強結合フェルミ気体におけるメゾスコピックスピン輸送
YS, H. Tajima, & S. Uchino, Phys. Rev. Research 2, 023152 (2020)

関野 裕太 (理研・仁科センター)

講演番号02(By Tajima-san) https://www.youtube.com/watch?v=vIhdQN7ttWE

Introduction
1. Quantum Transport

2. Ultracold Fermi gas

3. Spin vs mass currents

Tunable interaction

In a bulk

Fermi atoms (6Li, 40K,…) with
40K
Finite-temperature
unitary
Fermi
gas
two hyperﬁne state‣ Spin-1/2 particles interact in 3D with range r0 ⌧ kF 1 (pure s-wave scattering)

Quantum Hall eﬀect

Superconductivity

Polarization diﬀerence→ Spin current

Particle-number diﬀerence→ Mass current<latexit sha1_base64="mSypYCr3Vm3DL1GkzTtkQgyJypg=">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</latexit>
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Dilute &Only
atomicgas k , s-wave scattering length a, thermal wavelengthscales:F‣ coldlengthT<latexit sha1_base64="9G45aL5P66GngmbcOEto4nmlgyA=">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</latexit>
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StrongWeakn = 1013 to 1015 cm-1, T = 1μK to 10nKof the particles ExperimentIntroduction‣ Universal physics independent of the short-range structure
Regal & Jin PRL (2003)

B/w bulks

Ultracold atoms<latexit sha1_base64="7kxcO8+xeKdy+PWeJMKljToS5/A=">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</latexit>
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Unitary limit
100
0.3

−1

12

b

a

2

A crossover from BCS for (kF a) ∼ −∞−1
to BEC for (kF a) ∼ +∞Fermi liquid behavior above Tc

1/(kF a)Josephson junction

Cond. matt.

Quantum point contact

Pseudogap regime above Tc

k = 0.37

10
0.52

10−1

1

P(µ,T)/P0(µ,0)‣

Two-component (spin up/down) fermionic
atoms interactingwith aobtained
short-range as a function of
BCS-BEC
crossoverinteraction V0δ (r − r') characterized by a scattering length a .

T/TF,0E (EF)

0.66

Of particular interest is the unitary
Fermi gas (UFG) describing the limitof strongest interaction a → ∞or (kF a)−1 = 0

0.2
0

0.96

−1

1.1

0.1

−3

6
4

0.0

−2

Pair transport:

Spin singlet

2

1.25

10

Excitations of both atoms & pairs

No contribution from

0.81
−2

10

8

0.2

0.4

0.6

0.8

2

(kBT/µ)1.4

Multiple Andreev reﬂection

1.55
10−4

1
Sensitive
to single-particle
0
0.2
0.4
0.6
0.8
1
Ku et al, MIT (Science 2012):
Photoemission spectroscopy at unitarity
excitations !!
h/εFIG. S2. Gaussian fits to PES dataat (k a) = 0.1. a, The white circles indicate the centers
−3
0

0.5

0

Randeria and Taylor (2014)
Randeria and Taylor (2014)

1

1.5

2

−2

k (kF)−1
0
E (E )

Husmann, Uchino et al., Science (2015)

F

Equation of state is well describedfrom weighted
fits to EDCs
at fixedk. The white line shows the free-particle dispersion,
Sagi
et al,gaussian
Boulder(PRL2015):
by Fermi liquid theory(Strongly correlated)Spinbalanced
Highlypolarized
E
=
k
.
b,
Individual
EDCs
(blue
points)areshown,
along
with
the
fittedgaussians(redlines).
Backbendingabove
T⇤c.in photoemission
Pseudogap
T<
T<T• A challenging non-perturbative many-bodyproblem regime: pairing without condensationcHere, each EDC is individually normalized to have the same area, as in Ref. [12]. A solid black
spectroscopy
There
is
a
long-runningdebateabout
its
existence
unitaryline marksin
E =theE . Redstars show the limit.
center of each gaussian, and the dashed red line is a guide
LENS: Valtolina et al., Science (2015)
Nascimbene et al, Paris (PRL 2011):!15
See our review: S. Jensen, CG, Y. Alhassid arXiv:1807.03913 to the eye.
15
Spin response compatible
with Fermi
liquid behavior
Unitary limit
Many interesting properties: universality, scale invariance,…

FF,0Sagi, et al. (PRL 2015)‣

ETH: Brantut et al., Science (2012)

1

Josephson ocsillation

2

Valtolina et al., Science (2015)<latexit sha1_base64="RRGgfPkdVxuRlRRunB1XOP0Dz/U=">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</latexit>
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F

