大阪市大NITEP
レーザー量子・原子核合同MONTHLY MEETING：第2回

冷却フェルミ原子系で実現される少数系と多体系の概観

話題提供者：堀越 宗一
2019/07/16

Background : Understanding fundamental physics
underlying various quantum systems
SizeSingle𝐪𝐪

𝚲𝐊𝚵

Few-body𝐪𝐧Quark

Hadron𝐪𝐪𝐪

Many-body𝐪𝐪𝐪ee↑↑↑↓↓↓↓↓↑↑↓↑↑𝐪↑↓↑↓𝐪𝐪

Atom↑↓
↑↓

↑↓
↓
↑
↑↑↓
↓
↓↓
↓
↑
↑↑
↓
↑e𝐪𝐪

Quark matteree

Moleculee e𝐪eee↓↑↓

Diquark, Pentaquark a-particle, Hoyle state𝐪↑↓↓↑↑↑↓↓↓↓↓↑↑↑↑↓𝐩↑↓↑↓𝐪e↑↓↓↑↑↑↓↓↓↓↓↑↑↑↓↑e e↑↓↓↑↑↑↓↓↓↓↓↑↑↓↑↑↑↓↑↓↑↑↓ ↓↓↓↑↓↑↓↑↑e e ee↑↓↓↑↑↑↓↓↓↑↑e ↓↑↓↑↓ee e↑↓↓e↑↑↑↓↓↑↓↑↓↑↓↓↑↓↓↑↓↓↑↓↓↑
↓
↑↑↑
↓
↑
↑
ee

Atomic nucleus
Neutron star

Solid, Liquid, Gas

Background : Understanding fundamental physics
underlying various quantum systems
Cluster science : Evolution from Few-body to Many-body
Van der Waals force

Covalent bond

Mass spectrum of Xe clusters

Mass spectrum of C clusters

Metallic bond

[ J. Chern. Phys. 91, 5940 (1989) ][ W.D.Knight, Phys.Rev.Lett. 52, 2141 (1984) ]Van der Waals force
Bound state of cold atoms[ J. Chem. Phys. 81, 3322 (1984) ]Nuclear force
Binding energy of nucleon

Hydrogen bond
Water clusters[ J. Phys. Chem. A, 114, 11725 (2010) ]Magic number[ Particles and Nuclei ]Clusters in the phase diagram
Helium3 atom (fermion)

HadronColor superconductor

Solid𝐪𝒈𝒈𝒈𝐪𝐪 𝐪𝐪𝐪 𝐪𝐪

LiquidPressure [a.u.]Superfluid

CPSuperfluid

Neutron stars

Pressure [a.u.]Supercritical fluid𝚲Liquid𝐩CP

Gas

Temperature [a.u.]

𝐪𝒈𝐩𝐝

Temperature [a.u.]
No solid phase?

𝐪QGP𝐭

Gas
atomic nucleus

𝒈𝐪𝐧𝜶𝜶 𝜶𝐪𝐪 𝐪𝜶
↑↓↑↓
↑↓↓↑↓
↓↑↓↑
↓↑↑

𝒈𝐪 𝐪𝐪 𝐪𝐪𝐪𝐪𝐪𝐧

Plasma𝐪 𝐪𝐪 𝐪

𝚵𝐊

𝒈𝐪 𝐪𝐪𝐪Background : Understanding fundamental physics
underlying various quantum systems
Bose-Einstein condensation (BEC)
Superconductor

Liquid Helium

Ultracold atomic gasbosonsfermions

Neutron starPhase diagram
Quark system

Electron-Hole system

Coexistence[ Carlos A. R. Sá de Melo; Physics Today 2008, 61, 45-51]
M. Omachi, From Neutron Star winter school 2016

Gordon Baym, Nuclear Physics A 956 (2016) 1–10

Phase diagram
Cold atom system
Spin-1/2 Fermi system

Impurities in a condensate spin-1/2
Fermi system

𝑻/𝑻𝑭
Fermi gas
Thermal molecular
Molecular BEC
BCS superfluid𝟏
𝒌𝑭 𝒂

BEC limit
Unitarity limit
BCS limit(strong coupling)(weak coupling)

