INTERNATIONAL SCHOOL OF PHYSICS "ENRICO FERMI" Varenna, July 1st - July 11th 2008 " QUANTUM COHERENCE IN SOLID STATE SYSTEMS "

Sandro Stringari

University of Trento

1

Introduction to Bose-Einstein condensation

4. STRONGLY INTERACTINGATOMIC FERMI GASES

CNR-INFMCNR-INFM

Quantum statistics and temperature scales

3/1)83.0( NTkCB

!h= 3/1)6( NTkFB

!h=

2

Bosons and Fermions are not separate worlds

interaction

3

• atoms collide in open channel at small energy• same atoms in different hyperfine states form a bound state in closed channel• coupling through hyperfine interactions between open and closed channel • if open and closed channel have different magnetic moment ⇒ magnetically tunable ΔE= Δ µ × B

Resonance when bound-state and continuum become degenerate

!!"

#$$%

&

'

(+=

0

1BB

aa bg

Interactions can be tuned thanks toavailability of Feshbach resonance

4

Many-body aspects

BCS regimeunitary limit

BEC regime

5

Theory of superfluidity for and well established(BCS theory). Key results for uniform gases:

- Pairing between spin-up and spin-down atoms in momentum space (Cooper pairing)

- BCS critical temperature:

- BCS theory valid if

- Example . BCS critical temperature is too low even smaller than oscillator temperature ( ) with).

1|| <<akF

0<a

BCS theory (a<0)

ak

FC

FeTT2/

28.0!

=

1|| <<akF

FBCSFTTak4

102.0!

="!=

FhoT2

10!

"#h 510!N

(Gorkov, Melik-Barkhudarov, 1961)

6

Condition for having dilute molecular gas requires that distance between molecules be much larger than molecular size:

Such a molecular gas behaves like a dilute gas of bosons and exhibits BEC

BEC gas of molecules (a>0): calculation of

ad >>

Value of BEC critical temperature for a gas of bosons of mass 2m (molecules) directly related to value of Fermi energy of fermions of mass m.

Critical temperature for superfluidity is much higher in BECthan in BCS side where it is exponentially small.

3/194.0 !" NT

hoBECh=

3/1)6( !" NThoF

h=

FBECTT 5.0=

CT

High Tc superfluidity

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Experimental production of BEC molecules emerging from Fermi sea

JILA 2003: 40K2

Two routes have become available (2003):

- Cooling of two spin species Fermi gas with a<0. System reaches quantum degeneracy ( ), not enough for BCS. Further cooling is inhibited by Pauli blocking. Scattering length is then tuned through Feshbach resonance until BEC molecules are observed (Jila, Ens))

- Cooling of two spin species Fermi gas with a>0. Molecules are formed, further cooling yields BEC of molecules (Innsbruck, Mit).

Innsbruck 2003: 6Li2:

FTT 1.0!

8

Some questions concerning the BEC-BCS crossover

- search for many-body theory describing the transition

- interaction between molecules on BEC side

- what happens at unitarity ? (neither molecules, nor Cooper pairs)

- can we probe superfluidity at the BEC-BCS crossover ?

- effects of spin polarization on superfluids ?

1>>akF

9

Important effort in recent few years to provide improved many-bodyschemes (Holland, Griffin, Timmermans, Strinati, Stoof ….),including numerical quantum Monte Carlo approaches (Carlson,Giorgini, Prokofeev, Svistunov, Troyer)

First theoretical approach developed by Leggett (1980), Nozieres and Schmitt-Rink (1985) generalizes gap equation of BCS theory to the whole resonance region (BCS mean field)

Theory predicts (Randeria 1993):

- critical temperature and equation of state as a function of dimensionless parameter - formation of molecules with energy on the BEC side - molecules on the BEC side interact with scattering length

Many body theory of the BEC-BCS crossover

0.4

0.2

0

0

1-1

BEC

BCS

T C /

T F

-1/kFa0

akF

22/mah 1<<ak

F

aaM

2=

10

BCS mean field

Equation of state along the BEC-BCS crossover

Monte Carlo(Astrakharchicket al., 2004)

BEC BCS 0>µ

ideal Fermi gas

Energy is always smaller than ideal Fermi gas value. Attractive role ofinteraction along BCS-BEC crossover

Nnnn

0<µ

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BCS

BEC

(Randeria et al. 1993 extension of BCS mf theory to finite T)

QMC simulation(Burovskii, Prokofeev, Svistunov and Troyer 2006)

QMC result for critical temperature at unitarity recently confirmed by MIT experiment with polarized samples !

