Superfluidity in atomic Fermi gasesSuperfluidity in atomic Fermi gases

Luciano ViveritLuciano Viverit

University of Milan and CRS-BEC INFM TrentoUniversity of Milan and CRS-BEC INFM Trento

CRS-BEC inauguration meeting and Celebration of Lev Pitaevskii’s 70th birthday

OutlineOutline

• Why superfluidity in atomic Fermi gases?

• Some ways to attain superfluidity

• How to detect superfluidity and current experimental developments

• Vortices

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

1) Superfluidity in Superfluidity in dilutedilute gases gases

• Gorkov and Melik-Barkhudarov, JETP 13,1018 (1961)• Stoof, Houbiers, Sackett and Hulet, PRL 76, 10 (1996)• Papenbrock and Bertsch, PRC 59, 2052 (1999)• Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000)

Test ground for various theories:Test ground for various theories:

a < a < 00 ; k ; kF F || a| <<a| <<11

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

2) Detailed study of effective interactions in medium and Detailed study of effective interactions in medium and consequences on pairingconsequences on pairing

• Berk and Schrieffer, PRL 17, 433 (1966) (superconductors)• Schulze et al., Phys. Lett. B375, 1 (1996) (neutron stars)• Barranco et al., PRL 83, 2147 (1999) (nuclei)• Combescot, PRL 83, 3766 (1999) (atomic gases)• LV, Barranco, Vigezzi and Broglia, work in progress

a < a < 00 ; k ; kF F || a| a| ~~ 11

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

3) Boson enhanced pairing in Bose-Fermi mixturesBoson enhanced pairing in Bose-Fermi mixtures

• Bardeen, Baym and Pines, PRL 17, 372 (1966) (3He-4He)• Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000)• Bijlsma, Heringa and Stoof, PRA 61, 053601 (2000)• LV, PRA 66, 023605 (2002)

+

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

3) Boson enhanced pairing in Bose-Fermi mixturesBoson enhanced pairing in Bose-Fermi mixtures

• Bardeen, Baym and Pines, PRL 17, 372 (1966) (3He-4He)• Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000)• Bijlsma, Heringa and Stoof, PRA 61, 053601 (2000)• LV, PRA 66, 023605 (2002)

+

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

4) BCS-BEC crossoverBCS-BEC crossover

• Leggett (1980)• Nozières and Schmitt-Rink, JLTP 59, 195 (1985)• Sà de Melo, Randeria and Engelbrecht, PRL 71, 3202 (1993)• Pieri and Strinati, PRB 61, 15370 (2000)

1/ 1/ kkFFa a - -∞∞ 1/ 1/ kkFFaa ++∞∞

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

4a) Resonance superfluidityResonance superfluidity

• Holland, Kokkelmans, Chiofalo and Walser PRL 87, 120406 (2001)• Ohashi and Griffin, PRL 89, 130402 (2002)

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

4a) Resonance superfluidityResonance superfluidity

• Holland, Kokkelmans, Chiofalo and Walser PRL 87, 120406 (2001)• Ohashi and Griffin, PRL 89, 130402 (2002)

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

5) Superfluid-insulator transition in (optical) latticesSuperfluid-insulator transition in (optical) lattices

• Micnas, Ranninger and Robaszkiewicz RMP 62, 113 (1990) (High Tc)• Hofstetter, Cirac, Zoller, Demler and Lukin PRL 89, 220407 (2002)

Why superfluidity in atomic Fermi gases?Why superfluidity in atomic Fermi gases?

5) Superfluid-insulator transition in (optical) latticesSuperfluid-insulator transition in (optical) lattices

• Micnas, Ranninger and Robaszkiewicz RMP 62, 113 (1990) (High Tc)• Hofstetter, Cirac, Zoller, Demler and Lukin PRL 89, 220407 (2002)

Ways to attain superfluidityWays to attain superfluidity

1) BCS in a uniform BCS in a uniform dilutedilute gas ( gas (aa<0, <0, kkFF|a||a|<<1)<<1)

Gap equation at TGap equation at Tc,0c,0:

Number equation at TNumber equation at Tc,0c,0:

wherewhere .

• Sà de Melo, Randeria and Engelbrecht, PRL 71, 3202 (1993)• Stoof, Houbiers, Sackett and Hulet, PRL 76, 10 (1996)

If kF|a| kF|a|<<1<<1 solutions:

Now include the effects of induced interactions to second order in a (important also in the dilute limitimportant also in the dilute limit)

Now include the effects of induced interactions to second order in a (important also in the dilute limitimportant also in the dilute limit)

a

0

a

a a

a0

0

=0 =0

=0 ~ c (kFa)2; c>0

Since kF|a|<<1

then

By carrying out detailed calculations one finds

and thus

• Gorkov and Melik-Barkhudarov, JETP 13,1018 (1961)• Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000)

Formula ~ valid also in trap ifFormula ~ valid also in trap if

Practical problemPractical problem::

If kF|a|<<1 then

Best experimental performances with present techniques

Not enough if the gas is dilute!Not enough if the gas is dilute!

• Gehm, Hemmer, Granade, O’Hara and Thomas, e-print cond-mat/0212499• Regal and Jin, e-print cond-mat/0302246

Idea AIdea A:: let kF|a| approach 1 (but still kF|a|<1)

• Combescot, PRL 83, 3766 (1999) (=2kF a /)

•

WHY??WHY??

