冷却フェルミ原子気体のbcs-becクロスオーバー領...

冷却フェルミ原子気体のBCS-BECクロスオーバー領域における擬ギャップ状態

東京理科大学　土屋俊二共同研究者

大橋洋士, 渡邉亮太 (慶大理工)

PRA, 80, 033613 (2009)PRA, 82, 033629 (2010)PRA, 82, 043630 (2010)arXiv:1107.2915 (2011)

Pseudogap state of ultracold Fermi gases in the BCS-BEC crossover regime

Outline of talk:• Introduction : BCS-BEC crossover and pseudogap• Model and formalism : Nozieres-Schmitt-Rink theory• Pseudogap in a homogeneous Fermi gas : single-

particle spectral weight and density of states above Tc, pseudogap temperature and phase diagram

• Summary 1• Photoemission spectroscopy for a Fermi gas• Pseudogap signature in photoemission spectra of a

trapped Fermi gas• Summary 2

BCS-BEC crossover

weak-couplingstrong-coupling

BCSBEC

BCSBEC

Theory:Eagles,Leggett,Nozieres-Schmitt-Rink

Tc

40K

pair condensation

BCS

BECTc

Regal et al. PRL (2004)

BCS-BEC crossover in atomic Fermi gas

Tc/TF ∼ 0.08− 0.2 10−4 − 10−2

high-Tc superfluid!

Feshbach resonances allow one to control the interatomic interaction

Feshbach resonance

molecules form when

40Ks-wave scattering length tunable by magnetic field

resonance bound state

40K|9/2,−9/2+ |7/2,−7/2

|9/2,−9/2+ |9/2,−7/2

∆0

Evolution of Tc in BCS-BEC crossover

BCS state

BCS transition temperature

Cooper pair formation at TBCS

: gap at T=0TBCS =8γ

πe2εF e

π2kF as =

γ

π∆0

(pair biding energy)

TBCSTc

BEC BCS

T = TBEC

BEC BCS

λdB =

2π2

mkBT

1/2

∝ 1√T

thermal de Broglie length

λdB d (interatomic distance) at


Tc TBCS

Size of pair shrinks as increasing the interaction

λdB < d

λdB dnecessary to lower T to achieve


condition for BEC is not satisfied at TBCS :

Note that pair formation and condensation occurs at TBCS simultaneously because of large overlapping pairs

BEC BCS

TBCS

T > Tc

Pseudogap in high-Tc cuprates

Pseudogap observed in ARPES, tunneling, NMR, etc.

Renner et al. PRL (1998)

Hole doping

Tem

pera

ture

(K)

T*

PG

Bi2Sr2CaCuO8+

Origin of pseudogap • incoherent preformed pairs• strong AF spin fluctuations• hidden order etc.

TheoryYanase, Randeria, Strinati, Varma, Metzner ...

AFMSC

T < Tc

pair formation temperature

TBCS

TBCS Pseudogap is considered to exist in the temperature region due to the presence of incoherent non-condensed pairs

Tc T TBCS

BCSBEC

Tc PG ?

Pseudogap in atomic Fermi gases

TBCS

BCSBEC

Tc PG ?

Pseudogap in atomic Fermi gases

We calculate the density of states and spectral weight including strong coupling effects associated with pairing fluctuations to directly identify the pseudogap

Useful to clarify the possibility of preformed pair scenario for the pseudogap of high-Tc cuprates

Outline of talk:• Introduction : BCS-BEC crossover and pseudogap• Model and formalism : Nozieres-Schmitt-Rink theory• Pseudogap of a homogeneous Fermi gas : single-




Model

Single-channel model

Two-component Fermi gas : pseudospin

s-wave scattering length : tunable by Feshbach resonance

strong-coupling

weak-coupling unitarity

attractive interaction

Tc and are determined in a self-consistent manner by solving the coupled equations of Thouless criterion and number equation.

BCS theory implicitly assumes that all the Cooper pairs are Bose condensed in the same q=0 state. Cooper pairs with finite q are ignored. (q center of mass momentum)

NSR theory includes effects of pairs outside the condensate with q>0 involved in the pairing fluctuation diagrams.

In the BEC region (as >0), pairs become stable two-particle bound states which can occupy finite momentum states. As T increases, more and more pairs leave the condensate. BCS theory fails to describe these situations.

Nozieres-Schmitt-Rink theory

µ = εF( in the BCS theory)

Tc : Thouless criterion

BCS gap equation

pairing fluctuations

Many-body T-matrix

+=

Thouless criterionat

δΩ =

N = −∂Ω∂µ

Ω = Ω0 + δΩ Ω0

: number equation

Correction from pairing fluctuations (left out in BCS)

: free Fermion

: number of pairs with q>0

+ + +...

