Martti M. Salomaa (Helsinki Univ. of Tech.) 1. 2. 3.
Phase Phase contrast contrast images images (in (in situ) situ) Time Time of of flight flight (TOF) (TOF) images images BEC =
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 ±
Observing statistics Hulet et al., Science (2001) Bosons in situ image Fermions Condensation in real & momentum space! Fermi degeneracy
: 4 He, 3 He,, N 0 << N 0 O( ) c2 10 12 radians/sec (Khalatnikov, 1961) BEC N 0 N, 0 O(μm) c2 10 3-10 5 radians/sec 2 Kosterlitz-Thouless BEC-BCS
k = Bogoliubov
Collective excitations of of BEC BEC Dipole excitation (center-of-mass motion) Quadrupole excitation Stamper-Kurn et al., PRL 81, 500 (1999)
s ( ) (Heisenberg ) : Gross-Pitaevskii (GP) Hartree-Fock Self-consistent potential : Gorkov 1 = (r 1, 1 )
, an 1/3 << 1, a s n (, (, ) ) Schrodiger
=
= 1 2 1 1 Schrodiger
, (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
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)
Jin et al., PRL 77, 420 (1996) Breathing mode Quadrupole mode T/T c < 0.4
1 (1) Landau : q 1 + q q q 1 + q (2) Beliaev q 1 q + q q 1 active
m (m = q ) Beliaev :, Beliaev Landau T D. D. S. S. Jin Jin et et al., al., PRL PRL (1997) (1997) T 4
k = Bogoliubov
: L. Onsager (1949) BEC ( ) w : (r = 0)
2 ( ) (l=2, )
Madison et al., PRL ( 00) 2 2
T = 0 + core-mode z z : q z z q = -1 mode + - Kelvin Kelvin
( ) Biot-Savart : : ( : ) Kelvin : W. Thomson (1880) c.f.,
(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 )
: Bretin et al., PRL (2003) q = +2 q = 2 q = +2 +2 = 24 s -1 q = 2-2 = 57 s -1 : -2 2 +2 : +2 (0)
q = 2 q = +2 Beliaev?
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
Counter-rotating Counter-rotating mode mode Co-rotating Co-rotating mode mode Beliaev Landau damping
Counter-rotating Counter-rotating mode mode Co-rotating Co-rotating mode mode Kelvin mode phonon mode counter-rotating mode Kelvin mode Beliaev
( ) ( ) (Landau ) Counter-rotating mode Co-rotating mode (Kelvin mode) Kelvin mode Beliaev Landau Beliaev Beliaev : : TM et al, PRL 90, 180401 ( 03) Spectroscopic method
: Tkachenko k TK :, : : Baym and Chandler (1983), Sonin (1987) Tkachenko (c ) phonon c: Phonon velocity (compressibility), > ck, (Landau )
Coddington et al., PRL 91, 100402 (2003), inward flow ( / ), fluid flow Top View Resulting fluid flow Magnus force Fluid flow Magnus force, transverse
after 500 ms 1650 ms =
1st TK mode 1st BR mode Tkachenko 2nd breathing 1st breathing TM et al., PRL ( 04)
(i) (1) (Tkachenko) (2) (ii) ( ) (iii) 6-fold symmetric Tkachenko mode Coddington et al., PRL 2003 c2 = r Tkachenko mode (?)
k = Bogoliubov
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. ~ 20-100 Hz N ~ 10 6 E F ~ 100 nk -1 μk T ~ 0.05-0.1 E F 1/k F ~ 0.3 μm TF radius ~ 100 μm
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 x10 3 20 10 0-10 Scattering length a -1 650 Atoms form stable molecules 834.15 Magnetic Field: B [G] -20 BCS-BEC Crossover
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, 15153 (1997)
Vortex lattices in in the BEC-BCS crossover Zwierlein, Abo-Shaeer, Schirotzek, Schunck, and Ketterle, Nature 435, 1047 (2005)
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
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
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
BEC BCS, BEC p ( s ) 3 He Majorana zero modes 2 dimensional p-wave SF