axel pelster 1. introduction 2. dipolar bose-einstein condensates...

On the Dipolar Dirty Boson ProblemAxel Pelster

1. Introduction

2. Dipolar Bose-Einstein Condensates

3. Bose-Einstein Condensates with Weak Disorder

4. Dipolar Bose-Einstein Condensates with Weak Disorder

5. Summary and Outlook

1

1.1 Identical Quantum Particles

Bosons:

• integer spin

• symmetric wave function

Fermions:

• half-integer spin

• anti-symmetric wave function

2

1.2 Time-of-Flight Absorption Pictures

JILA (1995): 8737Rb , N=20 000 , ω1 = ω2 = ω3/√8 = 2π× 120 Hz

3

1.3 Periodic Table of Elements

4


1. Introduction





5

2.1 Trapping and Interaction Potentials

• Harmonic trap and aspect ratios:

Utrap(x) =M

2

(ω2xx

2 + ω2yy2 + ω2zz

2), λx =

ωzωx, λy =

ωzωy

• Interaction potential: Vint(x) = gδ(x) + Vdd(x)

g =4π~2asM

, Vdd(x) =Cdd

4π|x|3[1− 3 cos2 θ

]

-�

Repulsion

6

?

Attraction�

��

��

��

��

θ

6

2.2 Magnetic versus Electric Dipolar Systems

• Magnetic systems:– Interaction strength: CBdd = µ0m

2, with m ∼ 1 to 10 µB– Realized samples

Boson: 52Cr Griesmaier et al., PRL 94, 160401 (2005)Boson: 87Rb Vengalattore et al., PRL 100, 170403 (2008)Fermion: 53Cr Chicireanu et al., PRA 73, 053406 (2006)Both: Dy Lu et al., PRL 104, 063001 (2010); PRL 107, 190401 (2011)

– Observed effects: magnetostriction (Cr), Bose-nova explosion (Cr)

• Electric systems:– Interaction strength: CEdd = 4πd

2, with d ∼ 1 Debye– Realized samples

Fermion: 40K87Rb Ospelkaus et al., Science 32, 231 (2008)Boson: 41K87Rb Aikawa et al., NJP 11, 055035 (2009)

– Observed effects: thermalization (40K87Rb)

• Ratio: CBdd/CEdd ≈ 10−47

2.3 Field Theoretical Description

• Hamilton operator: Ĥ = Ĥ0 + Ĥint

• Free Hamiltonian:

Ĥ0 =

∫

d3x Ψ̂†(x, t)

[

−~2∇22M

+ Utrap(x)

]

Ψ̂(x, t)

• Interaction Hamiltonian:

Ĥint =1

2

∫

d3x

∫

d3x′ Ψ̂†(x, t)Ψ̂†(x′, t)Vint (x− x′) Ψ̂(x′, t)Ψ̂(x, t)

• Bosonic commutation relations:[

Ψ̂(x, t), Ψ̂†(x′, t)]

= δ(x−x′),[

Ψ̂†(x, t), Ψ̂†(x′, t)]

=[

Ψ̂(x, t), Ψ̂(x′, t)]

= 0

• Bogoliubov prescription:

Ψ̂(x, t) → Ψ(x, t) + δψ̂(x, t), Ψ̂†(x, t) → Ψ∗(x, t) + δψ̂†(x, t)

8

2.4 Gross-Pitaevskii Theory

• Heisenberg equation:

i~∂

∂tΨ̂(x, t) =

[

Ψ̂(x, t), Ĥ]

• Zeroth order: Ψ̂(x, t) → Ψ(x, t), Ψ̂†(x, t) → Ψ∗(x, t)

• Gross-Pitaevskii equation:

i~∂Ψ(x, t)

∂t=

[

−~2∇22M

+Utrap(x)+

∫

d3x′Ψ∗(x′, t)Vint(x−x′)Ψ(x′, t)]

Ψ(x, t)

• Action principle:

δ

∫

d3x

t2∫

t1

dtΨ∗(x, t)

[

i~∂

∂t−H(x, t)

]