• Exchange of density and spin Exchange of density and spin collectivecollective modes (higher orders in modes (higher orders in kFa than previously than previously

considered) andconsidered) and

• Fragmentation of single particle levelsFragmentation of single particle levels

both both stronglystrongly influence influence TTcc! !

So what?So what?

• Answer difficult, no completely reliable theory

• Answer interesting for several physical systems

• LV, Barranco, Vigezzi and Broglia, work in progress

• We wait for experiments ...

Idea BIdea B: BCS-BEC crossover: BCS-BEC crossover

Gap equation at TGap equation at Tc,0c,0:

Number equation at TNumber equation at Tc,0c,0:

Back to BCS equations.Back to BCS equations.

Solution inSolution in 1/ 1/ kkFFa a -∞ -∞

Including gaussian fluctuations in Including gaussian fluctuations in about the about the saddle-point:saddle-point: • Sà de Melo, Randeria and Engelbrecht, PRL 71, 3202 (1993)

BEC criticaltemperature

Superfluid transition in unitarity limit (kFa)predicted at

BUTBUT

• Exchange of density and spin modes, andExchange of density and spin modes, and

• Fragmentation of single particle levelsFragmentation of single particle levels

not included in the theory.not included in the theory. Then:Then:

?Strong interaction between theory and experiments needed.Strong interaction between theory and experiments needed.

What is happening with experiments?What is happening with experiments?

• O’Hara et al., Science 298, 2179 (2002) (Duke)• Regal and Jin, e-print cond-mat/0302246 (Boulder)• Bourdel et al., eprint cond-mat/0303079 (Paris)• Modugno et al., Science 297, 2240 (2002) (Firenze)• Dieckmann et al., PRL 89, 203201 (2002) (MIT)

Two component Fermi gas at Two component Fermi gas at T T ~ 0.1 ~ 0.1 TTF F in unitarity in unitarity

conditions (conditions (kkFFaa ±∞±∞).).

But is it?But is it?

According to theory the gas could be superfluid.

ProblemProblem: How do we detect superfluidity?How do we detect superfluidity?No change in density profile (at least in w.c. limit)No change in density profile (at least in w.c. limit)

Suggestion 1Suggestion 1: Look at expansion.Look at expansion.

• Menotti, Pedri and Stringari, PRL Menotti, Pedri and Stringari, PRL 8989, 250402 (2002), 250402 (2002)

Ei / E

ho=0

Ei / E

ho=-0.4

TheoryTheory

ExperimentExperiment

Ei / E

ho>0

Ei / E

ho=0

Ei / E

ho<0

ProblemProblem:

If the gas is in the hydrodynamic regime thenexpansion of normal gas = expansion of superfluid.Suggestion 1 cannot distinguish.

Suggestion 2Suggestion 2: Rotate the gas to see quantization of angular momentum.

• Normal hydrodynamicNormal hydrodynamic gas can sustain rigid body rotation

• SuperfluidSuperfluid gas can rotate only by forming vortices (because of irrotationality)

Superfluid vortex structure. Superfluid vortex structure. Simple modelSimple model

Vortex velocity field

Kinetic energy (per unit volume)

Condensation energy (per unit volume)

By imposing

one finds:

where

Vortex energy in a cylindical bucket of radius Rc

Vortex corenormal matter

Rc

Vortex energy in a cylindical bucket of radius Rc

Vortex energy in a cylindical bucket of radius Rc

Factor 1.36 model dependent. Let then

• Bruun and LV, PRA 64, 063606 (2001)

From microscopic calculation ...From microscopic calculation ...

• Nygaard, Bruun, Clark and Feder, e-print cond-mat/0210526

kFa=-0.43

kFa=-0.59

D=2.5

Above formula for v with D=2.5

The critical frequency for formation of first vortex is thus

since

~ h per particle.-

In unitarity limit one expects:

and thus

Very recent microscopic result ...

Density

• Bulgac and Yu, e-print cond-mat/0303235

Vortex forms if

In dilute limit this means

which is fulfilled if

In unitarity limit it reads

In trapsIn traps

Rough estimateRough estimate for for c1c1 in unitarity limit in trapin unitarity limit in trap ((CC=1)=1)

In the case of Duke experiment one finds

No angular momentum transfer to the gas for stirringNo angular momentum transfer to the gas for stirringfrequencies below frequencies below c1c1 if the gas is superfluid! if the gas is superfluid!

Example with bosons:

• Chevy, Madison, Dalibard, PRL 85, 2223 (2000)

How can one do the experiment?How can one do the experiment?e.g. e.g. Lift of degeneracy of quadrupolar modeLift of degeneracy of quadrupolar mode

Normal hydrodynamicNormal hydrodynamic

for arbitrarily small stirring frequency .

SuperfluidSuperfluid

only if <lz> h/2, ( > c1) and zero otherwise.-

Splitting for a single vortexSplitting for a single vortex

For fermionsFor fermions

then

Again in the case of Duke experiment one finds

ConclusionsConclusions

I showed various reasons why superfluidity in atomic gases is very interesting and important

I illustrated recent experimental developments

I showed how superfluidity can be detected by means of the rotational properties of the gas (vortices)

I pointed out several open questions which have to be addressed in the next future