What have we learned so far about dilute Fermi gases?What have we learned so far about dilute Fermi gases?

AurelAurel BulgacBulgacUniversity of Washington, SeattleUniversity of Washington, Seattle

These slides will be posted shortly athttp://www.phys.washington.edu/~bulgac/

What is the What is the Holy GrailHoly Grail of this field?of this field?

FermionicFermionic superfluiditysuperfluidity!!

• Dilute atomic Fermi gases Dilute atomic Fermi gases TTcc ≈≈ 1010--1212 –– 1010-9 eVeV

Liquid Liquid 33He He TTcc ≈ 1010--77 eVeV

Metals, composite materials Metals, composite materials TTcc ≈ 1010--3 3 –– 1010--22 eVeV

Nuclei, neutron stars Nuclei, neutron stars TTcc ≈ 101055 –– 101066 eVeV

•• QCD color superconductivity QCD color superconductivity TTcc ≈ 10107 7 –– 10108 8 eVeV

Superconductivity and Superconductivity and superfluiditysuperfluidity in Fermi systemsin Fermi systems

units (1 eV ≈ 104 K)

• 1913 1913 KamerlinghKamerlingh OnnesOnnes

•• 1972 1972 BardeenBardeen, Cooper and , Cooper and SchriefferSchrieffer

•• 1973 1973 EsakiEsaki, , GiaeverGiaever and and JosephsonJosephson

•• 1987 1987 BednorzBednorz and Mullerand Muller

•• 1996 Lee, 1996 Lee, OsheroffOsheroff and Richardsonand Richardson

•• 2003 2003 AbrikosovAbrikosov, , GinzburgGinzburg and Leggettand Leggett

Memorable years in the history of Memorable years in the history of superfluiditysuperfluidity and and superconductivity of Fermi systemssuperconductivity of Fermi systems

Robert B. Laughlin, Nobel Lecture, December 8, 1998Robert B. Laughlin, Nobel Lecture, December 8, 1998

Topics to be covered Topics to be covered (not necessarily in this order):

• A single atom in magnetic fieldA single atom in magnetic field•• Two atoms in magnetic fieldTwo atoms in magnetic field•• Atomic trapsAtomic traps•• Basic parameters of Basic parameters of fermionicfermionic dilute atomic clouds dilute atomic clouds •• A review of a number of key experimental resultsA review of a number of key experimental results•• What theory tells us so far? What theory tells us so far?

1995 BEC was observed.2000 vortices in BEC were createdthus BEC confirmed un-ambiguously.

In 1999 DeMarco and Jin created a degenerate atomic Fermi gas.2002 O’Hara, Hammer, Gehm, Granada and Thomas observed expansion of a Fermi cloud compatible with the existence of a superfluid fermionic phase.2003 Jin’s, Grimm’s, Ketterle’s groups and othersultracold molecules, mBEC from Fermi gas2004 Jin’s group (and bit later Ketterle’s group too) announces the observation of the resonance condensation of fermionic atomic pairs ?Grimm’s group reports measurements of the gapThomas’ group reports measurements of the specific heat

HISTORY

One One fermionicfermionic atom in magnetic fieldatom in magnetic field

Regal and Jin Phys. Rev. Lett. 90, 230404 (2003)

Feshbach resonance

BSSVSSa

V

VPrVPrVVVpH

nzn

eze

Znehfhf

dZi

i

hfi

r

)(,

)()()(2

2

1100

2

1

2

γγ

µ

−=⋅=

+++++= ∑=

Tiesinga, Verhaar, StoofPhys. Rev. A47, 4114 (1993)

Atomicseperation

Energy

Eres

Eth

Channel coupling

Atomicseperation

Energy

Eres

Eth

Köhler, Gasenzer, Jullienne and BurnettPRL 91, 230401 (2003), inspired byBraaten, Hammer and Kusunokicond-mat/0301489

NB The size of the “Feshbach molecule”(closed channel state) is largely B-independenand smaller than the interparticle separation.

