Bose gas in Flatland Berezinskii-Kosterlitz-Thouless Physics in an - PowerPoint PPT Presentation

Bose gas in Flatland Berezinskii-Kosterlitz-Thouless Physics in an Atomic Gas Zoran Hadzibabic * Peter Kruger, Marc Cheneau, Baptiste Battelier, and Jean Dalibard Ecole Normale Superieure, Paris *now at: University of Cambridge Collège de France

Outline of the talk Bose gases in 2D Berezinskii-Kosterlitz-Thouless transition Homogeneous vs. trapped & ideal vs. interacting gas Critical point of an interacting 2D gas BEC vs. BKT Vortices and quasi-long-range coherence

Long-range order in reduced dimensionality more vulnerable to fluctuations, disorder… c.f. classical transport: 3D - easy 1D - impossible 2D - marginal

BEC, coherence, and superfluidity in 2D Homogeneous 2D Bose fluid in the thermodynamic limit No BEC in an ideal gas No true long-range order in an interacting gas at finite T (Mermin-Wagner-Hohenberg theorem) But still a superfluid transition at finite T Bishop and Reppy (1978), superfluidity in liquid He films : “universal jump in Torsion oscillation period (ns) superfluid density” pendulum at T = T c shift of the 1.0 adsorbed He film T (K) 0 1.1 1.0 1.2

Berezinskii & Kosterlitz – Thouless (1971-73) Phase transition without spontaneous symmetry breaking T c 0 T normal superfluid exponential decay of g 1 algebraic decay of g 1 ( λ – thermal wavelength) Unbinding of Bound vortex- vortex pairs Proliferation of antivortex pairs free vortices

(Ideal gas) In a harmonic trap… 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 harmonic trapping make 2D boring? What about interactions?

The effect of (weak) interactions on BEC 3D harmonic trap: Repulsive interactions slightly decrease the central density, for given N and T For an ideal gas, the central density at condensation point is: (semi-classical) Just put in a bit more atoms to obtain the needed 2D harmonic trap: The same procedure completely fails: where

The effect of (weak) interactions on BEC 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… …BEC suppressed, expect BKT (?)

How to make an ultracold 2D Bose gas 3D BEC + 1D optical lattice 2 independent 2D clouds (no tunnelling) 10 5 atoms/plane plane thickness: 0.2 μ m, separation: 3 µm (other 2D experiments at MIT, Innsbruck, Oxford, Florence, NIST, Heidelberg etc.) Why 2 planes? Crucial info in the phase of Ψ , and accessible in an interference experiment

2. Critical point of an interacting 2D Bose gas P. Krüger, Z. H. and J. Dalibard, cond-mat/0703200

Phase transition in a 2D atomic gas Fix the temperature T Vary the atom number N Bimodal distribution for N > N C 120 Similar signature to 3D BEC Number of atoms in the core (x 10 3 ) 100 T = 92 (6) nK Dense core follows the 80 � � � � Thomas-Fermi law � � � 60 in time-of-flight expansion, N c = 85 000 40 characteristic of 20 superfluid hydrodynamics 0 0 0 50 50 100 100 150 150 200 200 250 250 � � � � � � Total atom number (x 10 3 )

Critical atom number vs. T 150 Critical atom number (x 10 3 ) Fit using 100 50 ideal gas BEC 0 0 40 80 120 T (nK) 5.3 times larger than the ideal gas BEC prediction!

Can it be the Kosterlitz-Thouless critical point? is universal and elegant, but not the whole story Total critical density depends on microscopics (long standing problem!) Fisher & Hohenberg + Prokof’ev et al.: dimensionless interaction strength For our setup: λ = ± n 2 8 . 6 0 . 8 Extract from the experiment: c (in the center of the cloud)

Critical atom number vs. T BKT + LDA + experimentally observed Gaussian profiles: Critical atom number (x 10 3 ) 150 100 BKT + LDA fit 50 ideal gas BEC 0 0 40 80 120 T (nK) Not bad…

Equation of state? Bimodal distribution fitted well by Gaussian + Thomas-Fermi …but why?

3. Coherence of an interacting 2D Bose gas Z. H., P. Krüger, M. Cheneau, B. Battelier, S. Stock, and J. Dalibard Phys. Rev. Lett. 95 , 190403 (2005) Nature 441 , 1118 (2006) cond-mat/0703200 + Schweikhard, Tung and Cornell, cond-mat/0704.0289 Shlyapnikov-Gangardt-Petrov, Holtzman et al., Kagan et al. , Theory: Stoof et al. , Mullin et al. , Simula-Blackie, Hutchinson et al. Polkovnikov-Altman-Demler

Interference of two 2D gases z z Time of flight y x x The interfering part coincides with the central part of the bimodal distribution

Bimodality and interferences 120 Interference amplitude (arb.units) atoms in TF part of distribution interference amplitude central component (x 10 3 ) 100 Number of atoms in the �� � � � 80 � � � � � � T = 92 (6) nK � � � � � � � 60 � � � � � � � 40 � � � 20 0 0 0 0 50 50 100 100 150 150 200 200 250 250 � � � � � � Total number of atoms (x 10 3 ) Within our accuracy, onsets of bimodality and interference coincide

Local vs. long-range coherence z Time of flight y x Phonons (“spin waves”) Vortices smooth phase variations sharp dislocations π cold hot 0 uniform phase 0

Free vortices in 2D clouds Fraction of images showing at least one dislocation in the central region: 40% 30% 20% 10% (Similar results at NIST) 0.5 0.75 1 Temperature control (arb.)

Long-range coherence Embedded in: The interference signal between and gives: x 0 z x

Long-range coherence Polkovnikov, Altman, Demler: Integrated contrast: x scales as: 0.50 "universal jump in superfluid density" drop in α from 0.5 to 0.25 (in an infinite uniform system) 0.25 0.5 0.75 1 temperature control (arb.un.)

Vortices vs. Correlations vs. Temperature first order coherence : vortices: 0.50 40% 2 = 30% λ n 4 S 20% 10% 0.25 0.5 0.75 1 0.5 0.75 1 temperature control (arb.un.) The onset of vortex proliferation coincides with the loss of quasi-LRO Z. Hadzibabic et al., Nature 441 , 1118 (2006) see also Schweikhard, Tung and Cornell, cond-mat/0704.0289 for KT in a lattice

So far in atomic Flatland… ( ) N , C T Phase transition with a critical point : C - eliminates conventional BEC - agrees quantitatively with BKT + LDA Direct visualization of free vortices: - coincides with loss of quasi-long-range order - supports the microscopic basis of the theory Open questions/future: Equation of state? Tune the interactions from g ~ 1 to g ~ 10 -4 Superfluidity – transport, dissipation? Resolve tightly bound vortex pairs in the superfluid state?

THE END