Outline - École Normale Supérieurecastin/shlyapnikov.pdf · Outline Introduction Dipole moment...
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Dipolar quantum gases
Gora ShlyapnikovLPTMS, Orsay, France
University of Amsterdam
OutlineIntroduction
Dipole moment
Progress in creating dipolar gases
Scattering problem. Ultracold limit
Dipolar BEC. Uniform case
Trapped dipolar BEC
Stability problem
Collaborations: P. Pedri (Orsay), L. Santos (Hannover), M.A. Baranov (Amsterdam), M. Lewenstein (Barcelona) . – p.1/42
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Dipolar quantum gases
Roton-maxon spectrum. Prehistory
Roton-maxon spectrum of a dipolar BEC
Thermal fluctuation
Roton instability and collapse
——————————————————————————
Dipolar Feri gas. Interaction problem
Superfluid pairing in a single-component dipolar Fermigas
Interesting problem to solve
Collaborations: P. Pedri (Orsay), L. Santos (Hannover), M.A. Baranov (Amsterdam), M. Lewenstein (Barcelona)
. – p.2/42
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Dipolar gas
Polar molecules, atoms with a large magnetic moment
Dipole-dipole interaction
Vd =~d1~d2R
2 − 3(~d1~R)(~d2
~R)
R5∼
1
R3
d1
d 2
Rlong-range, anisotropic
repulsion
attraction
Different physics compared to ordinary atomic ultracold gases. – p.3/42
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Dipole moment
Polar MoleculesCO 0.1D ≈ 0.05au
CsCl 10D ≈ 5au
Atoms with a large magnetic moment
Cr ⇒ 6µB
equivalent to d = 0.048D = 0.022auPresently a quatum gas of Cr bosons is created andBose-condensed (T. Pfau, Stuttgart)
Cr ⇒ Vd ∼ d2
r3 ∼ 10µK at r ∼ 200a0
and already exceeds the Van der Waals interaction
. – p.4/42
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Polar molecules
Buffer-gas cooling (Doyle, Harvard)
n ~ 10 cm8 −3
3He vapor
~~T 300mK
CaH
magnetic trap
———————————————————————Stark deceleration (Meijer, Berlin)
H3, ND3, CO, etc. n ∼ 108cm−3; T ∼ 10mK———————————————————————
Photoassociation(many labs; De Mille, Yalle)
R
U(R)
Rb Cs
small n, T 100µK
. – p.5/42
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Theoretical studies of dipolar gases
Bose gas
L. Santos, M. Lewenstein, P. Zoller, G. S.,C. Eberlein, S. Giovanazzi, O’Dell,K. Góral, M. Brewekyk, K. Rzatenski,S. Yi and L. You,S. Giovanazzi, A. Gorliz, T. Pfau.
Fermi gasM. A. Baranov, M. Lewenstein, et al.K. Góral, K. Rzatenski, M. Brewekyk
Bosons/Fermions in optical lattices
(L. Santos).... – p.6/42
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Dipole-diple scattering
d
d
r
Vd = d2
r3
(1 − 3 cos2 θrd
)
︸ ︷︷ ︸
∼Y20(θrd)
Wave function of the relative motion
ψin →∑
l,m ψkl(r)ilY ∗
lm(θkd, ϕkd)Ylm(θrd, ϕrd)
ψout →∑
l′,m′ ψkl′(r)il′Y ∗
l′m′(θkd, ϕkd)Yl′m′(θrd, ϕrd)
Scattering matrix ∼∫Y ∗
lmYl′m′Y20dΩrd
Vd couples all even l, and all odd l, but even and odd l aredecoupled from each other
. – p.7/42
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Radius of the dipole-dipole interaction
d
d
r
(
−~2
m∆ + Vd(~r))
ψ(~r) = ~2k2
m ψ(~r)
~2
mr2∗
= d2
r3∗
⇒ r∗ ≈ md2
~2
r ≫ r∗ → free relative motionr∗ ∼ 106 ÷ 102a0 polar moleculesr∗ ≈ 50a0 → chromium atoms
kr∗ ≪ 1 → Ultracold limit︸ ︷︷ ︸
T≪1mK for Cr
. – p.8/42
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Scattering amplitude I
d
d
r
V (~r) = U(~r) + Vd(~r)
f =∫ψ∗
ki(~r)V (~r)ei
~kf~rd3r
Ultracold limit kr∗ ≪ 1
Vd = 0 ⇒ f = g = 4π~2
m a————————————————————————
What Vd does?
