Clustering as a Window on the Hierarchical Structure of Quantum Systems--CLUSHIQ2022 (EMMI Workshop)--

Toward complete of the phase diagramof the BCS-BEC crossover
Osaka Metropolitan UniversityA, NITEPB
Munekazu HorikoshiA, B

Sendai, Japan, Oct 31- Nov 3, 2022

Introduction
Simplify the Hierarchical Structure of Quantum Systems

compositebosonscompositenessloosely bound clusters

simplify

spin-1/2
fermions

pairing fluctuations

particlenessIntroduction
compositenessHow does the interaction change?

compositebosonsCluster formation mechanism inthe many-body quantum systemspin-1/2
fermions

loosely bound clusters

pairing fluctuations

particlenessIntroduction
compositenessparticlenessIntroduction

compositenessparticleness

The phase diagram of spin-1/2 fermions with s-wave interactions
𝑇𝑇/𝑇𝑇𝐹𝐹

compositenessparticleness

BCS(weak coupling)

𝑇𝑇�𝑇𝑇�

∼0.4∼0.2

Superfluid
0Unitarity∼1𝑇𝑇𝑐𝑐1
𝑘𝑘𝐹𝐹 𝑎𝑎

BEC(strong coupling)Our approach to the cluster physics
𝑇𝑇/𝑇𝑇𝐹𝐹• Cluster formation region𝑇𝑇�𝑇𝑇�
Experimental determinationof the wholeEOS
on this phase diagram• Confinement region• Cluster-Cluster interactions∼0.4• Viscosity∼0.2

Superfluid
Universal quantum cluster physics
in the many-body (matter) system

BCS(weak coupling)

0Unitarity∼1𝑇𝑇𝑐𝑐1
𝑘𝑘𝐹𝐹 𝑎𝑎

BEC(strong coupling)Why EOS is necessary?
because the EOS tells us many physics• Cluster formation region• Confinement region
specific heat

𝑇𝑇/𝑇𝑇𝐹𝐹𝑇𝑇�

𝑇𝑇�

[Pieter van Wyk, et al., Phys. Rev. A 93, 013621 (2016)]• Cluster-Cluster interactions
compressibility, Virial coefficients
Poster 29 by K. Yoneda Poster 25 by H. Funaba∼0.4[D. Kagamihara, et al., Phys. Rev. A 106, 033308 (2022)][S. Endo and Y. Castin, Phys. Rev. A 92, 053624 (2015)]∼0.2• Viscosity
The EOS is required[K. Fujii and Y. Nishida, Phys. Rev. A 98, 063634 (2018)][D. Kagamihara, et al., J. Phys. Soc. Jpn. 89, 044005 (2020)]

Superfluid
BCS(weak coupling)

0Unitarity

∼1

𝑇𝑇𝑐𝑐
1
𝑘𝑘𝐹𝐹 𝑎𝑎

BEC(strong coupling)Why EOS is necessary?
𝑇𝑇/𝑇𝑇𝐹𝐹• composite boson interactions in vacuum
Virial expansion of the EOS
𝑃𝑃 = 2𝑇𝑇�

𝑇𝑇�

𝑘𝑘𝐵𝐵 𝑇𝑇2
3
43 (𝑧𝑧 + 𝐵𝐵2 𝑧𝑧 + 𝐵𝐵3 𝑧𝑧 + 𝐵𝐵4 𝑧𝑧 + ⋯ )
Λ 𝑇𝑇
∼0.4
∼0.2

Superfluid
BCS(weak coupling)

0Unitarity∼1𝑇𝑇𝑐𝑐1𝑘𝑘𝐹𝐹 𝑎𝑎

BEC
(strong coupling)Why EOS is necessary?
• composite boson interactions in the many-body system[D. Kagamihara, et al., Phys. Rev. A 106, 033308 (2022)]𝜅𝜅 𝑇𝑇 𝑇𝑇𝑐𝑐 =2−𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

2 (𝑈𝑈𝐵𝐵13−𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏+ 𝑈𝑈𝐵𝐵=2+ ⋯ )𝑛𝑛𝐵𝐵

𝑇𝑇/𝑇𝑇𝐹𝐹

1𝑇𝑇�

𝑇𝑇�

2𝑈𝑈eff 𝑛𝑛𝐵𝐵2

∼0.4
∼0.2

SuperfluidBCS(weak coupling)

0Unitarity∼1𝑇𝑇𝑐𝑐1
𝑘𝑘𝐹𝐹 𝑎𝑎

BEC(strong coupling)Previous experimental researches on the Phase diagram• EOS at the unitarity limit[MH, et al., Science 327, 442 (2010)][S. Nascimbène, et al., Nature 463, 1057 (2010)]

𝑇𝑇/𝑇𝑇𝐹𝐹[M. JH Ku, et al., Science 335, 563 (2012)]

