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
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
Evolution of Tc in BCS-BEC crossover
Tc TBCS
Size of pair shrinks as increasing the interaction
λdB < d
λdB dnecessary to lower T to achieve
Evolution of Tc in BCS-BEC crossover
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-
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
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-
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
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
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
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
Photoemission spectra above Tc
p/kF
p2A(p, ω)f(ω)