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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




Rosen-Morse Potential
Rectangular barrier
Conclusion, future directions
Potential Well
T=Tcritical
Temperature
The Gross-Pitaevskii
equation

1 d 2
2
i


V
(
x
)



|

|
.
2
t
2 dx
Stationary
1d
2

 V ( x)   |  |   E .
2
2 dx
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
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.
L  21 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K
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  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
The Problem
The Gross-Pitaevskii equation

 2  2
2
i

 (V ( x)  g  )  0
2
t
2m x
Time-independent GPE
1 d 2
2

 (  V ( x)  g  )  0
2
2 dx
With the boundary conditions
 ()   ()  1
2
2
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




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 BoseEinstein condensate by a double barrier within the Gross-Pitaevskii
equation”, xxx Physica Scripta (2010) (in press)
[email protected]
A different approach
The Problem
The Gross-Pitaevskii equation

 2  2
2
i

 (V ( x)  g  )  0
2
t
2m x
Time-independent GPE
1 d 2
2

 (  V ( x)  g  )  0
2
2 dx
With the boundary conditions
 ()   ()  1
2
2
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
  eikxu (x)
We have a quasi-linear eigenvalue problem for the potential
depth that we formulate in the following operator form
ˆ u  V u  gF (u)
H
L
0
where
u 1
2
F (u ) 
4 z ( z  1)
u
u()  u()  1
Reflectionless transmission
g=0
The linear part is the hypergeometric equation
u 2 F1 ( , 1   ; 1  ik ; z )
,where
 
1

2
1
 2V0
4
Reflectionless transmission if the condition u()  1 is satisfied
   n
The corresponding transmission resonances are then achieved for
VLn  
1
n( n  1)
2
As it is immediately seen, reflectionless transmission in the linear
case is possible only for potential wells !
Reflectionless transmission
g≠0
ˆ u  V u  gF (u)
H
L
0
u()  u()  1
Since the solution to the linear problem is known, it is straightforward to
apply the Rayleigh-Schrödinger perturbation theory
u  u NLn  u Ln  g u1  g 2u2  ...
V0  VNLn  VLn  gV1  g 2V2  ..
Then one obtains
1
1 ik
V1   z (1  z ) ik F (uLn ) uLn d z
Cn 0
The derived formula is highly accurate if g  0.25 and it provides
a rather good approximation up to g  (0.5  0.75) 
Resonance position shift is
approximately equidistant
• The dependence of V1  VNLn  VLn on k  2(  g ) is shown in Fig. 1.
•For each fixed k the separation between the curves is
approximately equidistant!
VNLn  VLn
n6
gn

1  2(   g )
• Remarkably simple structure
• In this case VNLn may be
positive – barriers!
n 1
Fig. 1. The nonlinear shift of the resonance position
vs. the wave vector .
Calculation of the integral
1
1 ik
V1   z (1  z ) ik F (uLn ) uLn d z
Cn 0
Note that for an integer n the function u Ln is a polynomial in z.
Hence, the integral can be analytically calculated for any given order n
g
VNL1  1 
1  2(   g )
VNL 2

9g 
1
2


 3 

7  1  2(   g ) 4  2(   g ) 
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




Rosen-Morse Potential
Rectangular barrier
Conclusion, future directions
Rectangular barrier
0 , x  0 and
V 
V0 , 0  x  1
x 1
• Transmission resonances in the linear case VLn  (   g ) 
•The shift V  V  g
NLn
Ln
 2n2
1

2
2
1
k
( Ln  1) Ln Ln d x ,where Cn  
Cn 0
2 2 2 n 2
The final result for the nonlinear resonance position
2
reads

g
3k 
2
VNLn  VLn  1  2 2 
4  n 
•The immediate observation is that for the rectangular barrier the
nonlinear shift of the resonance position is approximately
g
constant!
VNLn  VLn 
• An assymetric potential
4
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
 /(1    ) )
• … Stability of the resonances
2
2
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 RosenMorse 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
Thank You For Attention!