冷却原子気体，並びに固体物質における LOFF超流動研究の現状

岡山大学自然科学研究科町田一成

07.5.10. 東大

共同研究者　　　市岡優典，水島健，高橋雅裕

Outline1) General introduction to cold atom gases: BEC, BCS,

and crossover2) Resonace Fermionic superfluid with mismatched FS’s Possible realization of Fulde-Ferrell-Larkin-Ovchinniko

v state (FFLO)3) Microscopic calculation; 　 Bogoliubov-de Gennes (Bd

G) equation 4) Topological structure of vortex in FFLO; 　 physics of shift5) Condensed matter systems; superconductivity in Ce

CoIn5-- a heavy Fermion material; Quasi-classical Eilenberger formalism6) Conclusions

Magnetic confinement

“Laser cooling”

Trapped atomic gasesTrapped atomic gasesNeutral atoms: Li, Na, K, Rb, Cs, Cr, Yb, H, He*

⇒ Hyperfine spin F (e.g., 6Li atoms, F = 9/2, 7/2)

Trapping potentialTrapping potential

⇒　 3-dimensional harmonic trap

Typically, Axial symmetry

Inter-atomic interactionInter-atomic interaction

a: s-wave scattering length (e.g., a = 2.75nm in 23Na)

⇒ 　 By using Feshbach resonance, a → ±∞

trapping

cooling

imaging

What are statistics of Alkali atoms?

Alkali Atoms

Why Alkali’s?Strong transitions in optical/near IR:

Easily manipulated with lasers

Nuclear physics:Odd # neutrons + Odd # protons= Unstable

Alkali’s tend to be Bosons: odd p,e even n

Atom Isotope Abundance Half Life

HLiK

2640

0.01%8%0.01%

StableStable109 years

Only Fermionic Isotopes:

Composite Bosons: Made of even number of fermionsComposite Fermions: odd number of fermions

Condensation in real &

momentum space!

in situ image

Observing statisticsObserving statistics

Hulet et al., Science (2001)

Fermi degeneracy

BosonsBosons FermionsFermions

New Probes• in situ & TOF image (Density)• RF Spectroscopy (Tunneling current, density of states)• Noise Correlations

Settings• Low Dimension• Rotation• Optical lattices• Ring trap• Chips

Controls• Interactions• Population

States• Vortices (multiply-quantized & coreless vortices)• Soliton• Dipolar BEC• SF-Insulator Transition• Tonks-Girardeau gas & BKT phase• BCS-BEC crossover• Imbalanced Fermionic Superfluid

BEC: Li, Na, Rb, Yb, K, Cs, Cr, He, HFermionic SF: Li, K

Recent ProgressRecent Progress

Controlling InteractionControlling Interaction

“Feshbach resonance” between two lowest hf states“Feshbach resonance” between two lowest hf states

Scattering is dominated by bound state closest to threshold

Bound state (spin singlet)

En

ergy

Magnetic field: B B0

Spin triplet channel

Bound state energy is shifted relative to continuum

-1

0

1

834.15650-20

-10

0

10

20x10

3

Magnetic Field: B [G]

Atoms formstable molecules

bound state

Scatterin

g length

a

s-wave scattering length vs Magnetic field: 6Lis-wave scattering length vs Magnetic field: 6Li Zwierlein et al., Nature (2005)

Strong interactionsUniversality? Fraction of molecules?

Strong interactionsUniversality? Fraction of molecules?

← Molecular BEC BCS →

Dance AnalogyE

(Figures: Markus Greiner)

Tightly bound pairs

Every boy is dancing with every girl: distance between pairs greater than distance between people

Slow Dance

Fast Dance

Ultra cold atoms by laser cooling

Atomic Bose-Einsteincondensate (sodium)

Molecular Bose-Einsteincondensate (lithium 6Li2)

Pairs of fermionicatoms (lithium-6)

UniversalityOnly length-scale near resonance is density:

No microscopic parameters enter equation of state

Hypothesis: is Universal parameter -- independent of system

Binding energy: 2 MeV << proton mass (GeV)

Implications: Heavy Ion collisions, Neutron stars

Nuclear matter is near resonance!!

