MEDIATED 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 ATOMSExamples:

Weakly interacting bosons

Bose-Einstein condensation (BEC) (1995)

Strongly interacting bosons

Efimov trimers (2006)

ULTRA-COLD ATOMSExamples: Weakly interacting fermions

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 = −ℏ2

2𝑚

0.5672

𝑟2

𝑟

𝑔𝑔

Many-bodyBosons are created/absorbed“Exchange of virtual particles”

Yukawa potential (1930)

𝑉 𝑟 = −𝑔2𝑒−𝑚𝑟

𝑟

Three-bodyParticle 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

𝑔𝐵

impurity boson boson boson

𝑔 < 0 can be large 𝑔𝐵 > 0 is small

(attraction) (weak repulsion)

𝑛0𝑎𝐵3 ≪ 1

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

1

62

7

3

9

45

BEC of density 𝑛0

impurity|𝑔|

small largeresonant

Energy Bound state

𝑎 ≤ 0 𝑎 = ±∞ 𝑎 > 0

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

1

62

9

45

BEC of density 𝑛0

polaron|𝑔|

small largeresonant

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 largeresonant

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 largeresonant

Energy Bound state

𝑎 ≤ 0 𝑎 = ±∞ 𝑎 > 0

7

3

9

IMPURITIES IN A BOSE-EINSTEIN CONDENSATE

|𝑔|

small largeresonant

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 impurities

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

𝑔𝑔

excitation

The Bogoliubov excitations of the BEC can mediate a Yukawa potential

To second-order in perturbation theory:

𝑉 𝑟 ∝ −𝑔2𝑛0𝑒−𝑟 2/𝜉

𝑟𝜉 =

1

8𝜋𝑛0𝑎𝐵

BEC coherence length

for weak coupling 𝑔

polaron

polaron

POLARONIC INTERACTION

BEC of density 𝑛0

𝑔𝑔

excitation

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

for resonant coupling 𝑔

Non-perturbative!

𝑉 𝑟 ∝ −ℏ2

2𝑚

1

𝑟2

HAMILTONIAN

𝐻 =

𝑘

𝜖𝑘𝑏𝑘†𝑏𝑘 +

𝑔𝐵2𝑉

𝑘,𝑘′,𝑝

𝑏𝑘′−𝑝† 𝑏𝑘+𝑝

† 𝑏𝑘𝑏𝑘′ +

𝑘

𝜀𝑘𝑐𝑘†𝑐𝑘 +

𝑔

𝑉

𝑘,𝑘′,𝑝

𝑏𝑘′−𝑝† 𝑐𝑘+𝑝

† 𝑐𝑘𝑏𝑘′

Bosons Impurities Impurity-boson

𝜀𝑘=ℏ2𝑘2

2𝑀𝜖𝑘 =

ℏ2𝑘2

2𝑚

Bogoliubov approach 𝑏𝑘 = 𝑢𝑘𝛽𝑘 − 𝑣𝑘𝛽𝑘

†𝑏0 = 𝑁0 condensate

Bogoliubov excitation

𝑔

impurity boson

𝑔𝐵

boson boson

HAMILTONIAN

𝐸𝑘 = 𝜖𝑘(𝜖𝑘 + 2𝑔𝐵𝑛0)

𝐻 = 𝐸0 +

𝑘

𝐸𝑘𝛽𝑘†𝛽𝑘 +

𝑘

(𝜀𝑘 + 𝑔𝑛0)𝑐𝑘†𝑐𝑘 + 𝑁0

𝑔

𝑉

𝑘,𝑝

𝑢𝑝𝛽−𝑝† − 𝑣𝑝𝛽𝑝 𝑐𝑘+𝑝

† 𝑐𝑘 + ℎ. 𝑐.

+𝑔

𝑉

𝑘,𝑘′,𝑝

(𝑢𝑘′−𝑝𝑢𝑘′𝛽𝑘′−𝑝† 𝛽𝑘′ + 𝑣𝑘′−𝑝𝑣𝑘′𝛽𝑝−𝑘′𝛽−𝑘′

† )𝑐𝑘+𝑝† 𝑐𝑘

+𝑔

𝑉

𝑘,𝑘′,𝑝

(𝑢𝑘′−𝑝𝑣𝑘′𝛽𝑘′−𝑝† 𝛽

−𝑘′† + 𝑣𝑘′−𝑝𝑢𝑘′𝛽𝑝−𝑘′𝛽𝑘′)𝑐𝑘+𝑝

† 𝑐𝑘

Bogoliubov excitation energy

Yukawa (Fröhlich)

Efimov (Scattering)

𝑔′

impurity boson

excitation

𝑔′

impurity

boson

excitation

𝑔′′

impurity excitation

excitation

Bogoliubov approach 𝑏𝑘 = 𝑢𝑘𝛽𝑘 − 𝑣𝑘𝛽𝑘

†𝑏0 = 𝑁0 condensate

Bogoliubov excitation

Free excitations Dressed impurities

NON-PERTURBATIVE METHOD: TRUNCATED BASIS

Ψ =

𝑞

𝛼𝑞𝑐𝑞†𝑐−𝑞

† +

𝑞,𝑞′

𝛼𝑞,𝑞′𝑐𝑞†𝑐𝑞′† 𝛽

−𝑞−𝑞′† +

𝑞,𝑞′,𝑞′′

𝛼𝑞,𝑞′,𝑞′′𝑐𝑞†𝑐𝑞′† 𝛽

𝑞′′† 𝛽

−𝑞−𝑞′−𝑞′′† +⋯ |Φ⟩

BEC ground state

Impurity creation operator

Excitation creation operator

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

At resonance 𝑎 = ±∞

𝑟

0

0-50

𝜉

RESULT: POLARONIC POTENTIAL

Effective potential (Born-Oppenheimer) between polarons:

1

𝑎− 𝜅 +

1

𝑟𝑒−𝜅𝑟 +

8𝜋𝑛0𝜅2

= 0𝑉 𝑟 =ℏ2𝜅2

2𝜇 (𝑎𝐵 → 0)

For small 𝑎 ≲ 0

𝑉(𝑟)

𝑟

0

0-50

𝜉

−ℏ2

2𝜇

0.5672

𝑟2

Efimov

−ℏ2

2𝜇8𝜋𝑛0

2/3−2 × 𝑛04𝜋ℏ2

2𝜇𝑎 +

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

𝑔𝐵′

excitation boson

excitation

𝑔𝐵′′

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?