functional integral approach for trapped dirty...
TRANSCRIPT
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Functional Integral Approach for Trapped Dirty Bosons
Axel Pelster
1. On the Dirty Boson Problem
2. Functional Integral Approach
3. Perturbative Results
4. Hartee-Fock Mean-Field Theory:
Homogeneous Case
5. Hartee-Fock Mean-Field Theory:
Trapped Case
6. Outlook
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1.1 Overview of Set-Ups
• Superfluid Helium in Porous Media:
Crooker et al., PRL 51, 666 (1983)
• Laser Speckles:
Lye et al., PRL 95, 070401 (2005)
Clement et al., PRL 95, 170409 (2005)
• Wire Traps:
Kruger et al., PRA 76, 063621 (2007)
Fortagh and Zimmermann, RMP 79, 235 (2007)
• Localized Atomic Species:
Gavish and Castin, PRL 95, 020401 (2005)
Gadway et al., PRL 107, 145306 (2011)
• Incommensurate Lattices:
Damski et al., PRL 91, 080403 (2003)
Schulte et al., PRL 95, 170411 (2005)
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1.2 Laser Speckles: Controlled Randomness
Experimental Set-Up:
Fragmentation:
Lye et al., PRL 95, 070401 (2005)
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1.3 Wire Trap: Undesired Randomness
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
Kruger et al., PRA 76, 063621 (2007)
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Functional Integral Approach for Trapped Dirty Bosons
Axel Pelster
1. On the Dirty Boson Problem
2. Functional Integral Approach
3. Perturbative Results
4. Hartee-Fock Mean-Field Theory:
Homogeneous Case
5. Hartee-Fock Mean-Field Theory:
Trapped Case
6. Outlook
5
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2.1 Model SystemAction of a Bose gas:
A =
∫
~β
0
dτ
∫
d3x
ψ∗(x, τ)
[
~∂
∂τ− ~
2
2M∆+ U(x) + V (x)− µ
]
ψ(x, τ)
+g
2ψ∗(x, τ)2ψ(x, τ)2
Properties:
• harmonic trap potential: U(x) =M
2
3∑
i=1
ω2i x
2i
• disorder potential: V (x) ; bounded from below, i.e. V (x) > V0
• chemical potential: µ
• repulsive interaction: g =4π~2a
M
• periodic Bose fields: ψ(x, τ + ~β) = ψ(x, τ)
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2.2 Random Potential
Disorder Ensemble Average:
• =
∫
DV • P [V ] ,
∫
DV P [V ] = 1 , P [V < V0] = 0
Assumption:
V (x1) = 0 , V (x1)V (x2) = R(2)(x1 − x2)
Characteristic Functional:
exp
i
∫
dDx j(x)V (x)
= exp
∞∑
n=2
in
n!
∫
dDx1 · · ·∫
dDxnR(n)(x1, . . . ,xn) j(x1) · · · j(xn)
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2.3 Grand-Canonical Potential
Aim:
Ω = − 1
βlnZ
Z =
∮
D2ψ e−A[ψ∗,ψ]/~
Problem:
lnZ 6= lnZ
Solution: Replica Trick
Ω = −1
βlimN→0
ZN − 1
N
Parisi, J. Phys. (France) 51, 1595 (1990)
Mezard and Parisi, J. Phys. I (France) 1, 809 (1991)
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2.4 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 cumulants negligible in replica limit N → 0
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Functional Integral Approach for Trapped Dirty Bosons
Axel Pelster
1. On the Dirty Boson Problem
2. Functional Integral Approach
3. Perturbative Results
4. Hartee-Fock Mean-Field Theory:
Homogeneous Case
5. Hartee-Fock Mean-Field Theory:
Trapped Case
6. Outlook
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3.1 Bogoliubov Theory of Dirty Bosons
Assumptions:
homogeneous Bose gas: U(x) = 0
δ-correlated disorder: V (x) = 0 , V (x)V (x′) = Rδ(x−x′)
Condensate Depletion:
n0 = n− 8
3√π
√an0
3 − M2R
8π3/2~4
√
n0a
Superfluid Depletion:
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)
Graham and Pelster, Proc. 9th Int. Conf. Path Integrals, World Scientific (2008)
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3.2 Collective Excitations
Typical Values:
Lye et al., PRL 95, 070401 (2005)
σ = 10 µm
RTF = 100 µm
lHO = 10 µm
σ =σRTF
l2HO
√2≈ 7
σ
δω(σ)/δω
(0)
=⇒ Disorder effect vanishes in laser speckle experiment
Improvement:
laser speckle setup with correlation length σ = 1 µm
Clement et al., NJP 8, 165 (2006)
=⇒ Disorder effect should be measurable
Falco, Pelster, and Graham, PRA 76, 013624 (2007)
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3.3 Shift of Critical Temperature
~
N
1=6
T
N
1=6
T
(0)
~
R
0.01
0.02
0.03
0.04
1 2 3 4 5 6 7
solid: Gaussian
dashed: Lorentzian
Length Scale: lHO =
√
~
Mωg, ωg = (ω1ω2ω3)
1/3
Dimensionless Units: ξ =ξ
lHO, R =
R(
~2
Ml2HO
)2
l3HO
=M3/2R
~7/2ω1/2g
Timmer, Pelster, and Graham, EPL 76, 760 (2006)
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3.4 Superfluid Density as Tensor
• Linear response theory: pi = VM (nnijvnj + nsijvsj) + . . .
