ABOVE BARRIER TRANSMISSION OF BOSEEINSTEIN CONDENSATES IN GROSSPITAEVSKII

ABOVE BARRIER TRANSMISSION OF BOSE-EINSTEIN CONDENSATES IN GROSSPITAEVSKII APPROXIMATION H. A. Ishkhanyan, V. P. Krainov Moscow, 06. 07. 2010

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

Potential Well T=Tcritical Temperature

The Gross-Pitaevskii equation Stationary

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

1. Reflection from the step potential An atom moves slowly oppositely to the focused laser beam pph - р pph Atom р Fig. 1. Resonant light as a potential barrier for the atom. laser 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

1. 1 The step potential (Linear case) From the matching conditions one obtains

1. 2 The step potential 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

Zero interation First interation The whole solution

• 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 µ 0> V.

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

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

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

2. 1 Rectangular barrier When For example

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

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 onedimensional Rosen-Morse potential well in the Gross-Pitaevskii problem', JETP 136(4), 1 (2009).

Rosen-Morse potential The reflection coefficient is zero Example

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

Double-Delta potential

• 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

A different approach

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 onedimensional Rosen-Morse potential well in the Gross-Pitaevskii problem', JETP 136(4), 1 (2009).

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

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 !

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 rather good approximation up to and it provides a

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.

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!

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

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

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.

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

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 onedimensional 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 (2010). And in a "World Scientific" publishing’s book entitled “ Modern Problems of Optics and Photonics”.

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)

Hayk

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