BCS-BEC crossover

Yusuke Nishida, PRL 114, 115302 (2015)閾値近傍の普遍的物理エネルギー

閾値

束縛状態

粒子間距離

Background : Low energy scattering and loosely-boundstate around the zero energy
Shape resonanceFeshbach resonanceTwo channel model

Ikeda threshold rule

𝑬𝟎 ~𝑬𝒃

𝑬𝟎 ~𝑬′𝒃𝑬𝟎 ~𝑬∗• n-p (bound)• n-n (unbound)• cold atoms• Λ(1405)• α-cluster

𝑬𝑬

Close channel

Unbound𝒂<𝟎𝑬𝟎𝒅𝑬𝒃

𝑬𝑬𝟎 ′𝑬𝒃𝒅𝑬∗
𝑬𝟎|𝒆 >

𝒅Open channel

Bound𝒂>𝟎|𝒈 >
Resonance with own boundstateResonance with a boundstate in the other scatteringchannel

Resonance with an excitedstate of the compositeparticle

Background : Low energy scattering and loosely-boundstate around thezerop energy
nQuarkNucleonp

Excitation
𝑬𝑬n

Excitation
𝑬𝑬𝑬𝟎AtomB

M 𝒓

𝑬𝟎

Excitation

𝑬𝑬aa 𝒓ap
pnnpn
npnp pn

𝑬𝑬𝟎𝒓

Excitation

Excitation

Excitation

𝑬𝑬𝑬

𝑬𝟎𝒓

𝑬𝟎𝒓

𝑬𝟎𝒓

p
pnpnpnnnpn

Hadron(Barion)

Atomic nucleus

Molecule

Histrical success examples• In 1953, a nuclear theorist, H. Feshbach, studied control of scattering length
In 1998, the Feshbach resonance was demonstrated in cold atom experiment• In 1970, a nuclear theorist, V. Efimov, predicted 3-body bound state in
the condition of 𝑎 < 0In 2006, the Efimov trimer was confirmed in cold atom experiment• In 1999, a nuclear theorist, G. F. Bertsch, posed the problem“How does a system of Fermi particles with infinite s-wave scattering
length but vanishing interaction range behave?”
Since 2004, the unitary regime has been one of the hot topics
in cold atom experiment

Background : Quantum systems at ultralow temperature
Various particle systems

Electron, Hole𝐞𝐡Quark𝐪𝐪Hadron𝐧Atom

Similar quantum systems

𝝀𝑻 𝒅, ≫𝟏𝒓𝟎 𝒓𝟎

Details of particles
are suppressed𝐩boson fermion

𝝀𝑻𝒓𝟎𝒅𝒂𝒂≫𝟏𝒓𝟎

Unitary regime

Cold atoms
Dilute quantum gases

Momentum distribution

6Li (fermion)

Fermi superfluid

Solid (300K)

Heating
7Li (boson)

Hot gas (600K)
Cold gas (1mK)

Laser cooling Evaporative coolingBose-Einstein condensation

Quantum gas (100nK)

6Li atom
Internal degree of freedom of the atom
Lifetime of the states ≫ Experimental time (~30s)|𝟔 = |𝒎𝑺 = +𝟏/𝟐, 𝒎𝑰 = +𝟏
|𝟓 = |𝒎𝑺 = +𝟏/𝟐, 𝒎𝑰 = 𝟎|𝟒 = |𝒎𝑺 = +𝟏/𝟐, 𝒎𝑰 = −𝟏

Orbital：𝑳 = 𝟎Electro-spin：𝑺 = 𝟏/𝟐
F=3/22S

Nuclear-spin：𝑰 = 𝟏6Li atom

F=1/2|𝟑 = |𝒎𝑺 = −𝟏/𝟐, 𝒎𝑰 = −𝟏|𝟐 = |𝒎𝑺 = −𝟏/𝟐, 𝒎𝑰 = 𝟎|𝟏 = |𝒎𝑺 = −𝟏/𝟐, 𝒎𝑰 = +𝟏
Magnetic field

Length scales of our cold atomic system• Van der Waals length : 𝑹𝐯𝐝𝐰 =𝟏 𝟐𝝁𝑪𝟔𝟐ℏ𝟐

𝟏/𝟒𝑅vdw= 𝟏. 𝟕𝐧𝐦
𝛬T• Thermal length : 𝜦𝐓 =−𝟏/𝟑• Mean distance : 𝒏ℏ𝟐𝝅𝒎𝒌𝐁 𝑻~𝟏𝟎𝟎𝐧𝐦@𝟏𝛍𝐊~𝒌−𝟏
𝐅 ~𝟏𝟎𝟎𝐧𝐦• Potential size : 𝑳~𝟏𝟎𝝁𝐦