M. Ueda, Fundamentals and New Frontiers of Bose-Einstein Condensation (2010)
• Dipolar interaction at zero temperature:
=⇒ no anisotropic superfluidity
Lima and Pelster, PRA 84, 041604(R) (2011); PRA 86, 063609 (2012)
• Dipolar interaction at finite temperature:
=⇒ Directional dependence of first and second sound velocity
Ghabour and Pelster, PRA 90, 063636 (2014)
• Dipolar interaction and isotropic disorder at zero temperature:
Krumnow and Pelster, PRA 84, 021608(R) (2011)
Nikolic, Balaz, and Pelster, PRA 88, 013624 (2013)
• Condensate depletion larger than parallel superfluid depletion:
=⇒ Finite localization time
Graham and Pelster, IJBC 19, 2745 (2009)
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Functional Integral Approach for Trapped Dirty Bosons
Axel Pelster
1. On the Dirty Boson Problem
2. Functional Integral Approach
3. Perturbative Results
4. Hartee-Fock Mean-Field Theory:
Homogeneous Case
5. Hartee-Fock Mean-Field Theory:
Trapped Case
6. Outlook
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4.1 Order Parameters
Definition:
lim|x−x′|→∞
〈ψ(x, τ)ψ∗(x′, τ)〉 = n0
lim|x−x′|→∞
|〈ψ(x, τ)ψ∗(x′, τ)〉|2 = (n0 + q)2
Note:
q is similar to Edwards-Anderson order parameter of spin-glass theory
Hartree-Fock Mean-Field Theory:
Self-consistent determination of n0 and q for R(x− x′) = Rδ(x− x
′)
Phase Classification:
gas Bose glass superfluid
q = n0 = 0 q > 0, n0 = 0 q > 0, n0 > 0
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4.2 Hartree-Fock Results
Isotherm: T = const. Phase Diagram: µ = const.
disorder strength R = const.
R > Rc
R < Rc
µ
n
nBEC
nm
µMµmµBEC
gas
Bose glass
super-fluid
T
R
Rc
TcTBEC
gas
Boseglass
superfluid
first order
continuous
Graham and Pelster, IJBC 19, 2745 (2009)
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Functional Integral Approach for Trapped Dirty Bosons
Axel Pelster
1. On the Dirty Boson Problem
2. Functional Integral Approach
3. Perturbative Results
4. Hartee-Fock Mean-Field Theory:
Homogeneous Case
5. Hartee-Fock Mean-Field Theory:
Trapped Case
6. Outlook
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5.1 Self-Consistency Equations
n(x) = n0(x) + q(x) + nth(x)
[
−µ+ 2gn(x) + V (x)−DQ0(x)− gn0(x)−~2
2m∆
]
√
n0(x) = 0
q(x) = DΓ(
2− n
2
)
(
M
2π~2
)n/2n0(x) + q(x)
[−µ+ 2gn(x) + V (x)−DQ0(x)]2−n/2
Qm(x) = Γ(
1− n
2
)
(
M
2π~2
)n/2
[−i~ωm − µ+ 2gn(x) + V (x)−DQm(x)]1−n/2
nth(x) = limη↓0
∞∑
m=−∞
eiωmηQm(x)
β, ωm =
2πm
~β
Khellil and Pelster, JSM 063301 (2016)
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5.2 Quasi-1d Case, T=0
• Weak disorder: D = 0.016
• Stronger disorder: D = 0.386
• Thomas-Fermi radii:
Green =⇒ Variational
Red =⇒ Numerical
Blue =⇒ Analytical
Khellil, Balaz, and Pelster, NJP 18, 063003 (2016)
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5.3 Isotropic 3d Case, T=0
• Density profiles: d = 0.2
• Thomas-Fermi radii:
Red =⇒ Variational
Blue =⇒ Analytical
Khellil and Pelster, arXiv:1512.04870
Nattermann and Pokrovsky, PRL 100, 060402 (2008): dc = 0.115
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Functional Integral Approach for Trapped Dirty Bosons
Axel Pelster
1. On the Dirty Boson Problem
2. Functional Integral Approach
3. Perturbative Results
4. Hartee-Fock Mean-Field Theory:
Homogeneous Case
5. Hartee-Fock Mean-Field Theory:
Trapped Case
6. Outlook
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6.1 Bose-Einstein Condensation of Light
Set-Up
Result
Klaers, Schmitt, Vewinger, and Weitz, Nature 468, 545 (2010)
Pelster, Physik-Journal 10, Nr. 1, 20 (2011); Physik Journal 13, Nr. 3, 20 (2014)
Kopylov, Radonjic, Brandes, Balaz, and Pelster, PRA 92, 063832 (2015)
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6.2 Mapping Between Quantum Gas Experiments
• Time transformation:dτ(t)
dt= λ2(t)
Jackiw, Ann. Phys. 129, 183 (1980)
Cai, Inomata, and Wang, PLA 91, 331 (1982)
• Transformation formulas:
ˆψ(r, t) = e−iMn2~
λλr
2λD/2ψ(λr, τ(t))
V (r, t) = λ2V (λr, τ(t)) +Mr2
2λ3
(
1
λ2d
dt
)2
λ
U(r, r′, t) = [λ(t)]2U(λ(t)r, λ(t)r′, τ(t))
Wamba, Pelster, and Anglin, PRA 94, 043628 (2016)
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