Download Oksana A. Koval

JOINT INSTITUTE FOR NUCLEAR RESEARCH
UNIVERSITY CENTRE
MODELING OF BOUND
STATES OF QUANTUM
SYSTEMS IN
A TWO-DIMENSIONAL
GEOMETRY OF ATOMIC
TRAPS
Oksana A. Koval
Scientific supervisor
Prof. Vladimir S. Melezhik, DSc
Bogoliubov Laboratory of
Theoretical Physics
2014
ALUSHTA
OUTLINE:
1. Problem formulation
2. Bound state problem in a two-dimensional (2D) geometry
3. Atom scattering problem in a central field in 2D geometry
4. Numerical methods of finding the energy levels of atom
bound states and solution of the scattering problem in a
two-dimensional space
5. Discussion of results
6. Conclusions
The work was supported by the Russian Foundation for Basic Research, grant 14-02-00351
1. Problem formulation
H0 (  ,  )(  ,  )  V (  )  Vho (  ) (  ,  )  E(  ,  ),
where Vho (  ) 
 2  2
2
, 
 1.
H 0 (  , )


 1     1 2



   2 2  2 V (  )  Vho (  )   (  ,  )  2 E  (  ,  )

     





1
(  , ) 
  m (  )eim

m
 d2

1
m2  
2 2
im


2
E

U
(

)





(

)
e
0



m

2
2
2 
4
 
 m  d

1
1. Problem formulation
 d2

1 
2 2
 2  2 E  U (  )     2   (  )  0
4  

 d
U (  )  2V (  )
U 0 ,   0
U ( )  
0,   0
FIG. 1. Potential shape
2. Bound state problem in 2D geometry
 2 d2
1

2
2 2


(

)



(2
E




U
(

))
 (  )  0,

2


4

 d
 (0)   ()  0,

FIG. 2. Considered
potential:
potential well width
0  0.5
harmonic oscillator
frequency
ω = 1,
relative momentum
q = 0.01.
.
3. Atom scattering problem in a central
field in 2D geometry
 2 
d2
1 
(  )   k  U (  )  2   (  )  0
2
d
4  


 (  ) 
 A ,
 0
 (  ) 
 cos( 0 )  k  J 0 (k  )  tg ( 0 ) k  N 0 (k  )  ,
 
where k is the relative momentum,k  2E , J0 and N0 – corresponding Bessel and
Neumann functions,
А = const.
ctg ( 0 ) 
2   ka 

2
ln



O
(
k
)
 


  2 
that takes place at k  0, where k – relative momentum , а – scattering
length, γ - Euler's constant ,   0,5772156649
a
2


exp  ctg ( 0 )   
k
2

3. Atom scattering problem in a central
field in 2D geometry
1
The value L = 1/ ln  2  has the following four features:
 2a 
1
 0; L  
2
1
II )a 
 0; L  
2
III )a  ; L  0
IV )a  0; L  0
I )a 

2
FIG. 3. Range of values E and L= 1/ ln 1/ (2a )
Thus, the chosen value of L is continuously changed in the
range of ( ,  )

4. Numerical methods of finding the energy levels of
atom bound states and solution of the scattering
problem in a 2D space
For the calculations we have applied a uniform grid
 max  0


, k  0,1, 2...N  1
  k  0  kh, h 
N


 ( 
k
)



k
and seven-point finite-difference central scheme for the second order
derivative:
d 2 k 2 k 3  27 k 2  270 k 1  490 k  270 k 1  27 k  2  2 k 3
 
