Download Matter-wave solitons beyond 1D contact-interaction mean

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
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Dilute Bose gas at low T
Contact pseudopotential
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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:
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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 g<0, kinetic dispersion can balance attraction and arrest collapse.
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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 (g<0), no confinement
Posesses self-localized sech soliton solutions:
Bright soliton:
Healing length at x=0
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Chemical potential of a bright soliton
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Zero-temperature BEC solitons
Attractive interactions,
g0
No interactions,
matter wave dispersion
time
time
(self-focusing nonlinearity)
g 0
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Observation of BEC bright solitons
(1) Prepare BEC (static)
in the trap
(3) Turn off both the
trap and interactions
(Feshbach mechanism)
(2) Turn off the trap
and let evolve
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L. Khaykovich et al.
Science 296, 1290 (2002).
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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
- characteristic width of a 2D BEC wavefunction
is monotonic in
expansion
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collapse
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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
• The total dipolar interaction
is attractive at L< Lz and
repulsive at L> Lz. There is
a maximum in E(L, hence
no soliton.
• In order to stabilize 2D solitary waves in the PS
configuration, it is necessary to reverse dipoledipole 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 kinetic+contact terms, i.e.
•
Or, for
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E for confinement along the dipolar axis
z, gaussian ansatz, g=500
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Dipolar axis in the 2D plane
I. Tikhonenkov, B. A. Malomed, and AV, PRL 100, 090406 (2008)
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Dipolar axis in the 2D plane
For gd > 0 stable self trapping along the dipolar axis z:
y
z
x
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For gd > 0, what happens along x ?
y
y
z
z
x
x
Self trapping along x is enabled by the interplay of 1/Lx2
kinetic dispersion and -1/Lx dipolar attraction
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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
Deviation from /2 rotated soliton at t= /2
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Experimental realization
For g,gd > 0 :
•
52Cr
(magnetic dipole moment d=6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
of Physics, Zagreb Univesity
3Department of Chemistry, Ben-Gurion University
2Department
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What about quantum/thermal fluctuations ?
Trap OFF →
nonequilibrium dynamics
Prepared (static) BEC
partially condensed
Condensed particles
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Thermal cloud
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BEC-soliton dynamics affected by
(1) Thermal cloud (and vice versa)
(2) Condensate depletion during dynamics
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T=0 - Bogoliubov theory (ask Nir)
• Want to calculate zero temperature fluctuations.
• Separate:
condensate fluctuations
• retain quadratic fluctuation terms and add N0 constraint:
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T=0 - Bogoliubov theory
• Bogoliubov transformation:
v(x)
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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 soliton
Scattering 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
direct + exchange
pair production
no 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

Heisenberg eq. of motion for the Bose field operator
ˆ(x,t)
Fluctuations

separate, like before
Condensate
 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))
Condensate density
Normal noncondensate density
Anomalous noncondensate density
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Initial Conditions - static HFB solution in a
trap
Bose distribution
Fluctuations do not vanish even at T=0, quantum fluct.
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Dynamics - TDHFB equations
Initial conditions:
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System Parameters

 Quasi 1D geometry
 Parameters close to experiment:
x
 N = 2.2 104 7Li atoms
 ω = 4907 Hz ; a = 1.3 μm
 ωx = 439 Hz ;
ax = 4.5 μm
 Na3D = -0.68 μm
 TDHFB can be used only for limited time-scales:
 Tevolutionω << Tcollisionalω ~ 104
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TDHFB vs. GP
GPE evolution,
mechanical stability
Without interactions
matter wave dispersion
TDHFB: pairing
PRL 80,
180401 (2005)
Dynamical condensate
depletion
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Incoherent matter-wave solitons
Correlations
Mixture of condensed and
noncondensed atoms
Re μ(x1,x2,t=0)
Re μ(x1,x2,t)
Im μ(x1,x2,t)
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Number and energy conservation
Number conservation
Energy conservation
condensate fraction
condensate kinetic energy
thermal population
thermal cloud kinetic energy
total interaction energy
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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 Ξ zero-temperature BEC
Partially (in)coherent light in Kerr media Ξ partially condensed BEC
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