2. Strong correlations and quantum vortices for ultracold atoms in rotating lattices Murray Holland JILA (NIST and Dept. of Physics, Univ. of Colorado-Boulder)

3. Acknowledgments Stefano Giogini Marilu Chiofalo Rajiv Bhat Lincoln Carr John Cooper Rajiv Bhat Brandon Peden Ron Pepino Brian Seaman Jochen Wachter Special thanks: Erich Mueller

4. Outline 1.Released momentum distribution of a Fermi gas in the BCS-BEC crossover 2.Strongly interacting atoms in a rotating optical lattice [Two recent papers: look on cond-mat or PRL]

11. Dashed: before field ramp Solid: after field ramp Calculated momentum distribution at unitarity: homogeneous gas

12. Calculated release energy: homogeneous gas

13. Column integrated momentum distribution in a trap BCS BEC Solid: Theory Crosses: Experiment [Regal et al. PRL 95, 250404 (2005).]

14. Calculated release energy in a trap Blue: Leggett ground state + expansion Red circles: Regal/Jin experiment [PRL 95, 250404 (2005)] Green: two-body physics

15. Rotating a BEC Observation of vortex formation (Cornell, JILA) BEC rotated using  Stirring (ENS,JILA)  Two-component condensates ( JILA) Main features - Quantized vortices with depleted cores - Formation of vortex lattice Both have gotten close (  >0.99  ) to the Quantum Hall regime Can optical lattices help enter the strongly correlated QH regime?

16. The Quantum Hall effect in cold gases For a 2D system at  =  Very similar to one-body Hamiltonian for 2D electrons in a magnetic field

17. Ground State is the variational Laughlin state of general form N=5 atoms Contact interactions q=2

18. Quantum Phase Transitions Quantum Phases (Greiner et.al., 2001) a. Superfluid - Hopping dominates - Particles delocalized - Coherent - Number density on each site uncertain b. Mott Insulator -Onsite energy dominates -Particles localized -No phase correlation between sites -Integer number of particles on each site t/U

19. BECs in Optical Lattices Rotating Bose-Einstein Condensates BEC in a 2D rotating lattice

20. BEC in a rotating 2D lattice experimentally realized at JILA Scheme proposed by -J. Reijnders et al, PRL 93, 060401 (2004) -H. Pu et al, PRL 94, 190401 (2005) First experimental realization at JILA Lattice spacing ~ 10  Filling factor (particles/site)~10 3- 10 4 BEC in Rotating lattice schematic (Cornell, 2005)

21. Outline Theory Hamiltonian Methods Cross-checks Center depletion MI-SF phase diagram Results Interaction effects Symmetries QPTs Summary and future work

22. System described using modified Bose-Hubbard Hamiltonian Particle field operator expanded using site specific annihilation/ creation operators and a Wannier basis Two approximations -Tight binding approximation (Only nearest neighbor terms considered) -Only lowest Bloch band occupied I. Hopping IV. RotationII. Chemical potential III. Interaction energy Hamiltonian in a rotating frame of reference

23. System described using modified Bose-Hubbard Hamiltonian I.Hopping: KE associated with particles hopping from one site to the next II.Chemical potential part of onsite energy III.Interaction part of onsite energy: Proportional to number of pairs IV.Rotation: Preferred hopping in one direction. K ij is a dimensionless geometric factor  indicates angular velocity of rotating frame. x i and y i are coordinates from center of rotation.  indicates angular velocity of rotating frame. x i and y i are coordinates from center of rotation.

24. Methodology: Diagonalize and examine ground state Write down many-body Hamiltonian KE, PE, interaction, rotation Second quantize Hamiltonian Modified Bose-Hubbard Hamiltonian Specify basis set Product basis using n Fock states per site Exactly diagonalize Examine groundstate Use energy eigenvalues and a toolbox of operators (current, number density and discrete rotational symmetry) to examine the ground state

25. Center depletion seen in odd lattice Axis of rotation goes through central site in a 3X3 lattice Center number density gets depleted once rotation sets in

26. Results Interaction restricts current flow Discrete rotational symmetry states Quantum phase transitions between states of different symmetry

27. Results Interaction restricts current flow Discrete rotational symmetry states Quantum phase transitions between states of different symmetry

28. Interaction restricts current flow Current decreases with increasing interaction as particles find it more difficult to cross each other Same currents for 1 and 3 particles indicate Fermionization Two-state approximation correctly describes strongly repulsive particles in a lattice Number of particles in lattice J 12 12 34

29. Results Interaction restricts current flow Discrete rotational symmetry states Quantum phase transitions between states of different symmetry

30. Discrete rotational symmetry Schematic for 4X4 lattice 4X4 square lattice: Each site can have at most one particle (strongly repulsive bosons) The ground state must be consistent with the symmetry of the lattice(4-fold here). A Discrete Rotational Operator R exists such that R commutes with the lattice potential and with H. Energy eigenstates are simultaneous eigenstates. Symmetry states are labeled by index m

31. Discrete changes in energy as system adopts higher rotational energy symmetries 1 Particle in 4X4 lattice Discrete jumps in energy derivative due to level crossings Discontinuous changes in ground state symmetry n=1

32.  =0.1: Rotation yet to enter the system Current Number density Number density and current  =0.2: Rotation sets in the center Current Number density Current Number density  =0.4: Number density moves out  =0.8: Direction of current in rotating frame changes Current Number density

33. Results Interaction restricts current flow Discrete rotational symmetry states Quantum phase transitions between states of different symmetry

34. Avoided level crossings for transitions between same symmetry states Symmetry of the system dependent on filling Particles prefer to be spread out 4 particles in 16 sites

35. Level crossings for transitions between different symmetry states Filling not commensurate with 4- fold symmetry of system Energy level crossings for many particles as function of a Hamiltonian parameter is a non-trivial result characterizing Quantum Phase Transitions 5 particles in 16 sites

36. Summary Our current work on rotation is at the junction of interesting areas of research - rotating BECs and BECs in optical lattices. Future work: Rotating lattices play a two fold role by enhancing interactions and restricting number of particles per vortex and may help to open up strongly correlated regimes (FQHE) for exploration. Momentum distribution measurements have provided detailed quantitative information revealing limitations and strengths of the Leggett ground state Shape independent, time-dependent theory with no adjustable parameters