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Superfluidity in atomic Fermi gases. Luciano Viverit. University of Milan and CRS-BEC INFM Trento. CRS-BEC inauguration meeting and Celebration of Lev Pitaevskii’s 70th birthday. Outline. Why superfluidity in atomic Fermi gases? Some ways to attain superfluidity

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slide1

Superfluidity in atomic Fermi gases

Luciano Viverit

University of Milan and CRS-BEC INFM Trento

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

slide2

Outline

  • Why superfluidity in atomic Fermi gases?
  • Some ways to attain superfluidity
  • How to detect superfluidity and current
  • experimental developments
  • Vortices
slide3

Why superfluidity in atomic Fermi gases?

Test ground for various theories:

  • 1) Superfluidity in dilute 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)

a < 0 ; kF |a| <<1

slide4

Why superfluidity in atomic Fermi gases?

  • 2) Detailed study of effective interactions in medium and
  • consequences 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 < 0 ; kF |a| ~1

slide5

+

Why superfluidity in atomic Fermi gases?

  • 3) Boson 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)
slide6

+

Why superfluidity in atomic Fermi gases?

  • 3) Boson 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)
slide7

Why superfluidity in atomic Fermi gases?

  • 4) BCS-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/ kFa  -∞1/ kFa+∞

slide8

Why superfluidity in atomic Fermi gases?

  • 4a) Resonance superfluidity
  • Holland, Kokkelmans, Chiofalo and Walser
  • PRL 87, 120406 (2001)
  • Ohashi and Griffin, PRL 89, 130402 (2002)
slide9

Why superfluidity in atomic Fermi gases?

  • 4a) Resonance superfluidity
  • Holland, Kokkelmans, Chiofalo and Walser
  • PRL 87, 120406 (2001)
  • Ohashi and Griffin, PRL 89, 130402 (2002)
slide10

Why superfluidity in atomic Fermi gases?

  • 5) Superfluid-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)
slide11

Why superfluidity in atomic Fermi gases?

  • 5) Superfluid-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)
slide12

Gap equation at Tc,0:

Number equation at Tc,0:

Ways to attain superfluidity

1)BCS in a uniform dilute gas (a<0, kF|a|<<1)

where .

slide13

If kF|a|<<1 solutions:

  • Sà de Melo, Randeria and Engelbrecht, PRL 71, 3202
  • (1993)
  • Stoof, Houbiers, Sackett and Hulet, PRL 76, 10 (1996)
slide14

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

slide15

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

0

0

=0

=0

a

a

a

a

=0

~ c (kFa)2; c>0

0

a

slide16

Since kF|a|<<1

then

By carrying out detailed calculations one finds

and thus

slide17

Gorkov and Melik-Barkhudarov, JETP 13,1018 (1961)

  • Heiselberg, Pethick, Smith and LV, PRL 85, 2418 (2000)

Formula ~ valid also in trap if

slide18

Practical problem:

If kF|a|<<1 then

Best experimental performances with present techniques

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

Not enough if the gas is dilute!

slide19

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

  • Combescot, PRL 83, 3766 (1999) (=2kF a /)
slide20

WHY??

  • Exchange of density and spin collective
  • modes (higher orders in kFa than previously
  • considered) and
  • Fragmentation of single particle levels
  • both strongly influence Tc!
slide21

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 ...
slide22

Idea B: BCS-BEC crossover

Back to BCS equations.

Gap equation at Tc,0:

Number equation at Tc,0:

slide24

Including gaussian fluctuations in  about the

saddle-point:

  • Sà de Melo, Randeria and Engelbrecht, PRL 71, 3202
  • (1993)

BEC critical

temperature

slide25

Superfluid transition in unitarity limit (kFa)

predicted at

BUT

  • Exchange of density and spin modes, and
  • Fragmentation of single particle levels
  • not included in the theory.Then:
slide26

?

Strong interaction between theory and experiments needed.

slide27

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 T ~ 0.1 TF in unitarity

conditions (kFa ±∞).

slide28

According to theory the gas could be superfluid.

But is it?

Problem: How do we detect superfluidity?

No change in density profile (at least in w.c. limit)

Suggestion 1: Look at expansion.

  • Menotti, Pedri and Stringari, PRL 89, 250402 (2002)
slide29

Theory

Ei / Eho=0

Ei / Eho=-0.4

Experiment

Ei / Eho>0

Ei / Eho=0

Ei / Eho<0

slide31

Problem:

If the gas is in the hydrodynamic regime then

expansion of normal gas = expansion of superfluid.

Suggestion 1 cannot distinguish.

Suggestion 2: Rotate the gas to see quantization

of angular momentum.

slide32

Normal hydrodynamic gas can sustain rigid body

  • rotation
  • Superfluid gas can rotate only by forming vortices
  • (because of irrotationality)
slide33

Superfluid vortex structure. Simple model

Vortex velocity field

Kinetic energy (per unit volume)

Condensation energy (per unit volume)

slide34

By imposing

one finds:

where

slide37

Vortex energyin a cylindical bucket of radius Rc

Factor 1.36 model dependent. Let then

  • Bruun and LV, PRA 64, 063606 (2001)
slide38

From microscopic calculation ...

  • Nygaard, Bruun, Clark and Feder, e-print cond-mat/0210526
slide39

Above formula for v

with D=2.5

D=2.5

kFa=-0.43

kFa=-0.59

slide41

In unitarity limit one expects:

and thus

Very recent microscopic result ...

slide42

D

Density

  • Bulgac and Yu, e-print cond-mat/0303235
slide43

In traps

Vortex forms if

In dilute limit this means

which is fulfilled if

In unitarity limit it reads

slide44

Rough estimate for c1in unitarity limit in trap(C=1)

In the case of Duke experiment one finds

slide45

No angular momentum transfer to the gas for stirring

frequencies below c1 if the gas is superfluid!

Example with bosons:

  • Chevy, Madison, Dalibard, PRL 85, 2223 (2000)
slide46

How can one do the experiment?

e.g. Lift of degeneracy of quadrupolar mode

Normal hydrodynamic

for arbitrarily small stirring frequency .

Superfluid

-

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

slide49

Conclusions

  • 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