Solitons and vortex rings are examples of
nonlinear wave phenomena, which maintain their shape during propagation.
The collisions of such waves were
recently observed in a Bose-Einstein condensate for the first time. The
experiment at Harvard University showed evidence of unexpected
shell-like structures.
Simulations give evidence that these structures are hybride objects
composed of soliton fronts and vortex rings.
The 1D Bose gas at zero temperature has many surprising properties that are very distinct from 3D Bose-Einstein condensates including the loss of phase coherence and a fermionic excitation spectrum with appearance of two branches of elementary excitations. We approach the subject by making use of an exact mapping of the 1D Bose system to an interacting Fermi gas and by applying fermionic density functional theory in a local density approximation. An appealing feature of this approach is that the approximations become exact in the limit of strong interactions in the bosonic system and thus both the fermionic excitation spectrum and phase coherence properties are recovered. A time-dependent version of this theory may be used to simulate the time-dependent dynamics as it may become important in narrow atomic guides, e.g. on atom chips. In this project, we can make a connection between well established approximation methods from the field of quantum chemistry and solid state physics to the study of 1D quantum degenerate gases. Another promising direction is the development of a many-body perturbation theory starting from the strongly interacting limit, which is equivalent to a non-interacting Fermi gas.
When attractive Bose-Einstein condensates are confined to cigar-shaped harmonic traps they can avoid collapse and show features previously known from non-linear fiber optics like modulational instability and bright solitons. Motivated by the recent creation of the first matter-wave bright soliton train [Strecker et al. Nature 417 150 (2002)], we study the effect of modulational instability in inhomogeneous density profiles and external trapping potentials. Also the dynamics of bright solitons in a harmonic trap is considered.
Phys.
Rev. Lett. 92, 040401 (2004)
Phys.
Rev. A. 70, 033607 (2004)
Levinson's theorem of potential scattering connects the number of bound states of a given potential to the phase shifts of scattering solutions. Excitations of a weakly interacting Bose condensate are described by the coupled Bogoliubov equations. Scattering solutions for a finite trapping potential describe the scattering of single, identical particles. Can Levinson's theorem be generalized and the number of bound collective excitations of a condensate be linked with the phase shifts of single-particle scattering? Particularly interesting situations occur for kinks or vortices in shallow traps as they can give rise to 'bound states in the continuum' of single particle scattering.
Phys. Rev. Lett. 91 , 070403 (2003)
What happens to vortices when they are put into
a
narrow channel? What is the 1D analog of a vortex? 
We study the effects of transverse confinement on vortices in a
repulsive,
elongated BEC. In a regime where the width of the elongated trap is
about
6 to 12 healing lengths, vortices show properties usually associated
with
solitons. In particular, their velocity may be associated with a
characteristic phase step and collision properties are soliton-like.
Thus we speak of solitonic
vortices. A connection can be made to the snake instability of soliton
stripes
which is also a mechanism that may be exploited to experimentally
create
solitonic vortices [1
]. A different way of producing solitonic vortices in a controlled
manner
is stirring in a toroidal trap [2
]. Properties are studied using 2 and 3D simulations of the
Gross-Pitaevskii equation but also using the method of image charges
which yields an exactly
solvable model of vortex dynamics.
[1] J. Brand and W. P. Reinhardt, Phys. Rev. A, 65 (2002) 043612
[2] J. Brand and W. P. Reinhardt, J. Phys. B: At. Mol. Opt. Phys., 34 (2001) L113-L119
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