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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