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Crete, July 2007 Summer School

on Bose-Einstein Condensation

Theory of interacting Bose and Fermi gases in traps

Sandro Stringari

1st lecture

Role of the order parameter

University of Trento

CNR-INFM

Bosons

- 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)

Fermions

- - 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 toBCS superfluidity

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

Bosons1-body density matrix and long-range order

(Bose field operators)

Relevant observables related to 1-body density:

- Density:

- Momentum distribution:

In uniform systems

Bosons

Long range order and eigenvalues of density matrix

BEC occurs when . It is then convenient to rewrite

density matrix by separating contribution arising from condensate:

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 .

Bosons

In bulk matter

Off-diagonal long range order

(Landau, Lifschitz, Penrose, Onsager)

or

Example of calculation

of density matrix in strongly

correlated superfluid:

liquid He4

(Ceperley, Pollock 1987)

Bosons

ORDER PARAMETER

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:

If (BEC) one can use Bolgoliubov approximation

(non commutativity unimportant for

most physical properties within 1/N approximation).

Order parameter

(gauge symmetry breaking)

Quantum and thermal

fluctuations

Bosons

Dilute Bose gas at T=0

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

Density

coincides with condensate density

Many-body Hamiltonian

Zero range potential

a =s-wave scattering length

Bosons

Energy can be written in the form

Variational procedure

yields equation for order parameter

(Gross-Pitaevskii, 1961)

Conditions for applicability of Gross-Pitaevskii equation

- diluteness: (quantum fluctuations negligible)
- low temperature (thermal fluctuations negligble)

Bosons

- 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:

from particles to waves:

photons

atoms

particle (energy)

wave (frequency)

GP eq. is non linear (analogy with non linear optics)

GP equation often called non linear “Schroedinger equation”

Equation for order parameteris not equation for wave function

Bosons

BEC in harmonic trap

Non interacting ground state

depends

on

Gaussian with width

Role of interactions

Using and as units of lengths and energy, and

GP equation becomes

normalized to 1

dimensionless Thomas-Fermi parameter

If

Non interacting ground state

If

Thomas-Fermi limit (a>0)

Bosons

In Thomas Fermi limit kinetic energy can be ignored

and density profile takes the form (for n>0)

Does not

depend on

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

with fixed by normalization. One finds

Thomas-Fermi condition implies

Bosons

Some conclusions concerning equilibrium profiles

a >0

non interacting

Thomas-Fermi parameter

drives the transition from non interacting to

Thomas-Fermi limit

wave function

column density

non interacting

Huge effects due to

interaction at equilibrium;

good agreement

with experiments

GP

exp: Hau et al, 1998

Bosons

Thomas-Fermi regime is compatible with diluteness condition

Gas parameter in the center of the trap

Thomas-Fermi

Diluteness

example:

Gross-Pitaevskii theory is not perturbative

even if gas is dilute (role of BEC)!

Fermions

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)

Order parameter is proportional to (pairing !!)

rather than to

Fermi field operator

Equation for order parameter follows from proper

diagonalization of many body Hamiltonian.

- - 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)

Fermions

Many-body Hamiltonian can be diagonalized if one

treats pairing correlations at the mean field level.

Order parameter

(Bogoliubov - de Gennes Eqs.)

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

Fermions

Diagonalization is analytic in uniform matter.

Hamiltonian takes the form of Hamiltonian of a gas of

independent quasi-particles with energy spectrum

Coupled equations for and are obtained by imposing

self-consistency condition for pairing field F(s) and value of density:

T=0 + extensions

to finite T:

Eagles (1969)

Leggett (1980)

Nozieres and

Schmitt-Rink (1985)

Randeira (1993)

BCS mean field equations

Fermions

What is BCS mean field theory useful for ?

Provides prediction for equation of state

and hence for compressibility

- 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)

Key role plaid by order parameter !!

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

Fermions

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

Scattering length a iskey interaction parameter of the theory:

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

Fermions

In the presence of Feshbach resonance

the value of a can be tuned

by adjusting the external magnetic field

At resonance a becomes infinite

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)

Fermions

Some key predictions :

- BEC regime ( )
- - Chemical potential (gas of independent molecules)
- Single particle gap (energy needed to break a molecule)

- BCS regime ( )
- - Chemical potential ( weakly interacting Fermi gas)
- Single particle gap
- (Gap coincides with order parameter
- and is exponentially small)

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 gas
- of molecules with binding energy .
- However these molecules interact with wrong scattering length
- correct value is Petrov et al, 2004))

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

Equation of state along the BEC-BCS crossover

BCS mean field

ideal Fermi gas

Nnnn

Monte Carlo

(Astrakharchick

et al., 2004)

BEC

BCS

Energy is always smaller than ideal

Fermi gas value. Attractive role of

interaction along BCS-BEC crossover

Fermions

- - 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

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

for ideal Fermi gas

Values of beta:

Mean field -0.4

Monte Carlo: -0.6

dimensionless interaction parameter

characterizing unitary regime

Fermions

Equation of state can be used to calculate density profiles using

Local density approximation:

For example at unitarity

- - 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)

non interacting Fermi gas

BEC

BCS

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

Summary: role of order parameter in superfluids

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)

General reviews on BEC and Fermi superfluidity

- - 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

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