Condensed matter physics in dilute atomic gases S. K. Yip Academia Sinica

possible to cool and trap dilute atomic gases Atoms used: 7 Li 23 Na 87 Rb 1 H He* Yb 6 Li 40 K Some typical numbers: number of atoms (final) (mostly) peak density n < cm - 3 distance between particles ~ 10 4 A (dilute gas) size of cloud ~  m temperatures: down to nK

nucleus + [ closed electronic shell ] + e _ nucleon N odd atom = boson N even fermion 7 Li 23 Na 87 Rb 1 H : Bosons 6 Li 40 K : Fermions f = I s = I + 1/2 hyperfine spin nuclear spin electron spin m f = - f, - f + 1, …., f - 1, + f (integers for bosons, half-integers for fermions) _ alkalis

6 Li s = 1/2, i = 1; f = 1/2, 3/2

Magnetic Trap (most experiments): magnetic moment - . B ( r ) | B ( r ) | increasing from the center trapped not trapped U r typically can trap only one species ( else loss due to collisions ) effectively scalar ( spinless ) particles

laser spin degree of freedom remains Optical Trap: U ( r ) = -  (  ) E 2 ( r,  ) 1 2  (  ) > 0 if red detuned ex g (  <  res ) atoms attracted to strong field region (c.f. driven harmonic oscillator)

Identical particles bosons fermions many particle wavefunctions: symmetric antisymmetric

can occupy the same single particle states at sufficiently low T macroscopic occupation Bose-Einstein Condensation BOSONS:

Macroscopic wavefunction (common to all condensed particles)  (r, t) (c.f Schrodinger wavefunction) Supercurrent: Phase gradient  supercurrent

Quantized vortices: well defined at any position r  If |  |  then  unique up to 2n   0  0 2 

Rotating superfluid: if  = constant, then not rotating (no current) rotating    constant but circulation quantized  quantized vortices  0  2  

[MIT, Science, 292, 476 (2001)]

FERMIONS:

FERMIONS Exclusion principle: T=0 Particles filled up to Fermi energy Normal Fermi gas (liquid) Generally NOT superfluid Momentum space: Fermi sphere Single species (can be done in magnetic traps):

FERMIONS T=0 Momentum space: Fermi sphere Two species: need optical trap still not much interesting unless interacting

6 Li s = 1/2, i = 1; f = 1/2, 3/2 }  (need optical trap)

Can be superfluid if attractive interaction : Cooper pairing (Bardeen, Cooper, Schrieffer; BCS) k  -k  all k’s near Fermi surface  Underlying mechanism for superconductivity (in perhaps all superconductors)

How to get strong enough attractive interaction in dilute Fermi gases Feshbach Resonances: B  (1 2) (others) B res

Hydridization  level repulsion

Lowering of energy  attractive interaction

B  Two particles no coupling:  continuum closed channel molecule continuum

with coupling: effective attractive interaction between   fermions Bound state

eventually BEC of molecules effective attractive interaction between   fermions Bound state BCS pairing

Smooth crossover from BCS pairing to BEC (Leggett 80)

Experimental evidence: [MIT, Nature, 435, 1047 (2005)] (resonance)

New possibilities: unequal population [c.f. superconductor in external Zeeman field: pair-breaking ] Smooth crossover is destroyed ! ( Pao, Wu, Yip; 2006) uniform superfluid state unstable in shaded region N BF homogenous mixture

Many potential ground states for the shaded region experiments suggest phase separation near resonance another likely candidate state: Larkin-Ovchinnikov/ domain-walls (c.f.  -junctions in SFS) not yet found Finite T phase diagram open question Interesting interacting system even when it is not superfluid (non-Fermi liquid behaviours?)

Many other topics not covered: atoms in periodic lattice (c.f. solid!) “random” potential multicomponent (spin) Bosonic superfluids low dimensional systems (e.g 1D) rapidly rotating Bose gas (maximum number of vortices ?) tunable parameters, often in real time and many more opportunities!!