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BEC Meeting 23 May 2006
CNRINFM BEC CENTER & Physics Department
Cold Atoms in Optical LatticesG. Orso
●Sound propagation and oscillations of a superfluid Fermi gas in a 1D optical lattice, PRA 2005
●Formation of molecules near a Feshbach resonance in 1D optical lattice, PRL 2005●Twobody problem in periodic potentials, PRA 2006
●Umklapp collisions and oscillations of a trapped Fermi gas, PRL 2004
2. twobody problem in optical lattices
1. how to detect BCS transition in Fermi gases ?
●Sound propagation and oscillations of a superfluid Fermi gas in a 1D optical lattice, PRA 2005
●Formation of molecules near a Feshbach resonance in 1D optical lattice, PRL 2005●Twobody problem in periodic potentials, PRA 2006
●Umklapp collisions and oscillations of a trapped Fermi gas, PRL 2004
2. twobody problem in optical lattices
1. how to detect BCS transition in Fermi gases ?
Superfluid Fermi gas in a 1D optical lattice, PRL 2005
+
=
1D tight optical lattice + weak harmonic trapping
Kohn's theorem does not apply
collisional regime: oscillations killed by umklapp collisions
Relaxation rate from quantum Boltzmann equation
Question: what happens in interacting Fermi gases ?
1/uk
1uk
=C2 a2
d22 tℏ
F TF
,F4 t
T /F
F /4 t=5
21
PRL 2004
superfluid phase: persistent Josephson oscillations why ? phonons protected by energy gap
Can write hydrodynamic equations as for BEC. Ingredients: effective mass and compressibility (from BCS theory)
1ms=
∫ ∂2
∂ k z2
n0kd k
∫ n0 kd kn0k=2F−k
Quasimomentum distribution
eff. mass is density dependent. Recover BEC limit for
BEC: n0k=nk1/msb=1/m*
F≪4 t
●study of dynamical instability●application to trapped gasesPRA 2005
V ext z=s E R sin2 z /d no confinement
in xy directions
s laser intensity, ER=ℏ22
2 m d2recoil energy, d lattice spacing (0.11 m)
Bound states in 1D optical lattice
interparticle interactions modeled via swave pseudopotential Ur =gr ∂r rr
r=r1−r2 relative distance g=4ℏ2 a
mswave scattering
length
Green function for relative motion
1g= ∂∂r
r GE r r=0 GE r =−∑nnr n0
En−E
=CM Rr r V ext r1V ext r2=V CM RV r r
e.g. harmonic potential V ext r =12
m2 r2
COM separates
R=r1r2
2centerofmass coordinate
Binding energy given by algebraic equation
what about optical lattices ? e.g. 1D lattice
c.o.m. and relative motion coupledV opt z1V opt z2≠V CM Z V r z
!!
r , Z =∫ dZ ' GE r , Z ; 0, Z ' g ∂∂ r 'r 'r ' , Z ' r '=0
Solution via 2particle Green's function
Y Z =g∫dZ ' K E Z , Z ' Y Z '
regular kernel
Binding energy given by integral equation in COM variable
binding energy depends on quasimomentum
quasimomentum Q is conserved
invariant under Z Zd
f QZd =ei Q d f QZ solutions are Bloch states
K E Z , Z '
COM and relative motion coupled
For given Q, integral equation has been solved ●numerically PRL 2005●analytically (tightbinding) PRA 2006
s=20
10
5
0
Binding energy versus inverse scattering length
dots: lattice as perturbation
harmonic trap
d /acr binding energy vanishes
(Q=0)
Q=qB
Q=0
Critical value of scattering length
anisotropic 3D regime4t
∣Eb∣≪4 t
qz
qz
Regimes in tight 1D optical lattices
3D regime●bound state spread over many lattice sites●binding energy takes universal form ∣E b∣~
ℏ2
m C 1a− 1acr 2
renormalization of coupling constantg gC
*
g≫∣Eb∣≫4 t quasi2D regime
anisotropic 3D regime
dimensional crossover
4t∣Eb∣≪4 t
qz
qz
Regimes in tight 1D optical lattices
3D regime●bound state spread over many lattice sites●binding energy takes universal form ∣E b∣~
ℏ2
m C 1a− 1acr 2
renormalization of coupling constantg gC
*
quasi2D regime
∣Eb∣=ℏ0 exp−2 ∣a∣●tunneling is irrelevant: effective harmonic oscillator ●binding energy approaches asymptotic value
0.29
0
harmonic oscillator length
dacr=−C ln ℏ02 t
yields formula for
ℏ/m0 1/2
acr numericst. b.
