1. Interacting Ultra Cold Atoms a brief overview Fei Zhou PITP, University of British Columbia at Quantum Nanoscience conference, Noosa Blue, Australia, Jan 23, 2006 Collaborators: I. Affleck (UBC), E. Demler (Harvard), Z. C. Gu (TsingHua), M. Snoek (Utrecht), C. Wu (UCSB), H. Zhai (TsingHua) $: Office of the Dean of Science, UBC NSERC, Canada Sloan foundation, New York

2. Ultra Cold atoms Many-body physics (condensed matter physics) Quantum information Storages and quantum computers Few body physics (Nuclear physics, Atomic physics) Field theories (emergent gauge fields, color superconductivity, Neutron star physics) Cosmology and gravity (Kimble mechanism, Unruh Radiation etc)

3. Topological quantum computer (Kitaev, 97)

4. Bosons in optical lattices S=0 bosons S=1 bosons

5. S=0 bosons in lattices In (a) and (b), one boson per site. t is the hopping and can be varied by tuning laser intensities of optical lattices; U is an intra-site interaction energy. In a Mott state, all bosons are localized. M. P. A. Fisher et al., PRB 40, 546 (1989); On Mott states in a finite trap, see Jaksch et al., PRL. 81, 3108-3111(1998). U Mott states ( t << U) Condensates (t >>U)

6. Phase diagrams  E(k,x) n=3 n=2 n=1 x n x 2 1 Atomic Mott states in a trap t n=1 SF or BEC n=2 n=3 n=4 n  Small t Large t

7.   Interacting S=1 bosons Stamper-Kurn et al., 98. Ho, 98; Ohmi & Machida, 98; Law,98.

8. Condensates of S=1 bosons (sodium type) (d>1) N(Q) Q x y z   (Zhou, 01)

9. Half vortices in BECs of sodium atoms In a half vortex, each atom makes a  spin rotation; a half vortex carries one half circulation of an integer vortex. A half vortex ring is also a hedgehog.  circulation y  spin rotation Z x y x The vortex is orientated along the z-direction; the spin rotation and circulating current occur in an x-y plane. z ring

10. Each site is characterized by two unit vectors, blue and red ones. a) nematic BECs (nBEC); b) Nematic mott insulators (NMI); c) Spin singlet mott insulators (SSMI). Mott states of Spin-One Bosons

11. Nematic-spin singlet transitions (Mott Insulators)  vs.  (proportional to hopping) is plotted here. (Snoek and Zhou, 03; Demler, et al., 03; Demler and Zhou, 02)  SSMI NMI

12. Fermions S=1/2 fermions in Optical Lattices S=3/2 fermions, quintet pairing, exotic vortices studied (Wu, Hu and Zhang, 2003-2006). Feshbach resonances with population difference (Experiments: MIT group, the Rice University’s Group and JILA group; Theory effeorts: Son and Stephanov, 2005; Pao et al.,2005; Sheehy and Radzihovsky; Gu, Warner and Zhou; …….) Lattice Feshbach resonances (Stability of Mott states and invasion of superfluidity, factorized superfluids in 1D; Wu, Gu and Zhou, 2005-2006) And more…...

13. S=1/2 Fermions in optical lattices (small band width) Neel Ordered Spin liquidsNeel ordered only at T=0

14. S=1/2 femions across Feshbach resonances B E (6Li) F=3/2 F=1/2 Resonances between state 1 of |1/2,1/2> and state 2 of |1/2,-1/2>. Only electron spins shown

15. Superfluids near Feshbach Resonances Binding energy B

16. The Chemical potential and Mol. Fraction at resonance For y <<1, at FbR the many-body states are INDEPENDENT of both two body parameters such as the bg scattering length, the magnetic moments and the many-body parameter: the fermi momentum. Wide resonance (Ho and Diener, 04)

17. Energy splitting and population imbalance A conventional quantum statistical system I E(k) k k Cold atoms

18. Energy Landscape 1: Negative Scattering Length (N fixed) (Gu, Warner and Zhou, 05)

19. Energy Landscape 2: positive scattering length

20. Energy Landscape 3: Near resonance

21. Phase Separation in a Constrained Subspace (i.e. population imbalance is conserved) M-H curve for a global ground state Critical population imbalance Phase separated states I M M I 1 N Gapless SF SF+N Gapless SF + N Negative scattering length Positive scattering length

22. Zwierlein et al., 2005 ; Also studied by the Rice group.

23. Superfluids of polarized fermi gases Resonances take place along the blue dashed line (in the “universal regime”). ( Son et al., 2005; also see Sheehy and Radzihovsky, 2005) LOFF Partially polarized F.L. Fully polarized F.L. SF + Fermi sea Splitting between two chemical potentials inverse of scattering length (p, -p+Q)

24. many important and exciting new issues in many-body cold atomic matter (magnetic superfluids & Mott states, topological phases, superfluids with population imbalance etc). Cold atomic matter might also be applied to understand various fundamental concepts/issues in other fields. There are a lot we can learn about/from cold atoms. Summary

25. Quantum Information storage (?)

26. A 3-bit Hamming code [3,1,3]

27. Quantum Error Correction Code Code Space Error Space Requirement for a QECC 1) Errors in different code words are distinguishable; 2) Subspace of errors are indistinguishable so that there will be no information leakage.

28. The Chemical potential and Mol. Fraction at resonance For y <<1, at FbR the many-body states are INDEPENDENT of both two body parameters such as the bg scattering length, the magnetic moments and the many-body parameter: the fermi momentum. Wide resonance (Ho and Diener, 04)

29. Ising symmetries

30. A usual superfluid with a thin fermi shell (b) of qausiparticles is unstable; the shell can be deformed into a crecent (a) by a moving condensate.

31. Absorption images of interference patterns as the laser intensity is increased (from a to h). (a-d) BECs and (g-h) Mott insulating states. (Greiner et al, 2002)