1. Jairo Sinova 19 th of September 2002 Spinning a BEC away: quantum fluctuations, rotating BECs and 2D vortex matter Reference: J. Sinova et al, Phys. Rev. Lett. 89, 030403 (2002) J. Sinova et al, cond-mat/0209374

2. Outline 1.BEC: basics, its making, directions, basic theory of condensation and quantum fluctuations 2.Rotating the BEC: vortex formation, nucleation, and decay (experiments) 3.Rapidly-rotating weak-interacting limit: QHE for bosons 4.Quantum fluctuations and Bogoliubov theory in the fast rotating limit 5.Quantum field theory of vortex lattice 6.Conclusions and outlook

3. dB ~ n -1/3 + fermions BEC: in the beginning A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001). non-interacting particles: dB  n -1/3 particles can be treated classically dB ~ n -1/3 + bosons increase the statistical tendency to condense deplete part of the condensate through quantum fluctuations (zero-point motion) single macro-occupied state still OK if interactions weak enough (not the case for 4 He) Effects of weak interactions in a BEC N.N. Bogoliubov S. Bose A. Einstein –statistically, bosons tend to “cluster”  single state macro-occupied  Fermi sea –statistically fermions repel (Pauli exclusion prin.)

4. BEC with ultra-cold atoms: "From a certain temperature on, the molecules condense without attractive forces, that is, they accumulate at zero velocity. The theory is pretty but is there also some truth to it ?" - Albert Einstein 1.BEC predicted in 1924 by Bose and Einstein 2. superfluid 4 He discovered (1938) Key steps to BEC with a cold gas of atoms: 1 st Trap low-energy atoms: optical molasses with laser traps. Still not cold enough! http://www.colorado.edu/physics/2000/bec/index.html 1997 Chu, Cohen, Phillips abbreviated history of its making 2 nd Trap those cold atoms in a magnetic trap and do evaporative cooling: T~ nK!! Cornell, Ketterle, Weiman 2001

5. BEC: new directions BEC as the most tunable many-body system (condensed matter physics,...) J.R. Anglin and W. Ketterle, Nature 416, 211 (2002) what can you do with a gaseous BEC?what can’t you do with a gaseous BEC? Atom Lasers, QM benchmark (atomic optic physics,...) coherent states precision studies QM testing (images from MIT group) (Rice group) collapse dynamics quantum phase transition superfluidity, QM vortices

6. BEC theory I: mean field T<<T c H-F ansatz: Gross-Pitaevskii equation: Func. min. of dilute limit Statistical field theory formulation: coherent-state path integral An exact representation of the many-body problem Action:

7. BEC theory II: gaussian fluctuations (Bogoliubov App.) consider small fluctuations around the GP ground state Action:  ck for small k   (k), free particle, for large k Dispersion: less than 1% in most BECs (for 4 He it is 90%!) Quantum depleted fraction (T=0):

8. Outline 1.BEC: basics, its making, directions and possibilities 2.Rotating the BEC: vortex formation, nucleation, and decay (experiments) 3.Rapidly-rotating weak-interacting limit: QHE for bosons 4.Quantum fluctuations and Bogoliubov theory in the fast rotating limit 5.Quantum field theory of vortex lattice 6.Conclusions and outlook

9. Rotation in QM: vortices classical object quantum coherent fluid n  0 vortices

10. Rotating BEC’s: experiments I if not stirred rapidly enough no vortex: no quadrupole surface mode can be excited K.W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Vortex Formation of a Stirred BEC, Phys. Rev. Lett 84, 806 (2000) large optical spoon stirring fast enough: vortex lattice nucleation

11. Rotating BEC’s: experiments II MIT vortex lattice decay and nucleation groupssome highlights Paris single vortex decay and upper critical rotation Oxford vortex nucleation and decay JILA vortex deformation fast rotating regime MANY interesting questions: 1-How are vortices nucleated — is there other ways besides surface excitations? 2-How do the vortices interact and how do they form the vortex arrays? 3-What is the stability of these vortex arrays and lattices? 4-How do quantum fluctuation affect the vortex-lattice state (rapidly rot. limit, QHE) ? 5-How do the vortex lattices and individual vortices decay? 6-Are the observed effects dynamic or equilibrium dominated?

