Durham Atomic and Molecular Physics 
IntroductionWe are investigating the phenomenon of bright matterwave solitons, which have been observed in atomic BoseEinstein condensates with attractive interatomic interactions. Most importantly, we are trying to gain a deeper understanding of both their formation  which typically occurs in the aftermath of a rapid switch from repulsive to attractive atomic interactions  and their subsequent dynamics. In particular, we are interested in investigating the role of the relative relative phase, investigating the role of noncondensate atoms at finite temperatures, and developing beyondmeanfield descriptions of BEC dynamics that are applicable not only to our own investigations, but also to those of researchers around the world. Bright matterwave solitons in atomic BECsThe bright matterwave solitons we study are fundamentally related to the soliton solutions of the onedimensional nonlinear Schrödinger equation (NLSE)
The NLSE possesses a spectrum of soliton solutions in which the effects of dispersion are exactly countered by the nonlinearity. The general form of these soliton solutions depends on the sign of the nonlinearity: for positive nonlinearities there are dark or grey soliton solutions, which take the form of notches in the density, and for negative nonlinearites there are bright solition solutions, which take the form of peaks in the density. The standard meanfield description of an atomic BEC at zero temperature is the GrossPitaevskii equation (GPE)
for the condensate wavefunction φ. The nonlinearity can be written in terms of the atomic swave scattering length, a_{s}:
where a_{s} is negative for attractive, and positive for repulsive, interatomic interactions. The GPE is very similar to the NLSE; indeed, the GPE for BEC which is trapped very tightly in two dimensions but left free in the third can be reduced to exactly the NLSE. While either the addition of a confining potential in the third dimension or the relaxation of the confining potentials either of the first two dimensions breaks this equivalence and eliminates formal soliton solutions (as the integrability of the NLSE is destroyed by these modifications), the GPE continues to display similar solitarywave behaviour in these situations. Hence the term bright matterwave soliton. Remaining theoretical issuesThe main questions we aim to address with our research stem from several atomic BEC experiments in which bright solitons were produced by rapidly switching the interactions in a trapped BEC from repulsive to attractive using a magnetic Feshbach resonance. Under the right conditions this produces multiple bright matterwave solitons, which then interact within the trap. The interaction forces occurring among bright solitons in the NLSE are known analytically, and the interaction forces among bright matterwave solitons can be studied numerically using the GPE. Previous theoretical work has shown that, if one takes the GPE to be a good quantitative description of the dynamics in these experiments, one can infer from the experimentally observed interactions of bright matterwave soltions that the bright matterwave solitons are always created with specific relativephase relations: namely, all adjacent solitons are out of phase by an amount
in which case they interact repulsively. However, it is unclear how such a phase relation arises from the process of soliton production  a process which remains largely unstudied. Moreover, recent theoretical research suggests that accounting for finitetemperature effects (achieved through the inclusion of quantum noise using the truncated Wigner method in this case) renders bright matterwave soliton interactions repulsive regardless of relative phase. Clearly, more insight is needed in these areas, and our research aims to provide this. The scope of our research
By combining our work on these three strands, we hope ultimately to gain a more comprehensive understanding of the processes governing bright matterwave soliton formation and dynamics than currently exists.

Department of Physics, Durham University  Tel +44 (0)191 33 43520 
Rochester Building, Science Laboratories  Fax +44 (0)191 33 45823 
South Road, Durham DH1 3LE  
United Kingdom  © Simon A Gardiner, Durham University 2005 