1. 1 Vortex configuration of bosons in an optical lattice Boulder Summer School, July, 2004 Congjun Wu Kavli Institute for Theoretical Physics, UCSB Ref: Phys. Rev. A 69, 43609 (2004). Collaborators: H. D. Chen, J. P. Hu, S. C. Zhang

2. 2 Topological defect of the superfluid (SF) order parameter. Motivation (I) : vortices in rotating traps P. Engels, et al. PRL. 89, 100403 (2002) Weak coupling vortex: particle density is suppressed in the core. Time of flight image: x SF orderdensity

3. 3 Motivation (II) : SF- Mott-Insulator transition How about the vortex near the SF-Mott insulator (MI) transition? Superfluid Mott insulator M. Greiner et al., Nature (London) 415, 39 (2002 ).

4. 4 S. Tung, V. Schweikhard, and E. A. Cornell, cond-mat/0607697 Rotating optical lattice through holographic method

5. 5 S. Tung, V. Schweikhard, and E. A. Cornell, cond-mat/0607697 Vortex pinning to the lattice potential Weak optical potential, square optical lattice. How about in the strong coupling limit (near SF-MI) transition?

6. 6 Bosons in optical lattices without rotation N=2.00 N=2.05 N=1.95 Lobes of commensurate MI phases. Particle-like, hole-like, and particle-hole symmetric SF phases. M. P. A. Fisher et al., Phys. Rev. B 40, 546 (1989).

7. 7 (t/U=0.02) Superfluid vortex with nearly Mott-insulating core =1.95 =2.05 Hole-like vortex particle-like vortex (t/U=0.02) C. Wu, et al., Phys. Rev. A, 69, 43609 (2004); cond-mat/0211457.

8. 8 Bosons in rotating optical lattices Vector potential for Coriolis force : Neglect the trapping potential and the centrifugal potential. Josephson junction array in the magnetic field at low temperatures.

9. 9 Mean field approximation: Guzwiller Decouple to single site problems: The MF ground state wavefunction: The spatially dependent SF order: Valid at the small t/U.

10. 10 Competition between SF and MI phases t eff /U is frustrated as approaching the vortex core vortex core is more strongly correlated than the bulk area. superfluid vortex with Mott-insulating core.

11. 11 Evolution of the vortex particle density distribution Core particle density approaches the nearest integer number. Real space superfluid- Mott phase transition. Vortex core with maximum particle density in the hole-like case.

12. 12 Evolution of the vortex particle density distribution

13. 13 Strong to weak coupling vortex configuration t/U= Particle density profile along a cut passing the central plaquette. =1.95

14. 14 Phase diagram with next nearest repulsion Charge density wave (CDW) and Super-solid phases appear at the half-integer filling and small t/U. Map to the spin 1/2 Heisenberg model with the Ising or XY anisotropy.

15. 15 Vortex with the CDW core (“meron”) W/U=0.1, t/U=0.023 and =1.5. Similar to the antiferromagnetic vortex core in underdoped high T c superconductors.

16. 16 Possible experiments Vortex core size on the optical lattice is about 2 microns. Focus the probe laser beam to this size and scan the lattice to determine the particle density distribution. Raman photon-association to detect the percentage of Mott region. (D. J. Heinzen’s group) The Josephson junction array system in the magnetic field. From the electrical field distribution, it is possible to determine the vortex charge.

17. 17 Summary The vortex core is a more strongly coupled region compared to the bulk area. Near the SF-MI transition, the vortex core is nearly Mott insulating and the core particle density approaches the nearest commensurate value. As t/U increases, the vortex evolves from the strong coupling configuration to weak coupling one.