Bogoliubov-de Gennes Study of Trapped Fermi Gases

Han PuRice University

(INT, Seattle, 4/14/2011)

Leslie BaksmatyHong LuLei Jiang

Randy Hulet Carlos Bolech

Imbalanced Fermi mixturesImbalanced Fermi mixtures

• 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

• 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

Observation: Phase separation

Superfluid core with polarized halo

MW. Zwierlein, A. Schirotzek, C.H. Schunck, and W, Ketterle: Science 311, 492-496 (2006)

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

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.

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

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

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

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

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

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

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

SF

LONN

3 classes of states3 classes of states

SF LO NN

Increasing energy

Density profiles (N=50,000)Density profiles (N=50,000)

n↑

n↓

Upclose on the LO stateUpclose on the LO state

Pei, Dukelsky and Nazarewicz, PRA 82, 021603 (2010)

Bulgac and Forbes, PRL 101, 215301 (2008)

Robustness of the density oscillationRobustness of the density oscillation

homogeneous trapped

Orso, PRL (2007); Hu et al., PRL (2007)

FFLO in 1DFFLO in 1D

Liao et al., Nature 467, 567 (2010)

Experiment in 1D (Hulet group)Experiment in 1D (Hulet group)

3D

t

1D

……

t

X

3D 1D

Dimensional crossover: 3D – 1DDimensional crossover: 3D – 1D

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

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

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

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.

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

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

Magnetic impurity induced FFLO state (3D)

Magnetic impurity induced FFLO state (3D)

• Two component Fermi gas offers very rich physics.

• Effects of trapping confinement.

• Flexibility of atomic system provides opportunities of studying exotic pairing mechanisms.

ConclusionConclusion

• “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