MEDIATED INTERACTIONS:FROM YUKAWA TO EFIMOV
NATIONAL TAIWAN UNIVERSITY
δΈθ―ζ°ε 106εΉ΄11ζ10ζ₯
Pascal Naidon, RIKEN
ULTRA-COLD ATOMS
Alkali metal
oven
vapour
Vacuum chamber
Dilute and cold gas of atoms
π = 10β6 βΌ 10β9πΎ
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)
π π = ββ2
2π
0.5672
π2
π
ππ
Many-bodyBosons are created/absorbedβExchange of virtual particlesβ
Yukawa potential (1930)
π π = βπ2πβππ
π
Three-bodyParticle always thereβExchange of a real particleβ
MEDIATED INTERACTIONS
IMPURITIES IN A BOSE-EINSTEIN CONDENSATE
BEC of density π0
impurity
π
Interactions
Neglect direct interactions between impurities
ππ΅
impurity boson boson boson
π < 0 can be large ππ΅ > 0 is small
(attraction) (weak repulsion)
π0ππ΅3 βͺ 1
IMPURITIES IN A BOSE-EINSTEIN CONDENSATE
1
62
7
3
9
45
BEC of density π0
impurity|π|
small largeresonant
Energy Bound state
π β€ 0 π = Β±β π > 0
IMPURITIES IN A BOSE-EINSTEIN CONDENSATE
1
62
9
45
BEC of density π0
polaron|π|
small largeresonant
Energy Bound state
π β€ 0 π = Β±β π > 0
7
3
IMPURITIES IN A BOSE-EINSTEIN CONDENSATE
1
6
2
4
5
BEC of density π0
polaron |π|
small largeresonant
Energy Bound state
π0π < 0
π β€ 0 π = Β±β π > 0
7
3
9
IMPURITIES IN A BOSE-EINSTEIN CONDENSATE
1
6
2
4
5
BEC of density π0
polaron |π|
small largeresonant
Energy Bound state
π β€ 0 π = Β±β π > 0
7
3
9
IMPURITIES IN A BOSE-EINSTEIN CONDENSATE
|π|
small largeresonant
Energy Bound state
Bose polaron recently observed
JΓΈrgensen et al, PRL 117, 055302 (2016)
Ming-Guang Hu et al, PRL 117, 055301 (2016)
π β€ 0 π = Β±β π > 0
87Rb + 40K
39K|β1β© + 39K|0β©
BEC impurities
BEC impurities
polaron
polaron
POLARONIC INTERACTION
BEC of density π0
ππ
excitation
The Bogoliubov excitations of the BEC can mediate a Yukawa potential
To second-order in perturbation theory:
π π β βπ2π0πβπ 2/π
ππ =
1
8ππ0ππ΅
BEC coherence length
for weak coupling π
polaron
polaron
POLARONIC INTERACTION
BEC of density π0
ππ
excitation
The Bogoliubov excitations of the BEC can also mediate an Efimov potential
for resonant coupling π
Non-perturbative!
π π β ββ2
2π
1
π2
HAMILTONIAN
π» =
π
ππππβ ππ +
ππ΅2π
π,πβ²,π
ππβ²βπβ ππ+π
β ππππβ² +
π
ππππβ ππ +
π
π
π,πβ²,π
ππβ²βπβ ππ+π
β ππππβ²
Bosons Impurities Impurity-boson
ππ=β2π2
2πππ =
β2π2
2π
Bogoliubov approach ππ = π’ππ½π β π£ππ½π
β π0 = π0 condensate
Bogoliubov excitation
π
impurity boson
ππ΅
boson boson
HAMILTONIAN
πΈπ = ππ(ππ + 2ππ΅π0)
π» = πΈ0 +
π
πΈππ½πβ π½π +
π
(ππ + ππ0)ππβ ππ + π0
π
π
π,π
π’ππ½βπβ β π£ππ½π ππ+π
β ππ + β. π.
+π
π
π,πβ²,π
(π’πβ²βππ’πβ²π½πβ²βπβ π½πβ² + π£πβ²βππ£πβ²π½πβπβ²π½βπβ²
β )ππ+πβ ππ
+π
π
π,πβ²,π
(π’πβ²βππ£πβ²π½πβ²βπβ π½
βπβ²β + π£πβ²βππ’πβ²π½πβπβ²π½πβ²)ππ+π
β ππ
Bogoliubov excitation energy
Yukawa (FrΓΆhlich)
Efimov (Scattering)
πβ²
impurity boson
excitation
πβ²
impurity
boson
excitation
πβ²β²
impurity excitation
excitation
Bogoliubov approach ππ = π’ππ½π β π£ππ½π
β π0 = π0 condensate
Bogoliubov excitation
Free excitations Dressed impurities
NON-PERTURBATIVE METHOD: TRUNCATED BASIS
Ξ¨ =
π
πΌπππβ πβπ
β +
π,πβ²
πΌπ,πβ²ππβ ππβ²β π½
βπβπβ²β +
π,πβ²,πβ²β²
πΌπ,πβ²,πβ²β²ππβ ππβ²β π½
πβ²β²β π½
βπβπβ²βπβ²β²β +β― |Ξ¦β©
BEC ground state
Impurity creation operator
Excitation creation operator
Coupled equations for πΌπ and to πΌπ,πβ² Effective 3-body equation
At resonance π = Β±β
π
0
0-50
π
RESULT: POLARONIC POTENTIAL
Effective potential (Born-Oppenheimer) between polarons:
1
πβ π +
1
ππβπ π +
8ππ0π 2
= 0π π =β2π 2
2π (ππ΅ β 0)
For small π β² 0
π(π)
π
0
0-50
π
ββ2
2π
0.5672
π2
Efimov
ββ2
2π8ππ0
2/3β2 Γ π04πβ2
2ππ +
π2
π
Yukawa
POSSIBLE EXPERIMENTAL OBSERVATIONS
Heavy impurities in a condensate of light bosons (e.g. 133Cs + 7Li, Yb + 7Li)
β’ Polaron RF spectroscopy: mean-field shift with the impurity density
β’ Loss by recombination: shift of the loss peak with the condensate density
OUTLOOK
Strong interaction between bosons:
πβ²
impurity boson
excitation
πβ²β²
impurity excitation
excitation
Yukawa
Scattering
ππ΅β²
excitation boson
excitation
ππ΅β²β²
excitation excitation
excitation
Belyaev Scattering