march 12-14 (2015) 「中性子星核物質」研究会：京...

Introduction: cold Fermi gas and BCS-BEC crossover

Department of Physics, Keio University, Japan

March 12-14 (2015) 「中性子星核物質」研究会：京都大学（基研）

Specific heat in the BCS-BEC crossover regime of an ultracold Femi gas

Yoji Ohashi （公募班）

Collaborators:

P. van Wyk (M2)

Summary

Specific heat

Determination of “pairing-fluctuation” regime

Formulation: Gaussian fluctuation (NSR) theory

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キックオフ(理研）2012/10/26-27

第2回ウインタースクール＆研究会(理研）2013/12/25-28

公募キックオフ(東北大）2013/9/12-13

理論班研究会(西浦）2013/3/3

B03(?) 国際workshop（ポハン）2014/5/12-13

第3回研究会（熱川）2014/9/23-25

NO PHOTO (?)

第3回ウインタースクール（阪大）2014/12/22-23 （京大）2015/3/12-14

掲載予定（？）

Introduction: Ultracold Fermi gas physics

Homepage of Gonokami Lab,

University of Tokyo

6Li Fermi gas cloud

trap potential

Fermi atoms (6Li, 40K) are trapped in a magnetic/optical

potential, and are cooled down to <O(mK), where various

quantum phenomena can be observed, such as superfluidity.

Introduction: superfluid 40K Fermi gas (2004)

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

4 2/ ~ 0.08 0.2 10 10 (metal)c FT T

9 / 2, 7 / 2 9 / 2, 9 / 2

0.35FT Km510N

superconductivity Fermi gas superfluid

9 / 2, 7 / 2

9 / 2, 9 / 2

Cooper pair

Feshbach resonance: pairing mechanism of superfluid Fermi gas

5

Fermi atom

tunable by magnetic field

2 1

2effV g

molecule

tunable pairing interaction

(Timmermans (2001), Holland (2001))

BCS-BEC crossover

Bose-Einstein condensation (BEC) of an ideal Bose gas

( )

1

1BN

e m

q

q

/ cT T

/B cTm

BEC transition

Thermal de Broglie length: 2

TmT

h

quantum (statistical)

size of a particle

cT T cT T

Tl

~T l

The BEC occurs, when the quantum size of a particle reaches the interparticle distance.

boson

Essence of BCS-BEC crossover

Essence of BCS-BEC crossover phenomenon

weak U strong U

T

BCS BEC

Tc

cTstrong U

weak U

Pairing interaction between Fermi atoms0

Phase diagram of ultracold Fermi gas

weak U strong U

T

BCS BEC

Superfluid phase

Fermi atoms Bose molecules

Tc

pairing interaction between Fermi atoms0

cTstrong U

weak U


T

BCS BEC

Superfluid phase


Tc

00

1( )F sk a

1( ) 1F sk a 1( ) 1F sk a

Neutron star

計画研究B03


T

BCS BEC

Superfluid phase


Tc

00

1( )F sk a

1( ) 1F sk a 1( ) 1F sk a

Neutron star

計画研究B03

My work


T

BCS BEC

Superfluid phase


Tc

00

1( )F sk a

1( ) 1F sk a 1( ) 1F sk a

pairing fluctuations

BCS-like gap in the density of states (pseudo-gap)

Suppression of spin excitations (spin-gap)

“preformed Cooper-pair” formation

Determination of “crossover” temperature T*

1( ) 0F sk a

pseudogap!

spin gap

cT

density of states spin susceptibility

JILA, Nature 454, 744 (2008).

~ cT T40K

sin

gle

-pa

rtic

le e

xci

tati

on

en

ergy

experiment theory

Tsuchiya, Watanabe, Ohashi,

PRA 82 (2010) 033629

1( ) 0F sk a

6Li

Kashimura, Watanabe, Ohashi, PRA 86 (2012) 043622

Experiemtal data: Sanner et.al, PRL 106 (2011) 010402

1(( ) 0.35)c F sT T k a

theory

*T

Determination of “crossover” temperature T*

/ FT T

1( )F sk a

BCS BEC

cT

*T


Superfluid phase

Fermi atoms start to form preformed

Cooper pairs (bosons) .

Today’s talk

We study specific heat Cv in the BCS-BEC crossover regime of an ultracold

Fermi gas. We show that Cv is a useful quantity to determine T** which

physically distinguish between the Bose gas regime and fluctuation regime.

𝑇c𝑇/𝑇c

𝐶𝑉/𝑁

0.0 2.01.0𝑇/𝑇c

𝐶𝑉/𝑁

weak-coupling BCS ideal Bose gas

M. Ku et al. Science 335, 563 (2012)

6Li

unitarity limit

free Fermi gas

22~VC E E Assessment of strong-coupling theory to evaluate EoS

Formulation: Gaussian fluctuation (NSR) theory

：two atomic hyperfine states = pseudospin ↑,↓

U : effective pairing interaction associated with the F.R.

U

uniform gas is assumed.

Feshbach resonance We treat U as a tunable parameter.

(Minimal theory to describe the whole BCS-BEC crossover region)

† † †

/ 2 / 2 ' / 2 ', , '

( )p p p p q p q p q pp p p q

H c c U c c c c

m

11 tanh

2( ) 2U

T

m

m

p

p p

Formulation: Tc in the BCS-BEC crossover region

thermodynamic potential:0 NSR

U

pairing fluctuations (Gaussian fluctuation level)

Tc: Thouless criterion

( , ) q U

.....

pole at q = = 0

free

,

log 1 ( , )n

n

i

n

q

T e UN N

m

m

q

pair correlation function

Formulation: specific heat in the crossover region (T>Tc)

Specific heat:0 NSR

U

pairing fluctuations

Legendre transformation: E

,

v

V N

EC

T

𝐸 = 𝐸0 + 𝑇

𝒒,𝑖𝜈𝑛

𝑈

1 − 𝑈Π 𝒒, 𝑖𝜈𝑛𝜇𝜕Π 𝒒, 𝑖𝜈𝑛𝜕𝜇

+ 𝑇𝜕Π 𝒒, 𝑖𝜈𝑛𝜕𝑇

pair correlation function

Comparison with recent experiment on a 6Li Fermi gas

6Li (MIT 2012)

This work

free Fermi gas

The experimental result is affected by finite temperature resolution,

which may somehow smear the singularity around Tc.

SummarySpecific heat and effects of pairing fluctuations

We have discussed specific heat in the BCS-BEC crossover regime of an ultra-cold

Fermi gas. Within the framework of the NSR-theory, we calculated this

thermodynamic quantity as a function of temperature above Tc. From the temperature

dependence, we determined the Bose gas regime, as well as the region where the

system is dominated by fluctuating preformed pairs, in the phase diagram of an ultra-

cold Fermi gas.

march 12-14 (2015) 「中性子星核物質」研究会：京...

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