FROM YUKAWA TO EFIMOV M :

M EDIATED INTERACTIONS : FROM YUKAWA TO EFIMOV

NATIONAL TAIWAN UNIVERSITY 中華民國 106年11月10日

Pascal Naidon, RIKEN

ULTRA-COLD ATOMS

Alkali metal

oven

vapour

Vacuum chamber

Dilute and cold gas of atoms

𝑇 = 10⁻⁶ ∼ 10⁻⁹𝐾

Fully quantum many-body systems Quantum Field Theory

Interactions are controllable Non-perturbative regime

ULTRA-COLD ATOMS

Examples:

Bose-Einstein

condensation (BEC) (1995)

Strongly interacting bosons

Efimov trimers (2006)

ULTRA-COLD ATOMS

Examples:

Bardeen-Cooper- Schrieffer (BCS) pairing (2004)

BCS-BEC crossover

Unitary Fermi gas Like Neutron

star

Strongly interacting fermions

Bose-Einstein condensate (BEC) of dimers

Low viscosity like QGP

Efimov potential (1970)

𝑉 𝑟 = − ℏ

2𝑚

0.567

𝑟

𝑟

𝑔 𝑔

Many-body

Bosons are created/absorbed

“Exchange of virtual particles”

Yukawa potential (1930)

𝑉 𝑟 = −𝑔

𝑒

𝑟

Three-body

Particle always there

“Exchange of a real particle”

MEDIATED INTERACTIONS

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

BEC of density 𝑛₀ impurity

𝑔 Interactions

Neglect direct interactions between impurities

𝑔_𝐵

impurity boson boson boson

𝑔 < 0 can be large 𝑔_𝐵 > 0 is small

(attraction) (weak repulsion) 𝑛₀𝑎_𝐵³ ≪ 1

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

1

6

2

7 3

9 5 4

BEC of density 𝑛₀

impurity |𝑔|

small resonant large

Energy

Bound state

𝑎 ≤ 0 𝑎 = ±∞ 𝑎 > 0

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

1

6

2

9 5 4

BEC of density 𝑛₀

polaron |𝑔|

small resonant large

Energy

Bound state

𝑎 ≤ 0 𝑎 = ±∞ 𝑎 > 0

7 3

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

1

6

2 4 5

BEC of density 𝑛₀

polaron |𝑔|

small resonant large

Energy

Bound state

𝑛₀𝑔 < 0

𝑎 ≤ 0 𝑎 = ±∞ 𝑎 > 0

7 3

9

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

1

6

2 4 5

BEC of density 𝑛₀

polaron |𝑔|

small resonant large

Energy

Bound state

𝑎 ≤ 0 𝑎 = ±∞ 𝑎 > 0

7 3

9

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

|𝑔|

small resonant large

Energy

Bound state

Bose polaron recently observed

Jørgensen et al, PRL 117, 055302 (2016)

Ming-Guang Hu et al, PRL 117, 055301 (2016)

𝑎 ≤ 0 𝑎 = ±∞ 𝑎 > 0

87Rb + ⁴⁰K

39K_|−1〉 + ³⁹K_|0〉

BEC ^impu

rities

BEC ^impurities

polaron

polaron

POLARONIC INTERACTION

BEC of density 𝑛₀

for weak coupling 𝑔

polaron

polaron

POLARONIC INTERACTION

BEC of density 𝑛₀

for weak coupling 𝑔

polaron

polaron

POLARONIC INTERACTION

BEC of density 𝑛₀

for weak coupling 𝑔

polaron

polaron

POLARONIC INTERACTION

BEC of density 𝑛₀

𝑔

excitation 𝑔

The Bogoliubov excitations of the BEC can mediate a Yukawa potential

To second-order in perturbation theory:

𝑉 𝑟 ∝ −𝑔

𝑛

𝑒

𝑟

1 8𝜋𝑛₀𝑎_𝐵

BEC coherence length

for weak coupling 𝑔

polaron

polaron

POLARONIC INTERACTION

BEC of density 𝑛₀

𝑔

excitation 𝑔

The Bogoliubov excitations of the BEC can also mediate an Efimov potential

for resonant coupling 𝑔

Non-perturbative!

𝑉 𝑟 ∝ − ℏ

2𝑚

1

𝑟

HAMILTONIAN

𝐻 = ෍

𝑘

𝜖_𝑘𝑏_𝑘^†𝑏_𝑘 + 𝑔_𝐵

2𝑉 ෍

𝑘,𝑘^′,𝑝

𝑏_𝑘′−𝑝

† 𝑏_𝑘+𝑝^† 𝑏_𝑘𝑏_𝑘^′ + ෍

𝑘

𝜀_𝑘𝑐_𝑘^†𝑐_𝑘 + 𝑔

𝑉 ෍

𝑘,𝑘^′,𝑝

𝑏_𝑘′−𝑝

† 𝑐_𝑘+𝑝^† 𝑐_𝑘𝑏_𝑘^′

Bosons Impurities Impurity-boson

𝜀_𝑘=ℏ²𝑘² 𝜖_𝑘 = ℏ²𝑘² 2𝑀

2𝑚

Bogoliubov

approach 𝑏_𝑘 = 𝑢_𝑘𝛽_𝑘 − 𝑣_𝑘𝛽_𝑘^†

𝑏₀ = 𝑁₀ condensate

Bogoliubov excitation

𝑔

impurity boson

𝑔_𝐵

boson boson

HAMILTONIAN

𝐸_𝑘 = 𝜖_𝑘(𝜖_𝑘+ 2𝑔_𝐵𝑛₀)