𝑘F−1

𝐿𝑅vdw

Length scales
𝛬T

Locally homogeneousHierarchy : 𝑹𝐯𝐝𝐰 ≪ 𝜦𝑻 , 𝒌−𝟏
𝑭 ≪𝑳

𝑘𝐹−1Low energy
Dilute𝐿

Short-range potential

interaction is basically s-wave

Hydrodynamicity : 𝟎 <Ballistic

𝑫𝐚𝐱𝐢 𝑫𝐫𝐚𝐝,𝒍𝐦𝐟𝐩 𝒍𝐦𝐟𝐩

mean-free-path : 𝒍𝐦𝐟𝐩 ~𝒗𝚪𝐜𝐨𝐥 (𝒏, 𝒂)< ∞Hydrodynamic𝒏↑ −𝒏↓Control knobs : 𝑻, 𝝁, 𝜹𝒏 =, 𝒂, 𝒓𝒆 , 𝑫, 𝑩, 𝑭, 𝜹𝒎, …
𝒏↑ +𝒏↓フェッシュバッハ共鳴
2S
F=1/26Li原子(フェルミオン)― a12― a23― a134000散乱長 [𝒂𝟎 ]

|𝟑 >
|𝟐 >
|𝟏 >

2000

0-2000-4000
0

200

400

600

8001000

磁場 [𝐆𝐚𝐮𝐬𝐬][ G. Zürn, Phys. Rev. Lett. 110, 135301 (2013) ][P. Naidon,, Phys. Rev. Lett. 105, 023201 (2010) ]位相シフトと散乱長（簡単に復習）

２粒子の相対運動のS.E. ：1𝑚

𝑳22𝑝𝑟 + 𝑟 2+ 𝑉 𝑟 𝛹 𝑟, 𝜃, 𝜙 = 𝐸𝛹 𝑟, 𝜃, 𝜙

角度方向と半径方向に変数分離 ：𝛹 𝑟, 𝜃, 𝜙 = 𝜓𝑙 𝑟 𝑌𝑙𝑚 𝜃, 𝜙

半径方向のS.E. ：ℏ2𝑚𝑑22 𝑑𝑙 𝑙+1− 𝑑𝑟 2 − 𝑟 𝑑𝑟 + 𝑟 2+ 𝑉 𝑟 𝜓𝑙 𝑟 = 𝐸𝜓𝑙 𝑟
𝐶1

ポテンシャルサイズより遠方での一般解：𝜓𝑙 𝑟 ≅ 𝑘𝑟 sin 𝑘𝑟 + 𝜂𝑙 − 2 𝑙𝜋 , for 𝑟 ≫ 𝑟0𝐶

ポテンシャルサイズより遠方でのs波散乱 ： 𝜓 𝑟 ≅ 𝑘𝑟 sin 𝑘𝑟 + 𝜂0 , for 𝑟 ≫ 𝑟0
11s波の散乱長、有効長と位相シフトとの関係 ： cot 𝜂0 = − 𝑎𝑘 + 2 𝑟e 𝑘s波散乱長の変化 ： Shape resonance
井戸型ポテンシャル
３次元井戸型ポテンシャルの幅(𝑅)と深さ(𝑉0 )に対する散乱長の変化
tan 𝑉0 𝑅𝑎=1−𝑅𝑉0 𝑅
𝝍 𝒓𝒖 𝒓 = 𝒓𝝍 𝒓

黒点線：ポテンシャルが無い場合
赤実線：ポテンシャルがある場合

𝒓𝒓

𝑽𝟎𝑹

束縛状態

位相シフト：𝜹𝒔s波散乱長の変化 ： Shape resonance
入射エネルギーと束縛状態の共鳴散乱𝑎

井戸の深さ𝑎>0𝑎=0
𝑅𝑎<00𝑎 → −∞新しい束縛状態

そうは言っても粒子間ポテンシャルの形は容易には変えられない

𝑉0 𝑅調和ポテンシャル中の二粒子の相互作用に依存した固有値
BCS領域

ユニタリー極限

BEC領域
分子にならなければ斥力相互作用
これがa>0→斥力と誤解する原因
ポテンシャルは常に引力

固有値

理想気体
非束縛状態
弱い引力（多体系ならば多体効果で対形成する）

束縛(分子)状態
強い引力

1/a[Yvan Castin, arXiv:1103.2851v2 (2011)]