0

0 0

( ) 1 ( )

P r r aP r r r

>∝ >>

<

Halo Halo dimerdimer

11 12 2

21 22

211

222 0 0 0

0 0 0 00 0

012 1 1

21

2 20

u uv v

u u

2( ) sin , u ( ) sin( ), v ( ) ( )

( ) ( ), 2

max( , )exp( ) sin( )u( ) sin( ) ,

H Vk

V H

H k

H

rr r kr r rr r

rV r g r r r

r r rg ikr krr krrk

φ κ φ

πφ φ

δ

κ>> <

<

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦⎧ =⎪⎨

=⎪⎩

= = ∝

= − =

== +

−

0

1

2 20

21

1

2 22

02 2 2 21 1

R2 2

00

min( , )

1

u( ) exp( ) sin ( ),

1 d v( ) = = O( ) ( 1)

d u( ) d sin ( ) O( ) O( ) R

r

r r

ka g r

r i k r a r r

k ar r rk a g r g r

r r r kr R r

κ

δ

δ

⎧⎨ =⎩

⎧ −− =⎪

⎨⎪ = − >⎩⎧

≤⎪ +⎪⎨⎪ ≈ + =⎪⎩

∫

∫ ∫ Open channel wf (triplet)

Closed channel wf (singlet)

Closed channelOpen channel

0

1/ 46

0 2

30 0 0

2 2 00 2

0

ln u( ) 1

1, , O( ) 12O( ) O( )

2

r r

F

B

F

d rdr a

C mr

nr a r k rB mg

r R

µκ

λ

=

= −

⎛ ⎞≈ ⎜ ⎟⎝ ⎠

= =

≈

Halo dimersParticles in a pair spend most of the time outside the interaction zone, in the triplet state.

Optical trap for evaporative coolingborrowed from R. Grimm

special feature #1precise controlof laser power

10 W → few 100µW

special feature #1precise controlof laser power

10 W → few 100µW

special feature #2: axial magnetic confinement- spatial compression at very weak optical traps- perfectly harmonic !!- precisely known trap frequency for weak optical trap

special feature #2: axial magnetic confinement- spatial compression at very weak optical traps- perfectly harmonic !!- precisely known trap frequency for weak optical trap

νz = 24.5 Hz @ 1kG

Umagn

What is in a trap?What is in a trap?• Typically about 10Typically about 1055--10106 6 atoms divided 50atoms divided 50--50 among 50 among the lowest two hyperfine statesthe lowest two hyperfine states

•• Due to the high diluteness atoms in the same hyperfineDue to the high diluteness atoms in the same hyperfinestate do not interact with one anotherstate do not interact with one another

•• Atoms in different hyperfine states experience interactions Atoms in different hyperfine states experience interactions only in sonly in s--wave.wave. The strength of this interaction is fully tunable!The strength of this interaction is fully tunable!

Who does experiments?Who does experiments?

•• Jin’s group at Boulder Jin’s group at Boulder •• Grimm’s group in InnsbruckGrimm’s group in Innsbruck•• Thomas’ group at DukeThomas’ group at Duke•• Ketterle’sKetterle’s group at MIT group at MIT •• Salomon’s group in ParisSalomon’s group in Paris•• Hulet’sHulet’s group at Ricegroup at Rice

F

z

r

1 3

h y p e r f in e s p li t t in g 8 0 M H z F e rm i e n e rg y 2 0 5 0 k H z T e m p e ra tu re T 0 2

2 5 H z t r a p f r e q u e n c ie s c lo u d h a s c ig a r s h a p e7 5 0 H z

n u m b e r d e n s i ty n 1 0 a t

Fε ενν⎧⎪⎨⎪⎩

≈

3

0

5 6

0 030

o m s /c m ty p ic a l a to m s e p a ra t io n 5 0 0 0

N u m b e r o f a to m s N 1 0 1 0

r a d iu s o f in te r a c t io n r 1 0 0 d i lu te n e s s n r 1

a

a

Typical parametersTypical parameters

In dilute Fermi systems only very few characteristics are relevant.