Ψ(x, t) = 0

• Action extremalization through suitable ansatz for Ψ(x, t)

9

2.5 Bogoliubov-de Gennes Theory (Stationary)

• Bogoliubov transformation:

δψ̂(x) =∑

ν

′ [Uν(x)α̂ν + V∗ν(x)α̂†ν

], δψ̂†(x) =

∑

ν

′ [U∗ν (x)α̂†ν + Vν(x)α̂ν

]

• Diagonalized Hamiltonian:

Ĥ ′ = E′G +∑

ν

′ενα̂

†να̂ν

• Bogoliubov-de Gennes equations:

[εν −HFl(x)]Uν(x) =∫

d3x′Vint

(

x−x′)[

Ψ(x′)Ψ(x)Vν(x

′)+Ψ

∗(x

′)Ψ(x)Uν(x

′)]

− [εν +HFl(x)]Vν(x) =∫

d3x′Vint

(

x−x′)[

Ψ∗(x

′)Ψ

∗(x)Uν(x

′)+Ψ(x

′)Ψ

∗(x)Vν(x

′)]

HFl(x) = −~2∇22M

+ Utrap(x)− µ+∫

d3x′Ψ∗(x′)Vint (x− x′)Ψ(x′)

10

2.6 Trapped Dipolar Bose Gases: MF Theory

• Action:

A[n0, χ] = −∫

dt

∫

d3xn0(x, t)

{

M

[

χ̇(x, t) +∇χ(x, t)2

2

]

+ ETF(x, t)}

ETF(x, t) = Utrap(x) +1

2gn0(x, t) +

1

2

∫

d3x′ Vdd(x− x′)n0(x′, t)

• Variational ansatz:

n0(x, t) = n0(t)

[

1−∑

i

x2iR2i (t)

]

, χ(x, t) =1

2

∑

i

αi(t)x2i

• Equations of motion:MR̈i = −

∂

∂RiEMF (Rx, Ry, Rz) , ǫdd =

Cdd3g

EMF (Rx, Ry, Rz) =M

2

∑

j

R2jω2j +

15Ng/(4π)

RxRyRz

[

1− ǫddf(RxRz,RyRz

)]

O’Dell et al., PRL 92, 250401 (2004) ; Eberlein et al., PRA 71, 033618 (2005)

11

2.7 Anisotropy Function

0

5

10

0

5

10

-2

-1

0

1

f(x, y)

x

y

0 2 4 6 8 10 12 14-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

x

fs(x)

• Shift of critical temperature:Glaum, Pelster, Kleinert, and Pfau, PRL 98, 080407 (2007)

Glaum and Pelster, PRA 76, 023604 (2007)

• Dipolar Fermi Gas:Lima and Pelster, PRA 81, 021606(R) (2010); PRA 81, 063629 (2010)

12

2.8 Trapped Dipolar Bose Gases: MF Results• Stability diagram and aspect ratio:

0 1 2 3 4 5 60

2

4

6

8

10

ǫdd

Unstable

Metastable

Stable

λx = λy0 5 10 15 20 25 30

0

1

2

3

4

5

6R̃xλxR̃z

ǫbdd

λx = λy

O’Dell et al., PRL 92, 250401 (2004) ; Eberlein et al., PRA 71, 033618 (2005)

• Low-lying oscillations and time-of-flight:

0 1 2 3 40

2

4

6

8

rq

Ωrq

+

Ω+

−

Ω−

ǫbdd

λx = λy = 4

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

Bismut et al., PRL 105, 040404 (2010) Stuhler et al., PRL 95, 150406 (2005)

13

2.9 Trapped Dipolar Bose Gases: Beyond MF

• Semiclassical approximation:

εν → ε (x,k) , Uν → U (x,k) eik·x, Vν → V (x,k) eik·x

• BdG equations:[

ε (x, k) − ~2k2

2M

]

U (x, k) =√

n0(x)

∫

d3x′ Vint(

x−x′)

√

n0(x′)

[

V(x′, k)+U (x′, k)]

eik·

(

x′−x)

−[

ε (x, k) +~2k2

2M

]

