1. Quantum Gases: Past, Present, and Future Jason Ho The Ohio State University Hong Kong Forum in Condensed Matter Physics: Past, Present, and Future HKU and HKUST, Dec 18-20

2. Where we stand What’s new Fundamental Issues Challenges

3. A decade since discovery of BEC : Still expanding rapidly Discoveries of new systems, new phenomena, and new technique keep being reported in quick succession. Highly interdisciplinary -- (CM, AMO, QOP, QI, NP) New Centers and New Programs formed all over the world. England, Japan, Australia, CIAR, US ( MURI&DARPA ) Puzzling phenomena being to emerge in fermion expts Worldwide experimental effort to simulate strongly correlated CM systems using cold atoms

4. Bosons and Fermions with large spins F=I+J alkali atoms Hyperfine spin J=1/2 I J e Spin F=1, F=2 bosons: Spin F=1/2, 3/2, 5/2, 7/2, 9/2 fermions

5. B Magnetic trap Spinless bosons and fermions Atoms “lose” their spins!

6. B Magnetic trap Mixture of quantum gases: D.S. Hall, M.R. Matthews, J. R. Ensher, C.E. Wieman, and E.A. Cornell PRL 81, 1539 (1998) Pseudo-spin 1/2 bosons: Ho and Shenoy, PRL 96

7. Optical trapping: Focused laser BEC or cold fermions All spin states are trapped, Spin F=1, F=2 bosons: Spin F=1/2, 3/2, 5/2, 7/2, 9/2 fermions T.L.Ho, PRL 1998

8. Quantum Gases Atomic PhysicsCondensed Matter Physics Quantum Optics Nuclear Physics Quantum Information BEC High Energy Physics

9. Quantum Gases B BB BF FF F 3D 2D 1D 0D single trap lattice stationary fast rotating U(1) Magnetic trap, spins frozen S0(3) Optical trap, spins released systemenvironmentssymmetryinteraction

10. 1996 Discovery of BEC! 1997 Mixture of BEC and pseudo spin-1/2 Condensate interference collective modes solitons 1998 Spin-1 Bose gas (Super-radiance) Bosanova Bragg difffration, super-radience, Superfluid-Mott oscillation 1999 Low dimensional Bose gas (Vortices in 2-component BEC) 2000 (Vortices in BEC, Slow light in BEC) 2001 Fast Rotating BEC, Optical lattice, BEC on Chips 2002 Quantum degenerate fermions (Spin dynamics of S=1/2 BEC, Coreless vortex in S=1 BEC, evidence of universality near resonance) 2003 Molecular BEC, (Spin dynamics of S=1 BEC, noise measurements) 2004 Fermion pair condensation! (pairing gap, collective mode) BEC-BCS crossover, 2005 Vortices in fermion superfluids, discovery of S=3 Cr Bose condensate, observation of skymerion in S=1 Bose gas. 2006 Effect of spin asymmetry and rotation on strongly interacting Fermi gas. Boson-Fermion mixture in optical lattices.

11. New Bose systems: “spin”-1/2, spin-1, spin-2 Bose gas, Molecular Bose gas. (BEC at T=0) Un-condensed Bose gas: Low dimensional Bose gas, Mott phase in optical lattice Strongly Interacting quantum gases: Atom-molecule mixtures of Bosons near Feshbach resonance Fermion superfluid in strongly interacting region Strongly interacting Fermions in optical lattices Possible novel states: Bosonic quantum Hall states, Singlet state of spin-S Bose gas, Dimerized state of spin-1 Bose gas on a lattice. Fermion superfluids with non-zero angular momentum

12. Often described as experimental driven, but in fact theoretical ideas are crucial. Bose and Einstein, Laser cooling, Evaporative cooling

13. What is new ? A partial list: Bosons and Fermions with large spins Fast Rotating Bose gases Superfluid Insulator Transition in optical lattices Strongly Interacting Fermi Gases

14. Question: How do Bosons find their ground state?

15. Conventional Bose condensate : all Bosons condenses into a single state. How do Bosons find their ground state? Question:

16. What happens when there are several degenerate state for the Bosons to condensed in? G: Number of degenerate statesN: Number of Bosons

17. What happens when there are several degenerate state for the Bosons to condense in? G: Number of degenerate statesN: Number of Bosons Pseudo-spin 1/2 Bose gas: G =2

18. Spin-1 Bose gas : G=3, G<<N G: Number of degenerate statesN: Number of Bosons

19. Spin-1 Bose gas : G=3, G<<N Bose gas in optical lattice: G ~N G: Number of degenerate statesN: Number of Bosons

20. Spin-1 Bose gas : G=3, G<<N Bose gas in optical lattice: G ~N Fast Rotating Bose gas: G>>N G: Number of degenerate statesN: Number of Bosons