k = 0 → g =∫ψ∗
0(~r)(U(~r) + Vd(~r))d3r = const; r . r∗
g may depend on d and comes from all even l
. – p.9/42
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Scattering amplitude II
k 6= 0
f =∫ψ∗
ki(~r)V (~r)ei
~kf~rd3r
r . r∗ → put k = 0 → g
r ≫ r∗ → ψki= ei
~ki~r
f =∫Vd(~r)e
i~q~rd3r −→ 4πd2
3 (3 cos2 θqd − 1); ~q = ~kf − ~ki
f = g + 4πd2
3 (3 cos2 θqd − 1)
. – p.10/42
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Dipolar BEC I
uniform gas
H =∫d3
[
ψ†(~r)(
− ~2
2m∆)
ψ(~r) + 12gψ
†(~r)ψ†(~r)ψ(~r)ψ(~r)
+12
∫d3r′ψ†(~r)ψ†(~r′)Vd(~r − ~r′)ψ(~r′)ψ(~r)
]
————————————————
Bogoliubov approach ψ = ψ0 + δΨ → biliniear hamiltonian
HB = N2
2V g +∑
k
[~
2k2
2m a†kak + n(
g + 4πd2
3
(3 cos2 θk − 1
))
a†kak
n2
(
g + 4πd2
3
(3 cos2 θk − 1
)) (
a†ka†−k + aka−k
) ]
. – p.11/42
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Dipolar BEC II
Excitation spectrum
ǫk =
√
E2k + 2Ekn
(
g + 4πd2
3 (3 cos2 θk − 1))
g > 4πd2
3 → dynamically stable BEC
g < 4πd2
3 → complex frequencies at small k
and cos2 θk <13 → collapse
. – p.12/42
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Trapped dipolar BEC
z
ρ
Cylindrical trapVh = m
2
(ω2
ρρ2 + ω2
zz2)
Gross-Pitaevskii equation[
− ~2
2m∆+Vh(~r)+ gψ20 +
∫ψ0(~r
′)2Vd(~r−~r′)d3r′
]
ψ0(~r) = µψ0(~r)
Important quantityVeff = g
∫ψ4
0(~r)d3r +
∫ψ2
0(~r′)Vd(~r − ~r′)ψ2
0(~r)d3rd3r′
ωρ ωz
N c Veff > 0 or Veff < 0 and |V | < ~ω
g = 0 → N < Nc → suppressedlow k instability(Santos et.al, 2000)
. – p.13/42
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It is sufficient?
Veff< 0
N r*
l max
ωz
ω ρ= 1
z
ρ
R
R
Trap Cloud
1
2.1
12
N r*
l max
ωz
ω ρ> 0.4
Rz
ρRTrap Cloud
0.1
Veff > 0
. – p.14/42
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Stability problem
z
Dipolar BEC
〈Vd〉 =∫n0(~r
′)Vd(~r′ − ~r)d3r = −d2 ∂2
∂z2
∫ n0(~r′)
|~r−~r′|d3r
Vd = −d2 ∂2
∂z2
1|~r−~r′| −
4πd2
3 δ(~r − ~r′)
Large N ⇒ Thomas-Fermi BEC
n0 = n0 max
(
1 − z2
R2z− ρ2
R2ρ
)
Eberlein et. al (2005)
g > 4πd2
3 → stable at any N
. – p.15/42
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Example
π4 d2
3gε =
0.2
0.2 0.4 0.6 0.8
1
2
10
= 2
= 1
= 0.2
ωωz
ρ
ωω
ωω
z
ρ
z
ρ
. – p.16/42
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Experiment with Cr
g > 4πd2
3
(µ = 6µB!)