2
d
180h2
As a result, the equations of the bound state and scattering problems
can be written in the following general form:
Ak  k 3  Bk  k 2  Ck  k 1  Dk  k  Ek  k 1  Fk  k  2  Gk  k 3  0
4. Numerical methods of solution of the
scattering problem in a two-dimensional space
A good agreement between the numerical results and the analytical
ones for the potential well has been obtained:
U 0 ,   0
U ( )  
0,   0
tg ( 0 ) 
kJ1 (kr ) J 0 (kin r )  kin J1 (kin r ) J 0 ( kr )
;
kJ 0 (kin r ) N1 (kr )  kin J1 (kin r ) N 0 (kr )
kin  (2 E  U 0 ), E  U 0
k  2E
5. Discussion of Results
Solid bold line - the
dependence of energy
calculated in units of
harmonic oscillator
frequency 𝜔 on the value
of 𝐿 ≡ 1/ ln(0.5/𝑎2),
solid thin line - the
analytical solution [1],
dashed lines –
asymptote of the
corresponding levels in
case of 𝜔 = 1. The
lower curve corresponds
to the ground level and
the other curves
correspond to the
excited states of the
system
FIG 4. Dependence of the calculated energy spectrum E of the bound states
and obtained analytical curves 𝐸(𝐿) [1] on the values L = 1/ ln(0.5/𝑎2) at
the oscillator frequency 𝜔 = 1.
5. Discussion of Results
A good convergence of the computing scheme depending on the number of N
points on the radial variable has been obtained.
FIG. 5. Convergence in the number of 𝑁 → ∞ grid points in the
radial variable for the scattering length 𝑎(𝑉0).
5. Discussion of Results
FIG. 6 Dependence of the energies 𝐸 of bound states on the
frequency 𝜔 of the optical trap.
•
The dependence of the
bound state energies on
the parameter trap –
harmonic
oscillator
frequency
𝜔
(𝜔
=1,3,5,7,9)
has
been
numerically
investigated.
•
The energy of the
ground
bound
state
increases substantially
with increasing 𝜔 and
the energies of the
excited
states
rise
insignificantly.
6. Conclusions
The computational scheme was successfully constructed.
A good agreement with the analytical results of work [1]
as verification of the algorithm was obtained.
The numerical algorithm can be easily applied to a more
realistic Lennard-Jones potential in future investigations.
6. References
1.Busch, Th., Two Cold Atoms in a Harmonic Trap. / Th. Busch, B.-G. Englert, K.
Rzazewski, M. Wilkens. // Foundation of Physics. – 1998. – Vol. 28. – P.549-559.
2. Koval E.A., Koval O.A., Melezhik V.S., Anisotropic quantum scattering in two
dimensions/ PhysRevA.89.052710 (2014)
3. Kоваль О.А., Коваль Е.А., Моделирование связанных состояний квантовых систем
в двумерной геометрии атомных ловушек, Вестник РУДН – Серия «Математика,
информатика, физика» - №2 – стр.369 - 374.
1.Busch, Th., Two Cold Atoms in a Harmonic Trap. / Th. Busch, B.-G. Englert, K.
Rzazewski, M. Wilkens. // Foundation of Physics. – 1998. – Vol. 28. – P.549-559.
Thanks for your attention
5. Discussion of Results
The objective was achieved for the superposition of
potential well (Fig. 1) and harmonic oscillator that has been
illustrated in Fig. 2. We have considered narrow (width of the
potential well 𝜌0 = 0.5) and deep (depth 𝑉0 = −200) potential
well, which models zero-radius potential 𝑉0𝛿(𝜌) considered in
article [1] at 𝜌0 → 0, 𝑉0 → ∞.
The calculations were performed at the following
parameters:
𝜌𝑁 = 60 and k = 0.01 on the nested radial
variable grids 𝑁 → ∞.
4. Numerical methods of solution of the
scattering problem in a two-dimensional space
In order to solve this problem the algorithm that employed the idea
of recurrence relations for the sweep method for a seven-diagonal band
matrix has been used:
 k  pk 1 k 1  qk 1 k  2  sk 1 k 3  rk 1 ,  k , k  0, N  1
where coefficients pk , qk , sk , rk were
of the system of linear equations.
expressed in terms of the coefficients
Reverse sweep was done by applying the derived recurrence
relation:
 N  rN 1.
Then, using the sweep method we have calculated tg ( 0 )