Ultracold Scattering Processes in Three-Atomic Helium Systems
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Transcript of Ultracold Scattering Processes in Three-Atomic Helium Systems
Ultracold Scattering Processes in Three-Atomic Helium Systems
E.A. Kolganova (JINR Dubna)
A.K. Motovilov (JINR Dubna)
W.Sandhas (Bonn)
FB18, Santos, Brazil, August 25, 2006 Werner Sandhas (Bonn)2
Outline
Overview - experiment and theory (two-body, three-body)
Formalism (Faddeev equations, Hard-Core model)
Results
three-body bound states (4He3 and 3He4He2)
scattering (phase shifts and scattering length)
Conclusion
FB18, Santos, Brazil, August 25, 2006 Werner Sandhas (Bonn)3
First observation by Luo et al. (1993) and Schöllkopf, Toennies (1994)
First measurement of the bond length by Grisenti et al.(2000)
Estimation of the binding energy and scattering length
Two-body, experiment4He - 4He
o0.3 80.2 181.1 mK 104 Ad scl
-71 mK 10 eV
o
52 4AR
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Two-body, theory
Potential models: Aziz et al. – HFD-B (1987), LM2M2 (1991),Tang et al. – TTY (1995)
4He2
( ), the dimer wave function
d r
4He – 4He potential (TTY)
where and
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Potential models: Aziz et al. – HFD-B (1987), LM2M2 (1991)
Two-body, theory4He - 4He
Tang et al. – TTY (1995)
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Three-body, experiment and theory bound states
Experiment – Toennies et al. JCP 104, 1155 (1996), JCP 117, 1544 (2002)
4He - 4He - 4He
Theory – Variational methods
Hyperspherical approach
Faddeev equations
Egs 126 mK Eex 2.28 mK
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Three-body, theory formalism
[4] - L.D.Faddeev,S.P.Merkuriev, 1993, Quantum scattering theory for several particles
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At L=0 the partial angular momentum corresponds both to the dimer and an additional atom. stand to the standard Jacobi variables.
Three-body, theory formalism
Faddeev integro-differential equations after angular partial-wave analysis
4He2 - 4He
2 2
2 2 2 2
1 1( 1) ( , )
( ) ( , ),
0,
l
l
l l E F x yx y x y
V x x y x c
x c
,x y
'
1
' '
1
( , ) ( , ) ( , , ) ( ', '),l l ll ll
x y F x y d h x y F x y
The kernel depend only on hyperangles - see L.D.Faddeev,S.P.Merkuriev, 1993.
l
2 2 1/ 2 2 2 1/ 2 ˆ ˆ' (1/ 4 3/ 4 3 / 2 ) , ' (3/ 4 1/ 4 3 / 2 ) ,x x y xy y x y xy x y
'llh
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Three-body, theory formalism
Boundary conditions
4He2 - 4He
Here, is the dimer wave function, stands for the dimer energy,d
1/ 20 0
1/ 2
( , ) ( )exp( )[ ( )]
exp( )[ ( ) ( )]
l l d t d
tl
F x y x i E y a o y
i EA o
2 2 , arctan( / )x y x y
(as and/or )y
d
0 0( , ) ( , ) 0,l x l yF x y F x y
'
1
' '
1
( , ) ( , ) ( , , ) ( ', ') 0,l x c l ll ll
x y F c y d h c y F x y
Hard-core boundary conditions:
The asymptotic condition for the helium trimer bound states
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Three-body, theory formalism
Boundary conditions
4He2 - 4He
1/ 20
/ 2
0
1
( )
(
( , ; ) ( ){sin( ) exp( )[ ( )]}
exp( )[ ( )].)
d
l
l l a pF x y p x py ip o y
EoA
y
i
2
0
Here is the dimer wave function, stands for the scattering energy given by with the dimer energy, and is the relative momentum conjugate
to the variable . The coefficient ( ) is nothing
d
d d
EE p p
y a p
but the elastic scattering amplitude,while the functions ( ) provides us, at 0, with the corresponding partial-wave Faddeev breakup amplitudes. The scattering length is given by
lA E
0
03lim
2
( )sc
p
a pl
p
The asymptotic condition for the partial-wave Faddeev components of the (2 + 1 2 + 1 ; 1 + 1 + 1) scattering wave function reads, (as and/or )y
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Three-body, theory bound state
Bound state calculations 4He3
Esry,Lin,Greene (1996)
Barnett, Walley (1993)Roudnev, Yakovlev (1999)
Panharipande et al. (1983)
Motovilov,Kolganova, Sandhas,Sofianos (1997,2001)
Carbonell,Gignoux, Merkuriev (1993)
Cornelius, Gloeckle (1986)
Lewerenz (1997)
Nielsen,Fedorov,Jensen (1998)
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Three-body, theory bound state
Bound state calculations4He3
Esry,Lin,Greene (1996)Nakaichi-Maeda, Lim (1983)
Roudnev, Yakovlev (2000)
Blume,Greene (2000)
Motovilov,Kolganova, Sandhas,Sofianos (1997,2001)
Cornelius, Gloeckle (1986)
Barletta, Kievsky (2001)
Nielsen,Fedorov,Jensen (1998)
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Three-body, theory scattering
Scattering Length calculations
Braaten,Hammer (2003)
Penkov (2003)
Roudnev (2003)Blume,Greene (2000)
2004,2005 1998Motovilov,Kolganova,Sandhas
4He - 4He2
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Three-body, theory scattering state
4He 4He2
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Three-body, theory scattering state
4He 4He2
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Three-body, theory bound state
3He 4He2
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Three-body, theory scattering3He -
4He2
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Three-body, theory scattering
Phase shifts calculations using Faddeev differential equations
4He - 4He2
Roudnev (2003)
Kolganova,Motovilov(1998,2001)
Zero-range model - Hammer et al. (1999,2003), Penkov(2003)
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Three-body, theory scattering
3He - 4He2
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Three-body, theory scattering
Partial wave-function for HFD-B potential at E=1.4mK
4He2 - 4He
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Conclusions
We employed formalism which is suitable for three-body atomic systems interacting via hard-core potentials. This method lets us calculate bound states and scattering observables.
Scattering length and phase shifts for the helium three-atomic systems have been calculated.
It was demonstrated how the Efimov states emerge from the virtual ones when decreasing the strength of the interaction.