290 likes | 459 Views
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.
E N D
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
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
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).
"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:
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?
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
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.)
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
classical object quantum coherent fluid n0 vortices Rotation in QM:vortices
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
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
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
Beff in z-dirwithc =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
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
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
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
This is why not 5% of equations of derivation shown but with some patience …
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
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 Eqq2, so (-0) diverges! No BEC at T=0!!! • Finite systems finite (-0) ln (NV)
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
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
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)
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
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
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.
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
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)
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