Evolution from a Bose-Einstein condensate to a Tonks-Girardeau gas: An exact diagonalization study

We study ground state properties of spinless, quasi one-dimensional bosons which are confined in a harmonic trap and interact via repulsive delta-potentials. We use the exact diagonalization method to analyze the pair correlation function, as well as the density, the momentum distribution, different contributions to the energy and the population of single-particle orbitals in the whole interaction regime. In particular, we are able to trace the fascinating transition from bosonic to fermi-like behavior in characteristic features of the momentum distribution which is accessible to experiments. Our calculations yield quantitative measures for the interaction strength limiting the mean-field regime on one side and the Tonks-Girardeau regime on the other side of an intermediate regime.

Figure 1

Particle density of five bosons for different interaction strengths $U_{1D}$ along the axial direction $(x=y=0)$. The density becomes flatter and broader with $U_{1D}$. In the strong interaction regime it develops oscillations. At $U_{1D}=20\hslash {\omega }_z$ the density of the bosons perfectly agrees with the density of noninteracting fermions.

Figure 2

Pair correlation function of five bosons for different interaction strengths $U_{1D}$ along the axial direction $(x=y=0)$. One particle is fixed at position $z=0$. The distribution flattens and forms a hole at coinciding particle positions, $z=z^{\prime }$. The correlation function clearly indicates the transition between the three regimes (see text).

Figure 3

Evolution of various contributions to the total energy, $E_{\mathrm{tot},z}$, of five bosons with increasing interaction strength $U_{1D}$. The energies evolve towards the accordant energies of non-interacting fermions [23]. The minimum of the kinetic energy coincides with the onset of significant correlations and therefore marks the limit of the MF regime. By contrast $E_{\mathrm{kin},z}^{\mathrm{MF}}$ decreases in the whole interaction regime, see inset.

Figure 4

Momentum distribution of five bosons for different interaction strengths $U_{1D}$ $(p_x=p_y=0)$. The central peak reaches its maximum height at $U_{1D}=3\hslash {\omega }_z$, see inset. At $U_{1D}=20\hslash {\omega }_z$ the momentum distribution has developed the typical form of a harmonically trapped TG gas [29] which is completely different from the distribution of noninteracting fermions.

Figure 5

Occupation number distribution of five bosons for different interaction strengths $U_{1D}$. With increasing $U_{1D}$ the bosons leave the ground state and occupy excited states. Single-particle states with even parity are comparatively stronger populated than those with odd parity.