2D Solitons in Dipolar BECs

1
2D Solitons in Dipolar BECs

• 1I. Tikhonenkov, 2B. Malomed, and 1A. Vardi
• 1Department of Chemistry, Ben-Gurion University
• 2Department of Physical Electronics, School of
  Electrical Engineering, Tel-Aviv University

2
Dilute Bose gas at low T
3
Gross-Pitaevskii description

• Lowest order mean-field theory

Condensate order-parameter
Gross-Pitaevskii energy functional

• minimize EGP under the constraint

Gross-Pitaevskii (nonlinear Schrödinger) equation
4
Variational Calculation

• Evaluation of the EGP in an harmonic trap, using
  a gaussian solution with varying width b.
• Kinetic energy per-particle varies as 1/b2 -
  dispersion.
• Nonlinear interaction per-particle varies as gn -
  g/b3 in 3D, g/b in 1D.
• In 1D with glt0, kinetic dispersion can balance
  attraction and arrest collapse.

5
Solitons

• Localized solutions of nonlinear differential
  equations.
• Result in from the interplay of dispersive terms
  and nonlinear terms.
• Propagate long distances without dispersion.
• Collide without radiating.
• Not affected by their excitations.

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Zero-temperature BEC solitons

• NLSE in 1D with attractive interactions (glt0),
  no confinement

Posesses self-localized sech soliton solutions
Bright soliton
Healing length at x0
Chemical potential of a bright soliton
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Zero-temperature BEC solitons
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Observation of BEC bright solitons
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Observation of BEC solitons
Dark solitons by phase imprinting J. Denschlag
et al., Science 287, 5450 (2000).
Bright solitons L. Khaykovich et al. Science
296, 1290 (2002).
Bright soliton train K. E. Strecker et al.,
Nature 417, 150 (2002).
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Instability of 2D solitons without
dipolar-interaction
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Dipole-dipole interaction
?????vacuum permittivity d - magnetic/electric
dipole moment
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Units
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2D Bright solitons in dipolar BECs P. Pedri and
L. Santos, PRL 95, 200404 (2005)
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Manipulation of dipole-dipole interaction

• In order to stabilize 2D solitary waves in the PS
  configuration, it is necessary to reverse
  dipole-dipole behavior, so that side-by-side
  dipoles attract each other and head-to-tail
  dipoles repell one another.

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Manipulation of dipole-dipole interaction S.
Giovanazzi, A. Goerlitz, and T. Pfau, PRL 89,
130401 (2002)

• The magnetic dipole interaction can be tuned,
  using rotating fields from Vd at ???, to -Vd/2
  at ??????
• The maximum becomes a minimum and 2D bright SWs
  can be found, provided that the dipole term is
  sufficiently strong to overcome the
  kineticcontact terms, i.e.
• Or, for

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E????? for confinement along the dipolar axis z,
gaussian ansatz, g500
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Dipolar axis in the 2D planeI. Tikhonenkov, B.
A. Malomed, and AV, PRL 100, 090406 (2008)
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Dipolar axis in the 2D plane
For gd gt 0 stable self trapping along the dipolar
axis z
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For gd gt 0, what happens along x ?
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E??????? for confinement perpendicular to the
dipolar axis
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3D Propagation and stability
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Driven Rotation
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Experimental realization
For g,gd gt 0

• 52Cr (magnetic dipole moment d6?B)
• Dipolar molecules (electric dipole of 0.1-1D)

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Conclusions

• 2D bright solitons exist for dipolar alignment in
  the free-motion plane.
• For this configuration, no special tayloring of
  dipole-dipole interactions is called for.
• The resulting solitary waves are unisotropic in
  the 2D plane, hence interesting soliton collision
  dynamics.

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Incoherent matter-wave Solitons

• 1,2H. Buljan, 1M. Segev, and 3A. Vardi
• 1Department of Physics, The Technion
• 2Department of Physics, Zagreb Univesity
• 3Department of Chemistry, Ben-Gurion University

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What about quantum/thermal fluctuations ?
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T0 - Bogoliubov theory (ask Nir)

• Want to calculate zero temperature fluctuations.
  
• Separate

• retain quadratic fluctuation terms and add N0
  constraint

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T0 - Bogoliubov theory

• Bogoliubov transformation

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Bogoliubov spectrum of a bright soliton

• linearize about a bright soliton solution

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Bogoliubov spectrum of a bright
solitonScattering without reflection

• Transmittance

• Bogoliubov quasiparticles scatter without
  reflection on
• the soliton (B. Eiermann et al., PRL 92,
  230401 (2004),
• S. Sinha et al., PRL 96, 030406 (2006)).

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Limitations on Bogoliubov theory

• The condensate number is fixed - no backreaction
• The GP energy is treated separately from the
  fluctuations

pair production
direct exchange
Due to exchange energy in collisions between
condensate particles and excitations, it may be
possible to gain energy By exciting pairs of
particles from the condensate !
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TDHFB approximation

• separate, like before

• retain quadratic terms in the fluctuations, to
  obtain coupled equations for

Condensate order-parameter
Pair correlation functions - single particle
normal and anomalous densities
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TDHFB approximation
(e.g., Proukakis, Burnett, J. Res. NIST 1996,
Holland et al., PRL 86 (2001))
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Initial Conditions - static HFB solution in a trap
Fluctuations do not vanish even at T0, quantum
fluct.
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Dynamics - TDHFB equations
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System Parameters
?

• Quasi 1D geometry

x

• Parameters close to experiment

• N 2.2 104 7Li atoms

• ?? 4907 Hz a? 1.3 µm
• ?x 439 Hz ax 4.5 µm

• Na3D -0.68 µm

• TDHFB can be used only for limited time-scales

• Tevolution?? ltlt Tcollisional?? 104

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TDHFB vs. GP
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Incoherent matter-wave solitons
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Number and energy conservation
condensate fraction
thermal population
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Conclusions

• Dynamics of a partially condensed Bose gas
  calculated via a nonlinear TDHFB model
• Noncondensed particles (thermal/quantum) affect
  the dynamics of BEC solitons
• Pairing instability - dynamical depletion of a
  BEC with attractive interactions
• Incoherent matter-wave solitons constituting both
  condensed and noncondensed particles
• Analogy with optics
• Coherent light in Kerr media ?
  zero-temperature BEC
• Partially (in)coherent light in Kerr media ?
  partially condensed BEC