21. Effect of spin degeneracy on BEC Only the lowest harmonic state is occupied => a zero dimensional problem Spin-1 Bose Gas Effect of spin degeneracy on BEC A deep harmonic trap

22. Spin-1 Bose Gas Spin dynamics of spin-1 Bose gas A deep harmonic trap Hilbert space

23. Effect of spin degeneracy on BEC Spin-1 Bose Gas Effect of spin degeneracy on BEC A deep harmonic trap Under spin rotation, rotates like a 3D Cartesean vector. : 3D rotation

24. Conventional condensate : C>0

25. Exact ground state : C>0 = Ho and Yip, PRL, 2004

26. Average the coherrent state over all directions Relation between singlet state and coherent state x y z Because The system is easily damaged

27. Transformation of singlet into coherent states as a function of External field and field gradient: If the total spin is non-zero Bosonic enhancement

28. Transformation of singlet into coherent states as a function of External field and field gradient: If the total spin is non-zero Bosonic enhancement

29. Transformation of singlet into coherent states as a function of External field and field gradient: If the total spin is non-zero

30. Transformation of singlet into coherent states as a function of External field and field gradient: If the total spin is non-zero With field gradient

31. S=2 Cyclic state S=3 Spin biaxial Nematics

32. A geometric representation : Generalization of Barnett et.al. PRL 06 & T.L.Ho, to be published

33. Cycle Tetrahedron S=2 Cubic S=4 Octegonal S=3 Icosahedral S=6 T.L. Ho, to be published

34. Rotating the Bose condensate Generating a rotating quadrupolar field using a pair of rotating off-centered lasers condensate K. W. Madison, F. Chevy, W. Wohlleben, J. Dalibard PRL. 84, 806 (2000)

36. The fate of a fast rotating quautum gas : Superfluidity ----> Strong Correlation Vortex lattice Overlap => Melting Quantum Hall Boson Fermion Normal Quantum Hall In superconductors

37. as Rotating quantum gases in harmonic traps Electrons in Magnetic field trap external rotation A remarkable equivalence

38. , n>0, m>0. m E No Rotation : Two dimensional harmonic oscillator

39. , n>0, m>0. As Angular momentum states organize into Landau levels !, m E

40. m E

41. m E condensate Mean field quantum Hall regime: in Lowest Landau level

42. m E Strongly correlated case: interaction dominated

43. E. Mueller and T.L. Ho, Physical Rev. Lett. 88, 180403 (2002)

44. Simulate EM field by rotation:Eric Cornell’s latest experiment cond-mat/0607697 TL Ho, PRL 87, 060403(2001) V. Schweikhard, et.al. PRL 92, 040404 (2004) (JILA group, reaching LLL)

45. Boson + Fermion Fermion quantum Hall

46. Strongly interacting Fermi gases

47. Cooling of fermions Pioneered by Debbie Jin Motivation: To reach the superfluid phase Depends only on density For weakly interacting Fermi gas To increase Tc, use Feshbach resonance, since Holland et.al. (2001)

48. Dilute Fermi Gas Normal Fermi liquid Weak coupling BCS superfluid : S-wave scattering length Weak coupling

49. Dilute Fermi Gas Normal Fermi liquid Weak coupling BCS superfluid : S-wave scattering length What Happens?

50. Key Properties: Universality (Duke, ENS) Evidence for superfluid phase: Projection expt: Fermion pair condensataion -- JILA, MIT Specific heat -- Duke Evidence for a gap -- Innsbruck Evidence for phase coherence -- MIT BEC -- BCS crossover is the correct description Largest Origin of universality now understood

51. BCS Molecular BEC Universality : A statement about the energetics at resonance

52. How Resonance Model acquire universality has to hybridize with many pairs. If is large -- strong hybridization, then has relatively little weight in the pair! Small effect of means universality !

53. Two channel Model Single Channel model:

54. Origin of universality Scattering amplitude: (from both single and two channel model) r = effective range Question: what happen to scattering on Fermi surface Wide resonance Narrow resonance Bruun & Pethick PRL 03 Petrov 04 Diener and Ho 04 Strinati et.al 04 Eric Cornell, email

55. In two channel model: Small closed channel contribution pair size are given by interparticle spacing single channel description ok universal energy density

56. Current Development: Unequal spin population Rotation

57. c To quantum Hall regime Melting of vortex lattice Single vortex

58. Other possible Fermion superfluids: P-wave Fermion superfluids. B B o a>0 a<0 Molecular condensate Fermion Superfluid Ho and Diener, to appear in PRL Optiuum phase

