Andreev Pavel A
Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russian Federation and Faculty of Physics, Mathematics and Natural Sciences, Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow 117198, Russian Federation.
Chaos. 2021 Feb;31(2):023120. doi: 10.1063/5.0036511.
Traditional quantum hydrodynamics of Bose-Einstein condensates (BECs) is restricted by the continuity and Euler equations. The quantum Bohm potential (the quantum part of the momentum flux) has a nontrivial part that can evolve under quantum fluctuations. The quantum fluctuations are the effect of the appearance of particles in the excited states during the evolution of BEC mainly consisting of the particles in the quantum state with the lowest energy. To cover this phenomenon in terms of hydrodynamic methods, we need to derive equations for the momentum flux and the current of the momentum flux. The current of the momentum flux evolution equation contains the interaction leading to the quantum fluctuations. In the dipolar BECs, we deal with the long-range interaction. Its contribution is proportional to the average macroscopic potential of the dipole-dipole interaction (DDI) appearing in the mean-field regime. The current of the momentum flux evolution equation contains the third derivative of this potential. It is responsible for the dipolar part of quantum fluctuations. Higher derivatives correspond to the small scale contributions of the DDI. The quantum fluctuations lead to the existence of the second wave solution. The quantum fluctuations introduce the instability of the BECs. If the dipole-dipole interaction is attractive, but being smaller than the repulsive short-range interaction presented by the first interaction constant, there is the long-wavelength instability. There is a more complex picture for the repulsive DDI. There is the small area with the long-wavelength instability that transits into a stability interval where two waves exist.
玻色 - 爱因斯坦凝聚体（BECs）的传统量子流体动力学受连续性方程和欧拉方程的限制。量子玻姆势（动量通量的量子部分）有一个非平凡部分，它可以在量子涨落下演化。量子涨落是在主要由处于最低能量量子态的粒子组成的BEC演化过程中，激发态粒子出现的效应。为了用流体动力学方法描述这种现象，我们需要推导动量通量和动量通量流的方程。动量通量流演化方程包含导致量子涨落的相互作用。在偶极BECs中，我们处理的是长程相互作用。其贡献与平均场 regime 中出现的偶极 - 偶极相互作用（DDI）的平均宏观势成正比。动量通量流演化方程包含该势的三阶导数。它负责量子涨落的偶极部分。更高阶导数对应于DDI的小尺度贡献。量子涨落导致第二波解的存在。量子涨落引入了BECs的不稳定性。如果偶极 - 偶极相互作用是吸引性的，但小于由第一相互作用常数表示的排斥性短程相互作用，则存在长波长不稳定性。对于排斥性DDI，情况更为复杂。存在一个小区域具有长波长不稳定性，它会过渡到存在两波的稳定区间。