Two-body bound state determined by S-wave scattering length𝒎𝟐𝒎𝟏

Two particles can realize loosely bound state in vacuum for 𝑎 > 0
Wave function of the bound state： 𝜓𝑏 𝑟 = 𝒓1 − 𝒓2∞

Radius：𝑟 =0=4𝜋𝑟 2 𝑟 𝜓𝑏 𝑟

ℏ2Binding energy： 𝐸𝑏 𝑎 = − 2𝑚 𝑎2𝑟
𝑚1 𝑚2
Reduced mass： 𝑚𝑟 = 𝑚 + 𝑚
121𝑟
exp − 𝑎2𝜋𝑎2 𝑑𝑟 =𝑟𝑎2

Measurment of binding energy
Rf spectroscopy

Transition rate to |3>[ C. Chin, et al., Science 305, 1128 (2004) ]Temperature|𝟑 >Rf pulse

bound-free transition𝑬𝑩free-free transition|𝟐 >Bound state|𝟏 >Internal state of 6Li
Radio frequency [kHz]重ね合わせ状態の波動関数の測定
フェッシュバッハ分子(6Li原子) :ほとんどこの状態→普遍的𝜓 𝑟 = 𝑍 ⋅ 𝜓𝑚 𝑟 |𝜓𝑣=38 (𝑆) > + 1 − 𝑍 ⋅分子度：𝑍 ≃ 1 +2𝑔𝑎𝑚
𝑚3/24𝜋ℏ3−11𝑟
exp − 𝑎2𝜋𝑎𝑟|𝜙𝑎 (𝑇) >粒子の詳細に依存(非普遍性)

𝐸𝑏𝑔𝑎𝑚 : atom-molecule coupling constant𝑽∗𝒔 (𝒓)|𝒔∗ >[ R.A. Duine, Physics Reports 396, 115 (2004) ]𝑬𝜸𝑽𝒔 (𝒓)
|𝒔 >

𝚫𝑬 𝑩𝟎
[ G. B. Partridge, Phys. Rev. Lett. 95, 020404 (2005) ]

𝑬𝒃𝐬
|𝒕 >

𝑽𝒕 (𝒓)

𝒓𝟎𝒓

フェッシュバッハ分子(クラスター)の安定度分子のロス(非弾性散乱)レート：[ D. S. Petrov, PRL 93, 090404 (2004) ]e𝑑𝑛𝑑= −Γ𝑑𝑑 𝑛𝑑2 , Γ𝑑𝑑 ∝ 𝑎−2.55𝑑𝑡𝑑𝑛𝑑
分子－フェルミ粒子：= −Γ𝑓𝑑 𝑛𝑑 𝑛𝑓 , Γ𝑓𝑑 ∝ 𝑎−3.33𝑑𝑡••

3𝑙vdwの領域にどのくらい3粒子の波動関数の振幅があるかee↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑ee𝑎

分子－分子：

Γ𝑑𝑑ee

2007年12月の我々のデータe↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑e

パウリ効果が強く働き、3体衝突を抑制し、クラスターを安定化させているe720Gee

↑↓
↑ ↓↑
↑↓ ↓↓
↓↑↑ ↓↓↑ ↑↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑e

e

e𝑙vdw
740Ge
↑↓
↑ ↓↑
↑↓ ↓↓
↓↑↑ ↓
↓↑ ↑e

760G

𝐶𝑑𝑑

∝ 𝑎−2.6±0.4780Geクラスター間相互作用(弾性散乱)
パウリ効果が強く働くBe↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑Beeeeee↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑eee↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑eee

↑↓
↑ ↓↑
↑↓ ↓↓
↓↑↑ ↓↓↑ ↑ee𝑎𝑑𝑓 ≈ 1.18𝑎
neutron-deuteron scatteringと類似

𝑎eeee

↑↓
↑ ↓↑
↑↓ ↓↓
↓↑↑ ↓↓↑ ↑↑↓↑ ↓↑
↑↓ ↓↓
↓↑↑ ↓
↓↑ ↑eeee↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑eeeee

