May 13-14 (2011) GriffinFest, Toronto, Canada

Yoji Ohashi1,3, Takashi Kashimura1 , and Shunji Tsuchiya2,3

Formation of Magnetic Impurities, π-Junctions, and a Spontaneous Current State

in a Superfluid Fermi Gas

introductionidea to introduce magnetic impurities to a superfluid Fermi gas

physical properties around magnetic impurity

superfluid/ferromagnet/superfluid-junction,and spontaneous current state

1:Department of Physics, Keio University, Japan

local density of stateslocalized excited (bound) states

2:Department of Physics, Tokyo University of Science, Japan3:CREST (JST), Japan

1991.4~1994.3 (PhD course : Tokyo Institute of Technology)“Antiferromagnetic spin fluctuations in high-Tc cuprates”

1991.4~1994.3 (PhD course : Tokyo Institute of Technology)“Antiferromagnetic spin fluctuations in high-Tc cuprates”

1991~1994 (PhD course : Tokyo Institute of Technology)1994.4~1995.3 (PD: Osaka University)“Boundary effects on d-wave superconductors”

1991~1994 (PhD course : Tokyo Institute of Technology)1994.4~1995.3 (PD: Osaka University)“Boundary effects in d-wave superconductors”

1991~1994 (PhD course : Tokyo Institute of Technology)1994.4~1995.3 (PD: Osaka University)1995.4~2006.3 (Tsukuba University)“Carlson-Goldman mode and Josephson plasma in high-Tc cuprates”

1991~1994 (PhD course: Tokyo Institute of Technology)1994.4~1995.3 (PD: Osaka University)1995.4~2006.3 (Tsukuba University)“Carlson-Goldman mode and Josephson plasma in high-Tc cuprates”

1991~1994 (PhD course : Tokyo Institute of Technology)1994.4~1995.3 (PD: Osaka University)1995.4~2006.3 (Tsukuba University)

2001.8~2002.6 (visiting researcher, Toronto University)“BCS-BEC crossover in a gas of Fermi atoms with a Feshbachresonance”

2004.9~2004.11 (visiting researcher, 　　　　　　　　　　Toronto University)2006.4~ (Keio University)2011.5.13 and 14 (Toronto!)

May 13-14 (2011) GriffinFest, Toronto, Canada

Yoji Ohashi1,3, Takashi Kashimura1 , and Shunji Tsuchiya2,3

Formation of Magnetic Impurities, π-Junction, and Spontaneous Current State

in a Superfluid Fermi Gas

IntroductionIdea to introduce magnetic impurities to a superfluid Fermi gas

Properties around pseudo-magnetic impurity

superfluid/ferromagnet/superfluid-junction,π-junction, and spontaneous current state

1:Department of Physics, Keio University, Japan

density of stateslocalized excited (bound) states

2:Department of Physics, Tokyo University of Science, Japan3:CREST (JST), Japan

Superfluid 40K and 6Li Fermi gasesSuperfluid 40K and 6Li Fermi gases

“quantum simulator” to study various physical properties of superconductivity

C. A. Regal, et al. PRL 92 (2004) 040403.

| 9 / 2, 7 / 2− >

| 9/ 2, 9/ 2− >

40KBCS-BEC crossover by tunable interaction

optical lattice

Fermi SF Bose SF

strong-coupling effects beyond mean-field theorypseudogap (preformed pairs)

population imbalance

band effect

Hubbard model and strong correlation

superconductivity under magnetic fieldFFLO

low-dimensional effect

“Magnetic” effect“Magnetic” effect

superfluid Fermi gas

metallicsuperconductivity

C. A. Regal, et al. PRL 92 (2004) 040403.

| 9 / 2, 7 / 2− >

| 9/ 2, 9/ 2− >

40KCrow et al., Phys. Lett. 21, 378 (1966).

La3-xGdxIn

magnetic impurity concentration

0c

c

TT

“magnetic”impurity

?“spin”“pseudospin”

Magnetic effects on superconductivityMagnetic effects on superconductivity

spin

fluctuations

pairing mechanism depairing effect

magnetic impurity

Cooper pair

High-Tc cuprates: antiferromagnetic spin fluctuations

Superfluid liquid 3He: ferromagnetic spin fluctuations

pair-breaking effect by magnetic impuritiespair-breaking effect by magnetic impurities

Satori, Shiba (1992) Shiba(1968) Abrikosov, Gor’kov (1961)

bound statebelow the gap

competition between SC and Kondo effect

suppression of SC state by magnetic scattering

gapless SC

Nonmagnetic impurities do not affect s-wave SC.(Anderson’s theorem)

Kondosinglet

spindoublet

0.3KT=

∆

boun

d st

ate

ener

gy /Δ

/KT ∆

Kondosinglet

impurity band

supe

rcon

duct

ing

DO

S0c

c

TT

2 0( 1) (0) /ii p mp cm S S u N Tρ × +

π-junction π-junction Insulator ferromagnet

SC SC SCSC

( )x∆

x

“0”-junction

( )x∆

x0 0

Al/Al2O3/PdNi

“π”-junction

Nb Nb

T. Kontos et al., PRL 89, 137007 (2002)

( )x∆

SC

ferromagnet

π-junction π-junction Insulator ferromagnet

SC SC SCSC

( )x∆

x

“0”-junction

( )x∆

x0 0

Al/Al2O3/PdNi

“π”-junction

Nb Nb

T. Kontos et al., PRL 89, 137007 (2002)

( )x∆

SC

F SC

We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.

