Solitons and shock waves in Bose-Einstein condensates

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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
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Gross-Pitaevskii equation

• Dynamics of a dilute condensate is described
• by the Gross-Pitaevskii equation

where
3
is the atom-atom scattering length,
is number of atoms in the trap.
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Cigar-shaped trap
or
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If
then transverse motion is frozen and the
condensate wave function can be factorized
where is a harmonic oscillator
ground state function of transverse motion
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The axial motion is described by the equation
where
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Disc-shaped trap
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Quasi-one-dimensional expansion

• Hydrodynamic-like variables are introduced by

where
is density of condensate and
is its velocity.
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In Thomas-Fermi approximation the
stationary state is described by the distributions
where
is axial half-length of the condensate.
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After turning off the axial potential the
condensate expands in self-similar way
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Analytical solution is given by
where
has an order of magnitude of the sound
velocity in the initial state
is the density of the condensate.
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Shock wave in Bose-Einstein condensate

• Let the initial state have the density
  distribution

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A formal hydrodynamic solution has wave breaking
points
Taking into account of dispersion effects leads
to generation of oscillations in the regions
of transitions 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 y0
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Analytical theory of shocks

• The region of oscillations is presented as a
• modulated periodic wave

where
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The parameters
change
slowly along the shock. Their evolution is
described by the Whitham modulational equations
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Solution of Whitham equations has the form
where functions
are determined by the
Initial conditions. This solution defines
implicitly
as functions of
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into periodic solution gives
Substitution of
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
which develop eventually into two soliton trains
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Analytical form of each emerging soliton is
parameterized by an eigenvalue
where
can be found with the use of the
generalized Bohr-Sommerfeld quantization rule
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Formation of solitons in BEC with attractive
interaction
Solitons are formed due to modulational
instability. If initial distribution of density
has sharp fronts, then Whitham 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!