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.