Theory of interacting Bose and Fermi gases in traps

Sandro Stringari

University of Trento

Crete, July 2007 Summer School on Bose-Einstein Condensation

CNR-INFM

1st lecture

Role of the order parameter

Quantum statistics and temperature scales

3/194.0 NTk CB 3/1)6( NTk FB

- When T tends to 0 a macroscopic fraction of bosons occupies a single particle state (BEC)

- Wave function of macroscopically occupied single particle state defines order parameter

- Actual form of order parameter depends on two-body interaction (Gross-Pitaevskii equation)

- In the absence of interactions the physics of fermions deeply differs from the one of bosons (consequence of Pauli principle)

- Interactions can change the scenario in a drastic way: - pairs of atoms can form a bound state (molecule) and give rise to BEC - pairing can affect the many-body physics also in the absence of two-body molecular formation (many-body or Cooper pairing) giving rise to BCS superfluidity

Bosons

Fermions

First lecture

Theory of order parameter for both Bose and Fermi gases.Microscopic nature of order of parameter (and corresponding equations) very different in the two cases

Second lecture

Unifying approach to dynamics of interacting Bose and Fermi gases in the superfluid regime. Structure of equations of superfluid dynamics (irrotational hydrodynamics) in the macroscopic regime is the same for fermions and bosons

1-body density matrix and long-range order

)'(ˆ)(ˆ)',()1( rrrrn

(Bose field operators)

/)1(3 )2/),2/()2()( ipsesRsRndRdspn

),()( )1( rrnrn

Relevant observables related to 1-body density:

- Density:

- Momentum distribution:

/)1()1( )(1)()',( ipsepdpnV

snrrnIn uniform systems

Bosons

'rr 0

For large N the sum can be replaced by integral which tends to zero at large distances. Viceversa contribution from condensate remains finite up to distances fixed by size of

BEC and long range order: consequence of macroscopic occupation of a single-partice state ( ) . Procedure holds also in non uniform and in strongly interacting systems .

Long range order and eigenvalues of density matrix

)()'()',(' )1( rnrrrndr iii )'()()',( *)1( rrnrrn iiii

BEC occurs when . It is then convenient to rewritedensity matrix by separating contribution arising from condensate:

)'()()'()()',( *00

*00

)1( rrnrrNrrn iiii

10 Nno

Bosons

10 N

Off-diagonal long range order (Landau, Lifschitz, Penrose, Onsager)

Example of calculation of density matrix in strongly correlated superfluid: liquid He4(Ceperley, Pollock 1987)

Vr 1)(0

CTT

VNnsn s

00

)1( )( In bulk matter

Bosons

)(~)()( 0 pnpNpn or

Diagonalization of 1-body density matrix permits to identify single particle wave functions . In terms of these functions one can write field operator in the form:

iiiararr ˆ)(ˆ)()(ˆ

000

i

If (BEC) one can use Bolgoliubov approximation(non commutativity unimportant for most physical properties within 1/N approximation).

10 N000 ˆ,ˆ Naa

1]ˆ,ˆ[ 00 aa

Bosons

ORDER PARAMETER

)(ˆ)()(ˆ rrr

)()(ˆ)( 00 rNrr iiiarr ˆ)()(ˆ

0

Order parameter(gauge symmetry breaking)

Quantum and thermalfluctuations

)(ˆ)(ˆ)(ˆ)(ˆ2

)(ˆ)(2

)(ˆ 22

rrrrdrgrrVm

rdrH ext

Bosons

Many-body Hamiltonian

mag /4 2

Basic assumption: Almost all the particles occupy a single particle state (no quantum depletion; no thermal depletion)

Field operator can be safely replaced by classical field

),(ˆ),(),(ˆ trtrtr Density coincides with condensate density

2),( tr

),'(ˆ),,(ˆ),( trtrtrn

Zero range potential

a =s-wave scattering length

Dilute Bose gas at T=0

2422

2

21)(

2 grV

mdrNE ext

Energy can be written in the form

Variational procedure

yields equation for order parameter (Gross-Pitaevskii, 1961)

)()()]()(2

[ 22

rrrgnrVm ext

Bosons

0/)( * NE

Conditions for applicability of Gross-Pitaevskii equation

- diluteness: (quantum fluctuations negligible)

- low temperature (thermal fluctuations negligble)

13 na

CTT

HE

- Gross-Pitaevskii (GP) equation for order parameter plays role analogous to Maxwell equations in classical electrodynamics.- Condensate wave function represents classical limit of de Broglie wave (corpuscolar nature of matter no longer important)

Important difference with respect to Maxwell equations:GP contains Planck constant explicitly. Follows from different dispersion law of photons and atoms:

mpE

cpE

2/2

mk

ck

2/2

Ekp ,

photons

atoms

particle (energy) wave (frequency)

from particles to waves:

GP eq. is non linear (analogy with non linear optics)GP equation often called non linear “Schroedinger equation” Equation for order parameter is not equation for wave function

