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  • Universal clustering and the Efimov effect

    Pascal Naidon, RIKEN

    Clustering as a window on the hierarchical structure of quantum systems Kick-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

    En er

    gy What are universal clusters?

    two-body bound states

  • What are universal clusters?

    Binding in classical systems

    𝑔

    −𝑔

    −3𝑔

    three-body bound states

    En er

    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

    En er

    gy

    Binding in quantum systems

    two-body bound states

  • What are universal clusters?

    𝑔

    −𝑔

    −3𝑔

    three-body bound states

    En er

    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

    En er

    gy

    Binding in quantum systems

    Universal region

    two-body bound states

    Two-body resonances ⟹ unitarity points, universality and scale invariance

    Deuteron np 𝑎 = 5.4 fm

    1 fm

    Helium-4 dimer 𝑎 = 10 nm 0.5 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

    En er

    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

    En er

    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

    En er

    gy

    Inverse scattering length 1/𝑎

    𝐸2 = − 1

    𝑚𝑎2

    t3

    t4

    t5

    Predicted in 1970

    Caesium-133

    Lo ss

    − 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

    En er

    gy

  • Four bosons

    Inverse scattering length 1/𝑎

    Lo ss

    Caesium-133

    En er

    gy

  • Five bosons

    Inverse scattering length 1/𝑎

    En er

    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

    En er

    gy

  • Two different bosons

    Inverse scattering length 1/𝑎

    unfavoured

    favoured

    En er

    gy

  • Review papers

  • Projects

    • Universal clusters with nonzero angular momentum

    • Universal clusters in many-body systems

  • Inverse scattering length 1/𝑎

    unfavoured

    favoured

    En er

    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

    En er

    gy

  • Inverse scattering length 1/𝑎

    For 𝑀

    𝑚 > 12.9

    𝐿 = 1

    Kartavtsev-Malykh Universal trimers

    En er

    gy

    Universal clusters with 𝐿 ≠ 0

  • Inverse scattering length 1/𝑎

    For 𝑀

    𝑚 > 12.9

    𝐿 = 1

    Kartavtsev-Malykh Universal trimers

    En er

    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

    En er

    gy

    Inverse scattering length 1/𝑎

    t3

    t4

    t5

    𝐿 = 0 trimer

    dimer

    Universal clusters with 𝐿 ≠ 0

    3 identical bosons

    𝐿 = 2 trimer

    𝐿 = 2 tetramer

    Universal clusters with two units of angular momentum May 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

    En er

    gy

    Inverse scattering length

  • Clusters in many-body systems

    polaron

    dimer

    En er

    gy

    Inverse scattering length

  • Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

  • Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

  • Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

  • Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

  • polaron

    Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

  • polaron

    Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

  • polaron

    Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

  • polaron

    Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

    “Artificial nucleons”!

    Yukawa

  • polaron

    Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

    trimer

    trimer

    Yukawa

    Efimov

  • polaron

    Clusters in many-body systems

    dimer

    En er

    gy

    Inverse scattering length

    polaron

    bipolaron

    bipolaron

    Efimov

    Yukawa

    “Artificial deuteron”

  • polaron

    Clusters in many-body systems

    dimer

    En er

    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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