Fractional Quantum Hall states in optical lattices

1. Fractional Quantum Hall states in optical lattices Eugene Demler Collaborators: Anders Sorensen Mikhail Lukin

2. Bose-Einstein Condensation Cornell et al., Science 269, 198 (1995) Ultralow density condensed matter system Interactions are weak and can be described theoretically from first principles

3. New Era in Cold Atoms Research Focus on systems with strong interactions • Optical lattices • Feshbach resonances • Rotating condensates • One dimensional systems • Systems with long range dipolar interactions

4. Lindeman criterion suggests that the vortex lattice melts when . Cooper et al., Sinova et al. Vortex lattice in rotating condensates Pictures courtesy of JILA http://jilawww.colorado.edu/bec

5. Composite fermions Moore-Read Laughlin Vortex lattice Read-Rezayi QH states in rotating condensates Fractional quantum Hall states have been predicted at fast rotation frequencies: Wilkin and Gunn, Ho, Paredes et.al.,Cooper et al,…

6. QHE in rotating BEC It is difficult to reach small filling factors - scattering length Current experiments: Schweikhard et al., PRL 92:40404 (2004) Small energies in the QH regime require very low temperatures This work: Use optical lattices

7. Atoms in optical lattices Theory: Jaksch et al. PRL 81:3108(1998) Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004);

8. Bose Hubbard Model. Mean-field Phase Diagram. M.P.A. Fisher et al., PRB40:546 (1989) N=3 Mott 4 Superfluid N=2 Mott 0 2 N=1 Mott 0 Superfluid phase Weak interactions Mott insulator phase Strong interactions

9. Mott insulator Superfluid t/U Superfluid to Insulator Transition Greiner et al., Nature 415 (02)

10. Outline How to get an effective magnetic field for neutral atoms 2. Fractional Quantum Hall states of bosons on a lattice 3. How to detect the FQH state of cold atoms

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

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

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

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

15. 99.98% 95% Hall states in a lattice Is the state there?  Diagonalize H (assume J « U = ∞, periodic boundary conditions) N=2N  N=2  N=3  N=4  N=5 Dim(H)=8.5·105 ?

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

17. Superfluid Mott Hall Detection Ideally: Hall conductance, excitations Realistically: expansion image

18. Future • - Quasi particles • Exotic states • - Magnetic field generation Conclusions • Effective magnetic field can be created for cold • neutral atoms in an optical lattice • Fractional Quantum Hall states can be realized • with atoms in optical lattices • Detection remains an interesting open problem

19. y x Two dimensional rotating BEC Single particle Hamiltonian System rotating at frequency System at rest E E’ -2 -1 1 2 -2 -1 0 1 2 0