Bosons

BEC in harmonic trap 22

21 rmV hoext

Non interacting ground state

)/exp()( 22hoarrn Gaussian with width

hoho ma

depends on

Role of interactions

Using and as units of lengths and energy, and

GP equation becomes

)~(~~2)~(~)]~(~)/(8~~[ 222 rrraNar ho

If

If 1/ hoaNa

1/ hoaNa Non interacting ground state

Thomas-Fermi limit (a>0)

hoa ho 2/32/1~hoaN

normalized to 1

dimensionless Thomas-Fermi parameter

Bosons

In Thomas Fermi limit kinetic energy can be ignored and density profile takes the form (for n>0)

))((1)( 0 rVg

rn ext

Thomas-Fermi radius R is fixed by condition of vanishing density

with fixed by normalization. One finds 220 2

1 Rm ho

Thomas-Fermi condition implies1/ hoaNa

0

5/20 )15(

21

hoho a

aN

hoho aR ,0

5/1)15(ho

ho aaNaR

Does not depend on

Bosons

Thomas-Fermi parameter drives the transition from non interacting to Thomas-Fermi limit

Some conclusions concerning equilibrium profiles

Huge effects due to interaction at equilibrium;good agreementwith experiments

non interacting

hoaNa /

a >0

non interacting

wave function

column density

exp: Hau et al, 1998

hoaNa /

GP

Bosons

Thomas-Fermi regime is compatible with diluteness condition

Gas parameter in the center of the trap

5/126/133 )(1.0hoaaNa

gna

Thomas-Fermi Diluteness

1/ hoaNa 1/6/1 hoaaN

example:

63 10,10/ Naa ho

310/ hoaNa 26/1 10/ hoaaN

Gross-Pitaevskii theory is not perturbativeeven if gas is dilute (role of BEC)!

Bosons

Microscopic approach to superfluid phase is much more difficult in Fermi than in Bose gas (role of the interaction and of single particle excitations is crucial to derive equation for the order parameter)

Fermions

Order parameter is proportional to (pairing !!)rather than to

ˆˆ ˆ

Equation for order parameter follows from proper diagonalization of many body Hamiltonian.

)(ˆ)'(ˆ)'(ˆ)(ˆ)'(')(ˆ2

)(ˆ 22

rrrrrrVdrdrrVm

rdrH ext

- Interaction at short distances is active only in the presence of two spin species (consequence of Pauli principle)

- ( ) regularized potential (Huang and Yang 1957) (needed to cure ultraviolet divergencies, arising from 2-body problem)

mag /4 2rrrgrV )/)(()(

Fermi field operator

Many-body Hamiltonian can be diagonalized if one treats pairing correlations at the mean field level.

)(11)(4

)2/(ˆ)2/(ˆ),(

)'()2/(ˆ)2/(ˆ)()(

..)(ˆ)(ˆ)()(ˆ)'(ˆ)'(ˆ)(ˆ)'('

2

0

soas

RmsrsrsRF

sFgsrsrsdsVr

chrrrdrrrrrrrVdrdr

s

- Mean field Hamiltonian is bi-linear in the field operators - can be diagonalized by Bogoliubov transformation which transforms particle into quasi particle operators

)()(

)()(

)()(

0*

0

rvru

rvru

HrrH

i

ii

i

i

)()2/( 220 rVmH ext

(Bogoliubov - de Gennes Eqs.)

Fermions

Order parameter

ˆˆˆ vu

Diagonalization is analytic in uniform matter. Hamiltonian takes the form of Hamiltonian of a gas of independent quasi-particles with energy spectrum

2222 )2/( mkk

Coupled equations for and are obtained by imposing self-consistency condition for pairing field F(s) and value of density:

)21(

)2(1

4 2232kk

mkda

m

))2/(1

)2(1 22

3k

mkkdn

BCS mean field equations

Fermions

T=0 + extensions to finite T:

Eagles (1969)Leggett (1980)Nozieres andSchmitt-Rink (1985)Randeira (1993)

What is BCS mean field theory useful for ?

Provides prediction for equation of state

and hence for compressibility

)(n

)(2 nn

nmc

- Predicts gapped quasi-particle excitation spectrum

- According to Landau’s criterion for critical velocity

occurrence of gap implies superfluidity (absence of viscosity and existence of persistent currents)

pv p

pcr

min

Key role plaid by order parameter !!