(T. Pfau, Stuttgart) BEC (n ∼ 1014cm−3)
effect of the dipole-dipole interaction (small)
z
y
time of flight
z
y
time of flight
z
y
time of flight
n r( )
B
m
Fdd( )r
. – p.17/42
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0 2 4 6 8 10 12 140
0.5
1
1.5
2
2.5
3
3.5
4
time of flight [ms]
aspect ra
tio
/RyRz
z
y
m Bm, along :B y
m, along :B zm B
. – p.18/42
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Question
What happens at
g < gd =4πd2
3?
. – p.19/42
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Pancake dipolar BEC
Thomas-Fermi in the z direction
Extreme pancake (ωρ = 0)
z
ρl0 =
(~
mωz
)1/2
Vd + g (short-range) g > 0
q l0
hωz
ε
~1
g/gd = 1.06 µ/~ωz = 46g/gd = 0.94 µ/~ωz = 53
The reason for the roton structure⇒ decrease of the interaction amplitude
. – p.20/42
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Roton-maxon structure
ε
q
q
Roton minimum can be but at zeroInstability!
gg
c
d
hωµ
rotonization
instability
(L. Santos et al., 2003). – p.21/42
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Problem
Consider a Thomas-Fermi dipolar BEC, with g > 0
z
Find the TF density profile and the Bogoliubov excitations.
Obtain the stability diagram.
. – p.22/42
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Ideas from superfluid 4He
Why the roton-maxon structure?R. Feynman
ǫq =q2
2mS(q, ǫ)
Ene
rgy
(K)
Wavenumber (nm )−1
rotons
phonons
svp
20 bar
0
2
4
6
8
10
12
14
16
0 5 10 15 20 25
How to put the roton minimum higher (*) or lower (**)?
What happens? (S.Balibar, P. Nozieres, L. Pitaevskii)
(*) negative pressure (acoustic pulses)(**) increase the pressure
Metastable liquid
(**) ⇒ supersolid (density wave), loss of superfluidity, or?. – p.23/42
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Prehistory
L.P. Pitaevskii(1981)
v>vc
4 Hev=vc
ε
k
v > vc ⇒ Density waven
x
. – p.24/42
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Roton Instability in dipolar BEC
Extreme pancake BEC in the TF regime along z
ε
q
q
~1
Supersolid state with a period q−1r ∼ l0 ≪ Lz is collapsing
The main reason is the formation of localcollapsing small-size bubbles
. – p.25/42
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Toy 2D model at T=0
z
ρ
dl0 =
(~
mωz
)1/2
ϕ0(z) =1
π1/4l1/20
exp
−z2
2l20
short range interaction (g) + dipole-dipole
Consider 0 < g ≪4πd2
3. Then, for ql0 ≪ 1
V~q~p = g(1 − C|~q − ~p|)
where
C = 2πd2
gl0
. – p.26/42
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Spectrum
H =∑
~k
~2k2
2ma†~ka~k
+g
2
∑
~k,~q,~p
(1 − C|~q − ~p|l0)a†~k+~q
a†~k−~qa~k+~p
a~k−~p
ε
k
k
k
k2
ǫ2k = E2k + 2µEk(1 − Ckl) consider kl ≪ 1
kr,m =
[
32C
µ~ω0
1 ±
√
1 − 89
(~ω0
µ
)21
C2
]
1l0
. – p.27/42
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Rotonization
(~ω0
µ
)1/2≥ C ≥
√8
3
(~ω0
µ
)1/2
The roton minimum touches zero for
C =
(~ω0
µ
)1/2
≫ 1
kr∗ = 2
(µ
~ω0
)1/21
l0=
(mµ
~2
)1/2≪
1
l0
(twice the inverse healing length)
Cooper, Komineas
Pedri, Shlyapnikov.. – p.28/42
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Instability. Supersolid state ?
C >(
~ω0
µ
)1/2⇒ new ground state
Supersolid Ψ0
It is iportant that the expected periodof the supersolid state is ∼ l∗ ≫ l0
ψ0 = A0 +∑
iAi exp
i~ki~ρ
standing wavetriangular latticequadratic lattice
Imaginary time evolution ⇒ Collapse!. – p.29/42
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Fluctuations I
Assume we are still in the regime of a uniform BEC
C <(
~ω0
µ
)1/2
Density fluctuations
Quantum fluctuations ⇒ important only for the roton energyǫr extremely close to zero
Thermal fluctuations at T ≫ ǫr(δn
n
)2
=
∫1
n
d2k
(2π)2Ek
ǫk
1
exp(ǫk/T ) − 1≈T
ǫr
2
nξ2
large small
. – p.30/42
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Fluctuations II
~2
mξ2 = µ
⇓(
δnn0
)2≈ T
εr
mg
~2︸︷︷︸
small parameter for a 2D weakly interacting gas
The bosons can remain non-condensed even at T muchsmaller than TKT . Even at T < µ
What kind of state we get?