Phase sepration in a superfuid Fermi gas with population imbalance

pseudo-spin↑

↑－↓ > 0

pseudo-spin↓

N NSF

G. Partridge et al., PRL 97, 190407 (2006)

↑↓

↑ ↑

We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.

pseudo-spin↑

pseudo-spin↓

N NSF

　　　　　　　Super

fluidSuperfluid

Superfluid

Superfluid

Ferrom

agnetFerro

magnet

　　　　　

N SFSF

G. Partridge et al., PRL 97, 190407 (2006)

↑↓

↑ ↑↑↓↑↓ ↑

( )x∆

x

π-junction

Phase sepration in a superfuid Fermi gas with population imbalance

↑－↓ > 0

We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.

Phase sepration in a superfuid Fermi gas with population imbalance

pseudo-spin↑

pseudo-spin↓

N NSF

G. Partridge et al., PRL 97, 190407 (2006)

↑↓

↑ ↑↑

polarized magnetic impurities↑－↓ > 0

↑↑

↑

small nonmagneticimpurity potentials

We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.We theoretically discuss an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas.

　　　　　　　Super

fluidSuperfluid

Superfluid

Superfluid

Ferrom

agnetFerro

magnet

　　　　　

N SFSF↑↓↑↓ ↑

To examine these ideas in a simple manner, we consider a Hubbardmodel at T=0. We self-consistently determine the order parameter, particle density, and polarization, around impurity potential and barrier within the mean-field level. We examine the possibilities of magnetization of impurities, SFS-, and π-junction.

To examine these ideas in a simple manner, we consider a Hubbardmodel at T=0. We self-consistently determine the order parameter, particle density, and polarization, around impurity potential and barrier within the mean-field level. We examine the possibilities of magnetization of impurities, SFS-, and π-junction.

Phase sepration in a superfuid Fermi gas with population imbalance

↑↑↑

↑

FormulationFormulation t−

U−

two-component 2D Fermi gas ,σ =↑ ↓

N N↑ ↓>population imbalance

†

, ,.i j ii i

i ji

i iH t c c h c U n n V n N Nσ σ σ

σ

µ µ↑ ↓ ↑ ↑ ↓ ↓⎡ ⎤= − + − + − −⎣ ⎦∑ ∑ ∑

“nonmagnetic” potential

impurity junction trap

[ ]iV t

FormulationFormulation

† † †

, ,

,

. . .i ii j i i i

i j i i

i i Gi

H t c c h c c c h c U n n

V n N N E

σ σ σ σσ

σσ

µ µ

↑ ↓ −

↑ ↑ ↓ ↓

⎡ ⎤⎡ ⎤= − + − ∆ + −⎣ ⎦ ⎣ ⎦

+ − − +

∑ ∑ ∑

∑

We solve the BdG equations to self-consistently determine for given .

, ,i inσ σµ∆Nσ

0.36i in n↑ ↓+ ≅

phase diagram of 2D uniform Hubbard model

N NP

N N↑ ↓

↑ ↓

−=

+

Formation of pseudo-magnetic impurityFormation of pseudo-magnetic impurity2

0 2 2( )iimp

V V Γ=

− + ΓR R[ ]t 0 / 0.25/ 1/ 6

V tt

U t

=

Γ ==

∆ is not destroyed by impuritypotential (Anderson’s theorem).∆ is not destroyed by impuritypotential (Anderson’s theorem).

∆ is remarkably damaged by impuritypotential.∆ is remarkably damaged by impuritypotential.

300, 300N N↑ ↓= = 30 , 301 0N N↑ ↓= =per 41 41sites× per 41 41sites×

Formation of pseudo-magnetic impurityFormation of pseudo-magnetic impurity

Local magnetizationz i iS n n↑ ↓= −

5N∆ =1N∆ =0N N N↑ ↓∆ = − =

Nonmagnetic potential is magnetized by excess atoms!Nonmagnetic potential is magnetized by excess atoms!

Magnetization of nonmagnetic potentialMagnetization of nonmagnetic potential

magnetic impurity in metal“pseudo”-magnetic impurity

Strong electron correlation excludes double occupancy of ↑ and↓spins.

To minimize the condensation energy loss by the depairing effect, excess atoms are localized around the region where ∆ is small from the begining.

Coulombrepulsion( )x∆

impV

z zH J S σ= −(classical spin)

xH J σ= − ⋅S

(quantum spin)(no exchange term)

Bound states by off-diagonal pair-potential wellBound states by off-diagonal pair-potential well

5N∆ =

( )V x

bound states

potential well V(R)pair-potential well ∆(R)

We can expect that this well structure of off-diagonal pair potential ∆(Rx,Ry) induces bound states around the impurity potential, as in the case of ordinary (diagonal) potential well V(Rx,Ry) .

