1. Atoms in optical lattices and the Quantum Hall effect Anders S. Sørensen Niels Bohr Institute, Copenhagen

2. Intro Atomic physics: simple, well understood Extremely good experimental control of atoms (lasers) => Let us try to use atoms as a tool to solve other peoples problems BEC with cold atoms

3. What they do 1. Cool and trap atoms with lasers 2. Atom = magnet => trap with magnetic fields 3. Evaporative cooling => BEC 4. Release from trap; look at velocity distribution.

4. Features 1. Many body system with well known simple properties V ij = g  (r i -r j ) 2. Properties highly tunable (in real time)

5. Optical trapping Dielectric attracted into electric field +Q -Q  F Laser beam attracts dielectrics, cells, molecules, atoms....

6. Low D condensates 1D condensates 2D condensates A. Görlitz et al., Phys. Rev. Lett. 87, 130402 (2001)

7. But... Condensates are simple Mean field theory: Interactions among particles are weak Not very challenging theoretically Strong correlations, strong interactions => challenging

8. Strong Interactions 1: Rapid rotation

9. Rotating condensates MIT Vortices:

10. Quantum Hall in rotating BEC Wilikin and Gunn, Ho, Paredes et al.,.......  rot ~  vib => N Votices ~N Atoms => Fractional quantum Hall MzMz 012-2 E No Rotation MzMz 012-2 E With Rotation Rotation: Many degenerate states => Interactions dominate Interactions still very weak

11. Strong Interactions 2: Feshbach Resonances

12. Feshbach resonances V ij = g  (r i -r j ) => Change g r i -r j E Bound state Move bound state up and down => Dramatically change interaction

13. Feshbach resonances Bosons: three body loss => no good Fermions: VERY nice experiments (cooling harder for fermions)

14. Strong Interactions 3: Optical lattices

15. Two lasers => Standing wave Atoms trapped in planes 4 lasers = > atoms trapped in tubes 6 lasers => cubic lattice

16. Optical lattices Atomic potential J U Tunneling: J~exp(-I....) => can be tuned V 0 ~I (Bose-Hubbard model) Strong interactions: atoms confined to small volume => U Big

17. State preparation (Bose-Hubbard model) Load atoms in to lattice, cool, look at ground state => doesn’t work; can’t cool in lattices  E<<V 0 V0V0 Load cold atoms into lattice. Adiabatic loading => Constant Entropy

18. Mott insulator (Bose-Hubbard model) J>>U U>>J SuperfluidOne atom at each site J~U Quantum phase transition Load BEC Have been done in 1, 2, and 3D

19. Detection Velocity distribution = Fourier transform of density matrix ~ Probes long range order of off-diagonal elements Superfluid Mott Not the most convincing probe (did also probe excitation spectrum + density correlations)

20. Tonks Giradeau Gas One dimensional Bose gas, strong interactions ~ non interacting fermions Tune lattice potential => go from one regime to the other

21. Achievements - Bosons Mott insulator Tonks Giradeau “Entangling operations” Collapse and revival of matter wave field Spin dynamics Molecule formation Several experiments with weaker interactions

22. Fermions Harder to work with experimentally. Cooling harder (use Bosons to cool). Fermi degenerate gas loaded into lattice, observed Fermi surface, dynamics, interactions. Confinement induced change of collision properties (molecules always bound) More experiments underway

23. Extensions Now: atoms with a few spin states jumping around in lattice Extensions: Magnetism Bose-Fermi mixtures Quantum Hall Three particle interactions...... May or may not be feasible

24. Magnetism Mott regime U >> J Atoms have spin (several internal states) Interaction dependent on internal state (or use spin dependent tunneling) Include virtual processes:

25. Fractional Quantum Hall states in optical lattices Collaborators: Harvard Physics Eugene Demler Mikhail D. Lukin Mohammad Hafezi Martin Knudsen (NBI)

26. Fractional quantum Hall effect Tsui, Störmer, and Gossard, PRL 48, 1561 (1982) V/I= (2D)

27. Theory Magnetic flux:  = B · A = N  ·  0  0 = h/e Laughlin: if N  =m ·N  incompressible quantum fluid Quasi particles: charge e/m, anyons Particle+m fluxes  composite particle (boson)  condenses Goal: produce these states for cold bosonic atoms (m=2) Energy gap to excited state ∆E. Phase transition k B T~ ∆E

28. Requirements/outline 1. Effective magnetic field 2. What does the lattice do? 3. How do we get to the state? 4.How do we detect it?

29. Magnetic field See also Jaksch and Zoller, New J. Phys. 5, 56 (2003) 1. Oscillating quadropole potential: V= A ·x·y ·sin(  t) 2. Modulate tunneling x y

30. Magnetic field See also Jaksch and Zoller, New J. Phys. 5, 56 (2003) 1. Oscillating quadropole potential: V= A ·x·y ·sin(  t) 2. Modulate tunneling x y

31. Magnetic field See also Jaksch and Zoller, New J. Phys. 5, 56 (2003) 1. Oscillating quadropole potential: V= A ·x·y ·sin(  t) 2. Modulate tunneling Proof:  : Flux per unit cell 0≤  ≤1

32. Lattice: Hofstadter ButterflyE/J  ~B Particles in magnetic field Continuum: Landau levels B E Similar  « 1

33. Hall states in a lattice Is the state there?  Diagonalize H (assume J « U = ∞, periodic boundary conditions) 99.98% 95% Dim(H)=8.5·10 5 ?  N=2  N=3  N=4  N=5 N=2N 

34. Energy gap  N=2  N=3  N=4  N=5 N=2N 

35. Making the state Adiabatically connect to a BEC  Quantum Hall BEC Mott-insulator ?

36. Making the state U0U0 4 Atoms, 6  6 lattice,  =2/9=0.222 U 0 /J Overlap 98% U 0 /J

37. Detection Ideally: Hall conductance, excitations Realistically: expansion image Hall SuperfluidMott

38. Requirements/outline 1. Effective magnetic field 2. What does the lattice do? 3. How do we get to the state? 4.How do we detect it? Conclusion (1) Future - Quasi particles - Exotic states - Magnetic field generation

39. Conclusion Ultra cold atoms: Flexible many body system with well understood and controllable parameters Beginning to enter into the regime of strong coupling strong correlations: lattices, Feshbach resonances More complex system can be engineered Open question how much is feasible Quantum Hall: tunneling only turned on at short instances => reduced energy gap, super lattice hard. Not very near future