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arXiv:2007.15045 · Laser Physics Letters

Hydrodynamics of the atomic Bose–Einstein condensate beyond the mean-field approximation

Pavel A Andreev

Published Apr 19, 2021• 5 citations
arXivHTMLPDF

Abstract

Abstract Several hydrodynamic models of the atomic Bose–Einstein condensate (BEC) obtained beyond the mean-field approximation are discussed together from a single point of view. All these models are derived from the microscopic quantum description. The derivation is made within the many-particle quantum hydrodynamics method suggested by L Kuz’menkov. The derivation is demonstrated and discussed for the mean-field regime revealing the Gross–Pitaevskii equation as the simplest illustration. It appears in the first order by the interaction radius. Generalization of the hydrodynamic Euler equation obtained in the third order by the interaction radius is discussed. It includes the contribution of the isotropic short-range interaction (SRI) presented by the third space derivative of the square of concentration. The Euler equation also includes the contribution of the anisotropic part of the SRI proportional to the second order spherical function. A systematic account of the quantum fluctuations in terms of the many-particle quantum hydrodynamics method requires the extension of the set of hydrodynamic equations from the couple continuity and Euler equations to the set of four equations which also includes the pressure evolution equation and the evolution equation for the third rank tensor of the pressure flux. The pressure evolution equation contains no interaction contribution in the first order by the interaction radius. The source of the quantum fluctuations is in the interaction caused term existing in the third rank tensor evolution equation which is obtained in the first order by the interaction radius. The presented models are considered for the single-component BECs.

