Transcript Document
Bogoliubov-de Gennes Study of Trapped Fermi Gases
Han Pu Rice University
(INT, Seattle, 4/14/2011) Randy Hulet Carlos Bolech Leslie Baksmaty Hong Lu Lei Jiang
Imbalanced Fermi mixtures
Fulde-Ferrel-Larkin-Ovchinnikov instability
• BCS Cooper pairs have zero momentum • Population imbalance leads to finite-momentum pairs • FFLO instability results in textured states
Experiments on spin-imbalanced Fermi gas
• 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)
Observation: Phase separation
MW. Zwierlein, A. Schirotzek, C.H. Schunck, and W, Ketterle: Science 311, 492-496 (2006) Superfluid core with polarized halo
Experimental results
n ↑ n ↓ n ↑ n ↓ Hulet High T Ketterle Salomon Low T MIT/Paris data are consistent with Local Density Approximation (LDA) Rice data (low T) strongly violates LDA.
Surface Tension
• Phase Coexistence -> Surface Tension
1 mm 60 m m Aspect Ratio of Cloud: 50:1 Aspect Ratio of Superfluid: 5:1 Data: Hulet Surface tension causes density distortion Effects of surface tension more important in smaller sample.
Breakdown of LDA
P=0.14
P=0.53
P
N
N
N
N
LDA LDA + surface tension P=0.72
De Silva, Mueller, PRL 97, 070402 (2006) Data points from Rice experiment.
Surface tension
2 2
m n s
4/3 Optimal value that fits data: : 3 ~ 4 However, from microscopic theoretical calculation: : 0.15
PRA 79, 063628 (2009)
Solving BdG equations
H
s
2 1 2
m
r
2 2
r
z
2
z
2 Choose
T
and
N
take initial guesses of m ,
r
diagonalize the matrix
H s
* m
H
s
m adjust m compute new until:
N
( ) until the input and output m ,
r
Effect of trap anisotropy: N=200, P=0.4
2.5
2.0
1.5
1.0
0.5
Density along z-axis 2.0
1.5
1.0
0.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4
z Z TF 0.2 0.4 0.6 0.8 1.0 1.2 1.4
z Z TF 2.0
1.5
1.0
0.5
2.5
2.0
1.5
1.0
0.5
Density along r-axis 2.0
1.5
1.0
0.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4
r Z TF 0.05 0.10 0.15 0.20 0.25 0.30
r Z TF 2.0
1.5
1.0
0.5
0.2
0.4
0.6
0.8
1.0
z Z TF 0.008
0.016
0.024
r Z TF 1.0
0.8
0.6
0.4
0.2
Gap along z-axis 0.2 0.4 0.6 0.8 1.0 1.2 1.4
z Z TF AR=1 0.8
0.6
0.4
0.2
0.2 0.4 0.6 0.8 1.0 1.2 1.4
z Z TF AR=5 0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
z Z TF AR=50
Quasi-1D system: N=200, AR=50
2.0
1.5
1.0
0.5
2.0
1.5
1.0
0.5
Density along z-axis
0.2
0.4
0.6
0.8
1.0
z Z TF 2.0
1.5
1.0
0.5
0.2
0.4
0.6
0.8
1.0
z Z TF 2.0
1.5
1.0
0.5
0.2
0.4
0.6
0.8
1.0
z Z TF
Density along r-axis
0.008
0.016
0.024
r
Z TF 2.0
1.5
1.0
0.5
0.008
0.016
0.024
r
Z TF 2.0
1.5
1.0
0.5
0.008
0.016
0.024
r
Z TF 0.8
0.6
0.4
0.2
Gap along z-axis
0.2
0.4
0.6
0.8
1.0
z Z TF
P=0.2
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
z Z TF
P=0.4
0.5
0.2
0.4
0.6
0.8
1.0
z Z TF
P=0.7
BdG vs. LDA: N=200, AR=50, P=0.6
Gap along z-axis Gap along r-axis n ↑ n ↓ n ↑ n ↓
N
~200,000
Going to higher N
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.
3 classes of states
NN SF LO
Density profiles (N=50,000)
SF LO Increasing energy NN
Upclose on the LO state
n ↓ n ↑
Robustness of the density oscillation
Bulgac and Forbes, PRL
101
, 215301 (2008) Pei, Dukelsky and Nazarewicz, PRA
82
, 021603 (2010)
FFLO in 1D
homogeneous trapped Orso, PRL (2007); Hu
et al.
, PRL (2007)
Experiment in 1D (Hulet group)
Liao
et al
., Nature
467
, 567 (2010)
3D 3D
Dimensional crossover: 3D – 1D
t X t 1D 1D
Model for single impurity in Fermi superfluidity
H
H
0
H imp H imp
drU
U
U
H 0 is BCS mean field Hamiltonian
u
u
for contact potential a
1
e
x a
2 2
for gaussian potential u
:
impurity strength
:
u
u
.
Magnetic impurity
:
u
u
.
BdG and T matrix methods BdG
method gives numerical results for single impurity in harmonic trap. BdG solves self-consistently a set of coupled equations
Eu
(
r
) (
k
)
u
(
r
)
i
y
v
(
r
)
U
(
r
)
u
(
r
),
Ev
(
r
) (
k
)
v
(
r
)
i
y
u
(
r
)
U
(
r
)
v
(
r
)
T-matrix
gives exact solutions for localized contact impurity without trap.
G
(
k
,
k
' ;
w
)
G
0 (
k
;
w
) (
k
k
' )
G
0 (
k
;
w
)
T
(
w
)
G
0 (
k
' ;
w
) Contact potential: T matrix only depends on energy.
Localized non-magnetic impurity in 1D trap BdG
results with impurity without impurity
T matrix
results Bound state occurs when T -1 (w)=0
E
0 ( m
mu
0 2 ) 2 2 2 2
u
u
0
u
u
0 0 .
02
E F
0
z TF
Localized magnetic impurity
u
u
u
0 0
BdG
results
T matrix
results Bound state energy inside the gap 0
E
0 0 1 1 ( (
u
0
N u
0
N
0 0 This bound state is below the bottom of quasiparticle band.
/ 2 ) 2 / 2 ) 2
Density and gap profiles for localized magnetic impurity
Spin up Spin down What if we increase impurity width and strength
?
Magnetic impurity induced FFLO state
Impurity: Gaussian potential Spin up Spin down
u
0
a
0 .
12 0 .
2
z TF
, 0 .
4 , 1 .
0
E F z TF
Magnetic impurity induced FFLO state (3D)
Conclusion
• Two component Fermi gas offers very rich physics.
• Effects of trapping confinement.
• Flexibility of atomic system provides opportunities of studying exotic pairing mechanisms.
References
•
“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