Transcript Diapositiva 1 - Foundation for Research and Technology

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
Quantum statistics and temperature scales
kBTC  0.94  N1/ 3
kBTF   (6N )
1/ 3
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 to BCS 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
Bosons
1-body density matrix and long-range order
ˆ  (r )
ˆ (r ' )
n (1) (r , r ' )  
(Bose field operators)
Relevant observables related to 1-body density:
- Density:
n(r )  n (r, r )
(1)
- Momentum distribution:
n( p )  (2 ) 3  dRds n (1) ( R  s / 2), R  s / 2)e  ips / 
In uniform systems
1
n (r , r ' )  n ( s)   dpn ( p)eips / 
V
(1)
(1)
Bosons
Long range order and eigenvalues of density matrix
(1)
dr
'
n
(r , r ' ) i ( r ' )  ni i (r )

n(1) (r, r ' )  i nii* (r )i (r ' )
BEC occurs when no  N0  1 . It is then convenient to rewrite
density matrix by separating contribution arising from condensate:
n(1) (r, r ' )  N00* (r )0 (r ' )  i 0 nii* (r )i (r ' )
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 r  r' fixed by size of  0
BEC and long range order: consequence of macroscopic
occupation of a single-partice state ( N0  1 ) .
Procedure holds also in non uniform and in strongly
interacting systems .
Bosons
In bulk matter
or
0 (r ) 
1
V
~( p)
n( p)  N0 ( p)  n
(1)
n ( s ) s 
N0
 n0 
V
Off-diagonal long range order
(Landau, Lifschitz, Penrose, Onsager)
T  TC
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 i.
ˆ (r )   (r )aˆ 
In terms of these functions one can

0
0
i 0
write field operator in the form:

If N0  1 (BEC) one can use Bolgoliubov approximation
(non commutativity [aˆ0 , aˆ0 ]  1 unimportant for
most physical properties within 1/N approximation).
 i (r )aˆi
aˆ0 , aˆ0  N 0
ˆ (r )  (r )  
ˆ (r )

ˆ (r )  N  (r )
(r )  
0 0
Order parameter
(gauge symmetry breaking)
ˆ (r )    (r )aˆ

i
i
i 0
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

ˆ (r, t )   (r, t )  
ˆ (r , t )

Many-body Hamiltonian

ˆ  (r, t ), 
ˆ (r' , t ) 
n(r , t )  
Density
coincides with condensate density
 (r , t )
2
Zero range potential
2

ˆ

g
2
ˆ
ˆ  (r )
ˆ  (r )
ˆ (r )
ˆ (r )
H   dr (r ) 
  Vext (r ) (r )   dr
2
 2m


g  4 2 a / m
a =s-wave scattering length
Bosons
Energy
E  H  can be written in the form
 2
1
2
2
4
2
E  N   dr
  Vext (r )   g     
2
 2m

Variational procedure
 ( E  N ) / *  0
yields equation for order parameter
(Gross-Pitaevskii, 1961)
2 2
[
  Vext (r )  gn(r )] (r )   (r )
2m
Conditions for applicability of Gross-Pitaevskii equation
- diluteness:
na3  1 (quantum fluctuations negligible)
- low temperature T  TC (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:
p  k , E  
photons
E  cp
  ck
atoms
E  p 2 / 2m
  k / 2m
particle (energy)
wave (frequency)
2
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
Vext 
1
2 2
mho
r
2
Non interacting ground state
n(r )  exp(r / a )
2
2
ho
Gaussian with width aho 

m ho
depends
on 
Role of interactions
Using
aho
and
ho
~
as units of lengths and energy, and   N 1/ 2a3/ 2
ho
GP equation becomes
normalized to 1
~ 2 ~2
~2 ~ ~ ~
~ ~
~
[  r  8 ( Na / aho ) (r )](r )  2(r )
dimensionless Thomas-Fermi parameter
If Na / aho  1
Non interacting ground state
If Na / aho  1
Thomas-Fermi limit (a>0)
Bosons
In Thomas Fermi limit kinetic energy can be ignored
and density profile takes the form (for n>0)
1
n(r )  ( 0  Vext (r ))
g
Does not
depend on 
Thomas-Fermi radius R is fixed by condition of vanishing density
1
2
 0  mho2 R 2
with
0
fixed by normalization. One finds
1
a 2/5
0  ho (15N
)
2
aho
Thomas-Fermi condition
a 1/ 5
R  aho (15N
)
aho
Na / aho  1implies
0  ho , R  aho
Bosons
Some conclusions concerning equilibrium profiles
a >0
non interacting
Thomas-Fermi parameter
Na / aho drives the transition
from non interacting to
Thomas-Fermi limit
wave function
Na / aho
Huge effects due to
interaction at equilibrium;
good agreement
with experiments
exp: Hau et al, 1998
column density
GP
non interacting
Bosons
Thomas-Fermi regime is compatible with diluteness condition
Gas parameter in the center of the trap
na 
3

g
a  0.1( N
3
1/ 6
Thomas-Fermi
Diluteness
Na / aho  1
example:
a 12 / 5
)
aho
N 1/ 6 a / aho  1
a / aho  103 , N  106
Na / aho  10
3
N 1/ 6a / aho  102
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
ˆ 
rather than to  
ˆ
ˆ 

