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!