BCS transition in 1D optical lattice
QUESTION: how is Tc modified by 1D lattice ?
1geff
=P∫ d3 q
231q
1exp q /T c
01
q=ℏ2 q p
2
2 mqz− dispersion relation with =F
effective coupling constant for Cooper pairs related to 2body scattering amplitude
geff
1geff
=m
4ℏ2ℜ[ 1f scE=2 ]
incident 2particle state
Scattering amplitude
f scE =a∫ dZE 0, Z ∂r rr , Z r=0
E r , Z =ei q p⋅r pqz z1−qzz2
E=ℏ2 q p
2
m2qz energy
Bloch statesqzz
Y Z ≡∂r rr , Z r=0
Y Z =E 0, Z g∫dZ ' K E Z , Z ' Y Z '
Tightbinding solution
Y Z ~A∑ j w2Z− j d w z=
11/41/2
exp [ −z2
22]
f sc E =aC
1−a /acra /2E
describes dimensional crossoverE
=i arccos(1 E/4t) (E8t)
ansatz:
Wannier state
anisotropic 3D regime (E 8t)
f scE =C
1/a−1/acri C E m /ℏ*
shift of Feshbach resonance
1geff
=m
4ℏ2 C 1a− 1acr density independent e.g. dilute BEC in optical lattices ∣a∣≪∣acr∣ geff≃Cg
lattice enters through effective mass and coupling constant
4 t1
geff=
m4ℏ2 C 1a 12 ln ℏ2 411−4 t /2
density dependent
change in behaviour occurs exactly at =4 t
f scE =C
1a
1
2 [ ln ℏE i]=0.29 quasi2D regime (E 8t)
Results for transition temperature
F4 t
T c0=
2
e−F F /4 t F exp [ −2 qzF C 1∣a∣− 1∣acr∣]3D regime T c0=0.61F exp [ −2 qzF C 1∣a∣− 1∣acr∣]F≪4 t
F4 t
T c0=
2 11− 4 tF F 4 t exp [− 2 1∣a∣− 1∣acr∣]quasi 2D regime
F≫4 tT c
0=0.43F ℏexp−2 ∣a∣KosterlitzThouless
Gorkov's corrections
Gorkov's corrections reduce mean field result T c /T c01
1/e
Van Hove singularity
Quantum phase model
SuperfluidMott Insulator QPT
●zero temperature●discs are finite size and tunneling is weak
t
phase of order parameter
Question: what is critical tunneling rate for transition to Mott phase ?
N j=NN j
in each disc relevant variables arenumber of atoms
j
HYDRODYNAMIC APPROACH
HQP=∑ j E c /2N j2−E J cos j1− j
[N j , j ]=1
Ec charging energy E J Josephson energy
(N 1)
weakly interacting BEC BCS superfluid
E J=t2
FN
E c0.81 E J superfluidE c0.81 E J Mott insulator
tc~FN
Ec=2
N
E JB=t N
tcB~
N 2
BradleyDoniach, PRB (1984)
Complete analogy with BEC:
...but Josephson term is different !
Cooper pairs single atom
3body losses quenched by Fermi statistics
Question: can this QPT be achieved with cold gases ?Answer: only if critical tunneling rate large compared to atom loss rate due to 3body recombination
● BEC: NO, unless● BCS superfluids: YES, up to N~103−104
N~1−5
Ultracold gases
Conclusions
umklapp collisions in Fermi gasespropagation of sound in superfluid Fermi gases twobody problem BCS transition temperature SuperfluidMott insulator QPT in Fermi superfluids
Conclusions
umklapp collisions in Fermi gasespropagation of sound in superfluid Fermi gases twobody problem BCS transition temperature SuperfluidMott insulator QPT in Fermi superfluids