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


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


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


Potential models: Aziz et al. – HFD-B (1987), LM2M2 (1991)

Two-body, theory4He - 4He

Tang et al. – TTY (1995)


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


Three-body, theory formalism

[4] - L.D.Faddeev,S.P.Merkuriev, 1993, Quantum scattering theory for several particles


At L=0 the partial angular momentum corresponds both to the dimer and an additional atom. stand to the standard Jacobi variables.


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



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



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


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)



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)


Three-body, theory scattering

Scattering Length calculations

Braaten,Hammer (2003)

Penkov (2003)

Roudnev (2003)Blume,Greene (2000)

2004,2005 1998Motovilov,Kolganova,Sandhas

4He - 4He2


Three-body, theory scattering state

4He 4He2



3He 4He2


Three-body, theory scattering3He -

4He2



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)



3He - 4He2



Partial wave-function for HFD-B potential at E=1.4mK

4He2 - 4He


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.

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