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Jairo Sinova 19 th of September 2002

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

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Jairo Sinova 19 th of September 2002

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

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

  3. non-interacting particles: • dB n-1/3particles can be treated classically S. Bose  single state macro-occupied • dB~ n-1/3 + bosons • statistically, bosons tend to “cluster” dB~ n-1/3 + fermions  Fermi sea • statistically fermions repel (Pauli exclusion prin.) A. Einstein Effects of weak interactions in a BEC • 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 4He) N.N. Bogoliubov BEC:in the beginning A.J. Leggett, Rev. Mod. Phys. 73, 307 (2001).

  4. "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  • BEC predicted in 1924 by • Bose and Einstein 1997 • 1st Trap low-energy atoms: optical molasses • with laser traps. Still not cold enough! http://www.colorado.edu/physics/2000/bec/index.html Chu, Cohen, Phillips 2001 • 2nd Trap those cold atoms in a magnetic trap and • do evaporative cooling: T~ nK!! Cornell, Ketterle, Weiman BEC with ultra-cold atoms: abbreviated history of its making 2. superfluid 4He discovered (1938) Key steps to BEC with a cold gas of atoms:

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

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

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

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

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

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

  11. 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? vortex deformation fast rotating regime Rotating BEC’s: experiments II groups some highlights single vortex decay and upper critical rotation Paris MIT vortex lattice decay and nucleation Oxford vortex nucleation and decay JILA

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

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

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

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

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

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

  18. collective excitation spectrum Quadratic dispersion of collective mode! 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 2nd order +diagonalize using Bogoliubov transformation

  19. Bogoliubov approx: fraction outside condensate Consequences of E(q) q2 • BEC, no rotation:E(q) q finite • condensate fraction 0 … but here, q const. and Eqq2, so (-0) diverges! No BEC at T=0!!! • Finite systems  finite (-0)  ln (NV)

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

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

  22. Quantum (T=0) fluctuation of vortex positions • (combine B.A. and eff. field theory): • 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)] Melting of the vortex lattice • No divergent fluctuations of (G) in B.A. • (density-wave order parameter of vortex lattice)

  23. Summary of results in rotating BECs • LLL: Rapid rotation, weak interactions • No BEC in rapidly rotating 2D Bosons • …in thermodynamic (Nv) limit • E(q)1/q2  (- 0) ln(NV) • 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. A growing field • Why is BEC interesting to CM: spherical cow of many-body systems GP Eq. ~80% of literature Financial support by work done in collaboration with: A. H. MacDonald C. B. Hanna J. C. Diaz-Velez OUTLOOK: take home message • 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 Hc2 BUT: • Bosons here are not charged so effective field does not get screened by currents. Vortex physics cleaner. • 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. Quantum Hall Regime ng  h; kBT < h _ _ Rotating Trap Parameters Leggett: Rev. Mod. Phys. 73, 307 (2001) • Mean-Field Energy: n g = 2 kHz • Transition Temperature: kTc = 10 kHz • Trap Frequency: 0  100 Hz • Rotation Frequency:   0-100 Hz

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

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

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