• These systems are typically very cold

• A dilute Fermi system is degenerate and the fastest particle has a momentum of the order of the Fermi momentum

• The wave functions are basically constant over the interactionvolume of two particles and thus they cannot “see” any details,except the scattering length typically.

2 1 / 3 2 1 / 3(6 ) (6 )

2

Fp n nnn n

π π↑ ↓

↑ ↓

= =

= =

m

p εk

TT FFB

FF 2

,2

==

1/a

T

a0shallow 2-body bound state

halo halo dimersdimers

BCS BCS SuperfluidSuperfluid

High T, normal atomic (plus a few molecules) phase

?Molecular BEC and/orMolecular BEC and/orAtomic + Molecular Atomic + Molecular SuperfluidsSuperfluids

Expected phases of a two species dilute Fermi system BCS-BEC crossover

weak interactionweak interaction weak interactionsweak interactions

Strong interactionStrong interaction

F

akπ

≤

BCS → BEC crossover

If a>∆

=

BertschBertsch ManyMany--Body X challenge, Seattle, 1999Body X challenge, Seattle, 1999

What are the ground state properties of the manyWhat are the ground state properties of the many--body system composed of body system composed of spin ½ fermions interacting via a zerospin ½ fermions interacting via a zero--range, infinite scatteringrange, infinite scattering--length contactlength contactinteraction. interaction.

In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not.

- systems of bosons are unstable (Efimov effect)- systems of three or more fermion species are unstable (Efimov effect)

• Baker (winner of the MBX challenge) concluded that the system is stable.See also Heiselberg (entry to the same competition)

• Carlson et al (2003) Fixed-Node Green Function Monte Carloand Astrakharchik et al. (2004) FN-DMC provided best the theoretical estimates for the ground state energy of such systems.

• Thomas’ Duke group (2002) demonstrated experimentally that such systemsare (meta)stable.

Experimental signatures/predictions of the large size pairs/ halo dimer modelversus fermion-boson model:

1) Near the Feshbach resonance the pair is in triplet statein agreement with Grimm’s group experiments (fermion-boson model predicts a singlet state)

2) Particle loss is consistent with large spatial size pairsin agreement with Grimm’s and Jin’s groups experiments(fermion-boson model is consistent with small spatial size pairs)

3) Spatial size of the cloud in agreement with Grimm’s group experiment(fermion-boson model disagrees with experiment)

4) Frequencies and frequency shift of the frequencies of the collective oscillations(if at the Feshbach resonance the system would be made of small size pairs the frequency would be higher and the frequency shift would be much smaller than observed in experiments, see Pitaevskii and Stringari, PRL 81, 4541 (1998), Braaten and Pearson, PRL 82, 255 (1999) )

Theory is now in such a state that it can Theory is now in such a state that it can make verifiable or falsifiable predictions.make verifiable or falsifiable predictions.

30

63 (0) 1128

M

M

n rδω πω

≈

Ohashi, Levico 2004

Closed (singlet) channel

Open (triplet) channel

Atomicseperation

Energy

Eres

Eth

Chang et al. physics/0404115

Astrakharchik et al, cond-mat/0406113

Even though two atoms can bind, there is no binding among dimers!

Fixed node GFMC results, J. Carlson et al. (2003)

⎟⎟⎠

⎞⎜⎜⎝

⎛=∆

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛=∆

akmk

e

akmk

e

F

FBCS

F

FGorkov

2exp

28

2exp

22

22

2

223/7

π

π

Fixed node GFMC results, J. Carlson et al. (2003)

S. Jochim et al. PRL 91, 240402 (2003)

1)2)

Electron spin along B

Electron spin opposite B

see also Petrov et al. cond-mat/0309010Regal et al. PRL 92, 083201 (2004)

Ohashi and Griffin, Phys. Rev. Lett. 89, 130402 (2002) Bruun, cond-mat/0401497

The fermion-boson model predict that at resonance an atomic Fermi cloud consists predominantly of molecules in the closed channel (singlet), which thus have an almost vanishing magnetic moment.