V(x, k) =√

n0(x)

∫

d3x′Vint

(

x−x′)√

n0(x′)

[

V(x′, k)+U (x′, k)]

eik·

(

x′−x)

• Local density approximation for exchange term:√

n0(x)

∫

d3x′[

gδ(

x−x′)

+Vdd(

x−x′)]

√

n0(x′)q(

x′, k

)

eik·(x′−x)

≈ ξ (x, k) q (x, k)

with ξ (x,k) = gn0(x)[1 + ǫdd

(3 cos2 θ − 1

)]

Timmermans, Tommasini, and Huang, PRA 55, 3645 (1997)

14

2.10 Trapped Dipolar Bose Gases: Beyond MF

• Bogoliubov spectrum:

ε2 (x,k) =

[~2k2

2M+ ξ (x,k)

]2

− ξ2 (x,k)

• Bogoliubov amplitudes:

U (x,k)2 − 1 = V (x,k)2 = 12

[~2k2

2M + ξ (x,k)

ε (x,k)− 1

]

• Depletion: ∆NN

=5√π

8Q3(ǫdd)

√

n(0)a3s

• Energy: ∆E = 5√π

8gn(0)Q5(ǫdd)

√

n(0)a3s0.0 0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

2.5

Ql(x)

x

• Beyond mean-field equations of motion:

MR̈i = −∂

∂Ri

[

EMF (Rx, Ry, Rz) +7

N∆E (Rx, Ry, Rz)

]

15

2.11 Trapped Dipolar Bose Gases: Beyond MF Results

• Aspect ratio in equilibrium:

κ ≡ RxRz = κ0 (1 + δκ)

δ̃κ = 105√π

32

√

a3sn(0) ≈ 0.06 0.0 0.5 1.0 1.5 2.0 2.5 3.005

10

15

20δκ

δ̃κ

ǫb

dd= 0.97

ǫb

dd= 0.95

ǫb

dd= 0.8

λx = λy

• Low-lying oscillations: typical for Cr

Ω = Ω0 (1 + δΩ)

δ̃Ω = 63√π

128

√

a3s n(0) ≈ 0.01 0 1 2 3 4 5-0.20.0

0.2

0.4

0.6

0.8

1.0

1.2δΩδ̃Ω ǫbdd = 0

ǫbdd = 0.16, as = 100a0

MonopoleQuadrupole

λx = λy

• Time-of-flight: expected for Dy

κ(t) =Rx(t)Rz(t)

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0κ(t)

λx = λy = 0.5

ǫbdd

= 0.9, as = 150a0

Quantum CorrectedMean Field

ωxt

Lima and Pelster, PRA 84, 041604(R) (2011); arXiv:1111.0900

16


1. Introduction





17

3.1 Overview of Set-Ups

• Superfluid Helium in Porous Media: persistence of superfluidityReppy et al., PRL 51, 666 (1983)

• Laser Speckles: controlled randomnessBilly et al., Nature 453, 891 (2008)

• Wire Traps: undesired randomnessKrüger et al., PRA 76, 063621 (2007)

Fortàgh and Zimmermann, RMP 79, 235 (2007)

• Localized Atomic Species: theoretical suggestionGavish and Castin, PRL 95, 020401 (2005)

• Incommensurate Lattices: quasi-randomnessRoati et al., Nature 453, 895 (2008)

18

3.2 Laser Speckles

global condensate vanishesLye et al., PRL 95, 070401 (2005)

19

3.3 Wire Trap

0 100 200

−5

0

+5

Longitudinal Position (µm)

∆B/B

(10

−5 )

−20 0 20∆B/B (10−6)

arbi

trar

y un

its

Distance: d = 10 µm Wire Width: 100 µm

Magnetic Field: 10 G, 20 G, 30 G Deviation: ∆B/B ≈ 10−4

Krüger et al., PRA 76, 063621 (2007)

Fortàgh and Zimmermann, RMP 79, 235 (2007)

20

3.4 Model System

Action of a Bose Gas:

A =∫ ~β

0

dτ

∫

d3x

{

ψ∗[

~∂

∂τ− ~

2

2M∆+U (x) + V (x)− µ

]

ψ +g

2ψ∗2ψ2

}

Properties:

• harmonic trap potential: U(x) = M2ω2x2

• disorder potential: V (x)

V (x1) = 0 , V (x1)V (x2) = R(x1 − x2) , . . .

