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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
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
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)
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
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
Rosen-Morse Potential Rectangular barrier
Conclusion, future directions
0))((2
2
2
22
gxVxmt
i
The Gross-Pitaevskii equation
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
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
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)
• 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 Gross-Pitaevskii equation
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
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
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!