corresponding eigenvalues $\lambda_i$ giving the respective occupation numbers $$\rho_1(x,y) = \sum_i \lambda_i \phi_i^*(x) \phi_i(y),
\label{eq:naturalorbitals}$$ where $\sum \lambda_i=N$. For a non-interacting Bose gas at zero temperature only the lowest lying orbital is occupied, $\lambda_0=N$, which corresponds to a completely coherent and superfluid system. Conversely, a ground state wavefunction with $\lambda_0=1$ corresponds to a completely incoherent gas where the $N$ lowest orbitals are equally occupied. The occupation number of the lowest lying eigenstate of the RSPDM can therefore be used as a measure of the coherence of the system. In fact, for a TG gas in free space it is known that the lowest lying orbital has an occupation proportional to $\sqrt{N}$, while the occupation of orbitals with $i\gg N$ tends to zero [@Lenard1964; @Forrester2003]. Clearly such a state is not strictly superfluid, however it is also not completely incoherent, as $\lambda_0$ is of the order $\sqrt{N}$. It therefore does possess some superfluid properties. For simplicity, we will refer to phases displaying finite coherence as superfluid in the rest of this manuscript. In the pinned phase, however, each particle is highly localised which destroys the coherence between individual wells once tunneling is sufficiently suppressed. The system is therefore reduced to $N$ 