• chemical potential: µ

• repulsive interaction: g = 4π~2a

M

• periodic Bose fields: ψ(x, τ + ~β) = ψ(x, τ)

21

3.5 Grand-Canonical Potential

Aim:

Ω = − 1βlnZ

Z =∮

DψDψ∗ e−A[ψ∗,ψ]/~

Problem:

lnZ 6= lnZ

Solution: Replica Trick

Ω = −1β

limN→0

ZN − 1N

22

3.6 Replica Trick

Disorder Averaged Partition Function:

ZN =∮

{N∏

α′=1

D2ψα′

}

e−∑Nα=1 A([ψ∗α,ψα])/~ =

∮{

N∏

α=1

D2ψα

}

e−A(N)/~

Replicated Action:

A(N) =∫

~β

0

dτ

∫

dDx

N∑

α=1

{

ψ∗α(x, τ)

[

~∂

∂τ− ~

2

2M∆ + U(x) − µ

]

ψα(x, τ)

+g

2|ψα(x, τ)|4

}

+∞∑

n=2

1

n!

(−1~

)n−1 ∫ ~β

0

dτ1 · · ·∫ ~β

0

dτn

∫

dDx1 · · ·∫

dDxn

×N∑

α1=1

· · ·N∑

αn=1

R(n)

(x1, . . . , xn)∣

∣ψα1(x1, τ1)∣

∣

2 · · · |ψαn(xn, τn)|2

=⇒ Disorder amounts to attractive interaction for n = 2=⇒ Higher-order disorder cumulants negligible in replica limit N → 0

23

3.7 Condensate Density

Assumptions:

homogeneous Bose gas: U(x) = 0

δ-correlated disorder: R(x) = Rδ(x)

Bogoliubov Theory:

background method: ψα(x, τ) = Ψα + δψα(x, τ)

replica symmetry: Ψα =√n0

Result: n0 = n−8

3√π

√an0

3 − M2R

8π3/2~4

√n0a

Huang and Meng, PRL 69, 644 (1992)

Falco, Pelster, and Graham, PRA 75, 063619 (2007)

24

3.8 Superfluid Density

Galilei Boost:

∆A =∫ ~β

0

dτ

∫

d3xψ∗(x, τ)u~

i∇ψ(x, τ)

dΩ = −S dT − p dV −N dµ− p du

p = − ∂Ω(T, V, µ,u)∂u

∣∣∣∣T,V,µ

=MV nn u+ . . .

Result: ns = n− nn = n−4

3

M2R

8π3/2~4

√n0a

Huang and Meng, PRL 69, 644 (1992)

Falco, Pelster, and Graham, PRA 75, 063619 (2007)

25

3.9 Collective Excitations

Typical Values:

Inguscio et al., PRL 95, 070401 (2005)

ξ = 10 µmRTF = 100 µmlHO = 10 µm

}

ξ̃ =ξRTF

l2HO√2≈ 7

~�

Æ! 0(�)=Æ!0(0)

54321010:80:60:40:20

n = 0, l = 1

n = 0, l = 2

=⇒ Disorder effect vanishes in laser speckle experiment

Improvement:

laser speckle setup with correlation length ξ = 1 µm

Aspect et al., NJP 8, 165 (2006)

=⇒ Disorder effect should be measurableFalco, Pelster, and Graham, PRA 76, 013624 (2007)

26


1. Introduction





27

4.1 Disorder Induced Depletions

• Condensate:

n− n0n

=

∫d3k

(2π)3R(k)

[~2k2/2M + 2nVint(k)]2 + . . .

• Superfluid:

nδij − nS,ijn

=

∫d3k

(2π)34R(k)kikj

k2 [~2k2/2M + 2nVint(k)]2 + . . .