Tune quark masses: drive QCD to resonance

Implications: Lattice QCD calculationsBraaten and Hammer, Phys. Rev. Lett. 91, 102002 (2003)

pion mass (140 MeV)

Bertsch: Challenge problem in many-body physics (1998): ground state of resonant gas

Calculations

Fixed Node Greens Function Monte CarloJ. Carlson, S.-Y Chang, V. R. Pandharipande, and K. E. SchmidtPhys. Rev, Lett. 91, 050401 (2003)

Linked Cluster ExpansionG. A. Baker, Phys. Rev. C 60, 054311 (1999)

Lowest Order Constrained Variational MethodH. Heiselberg, J. Phys. B: At. Mol. Opt. Phys. 37, 1 (2004)

No systematic expansion

Ladder (Galitskii) approximationH. Heiselberg, Phys. Rev. A 63, 043606 (2003)

Mean field theoryEngelbrecht, Randeria, and Sa de Melo, Phys. Rev. B 55, 15153 (1997)

Resumation using an effective field theorySteele, nucl-th/0010066

Experiments:Duke: -0.26(7) ENS: -0.3 JILA: -0.4 Innsbruck: -0.68(1)

Fixed Node Diffusion Monte CarloG. E. Astrakharchik, J. Boroonat, J. Casulleras, and S. Giorgini,Phys. Rev. Lett. 93, 200404 (2004)

Superfluidity near resonanceE

B

Molecules:

Bosons -- condense, form superfluid

Atoms:Fermions with attractive interactions -- pair (cf BCS) form superfluid

Theory: continuously deform one into other; BCS-BEC crossover

Superfluidity: Needs bosons which condense

Superfluidity: Needs bosons which condense

Leggett, J. Phys. (Paris) C7, 19 (1980)P. Nozieres and S. Schmitt-Rink, J. Low Temp Phys. 59, 195 (1985)

Superfluidity near resonanceE

B

B

BCSBEC

All properties smooth across resonance

Pairs shrink

Fermionic Superfluidity withImbalanced Spin Populations

ExperimentsExperiments • Zwierlein et al., Science (2006); Nature (2006)• Zwierlein et al., Science (2006); Nature (2006)

TOF images (after expansion)TOF images (after expansion)

100nK100nK50nK50nK

350nK350nK 260nK260nK 190nK190nK

70nK70nK

70nK70nK

SF or Normal ?

Bimodal structure in minority component ⇒ 　 the dense in the central area marks the onset of the condensation!

50-50% BCS ?Dashed lines: normal fermions(TF approximation)

In situ & reconstructed 3D imagesIn situ & reconstructed 3D images

Columnar densitiesColumnar densities

Absorption images

Empty core! (P < Pc ~ 0.8)　⇒　 locally 50-50% BCS pairing

Reconstructed 3D profiles from the integrated 2D distributions

Reconstructed 3D profiles from the integrated 2D distributions

⇒　 Only assuming the axial symmetry

Cross section

Observation of quantum phase transitionObservation of quantum phase transition

Critical population imbalance Pc

Critical difference in “Fermi energies”Critical difference in “Fermi energies”

Summary of MIT experimentsSummary of MIT experiments

1. Density profile (integrated & cross-section profiles)Bimodal structure ⇒ Observation of fermionic superfluidity 　 with mismatched spin populationPhase separation like profile

2. Direct observation of quantum phase transitionCritical population imbalance ~ pairing gap

Question

Is the density in the superfluid always 50-50%?

⇒ SF-N phase separation or other exotic pairing?

On resonance a diverges

Only remaining energy scales are EF1 and EF2

Condition for breakdown universal constant ·

will relate EF1 to EF2 and thus pick out a

universal number mismatch for breakdown in a harmonic trap:

= 70(3) %

Universal Physics

The critical imbalance is a measure of the unitary interaction strength!

A condensate emerges from the Fermi sea

Critical Imbalancec= 71(3)%

Increase atom number of smaller cloud:

FFLO and -shift physics

magnetization

(z)

Fulde & Ferrell, PR 135, A550 (1964) Larkin & Ovchinnikov, JETP 20, 762 (1965)

(k↑,-k+q↓): spatially inhomogeneous pairing field

Cooper pairing (k,-k)Cooper pairing (k,-k+q)