𝐻 = 𝐸₀ + ෍

𝑘

𝐸_𝑘𝛽_𝑘^†𝛽_𝑘 + ෍

𝑘

(𝜀_𝑘 + 𝑔𝑛₀)𝑐_𝑘^†𝑐_𝑘 + 𝑁₀ 𝑔 𝑉෍

𝑘,𝑝

𝑢_𝑝𝛽_−𝑝^† − 𝑣_𝑝𝛽_𝑝 𝑐_𝑘+𝑝^† 𝑐_𝑘 + ℎ. 𝑐.

+𝑔

𝑉 ෍

𝑘,𝑘^′,𝑝

(𝑢_𝑘^′_−𝑝𝑢_𝑘^′𝛽_𝑘′−𝑝

† 𝛽_𝑘^′ + 𝑣_𝑘^′_−𝑝𝑣_𝑘^′𝛽_𝑝−𝑘^′𝛽_−𝑘† ′

)𝑐_𝑘+𝑝^† 𝑐_𝑘

+𝑔 𝑉 ෍

𝑘,𝑘^′,𝑝

(𝑢_𝑘^′_−𝑝𝑣_𝑘^′𝛽_𝑘′−𝑝

† 𝛽_−𝑘† ′ + 𝑣_𝑘^′_−𝑝𝑢_𝑘^′𝛽_𝑝−𝑘^′𝛽_𝑘^′)𝑐_𝑘+𝑝^† 𝑐_𝑘

Bogoliubov excitation energy

Yukawa (Fröhlich)

Efimov (Scattering)

𝑔^′

impurity boson excitation

𝑔^′

impurity

boson

excitation

𝑔^′′

impurity excitation excitation

Bogoliubov

approach 𝑏_𝑘 = 𝑢_𝑘𝛽_𝑘 − 𝑣_𝑘𝛽_𝑘^†

𝑏₀ = 𝑁₀ condensate

Bogoliubov excitation

Free excitations Dressed impurities

NON-PERTURBATIVE METHOD: TRUNCATED BASIS

Ψ = ෍

𝑞

𝛼_𝑞𝑐_𝑞^†𝑐_−𝑞^† + ෍

𝑞,𝑞^′

𝛼_𝑞,𝑞^′𝑐_𝑞^†𝑐_𝑞†′𝛽_{−𝑞−𝑞}† ′ + ෍

𝑞,𝑞^′,𝑞^′′

𝛼_𝑞,𝑞^′_,𝑞^′′𝑐_𝑞^†𝑐_𝑞†′𝛽_𝑞†′′𝛽_{−𝑞−𝑞}′−𝑞^′′

† + ⋯ |Φ〉

BEC ground state Impurity creation

operator

Excitation creation operator

Coupled equations for 𝛼_𝑞 and to 𝛼_𝑞,𝑞^′ Effective 3-body equation

At resonance 𝑎 = ±∞

𝑟

0

-500

𝜉

RESULT: POLARONIC POTENTIAL

Effective potential (Born-Oppenheimer) between polarons:

1

𝑎 − 𝜅 + 1

𝑟 𝑒

+ 8𝜋𝑛

𝜅

= 0

𝑉 𝑟 = ℏ²𝜅²

2𝜇 (𝑎_𝐵 → 0)

For small 𝑎 ≲ 0

𝑉(𝑟)

𝑟

0

-50 0

𝜉

−ℏ² 2𝜇

0.567² 𝑟²

Efimov

−ℏ²

2𝜇 8𝜋𝑛₀ ^2/3

−2 × 𝑛₀4𝜋ℏ²

2𝜇 𝑎 + 𝑎² 𝑟

Yukawa

POSSIBLE EXPERIMENTAL OBSERVATIONS

Heavy impurities in a condensate of light bosons (e.g. ¹³³Cs + ⁷Li, Yb + ⁷Li)

• Polaron RF spectroscopy: mean-field shift with the impurity density

• Loss by recombination: shift of the loss peak with the condensate density

OUTLOOK

Strong interaction between bosons:

𝑔^′

impurity boson excitation

𝑔^′′

impurity excitation excitation

Yukawa

Scattering

𝑔_𝐵^′

excitation boson excitation

𝑔_𝐵^′′

excitation excitation excitation

Belyaev Scattering

CONCLUSION

• A Bose-Einstein condensate of atoms can mediate interactions that go from weak Yukawa-type to strong Efimov-type.

arXiv:1607.04507

• Fermionic impurities in a BEC : atomic analogues of nucleons and mesons

• Analogues of quarks and gluons?