DOI
10.1088/1612-202x/abf5cd

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Abstract Several hydrodynamic models of the atomic Bose–Einstein condensate (BEC) obtained beyond the mean-field approximation are discussed together from a single point of view. All these models are derived from the microscopic quantum description. The derivation is made within the many-particle quantum hydrodynamics method suggested by L Kuz’menkov. The derivation is demonstrated and discussed for the mean-field regime revealing the Gross–Pitaevskii equation as the simplest illustration. It appears in the first order by the interaction radius. Generalization of the hydrodynamic Euler equation obtained in the third order by the interaction radius is discussed. It includes the contribution of the isotropic short-range interaction (SRI) presented by the third space derivative of the square of concentration. The Euler equation also includes the contribution of the anisotropic part of the SRI proportional to the second order spherical function. A systematic account of the quantum fluctuations in terms of the many-particle quantum hydrodynamics method requires the extension of the set of hydrodynamic equations from the couple continuity and Euler equations to the set of four equations which also includes the pressure evolution equation and the evolution equation for the third rank tensor of the pressure flux. The pressure evolution equation contains no interaction contribution in the first order by the interaction radius. The source of the quantum fluctuations is in the interaction caused term existing in the third rank tensor evolution equation which is obtained in the first order by the interaction radius. The presented models are considered for the single-component BECs.
Abstract Ultracold Bose atoms is the physical system existing at the small finite temperatures, where the quantum and nonlinear phenomena play crucial role. Bosons are considered to be composed of two different fluids: the Bose–Einstein condensate and the normal fluid (the thermal component). The extended hydrodynamic models are obtained for each fluids, where the pressure evolution equations and the pressure flux third rank tensor evolution equations are obtained along with the continuity and Euler equations. It is found that the pressure evolution equation contains zero contribution of the short-range interaction. The pressure flux evolution equation contains the interaction which simplifies to the quantum fluctuations in the zero temperature limit. The structure of the third rank tensor describing this interaction is obtained in the regime of small temperature and weak interaction. The model is derived via the straightforward calculation of evolution of macroscopic functions using the microscopic many-particle Schrodinger equation in the coordinate representation. Finally, the two-fluid hydrodynamics is constructed in form of four equations for each fluid in order to give model describing the quantum fluctuations in BEC and the thermal effects in the normal fluid.
We describe recent development of quantum hydrodynamics for ultracold Bose particle studying and consider different kinds of interactions. The method of derivation of equations describing the evolution of the neutral Bose particle system at low temperatures is described. Despite the fact that we consider the neutral particles we account the short-range interaction between particles. We consider the particles in the Bose-Einstein condensate (BEC) state. This method is called the method of quantum hydrodynamics, because natural for of the quantum mechanics rewritten in terms of material fields of observable quantities in three dimensional space is the set of equations, which look like the hydrodynamics equations. It can be shown that from the quantum hydrodynamics equations can be derived macroscopic non-linear Schrodinger equation. Most famous non-linear Schrodinger equation is the Gross-Pitaevskii (GP) equation, which contains nonlinearity of the third degree. There are generalizations of the GP equation. New term appears in the GP equation at account of the three-particle interaction. This term contains nonlinearity of the fifth degree. At more detailed account of the two particle interaction we come to the nonlocal non-linear Schrodinger equation. This equation contains spatial derivatives of the order parameter in the non-linear terms caused by the interaction. In this terminology the GP equation corresponds to the first order by the interaction radius. For the BEC of the neutral particles with anisotropic long-range dipole-dipole interaction the generalization of the GP equation was also suggested. Detailed analyses of the applicability conditions shows that this equation valid for the system of dipoles parallel to each other, which do not change their direction, and where the dipole-dipole interaction interferences translational motion of particles.
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.
Abstract A quantum hydrodynamic model is developed for axial symmetric anisotropic short-range interactions. The quantum stress tensor represents the interaction and is derived up to the third order by the interaction radius. The first order by the interaction radius contains the isotropic part only. This leads to interaction in the Gross–Pitaevskii approximation. Terms existing in the third order by the interaction radius are caused by the isotropic and nonisotropic parts of the interaction, and each of them introduces an interaction constant. Therefore, three interaction constants are involved in the model. Atoms, except for alkali and alkali-earth atoms, can have an anisotropic potential of interaction (this has particularly been demonstrated for the lanthanides). Short-wavelength instability caused by nonlocal terms appears in the Bogoliubov spectrum, and the conditions for stable and unstable behavior are described. Bright solitons in a repulsive Bose–Einstein condensate (BEC) are studied under the influence of anisotropic short-range interaction in the BEC of one species. The area of existence of bright solitons corresponds to the area of instability of the Bogoliubov spectrum. Approximate reduction of nonlocal nonlinearity to quintic nonlinearity in the bright soliton regime is also demonstrated.
Abstract We develop a novel model of the magnetized spin-1 Bose–Einstein condensate (BEC) of neutral atoms, using the method of many-particle quantum hydrodynamic (QHD) and propose an original derivation of the system of continual equations. We consider bosons with a spin-spin interaction and a short range interaction in the first order in the interaction radius, on the of basis of the self-consistent field approximation of the QHD equations. We demonstrate that the dynamics of the fluid velocity and magnetization is determined by a nontrivial modification of the Euler and Landau–Lifshitz equation, and show that a nontrivial modification of the spin density evolution equation contains the spin torque effect that arises from the self-interactions between spins of the bosons. The properties of the dispersion spectrum of collective excitations are described. We obtain the new contribution of the self-interaction of spins in the spin wave spectrum together with the influence of an external magnetic field and spin-spin interactions between polarized particles. The anisotropic spin wave instability is predicted.
A finite temperature hydrodynamic model is derived for the spin-1 ultracold bosons by the many-particle quantum hydrodynamic method. It is presented as the two fluid model of the Bose–Einstein condensate (BEC) and normal fluid. The continuity, Euler, spin evolution, and nematic tensor evolution equations are derived for each fluid. The linear and quadratic Zeeman effects are included. Scalar and spin–spin like short-range interactions are considered in the first order by the interaction radius. Obtained hydrodynamic equations are also represented as the set of two nonlinear Pauli equations. The spectrum of the bulk collective excitations is considered for the ferromagnetic phase in the small temperature limit. The spin wave is not affected by the presence of the small temperature in the described minimal coupling model, where the thermal part of the spin-current of the normal fluid is neglected. The two sound waves are affected by the spin evolution in the same way as the change of spectrum of the single sound wave in BEC, where speed of sound is proportional to g1+g2 with gi as the interaction constants.