4つのフェルミオンから成るα粒子はクラスターを構成できる。
フェッシュバッハ分子との違いは何か？

ee↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑↑↓↑ ↓↑↑↓ ↓↓↓↑↑ ↓↓↑ ↑eee𝑎𝑑𝑑 ≈ 0.60𝑎
クラスター間は斥力相互作用
よって分子BEC状態は安定
クラスター同士の束縛状態はない[ D. S. Petrov, PRL 93, 090404 (2004) ][ J. Levinsen, PRA 73, 053607 (2006) ]𝒂 > 𝟎 is repulsive or attractive ?
Repulsive core𝑟𝒂>𝟎𝑟
Attractive well𝒂>𝟎

相互作用ポテンシャルと散乱長• Grand canonical Hamiltonian :ℋ − 𝜇𝒩 =𝜎ℏ2𝛻𝛹𝜎† 𝒓 𝛻 𝛹𝜎 𝒓 − 𝜇 𝑑𝒓2𝑚+𝛹↑† 𝒓↑ 𝛹↓† 𝒓↓ 𝑈int 𝒓↑ − 𝒓↓ 𝛹↓ 𝒓↓ 𝛹↑ 𝒓↑ 𝑑𝒓↑ 𝑑𝒓↓

デルタ型擬ポテンシャル

Non-universal

𝑈int 𝒓↑ − 𝒓↓𝜎ℏ2= − 𝑔 𝑎 𝛿(𝒓↑ − 𝒓↓ )𝑚

1
11=−+𝑔 𝑎4𝜋𝑎 𝑉Universalℋ − 𝜇𝒩 =Detail of the particle

𝑘1>0𝑘2ℏ2𝛻𝛹𝜎† 𝒓 𝛻 𝛹𝜎 𝒓 − 𝜇 𝑑𝒓2𝑚
ℏ2− 𝑔(𝑎)𝑚𝛹↑† 𝒓 𝛹↓† 𝒓 𝛹↓ 𝒓 𝛹↑ 𝒓 𝑑𝒓

Renormalized coupling constantMany-body Hamiltonian and thermodynamics
Interaction• Grand canonical Hamiltonian :ℋ − 𝜇𝒩 =𝜎ℏ2ℏ2†𝛻𝛹 𝒓 𝛻𝛹𝜎 𝒓 − 𝜇 𝑑𝒓 − 𝑔(𝑎)2𝑚 𝜎𝑚𝛹↑† 𝒓 𝛹↓† 𝒓 𝛹↓ 𝒓 𝛹↑ 𝒓 𝑑𝒓• Eigenstate and eigenvalue of state i : 𝐾𝑖 = Ψ𝑖 ℋ − 𝜇𝒩 Ψ𝑖∞• Grand partition function : 𝑍𝐺 =𝑖=0

𝐾𝑖 Excitation
exp −𝑘𝐵 𝑇𝑔 𝑎 ≥0𝑑𝑔 𝑎𝑔2 𝑎=𝑑𝑎−14𝜋• Thermodynamic potential : Ω 𝑉, 𝑇, 𝜇, 𝑎−1 = −𝑘𝐵 𝑇ln𝑍𝐺• Thermodynamic relation :Hellmann–Feynman theorem𝜕(ℋ − 𝜇 𝒩)
𝐾𝑖
𝜕𝐾𝑖 (𝑎−1 )𝐾𝑖ΨΨ
exp−𝑖𝑖𝑖
𝑘𝐵 𝑇𝜕𝑎−1𝑖 𝜕𝑎 −1 exp − 𝑘 𝑇𝜕Ω
𝐵==𝐾𝑖
𝐾𝑖
𝜕𝑎−1 𝑉,𝑇,𝜇exp−exp−𝑛𝑖𝑘𝐵 𝑇𝑘𝐵 𝑇ℏ2≡−𝐶 𝒓 𝑑𝒓4𝜋𝑚Contact densityFew-body physics and Virial coefficients∞𝑁• Grand partition function : 𝑍𝐺 𝑉, 𝑇, 𝜇, 𝑎−1 =𝜇𝑍𝑁 𝑉, 𝑇, 𝑁, 𝑎−1 𝑒 𝑘𝐵 𝑇
𝑁=0𝜇𝜇𝜇2
3= 1 + 𝑍1 𝑒 𝑘𝐵𝑇 + 𝑍2 𝑒 𝑘𝐵𝑇 + 𝑍3 𝑒 𝑘𝐵𝑇 + ⋯Partition function∞

