bogoliubov-de gennes study of trapped fermi gases
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Bogoliubov-de Gennes Study of Trapped Fermi Gases. Han Pu Rice University (INT, Seattle, 4/14/2011). Leslie Baksmaty Hong Lu Lei Jiang. Randy Hulet Carlos Bolech. Imbalanced Fermi mixtures. Fulde-Ferrel-Larkin-Ovchinnikov instability. BCS Cooper pairs have zero momentum - PowerPoint PPT PresentationTRANSCRIPT
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Bogoliubov-de Gennes Study of Trapped Fermi Gases
Han PuRice University
(INT, Seattle, 4/14/2011)
Leslie BaksmatyHong LuLei Jiang
Randy Hulet Carlos Bolech
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Imbalanced Fermi mixturesImbalanced Fermi mixtures
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• BCS Cooper pairs have zero momentum
• Population imbalance leads to finite-momentum pairs
• FFLO instability results in textured states
Fulde-Ferrel-Larkin-Ovchinnikov instabilityFulde-Ferrel-Larkin-Ovchinnikov instability
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• Rice (Hulet Group)– Science 311, 503 (2006)– PRL 97, 190407 (2006)– Nuclear Phys. A 790, 88c (2007)– J. Low. Temp. Phys. 148, 323 (2007)– Nature 467, 567 (2010)
• MIT (Ketterle Group)– Science 311, 492 (2006)– Nature 442, 54 (2006)– PRL 97, 030401 (2006)– Science 316, 867 (2007)– Nature 451, 689 (2008)
• ENS (Salomon Group)– PRL 103, 170402 (2009)
Experiments on spin-imbalanced Fermi gasExperiments on spin-imbalanced Fermi gas
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Observation: Phase separation
Superfluid core with polarized halo
MW. Zwierlein, A. Schirotzek, C.H. Schunck, and W, Ketterle: Science 311, 492-496 (2006)
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High T Low T
Hulet
Ketterle
n↑
n↓
n↑ - n↓
MIT/Paris data are consistent with Local Density Approximation (LDA)Rice data (low T) strongly violates LDA.
Experimental resultsExperimental results
Salomon
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Surface Tension• Phase Coexistence -> Surface Tension
1 mm
60 m
Aspect Ratio of Cloud: 50:1
Aspect Ratio of Superfluid: 5:1
Data: Hulet
Surface tension causes density distortionEffects of surface tension more important in smaller sample.
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De Silva, Mueller, PRL 97, 070402 (2006)Data points from Rice experiment.
P=0.14
P=0.53
P=0.72
LDA
LDA + surface tension
Breakdown of LDABreakdown of LDA
N NP
N N
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24/3
2 snm
Optimal value that fits data:
: 3 ~ 4
However, from microscopic theoretical calculation: : 0.15
PRA 79, 063628 (2009)
Surface tensionSurface tension
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take initial guesses of
, ( ), ( )n r r *
diagonalize the matrixs
s
H
H
adjust
until:
( )N dr n r
until the input and output
, ( ), ( ) convergen r r
Choose T and N
compute new
( ), ( )n r r
2 2
2 2 2 2
/ (2 ) ( , )
1( , )
2
s
r z
H m V r z
V r z m r z
Solving BdG equationsSolving BdG equations
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0.2 0.4 0.6 0.8 1.0 1.2 1.4z�Z
TF
0.5
1.0
1.5
2.0
2.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4r �Z
TF
0.5
1.0
1.5
2.0
2.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4z�Z
TF
0.5
1.0
1.5
2.0
0.05 0.10 0.15 0.20 0.25 0.30r �Z
TF
0.5
1.0
1.5
2.0
0.2 0.4 0.6 0.8 1.0z�Z
TF
0.5
1.0
1.5
2.0
0.008 0.016 0.024r �Z
TF
0.5
1.0
1.5
2.0
0.2 0.4 0.6 0.8 1.0z�Z
TF- 0.2
0.2
0.4
0.6
0.8
0.2 0.4 0.6 0.8 1.0 1.2 1.4z�Z
TF
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0 1.2 1.4z�Z
TF
0.2
0.4
0.6
0.8
AR=1 AR=5 AR=50
Density along z-axis
Density along r-axis
Gap along z-axis
Effect of trap anisotropy: N=200, P=0.4Effect of trap anisotropy: N=200, P=0.4
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0.2 0.4 0.6 0.8 1.0z�Z
TF
0.5
1.0
1.5
2.0
0.008 0.016 0.024r �Z
TF
0.5
1.0
1.5
2.0
0.2 0.4 0.6 0.8 1.0z�Z
TF
0.2
0.4
0.6
0.8
0.2 0.4 0.6 0.8 1.0z�Z
TF
0.5
1.0
1.5
2.0
0.008 0.016 0.024r �Z
TF
0.5
1.0
1.5
2.0
0.2 0.4 0.6 0.8 1.0z�Z
TF- 0.2
0.2
0.4
0.6
0.8
0.2 0.4 0.6 0.8 1.0z�Z
TF
0.5
1.0
1.5
2.0
0.008 0.016 0.024r �Z
TF
0.5
1.0
1.5
2.0
0.2 0.4 0.6 0.8 1.0z�Z
TF
- 0.5
0.5
Density along z-axis
Density along r-axis
Gap along z-axis
P=0.2 P=0.4 P=0.7
Quasi-1D system: N=200, AR=50Quasi-1D system: N=200, AR=50
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Gap along z-axis Gap along r-axis
LD
AB
dG
n↑ n↓n↑ - n↓
N~200,000
BdG vs. LDA: N=200, AR=50, P=0.6BdG vs. LDA: N=200, AR=50, P=0.6
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BdG equation is very nonlinear, it may support many stationary states.
