A variationally optimised basis allows an accurate description of the quantum
behaviour of ultra-cold atoms, even in the strongly correlated regime. A rescaling
scheme corrects discrepancies caused by using a reduced Hilbert space. This
approach also allows the modelling of experimentally realizable double-well potentials,
which still reveals the maximally-entangled states seen in fixed basis models.
Time dynamics of these double-well systems show macroscopic tunnelling between
wells for bosons with a sufficient interaction strength.
The many-body problem of interacting bosons in the highly-correlated regime
is difficult. The number of basis states needed to describe this quantum system
accurately quickly grows beyond computational reach. Rescaling the interaction
strength proves a simple and effective method of calculating exact eigenvalues in a
reduced Hilbert space.
Bosonic systems in the double-well potential are investigated next. First, how
different eigen-states depend on the interaction strength is examined. The variationally
optimised method has advantages over a standard fixed basis method with
the ability to model experimentally viable systems and explore more stronglycorrelated
regimes. Secondly, tunnelling dynamics in the double well are studied,
specifically for a system where all particles initially occupy a single well. Oscillations
corresponding to collective tunnelling between wells are found in regimes
where there are zero interactions or bosons lie in a maximally-entangled state.
What governs the dynamics outside these two regimes is also considered.