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

ABOVE BARRIER TRANSMISSION OF BOSE-EINSTEIN

CONDENSATES IN GROSS-PITAEVSKII APPROXIMATION

Moscow, 06.07.2010

H.A. Ishkhanyan, V.P. Krainov

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

Temperature

T=Tcritical

Potential Well

The Gross-Pitaevskii equation

22

2

1( ) | | .

2

di V x

t dx

22

2

1( ) | | .

2

dV x E

dx

Stationary

1. Reflection from the step potential

An atom moves slowly oppositely to the focused laser beam

Resonant laser presents an one-dimensional potential barrierHartree approximation. Resonant impulse

р

pph

pph - р

Atom

laserFig. 1. Resonant light as a potential barrier for the atom.

For real optical laser frequency and mass of atom the kinetic energy is of the order of K1

frequency of transition to the first excited state 21 L

1.1 The step potential

From the matching conditions one obtains

(Linear case)

1.2 The step potentialThe 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 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.

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

0))((2

2

2

22

gxVxmt

i


The Problem

Time-independent GPE

0))((2

1 2

2

2

gxV

dx

d

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).

1)()(22

With the boundary conditions

Rosen-Morse potential

Example

The reflection coefficient

is zero

Double-Delta potential

• H.A. Ishkhanyan and V.P. Krainov, Phys. Rev. A (2009)

• H.A. Ishkhanyan and V.P. Krainov, JETP 136(4), 1 (2009)

• H.A. Ishkhanyan and V.P. Krainov, Laser Physics 19(8), 1729 (2009)

[email protected]

• 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)

A different approach

0))((2

2

2

22

gxVxmt

i


The Problem

Time-independent GPE

0))((2

1 2

2

2

gxV

dx

d

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).

1)()(22

With the boundary conditions

)(xueikxThe solution

A bit of mathematics

We have a quasi-linear eigenvalue problem for the potential depth that we formulate in the following operator form

)(ˆ0 ugFuVuH L 1)()( uu

uzz

uuF

)1(4

1)(

2

where

Reflectionless transmissiong=0

);1;1,(12 zikFu

Reflectionless transmission if the condition is satisfied1)( u

n

)1(2

1 nnVLn

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 !

The linear part is the hypergeometric equation

024

1

2

1V,where

)(ˆ0 ugFuVuH L 1)()( uu

Since the solution to the linear problem is known, it is straightforward to apply the Rayleigh-Schrödinger perturbation theory

1

0

1 )()1(1

zduuFzzC

V LnLnikik

n

..22

10 VggVVVV LnNLn

Then one obtains

The derived formula is highly accurate if and it provides a rather good approximation up to )75.05.0( g

Reflectionless transmissiong≠0

...22

1 uguguuu LnNLn

25.0g

• The dependence of onLnNLn VVV 1 )(2 gk

Fig. 1. The nonlinear shift of the resonance position vs. the wave vector .

Fig. 1. The nonlinear shift of the resonance position vs. the wave vector .

)(21 g

ngVV LnNLn

Resonance position shift is approximately equidistant

•For each fixed the separation between the curves is approximately equidistant!

is shown in Fig. 1.

• Remarkably simple structure

• In this case may be positive – barriers!

NLnV

6n

1n

k

Note that for an integer n the function is a polynomial in z. Lnu

Hence, the integral can be analytically calculated for any given order n

)(2111 g

gVNL

)(24

2

)(21

1

7

932 gg

gVNL

A remarkable observation is that the formula for the first resonance, interestingly, turns out to be exact!

Calculation of the integral

1

0

1 )()1(1

zduuFzzC

V LnLnikik

n

Rectangular barrier

10,

1and0,0

0 xV

xxV

2)(

22ngVLn

• Transmission resonances in the linear case

1

0

2)1( xd

C

gVV LnLnLn

nLnNLn

22

2

22

1

n

kCn

•The shift

The final result for the nonlinear resonance position reads

22

231

4 n

kgVV LnNLn

•The immediate observation is that for the rectangular barrier the nonlinear shift of the resonance position is approximately constant!

,where

• An assymetric potential4

gVV LnNLn

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

)1/(22

• 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 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).

PublicationsSome parts of the problem are already published

• 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”.

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 Hayk IshkhanyanIshkhanyan

Thank You For Attention!

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

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