12. Outline 1.BEC: basics, its making, directions and possibilities 2.Rotating the BEC: vortex formation, nucleation, and decay (experiments) 3.Rapidly-rotating weak-interacting limit: QHE for bosons 4.Quantum fluctuations and Bogoliubov theory in the fast rotating limit 5.Quantum field theory of vortex lattice 6.Conclusions and outlook

13. Rotating BEC’s – H eff How to treat a rotating system?: go to rotating frame B eff in z-dir with  c = 2  reduced radial confinement 2D bosons + effective B field Bosonic QHE Rapid rotating limit

14. QHE 101: 2D particles in a strong B field f is analytic in z : zeros of f are the vortices of state  LLL and their positions determine f completely! n=0 LLL Landau levels are macroscopically degenerate Lowest Landau Level approx. B=0 E k

15. Conclusion  N/N V < 6 Vortex Fluid N/N V > 6 Vortex Lattice Theory studies – exact diagonalization N.R. Cooper, N.K. Wilkin, and J.M.F. Gunn, Phys. Rev. Lett. 87, 120405 (2001)

16. Outline 1.BEC: basics, its making, directions and possibilities 2.Rotating the BEC: vortex formation, nucleation, and decay (experiments) 3.Rapidly-rotating weak-interacting limit: QHE for bosons 4.Quantum fluctuations and Bogoliubov theory in the fast rotating limit 5.Quantum field theory of vortex lattice 6.Conclusions and outlook

17. 5% of equations of derivation shown This is why not but with some patience …

18. LLL Bogoliubov theory of vortex lattices in the unconfined limit 1. GP solution: Abrikosov vortex lattice, use magnetic Bloch-state representation 2. Bogoliubov approx.= fluctuations to 2 nd order +diagonalize using Bogoliubov transformation Quadratic dispersion of collective mode! collective excitation spectrum

19. Consequences of E(q)  q 2 Bogoliubov approx: fraction outside condensate BEC, no rotation: E(q)  q  finite condensate fraction 0 … but here,  q const. and E q  q 2, so ( - 0 ) diverges! No BEC at T=0!!! Finite systems  finite ( - 0 )  ln (N V )

20. Outline 1.BEC: basics, its making, directions and possibilities 2.Rotating the BEC: vortex formation, nucleation, and decay (experiments) 3.Rapidly-rotating weak-interacting limit: QHE for bosons 4.Quantum fluctuations and Bogoliubov theory in the fast rotating limit 5.Quantum field theory of vortex lattice 6.Conclusions and outlook

21. Effective LLL field theory: vortex positions LLL:  completely determined by location of zeros (vortices) z i =x i +iy i : vortex lattice sites u i =u xi +u yi : fluctuations about z i After Fourier-transforming: Results from  S: quadratic dispersion: Positional LRO: quasi-ODLRO: Expand to quadratic order: (LLL  KE is constant) LLL limit  0 limit

22. Melting of the vortex lattice No divergent fluctuations of  (G) in B.A. (density-wave order parameter of vortex lattice) Lindemann criterion:melting at ~8 Exact diagonalizations: melting at ~6 [N.R. Cooper, N.K. Wilkin, and J.M.F. Gunn, Phys. Rev. Lett. 87, 120405 (2001)] Quantum (T=0) fluctuation of vortex positions (combine B.A. and eff. field theory):

23. Summary of results in rotating BECs LLL: Rapid rotation, weak interactions No BEC in rapidly rotating 2D Bosons …in thermodynamic (N v  ) limit E(q)  1/q 2  ( - 0 )  ln(N V ) Algebraic-decaying quasi-ODLRO at T=0 Two approaches: Quantum Theory of Vortex Lattice State Bogoliubov approx. in LLL Melting of vortex lattice ~8 (Exact diagonalizations give ~6) J. Sinova et al, Phys. Rev. Lett. 89, 030403 (2002) J. Sinova et al, cond-mat/0209374

24. OUTLOOK: take home message A. H. MacDonald C. B. Hanna Financial support by work done in collaboration with: J. C. Diaz-Velez GP Eq. ~80% of literature A growing field Why is BEC interesting to CM: spherical cow of many-body systems Simplicity: possibility of full understanding of outstanding problems in CM Many open issues (in rot. BECs) : Vortex decay (T vs QM) Vortex formation and interactions Meta-stability: “superfluidity” Multi-component systems: Skyrmion physics

25. SIMILAR TO TYPE-II SC WITH H CLOSE TO H c2 BUT: 1.Bosons here are not charged so effective field does not get screened by currents. Vortex physics cleaner. 2.In a superconductor there are other degrees of freedom around - bound states in vortex cores, phonons etc. Inelastic interactions with these other degrees of freedom cause the system to behave classically - quantum coherence effects are lost.

26. Rotating Trap Parameters Mean-Field Energy: n g = 2 kHz Transition Temperature: kT c = 10 kHz Trap Frequency:  0  100 Hz Rotation Frequency:   0-100 Hz Leggett: Rev. Mod. Phys. 73, 307 (2001) Quantum Hall Regime ng  h  ; k B T < h  _ _

28. Spinning a BEC away: quantum fluctuations, rotating BECs and 2D bosonic vortex matter Jairo Sinova 3 th of September 2002 Financial support by Reference: J. Sinova et al, Phys. Rev. Lett. 89, 030403 (2002)

29. Rotating BEC’s: experiments II JILA Rotating freq. is 98% of trap frequency ! stirring methodsgroupssome highlights large optical spoon several small optical spoons stir before BEC MIT vortex lattice decay and nucleation Paris Oxford single vortex decay and upper critical rotation large magnetic spoonvortex nucleation and decay vortex deformation fast rotating regime