1. Cold Gases Meet Condensed Matter Physics Cold Gases Meet Condensed Matter Physics C. Salomon Laboratoire Kastler Brossel, Ecole Normale Supérieure & UPMC, Paris, FR EMMI, GSI, Darmstadt, July 17th, 2008

2. 1 K 100 K10 4 K10 6 K 1 mK 1  K T this room sun surface sun center liquid N 2 liquid He cold atomic gases Dilute, but interacting systems Typical density: Interatomic distancerange of interatomic potentials quantum of the motion in the trap thermal energy Equilibrium properties and dynamics are governed by interactions quark-gluon plasma 10 12 K Temperature scale of cold gases

3. Bose-Einstein statistics (1924) Bose-Einstein condensate Bose enhancement Quantum gases in harmonic traps Dilute gases: 1995, JILA, MIT Fermi-Dirac statistics (1926) Fermi sea Pauli Exclusion Dilute gases: 1999, JILA

4. Dilute gaz at temperature T confined in harmonic trap : Condensation threshold: n 0 : central density Liquid Helium : 10 27 atoms/m 3 n 0 -1/3 = 10 ÅT ~1 K Gaseous condensate: 10 19 atoms/m 3 n 0 -1/3 = 0.5  mT ~1  K Quantum gases : orders of magnitude

5. Camera CCD in situ: cloud size After switching off the trap: momentum distribution Absorption imaging

6. Bose-Einstein condensate and Fermi sea 10 4 Li 7 atoms, in thermal equilibrium with 10 4 Li 6 atoms in a Fermi sea. Quantum degeneracy: T= 0.28  K = 0.2(1) T C = 0.2 T F Lithium 7 Lithium 6 2001 ENS

7. k/k F 1 n(k) 1 k/k F 1 n(k) 1 Interest of cold gases Diluteness: atom-atom interactions described by 2-body (and 3 body) physics At low energy: a single parameter, the scattering length Control of the sign and magnitude of interaction Control of trapping parameters: access to time dependent phenomena, out of equilibrium situations, 1D, 2D, 3D Simplicity of detection Comparison with theory: Gross-Pitaevskii, Bose and Fermi Hubbard models, search for exotic phases, disorder effects,… Link with condensed matter (high Tc superconductors, magnetism in lattices), nuclear physics, high energy physics (quark-gluon plasma), and astrophysics (neutron stars) Towards quantum simulation with cold atoms « a la Feynman »

8. k/k F 1 n(k) 1 k/k F 1 n(k) 1 This talk 3 examples of the physics of strongly correlated systems Tuning atom-atom interactions: 1) Superfluidity in strongly interacting Fermi gas: a high Tc system Tuning the trapping potential: 2) The Mott-Insulator transition in a 3D periodic potential 3) Superfluidity in 2D Bose gas: the Berezinski-Kosterlitz-Thouless transition

9. Fermi superfluid and Bose-Einstein condensate of Molecules Fermions with two spin states with attractive interaction BEC of molecules BCS fermionic superfluid Dilute gases: Feshbach resonance Interaction strengthBound state No bound state

10. k/k F 1 n(k) 1 k/k F 1 n(k) 1 Tuning atom-atom interactions

11. At low T, interactions are characterized by the s wave scattering length a C6C6 n bound states n+1 bound states n-1 bound states 1 a / b |U> + |U> |U’> + |U’> r A Fano-Feshbach resonance brings a new (closed) channel in the collision process, and it “ mimicks ” the entrance of a new bound state. Atom-atom interactions

12. E B =-h 2 /ma 2 Lithium 6 Feshbach resonance

13. BCS phase condensate of molecules BEC-BCS Crossover No bound state Bound state interacting fermions interacting fermions

14. Optical trap Experimental approach Experimental approach Cooling of 7Li and 6Li 1000 K: oven 1 mK: laser cooling 10  K: evaporative cooling in magnetic trap Tuning the interactions in optical trap Evaporation in optical trap

15. Condensates of molecules JILA: 40 K 2 6 Li 2 :Innsbruck ENS 6 Li 2 MIT 6 Li 2 7 Li Also Rice, Duke, Tokyo, Swinburne, 6 Li 2

16. Experiments on BEC-BCS crossover Interaction between molecules and lifetime of molecules Unitarity regime: k F a>>1 ? Probing fermionic superfluidity Momentum distribution of particles Superfluidity with imbalanced Fermi spheres Can we measure the excitation gap ? JILA, Innsbruck, MIT, ENS, Rice, Duke, Swinburne, Tokyo,… A high temperature superfluid: T c =0.2 T F

17. Balanced Fermi gas ( ) x numerical factor ExperimentENS ( 6 Li)0.42(15) Rice ( 6 Li)0.46(5) JILA( 40 K)0.46(10) Innsbruck ( 6 Li)0.27(10) Duke ( 6 Li)0.51(4) TheoryBCS0.59 Astrakharchik0.42(1) Perali0.455 Carlson0.42(1) Haussmann0.36 Determination of  First example of a quantum simulator ! Universal equation of state of the unitary Fermi gas

