Construction of Analytical Many-Body Wave Functions for Correlated Bosons in a Harmonic Trap

We develop an analytical many-body wave function to accurately describe the crossover of a one-dimensional bosonic system from weak to strong interactions in a harmonic trap. The explicit wave function, which is based on the exact two-body states, consists of symmetric multiple products of the corresponding parabolic cylinder functions and respects the analytically known limits of zero and infinite repulsion for arbitrary number of particles. For intermediate interaction strengths we demonstrate that the energies, as well as the reduced densities of first and second order, are in excellent agreement with large scale numerical calculations.

Figure 1

(a) Wave function of the relative motion for two particles ${\Psi }_{\mathrm{CP}}(r\equiv x_1-x_2)$ and several strengths of the interaction ($\mu =0,0.5,0.8,1$) and comparison of $\mu =1$ with the fermionic antisymmetric function. Contour plots of the density distribution $|\Psi (x_1,x_2,x_3){|}^2=0.01$ for $N=3$ using ${\Psi }_r={\Psi }_{\mathrm{CP}}$ with (b) $\mu =0.2$, (c) $\mu =0.5$. (d) The two-body correlation function (density) for strong interaction $g=20.0$.

Figure 2

(a) Total energies as a function of the interaction strength $g$ obtained numerically (marks) and analytically (lines) via the CPWF, for $N=3,4,5$ (exact diagonalization), 6 (multiconfiguration time-dependent Hartree method). Analytically and numerically obtained one-body densities $\rho (x)=\rho (-x)$ for (b) $N=3$, (c) $N=4$, and (d) $N=5$ particles at $g=0.2,2.0,10.0$. For $g=2.0$ variational results from CPWF and the Thomas-Fermi curve are shown.