(pairing !!)
Fermi field operator
Equation for order parameter follows from proper
diagonalization of many body Hamiltonian.
2

ˆ

2
ˆ
ˆ  (r )
ˆ  (r ' )
ˆ (r ' )
ˆ (r )

H   dr (r ) 
  Vext  (r )   drdr'V (r  r ' )




2
m



- Interaction at short distances is active only in the presence of two spin
species (consequence of Pauli principle)

2
- V (r )  g (r )( / r )r ( g  4 a / m ) 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.
ˆ  (r )
ˆ  (r ' )
ˆ (r ' )
ˆ (r )   dr(r )
ˆ  (r )
ˆ  (r )  h.c.
drdr
'
V
(
r

r
'
)









ˆ ( r  s / 2) 
ˆ (r  s / 2)   g ( sF )'
(r )    dsV ( s ) 
s 0


ˆ (r  s / 2)
ˆ (r  s / 2)  m ( R ) 1  1   o( s )
F ( R, s )  


4 2
s a
Order parameter
- Mean field Hamiltonian is bi-linear
in the field operators
- can be diagonalized by Bogoliubov
transformation which transforms
particle into quasi particle operators
ˆ  u
ˆ  v
ˆ

(Bogoliubov - de Gennes Eqs.)



(r )  ui (r ) 
 H0
 ui (r ) 
 * 
     i   
  (r )  H 0  vi (r ) 
 vi (r ) 

2
2
H0  ( / 2m)  Vext (r )  
Fermions
Diagonalization is analytic in uniform matter.
Hamiltonian takes the form of Hamiltonian of a gas of
independent quasi-particles with energy spectrum
 k  2  ( 2 k 2 / 2m   ) 2
Coupled equations for  and  are obtained by imposing
self-consistency condition for pairing field F(s) and value of density:
 1
m
m
1
  dk
( 2 2
)
2
3
4 a
(2 )  k
2 k

n   dk
 ( 2 k 2 / 2m   ) 
1 
) 
3 
(2 ) 
k

1
BCS mean field equations
T=0 + extensions
to finite T:
Eagles (1969)
Leggett (1980)
Nozieres and
Schmitt-Rink (1985)
Randeira (1993)
Fermions
What is BCS mean field theory useful for ?
Provides prediction for equation of state
 (n)
mc 2  n

 ( n)
n
and hence for compressibility
- Predicts gapped quasi-particle excitation spectrum
 p  2  ( p 2 / 2m   ) 2
- According to Landau’s criterion
for critical velocity
vcr  min p
 0
p
p

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
2
F 
(3 2 n) 2 / 3
2m
Equation of state, order parameter and excitation spectrum
depend on dimensionless combination
kF a
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 a  r
0
Scattering length a is key 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  2 / m a2 are formed
If size of molecules is much smaller than average distance between
molecules k F a  1 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 ( a  0 ; kF a  1 )
- Chemical potential    2 / ma2 (gas of independent molecules)
- Single particle gap  gap   2 / ma2 (energy needed to break a molecule)
BCS regime ( a  0 ; k a  1 )
F
- Chemical potential    F ( weakly interacting Fermi gas)
- Single particle gap     8e2 exp( / 2k a)  0
gap
F
F
(Gap coincides with order parameter
and is exponentially small)
Fermions
Many-body aspects (BEC-BCS crossover)
BEC regime
unitary limit
BCS regime
Fermions
2003: Molecular Condensates
JILA: 40K2
MIT
6Li
2
6Li :Innsbruck
2
ENS
6Li
2
6Li
2
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 gas
of molecules with binding energy  2 / m a2 .
However these molecules interact with wrong scattering length
correct value is
aM  2a
aM  0.6a 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   0
BCS   0
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
  gn (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 kF a  1 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 k F r0  1(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):
 
2
Atomic chemical potential  
6 2
2m

2/3
(1   )n
dimensionless interaction parameter
characterizing unitary regime
2/3
 0
for ideal Fermi gas
Values of beta:
Mean field -0.4
Monte Carlo: -0.6
Fermions
Equation of state can be used to calculate density profiles using
Local density approximation:

0   (n)  Vext (r )
For example at unitarity

n(r )

8 N  r 
1  2 
n( r ) 
3 
 R  R 
R  (1   )1/ 4 aho N 1/ 6
2
3/ 2
- 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
  0.7
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:
2
- density profiles in dilute BEC gases n( r )   ( r ) (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