solitons and shock waves in bose-einstein condensates a.m. kamchatnov*, a. gammal**, r.a....
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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
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
sa
,4 2
m
ag s
is the atom-atom scattering length,
,r|| 2 Nd
is number of atoms in the trap.N
Cigar-shaped trap
zyx 1
z
02Z
or
2a
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
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
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
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.
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.
After turning off the axial potential the condensateexpands in self-similar way:
0Z0Z
maxv tmaxv t
0t
1zt
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
Shock wave in Bose-Einstein condensate
Let the initial state have the density distribution
12 vv
1v
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.
Numerical solution of 2D Gross-Pitaevskii equation
Density profiles at y=0
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
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
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
Substitution of ( , )i z t into periodic solution gives
profile of dissipationless shock wave:
Formation of dark solitons
Let an initial profile of density have a “hole”
After wave breaking two shocks are formed whichdevelop eventually into two soliton trains:
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
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.
Results of 3D numerics
1D cross sections of density distributions
Whitham theory
Thank you for your attention!