Cooper pairing has a non-vanishing center-of-mass momentum q

• Superfluid phase in unequal mixture of two species with mismatched FS’s

Introduction to FFLO

zexp(iqz) FF state

zsin(qz) LO state

Physics ofshift

Order parameter changes signWhen connecting two ground states

Doubly degenerate ground state

Midgap state

K. Machida and H. Nakanishi, PRB30,122 (1984).

THEORETICAL FRAMEWORK: Mean-field theory

• Self-consistent condition: Pairing field & particle density

Bogoliubov-de Genns (BdG) equation

FFLO State in Uniform SystemFFLO State in Uniform System

-phase shift　→　“ mid-gap states” with zero-energy-phase shift　→　“ mid-gap states” with zero-energy

Pairing field |(r,z)|

①

① Local Density of States (LDOS): N(z, E)

up-spin down-spin

Zeeman splitting2 = 1.0

Local magnetization: ↑ - ↓

-phase shift

FFLOnodal plane

+(r)

-(r)

-shift

P= 0.34P= 0.34

BCS

SF

FFLO

N

Ground state at T=0Ground state at T=0

Density (r)

Pairing field (r)

g=-1.5 　⇒　 (0)/EF(0) =0.35 and Pc = 0.62

Shell structure!Shell structure!

FFLOnodal plane

Pairing fieldPairing field

Spatial Profiles of Ground State in finite PSpatial Profiles of Ground State in finite P

Local “magnetization”Local “magnetization”

• Ground state in nonzero P at T = 0 　⇒　 Spatially modulated “FFLO-like” pairing state• With increasing P, the area of suppressed polarization shrinks toward the trap center. i.e., the stable region of “BCS” pairing shrinks, BUT, the FFLO oscillation emerges outside region, which allows the coexistence with excess atoms 　⇒　 NOT simple BCS-Normal Phase separated state!

0(0)/EF(0) = 0.32

Condensation radiiCondensation radii

Locally equal population　⇒　 BCS pairingLocally equal population　⇒　 BCS pairing

FFLO modulated pairing⇒ SF is still robust!

FFLO modulated pairing⇒ SF is still robust!

Pc

Depletion throughcondensate!

Difference Profiles on ResonanceDifference Profiles on Resonance

~ Pc = 0.7

Zwierlein et al., Science (2006); Nature (2006)Zwierlein et al., Science (2006); Nature (2006)

Phase diagram at T = 0Phase diagram at T = 0

BdG in trapped systemBdG in trapped system

“FFLO” ⇒ Pairing state which changes sign“BCS” ⇒ Pairing state having a definite sign

On resonance?

Zwierlein et al., Science ‘06Zwierlein et al., Science ‘06

Critical population imbalance & pair potential⇒ 　 Linear relation in WC limit

Critical population imbalance at T = 0

Pc = 0.57

Critical temperature at P = 0

c0/ ~ 5.8

(0)/EF(0) =0.32

Generic phase diagram e.g., CDW, SDW, and stripe phase etc. ⇒ Transition from C (BCS) to IC (FFLO) phases

Generic phase diagram e.g., CDW, SDW, and stripe phase etc. ⇒ Transition from C (BCS) to IC (FFLO) phases

Phase diagramPhase diagram

Lifshitz (Leung) pointTL ~ 0.6Tc0

Lifshitz (Leung) pointTL ~ 0.6Tc0

Tc curve for BCS-Normal phase transition obtained from the gap equation

Superfluidity of a two-component Fermi gas with asymmetric spin densitiesbased on the microscopic theory approaching from the WC towards SC limits.

1. Superfluid state of unequal mixture at T = 0

• The stable region of “BCS” pairing shrinks toward the trap center, with increasing P, while the “FFLO” pairing emerges in the outer region.• The strong suppression of the local “magnetization” ⇒ 　 direct evidence of “superfluidity”.

2. Stable superfluid state in finite T’s and phase diagram

• “FFLO” pairing is favored in large P and low T’s• The T-dependence of Pc(T) is observable in the experiment!

Especially, enhancement of Pc in low T region• Generic phase diagram e.g., (i) Double-phase transition (BCS ⇒ FFLO ⇒ Normal), (ii) Two second-order phase transition lines merge in the L point with TL ~ 0.6 Tc0.

