ABOVE BARRIER TRANSMISSION OF BOSE-EINSTEIN CONDENSATES IN GROSS-PITAEVSKII APPROXIMATION

1. ABOVE BARRIER TRANSMISSION OF BOSE-EINSTEIN CONDENSATES IN GROSS-PITAEVSKII APPROXIMATION H.A. Ishkhanyan, V.P. Krainov Moscow, 06.07.2010

2. Outline • Introduction • Reflection from the step potential • The rectangular barrier • Rosen-Morse Potential • Double delta barrier • A different approach • Rosen-Morse Potential • Rectangular barrier • Conclusion, future directions

3. Bose-Einstein Condensation Atom Trapping and Cooling Potential Well T=Tcritical Temperature

4. The Gross-Pitaevskii equation Stationary

5. Outline • Introduction • Reflectionfrom the steppotential • The rectangularbarrier • Rosen-Morse Potential • Double delta barrier • A differentapproach • Rosen-Morse Potential • Rectangularbarrier • Conclusion, future directions

6. 1. Reflection from the step potential An atom moves slowly oppositely to the focused laser beam pph - р pph Atom р laser Fig. 1. Resonant light as a potential barrier for the atom. frequency of transition to the first excited state Hartree approximation. Resonant impulse Resonant laser presents an one-dimensional potential barrier For real optical laser frequency and mass of atom the kinetic energy is of the order of

7. 1.1 The step potential (Linear case) From the matching conditions one obtains

8. 1.2The steppotential The stationary Gross-Pitaevskii equation In the left region we do not have such a simple solution, so we use the multiscale analysis Considering only linear terms with respect to a

9. Zero interation First interation The whole solution

10. When m increases, the role of nonlinearity diminishes • Oppositely, for repulsive nonlinearity transmission through barrier begins not when µ = V, but for the definite energy µ0> V.

11. An example The probability density The phase of wave function H.A. Ishkhanyan and V.P. Krainov, Laser Physics 19(8), 1729 (2009)

12. Outline • Introduction • Reflectionfrom the steppotential • The rectangularbarrier • Rosen-Morse Potential • Double delta barrier • A differentapproach • Rosen-Morse Potential • Rectangularbarrier • Conclusion, future directions

13. An example The probability density The phase of wave function H.A. Ishkhanyan and V.P. Krainov, Laser Physics 19(8), 1729 (2009)

15. 2.1 Rectangular barrier When For example

16. Outline • Introduction • Reflectionfrom the steppotential • The rectangularbarrier • Rosen-Morse Potential • Double delta barrier • A differentapproach • Rosen-Morse Potential • Rectangularbarrier • Conclusion, future directions

17. The Problem The Gross-Pitaevskii equation Time-independent GPE With the boundary conditions The case of the first resonance • H.A. Ishkhanyan and V.P. Krainov, 'Resonance reflection by the one-dimensional Rosen-Morse potential well in the Gross-Pitaevskii problem', JETP 136(4), 1 (2009).

18. Rosen-Morse potential The reflection coefficient is zero Example

19. Outline • Introduction • Reflectionfrom the steppotential • The rectangularbarrier • Rosen-Morse Potential • Double delta barrier • A differentapproach • Rosen-Morse Potential • Rectangularbarrier • Conclusion, future directions

20. Double-Delta potential

21. H.A. Ishkhanyan and V.P. Krainov, Laser Physics 19(8), 1729 (2009) • H.A. Ishkhanyan and V.P. Krainov, Phys. Rev. A (2009) • H.A. Ishkhanyan and V.P. Krainov, JETP 136(4), 1 (2009) • V.P. Kraynov and H.A. Ishkhanyan, “Resonant reflection of Bose-Einstein condensate by a double barrier within the Gross-Pitaevskii equation”, xxx Physica Scripta (2010) (in press) hishkhanyan@gmail.com

22. A different approach

23. The Problem The Gross-Pitaevskii equation Time-independent GPE With the boundary conditions The case of the first resonance • H.A. Ishkhanyan and V.P. Krainov, 'Resonance reflection by the one-dimensional Rosen-Morse potential well in the Gross-Pitaevskii problem', JETP 136(4), 1 (2009).

