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Quantum hydrodynamic theory of quantum fluctuations in dipolar Bose-Einstein condensate

Traditional quantum hydrodynamics of Bose-Einstein condensates (BECs) is restricted by the continuity and Euler equations. It corresponds to the well-known Gross-Pitaevskii equation. However, the quantum Bohm potential, which is a part of the momentum flux, has a nontrivial part with can evolve under the quantum fluctuations. To cover this phenomenon in terms of hydrodynamic methods we need to derive equations for the momentum flux, and the third rank tensor. In all equations we consider the main contribution of the short-range interaction (SRI) in the first order by the interaction radius. Derived hydrodynamics consists of four hydrodynamic equations. The third moment evolution equation contains interaction leading to the quantum fluctuations. It includes new interaction constant. The Gross-Pitaevskii interaction constant is the integral of potential, but the second interaction constant is the integral of second derivative of potential. If we have dipolar BECs we deal with a long-range interaction. Its contribution is proportional to the potential of dipole-dipole interaction (DDI). The Euler equation contains the derivative of the potential. The third rank tensor evolution equation contains the third derivative of the potential. The quantum fluctuations lead to existence of the second wave solution. Moreover, the quantum fluctuations introduce the instability of BECs. If the DDI is attractive, but being smaller then the repulsive SRI presented by the first interaction constant, there is the long-wavelength instability. For the repulsive DDI these is more complex picture. There is the small area with the long-wavelength instability which transits into stability interval, where two waves exist. There is the short-wavelength instability as well. These results are found for the DDI strength comparable with the Gross-Pitaevskii SRI.