ConclusionsConclusions

50-50% BCS core + FFLO pairing +

surrounded by fully polarized normal cloud

50-50% BCS core + FFLO pairing +

surrounded by fully polarized normal cloud

For the details, Machida, Mizushima, Ichioka, PRL (2006)

Side imaging

Top imaging

Vortex lattices in rotating Fermionic superflidVortex lattices in rotating Fermionic superflid

Quantized vortices in imbalanced superfluidQuantized vortices in imbalanced superfluidDirect observation of “superfluidity” in unequal mixture

⇒ 　 Quantized vortices induced by external rotation

P = 1 0.74 0.58 0.48 0.32 0.16 0.07 0

Zwierlein et al., Science 311, 492 (2006)

Vortex core structure in population imbalance

-----why vortex is visible in balance case ?-----why vortex is invisible in imbalance case ?

BCS ----- OP ----- (r)

BEC ------ OP----- n (r)

vortex visibility or invisibility

M. Takahashi, et al, PRL (2006)

Caroli-de Gennes-Matricon state & Quantum depletionCaroli-de Gennes-Matricon state & Quantum depletion

Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998) Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998)

Continuous 2 phase change around the singularity

Quasiparticles passing through vortex experience the -phase shift

↓Appearance of core-bound state

“Caroli-de Gennes-Matricon (CdGM) state”

-phase shift

Fermi level: EF

Local density of states(LDOS)

Vortexcenter

Lowest CdGM state (A) with finite amplitude at core ⇒ Positive shiftLowest CdGM state (A) with finite amplitude at core ⇒ Positive shift

P = 0

“Quantum depletion”

Caroli-de Gennes-Matricon state & Quantum depletionCaroli-de Gennes-Matricon state & Quantum depletion

Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998) Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998)

Lowest CdGM state ⇒ Discretization & Positive shift⇒ Unoccupied at low T

Total density with vortex at T = 0 (solid line)

vortex core

Vortex center

cf Majorana zero mode when chiral p wave px+ipy

Reduced quantum depletion inside core in imbalance caseReduced quantum depletion inside core in imbalance case

Vortex core structure at P = 0.3 and 0/EF = 0.32

(1) vortex with imbalance (2) vortex with balance (3) vortex free

Local “polarization”: m(r) = n↑(r) -n↓(r)⇒ Peak at core

DensitiesDensitiesPairing fieldPairing field

Reduced quantum depletion inside core in imbalance caseReduced quantum depletion inside core in imbalance case

Total densityTotal density

“Core filling factor”

SC

WC

F = n(0)/nmax

It is difficult to see the quantum depletion in imbalanced caseBut minority component core is still visible in density profile experiment.

Total@balanced case

Majority@P=0.3 Minority@P=0.3

Total@P=0.3

It is difficult to see the quantum depletion in imbalanced caseBut minority component core is still visible in density profile experiment.

Total@balanced case

Majority@P=0.3 Minority@P=0.3

Total@P=0.3

SummarySummary

Vortex core structure under population imbalance

Ref: Takahashi, Mizushima, Ichioka, Machida, PRL(2006)

• Core is filled in by majority component ⇒ difficult to see But minority component core is visible in density profile experiment This can be checked by in situ imaging (Zweirlein et al. 2006)• Local polarization shows a peak at vortex core• Splitting of CdGM states due to the resonance with mid-gap states at FFLO node

Topological structure of a vortex in FFLO superfluidTopological structure of a vortex in FFLO superfluidTM, Ichioka, Machida, PRL 95, 117003 (2005)Ichioka, Adachi, TM, Machida, preprint (2006)

What happens in quasiparticle structure if there exists FFLO nodal plane crossing vortex line ?

What happens in quasiparticle structure if there exists FFLO nodal plane crossing vortex line ?

• FFLO modulation vector // Vortex line

+(r)

FFLOnodal plane

Vortex line

-(r)

“-phase shift”

Splitting of CdGM statesdue to resonance with mid-gap (surface) states

• FFLO state (2D localization): 2-dimensional (planar) defects⇒　 2D localization of excess atoms at nodal plane

• Vortex state with BCS-pairing: 1-dimensional (line) defects⇒ 1D accumulation of excess atoms at vortex line

• Vortex state with FFLO modulated pairing→ ?

Paramagnetic moment　⇄　 Electronic state (local density of states)Paramagnetic moment　⇄　 Electronic state (local density of states)

Topological structure of the pair potential in the FFLO state

2 phase winding

Vortex line

phase shiftFFLO nodal plane

Topological structure of a vortex in FFLO superfluidTopological structure of a vortex in FFLO superfluidMizushima, Ichioka, Machida, PRL 95, 117003 (2005)

FFLO States With a Vortex Line

Missing of local magnetization at the intersection point ?Missing of local magnetization at the intersection point ?