59. Many quantum phenomenon observed: Condensate interference collective modes solitons Bosanova Bragg difffration, super-radience, Superfluid-Mott oscillation Engineering quantum states in optical lattices, vortices and spin-dynamics of spin-1/2 Bose gas, phase fluctuation in low dimensional Bose gas, spatial fragmention of BEC on chips, slow light in Bose gases, large vortex lattice, Skymerion vortices in spin-1 Bose gas, spin dynamics of spin-1 and spin-2 Bose gas, dynamics in optical lattices

60. Unique Capability for Lattice Quantum Gases Solid State environment without disorder Simulate electro-magnetic field by rotation Great Ease to change dimensionality Great Ease to change interactions Major Incentive: Observation of Superfluid-insulator transition -- a QPT in a strongly correlated system Realization of Fermion Superfluid using Feshbach resonance Exciting Prospects: Novel States due to unique degrees of freedom of cold atoms Bose and Fermion superfluids with large spin Quantum Hall state with large spin Lattice gases in resonance regime

61. 0 1 2 3

62. 0 1 2 3

63. 0 1 2 3

64. 0 1 2 3

65. 0 1 2 3

66. Superfluid : Mott :

67. Superfluid State : ++ ODLRO

68. Superfluid State : ++ ODLRO

69. Mott State

70. Resists addition of boson require energy U, hence insulating

71. Nature, 419, 51-54 (2002)

72. Figure 2 Absorption images of multiple matter wave interference patterns. These were obtained after suddenly releasing the atoms from an optical lattice potential with different potential depths V 0 after a time of flight of 15 ms. Values of V 0 were: a, 0 E r ; b, 3 E r ; c, 7 E r ; d, 10 E r ; e, 13 E r ; f, 14 E r ; g, 16 E r ; and h, 20 E r. M. Greiner et.al, Nature 415, 39 (2002) M. Greiner, O. Mandel. Theodor, W. Hansch & I. Bloch,Nature (2002) Observation of Superfluid-insulator transition

73. Phase diagram of Boson-Hubbard Model

75. Part IC Current experiments

76. I. Bloch, et.al, PRA72, 053606 (2005) Ketterle et.al, cond-mat/0607004 Esslinger, PRL 96, 180402 (2006) Sengstock et.al. PRL 96, 180403 (2006) Expts involving superfluid-insulator transitions: F-B mixture Fermions in optical lattice, 2 fermions per site

77. Band insulator 2 atoms per site 2 to 3 bands occupied 0 ETH Experiment: very deep lattice, less than two toms per site

80. 2 fermions Per site

81. Part I: Why cold atoms for condensed matter? A. Major developments in CM and Long Standing Problems B. The Promise of cold atoms C. Current experimental situation Part II: Necessary conditions to do strongly correlated physics: Quantum Degeneracy and method of detection: A.The current method of detecting superfluidity in lattices is misleading B.B. A precise determination of superfluidity => illustration of far from quantum degeneracy in the current systems. Part III: Solid state physics with ultra-cold fermions: A. Metallic and semi-conductor physics with cold fermions B. Studying semiclassical electron motions with cold fermions

82. Part II Necessary conditions for studying strongly correlated physics: * Quantum Degeneracy * Method of Detection: * Quantum Degeneracy

83. Condition for quantum degeneracy Condition for BEC :

84. Free space Lattice

85. Free space Quantum degeneracy Lowest temperature attainable: Optical lattice

86. I. Bloch, et.al, PRA72, 053606 (2005) Ketterle et.al, cond-mat/0607004 Esslinger, PRL 96, 180402 (2006) Sengstock et.al. PRL 96, 180403 (2006) Current method of identifying superfluidity: sharpness of n(k) F-B mixture Fermions in optical lattice, 2 fermions per site

87. However, a normal gas above Tc can also have sharp peak! Diener, Zhao, Zhai, Ho, to be published.

88. I. Bloch, et.al, PRA72, 053606 (2005) Ketterle et.al, cond-mat/0607004 Esslinger, PRL 96, 180402 (2006) Sengstock et.al. PRL 96, 180403 (2006) Current method of identifying superfluidity: sharpness of n(k) F-B mixture Fermions in optical lattice, 2 fermions per site

89. Part II Necessary conditions for studying strongly correlated physics: * Quantum Degeneracy * Method of Detection: * Method of Detection

90. An accurate method for detecting superfluidity: Visibility Reciprocal lattice vector Not a reciprocal lattice vector

91. DZZH, to be published T=0 visibility 2nd Mott shell

93. Main message: Current Experiments in optical lattice are far from quantum degeneracy Need new ways to cool down to lower temperature Need reliable temperature scale

94. Finite temperature effect becomes important More intriguing physics of quantum critical behavior can be expected