1. Dieter Jaksch, 21.9.2006 Irreversible loading of optical lattices Rotation of cold atoms University of Oxford Christopher Foot

2. Dieter Jaksch, 21.9.2006 Outline Superfluidity – tested by the response to rotation TOP trap  rotating elliptical potential Observation of the scissors mode Nucleation of vortices Superfluid gyroscope Ring trap for cold atoms Rotating optical lattice  artificial B-field

3. Dieter Jaksch, 21.9.2006 BEC 10 5 rubidium atoms. Temperature ~ 50 nK Density ~ 10 14 cm -3 Magnetic coils and vacuum cell TOP trap Time-orbiting potential

4. Dieter Jaksch, 21.9.2006 TOP trap

5. Dieter Jaksch, 21.9.2006 Shape of BEC in a TOP trap Pancake (oblate) rather than a cigar (prolate), or `baguette-shaped’ as in Ioffe traps.

6. Dieter Jaksch, 21.9.2006 u The velocity field  gradient u Hence velocity field is irrotational u Circulation around a closed contour is quantised Zero circulation = irrotational flow Non-zero circulation = vortices ħ ħ ħ Quantised circulation in a quantum fluid

7. Dieter Jaksch, 21.9.2006  22 Trap tilted adiabatically to angle  Trap suddenly rotated by -2  Cloud oscillates about new equilibrium position Excitation of scissors mode c.f. torsion pendulum

8. Dieter Jaksch, 21.9.2006 Described in book: Bose-Einstein Condensation Pitaevskii & Stringari Oxford University Press 2003 Scissors mode results

9. Dieter Jaksch, 21.9.2006 Types of flow

10. Dieter Jaksch, 21.9.2006 Rotation of the confining magnetic potential BEC Impart angular momentum using rotating elliptical potential

11. Dieter Jaksch, 21.9.2006 Nucleation of a single vortex

12. Dieter Jaksch, 21.9.2006  Critical frequency  c  2 u Line II : stability boundary for the quadrupole II branch.  Vortices nucleated below  c. Thresholds for vortex nucleation Eleanor Hodby et al = Rotation frequency maximum rotation freq.

13. Dieter Jaksch, 21.9.2006 Rotational also introduces ‘centrifugal’ term into the Hamiltonian Radial harmonic potential ‘Centrifugal’ term Radial trapping decreases as    

14. Dieter Jaksch, 21.9.2006  Critical frequency  c  2 u Line II : stability boundary for the quadrupole II branch.  Vortices nucleated below  c. Thresholds for vortex nucleation Eleanor Hodby et al = Rotation frequency maximum rotation freq.

15. Dieter Jaksch, 21.9.2006 Nucleation of a single vortex

16. Dieter Jaksch, 21.9.2006 Numerical simulation by Nilsen, McPeake & McCann, Queens University, Belfast Scissors mode + vortex = ‘Superfluid gyroscope’

17. Dieter Jaksch, 21.9.2006 Nucleation of an array of vortices Other experiments: ENS, MIT, JILA Nathan Smith Will Heathcote Chris Foot, Oxford

18. Dieter Jaksch, 21.9.2006 Observing the Tilting Mode ( side view of the vortex array )

19. Dieter Jaksch, 21.9.2006 Precession of angle of condensate with vortex lattice

20. Dieter Jaksch, 21.9.2006 Precession of angle of condensate with vortex lattice = 27.75 Hz = 8.4 ± 0.4 ħ

21. Dieter Jaksch, 21.9.2006 Outline Superfluidity – tested by the response to rotation TOP trap  rotating elliptical potential Observation of the scissors mode Nucleation of vortices Superfluid gyroscope Ring trap for cold atoms Rotating optical lattice  artificial B-field

22. Dieter Jaksch, 21.9.2006 RF-dressed magentic potentials Modify magnetic trap using RF radiation Proposed by: O. Zobay and B. Garroway, PRL 86 (2001), 1195-1198. Other Experiments: Helen Perrin, Paris Nord, France. Schmiedmayer Group: double well potential on an atom chip

23. Dieter Jaksch, 21.9.2006 x magnetic potential M F = -1 M F = 0 M F = +1 rf Modification of a magnetostatic trap by RF radiation Proposed by Zobay & Garraway PRL (2001) dressed-atom picture rf avoided crossings F=1 hyperfine level of Rb-87

24. Dieter Jaksch, 21.9.2006 x dressed-atom potential rf x magnetic potential M F = -1 M F = 0 M F = +1 rf Proposed by Zobay & Garraway PRL (2001) Modification of a magnetostatic trap by RF radiation F=1 hyperfine level of Rb-87

25. Dieter Jaksch, 21.9.2006 Contours of a quadrupole magnetic field div B  quadrupole coils ON |B|  constant Apply RF with TOP coils

26. Dieter Jaksch, 21.9.2006 Atoms trapped on a magnetic field contour B 0 rf = 0.3 G B 0 rf = 0.24 G B 0 rf = 0.18 G B 0 rf = 0.12 GB 0 rf = 0.06 G

27. Dieter Jaksch, 21.9.2006 z Weak radial confinement by the magnetic trap Squeeze atoms between two sheets of light Creates a thin sheet of atoms = 2D Bose gas BEC Two-dimensional trapping of Bose gas Physics of 2-D systems  z = 2 kHz   = 10 Hz

