Mean-field approach

The DMC results are compared with the predictions of mean-field theory which are obtained from the stationary Gross-Pitaevskii (GP) equation (1.123)

$$\left(-\frac{\hbar^2}{2m}\Delta+V_{ext}(\r)+g(N-1)\vert\Phi(\r)\vert^2\right)\Phi(\r)=\mu\Phi(\r) \;,$$ (3.3)

where $\Phi(\r)$ is the order parameter normalized to unity: $\int\vert\Phi(\r)\vert^2{d\vec r}=1$ and $g=4\pi\hbar^2a_{3D}/m$ is the coupling constant (1.86). Further, finite size effects have been taken into account in the GP equation by the factor $N-1$ in the interaction term [Esr97] (see, also, 1.113). In the case of anisotropic traps with $\lambda<1$, the GP equation (3.3) is expected to provide a correct description of the system if the transverse confinement is weak $a_{3D}/a_\perp\ll 1$, and if the mean separation distance between particles is much smaller than the healing length $1/n_{3D}^{1/3}\ll\xi$, where $\xi=1/\sqrt{8\pi n_{3D}a_{3D}}$ and $n_{3D}$ is the central density of the cloud. In terms of the linear density along $z$, $n_{1D}(z)=2\pi\int_0^\infty r_\perp n_{3D}(r_\perp,z)\,dr_\perp$, this latter condition reads $1/n_{1D}\ll a_\perp^2/a_{3D}$. If the mean separation distance between particles in the longitudinal direction becomes much larger than the effective 1D scattering length given by $a_\perp^2/a_{3D}$ [PSW00], the mean-field approximation breaks down because of the lack of off diagonal long range order.

The system enters the 1D regime when the motion in the radial direction becomes frozen. In this regime the radial density profile of the cloud is fixed by the harmonic oscillator ground state, resulting in a mean square radius which coincides with the transverse oscillator length $\sqrt{\langle r_\perp^2\rangle}=a_\perp$. Further, the energy per particle is dominated by the trapping potential and one has the condition $E/N-\hbar\omega_\perp\ll\hbar\omega_\perp$.