We study the effect of beyond-mean-field quantum-fluctuation (QF) Lee–Huang–Yang (LHY) and three-body interactions, with quartic and quintic nonlinearities, respectively, on the formation of a stable self-repulsive (positive scattering length $a$) and a self-attractive (negative $a$) self-bound dipolar Bose–Einstein condensate (BEC) droplet in free space under the action of two-body contact and dipolar interactions. Previous studies of dipolar droplets considered either the LHY interaction or the three-body interaction, as either of these interactions could stabilize a dipolar BEC droplet against collapse. We find that the effect of three-body recombination on the formation of a dipolar droplet could be quite large and for a complete description of the problem both the QF LHY and three-body interactions should be considered simultaneously, where appropriate. In the self-repulsive case for small $a$ and in the self-attractive case, no appropriate LHY interaction is known and only three-body interaction should be used, otherwise both beyond-mean-field interactions should be used. We consider a numerical solution of a highly-nonlinear beyond-mean-field model as well as a variational approximation to it in this investigation and present results for size, shape and energy of a dipolar droplet of polarized ¹⁶⁴Dy atoms. The shape is filament-like, along the polarization direction, and could be long, for a large number of atoms $N$, short for small $N$, thin for negative $a$ and small positive $a$, and fat for large positive $a$.

我们研究了超平均场量子涨落 (QF) Lee-Huang-Yang (LHY) 和三体相互作用（分别具有四次和五次非线性）对稳定自排斥（正散射）形成的影响长度$\mathrm{一个}$）和自我吸引（消极$\mathrm{一个}$）在两体接触和偶极相互作用的作用下，自由空间中的自结合偶极玻色-爱因斯坦凝聚（BEC）液滴。先前对偶极液滴的研究考虑了 LHY 相互作用或三体相互作用，因为这些相互作用中的任何一种都可以稳定偶极 BEC 液滴以防止塌陷。我们发现三体重组对偶极液滴形成的影响可能非常大，为了完整描述该问题，应在适当的情况下同时考虑 QF LHY和三体相互作用。在自我排斥的情况下为小$\mathrm{一个}$在自吸引的情况下，不知道适当的 LHY 相互作用，只应使用三体相互作用，否则应使用两种超平均场相互作用。我们在本研究中考虑了一个高度非线性的超平均场模型的数值解以及它的变分近似，并给出了极化¹⁶⁴ Dy 原子的偶极液滴的大小、形状和能量的结果。形状呈细丝状，沿偏振方向，可以很长，用于大量原子$ñ$, 小号的缩写$ñ$, 瘦为负$\mathrm{一个}$和小正$\mathrm{一个}$，而脂肪为大阳性$\mathrm{一个}$.