solitons and shock waves in bose-einstein condensates a.m. kamchatnov*, a. gammal**, r.a....
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![Page 1: Solitons and shock waves in Bose-Einstein condensates A.M. Kamchatnov*, A. Gammal**, R.A. Kraenkel*** *Institute of Spectroscopy RAS, Troitsk, Russia **Universidade](https://reader035.vdocument.in/reader035/viewer/2022062407/56649d565503460f94a34a8d/html5/thumbnails/1.jpg)
Solitons and shock waves in Bose-Einstein condensates
A.M. Kamchatnov*, A. Gammal**, R.A. Kraenkel***
*Institute of Spectroscopy RAS, Troitsk, Russia
**Universidade de São Paulo, São Paulo, Brazil
***Instituto de Física Teórica, São Paulo, Brazil
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Gross-Pitaevskii equation
Dynamics of a dilute condensate is described
by the Gross-Pitaevskii equation
22(r) | |
2 exti V gt m
where
)(2
)r( 222222 zyxm
V zyxext
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sa
,4 2
m
ag s
is the atom-atom scattering length,
,r|| 2 Nd
is number of atoms in the trap.N
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Cigar-shaped trap
zyx 1
z
02Z
or
2a
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If
10
Z
Nas
then transverse motion is “frozen” and the condensate wave function can be factorized
),(),(),r( tzyxt where is a harmonic oscillator ground state function of transverse motion:
( , )x y
.2
exp1
),(22
a
yx
ayx
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The axial motion is described by the equation
2 22 2 2
12
1| |
2 2 z Di m z gt m z
where2
1 2 2
2,
2s
D
agg
a ma
,am
2| | .dz N
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Disc-shaped trap
1,z
(r, ) ( ) ( , , ),t z x y t
2
1/ 4 1/ 2 2
1( ) exp( ),
2z z
zz
a a
22 2 2 2 2
2
1( ) | |
2 2 x y Di m x y gt m
2
2
2 2,
2s
Dzz
agg
maa
2| | .dxdy N
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Quasi-one-dimensional expansion
Hydrodynamic-like variables are introduced by
( , ) ( , ) exp ( ', ) ' ,zi
z t z t v z t dz
where ( , )z t is density of condensate and
( , )v z t is its velocity.
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In Thomas-Fermi approximation the stationarystate is described by the distributions
2
20 0
3( ) 1 ,
4
N zz
Z Z
0v
2 2 1/30 (3 )sZ Na a
where
is axial half-length of the condensate.
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After turning off the axial potential the condensateexpands in self-similar way:
0Z0Z
maxv tmaxv t
0t
1zt
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Analytical solution is given by2
2 2max max
3( , ) 1 ,
4
N zz t
v t v t
1,zt
max 02 zv Z where
has an order of magnitude of the sound velocityin the initial state:
max 12 ,s sv c a nm
2
1 ,n a n
is the density of the condensate.n
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Shock wave in Bose-Einstein condensate
Let the initial state have the density distribution
12 vv
1v
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A formal hydrodynamic solution has wave breaking points:
zTaking into account of dispersion effects leads to generation of oscillations in the regions oftransitions from high density to low density gas.
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Numerical solution of 2D Gross-Pitaevskii equation
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Density profiles at y=0
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Analytical theory of shocks
The region of oscillations is presented as amodulated periodic wave:
21 2 3 4
21 2 3 4 1 3 2 4
1( , ) ( )
4
( )( ) (2 ( )( ) , ),
z t
sn m
where
1 2 3 4( ) ,z t 1 2 3 4
1 3 2 4
( )( ).
( )( )m
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The parameters change( , ), 1, 2,3,4,i i z t i slowly along the shock. Their evolution is described by the Whitham modulational equations
( ) 0,i iit x
( ) 1 ,i ii
LV
L
,ii
,iV 1 3 2 4
( ).
( )( )
K mL
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Solution of Whitham equations has the form
( ) ( ), 1, 2,3,4,i ix t w i
where functions ( )iw are determined by the
Initial conditions. This solution defines implicitly
i as functions of , :x t
t const
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Substitution of ( , )i z t into periodic solution gives
profile of dissipationless shock wave:
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Formation of dark solitons
Let an initial profile of density have a “hole”
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After wave breaking two shocks are formed whichdevelop eventually into two soliton trains:
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Analytical form of each emerging soliton is parameterized by an “eigenvalue” n
2(0) 0
0 2 20
( , ) ,cosh [ ( 2 )]
n
n n
z tz t
where n can be found with the use of the
generalized Bohr-Sommerfeld quantization rule
21 1( ,0) ( ,0) , 0,1,2,...
2 2
n
n
z
n
z
v z z dz n n
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Formation of solitons in BEC with attractive interaction
22| |
2i g
t m
Solitons are formed due to modulational instability.If initial distribution of density has sharp fronts, thenWhitham analytical theory can be developed.
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Results of 3D numerics
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1D cross sections of density distributions
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Whitham theory
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Thank you for your attention!