Complicated energy landscape
For large N, starting from different initial configurations, the BdG solver may converge to different final states.
Going to higher NGoing to higher N
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SF
LONN
3 classes of states3 classes of states
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SF LO NN
Increasing energy
Density profiles (N=50,000)Density profiles (N=50,000)
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n↑
n↓
Upclose on the LO stateUpclose on the LO state
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Pei, Dukelsky and Nazarewicz, PRA 82, 021603 (2010)
Bulgac and Forbes, PRL 101, 215301 (2008)
Robustness of the density oscillationRobustness of the density oscillation
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homogeneous trapped
Orso, PRL (2007); Hu et al., PRL (2007)
FFLO in 1DFFLO in 1D
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Liao et al., Nature 467, 567 (2010)
Experiment in 1D (Hulet group)Experiment in 1D (Hulet group)
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3D
t
1D
……
t
X
3D 1D
Dimensional crossover: 3D – 1DDimensional crossover: 3D – 1D
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Model for single impurity in Fermi superfluidityModel for single impurity in Fermi superfluidity
2
2
0
( ) ( ) ( )
( ) ( )
1( )
:
: .
: .
imp
imp
x
a
H H H
H drU r r r
U r u r for contact potential
U r u e for gaussian potentiala
u impurity strength
Non magnetic impurity u u
Magnetic impurity u u
H0 is BCS mean field Hamiltonian
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BdG and T matrix methodsBdG and T matrix methods
)()()()()()(
),()()()()()(
rvrUruirvkrEv
rurUrvirukrEuy
y
T-matrix gives exact solutions for localized contact impurity without trap.
Contact potential: T matrix only depends on energy.
BdG method gives numerical results for single impurity in harmonic trap.
);'()();()'();();',( 000 wkGwTwkGkkwkGwkkG
BdG solves self-consistently a set of coupled equations
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Localized non-magnetic impurity in 1D trapLocalized non-magnetic impurity in 1D trap
222
20
0 )2
(
muE Bound state occurs when T-1(w)=0
TFF zEu
uuu
02.0
0
0
0
BdG results
T matrix results
with impurity
without impurity
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Localized magnetic impurityLocalized magnetic impurity
00 uuu
200
200
0
00 )2/(1
)2/(1
Nu
NuE
Bound state energy inside the gap
BdG results
T matrix results
This bound state is below the bottom of quasiparticle band.
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Density and gap profiles for localized magnetic impurityDensity and gap profiles for localized magnetic impurity
What if we increase impurity width and strength ?
Spin up Spin down
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Spin up Spin down
Magnetic impurity induced FFLO stateMagnetic impurity induced FFLO state
Impurity: Gaussian potential
TF
TFF
za
zEu
2.0
0.1,4.0,12.00
Impurity
strength
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Magnetic impurity induced FFLO state (3D)
Magnetic impurity induced FFLO state (3D)
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• Two component Fermi gas offers very rich physics.
• Effects of trapping confinement.
• Flexibility of atomic system provides opportunities of studying exotic pairing mechanisms.
ConclusionConclusion
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• “Concomitant modulated superfluidity in polarized Fermi gases”, Phys. Rev. A 83 023604 (2011)
• “Single impurity in ultracold Fermi superfluids”, arXiv:1010.3222
• “Bogoliuvob-de Gennes study of trapped spin-imbalanced unitary
Fermi gases”, arXiv:1104.2006
ReferencesReferences