We can expect that this well structure of off-diagonal pair potential ∆(Rx,Ry) induces bound states around the impurity potential, as in the case of ordinary (diagonal) potential well V(Rx,Ry) .

local density of states ρ↑(ω,R)local density of states ρ↑(ω,R)

/ tω

××

ρ ↑(ω

,R)

uniform Fermi gas

impurity

In-gap states appear around “pseudo-magnetic” impurity.In-gap states appear around “pseudo-magnetic” impurity.

BCS gap

( / 0.05)tγ =

yR

xR

30 , 301 0N N↑ ↓= =

N N↑ ↓=

bound state wavefunctions below the energy gapbound state wavefunctions below the energy gap2

particle| ( , ) |x yR RΨ

yR

xR

L=0 (s)

L=3 (f)L=2 (d)L=1(p)

lowest bound state

2nd lowest states 3rd lowest states 4th lowest states

The pair-potential well ∆(Rx,Ry) works as a cylindrical potential well, so that bound states can be classified by angular momentum L.The pair-potential well ∆(Rx,Ry) works as a cylindrical potential well, so that bound states can be classified by angular momentum L.

Superfluid/ferromagnet/superfluid (SFS) junctionSuperfluid/ferromagnet/superfluid (SFS) junction

yR

yR

xR

xR

( )V R

( )z i iS n n↑ ↓= −R

0.11N N

PN N

↑ ↓

↑ ↓

−= =

+

Population imbalance

/ 7U t =99N =

11N N N↑ ↓∆ = − =S SF

yR

( )zS R

0.11N N

PN N

↑ ↓

↑ ↓

−= =

+

( , )x yR R∆ ( , )x yR R∆

yRyR

xRxR

S SF

+ + +_

0-junction

π-phase in a superfluid Fermi gasπ-phase in a superfluid Fermi gas

/ 875.00E t = − / 874.78E t = −

π-junction

stability of π-junctionstability of π-junction

/z yS L

junction 0 junctionE Eπ − −−

π-junction

0-junction

1[( ) ]ytL −

The π-junction becomes stable when magnetization is large to some extent.The π-junction becomes stable when magnetization is large to some extent.

Effects of trap potential 2Effects of trap potential 2

＋ －

2D cigar trap ( , )z x yS R R

( , )x yR R∆

3228

40.10.001

1.06

x

y

B

NNU tV tV t

V tσ

↑

↓

=

=

==

=

==

spontaneous current state in a ring trap spontaneous current state in a ring trap

Because of the single-valueness of ∆, the π-junction twists the phase θ of the order parameter by π along the ring.

ieπ−∆ = ∆+∆

N N↑ ↓>

Spontaneous current

~ 0J θ∇ >

nonmagnetic potential barrier+

localized excess ↑-spin atoms| |

π-junction( )x∆

xR+

_ “corner-junction”

spontaneous current state in a ring trap spontaneous current state in a ring trap

( , ) (4 , 40)1N N↑ ↓ =

~δθ π

1D-ring trap

spontaneous flow

nonmagnetic potential barrier+

localized excess ↑-spin atoms| |

π-junction

ieπ−∆ = ∆+∆

N N↑ ↓>

( )x∆xR

+_

~ 0J θ∇ >

θ(x)

/π ( )| ( ) | i xx e θ∆

xR

SummarySummaryWe have discussed an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas. Using the phase separation of a polarized Fermi superfluid, we have shown that nonmagnetic potential is magnetized in the sense that some of excess atoms are localized around the potential.

We have discussed an idea to introduce magnetic impurities and magnetic junction to a superfluid Fermi gas. Using the phase separation of a polarized Fermi superfluid, we have shown that nonmagnetic potential is magnetized in the sense that some of excess atoms are localized around the potential.

pseudo-magnetic impurities

polarized classical spinsuppression of ∆ around impurity in-gap states

“quantum dot”gapless Fermi superfluid

FS Ssuperfluid/ferromagnet/superfluid junction

xRyR

( , )x yR R∆

+_

π-junction FFLO ( )FFLO x∆

πS

spontaneous supercurrent

SummarySummary

T. Kashimura, S. Tsuchiya, Y. Ohashi, Phys. Rev. A, 82, 033617 (2010)Y. Ohashi, Phys. Rev. A (2011), in press, cond-mat/1103.1942.T. Kashimura, S. Tsuchiya, and Y. Ohashi, in preparation

T. Kashimura, S. Tsuchiya, Y. Ohashi, Phys. Rev. A, 82, 033617 (2010)Y. Ohashi, Phys. Rev. A (2011), in press, cond-mat/1103.1942.T. Kashimura, S. Tsuchiya, and Y. Ohashi, in preparation

pseudo-magnetic impurities

polarized classical spinsuppression of ∆ around impurity in-gap states

“quantum dot”gapless Fermi superfluid

FS Ssuperfluid/ferromagnet/superfluid junction

xRyR

( , )x yR R∆

+_

π-junction FFLO ( )FFLO x∆

πS

spontaneous supercurrent