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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
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