Coherence and superfluidity in 2d Bose gases

1. Peter Krüger École Normale Supérieure Paris Coherence and superfluidity in 2d Bose gases

2. Outline • superfluidity in 2d • Bose-Einstein condensation or Berezinskii-Kosterlitz-Thouless transition • Measurement of the critical point • Transition to the ordered phase: the role of vortices

3. Order and dimensionality The role of (thermal) fluctuations increases in reduced dimensions In 1d and 2d, true long-range order cannot exist in a homogeneous system Mermin-Wagner-Hohenberg 1966 What about 2d superfluidity and BEC?

4. 1.0 T (K) 0 1.1 1.0 1.2 Superfluidity in 2d A 2D film of helium becomes superfluid at sufficiently low temperature (Bishop and Reppy, 1978) Torsion pendulum shift of the oscillation period (ns) adsorbed He film “universal” jump to zero of superfluid density at T = Tc

5. (Un)binding of vortex pairs BKT theory topological phase transition associated with the binding/unbinding of vortex pairs Tc 0 T normal superfluid exponential decay of g1 algebraic decay of g1 Proliferation of free vortices Bound vortex- antivortex pairs

6. BEC in 2d ? – The ideal Bose gas Homogeneous system: 3D: BEC occurs when the phase space density reaches 2D: no BEC for any phase space density In a harmonic trap: 3D: BEC occurs when 2D: BEC occurs when Does the trapping potential obscure the dimensionality difference?

7. Interactions 3D harmonic trap Repulsive interactions slightly decrease the central density, for given N and T. For an ideal gas, the central density at the condensation point is (semi-classical) A few atoms more lead to the proper 2D harmonic trap The same procedure completely fails:

8. Interactions Treat the interactions at the mean field level: where the mean field density is obtained from the self-consistent equation Two remarkable results • One can accommodate an arbitrarily large atom number. Badhuri et al • The effective frequency deduced from tends to zero when Holzmann et al Similar to a 2D gas in a flat potential… Does this mean there’s only BKT, even in the trap?

9. The experiment • A regular 3d BEC is split into two by super-imposing a blue detuned 1d optical lattice • Ramping up the lattice compresses the cloud into 2d  = 0.2 • 105 atoms/plane • plane thickness: 0.1mm • plane separation: 3 µm (lattice period @ small angle) • barriers broad and high  no tunneling Experiments also at MIT, Oxford, Innsbruck, Heidelberg, Florence, NIST, …

10. Measuring critical atom numbers Produce 2d cloud, wait for atom number to reduce (~ 10s) RF knife on to keep T constant

11. 80 60 40 20 0 0 20 40 60 80 100 120 140 Bimodal density profiles Optical density Optical density Bimodal fits (Gauss + TF) give info on numbers, sizes, temperature … Integrated optical density

12. Nc = 85 000 Onset of bimodality T = const. N0 (in thousands) Total atom number (in thousands) Clear threshold of bimodality as total atom number is varied

13. Interference FFT (1d) image kz z x x TF radii of density (DC peak) and interfering region (1st harmonic) coincide interference density x (microns)

14. Comparing critical numbers Interference amplitude (arb.units) N0 (in thousands) Total number of atoms (in thousands) Within our accuracy, onset of bimodality and interference agree

15. Temperature dependence of Nc Critical atom number (in thousands) Temperature (nK) critical atom numbers are systematically higher than ideal gas prediction (multiple planes taken into account)

16. Critical density: BKT universal jump in superfluid density from ρsλ2=0 to ρsλ2=4, but total density at transition depends on interactions: Fisher and Hohenberg for sufficiently weak interactions in our case Quantum Monte-Carlo calculations give C=380: Svistunov et al.

17. Ideal gas Critical density -> critical number ? Once the shape of the density distribution is known, the local density approximation allows to derive critical numbers data Gauss fit Gaussian profile fits data well, the peaked g3/2 distribution (ideal gas) is not compatible with data

18. Heuristic model Assuming a Gaussian density profile (empirically motivated) directly provides a prediction for the critical atom number in BKT theory:

19. Temperature dependence Critical atom number (in thousands) Temperature (nK) heuristic model explains data well: fit based on gives (4.9 expected from model)

20. Time of flight p z 0 y x uniform phase 0 A closer look at interference m 30 m vortex hot cold

21. Proliferation of thermal vortices • Drastic in-crease in sharp phase dislocations for high T • Ratio should go to 1, but only subset of vortices is detected Fraction of images showing at least one phase dislocation high T low T central contrast c0

22. Long range order First order correlation function The interference signal between and gives 0 x z x

23. x low T Exponent α no long range order quasi long range order Just below the transition, high T Just above the transition, decays exponentially decays algebraically like Local contrast Temperature dependence Integrated contrast: average over many images: Polkovnikov, Altman, Demler fit by

24. m 30 m Correlations and Vortices Loss of quasi long range order Proliferation of vortices Hadzibabic, Krüger, Cheneau, Battelier, Dalibard, Nature 2006

25. Conclusion • weakly interacting trapped 2d Bose gas undergoes phase transition • onset of bimodality and interference conincide • critical atom numbers are 5 times higher than expected for ideal gas • BKT theory with LDA and Gaussian density profiles explains deviation • low T phase behaves like 3d, except true long-range order is missing • loss of quasi-long-range order coincides with appearance of free vortices

26. The team Zoran Hadzibabic Jean Dalibard Marc Cheneau Baptiste Battelier Patrick Rath Tarik Yefsah Hadzibabic, Krüger, Cheneau, Battelier, Dalibard Nature 2006 Krüger, Hadzibabic, Dalibard submitted to PRL, arXiv cond-mat/0703200