ħω=0.568 x 10-12eV, a = -12.63nm (when finite)

40K (Fermi) atoms in a spherical harmonic trap

Effect of interaction, with and without weak and strong pairing correlations with fixed particle number, N = 5200.

0 2 4 6 8 10 12−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

r [µm]

U(r

) or

∆(r

) [1

0−10

eV

]

Mean−field and Pairing field

U∞

Ua

U0∞

U0 a

∆∞(r)

∆a(r)

0 2 4 6 8 10 120

1

2

3

4

5

6

7

8

9

r [µm]n(

r) [1

012

cm−3

]

Particle number density

no interaction mean−field + pairing mean−field + no pairing

Unpublished, fully self-consistent SLDA (Kohn-Sham generalized to pairing)calculation performed by Yongle Yu in July 2003.

3)

Ohashi, Levico 2004 Grimm, Levico 2004

The fermion-boson model predict that at resonance the size of the cloud is significantly smaller than the observed one.

3)

First soundSpherical

CollisionalregimeCompressionalCompressionalmodemode

Landau’s zero soundNormal Fermi fluidcollisionlessIncompressionalIncompressionalmodemode

Anderson-BogoliubovsoundSpherical

SuperfluidSuperfluidcollisionlesscollisionlessCompressionalCompressionalmodemode

Soundvelocity

Local shapeof Fermi surface

Elongated along propagation direction

s Fv v1

ss=

>

vskω =Sound in infinite Sound in infinite fermionicfermionic mattermatter

vv3F

s ≈

vv3F

s ≈

( ) ( )

( )

3

2 2

2

2 2 2 2 20

2

2

3 5 1( ) O5 2 3

0.44, 1, 1

21(0)

F

F F F

F

knm k a k a k a

m x y zU

Kk a

ς ιε ξ

ξ ς ι

ω λ

δω ςω ξ

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟= − − +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦≈ ≈ ≈

+ +=

=Perturbation theory result usingGFMC equation of state in a trap

Adiabatic regimeSpherical Fermi surface

Frequency shifts inthese modes mightcarry information about possible atom-halo dimer mixture

Hu et al. cond-mat/0404012, a semi-quantitative analysis (gap and chemical potential inaccurate) assuming a polytropic equation of state

For a more careful analysis, using GFMC equation of state in a trap see Bulgac and Bertsch, cond-mat/0404687

Grimm’s group experiment

Thomas’ group experiment

4)

Fermion-boson model would predict here

a0

0

0

0

( ) 1

174

7.8 0.6 G 202.10 0.07 G

bg

bg

wa B aB B

a a

wB

⎛ ⎞= −⎜ ⎟−⎝ ⎠

=

= ±= ±

Falco and Stoof, Phys. Rev. Lett. 92, 130401 (2004)

Theory declared full victory here!Regal, Greiner and Jin,Phys. Rev. Lett. 92, 040403 (2004)

Zwierlein et al.Phys. Rev. Lett. 92, 120403 (2004)

3

112 ( )m F

nnk a

⎡ ⎤−⎢ ⎥

⎢ ⎥⎣ ⎦

Fermion superfluidity, more specificaly superflowsuperflow,, has not yet been demonstratedunambiguously experimentally. There is lots of circumstantial evidence and facts in agreement with theoretical models assuming its existence.

Theory is able to make very precise predictions in this regime and the agreementwith experiment can be check quantitatively.

radio-frequency spectroscopy

high B-field

1

0

mI=-1

meas. of mol. bind. energy in 40KRegal et al., Nature 424, 47 (2003)

rf spectroscopy of 6Li: Gupta et al., Science 300, 1723 (2003)

rf

~80MHz

breaking moleculescosts energy

→molecular signal

up-shifted

breaking moleculescosts energy

→molecular signal

up-shifted

breaking pairscosts energy

→pair signalup-shifted

breaking pairspairscosts energy

→pairpair signalup-shifted

borrowed from R. Grimm

rf spectra in crossover regime

rf offsetborrowed from R. Grimm

temperature dependence of pairing

0,0

0,4

0,0

0,4

-20 0 20 400,0

0,4

0,0

0,4

(d)