28. Dieter Jaksch, 21.9.2006 Combined optical and magnetic trap = ring trap Contours of constant magnetic potential Light sheets confine atoms to plane z = const. z x x rf

29. Dieter Jaksch, 21.9.2006 Trapping potential: Static + RF fields

30. Dieter Jaksch, 21.9.2006 Ring shaped cloud of atoms (March 2007) Eileen Nugent & Chris Foot: application to persistent currents

31. Dieter Jaksch, 21.9.2006 Rotating atoms in the ring trap Detection of current using scheme proposed in “Superfluid toroidal currents in atomic condensates ”, E. Nugent, D. McPeake and J.F. McCann, Phys Rev A 68, 063606. Persistent current 1. Original plug 2. Deform plug 3. Rotate deformation Bill Phillip’s team at NIST, Gaithersburg have reported seeing a persistent current in a recent preprint

32. Dieter Jaksch, 21.9.2006 Outline Superfluidity – tested by the response to rotation TOP trap  rotating elliptical potential Observation of the scissors mode Nucleation of vortices Superfluid gyroscope Ring trap for cold atoms (persistent current) Rotating optical lattice  artificial B-field

33. Dieter Jaksch, 21.9.2006 Overview of cold atoms/molecules BEC Dilute quantum gases: Fermi gas Quantum fluids: superfluid helium Atoms in optical lattices: Physics of strongly correlated systems Cold molecules Condensed Matter Physics Quantum Information Processing

34. Dieter Jaksch, 21.9.2006 Simulation of Condensed Matter Systems Hamiltonian of atoms in optical lattice = Hamiltonian of CMP system E.g. Fractional Quantum Hall Effect

35. Dieter Jaksch, 21.9.2006 Mathematical equivalence of rotation on cold atoms and the effect of a magnetic field on charged particles (electrons) Coriolis force: F = 2m v x  Lorentz force: F = q(E  v x B ) q B eff ↔ 2m  For electron, q  e Cyclotron frequency,  c  eB  2  rot m 

36. Dieter Jaksch, 21.9.2006 Effective magnetic fields via rotation Neutral atom in rotating frame Electron under magnetic field

37. Dieter Jaksch, 21.9.2006 Energy levels of a rotating 2-D harmonic oscillator 0 at rest 0123-2-3 0 1 2 3 4 5 6 0123-2-3 0 1 2 3 4 5 6 +1 rotating        

38. Dieter Jaksch, 21.9.2006 Landau levels 0123-2-3 0 1 2 3 4 5 6  2D harmonic oscillator levelsDegenerate Landau levels Energy levels of 2D harmonic oscillator Near degeneracy as    . Interactions mix single particle states  strongly correlated multi-particle states

39. Dieter Jaksch, 21.9.2006 Fractional quantum Hall states FQHE states predicted in BEC at fast rotation frequencies: Wilkin and Gunn, Ho, Paredes et al., Cooper et al,… Read-RezayiMoore-Read Composite fermionsLaughlin Vortex lattice Zoo of strongly correlated states Lindemann criterion suggests that the vortex lattice melts when N  Number of atoms N v Number of vortices ( cf. filling factor, )

40. Dieter Jaksch, 21.9.2006 Optical lattice in the rotating frame

41. Dieter Jaksch, 21.9.2006 Atoms in a rotating lattice Phase shift from hopping around one lattice cell is   Theory: R. Palmer & D. Jaksch, Phys. Rev. Lett. 96, 180407 (2006)   eB eff d 2    h h/e  d  = flux through loop h/e  flux quantum

42. Dieter Jaksch, 21.9.2006 The Hofstadter butterfly 0 1 E A = Area

43. Dieter Jaksch, 21.9.2006 The Hofstadter butterfly 0 1 E B E n  B E

44. Dieter Jaksch, 21.9.2006 The Hofstadter butterfly 0 1 E R.N. Palmer and D. Jaksch, Phys. Rev. Lett. 96, 180407 (2006)

45. Dieter Jaksch, 21.9.2006 High-field FQHE The optical lattice setup allows to explore parameter regimes which are not accessible otherwise  beyond mimicking condensed matter

46. Dieter Jaksch, 21.9.2006 Experiment in Oxford Microscope for quantum matter.

47. Dieter Jaksch, 21.9.2006 Two-dimensional rotating optical lattice High NA lens Confinement along z by two sheets of laser light (not shown). Funded by ESF EuroQUAM programme

48. Dieter Jaksch, 21.9.2006 Movie of rotating lattice Movie prepared by Ross Williams, Oxford.

49. Dieter Jaksch, 21.9.2006 Summary Scissors mode and vortices Superfluidity Magnetic trap + rf = ring potential for atoms in the dressed state persistent current? Rotating optical lattice gives term in atomic Hamiltonian analogous to an applied magnetic field of a charged particle (e.g. electron) Highly correlated quantum states as in Fractional Quantum Hall Effect Other experiments along the way?

50. Dieter Jaksch, 21.9.2006 Acknowledgments People: Chris Foot Eileen Nugent Ross Williams Amita Deb Ben Sheard Ben Fletcher Ben Sherlock Min Sung Yoon Marcus Gildemeister Herbert Crepaz Sara Al-Assam* Funding: Engineering and Physical Sciences Research Council European Science Foundation *Jointly supervised by Dieter Jaksch