18. Rotating classical gas velocity field of a rigid body Rotating a quantum macroscopic object macroscopic wave function: In a place where, irrotational velocity field: The only possibility to generate a non-trivial rotating motion is to nucleate quantized vortices (points in 2D or lines in 3D) Feynman, Onsager Vortices now all have the same sign, imposed by the external rotation Direct proof of superfluidity: classical vs. quantum rotation

19. MIT 2005: Vortex lattices in the BEC-BCS Crossover [G] Magnetic Field Energy [MHz] M. Zwierlein A.. Schirotzek C. Stan C. Schunk P. Zarth W. Ketterle Science 05 Direct proof of superfluidity BEC side BCS side

20. Superfluidity in a freely expanding gas ! MIT 2005

21. 2: Creating artificial crystals

22. A laser beam (far detuned from resonance to avoid spontaneous emission) creates a conservative potential on the atoms: typical depth: microkelvin to millikelvin Optical lattice: periodic potential created by interfering beams Here period: 27  m Cs atoms in an optical lattice ENS, 1998 Periodic optical potential

23. tunnelling if tunnelling is large enough, the coherence between the microBECs at each site is maintained Time-of-flight experiment: time of flight optical lattice I. Bloch et al., 2002 release the atoms from the lattice let the various clouds overlap Constructive interference as for Bragg diffraction by a grating A Bose-Einstein condensate in a lattice

24. Munich 2002 V 0 = 10 Er V 0 = 13 Er V 0 = 16 Er Large lattice depth: repulsive interactions dominate over tunnelling coherence is lost! Bose-Hubbard problem Fisher et al. 1989, Jaksch et al. 1998 The system evolves to a state with a fixed number of atoms/site The superfluid – Mott insulator transition

25. First step: generate an interaction between adjacent sites? Use Pauli principle + on-site interactions: super-exchange Observed in a double well potential by the Mainz group (2007) Second step: achieve a low enough temperature in a lattice: : spin up : spin down Square lattice: N é el ordering Triangular lattice: frustration Use atoms with a large dipole (chromium, Stuttgart) Use ground state polar molecules Next challenge: produce antiferromagnetic order

26. 3: Cold gases in low dimensions

27. High T c superconductivity Quantum wells and MOS structures also Quantum Hall effect, films of superfluid helium, … Key words of two-dimensional physics: absence of true long range order (no BEC stricto sensu) existence of a new kind of phase transition (Kosterlitz-Thouless) No spin-statistics theorem, and existence of parastatistics: any-ons Non abelian physics: towards topological quantum computing ?? Two-dimensional Quantum Physics

28. T 0 TcTc superfluid normal 1971-73, 2D gas of bosons algebraic decay of g 1 exponential decay of g 1 Microscopic origin of this phase transition: quantized vortices Vortex: point where, around which rotates by Around a vortex: The Berezinski-Kosterlitz-Thouless mechanism

29. However: Superfluidity in 2 dimensions T 0 TcTc superfluid normal Berezinski and Kosterlitz –Thouless 1971-73 Bound vortex- antivortex pairs Proliferation of free vortices Unbinding of vortex pairs Superfluid transition observed with liquid helium films by Bishop-Reppy, 1978 Superfluidity in 2D: Berezinski-Kosterlitz-Thouless Mechanism

30. Producing a cold 2D gas Paris: superposition of a harmonic magnetic potential + periodic potential of a laser standing wave Two independent planes 2D experiments at MIT, Innsbruck, Oxford, Florence, Boulder, Heidelberg, Gaithersburg, Paris (Dalibard’s group) Time of flight Producing a 2 Dimensional Cold Gas

31. coldhot sometimes: dislocations! uniform phase 0 0  Dislocation = evidence for free vortices fraction of images showing a dislocation 0.5 0.751 0.1 0.2 0.3 0.4 temperature control (arb.units) ENS Dalibard group 2D physics revealed by matter-wave interferometry

32. New experimental methods: Image a many-body wavefunction with micrometer resolution Measure correlation functions Photoemission spectroscopy to measure Fermi surface and single particle excitations Time-dependent phenomena in 1, 2, and 3 D With cold atoms, one can simulate several many-body Hamiltonians Bosons, fermions, and mixtures Pairing with mismatched Fermi spheres, exotic phases Periodic potential or disordered (Anderson localization) Gauge field with rotation or geometrical phase Non abelian Gauge field for simulating the Hamiltonian of strong interactions in particle physics Quantum Hall physics and Laughlin states Prospects

34. BoBo B E kin a>0 a<0 BEC of Molecules Method BEC of Molecules Method Recipe : in region a<0, cool a gas of fermions below 0.2T F Slowly scan across resonance towards a>0 Typically : 1000 G to 770 G in 200 ms This produces molecules with up to 90% efficiency ! Reversible process ! Entropy is conserved. If T< 0.2 T F, BEC of molecules JILA ENS BEC of molecules