BCS limit

Thouless criterion (Gap equation)

Number equation

Number equation Gap equation

Tc =8γ

πe2εF e

π2kF as = TBCS

Ebind =1

ma2s

BEC limitideal Bose gas of N/2 bosons

molecular binding energy

Number equation

Thouless criterion (Gap equation)

Tc and exhibit smooth crossover from the BCS region to the BEC region

µc < 0

BEC

BCS

stable molecular bound state

in the BEC side

BCS

BEC

Tc and in NSR theory

µ(Tc)

µc

Single-particle Green’s function in T-matrix app.

Number equation Single-particle Green’s function

Self-energy couples the pairing fluctuations into the single-particle spectrum.

Self-energy

= + + + …

Density of states Spectral weight

BCS density of states and spectral weight

Destroying an atom involves destroying an excitation and at the same time creating one. This is the source of the negative energy pole of the SW.

: free Fermion

: Bogoliubov quasiparticle spectrum

(as < 0)

Density of states at Tc

BCS side BEC side

suppression of DOS near the Fermi level = Pseudgap

(as > 0)

Pseudogap is most remarkable at the unitarity limit and disappears in both the BCS and BEC limits

Spectral weight at TcBCS BECUnitarity

Double-peak structure (Levin,Strinati)

double-peak structure in SW yields the pseudogap in DOS

similar to BCS SW: pseudogap

G11(p, iωn) =u2

p

iωn − Ep+

v2p

iωn + Ep

∆2pg = −T

q,νn

Γq(iνn)

=1

(iωn − ξp)− ∆2

iωn+ξp

Σp(iωn) = T

q,νn

Γq(iνn)G0q−p(iνn − iωn)

T

q,νn

Γq(iνn)×G0−p(−iωn)

G0−p(−iωn)

Γq=0(iνn = 0) diverges at Tc (Thouless criterion)

BCS Green’s function

: hole Green’s func.

Simple picture for origin of pseudogap

pseudogap describes particle-hole coupling strength

Spectral weight at TcBCS BEC

Broadened peaks indicate short life time of quasiparticles

Unitarity

evolves as increasing interaction strength∆pg

Spectral weight at TcBCS BEC

sharp coherent upper peak

asymmetric upper and lower peaks

broad incoherent lower peak

Unitarity

atoms from dissociated molecules

hole-type excitation : many-body effect(completely absent in the BEC limit)

T* T**=

Density of states above Tc (BCS side)

Pseudogap in DOS disappears at T*, while the double peaks in SW merge into a single peak at T**.

T**

T*1.14Tc

1.03Tc

Pseudogap in DOS persists to higher temperatures than the double-peak structure.

Density of states above Tc (BCS side)

Pseudogap arises from broad suppressed single peak

T* > T**

Density of states at above Tc (BEC side)

Disappearance of double-peak structure at very high temperatures :

T*1.53Tc

T** 1.03Tc

T* T**=

T** εF

Density of states at above Tc (BEC side)

The double-peak structure persists to higher temperatures than the pseudogap.

lower broad peak is smeared out in density of states

T* < T**

Physical backgrounds of T* and T**

Double peaks merge into a single broad peak at T**.

BCS-type quasiparticles are not well-defined above T**, because of their short life time.

Pseudogap in DOS disappears at T*.Suppression of DOS around Fermi level indicates that fermionic degrees of freedom are transformed into bosonic ones.

Pairs are formed below T*

T* ≠T**　suggests that formation of pairs and BCS-type quasiparticles do not occur at the same temperature. (In the BCS theory, they both occur at Tc.)

Phase diagram of the BCS-BEC crossover

T*

T**

TBCS

Tc

Ebind

T*T**

Ebind

: pseudogap in DOS: double-peak structure in SW: molecular binding energy

BCS BEC

Phase diagram of the BCS-BEC crossover

T*

Tcsuperfluid

molecular Bose gas

pseudogap

T**normal Fermi gas

Since the pseudogap is a crossover phenomenon, the pseudogap temperature may depend on what we measure. T* : DOS T** : SW

BCS BEC

Summary : part 1• We investigated the pseudogap phenomenon of a

Fermi gas in the BCS-BEC crossover within the many-body T-matrix theory (NSR theory).

• We calculated the DOS and SW above Tc, and demonstrated that pseudogap indeed appears in these quantities in the BCS-BEC crossover region.

• In the BCS side (as<0), the pseudogap persists to higher temperatures than the double-peak structure in the spectral weight. (T* > T**)

• In the BEC side (as>0), the double-peak structure persists to higher temperatures than the pseudogap. (T* < T**)

• We have determined the pseudogap region in the BCS-BEC phase diagram.

Outline of talk:• Introduction : BCS-BEC crossover and pseudogap• Model and formalism : Nozieres-Schmitt-Rink theory• Pseudogap in a homogeneous Fermi gas : single-




Photoemission spectroscopyPhotoemission spectroscopy for electronic systems

ARPES for high-Tc cuprates

Matsui et al., PRL (2003)

Powerful technique to probe occupied single-particle states

Useful for determining quasiparticle spectrum, symmetry of order parameter etc.

