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

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 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 2

Landau's criterion for superfluidity ● Superfluid = no elementary excitation if v<v c =min(ε/|p|)→look at the dispersion relation 3

Landau's criterion for superfluidity ● Superfluid = no elementary excitation if v<v c =min(ε/|p|)→look at the dispersion relation ● Two candidates for superfluidity: Liquid He (b) and weakly-interacting Bose gas (a) ● Rk: interactions necessary ! 4

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 5

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 ? 6

Link between drag force and dynamic structure factor Astrakharchik and Pitaevskii, PRA 70 , 013608 (2004) ● Weak potential barrier U(r,t) stirred in the fluid + linear response theory → ● Let us put it in practice ! 7

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... 8

At equilibrium (no barrier) ● Lieb-Liniger model: contact interactions ● Dimensionless interaction strength: 9

→∞ : the Tonks-Girardeau gas γ ● 1D: Lieb-Liniger model for hard-core bosons → Tonks-Girardeau gas ≈ free fermions ! ● Statistical transmutation in 1D: 10

→∞ : the Tonks-Girardeau gas γ ● Dynamic structure factor at T=0: Backscattering process (umklapp point) 11 Fermi wavevector

Dynamic structure factor of a Tonks- Girardeau gas at finite temperature T=0.5T F T=0.1T F T=T F T=4T F 12

Drag force in the Tonks-Girardeau gas at finite temperature For a point-like ● Depend on T barrier (w=0): Integration line 13

Drag force in the Tonks-Girardeau gas at finite T ● F(T) decreases ! ● Tonks- Girardeau gas not superfluid ! T=0.1T F T=0 T=4T F T=4T F 14

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 →∞ 15

The Luttinger liquid model at γ >>1 ● Hydrodynamic approximation: ● φ : phase field ● Mode ● θ : counting field expansion → ● K : linked to compressibility ● V s : sound velocity 16

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=2k F ! 17

Comparison at finite T ● Need K(T), vs(T) at = +∞ γ ● Analytically : Sommerfeld expansion ● Numerically : sum rule and static structure factor. 18

Luttinger liquid Vs Tonks-Girardeau at finite T ● Dynamic structure Tonks-Girardeau (exact) Luttinger liquid factor at T=0.1T F ● Luttinger liquid : excellent approximation at low energy and finite T in the backscattering region, curvature not taken into account 19

Effect of T on the drag force, Luttinger liquid Vs Tonks-Girardeau L.L., T=0 L.L., T=0.1T F T.-G., T=0 T.G., T=0.1T F ● Excellent agreement at low velocity ! 20

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). 21

Finite barrier width in the Tonks- Girardeau gas ● The drag force fades out for a wide barrier at high velocity ! T w 22

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 2k F 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] 23

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 ? ● … 24

| End > < End| Thanks for your attention ! 25