PurposePseudogap regime

Finite-temperature unitary Fermi gas
Spin current @ a →1
Understanding of transport
properties
particles interact in 3D with range r0 ⌧ kF (pure s-wave scattering)‣ Spin-1/2

Intuitive interpretation2

0.3<latexit sha1_base64="mSypYCr3Vm3DL1GkzTtkQgyJypg=">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</latexit>
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1.5

T/TF,0<latexit sha1_base64="9G45aL5P66GngmbcOEto4nmlgyA=">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</latexit>
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ideal gaskF , s-wave scattering length a, thermal wavelength
Ispin ( = 0.33ε
10F,0) T
spinεF,0)for strongly interacting‣ Fermi
gases
Only lengthscales:YS, Tajima, & Uchino, PR
Research(2020)<latexit sha1_base64="T6ovWNSlJgZRlalj1jTR0nuXlLQ=">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</latexit>
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1

0.2

Transport

Excitation properties in bulks‣

Two-component (spin up/down) fermionic
atoms interactingwith aobtained
short-range as a function of
BCS-BEC
crossoverinteraction V0δ (r − r') characterized by a scattering length a .

Ispin/(

Universal physics independent of the short-range structure of the particles Experiment 0.1
Introduction
0.5
Pseudogap
regime above Tc

Fermi liquid behavior above Tc

1/(kF a)00<latexit sha1_base64="7kxcO8+xeKdy+PWeJMKljToS5/A=">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</latexit>
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0.2

0.4

0.6

0

0.8

h/εF,0

0

a

2

0.4

12

b

0.6

0.8

h/εF,0

1

0

10

0.2

1

k = 0.37

10

0.66
0.81

0
10−2

0.96

@h→0

−1

1
−2

10−3

0.5

0
Gspin/Gspin

−3
0

1.4
(kF,0a) = -0.5
1.55

1 0.6 1.5

10−4

2

−2

k (kF)

Randeria and Taylor (2014)
Randeria and Taylor (2014)

Unitary limit
Many interesting properties: universality, scale invariance,…

1.5

1.25

-1

0.8

8
6

Pseudogap
formation

4

Dos in the unitary limit

1.1ρα(ω)/ρ0

E (EF)

Of particular interest is the unitary
Fermi gas (UFG) describing the limitof strongest interaction a → ∞or (kF a)−1 = 0

P(µ,T)/P0(µ,0)

1
Spin
conductanceA crossover from BCS for (kF a)−1 ∼ −∞−1
to BEC for (kF a) ∼ +∞

0.52

10−1

−1
0
E (E )

1

1

F

2

T0.0 = Tc 0.2

0.4

0.6

0.8

2

(kBT/µ)= 0.25εF,0 (α = +)=0

Ku et al, MIT (Science 2012):(kF,0a) = 0
Photoemission spectroscopy at unitarity
0.5
FIG. S2. Gaussian fits to PES data at (kF a) 1 = 0.1. a, The white circles indicate theEquation
centersof state is well described
Sagi, et al.0.2(PRL 2015)(kF,0whitea)-1 line= 0.5
from weighted
fits to EDCs
at fixedk. Theshows the free-particle dispersion,
Sagi
et al,gaussian
Boulder(PRL2015):
by Fermi liquid theory-1

0.4

Suppression of Gspin= 0.25εF,0 (α = )E = k . b, Individual above
EDCs 0(bluepoints)shown,0.6
along withgaussians (red lines).
Backbending
photoemission
0 T
0.4
0.8 the fitted
1⇤0.2inare0-1.5 -1 -0.5
0
0.5
1.5
1
Pseudogap
regime:pairing
without
condensation
.
c
T
<
T
<
T
• A challenging non-perturbative many-body problemcHere, each EDC is individually normalized to T/Thave thesame area, as in Ref. [12]. A solid black
F,0
spectroscopy
ω/εF,0
There is a long-running debate about its existenceintheunitary
limit.
line marks E = E . Red stars show the center of each gaussian, and the dashed red line is a guide
Nascimbene et al, Paris (PRL 2011):!15
See our review: S. Jensen, CG, Y. Alhassid arXiv:1807.03913 to the eye.
15
Spin response compatible
with Fermi
liquid behavior YS, Tajima, & Uchino, PR Research(2020)