. – p.31/42
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Dipolar Fermi gas
What does th dipole-dipole interaction do in a Fermi gas?
Single component gas
The dipole-dipole scattering amplitude is independent of |k|at any orbital angular momenta alowed by the selectionrules.
Long-range cotribution ∼ d2
Short-range cotribution ∼ k︸ ︷︷ ︸
omit
Universal result for f. – p.32/42
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Physical picture
1k
< r*r
ψ
r
. – p.33/42
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Odd-l scattering amplitude
f = 12
∫ (
ei~ki~r − e−i~ki~r
)
Vd(~r)(
e−i~kf~r − ei~kf~r
)
d3r =
= 4πd2(cos2 θdq− − cos2 θdq+)
~q± = ~ki − ~kf
ei~k~r = 4π
∑∞l=0
∑lm=−l jl(kr)i
lYlm(r)Y ∗lm(k)
Partial amplitude f(limi; lfmf )
f(10; 10) = −6d2
5 cos θdkicos θdkf
. – p.34/42
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Superfluid p-wave pairing
nd2 ≪ EF
Analog of a → ∼ md2
~2 (r∗!)
∆ = g〈ψψ〉 ∼ EF exp(− 1
λ
); λ ∼ 1
kF r∗;
∆ ∝ EF exp− πEF
12nd2
Cooper pairs are superposition
of all odd angular momenta (for ml = 0)
. – p.35/42
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Transition temperature
nd2 ≪ EF
Tc = 1.44EF exp− πEF
12nd2
Baranov et.al (2002)
GM correction included
————————————————————————-
∆ → anisotropic ∝ sin(
π2 cos θ
)
θd Maximum in the direction of the dipoles.
Vanishes in the direction perpendicular tothe dipoles
. – p.36/42
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Distiguished features
Anisotropic gap → gapless excitations
in the direction perpendicular to
the dipoles. Damping rates etc.
Anisotropy → different from p-wave
superfluid B and A phases of 3He. In B
∆ is isotropic, and in A it vanishes only at
2 points on the Fermi shpere (θ = 0 and θ = π )
. – p.37/42
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Value of Tc
d ∼ 0.1 ÷ 1 D
ND3 ⇒ d = 1.5D equivalent to a = −1450A
Tc
EF→ 0.1 at n→ 1012cm−3
(Tc → 100nK)
HCN ⇒ d ≈ 3D equivalent to a = −7400A
. – p.38/42
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Problem
Consider a two-component dipolar Fermi gas ↑ and ↓
Discuss competition between s-wave and p-wave pairing
Construct the phase diagram
. – p.39/42
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Trapped Fermi gas
λ
Γ
1.2
1.0
0.8
0.60.5 1.0 1.5 2.0 2.5 3.0
−1
N=105
N=2*10
N=106
6
no BCS
BCS
0.3
0.2
0.4
/ρ ωzω
Eπ F
12nd 2
. – p.40/42
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Quasi 2D dipolar Fermi gas
ρ
zd
d+
−l0 =
(~
mωz
)1/2
ϕ0(z) =1
π1/4l1/20
exp
−z2
2l20
short range g < 0
H =∑
k~
2k2
2m
(
a†k↓ak↓ + a†k↑ak↑)
+
+|g|∑
k,p,q (−1 + C|~q − ~p|) a†k+q↑a†k−q↓ak−p↓ak+p↑
C = 2πd2
|g|l0. – p.41/42
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Interesting problem to solve
Interaction at the Fermi surface
|g|(−1 + CkF l0)
kF l0 <1C → 2πd2kF < |g| → superfluidity
kF l0 >1C → 2πd2kF > |g| → no superfluidity
Quantum transition to a normal state with increasing density
. – p.42/42