Han PuRice University

Collaborators:

Lei Jiang (NIST/JQI)

Hui Hu, Xia-Ji Liu (Swinburne)

Yan Chen (Fudan U.)

2013 Hangzhou Workshop on Quantum Matter

Localized Impurity in Ultracold Fermi Gas

Useful impurities

• Impurities can be used to probe physical properties of their host.

Quasi-particle DOS

S.H. Pan, et al., Nature, 403, 746(2000)

Zn impurity in BSCCO

Impurities in cold atoms

• Cold atoms are intrinsically clean.• Impurities or disorders can be engineered.

Anderson localization of atomic BEC

Inguscio group, nature 453, 895 (2008) Aspect group, nature 453, 891 (2008)

A local impurity in fermionic superfluid

2 2

0 0

†

,

/

, : BCS mean-field Hamiltonian

( ) ( ) ( )

contact impurity: ( ) ( )

1extended impurity: ( )

:

:

imp

imp

x a

H H H H

H d U

U u

U u ea

u impurity strength

Non magnetic impurity u u

Magnetic impu

r r r r

r r

r

:rity u u

BdG and T-Matrix method

)()()()()()(

),()()()()()(

rvrUruirvkrEv

rurUrvirukrEuy

y

T-matrix is convenient 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

Contact non-magnetic impurity

0

0

0

0.02 F TF

u u u

u E x

BdG results

with impurity

without impurity

LDOS at impurity site

Contact non-magnetic impurity

2220

0 22

muE

Bound state occurs when T-1(w)=0

T-matrix results

T-matrix

BdG

LDOS

x x

Contact 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

T-Matrix results

Yu-Shiba stateYu, Acta Phys. Sin. 21, 75 (1965)Shiba, Prog. Theor. Phys. 40, 435 (1968)

Contact magnetic impurity

BdG results

LDOS Superfluid gap

Extended magnetic impurity

Spin up Spin down

Impurity: Gaussian potential with width a

0 0.12, 0.4, 1.0

0.2F TF

TF

u E x

a x

Impurity strength

A strong magnetic impurity induces local spin imbalanceA controlled way of generating FFLO state

Jiang et al., PRA 83, 061604(R) (2011)

Impurity in Fermi gas with spin-orbit coupling

• 2D Fermi gas with Rashba spin-orbit coupling

Sau et al., PRL 104, 040502 (2010)

Topological superconducting state

Supports Majorana mode in the vortex excitation

Topological phase transition in Fermi gas with SOC

Liu et al., PRA 85, 021603(R) (2012)

Quasi-particle excitation spectrum in the presence of a single vortex

Non-topological Topological

Impurity in topological and non-topological host

Non-topological Topological

Incr

easi

ng im

purit

y st

reng

thLDOS near a contact magnetic impurity

Contact impurity in topological superfluid

Both magnetic and non-magnetic impurity induces mid-gap bound states.

Incr

easi

ng im

purit

y st

reng

th

As the impurity strength increases,the energy of the bound state approacha universal value: 2

0~ / FE

Gap profile

For sufficiently large impurity strength, the order parameter is suppressednear the impurity site.

Superfluid order parameter:

Mid-gap state at the edge

For sufficiently large impurity strength, the order parameter is suppressednear the impurity site.

Topological superfluid

0

20 0

/

/ /

F

F F

v

E k E

Alicea, Rep. Prog. Phys. 75, 076501 (2012)

Visualizing the bound state

†

,

u c v c

2 21( , ) ( ) ( )

2E u E E v E E

r

LDOS:

Bound states induced by extended impurity

Incr

easi

ng im

purit

y st

reng

th

Hu et al., PRL 110, 020401 (2013)

Measuring LDOS

3

RF-spectroscopy

†3 3

3 3

( ) ( ) ( )

' ( , '; ) ( ', ; ) ( )

dI

dt

d d A A f

r r r

r r r r r

If state 3 has a flat dispersion:

3 3

3

( ', ; ) ( ' )

( ) ( , )

A

I

r r r r

r r

Conclusions and References

Engineered impurities allow us to investigate a host of interesting quantum effects in ultracold atomic gases.