Critical temperature along the BEC-BCS crossover12

Behaviour near unitarity

At unitarity ( ) system is strongly correlatedproperties do not depend on value (even sign) of scattering length a UNIVERSALITY ( ) All lengths disappear from the calculation of thermodynamic functions.(Bertsch 2002)

Example: T=0 equation of state of uniform gas should exhibitsame density dependence as ideal Fermi gas(argument of dimensionality rules out different dependence):

( ) 3/23/222

)1(62

nm

!"µ +=h

1>>akF

10<<rk

F

0

0

!

=

"

" - ideal Fermi gas- at unitarity

Many-body calculation needed to determine value of ( ). is negative, reflecting attractive role of interaction. Equation of state can be used to determine density profiles, release energy and collective frequencies in Thomas-Fermi approximation.

! 6.0!"

!

13

-dependence of radii at unitarity

!

Equilibrium profile in local density approximation (isotropic trapping for simplicity)

( ) µ!"# =++ 223/23/222

2

1)1(6

2rmn

mho

h

Same result for density profile as for ideal gaswith replacement )1/(,1/ !µµ!"" +#+#

hoho

4/1)1( !+=idealRR

Simple rescaling for Thomas-Fermi radii with respect to ideal gas prediction !!

In ideal gashohohoideal

maNaR !/,)48( 6/1h==

14

Measurement of in situ column density: role of interactions(Innsbruck, Bartenstein et al. 2004)

7.0!"#

non interacting Fermi gas

BEC BCS

More accurate test of equation of state and of superfluidityavailable from study of collective oscillations (next lectures)

15

The quest for superfluidityin Fermi gases

Experimental probe of superfluidity in trapped Fermi gases

- expansion and aspect ratio- collective oscillations- pairing gap- thermodynamic functions- quantized vortices- spin polarization

16

Collective oscillations and hydrodynamic theory

In linear regime ( ) hydrodynamic equations

take the form

nnneq

!+=

))(( 02

2

nn

nnt

!µ

!"

"##=

"

"

-Surface modes are unaffected by equation of state.

- For isotropic trap one finds where is angular momentum

- surface mode is driven by external potential, not by surface tension

- Dispersion law differs from ideal gas value (interaction effect)

hol!! = l

hol!! =

Solutions of HD equations predictsurface and compression modes

Surface modes

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Compression modes

- Sensitive to the equation of state

- analytic solutions for collective frequencies available for polytropicequation of state [at unitarity (1/a=0) ]

- Example: radial compression mode in cigar trap

- At unitarity one predicts universal value

- For a BEC one finds

- First correction on BEC side due to Lee-Huang-Yang (LHY) eq. of state

beyond mean field effect (Pitaevskii, Stringari 1998)

!µ n"

!"

#$%

&++=+ ...)(

3

321

2

2/13

2

2

MM nangma '

µh

!= ""3

10

!= "" 2

])0(256

1051[2 3

naM

!"" += #

3/2=!

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Radial breathing mode at Innsbruck (2006)

MC equation of state (Astrakharchick et al., 2005)

BCS mean field(Hu et al., 2004)

83.13/10 =

includes LHY effect

does not include LHY effects

Measurement of collective frequencies provides accurate test of equation of state !!

19

Quantized vortices in Fermi gasesobserved along the BEC-BCS crossover (MIT, Nature June 2005, Zwierlein et al.)

Scattering length is suddenly ramped to small and positive values in order to increase visibility of vortex lines

20

Differently from BEC’sphase separation is not easily observed by imaging density profiles of Fermi gas

(bimodal distribution is absent at unitarity as well as in BCS

Phase separation can be nevertheless observed in spin polarized samples

Spin polarized Fermi superfluids!"

# NN

21

Density difference (phase contrast imaging, MIT 2006)

In superfluid phase

In polarized normal phase

!"= nn

!"> nn

!"# nn

Occurrence of phase separation in spin polarized Fermi gasobserved experimentally at unitarity (see also Rice exp)

22

Clogsto

Phase diagram of uniform matter at T=0

Interactions in normal phase play a crucial role in determining critical polarization.Example: neglecting interactions in normal phaseyields

Chandrasekher-Clogston limit at unitarity

1!CP

FFLO phase

!"

!"

+

#=

NN

NNP

39.0=CP

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Density jump at the interface

Exp: MIT (Shin et al. 2007)Theory: Trento (Lobo et al.1996)

Spin up density practically continuous at the interface

Spin down density exhibits jump at the interface

Based on MC equations of state for superfluid and polarized normal phaseTheory predicts critical polarization in excellent agreement with exps

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