly interacting 1D - - PowerPoint PPT Presentation

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Dynamic structure factor and drag force in a strongly interacting 1D Bose gas at finite temperature Guillaume Lang , Anna Minguzzi, Frank Hekking LPMMC, Grenoble, France 1 Overview: Criteria for superfluidity, dynamic structure factor

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Dynamic structure factor and drag force in a strongly interacting 1D Bose gas at finite temperature

  • Guillaume Lang, Anna Minguzzi, Frank Hekking

LPMMC, Grenoble, France

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Overview:

  • Criteria for superfluidity, dynamic structure factor

and drag force

  • The setup
  • Infinite interactions : the Tonks-Girardeau gas
  • Large interactions : the Luttinger liquid
  • Luttinger liquid Vs Tonks-Girardeau in the infinite

interaction regime

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Landau's criterion for superfluidity

  • Superfluid = no

elementary excitation if v<vc=min(ε/|p|)→look at the dispersion relation

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Landau's criterion for superfluidity

  • Superfluid = no

elementary excitation if v<vc=min(ε/|p|)→look at the dispersion relation

  • Two candidates for

superfluidity: Liquid He (b) and weakly-interacting Bose gas (a)

  • Rk: interactions necessary !
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Beyond Landau's criterion

  • Problem : probabilities of

excitations not taken into account

  • Solution : dynamic structure

factor S(q,ω)

  • New criterion : drag force F

such that Ė=-F·v, superfluid → F=0

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Beyond Landau's criterion

  • Problem : probabilities of

excitations not taken into account

  • Solution : dynamic structure

factor S(q,ω)

  • New criterion : drag force F such

that Ė=-F·v, superfluid → F=0

  • Both are measurable! Fabbri et al.

PRA 91, 043617 (2015), Meinert et al. arXiv:1505.08152v1 [cond-mat.quant-gas] 29 May 2015, Onofrio et al. PRL 85, 2228 (2000), Desbuquois et al. Nature Physics 8, 645-648 (2012), ...

  • Link between drag force and

dynamic structure factor ?

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  • Weak potential barrier U(r,t) stirred in the

fluid + linear response theory →

Link between drag force and dynamic structure factor

Astrakharchik and Pitaevskii, PRA 70, 013608 (2004)

  • Let us put it in practice !
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System, assumptions

  • 1D, N bosons,

homogeneous

  • Contact 2-body (strong)

interaction g

  • Size L→+∞, N/L=cst
  • Barrier = Gaussian laser

beam, waist w>0, low power, stirred at velocity v=cst

  • Temperature T>0
  • No spin, no magnetic

field...

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At equilibrium (no barrier)

  • Lieb-Liniger model:

contact interactions

  • Dimensionless

interaction strength:

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γ →∞ : the Tonks-Girardeau gas

  • 1D: Lieb-Liniger model for

hard-core bosons → Tonks-Girardeau gas ≈ free fermions !

  • Statistical transmutation in

1D:

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γ →∞ : the Tonks-Girardeau gas

  • Dynamic structure

factor at T=0:

Backscattering process (umklapp point)

Fermi wavevector

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Dynamic structure factor of a Tonks- Girardeau gas at finite temperature

T=0.1TF T=0.5TF T=TF T=4TF

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  • For a point-like

barrier (w=0):

Drag force in the Tonks-Girardeau gas at finite temperature

Integration line Depend on T

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Drag force in the Tonks-Girardeau gas at finite T

  • F(T) decreases !
  • Tonks-

Girardeau gas not superfluid !

T=0.1TF T=4TF T=4TF T=0

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Large but finite interactions

  • Lieb-Liniger model integrable → Bethe Ansatz
  • Our approach : we only need the low energy

behavior → effective Luttinger liquid model

  • Comparison with exact solution at γ

→∞

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The Luttinger liquid model at γ >>1

  • Mode

expansion →

  • φ: phase field
  • θ : counting field
  • K : linked to

compressibility

  • Vs : sound velocity
  • Hydrodynamic approximation:
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Luttinger liquid Vs Tonks-Girardeau

  • Validity of the model ?
  • At T=0, γ

→∞:

  • Luttinger liquid and

Tonks-Girardeau gas

  • Linearization ok at

low energy → near q=2kF !

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Comparison at finite T

  • Need K(T), vs(T) at

γ = +∞

  • Analytically :

Sommerfeld expansion

  • Numerically : sum

rule and static structure factor.

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Luttinger liquid Vs Tonks-Girardeau at finite T

  • Dynamic structure

factor at T=0.1TF

  • Luttinger liquid : excellent approximation at

low energy and finite T in the backscattering region, curvature not taken into account

Tonks-Girardeau (exact) Luttinger liquid

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Effect of T on the drag force, Luttinger liquid Vs Tonks-Girardeau

L.L., T=0 T.-G., T=0 L.L., T=0.1TF T.G., T=0.1TF

  • Excellent agreement at low velocity !
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Luttinger liquid at finite interactions

  • Analytical

expressions for S(q,ω) and F(v) involve Luttinger parameters →

  • We solve the Bethe

ansatz equations (here at T=0).

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Finite barrier width in the Tonks- Girardeau gas

  • The drag force fades out for a wide barrier

at high velocity !

T

w

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Conclusions

  • Drag force and dynamic structure factor

at finite temperature (and barrier width) at large interactions

  • T-dependence of Luttinger parameters
  • Comparison with exact solution →

validity of the Luttinger liquid model: S(q,ω) near 2kF at low energy, F(v) at low v→ describes well the backscattering region, still valid at low T !

  • Learn more on Phys. Rev. A 91 063619

(2015), or arXiv 1503.08038 [cond- mat.quant-gas]

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Outlook

  • Comparison with exact solution at finite γ

and experiments at low energy near the backscattering region

  • Drag force beyond linear response theory ?
  • Renormalization of the barrier ?
  • …
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| End > < End| Thanks for your attention !