Relevant parameters and DMC approach

The relevant parameters of the problem are the number of particles $N$, the ratio $a_{3D}/a_\perp$ of the scattering length to the transverse harmonic oscillator length $a_\perp=\sqrt{\hbar/m\omega_\perp}$ and the anisotropy parameter $\lambda=\omega_z/\omega_\perp$. For a given set of parameters we solve exactly, using the Diffusion Monte-Carlo method (Sec. 2.3), the many-body Schrödinger equation (2.6) for the ground state and we calculate the energy per particle and the mean square radii of the cloud in the axial and radial directions. Importance sampling is used through the Bijl-Jastrow trial wave function (2.37). For the one-body term, which accounts for the external confinement, we use a simple gaussian ansatz (2.41) $f_1(r_\perp,z)=\exp\{-\alpha_\perp r_\perp^2-\alpha_z z^2\}$, with $\alpha_\perp$ and $\alpha _z$ optimized variational parameters. The two-body term $f_2(r)$ accounts instead for the particle-particle interaction and is chosen using the same technique employed in Ref. [GBC99] for a homogeneous system. Of course, since DMC is an exact method, the precise choice of $\psi_T({\bf R})$ is to a large extent unimportant and the results obtained are not biased by the choice of the trial wave function^3.1.