Local Polarization: m = ↑ - ↓Pairing field: 0/EF = 0.1, = 0.5

Quasiparticles crossing the intersection point cannot experience phase shift

-shift

-shift

-shift

Neglecting the background potential: V = 0

VortexFFLO

Topology of FFLO vortex

0

0

+

-Two dimensional nodal plane: magnetization accumulation due to shift---planar defect m-sheet

vortex line: magnetization accumulation due to shift---line defect m-rod

Intersection point of node and vortex:＋ shift　 non-singularNo bound stateLocalized magnetization absent

NUMERICAL RESULTS: FFLO states

-phase shift　→　“ mid-gap states” with zero-energy-phase shift　→　“ mid-gap states” with zero-energy

Pairing field |(r,z)|

①

① Local Density of States (LDOS): N(z, E)

up-spin down-spin

Zeeman splitting2 = 1.0

Local magnetization: ↑ - ↓

0 = max[(r, z)] = 1.5

↑↓

≠ 0 ≠ 0

NUMERICAL RESULTS: Vortex states

①

Particle-hole asymmetry at the coreParticle-hole asymmetry at the core

① LDOS: N(r, E) ② LDOS: N(r=0, z=0, E)

②

0

/2

/2

Pairing field |(r,z)|: = 0

q = 1/2

Vortex without FFLO

Topology of FFLO vortex

Two dimensional nodal plane: magnetization accumulation due to shift---planar defect

vortex line: magnetization accumulation due to shift---line defect

Intersection point of node and vortex:＋ shift　 non-singularNo bound stateLocalized magnetization absent

• Numerical results: 0/ = 0.1, = 0.5

Pairing field |(r,z)|Local magnetization: ↑ - ↓

0

0

+

-

①

③

②

Pairing field |(r,z)|

② LDOS: N↑(r, E) ① LDOS: N↑(z, E)

FFLO modulation

ordinary vortex

③ LDOS: N(r=0, z=0, E)

Disappearance of magnetization ← splitting of bound state

Doppler shift

LDOS at the intersection point

Pauli paramagnetic effect on vortex lattice

Zeeman effect -----> up spin and down spin population imbalance

-------> Fulde-Ferrell-Larkin-Ovchinikov (FFLO)

CeCoIn5

Phase diagram in H vs T

CeCoIn5

準古典近似に基づく vortex 描像

Integrate out high frequency partsfrom Gorkov green functions

valid for long wave length description

reliable for quantitative predictions

cf BdG useful, but hard to handle

Quasiclassical Eilenberger theory

Free energy with paramagnetic effect

Normal state susceptibility

Average flux density

Internal field distribution

Self-consistent equation

Pairing potential

Vector potential

Total internal fieldParamagnetic contribution

Diamagnetic (super-current) contribution

Paramagnetic magnetization

Normal state magnetization

B(r)/B0

YNi2B2C

Nishida, et al, JPSJ 73, 3247 (04)

CeCoIn5 ---A heavy Fermion supercondutorA. Bianchi et al, PRL91, 187004 (2003).

FFLO

NMR experiement on CeCoIn5 Kakuyanagi et al, PRL 94, 047602 (2005)

FFLO

Distribution of magnetization in FFLO vortex

Calculation by BdGCalculation by quasi-classical Eilenberger eq.

M

Normal component

Nodal sheet

Conclusion Possible observation of FFLO state in resonance Fermion superfluids in cold atoms w

ith unequal populations for two species. Also possibly in CeCoIn5 in high field.

Topological structure of vortex in FFLO Missing of the magnetization at the intersection between nodal plane (sheet-like magnetization) and vortex line (rod-like magnetization)

-phase shift physics:doubly degenerate ground state yields a similar phenomena such as in incommensurate spin density wave states (Cr),

spin Peierls systems (CuGeO3, etc)

Refs. T. Mizushima, K.M. and M. Ichioka, PRL 94, 060404 (2005). T. Mizushima, K.M. and M. Ichioka, PRL 95, 117003 (2005).

FFLO vortex and spin polarization

vortex line

vortex line

nodal plane

nodal plane

FFLO in cigar shape geometry modulated spin polarization

bservable by species selective density profile experiment

Midgap state

Local density of states in FFLO

Midgap state

＋

＋