Quantum description of dark solitions in one-dimensional quantum gases : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
The main objective of this thesis is to explain, from the quantum-mechanical point
of view, the nature of dark solitons in one-dimensional cold-atom systems. Models
of bosons and fermions with contact interactions on a ring are exactly solvable via
the Bethe ansatz, and support so-called type-II elementary excitations. These have
long been associated with dark solitons of the Gross-Pitaevskii equation due to
the similarity of the dispersion relation, despite the completely different physical
properties of the states. Fully understanding this connection is our primary aim.
We begin by reviewing the Gross-Pitaevskii equation and its dark soliton solutions.
Next, we solve the mean-field problem of two coupled one-dimensional
Bose-Einstein condensates, with special emphasis on Josephson vortices and their
dispersion relation. Predictions are given for possible experimental detection. Then
we give a derivation that justifies a method for the extraction of the so-called missing
particle number from the dispersion relation of solitonic excitations.
A derivation of the finite Bethe ansatz equations for the Lieb-Liniger and Yang-
Gaudin models follows. These describe a single species of bosons and two component
fermions, respectively. We review the elementary excitations of the Lieb-Linger
model, and carry out a comprehensive study of the (much richer) excitations of
the Yang-Gaudin model. The thermodynamic limit Bethe ansatz equations for all
states of interest in both models are derived, and the missing particle number and
the closely-related phase-step are extracted from the dispersion relations. Next, we
develop a method for approximating the finite-system dispersion relation of solitonic
excitations from the thermodynamic limit results.
Finally, we show that the single particle density and phase profiles of appropriatelyformed
superpositions of type-II states with different momenta exhibit solitonic features.
Through this idea, the missing particle number and phase step extracted from
the dispersion relation gain physical meaning. Moreover, we use a convolution model
to extract the fundamental quantum dark soliton length scale across the range of
interactions and momenta. The insight gained in the bosonic case is used to make
inferences about dark solitons in the fermionic case. Furthermore, we study the
Hess-Fairbank effect in the repulsive Yang-Gaudin model and the fermionic super