Bright Soliton Theory: Research

Introduction

We are investigating the phenomenon of bright matter-wave solitons, which have been observed in atomic Bose-Einstein 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 non-condensate atoms at finite temperatures, and developing beyond-mean-field descriptions of BEC dynamics that are applicable not only to our own investigations, but also to those of researchers around the world.

Bright matter-wave solitons in atomic BECs

The bright matter-wave solitons we study are fundamentally related to the soliton solutions of the one-dimensional non-linear 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 mean-field description of an atomic BEC at zero temperature is the Gross-Pitaevskii equation (GPE)

for the condensate wavefunction φ. The nonlinearity can be written in terms of the atomic s-wave scattering length, as:

where as 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 solitary-wave behaviour in these situations. Hence the term bright matter-wave soliton.

Remaining theoretical issues

The 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 matter-wave 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 matter-wave 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 matter-wave soltions that the bright matter-wave solitons are always created with specific relative-phase 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 un-studied. Moreover, recent theoretical research suggests that accounting for finite-temperature effects (achieved through the inclusion of quantum noise using the truncated Wigner method in this case) renders bright matter-wave 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

  • Exploration of the role of modulational instability in soliton formation, in both mean-field (Gross-Pitaevskii) and finite-temperature descriptions.
  • Exploration of possible modifications to the 1-D Lieb-Liniger theory, and use of this theory to study quantum bright solitons.
  • Development and numerical implementation of a second-order, quantum field-theoretical description of finite-temperature BEC dynamics.

By combining our work on these three strands, we hope ultimately to gain a more comprehensive understanding of the processes governing bright matter-wave soliton formation and dynamics than currently exists.

Content © Thomas P Billam, Durham University 2009