1. MODELING OF BOUND STATES OF QUANTUM SYSTEMS IN A TWO-DIMENSIONAL GEOMETRY OF ATOMIC TRAPS Scientific supervisor Prof. Vladimir S. Melezhik, DSc 2014 ALUSHTA Oksana A. Koval Bogoliubov Laboratory of Theoretical Physics JOINT INSTITUTE FOR NUCLEAR RESEARCH UNIVERSITY CENTRE

2. 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

3. where Problem formulation 1. Problem formulation

4. FIG. 1. Potential shape Problem formulation 1. Problem formulation

5. Bound state problem in 2D geometry 2. Bound state problem in 2D geometry FIG. 2. Considered potential: potential well width harmonic oscillator frequency ω = 1, relative momentum q = 0.01.

6. that takes place at, where k – relative momentum, а – scattering length, γ - Euler's constant, 3. Atom scattering problem in a central field in 2D geometry 3. Atom scattering problem in a central field in 2D geometry where k is the relative momentum,, J 0 and N 0 – corresponding Bessel and Neumann functions, А = const..

7. The value L = has the following four features: Thus, the chosen value of L is continuously changed in the range of FIG. 3. Range of values E and L= 3. Atom scattering problem in a central field in 2D geometry 3. Atom scattering problem in a central field in 2D geometry

8. 4. Numerical methods of finding the energy levels of atom bound states and solution of the scattering problem in a 2D space 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 and seven-point finite-difference central scheme for the second order derivative: As a result, the equations of the bound state and scattering problems can be written in the following general form:

9. A good agreement between the numerical results and the analytical ones for the potential well has been obtained : 4. Numerical methods of solution of the scattering problem in a two-dimensional space 4. Numerical methods of solution of the scattering problem in a two-dimensional space

10. Solid bold line - the dependence of energy calculated in units of harmonic oscillator frequency on the value of ≡ 1/ ln(0.5/ 2 ), solid thin line - the analytical solution [1], dashed lines – asymptote of the corresponding levels in case of = 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 () [1] on the values L = 1/ ln(0.5/ 2 ) at the oscillator frequency = 1. 5. Discussion of Results 5. Discussion of Results

11. FIG. 5. Convergence in the number of → ∞ grid points in the radial variable for the scattering length ( 0 ).. A good convergence of the computing scheme depending on the number of N points on the radial variable has been obtained. 5. Discussion of Results 5. Discussion of Results

12. FIG. 6 Dependence of the energies of bound states on the frequency of the optical trap. 5. Discussion of Results 5. Discussion of Results The dependence of the bound state energies on the parameter trap – harmonic oscillator frequency ( =1,3,5,7,9) has been numerically investigated. The energy of the ground bound state increases substantially with increasing and the energies of the excited states rise insignificantly.

13. 6. Conclusions 6. Conclusions 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. The computational scheme was successfully constructed.

14. 6. References 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.

15. Thanks for your attention

16. 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 0 = 0.5) and deep (depth 0 = −200) potential well, which models zero-radius potential 0 () considered in article [1] at 0 → 0, 0 → ∞. The calculations were performed at the following parameters: = 60 and k = 0.01 on the nested radial variable grids → ∞. 5. Discussion of Results 5. Discussion of Results

17. where coefficients were expressed in terms of the coefficients of the system of linear equations. Then, using the sweep method we have calculated 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: 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: Reverse sweep was done by applying the derived recurrence relation: 4. Numerical methods of solution of the scattering problem in a two-dimensional space 4. Numerical methods of solution of the scattering problem in a two-dimensional space