Quasi-One-Dimensional Bose Gases with a Large Scattering Length

Bose gases confined in highly elongated harmonic traps are investigated over a wide range of interaction strengths using quantum Monte Carlo techniques. We find that the properties of a Bose gas under tight transverse confinement are well reproduced by a 1D model Hamiltonian with contact interactions. We point out the existence of a unitary regime, where the properties of the quasi-1D Bose gas become independent of the actual value of the 3D scattering length $a_{3\mathrm{D}}$. In this unitary regime, the energy of the system is well described by a hard-rod equation of state. We investigate the stability of quasi-1D Bose gases with positive and negative $a_{3\mathrm{D}}$.

Figure 1

$g_{1\mathrm{D}}$ [dashed line, Eq. (3)] and $a_{1\mathrm{D}}$ [solid line, Eq. (4)] as a function of $a_{3\mathrm{D}}$. A vertical arrow indicates the value of $a_{3\mathrm{D}}$ where $g_{1\mathrm{D}}$ diverges, $a_{3\mathrm{D}}^c/a_{\rho }=0.9684$. Horizontal arrows indicate the asymptotic values of $g_{1\mathrm{D}}$ and $a_{1\mathrm{D}}$, respectively, as $a_{3\mathrm{D}}\to \pm \infty$ ($g_{1\mathrm{D}}=-1.9368a_{\rho }\hslash {\omega }_{\rho }$ and $a_{1\mathrm{D}}=1.0326a_{\rho }$). Inset: $a_{3\mathrm{D}}$ as a function of the well depth $V_0$ for $V^{\mathrm{S}\mathrm{R}}(r)$.

Figure 2

3D DMC energy per particle calculated using $V^{\mathrm{H}\mathrm{S}}$ (diamonds) and $V^{\mathrm{S}\mathrm{R}}$ (asterisks), respectively, together with 1D DMC energy per particle calculated using $g_{1\mathrm{D}}$ [squares, Eq. (3)] and $g_{1\mathrm{D}}^0$ (plusses), respectively, as a function of $a_{3\mathrm{D}}$ [(a) linear scale; (b) logarithmic scale] for $N=5$ and $\lambda =0.01$. The statistical uncertainty of the DMC energies is smaller than the symbol size. Dotted and solid lines show the 1D energy per particle calculated within the local density approximation (LDA) for $g_{1\mathrm{D}}^0$ (using the LL equation of state) and for $g_{1\mathrm{D}}$, Eq. (3) (using the LL equation of state for $g_{1\mathrm{D}}>0$, and the hard-rod (HR) equation of state for $g_{1\mathrm{D}}<0$), respectively. A dashed horizontal line indicates the TG energy, and a vertical arrow the position where $g_{1\mathrm{D}}$, Eq. (3), diverges.

Figure 3

VMC energy per particle as a function of the variational parameter $\alpha$ for $N=5$, $\lambda =0.01$, and $a_{1\mathrm{D}}/a_{\rho }=1.0326$ (plusses), 2 (asterisks), 3 (diamonds), and 4 (triangles).