Wave functions of the super-Tonks-Girardeau gas and the trapped one-dimensional hard-sphere Bose gas

Abstract

Recent theoretical and experimental results demonstrate a close connection between the super-Tonks-Girardeau (STG) gas and a one-dimensional (1D) hard-sphere Bose (HSB) gas with hard-sphere diameter nearly equal to the 1D scattering length ${a}_{1\mathrm{D}}$ of the STG gas, a highly excited gaslike state with nodes only at interparticle separations $|{x}_{j\ensuremath{\ell}}|={x}_{\mathrm{node}}\ensuremath{\approx}{a}_{1\mathrm{D}}$. It is shown herein that when the coupling constant ${g}_{B}$ in the Lieb-Liniger interaction ${g}_{B}\ensuremath{\delta}({x}_{j\ensuremath{\ell}})$ is negative and $|{x}_{12}|\ensuremath{\geqslant}{x}_{\mathrm{node}}$, the STG and HSB wave functions for $N=2$ particles are not merely similar, but identical; the only difference between the STG and HSB wave functions is that the STG wave function allows a small penetration into the region $|{x}_{12}|<{x}_{\mathrm{node}}$, whereas for a HSB gas with hard-sphere diameter ${a}_{\mathrm{hs}}={x}_{\mathrm{node}}$, the HSB wave function vanishes when $|{x}_{12}|<{a}_{\mathrm{hs}}$. Arguments are given suggesting that the same theorem holds also for $N>2$. The STG and HSB wave functions for $N=2$ are given exactly in terms of a parabolic cylinder function, and for $N\ensuremath{\geqslant}2$, ${x}_{\mathrm{node}}$ is given accurately by a simple parabola. The metastability of the STG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.