𝑇𝑇�𝑇𝑇�

[C. Carcy, et al., Phys. Rev. Lett. 122, 203401 (2019)]• EOS in the BCS-BEC crossover atalmost zero temperature[S. Nascimbène, et al., Nature 463, 1057 (2010)][C. Sanner, et al., Phys. Rev. Lett. 106, 010402 (2011)][MH, et al., Phys. Rev. X 7, 041004 (2017)]• Share viscosity in the BCS-BECcrossover above 𝑻𝑻𝐜𝐜

∼0.4
∼0.2

Superfluid[E. Elliott, et al., Phys. Rev. Lett. 113, 020406 (2014)]

BCS[J. A. Joseph, et al., Phys. Rev. Lett. 115, 020401 (2015) (weak coupling)• Share viscosity at the unitarity limit

0Unitarity

∼1

𝑇𝑇𝑐𝑐
1𝑘𝑘𝐹𝐹 𝑎𝑎

BEC(strong coupling)Method to construct the EOS
This function is what we want to determine

EOS (pressure) : 𝑃𝑃 𝑇𝑇, 𝜇𝜇↑ , 𝜇𝜇↓ , 𝑎𝑎𝑠𝑠−1

for spin-balanced system(𝜇𝜇↑ = 𝜇𝜇↓ = 𝜇𝜇)
𝑓𝑓𝑃𝑃 = 𝑃𝑃⋮

Λ3𝑇𝑇

𝑘𝑘𝐵𝐵 𝑇𝑇𝜇𝜇𝑘𝑘𝐵𝐵 𝑇𝑇⋮𝑘𝑘𝐵𝐵 𝑇𝑇𝜇𝜇↑ 𝜇𝜇↓ ΛT,,= 3 𝑓𝑓𝑃𝑃,𝑘𝑘𝐵𝐵 𝑇𝑇 𝑘𝑘𝐵𝐵 𝑇𝑇 𝑎𝑎𝑠𝑠
Λ 𝑇𝑇
Λ3𝑇𝑇
𝑓𝑓𝑃𝑃 = 𝑃𝑃𝑘𝑘𝐵𝐵 𝑇𝑇

ΛT𝑎𝑎𝑠𝑠⋮𝜇𝜇𝑘𝑘𝐵𝐵 𝑇𝑇ΛT ≡2𝜋𝜋ℏ2𝑚𝑚𝑘𝑘𝐵𝐵 𝑇𝑇𝜅𝜅 𝑇𝑇 /𝜅𝜅0ΛT𝑎𝑎𝑠𝑠

𝑇𝑇/𝑇𝑇𝐹𝐹
1𝑘𝑘𝐹𝐹 𝑎𝑎𝑠𝑠

Method to construct the EOS
yexperimental observables

local density : 𝑛𝑛𝜎𝜎 𝒓𝒓

trapping potential : 𝑉𝑉 𝒓𝒓

scattering length : 𝑎𝑎𝑠𝑠 (𝐵𝐵)LDA𝑛𝑛↑ 𝑉𝑉 , 𝑛𝑛↓ 𝑉𝑉

𝑉𝑉

𝑃𝑃 𝑉𝑉 = − ∫+∞ 𝑛𝑛↑ 𝑉𝑉 + 𝑛𝑛↓ 𝑉𝑉 𝑑𝑑𝑑𝑑

Chemical potential at thebottom of the potential : 𝜇𝜇0𝜎𝜎
LDA : 𝜇𝜇𝜎𝜎 𝒓𝒓 = 𝜇𝜇0𝜎𝜎 − 𝑉𝑉(𝒓𝒓)𝑇𝑇, 𝑎𝑎𝑠𝑠zx𝑛𝑛↑ 𝑉𝑉

extension of the EOS

fitting with 𝑇𝑇, 𝜇𝜇0↑ , 𝜇𝜇0↓

𝑉𝑉by the known EOS
at high temperature region𝑛𝑛↓ 𝑉𝑉

Inspired by [S. Nascimbène, et al., Nature 463, 1057 (2010)]

𝛬𝛬3𝑇𝑇 (𝑇𝑇)
𝑃𝑃(𝑉𝑉)
𝑘𝑘𝐵𝐵 𝑇𝑇𝜇𝜇0↑ − 𝑉𝑉
𝑘𝑘𝐵𝐵 𝑇𝑇𝜇𝜇0↓ − 𝑉𝑉
𝑘𝑘𝐵𝐵 𝑇𝑇𝛬𝛬 𝑇𝑇 (𝑇𝑇)𝑎𝑎𝑠𝑠to lower temperature

Ultracold 6Li experiment @ Osaka

momentum distributionof molecules

ThermalMove to Osaka(2020/2/17)

BEC

First BEC
(2021/9/3)

First laser cooling(2020/8/7)

Stabilization and calibrationof the apparatus

Test measurement of the equation of state
Construction

Experimental apparatus
Column density : 𝑛𝑛�↑,↓ 𝑥𝑥, 𝑧𝑧 = �s-wave Feshbach resonances of 6Li
s-wave scattering length [𝑎𝑎0 ]𝐵𝐵res600

