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

Aurel BulgacAurel BulgacUniversity 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?

Fermionic superfluidity!Fermionic superfluidity!

• Dilute atomic Fermi gases TDilute atomic Fermi gases Tcc 1010-12-12 –– 1010-9 eV eV

Liquid Liquid 33He THe Tcc 1010-7-7 eVeV

Metals, composite materials TMetals, composite materials Tcc 1010-3 -3 – – 1010-2-2 eV eV Nuclei, neutron stars TNuclei, neutron stars Tcc 101055 – – 101066 eV eV

• QCD color superconductivity TQCD color superconductivity Tcc 10107 7 – – 10108 8 eV eV

Superconductivity and superfluidity in Fermi systemsSuperconductivity and superfluidity in Fermi systems

units (1 eV 104 K)

• 1913 Kamerlingh Onnes1913 Kamerlingh Onnes

• 1972 Bardeen, Cooper and Schrieffer1972 Bardeen, Cooper and Schrieffer

• 1973 Esaki, Giaever and Josephson1973 Esaki, Giaever and Josephson

• 1987 Bednorz and Muller1987 Bednorz and Muller

• 1996 Lee, Osheroff and Richardson1996 Lee, Osheroff and Richardson

• 2003 Abrikosov, Ginzburg and Leggett2003 Abrikosov, Ginzburg and Leggett

Memorable years in the history of superfluidity and Memorable years in the history of superfluidity 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 fermionic dilute atomic clouds Basic parameters of fermionic 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 created thus 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 others ultracold molecules, mBEC from Fermi gas 2004 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 gap Thomas’ group reports measurements of the specific heat

HISTORY

One fermionic atom in magnetic fieldOne fermionic atom in magnetic field

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

Feshbach resonance

BSSVSSa

V

VPrVPrVVVp

H

nzn

eze

Znehfhf

dZi

i

hfi

r

)(,

)()()(2

2

1100

2

1

2

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

Channel coupling

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

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

0

0 0

( ) 1

( )

P r r a

P r r r

Halo dimerHalo dimer

11 12 2

21 22

211

222 0 0 0

0 0 0 00 0

012 1 1

21

2 20

u u

v 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 r

r r

rV r g r r r

r r rg ikr krr kr

rk

0

1

2 20

21

1

2 22

02 2 2 21 1

R2 2

0

0

min( , )

1

u( ) exp( )sin ( ),

1 d v( ) = = O( )

( 1)

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

r

r r

k

a g r

r i k r a r r

k ar r r

k a g r g r

r r r kr R r

Open channel wf (triplet)

Closed channel wf (singlet)

Closed channelOpen channel

0

1/ 4

60 2

30 0 0

2 2 00 2

0

ln u( ) 1

1, , O( ) 1

2O( ) O( )

2

r r

F

B

F

d r

dr a

C mr

nr a r k r

B mg

r R

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

Optical trap for evaporative cooling borrowed 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

z = 24.5 Hz @ 1kG

Umagn

What is in a trap?What is in a trap?

• Typically about 10Typically about 105-5-10106 6 atoms divided 50-50 among atoms divided 50-50 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 s-wave.only in s-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’s group at MIT Ketterle’s group at MIT • Salomon’s group in ParisSalomon’s group in Paris• Hulet’s group at RiceHulet’s group at Rice

F

z

r

13

hyperfine splitting 80 MHz Fermi energy 20 50 kHz Temperature T 0 2

25 Hz trap frequencies cloud has cigar shape

750 Hz

number density n 10 at

F

3

0

5 6

0 030

oms/cm typical atom separation 5000

Number of atoms N 10 10

radius of interaction r 100

diluteness nr 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 n

nn n

m

p ε

kTT F

FB

FF 2

,2

1/a

T

a<0no 2-body bound state

a>0shallow 2-body bound state halo dimershalo dimers

BCS SuperfluidBCS Superfluid

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

ak

BCS → BEC crossover

If a<0 at T=0 a Fermi system is a BCS superfluid

Eagles (1969), Leggett (1980), Nozieres and Schmitt-Rink (1985), Randeria et al. (1993),…

F

F

FFF

F

F

kkak

akm

k

e

11 and 1|| iff ,

2exp

2

2 223/7

)( , and 5

344.0 ,

5

354.0 uperfluidnormal

FFFs

F OON

E

N

E

If |a|=∞ and nr031 a Fermi system is strongly coupled and its properties

are universal. Carlson et al. PRL 91, 050401 (2003)

If a>0 (ar0) and na31 the system is a dilute BEC of tightly bound dimers

23

2 2 and 1, where and 0.6 0

2f

b b bb

nn a n a a

ma

Bertsch Many-Body X challenge, Seattle, 1999Bertsch Many-Body X challenge, Seattle, 1999

What are the ground state properties of the many-body system composed of What are the ground state properties of the many-body system composed of spin ½ fermions interacting via a zero-range, infinite scattering-length contactspin ½ fermions interacting via a zero-range, infinite scattering-length 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 Carlo and 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 systems are (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 state in agreement with Grimm’s group experiments (fermion-boson model predicts a singlet state)

2) Particle loss is consistent with large spatial size pairs in 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) 1

128M

M

n r

Ohashi, Levico 2004

Closed (singlet) channel

Open (triplet) channel

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)

akm

k

e

akm

k

e

F

FBCS

F

FGorkov

2exp

2

8

2exp

2

2

22

2

223/7

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.

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)

Sound

velocity

Collisional

regime

Compressional Compressional modemode

SphericalFirst sound

SuperfluidSuperfluid

collisionlesscollisionless

Compressional Compressional modemode

Spherical

Anderson-Bogoliubov sound

Normal Fermi fluid

collisionless

Incompressional Incompressional modemode

Landau’s zero sound

Local shapeof Fermi surface

Elongated along propagation direction

s Fv v

1

s

s

vsk Sound in infinite fermionic matterSound in infinite fermionic matter

vv

3F

s

vv

3F

s

3

2 2

2

2 2 2 2 20

2

2

3 5 1( ) O

5 2 3

0.44, 1, 1

2

1

(0)

F

F F F

F

kn

m k a k a k a

m x y zU

Kk a

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

a<0a>0

0

0

0

( ) 1

174

7.8 0.6 G

202.10 0.07 G

bg

bg

wa B a

B B

a a

w

B

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

11

2 ( )mF

nn

k a

Fermion superfluidity, more specificaly superflow,superflow, has not yet been demonstrated unambiguously 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 agreement with 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 pairs pairs costs energy

→pairpair signalup-shifted

breaking pairs pairs costs 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 40

0,0

0,4

0,0

0,4

(d)

(c)

(a)

fra

ctio

na

l lo

ss 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 EaT5/2

Consistent with EbT2

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

*

u ( ) u ( ))

v ( ) v ( )( ) ( ( ) )k k

ik k

r rrE

r rr T U r

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 ikz

r i n ikz

r i r z

v v2

Oz - vortex symmetry axis

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

1V ( , ,0) V

2 2 2C

r y x r drm m

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 vortical motionextremely fast vortical motion.

s c

F F

v T

v 2 TF

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, -0.1, -0.5a = 0.3, 0.1, 0, -0.1, -0.5

Bosons with naBosons with na33 = 10 = 10-3-3 and 10 and 10-5-5

From Ketterle’s group

1/a

T

a<0no 2-body bound state

a>0shallow 2-body bound state halo dimershalo dimers

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.