24. A bit of mathematics The solution We have a quasi-linear eigenvalue problem for the potential depth that we formulate in the following operator form where

25. Reflectionless transmission g=0 The linear part is the hypergeometric equation ,where Reflectionless transmission if the condition is satisfied The corresponding transmission resonances are then achieved for As it is immediately seen, reflectionless transmission in the linear case is possible only for potential wells!

26. Reflectionless transmission g≠0 Since the solution to the linear problem is known, it is straightforward to apply the Rayleigh-Schrödinger perturbation theory Then one obtains The derived formula is highly accurate if and it provides a rather good approximation up to

27. Resonance position shift is approximately equidistant • The dependence of on is shown in Fig. 1. • For each fixed the separation between the curves is approximately equidistant! • Remarkably simple structure • In this case may be positive – barriers! Fig. 1. The nonlinear shift of the resonance position vs. the wave vector .

28. Calculation of the integral Note that for an integer n the function is a polynomial in z. Hence, the integral can be analytically calculated for any given order n A remarkable observation is that the formula for the first resonance, interestingly, turns out to be exact!

29. Outline • Introduction • Reflectionfrom the steppotential • The rectangularbarrier • Rosen-Morse Potential • Double delta barrier • A differentapproach • Rosen-Morse Potential • Rectangularbarrier • Conclusion, future directions

30. Rectangular barrier • Transmission resonances in the linear case • The shift ,where The final result for the nonlinear resonance position reads • The immediate observation is that for the rectangular barrier the nonlinear shift of the resonance position is approximately constant! • An assymetric potential

31. Results, Conclusions • Reflection coefficients of Bose-Einstein condensates from four potentials are obtained. In some cases the exact analytical solutions are obtained. • For the higher order resonances the onlinear shift of the resonance potential depth is determined within a modified Rayleigh-Schrödinger theory. • Resonance position shift is approximately equidistant in the case of R-M potential and constant for the rectangular barrier.

32. Future Directions • ... Other potentials, other governing equations (e.g., assymetric potential), • …Other types of nonlinearities (e.g., saturation nonlinearity ) • … Stability of the resonances

33. Publications Some parts of the problem are already published • H.A. Ishkhanyan and V.P. Krainov, 'Above-Barrier Reflection of Cold Atoms by Resonant Laser Light within the Gross-Pitaevskii Approximation', Laser Physics 19(8), 1729 (2009). • H.A. Ishkhanyan and V.P. Krainov, 'Resonance reflection by the one-dimensional Rosen-Morse potential well in the Gross-Pitaevskii problem', JETP 136(4), 1 (2009). • H.A. Ishkhanyan and V.P. Krainov, 'Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem', PRA 80, 045601 (2009). • H.A. Ishkhanyan, V.P. Krainov, and A.M. Ishkhanyan, Transmission resonances in above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential ', J. Phys. B 43, 085306 , J. Phys. B 43, 085306 (2010). And in a "World Scientific" publishing’s book entitled “ Modern Problems of Optics and Photonics”.

34. And some more are in press • H.A. Ishkhanyan, V.P. Krainov “Higher order transmission resonances in above-barrier reflection of ultra-cold atoms”, European Physical Journal D, xxx (2010)(in press) • H.A. Ishkhanyan “Higher order above-barrier resonance transmission of cold atoms in the Gross-Pitaevskii approximation”, Proc. of Intl. Advanced Research Workshop MPOP-2009, Yerevan, Armenia, xxx (2010) (in press). • V.P. Kraynov and H.A. Ishkhanyan “The reflection coefficient of Bose-Einstein condensate by a double delta barrier within the Gross-Pitaevskii equation”, xxx Laser Physics (2010) (submitted)

35. Hayk Ishkhanyan

36. Thank You For Attention!