• Isotropy: Vint(k) = Vint(|k|) and R(k) = R(|k|) yield

n− nS =4

3(n− n0)

28

4.2 Specialization

• Dipoles polarized along z-direction:

Vint(k) = g[1 + ǫdd

(3 cos2 θ − 1

)]

• Relative interaction strength:

ǫdd =Cdd3g

, Cdd =

{µ0m

2 magnetic dipoles4πd2 electric dipoles

• Gaussian correlated disorder:

R(k) = Re−σ2k2

29

4.3 Condensate Depletion

n− n0 = nHM f(

ǫdd,σ

ξ

)

Coherence length: ξ =

√

~2

4Mng

Huang-Meng depletion: nHM =(M

2π~2

)32√

π

2gnR

00.5

11.5

22.5

3 00.2

0.40.6

0.810

0.5

1

1.5

2

σξ

ǫdd

f(

ǫdd,σξ

)

0

0.4

0.8

1.2

1.6

2

Krumnow and Pelster, PRA 84, 021608(R) (2011)

30

4.4 Superfluid Depletion

n− nS = nHM

f⊥(

ǫdd,σξ

)

0 0

0 f⊥(

ǫdd,σξ

)

0

0 0 f||(

ǫdd,σξ

)

00.5

11.5

22.5

3 00.2

0.40.6

0.810

0.2

0.4

0.6

0.8

1

1.2

1.4

σξ

ǫdda)

f||

(

ǫdd,σξ

)

00.20.40.60.811.21.4

00.5

11.5

22.5

3 00.2

0.40.6

0.810

0.5

1

1.5

2

2.5

3

σξ

ǫddb)

f⊥

(

ǫdd,σξ

)

00.511.522.53


=⇒ Finite localization timeGraham and Pelster, IJBC 19, 2745 (2009)

31

4.5 Hydrodynamic Theory

• Two-fluid model: n = nS + nN , j = nSvS + nN vN︸︷︷︸=0

• Continuity equation: ∂∂tn+∇j = 0

• Euler equation: M ∂∂t

vS +∇(

µ+1

2Mv2S

)

= 0

• Linearization around equilibrium:nS = nS,eq + δnS(x, t) , vS = δvS(x, t)

n = neq + δn(x, t) , µ = µ (neq) +∂µ

∂n

∣∣∣∣eq

δn(x, t)

• Wave equation:

∂2

∂t2δn(x, t)− 1

M

∂µ

∂n

∣∣∣∣eq

∇[

nS,eq(x)∇δn(x, t)]

= 0

32

4.6 Speed of Sound

• Delta correlated disorder and dipole-dipole interaction:

c2(ǫdd, ϑ) =ng

M

[3ǫdd cos

2 (ϑ) + 1− ǫdd]+nHMg

Ms(ǫdd, ϑ)

• Special case of contact interaction: s(0, ϑ) = 53Giorgini, Pitaevskii, and Stringari, PRB 49, 12938 (1994)

00.2

0.40.6

0.81

−π −π/20

π/2 π

00.20.40.60.8

11.21.41.61.8

ǫdd

a) ϑ

c0(ǫdd,ϑ)√ng/m

0

0.4

0.8

1.2

1.6

00.2

0.40.6

0.81

−π −π/20 π/2

π

-1

0

1

2

3

ǫdd

b) ϑ

s (ǫdd, ϑ)

-10123


=⇒ Measurable via Bragg spectroscopy

33


1. Introduction





34

5.1 Summary

• Frozen Disorder Potential :arises both artificially (laser speckles) or naturally (wire trap)

• Bosons:local condensates in minima + global condensate + thermally excited

• Localization Versus Transport: disorder reduces superfluidity

• Phase Diagram: yet unknown for strong disorderNavez, Pelster, and Graham, APB 86, 395 (2007)

Graham and Pelster, IJBC 19, 2745 (2009)

• Anisotropic superfluidity at zero temperature:delicate interplay between isotropic disorder and dipolar interaction−→ characteristic direction dependent sound velocity−→ possible experimental detection: Bragg spectroscopy

