Martti M. Salomaa (Helsinki Univ. of Tech.)

Transcript

1 Martti M. Salomaa (Helsinki Univ. of Tech.)

2 Phase Phase contrast contrast images images (in (in situ) situ) Time Time of of flight flight (TOF) (TOF) images images BEC =

3 Trapped atomic gases Neutral atoms: Li, Na, K, Rb, Cs, Cr, Yb Hyperfine spin F (e.g., 6 Li atoms, F = 9/2, 7/2) Magnetic or Optical confinement Trapping potential 3-dimensional harmonic trap Typically, Axial symmetry Inter-atomic interaction a: s-wave scattering length (e.g., a = 2.75nm in 23 Na) By using Feshbach resonance, a ±

4 Observing statistics Hulet et al., Science (2001) Bosons in situ image Fermions Condensation in real & momentum space! Fermi degeneracy

5 : 4 He, 3 He,, N 0 << N 0 O( ) c radians/sec (Khalatnikov, 1961) BEC N 0 N, 0 O(μm) c radians/sec 2 Kosterlitz-Thouless BEC-BCS

6 k = Bogoliubov

7 Collective excitations of of BEC BEC Dipole excitation (center-of-mass motion) Quadrupole excitation Stamper-Kurn et al., PRL 81, 500 (1999)

8 s ( ) (Heisenberg ) : Gross-Pitaevskii (GP) Hartree-Fock Self-consistent potential : Gorkov 1 = (r 1, 1 )

9 , an 1/3 << 1, a s n (, (, ) ) Schrodiger

10 =

11 = Schrodiger

12 , (q ) (r,, z) z (q z ) : q z = 0 qq = 00 qq = = 1, 1, 1 1 qq = 2, 2, 2 2 Breathing mode Dipole mode Quadrupole mode

13 Collective excitations of of BEC BEC Jin et al., PRL 77, 420 (1996) Dipole excitation (center-of-mass motion) Breathing mode Quadrupole mode Quadrupole excitation (center-of-mass motion) Stamper-Kurn et al., PRL 81, 500 (1999)

14 Jin et al., PRL 77, 420 (1996) Breathing mode Quadrupole mode T/T c < 0.4

15 1 (1) Landau : q 1 + q q q 1 + q (2) Beliaev q 1 q + q q 1 active

16 m (m = q ) Beliaev :, Beliaev Landau T D. D. S. S. Jin Jin et et al., al., PRL PRL (1997) (1997) T 4

17 k = Bogoliubov

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19 : L. Onsager (1949) BEC ( ) w : (r = 0)

20 2 ( ) (l=2, )

21 Madison et al., PRL ( 00) 2 2

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23 T = 0 + core-mode z z : q z z q = -1 mode + - Kelvin Kelvin

24 ( ) Biot-Savart : : ( : ) Kelvin : W. Thomson (1880) c.f.,

25 (Kelvin) Beliaev : 1. ±2 Kelvin mode -1 (q z ) 2. q = -2 Kelvin mode q = -1 Counter-rotating quadrupole mode (q = -2 q -2 q z ) z ) (q = -1, -1, q q z ) z ) + (q = -1, -1, q q z ) z ) Kelvin mode Beliaev co-rotating mode (co-rotating mode )

26 : Bretin et al., PRL (2003) q = +2 q = 2 q = = 24 s -1 q = 2-2 = 57 s -1 : : +2 (0)

27 q = 2 q = +2 Beliaev?

28 z z (r-z ) Spectra at T = 70nK: 70% condensate fraction Co-rotating Mode: Decay rate: +2 ~ 0.05 r ~ 30s -1 Experiment: +2 ~ 24 ± 5 s -1 Counter-rotating Mode: Decay rate: -2 ~ 0.11 r ~ 65s -1 Experiment: +2 ~ 57 ± 10 s -1

29 Counter-rotating Counter-rotating mode mode Co-rotating Co-rotating mode mode Beliaev Landau damping

30 Counter-rotating Counter-rotating mode mode Co-rotating Co-rotating mode mode Kelvin mode phonon mode counter-rotating mode Kelvin mode Beliaev

31 ( ) ( ) (Landau ) Counter-rotating mode Co-rotating mode (Kelvin mode) Kelvin mode Beliaev Landau Beliaev Beliaev : : TM et al, PRL 90, ( 03) Spectroscopic method

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33 : Tkachenko k TK :, : : Baym and Chandler (1983), Sonin (1987) Tkachenko (c ) phonon c: Phonon velocity (compressibility), > ck, (Landau )

34 Coddington et al., PRL 91, (2003), inward flow ( / ), fluid flow Top View Resulting fluid flow Magnus force Fluid flow Magnus force, transverse

35 after 500 ms 1650 ms =

36 1st TK mode 1st BR mode Tkachenko 2nd breathing 1st breathing TM et al., PRL ( 04)

37 (i) (1) (Tkachenko) (2) (ii) ( ) (iii) 6-fold symmetric Tkachenko mode Coddington et al., PRL 2003 c2 = r Tkachenko mode (?)

38 k = Bogoliubov

39 Fermi Atoms: 6 Li 40 K up & down species: two different hyperfine states e.g. 6 Li Pairing of spin up and down fermions interacting via a tunable 2-body interaction: Feshbach Resonance Experiments: Jin (JILA) Ketterle (MIT) Grimm (Innsbruck) Hulet (Rice) Thomas (Duke) Salamon (ENS) Typical Numbers: Trap freq. ~ Hz N ~ 10 6 E F ~ 100 nk -1 μk T ~ E F 1/k F ~ 0.3 μm TF radius ~ 100 μm

40 s-wave scattering length vs vs Magnetic field: 6 6 Li Li 1 0 bound state BEC tightly bound molecules pair size BCS cooperative Cooper pairing pair size x Scattering length a Atoms form stable molecules Magnetic Field: B [G] -20 BCS-BEC Crossover

41 gap BCS BCS to to BEC BEC crossover at at T=0 T=0 A.J. Leggett, Karpacz Lectures (1980) BCS gap eqn. plus μ renormalization Qualitative theory Fermi energy chemical potential μ momentum distribution n(k) Binding energy μ = E b /2 = -1/2ma 2 Crossover: Engelbrecht, Randeria, and Sa de Melo, PRB 55, (1997)

42 Vortex lattices in in the BEC-BCS crossover Zwierlein, Abo-Shaeer, Schirotzek, Schunck, and Ketterle, Nature 435, 1047 (2005)

43 Caroli-de Gennes Matricon state Quasiparticles passing through vortex center experience the -phase shift Local Local density density of of states states -phase shift Core-bound state Caroli-de Gennes-Matricon (CdGM) state Density Density at at T = 00 unoccupied Vortex center Quantum depletion around Feshbach resonance occupied Quantum depletion BEC BCS

44 Vortex Lattice in in Balanced System BEC side: 1/k F a = 0.16 / = 0.7 Pair Pair potential potential Density Density Visualization Visualization of of vortices vortices

45 Vortex Lattice in in Balanced System Pair Pair potential potential BCS side: 1/k F a = -0.4 / = 0.7 Density Density Invisible Invisible vortices vortices

46 BEC BCS, BEC p ( s ) 3 He Majorana zero modes 2 dimensional p-wave SF