Date post: | 02-Jan-2016 |
Category: |
Documents |
Upload: | mohammad-freeman |
View: | 29 times |
Download: | 0 times |
Pascal Naidon
Microscopic origin and universality classes of the three-body parameter
Shimpei Endo
Masahito Ueda
The University of Tokyo
3 particles (bosons or distinguishable) with
resonant two-body interactions
• single-channel two-body interaction
• no three-body interaction
Zero-range condition with
Efimov attraction
R
x
𝑅2=23(𝑥2+𝑦2+𝑧 2)Hyperradius
The Efimov effect (1970)
y
z
1/𝑎
𝐸
The Efimov 3-body parameter
Th
ree-b
od
y
para
mete
r
Parameters describing particles at low energyScattering length a
𝑎
1/𝑎−
-
dimertrimer
Λ−1 3-body parameter
(2-body parameter)
Zero-range condition with
Efimov attraction
R
x
𝑅2=23(𝑥2+𝑦2+𝑧 2)Hyperradius
The Efimov effect (1970)
y
z
1/𝑎
𝐸
The Efimov 3-body parameter
Th
ree-b
od
y
para
mete
r
Parameters describing particles at low energyScattering length a
𝑎
1/𝑎−
-
dimertrimer
Λ−1 3-body parameter
(2-body parameter)
22.72
Microscopic determination?
Universality for atoms
r
short-range details
−1
𝑟6 van der Waals
Two-body potential
Effective three-body potential
Hyperradius R
short-range details
−1
𝑅2 Efimov
trip
let s
catt
. len
gth
4He
7Li
6Li
39K23Na
87Rb
85Rb
133Csno universality of the scattering length
𝑎−[𝑎
0]
𝑎−≈ −10𝑟 𝑣𝑑𝑊
universality of the 3-body parameter
Three-body with van der Waals interactionsPhys. Rev. Lett. 108 263001 (2012)J. Wang, J. D’Incao, B. Esry, C. Greene
𝑎−≈ −11𝑟 𝑣𝑑𝑊Three-body repulsion
at
Efimov
Lennard-Jones potentials supporting n = 1, 2, 3, ...10 s-wave bound states
Hyperradius R
Interpretation: two-body correlation
𝜓𝑘 =sin(𝑘𝑟 ¿−𝛿𝑘)¿
Asymptotic behaviour
Two-body correlation
𝜓𝑘
𝑉 (𝑟 )
Resonance
𝜓𝑘
𝑉 (𝑟 )
Interatomic separation r
Strong depletion
∼ 12𝑟𝑒=∫
0
∞
(𝜓 0 (𝑟 )2 −𝜓 0 (𝑟 )2 )𝑑𝑟𝑟0 ≈𝑟 𝑣𝑑𝑊
Interpretation: two-body correlation
squeezed
equilateral
elongated
Excluded configurations
induced deformation Efimov
Kinetic energy
cost due to
deformation
deformation
𝑟0
Exc
lud
ed
configura
tions
Confirmation 1: pair correlation model
FModel = FEfimov x j(r12) j(r23) j(r31) (product of pair
correlations)(hyperangular wave function)
3-body potential
𝑈 (𝑅 )= 𝜆𝑅2
+∫𝑑cos𝜃 𝑑𝛼|𝜕Φ𝜕𝑅|
2
Hyperradius R []
Ene
rgy
E [
] Pair model(for Lennard-Jones two-body interactions)
Exact
Efimov attraction
Reproduces the low-energy 2-body physics• Scattering length• Effective range• Last bound state• ….
Confirmation 2: separable model
𝑉=𝜉|𝜒 ⟩ ⟨ 𝜒∨¿Parameterised to reproduce exactly the two-body correlation at zero energy.
1/𝑎−
-
𝜒 (𝑞)=1−𝑞∫0
∞
(𝜓 0 (𝑟 ) −𝜓 0(𝑟 ))sin𝑞𝑟 𝑑𝑟
𝜉=4𝜋 ( 1𝑎
−2𝜋∫0
∞
|𝜒 (𝑞 )|2𝑑𝑞)
−1
Hyperradius R
Inte
gra
ted
pro
babili
ty
Hyperradius R
Energ
y
Confirmation 2: separable model
𝑉=𝜉|𝜒 ⟩ ⟨ 𝜒∨¿Parameterised to reproduce exactly the two-body correlation at zero energy.
𝑎−
𝑟𝑣𝑑𝑊 𝑎−=−10.86 (1)𝑟 𝑣𝑑𝑊
n
Exact
Pair model
Separable model
Confirmation 2: separable model
𝑉=𝜉|𝜒 ⟩ ⟨ 𝜒∨¿Parameterised to reproduce exactly the two-body correlation at zero energy.
S. Moszkowski, S. Fleck, A. Krikeb, L. Theuÿl, J.-M.Richard, and K. Varga, Phys. Rev. A 62 , 032504 (2000).
Other potentials
Potential
Yukawa -5.73 0.414
Exponential -10.7 0.216
Gaussian -4.27 0.486
Morse () -12.3 0.180
Morse () -16.4 0.131
Pöschl-Teller () -6.02 0.367
-6.55
-11.0
-4.47
-12.6
-16.3
-6.23
0.204
-0.366
0.472
0.173
0.128
0.350
Separable model
Exact calculations
at most 10% deviation
Summary
two-body correlation
three-body deformation
three-body repulsion
three-body parameter
universal
universal effective range
effective range
Two-body correlation universality classes
Power-law tails Faster than Power-law tails
Interparticle distance ()
Pro
bab
ility
densi
ty
0.0 1.0 2.0 3.0 4.0
Van der Waals
∝−1
𝑟 𝑛 (𝑛>3)
Step function correlation limit
Universal correlation𝜓 0 (𝑟 )=Γ (𝑛−1
𝑛−2 ) (𝑟 /𝑟𝑛)1/2 𝐽 1/(𝑛− 2)(2 (𝑟 /𝑟 𝑛)−(𝑛− 2)/2)
Separable model
Number of two-body bound states
Bin
din
g w
ave n
um
ber
at
unit
ari
ty3-body parameter in units of the two-body effective range
Atomic physics
Nuclear physics
?
(= size of two-body correlation)
𝜅=−0.2190 (1 )( 𝑟 𝑒
2 )−1
𝜅=−0.261 (1 )( 𝑟𝑒2 )−1
𝜅=−0.364 (1 )( 𝑟𝑒2 )− 1
P. Naidon, S. Endo, M. Ueda, PRL 112, 105301 (2014)
P. Naidon, S. Endo, M. Ueda, PRA 90, 022106 (2014)
Summary
The 3-body parameter is (mostly) determined by the low-energy 2-body correlation.Reason: 2-body correlation induces a deformation of the 3-body system.Consequences: the 3-body parameter• is on the order of the effective range.• has different universal values for distinct classes
of interaction
Obrigado pela sua attenção!