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Universal clustering and the Efimov effect
Pascal Naidon, RIKEN
Clustering as a window on the hierarchical structure of quantum systemsKick-off symposium
Tokyo Institute of Technology, 2018-11-20
D01 Emergence mechanism of hierarchical structure of matter studied by ab-initio calculations
Outline
• What are universal clusters and what is the Efimov effect?
• Projects
What are universal clusters?
(source: Takashi Nakamura)
Net colour = white
Net electric charge = 0
Universal clusters
𝑟
−𝑔
𝑉(𝑟)
Short-range attractive interaction potential
What are universal clusters?
Binding in classical systems
𝑔
−𝑔
Ener
gyWhat are universal clusters?
two-body bound states
What are universal clusters?
Binding in classical systems
𝑔
−𝑔
−3𝑔
three-body bound states
Ener
gy
two-body bound states
What are universal clusters?
𝑔
−𝑔
−3𝑔
Quantum fluctuations + interactions ⟹ critical strength 𝑔 (zero-point) and quantisation
three-body bound states
Ener
gy
Binding in quantum systems
two-body bound states
What are universal clusters?
𝑔
−𝑔
−3𝑔
three-body bound states
Ener
gy
Binding in quantum systems
Universal region
two-body bound states
Two-body resonances ⟹ unitarity points, universality and scale invariance
What are universal clusters?
𝑔
−𝑔
−3𝑔
three-body bound states
Ener
gy
Binding in quantum systems
Universal region
two-body bound states
Two-body resonances ⟹ unitarity points, universality and scale invariance
Deuteron np𝑎 = 5.4 fm
1 fm
Helium-4 dimer 𝑎 = 10 nm0.5 nm
Feshbach dimers from cold atoms
𝑎 ≫ 10 nm
1.7 nm
In nature In the laboratory
What are universal clusters?
𝑔
−3𝑔
three-body bound states
Ener
gy
Binding in quantum systems
Universal region
Two-body resonances ⟹ unitarity points, universality and scale invariance
two-body bound states
What are universal clusters?
𝑔
−3𝑔
three-body bound states
.
t1
t2
t3
t
4
Ener
gy
Binding in quantum systems
Universal region
Two-body resonances ⟹ unitarity points, universality and scale invariance
two-body bound states
t
5
.
t2
The Efimov effect
Ener
gy
Inverse scattering length 1/𝑎
𝐸2 = −1
𝑚𝑎2
t3
t4
t5
Predicted in 1970
Caesium-133
Loss
−1
𝑚
𝑑2
𝑑𝑅2−1
𝑚
𝛼
𝑅2− 𝐸3 𝜓 = 0
Efimov attraction
Discrete scale invariance
trimer
dimer
n
p n
triton
Helium-4 trimer
Lithium-6
Ener
gy
Four bosons
Inverse scattering length 1/𝑎
Loss
Caesium-133
Ener
gy
Five bosons
Inverse scattering length 1/𝑎
Ener
gy
Two different bosons
Inverse scattering length 1/𝑎
−1
𝑀
𝑑2
𝑑𝑅2−1
𝑚
1
𝑅2− 𝐸3 𝜓 = 0
unfavoured
favoured
M
m
−1
𝑚
𝑑2
𝑑𝑅2−1
𝑀
1
𝑅2− 𝐸3 𝜓 = 0
Ener
gy
Two different bosons
Inverse scattering length 1/𝑎
unfavoured
favoured
Ener
gy
Review papers
Projects
• Universal clusters with nonzero angular momentum
• Universal clusters in many-body systems
Inverse scattering length 1/𝑎
unfavoured
favoured
Ener
gy
Universal clusters with 𝐿 ≠ 0
Universal clusters with 𝐿 ≠ 0
Inverse scattering length 1/𝑎
For 𝑀
𝑚> 13.6
−1
𝑚
𝑑2
𝑑𝑅2+𝐿(𝐿 + 1)
𝑚𝑅2−1
𝑀
𝛼
𝑅2− 𝐸3 𝜓 = 0
−1
𝑀
𝑑2
𝑑𝑅2+𝐿(𝐿 + 1)
𝑀𝑅2−1
𝑚
𝛼
𝑅2− 𝐸3 𝜓 = 0
suppressed
𝐿 = 1
𝐿 = 1
Ener
gy
Inverse scattering length 1/𝑎
For 𝑀
𝑚> 12.9
𝐿 = 1
Kartavtsev-MalykhUniversal trimers
Ener
gy
Universal clusters with 𝐿 ≠ 0
Inverse scattering length 1/𝑎
For 𝑀
𝑚> 12.9
𝐿 = 1
Kartavtsev-MalykhUniversal trimers
Ener
gy
Universal clusters with 𝐿 ≠ 0
Zero-range theory: Borromean states are in principle possible!
We need to determine which conditions in the finite-range potentials leads to this possibility.
(ab initio 3-body in collaboration with Emiko Hiyama)
.t2
Ener
gy
Inverse scattering length 1/𝑎
t3
t4
t5
𝐿 = 0 trimer
dimer
Universal clusters with 𝐿 ≠ 0
3 identical bosons
𝐿 = 2trimer
𝐿 = 2tetramer
Universal clusters with two units of angular momentumMay be observed with helium atoms or cold atoms
(ab initio few-body in collaboration with Emiko Hiyama and Shimpei Endo)
Clusters in many-body systems
unbound
dimer
Ener
gy
Inverse scattering length
Clusters in many-body systems
polaron
dimer
Ener
gy
Inverse scattering length
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
“Artificial nucleons”!
Yukawa
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
trimer
trimer
Yukawa
Efimov
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
bipolaron
bipolaron
Efimov
Yukawa
“Artificial deuteron”
polaron
Clusters in many-body systems
dimer
Ener
gy
Inverse scattering length
polaron
bipolaron
bipolaron
polaron
tripolaron
Efimov
Can be studied theoretical with variational few-body methods
Can be observed with cold atoms
Efimov-induced Efimov effect!“artificial triton”
Conclusion
• Weakly bound clusters near the critical binding of two particles form an exotic yet universal class of states, where the Efimov attraction is a central paradigm.
• We want to explore these universal clusters with higher angular momentum and in many-body systems.
• They can be realised experimentally with cold atoms, and help us to understand general clustering mechanisms.
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