Condensed matter physics in

dilute atomic gases

S. K. YipAcademia Sinica

possible to cool and trap dilute atomic gases

Atoms used:

7Li 23Na 87Rb 1H He* Yb6Li 40K

Some typical numbers:

number of atoms 104 106 108

(final) (mostly)

peak density n < 10 14 cm-3

distance between particles ~ 104 A (dilute gas)

size of cloud ~ m

temperatures: down to nK

nucleus + [ closed electronic shell ] + e _

nucleon N odd atom = boson N even fermion

7Li 23Na 87Rb 1H : Bosons6Li 40K : Fermions

f = I s = I + 1/2hyperfine spin nuclear spin electron spin

mf = - f, - f + 1, …. , f -1, + f (integers for bosons, half-integers for fermions)

_

alkalis

6Li 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 ) = - ( ) E2 ( r , )1

2

( ) > 0 if red detunedex

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 samesingle 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:

)(2

** mi

J

ie|| 2||J

Phase gradient supercurrent

Quantized vortices:

)(r

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

6Li 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)

Bres

Hydridization level repulsion

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

eventuallyBEC 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

BFhomogenousmixture

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!!