𝑍𝑁 =exp −𝑖=0𝐸𝑁 𝑖 Excitation
𝜇𝑘𝐵 𝑇𝑘𝐵 𝑇High temperature region : z ≡ 𝑒≪1• Thermodynamic potential : Ω 𝑉, 𝑇, 𝜇, 𝑎−1 → −𝑘𝐵 𝑇𝑍1 (𝑧 + 𝐵2 𝑧 2 + 𝐵3 𝑧 3 + ⋯ 𝐵𝑛 𝑧 𝑛 )• Virial coefficients :𝐵2 = (𝑍2 − 𝑍12 /2)/𝑍1
𝐵3 = (𝑍3 − 𝑍1 𝑍2 − 𝑍13 /3)/𝑍1…
• Universal equation of state :𝑃𝑃ideal 𝜇𝜇= 𝑓𝑃→ 𝑒 𝑘𝐵 𝑇 + 𝐵2𝜇 ΛT (T),𝑘𝐵 𝑇𝑎
ΛT T𝑎𝑒𝜇2𝑘 𝑇
𝐵+ 𝐵3

ΛT T𝑎𝑒𝜇3𝑘 𝑇𝐵+⋯Few-body physics and higher Virial coefficients• Universal equation of state at high temperature :𝜇𝑃𝑃ideal 𝜇= 𝑒 𝑘𝐵 𝑇 + 𝐵21-particle

ΛT 𝑇
𝑎2-particle

𝑒𝜇2𝑘 𝑇
𝐵+ 𝐵3

ΛT 𝑇𝑎3-particle

𝑒𝜇3𝑘 𝑇
𝐵+ 𝐵4

ΛT 𝑇𝑎𝑒4-particle

𝜇4𝑘 𝑇𝐵+⋯2nd and 3rd Virial coefficients
2nd Virial coefficients

N. Sakumichi,
Phys. Rev. A 89, 033622 (2014)

3rd Virial coefficients

X. Leyronas,
Phys. Rev. A 84, 053633 (2011)「Virial expansion with Feynman diagrams」

Many-body Hamiltonian and thermodynamics• Thermodynamic relation :𝜕Ωℏ2=−
𝜕𝑎−1 𝑉,𝑇,𝜇4𝜋𝑚• Contact density : 𝐶 𝒓 =𝒈 𝒂 𝛹↑† 𝒓 𝛹↓† 𝒓

For homogeneous system

𝐶 𝒓 𝑑𝒓𝒈 𝒂 𝛹↓ 𝒓 𝛹↑ 𝒓

𝐶 𝒓 𝑑𝒓 → 𝐶𝑉

ℏ2• Gibbs-Duhem equation : 𝑑𝑃 = 𝑠𝑑𝑇 + 𝑛𝑑𝜇 +𝐶 𝑑𝑎−14𝜋𝑚ℱ = 𝜇𝑛 − 𝑃∴ 𝑷 = 𝑷(𝑻, 𝝁, 𝒂−𝟏 )ℏ2• Total differential of free energy density : 𝑑ℱ = −𝑠𝑑𝑇 + 𝑛𝑑𝜇 −𝐶 𝑑𝑎−14𝜋𝑚∴ 𝓕 = 𝓕(𝑻, 𝒏, 𝒂−𝟏 )Phase diagram for Spin-1/2 fermionss-wave contact interactionℋint 𝑈int 𝒓↑ − 𝒓↓

→ ℋint 𝑎

Tunable by Feshbach resonance
𝑻/𝑻𝑭

One-bodyℏ𝟐
𝑬𝑩 = −, only for 𝒂 > 𝟎𝒎𝒂𝟐
Thermal molecularFermi gas

Few-body

𝑻𝒄

Molecular Bose-Einstein condensate

BCS superfluid
BCS limit(weak coupling)

Unitarity limit

BCS-BEC crossover

𝟏/𝒌𝑭 𝒂BEC limit(strong coupling)