Direct measurements of spectral weight

Ω = hν − φ A(p,ω) = − 1π

ImGp(ω+)

Photoemission spectroscopy for atomic gases

Applied radio-frequency pulse transfers atoms to a third empty atomic state (no final state interaction for 40K)

Momentum-resolved rf current of atoms in the third state

Photoemission spectroscopy for Fermi gases in the BCS-BEC crossover D. Jin’s group (JILA)

involved in pairing

: rf detuning

I(p,Ω) ∝ A(p, ξp − Ω)f(ξp − Ω) spectral weight

Tc and of trapped Fermi gases

ρ(ω, r)

εF = (3N)1/3ωtr

BCS

BEC

Local density approximationµ→ µ− V (r) = µ(r)

effects of trapping potential

A(p,ω, r)spatial dependence

µ

BCS

BEC

V (r) = mω2trr

2/2

Local DOS : unitarity limit

r/RF r/RF

r/RF

Pseudogap disappears from the outer region of the cloud, because low particle density suppresses pair fluctuations.

ω/εFω/εF

ω/εF

RF =

2εF /(mω2) : T-F radius

ρ(ω,r

)/[m

k F/(

2π2)]

ρ(ω,r

)/[m

k F/(

2π2)]

Local DOS : unitarity limit

r/RF r/RF

r/RF

The pseudogap temperature is defined at r =0.

T* 1.07Tc

ω/εF

ω/εF

RF =

2εF /(mω2) : T-F radius

ω/εF

ρ(ω,r

)/[m

k F/(

2π2)]

ρ(ω,r

)/[m

k F/(

2π2)]

A(p,ω, r)εF

r

T µ(r)

Local SW : unitarity limitr = 0 r = 0.4RF r = RF

p/kF

T = Tc

1.65Tc

ω/ε

F

Increasing r and T give similar effects:or

1.20Tc T**

nF (r) = 2T

p,ωn

G0p(iωn, r)eiωnδ nB(r) = T

p,ωn

Gp(iωn, r)−G0

p(iωn, r)eiωnδ

Density distribution of pairs

Phase diagram of a trapped Fermi gas

: molecular binding energyEbind

T* : pseudogap in DOS: double-peak structure in SWT**

superfluid

molecular Bose gas

normal Fermi gas

pseudogap

T*T**

Tc

Ebind

BECBCS

ρ(ω)f(ω) =

p

Iave(p,Ω→ ξp − ω)/(2πt2F )

Calculation of photoemission spectrumrf-pulse is applied to the whole gas cloud

I(p,Ω, r) = 2πt2F A(p, ξp(r)− Ω, r)f(ξp(r)− Ω): momentum-resolved rf current

Iave(p,Ω) =1V

drI(p,Ω, r)

averaged occupied SW and DOS

A(p,ω)f(ω) = Iave(p,Ω→ ξp − ω)/(2πt2F )

A(p,ω)f(ω) = I(p,Ω→ ξp − ω)/(2πt2F ) homogeneous gas

Stewart et al. Nature (2008)

observed spectrum involves contributions from all spatial regions of the cloud.

(ω+

µ)/

ε F

p2A(p, ω)f(ω)

Photoemission spectrum at Tc

unitarity BEC regimep/kF

(kF as)−1 = −1 (kF as)−1 = 0

(kF as)−1 = 1

ξp ξp

Stewart et al. Nature (2008)

BCS regime

Photoemission spectrum at Tc

(kF as)−1 = 1

Averaged spectra have both the features of the pseudogapped spectrum in the center and the single particle spectrum at the edge.

BEC regime

(kF as)−1 = 1

Spatially averaged density of states

Theory curves agree well with the photoemission data of the Jin’s group.

(kF as)−1 0.15

Observation of pseudogap

Gaebler et al. Nature Phys.(2010)

back-bending peak = pseudogap!

Photoemission spectra above Tc

T = Tc T/Tc = 1.28 T/Tc = 1.84

(kF as)−1 = 0.2

p2A(p,ω)f(ω)

(ω+

µ)/

ε F

(kF as)−1 0.15


T = Tc T/Tc = 1.28 T/Tc = 1.84

(kF as)−1 = 0.2

p2A(p,ω)f(ω)

(ω+

µ)/

ε F

The back-bending peak disappears at T** reflecting the disappearance of the pseudogap in the trap center.The lower broad peak remaining at high temperatures is related to the universal behavior of Fermi gas with contact interaction (nothing to do with the pseudogap physics). Tan (2008), Randeria (2010)

Summary : part 2

• We calculated the photoemission spectrum of a trapped Fermi gas above Tc by including pairing fluctuations within the T-matrix approximation, as well as effects of a trap potential within the LDA.

• We showed that spatially inhomogeneous pair fluctuations produced by the trap potential is crucial to understand the observed photoemission spectrum.

• The quantitative agreement with the experimental data strongly supports that our theoretical approach correctly describes the pseudogap phenomena in a cold Fermi gas.

(ω+

µ)/

ε F


p/kF

p2A(p, ω)f(ω)

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