(c)

(a)

fract

iona

l los

s in

|2>

RF frequency offset (kHz)

(b)

T < 0.1 TF

≈ 0.2 TF

≈ 0.4 TF

≈ 0.5 TF

borrowed from R. Grimm

J. Thomas’ group at Duke

Consistent with E≈aT5/2

Consistent with E≈bT2

[ ] [ ]

[ ] [ ]

22

0 0

*k

0

2 23 5/3 *

2

1/3

( ) 2 v ( ) , ( ) 2 v ( )

( ) v ( )u ( )

E ( ) ( ) ( ) ( ) ( )2

( ) 1( ) ( ), ( )( ) ( )

( ) (

k c k c

k c

k kE E E E

kE E

eff

F

n r r r r

r r r

d r r x r n r x r rm m

x rx r r x r

m n r k r a

T U r

τ

ν

τ β ν

γν

µ

< < < <

< <

= = ∇

=

⎧ ⎫= + −∆⎨ ⎬

⎩ ⎭

∆ = − =

+ − ∆

∑ ∑

∑

∫

*

u ( ) u ( ))v ( ) v ( )( ) ( ( ) )

k ki

k k

r rrE

r rr T U r µ⎛ ⎞ ⎛ ⎞⎛ ⎞

=⎜ ⎟ ⎜ ⎟⎜ ⎟∆ − + −⎝ ⎠⎝ ⎠ ⎝ ⎠

Superfluid LDA (SLDA)

number and kinetic densities

anomalous density

Bogoliubov-de Gennes like equations.Correlations are however included by default!

Cutoff and position running coupling constant!

Divergent!

Vortex in fermion matter

kn

kn

u ( ) u ( )exp[ ( 1/ 2) ], n - half-integer

v ( ) v ( )exp[ ( 1/ 2) ]

( ) ( )exp( ), ( , , ) [cyllindrical coordinates]

r i n ikzr i n ikz

r i r z

α α

α α

ρ φρ φ

ρ φ ρ φ

+ −⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

∆ = ∆ =

( ) ( )v v2

Oz - vortex symmetry axis

Ideal vortex, Onsager's quantization (one per Cooper pair)

1V ( , ,0) V2 2 2C

r y x r drm mρ π

= − ⇐ ⋅ =∫

How can one put in evidence a vortexin a Fermi superfluid?

Hard to see, since density changes are not expected, unlike the case of a Bose superfluid.

However, if the gap is not small, one can expect a noticeable density depletion along the vortex core, and the bigger the gap the bigger the depletion, due to an extremely fast extremely fast vorticalvortical motionmotion.

s c

F F

v Tv 2 TFε

∆< ∝

NB Tc unknown in the strong coupling limit!

Number density and pairing field profiles

The depletion along the vortex coreis reminiscent of the correspondingdensity depletion in the case of a vortex in a Bose superfluid, when the density vanishes exactly along the axis for 100% BEC.

Extremely fast quantum vortical motion!

Local vortical speed as fraction of Fermi speed

Fermions with 1/kFermions with 1/kFFa = 0.3, 0.1, 0, a = 0.3, 0.1, 0, --0.1, 0.1, --0.50.5

Bosons with naBosons with na33 = 10= 10--33 and 10and 10--55

From Ketterle’sgroup

1/a

T

a0shallow 2-body bound state

halo halo dimersdimers

Phases of a two species dilute Fermi system BCS-BEC crossover

weak interactionweak interaction weak interactionsweak interactionsStrong interactionStrong interaction

“Before”- Life

Atoms form rather unstablecouples, spread over largedistances, with many othercouples occupying thesame space

After – Life

Atoms form very strong couples, mostly inert and at rest, halo dimers, widely separated from one another and weakly interacting

LIFE!LIFE!

Atoms form strong couples, living as in an apartment building, in close proximity of one another, and obviouslyanoying each other quite a bit

Conclusions:Conclusions:

The field of dilute atomic systems is going to be for many years to comeone of the most exciting fields in physics, with lots surprises at every corner.