700

800

900

1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0

𝑛𝑛↑,↓ 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 𝑑𝑑𝑑𝑑−∞

Binding energy [MHz]

10000
8000
6000
4000
2000
0-2000
-4000
-6000
-8000
-10000+∞~2mm𝒚𝒚

𝑁𝑁atom ~105 ~106𝒙𝒙𝒛𝒛

6Li atoms~240µm(resolution : 2µm)

An example of spin imbalance

External magnetic field : 𝐵𝐵 [Gauss]
Unitarity limit

Helmholtz coils
Probe laser

BCS(weak coupling)

BEC(strong coupling)

𝑎𝑎𝑠𝑠 (𝐵𝐵)0(balance)

Optical trap
20cm

Tc

Superfluid

Normal𝑇𝑇ℎ ≡ 𝜇𝜇↑ − 𝜇𝜇↓Typical data for a spin balanced gas
Column density : 𝑛𝑛� 𝑥𝑥, 𝑧𝑧 = �+∞

−∞

𝑥𝑥
𝑦𝑦𝑧𝑧

𝑛𝑛 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 𝑑𝑑𝑑𝑑

Number density

Pressure and compressibility

𝑛𝑛 𝑉𝑉

𝑃𝑃 𝑉𝑉 = − ∫+∞ 𝑛𝑛 𝑉𝑉 𝑑𝑑𝑑𝑑

Averaged column density

Inverse Abel
Local density : 𝑛𝑛(𝑟𝑟, 𝑧𝑧)

𝑟𝑟Trapping potential : 𝑉𝑉(𝑟𝑟, 𝑧𝑧)

𝑧𝑧

LDA𝑉𝑉𝜅𝜅 𝑇𝑇 =1 𝜕𝜕𝜕𝜕1 𝑑𝑑𝑛𝑛= 2𝑛𝑛 𝜕𝜕𝜕𝜕 𝑇𝑇 𝑛𝑛 𝑑𝑑𝑉𝑉

Test of data reliability forbalanced unitary and non-interacting Fermi gases
From thermodynamic relations, unitary Fermi gases and non-interacting Fermi gases obey𝜅𝜅 𝑇𝑇𝑃𝑃= 𝑔𝑔
𝜅𝜅0𝑃𝑃0Low

High𝑇𝑇, 𝑃𝑃𝑜𝑜 =23 1𝑛𝑛𝜀𝜀𝐹𝐹 𝑛𝑛 , 𝜅𝜅𝑜𝑜 =52 𝑛𝑛𝜀𝜀𝐹𝐹 𝑛𝑛Low

for any temperature

High𝑇𝑇LowHigh𝑇𝑇

Test of data reliability forbalanced unitary and non-interacting Fermi gases
From thermodynamic relations,𝜅𝜅 𝑇𝑇𝑃𝑃= 𝑔𝑔
𝜅𝜅0𝑃𝑃0, 𝜅𝜅 𝑇𝑇 =1 𝜕𝜕𝜕𝜕23 1, 𝑃𝑃𝑜𝑜 = 𝑛𝑛𝜀𝜀𝐹𝐹 𝑛𝑛 , 𝜅𝜅𝑜𝑜 =𝑛𝑛 𝜕𝜕𝜕𝜕 𝑇𝑇
52 𝑛𝑛𝜀𝜀𝐹𝐹 𝑛𝑛

Non-interacting fermi gas𝜅𝜅 𝑇𝑇 /𝜅𝜅0𝜅𝜅 𝑇𝑇 /𝜅𝜅0Unitary fermi gas

for any temperature

𝑃𝑃/𝑃𝑃0𝑃𝑃/𝑃𝑃0Behavior around Tc
Number density

4Unitary(2022/10/18)
Measure at Vdepth=2.9uKThis work
MIT experiment [Science 335, 563 (2012)]]
3rd Virial EOS
Tc

T/κ0𝜅𝜅 𝑇𝑇κ/𝜅𝜅
032

100Tc

12P/P0𝑃𝑃/𝑃𝑃0341 𝜕𝜕𝜕𝜕1 𝑑𝑑𝑛𝑛
𝜅𝜅 𝑇𝑇 ==𝑛𝑛 𝜕𝜕𝜕𝜕 𝑇𝑇 𝑛𝑛2 𝑑𝑑𝑉𝑉

We need numerical derivative of the dataSummary
Toward complete of the phase diagramof the BCS-BEC crossover

𝑇𝑇/𝑇𝑇𝐹𝐹The Quantum Simulator is ready𝑇𝑇�𝑇𝑇�

Three remained subjects
1. Discrepancy around Tc
2. Robust EOS construction process
3. Imaging condition for molecules∼0.4

We will show you cluster physics within thissimplified model from the EOS in near future.
BCS(weak coupling)

Discrepancy∼0.2

0Unitarity

Superfluid∼1𝑇𝑇𝑐𝑐
1𝑘𝑘𝐹𝐹 𝑎𝑎

BEC(strong coupling)