Results for uniform matter can be used in trapped gases using LDA

222 )2/( mpp

Fermions

0

When expressed in units of Fermi energy

Equation of state, order parameter and excitation spectrum

depend on dimensionless combination

This feature is not restricted to BCS mean field, but holds in general for broad resonances where the scattering length is the only interaction parameter determining the macroscopic properties of the gas

Holds if scattering length is much larger than effective range of the potential

3/222

)3(2

nmF

akF

Scattering length a is key interaction parameter of the theory:

Determined by solution of Schrodinger equation for the two-body problem

0ra

Fermions

When scattering length is positive weakly bound molecules of size a and binding energy are formed If size of molecules is much smaller than average distance between molecules the gas is a BEC gas of molecules

In opposite regime of small small and negative values of a size of pairs is larger than interparticle distance (Cooper pairs, BCS regime)

In the presence of Feshbach resonance the value of a can be tuned by adjusting the external magnetic field At resonance a becomes infinite

22 /ma

1akF

Fermions

Fermions

Some key predictions :

BEC regime ( )

- Chemical potential (gas of independent molecules)

- Single particle gap (energy needed to break a molecule)

1;0 aka F

22 /ma

22 /magap

BCS regime ( )

- Chemical potential ( weakly interacting Fermi gas)

- Single particle gap (Gap coincides with order parameter and is exponentially small)

1;0 aka F

F

0)2/exp(8 2 ake FFgap

Many-body aspects (BEC-BCS crossover)

BCS regimeunitary limit

BEC regime

Fermions

                                                          

  

2003: Molecular Condensates

JILA: 40K2

6Li2:Innsbruck

ENS6Li2

MIT6Li2

6Li2 7Li

Also Rice 6Li2

Fermions

- Basic many body features well accounted for by BCS mean field theory.

- However BCS mean field is approximate and misses important features

For example: on BEC side of resonance this theory correctly describes gasof molecules with binding energy .

However these molecules interact with wrong scattering length

correct value is Petrov et al, 2004))

22 /ma

aaM 2

aaM 6.0

Equation of state predicted by BCS mean field is approximate.

- Exact many-body calculatons of equation of state are now available along the whole BEC-BCS crossover using Quantum Monte Carlo techniques (Carlson et. al; Giorgini et al 2003-2004))- QMC calculations gives also access to gap parameter.

Fermions

BCS mean field

Equation of state along the BEC-BCS crossover

Monte Carlo(Astrakharchicket al., 2004)

Fermions

BEC BCS 0

ideal Fermi gas

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

Nnnn

0

- Behavior of equation of state is much richer than in dilute Bose gases where (Bogoliubov equation of state)

- Possibility of exploring both positive and negative values of scattering length including unitary regime where scattering length takes infinite value

gn

Fermions

Behaviour at resonance (unitarity)

- At resonance the system is strongly correlated but its properties do not depend on value of scattering length a (independent even of sign of a). UNIVERSALITY.

- UNIVERSALITY requires (dilute, but strongly interacting system) All lengths disappear from the calculation of thermodynamic functions (similar regime in neutron stars)

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

Atomic chemical potential 3/23/22

2

)1(62

nm

1akF

10 rkF

0 for ideal Fermi gas

dimensionless interaction parameter characterizing unitary regime

Values of beta:Mean field -0.4Monte Carlo: -0.6

Fermions

Equation of state can be used to calculate density profiles using Local density approximation:

For example at unitarity

)()(0 rVn ext

6/14/1

2/3

2

2

3

)1(

18)(

NaR

Rr

RNrn

ho

)(rn

- From measurement of density profiles one can determine value of interaction parameter

-Value of measurable also from release energy (ENS 2004) and sound velocity (Duke 2006) (see next lecture)

Fermions

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

7.0

non interacting Fermi gas

BEC BCS

Fermions

More accurate test of equation of state and of superfluidity available from study of collective oscillations (next lecture)

Key parameter of theory (Gross-Pitaevskii eqs. for BEC ) (Bogoliubov de Gennes eqs. for Fermi superfluids )

Directly related to basic features of superfluids: - density profiles in dilute BEC gases (easily measured) - gap parameter in Fermi superfluids (relevant for Landau’s criterion of superfluidity, measurable with rf transitions ?)

In both Bose and Fermi superfluids order parameter is a complex quantity.(modulus + phase). This lecture mainly concerned with equilibrum configurations where order parameter is real Phase of order parameter plays crucial role in the theory of superfluids: - accounts for coherence phenomena (interference) - determines superfluid velocity field: important for quantized vortices, solitons and dynamic equations (next lecture)

Summary: role of order parameter in superfluids

2)()( rrn

- Theory of Bose-Einstein Condensation in trapped gases F. Dalfovo et al., Rev. Mod. Phys. 71, 463 (1999)

- Bose-Einstein Condensation in Dilute Gases C. Pethick and H. Smith (Cambridge 2001)

- A. Leggett, Rev. Mod. Phys. 73, 333 (2001)

- Bose-Einstein Condensation L. Pitaevskii and S. Stringari (Oxford 2003

- Ultracold Fermi gases Proccedings of 2006 Varenna Summer School W. Ketterle, M. Inguscio and Ch. Salomon (in press)

- Theory of Ultracold Fermi gases S. Giorgini et al. cond-mat/0706.3360

General reviews on BEC and Fermi superfluidity