Here, spin transport in normal
phase is focused on‣

2<latexit sha1_base64="RRGgfPkdVxuRlRRunB1XOP0Dz/U=">AAACjXicbZHPTttAEMY3hhZIaSHlyGXVqBLiENlVKzhEFRIHeqSSE5CwidbrCWzZP2Z3nDay8hxc6WPxNl2nVksSRhrpm++3I83OZIUUDsPwqRWsrb96vbG51X6z/fbdzm7n/dCZ0nIYcCONvcyYAyk0DFCghMvCAlOZhIvs7rTmFxOwThgd47SAVLEbLcaCM/RWGo847dO4zuvD0W437IXzoKsiakSXNHE+6rR+JLnhpQKNXDLnrqKwwLRiFgWXMGsnpYOC8Tt2A1deaqbApdV86hn96J2cjo31qZHO3ecdFVPOTVXmXyqGt26Z1eaLLFPeppmROa3h4hQ4Pk4roYsSQfO/Q4xLSdHQej00FxY4yqkXjFvh/0H5LbOMo19iO9HwkxulmM6rZMLsrEoQfqFV1dAXs0XOzeQ/P/XFEo+ftccr3fiP6gpr6q8TLd9iVQw/9aKwF33/3D352txpk+yTD+SAROSInJBv5JwMCCf35IE8kt/BTvAl6AfN26DV9OyRhQjO/gCo+stv</latexit>
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F

Polaron regime

1. Tunneling Hamiltonian model YS, Tajima, & Uchino, PR Research(2020)

6μ−Minority atoms behave as Fermi polaronsh(↑ , L) (↓ , L)
α=+

α=−

Majority: α = +Reservoirs:

Majority atoms form a Fermi sea
10
Theory: Chevy PRA (2006); Combescot et al., PRL (2007);…
Exp: Schirotzek et al., PRL (2009);(↑ , R)
α=−Minority:

4

0.6

2

0.4

0

0.2

1.17kF,+
1

Koschorreck et al., Nature (2012); Kohstall et al., Nature (2012)

p/kF,+

2

3

0

1.5
1
0.5
0
-2

T = 0.21TF,0h = εF,0
-1

0

1

2

(ω+μ－)/εF,+

@ a→

3

4

@ a→

PHYSICAL REVIEW A 83, 063620 (2011)

RICHARD SCHMIDT AND TILMAN ENSS

4. Spin current

Schmidt & Enss, PRA (2011)2★
Polaron-molecule
transition

Molecule0
Ispin/(Gspin
F,0)polaronTunneling term:

0.8

-2
0

Repulsive

Contact interaction1ρ−(ω)/ρ0,+

h

μ

xR,0

2. Minority spectrum & DoS
2

μσ,j
μ+
xL,0

1. Highly polarized gas(ω μ−)/εF,+

ModelBroad repulsive branch &Broadened attractive polaron

T = 0.25TF,0

1.5(kF,0a)-1 = -0.5

DoS enhancement

1

in

0
0.5

(b)
(a)

Attractive00polaron

2. Kubo formula

0.5
0.2

0.4

0.6

h/ F,00.8

1Increase of the spin current

Summary
Density of states (DoS):

Two-terminal spin transport for strongly interacting Fermi gas
(d)

(c)

FIG. 2. (Color online) Spectral functions at zero momentum in dependence on 1/(kF a). (a), (b) Polaron spectral function A↓ (ω,p = 0).
(c), (d) Molecule spectral functions Aφ (ω,p = 0). In order to make the δ-function peak for the ground state visible we introduced an artificial
width of the quasiparticles of 0.007ϵF .

Fermi distribution func.:

flow as additional flowing couplings, or implicitly by using the
Katanin scheme [40] or rebosonization [41].
Until recently [22] most experiments with ultracold Fermi
gases have focused on the lower, attractive branch on the BEC
side (kF a)− 1 > 0. There exists, however, also the repulsive
polaron branch (solid green line) which corresponds to a higher
excited state of the ↓-atom interacting repulsively with the↑-Fermi sea. Our results for the energy of the repulsive branchagree with the weak-coupling results [18] for (kF a)− 1 ∼> 1.
In the strong-coupling regime our energies lie between the
result from the non-self-consistent T-matrix approach [21]and the MC results for square well potentials [24]. In thepolaron spectral function [cf. Fig. 2(b)], one can clearly discern
the attractive polaron branch as a very sharp peak at low

1. Small spin bias: Suppression of Ispin due to the pseudogap
063620-6

3. Extended T-matrix approx.(ETMA):[Kashimura, Watanabe, & Ohashi PRA (2012)]
−α

α

Σα

+

α =

=
−

Spin current measurement becomes a sensitive
probe to pseudogap & polarons

Future perspective:+Γ−Double line:Γ

2. Large spin bias: Enhancement of Ispin due to the polarons+ΓSingle line:
Dot:

Superﬂuid regime, low-energy polaron current, current noise