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arXiv:2102.12538 · Physics of Fluids

Quantum hydrodynamics of the spinor Bose–Einstein condensate at non-zero temperatures

Pavel A. Andreev, Ilia Mosaki, Mariya Iv. Trukhanova

Published Jun 1, 2021• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0053035

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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.
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.
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.
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.
We develop the complete hydrodynamical description of the spin-1 partially polarized Bose–Einstein condensate (BEC), based on the many-particle quantum hydrodynamic formalism. We derive the continuity equation, momentum balance equation, spin evolution and nematicity evolution equations considering the spin–spin dipole long-range interactions and short-range interactions between partially spin-polarized atoms of condensate. The nematic or quadrupolar tensor plays an important role in the experiments of spin-nematic squeezing. We predict the influence of the spin–spin dipole interactions and spin torque force on the dynamics of nematic tensor. The collective modes for phonons and magnons are predicted for the ferromagnetic phase of spinor condensate, where the spin–spin and short-range interactions modify the dispersion equation.
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.
We present a derivation of the quantum hydrodynamic (QHD) equations for neutral bosons. We consider the short-range interaction between particles. This interaction consist of a binary interaction U( ri, rj) and a three-particle interaction (TPI) U( ri, rj, rk) and the last one does not include binary interaction between particles. From QHD equations for Bose–Einstein condensate we derive a nonlinear Schrödinger equation. This equation was derived for zero temperature and contains the nonlinearities of the third and the fifth degree. Explicit form of the constant of the TPI is obtained. First of all, developed method we used for studying of dispersion of the linear waves. Dispersion characteristics of the linear waves are compared for different particular cases. We make comparison of the two-particle interaction in the third order by the interaction radius (TOIR) and TPI at the zero temperature. We consider influence of the temperature on the dispersion of the elementary excitations. For this aim we derive a system of the QHD equations at nonzero temperature. Obtained system of equation is an analog of the well-known two-fluid hydrodynamics. Moreover, it is generalization of the two-fluid hydrodynamics equations due to TPI. Explicit expressions of the velocities for the first and the second sound via the concentration of superfluid and noncondensate components is calculated.