• Quantum and thermal fluctuations necessitate 3-fluid model

35

5.2 Optical LatticesQuantum Phase Transition:

Greiner, Mandel, Esslinger, Hänsch, and Bloch, Nature 415, 39 (2002)

Theoretical Description:

• Bose-Hubbard Hamiltonian:ĤBH = −t

∑

〈i,j〉â†i âj +

∑

i

[U

2n̂i(n̂i − 1)− µn̂i

]

, n̂i = â†i âi

• Landau Theory: dos Santos and Pelster, PRA 79, 013614 (2009)• Ginzburg-Landau Theory:

Bradlyn, dos Santos, and Pelster, PRA 79, 013615 (2009)

36

5.3 Quantum Phase Diagram ( T = 0,D = 3)

• Blue line: Strong-coupling, Freericks and Monien, PRB 53, 2691 (1996)• Error bar: Extrapolated strong-coupling series• Black line: Mean-field theory, Fisher et al., PRB 40, 546 (1989)• Red line: Landau theory, dos Santos and Pelster, PRA 79, 013614 (2009)• Blue dots: QMC data, Capogrosso-Sansone et al., PRB 75, 134302 (2007)

Numerical extension of Landau theory to higher hopping orde rs:Teichmann et al., PRB 79, 224515 (2009)

=⇒ Convergence of hopping expansion for Landau theory37

5.4 Excitation Spectra near Phase Bouncary

Grass, dos Santos, and Pelster, PRA 84, 013613 (2011)

Bragg spectroscopy: Bissbort et al., PRL 106, 205303 (2011)

38

5.5 Disordered Bosons in LatticeBose-Hubbard Hamilton Operator:

ĤBH = −t∑

â†i âj +∑

i

[U

2n̂i(n̂i − 1) + (ǫi − µ) n̂i

]

• = ∏i∫ +∞−∞ • p(ǫi) dǫi , p(ǫi) =

{1/∆ ; ǫi ∈ [−∆/2,+∆/2]0 ; otherwise

Mean-Field Phase Diagram:

0 1 2

0.05

0.1

0.15

0 1 2

0.05

0.1

0.15

µ/Uµ/U

2dt/U2dt/U

MIMI

T = 0∆/U = 0.5

SF

0 1 2

0.05

0.1

0.15

0 1 2

0.05

0.1

0.15

SF

MIMIµ/Uµ/U

2dt/U2dt/U

BGBG

kT/U = 0.01∆/U = 0.5

Krutitsky, Pelster, and Graham, NJP 8, 187 (2006)

Stochastic Mean-Field Theory: Bissbort and Hofstetter, EPL 86, 50007 (2009)

39

5.6 Acknowledgement

Postdocs:

• Aristeu Lima (DFG)• Ednilson Santos (FAPESP)

PhD students:

• Javed Akram (DAAD)• Hamid Al-Jibbouri (DAAD)• Mahmoud Ghabour• Tama Khellil (DAAD)• Mohamed Mobarek (Egyp. Gov.)• Tao Wang (CSC)

Diploma students:

• Max Lewandowski (Potsdam)• Tobias Rexin (Potsdam)• Falk Wächtler (Potsdam)

Bachelor students:

• Tomasz Checinski (Bielefeld)• Christian Krumnow (FU Berlin)• Moritz von Hase (FU Berlin)• Carolin Wille (FU Berlin)• Nikolas Zöller (FU Berlin)

Volkswagen: Bakhodir Abdullaev , Abdulla Rakhimov (Tashkent)DAAD: Antun Balaz, Vladimir Lukovic, Branko Nikolic (Belgrade)Mentors: Robert Graham (Duisburg-Essen), Hagen Kleinert (FU Berlin)

40

5.7 Announcement

Wilhelm and Else Heraeus Seminar

Quo Vadis BEC? IV

organized by Carlos S á de Melo and Axel Pelster

Physikzentrum, Bad Honnef

August 21 – 25, 2012

applications possible at the beginning of 2012:http://users.physik.fu-berlin.de/˜pelster

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axel pelster 1. introduction 2. dipolar bose-einstein condensates...

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