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

Abstract

Finite temperature hydrodynamic model is derived for the spin-1 ultracold bosons by the manyparticle quantum hydrodynamic method. It is presented as the two fluid model of the BEC and normal fluid. The linear and quadratic Zeeman effects are included. Scalar and spin-spin like shortrange interactions are considered in the first order by the interaction radius. It is 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 are the interaction constants.

Hydrodynamic model of skyrmions and vorticities in the spin-1 Bose-Einstein condensate in ferromagnetic phase at finite temperatures

“…Definition (1) is the first step of derivation of the hydrodynamic equations from the microscopic many-particle Schrodinger equation in the coordinate representation by the quantum hydrodynamic method suggested by L. S. Kuz'menkov and S. G. Maksimov [30], [31], [32]. This method is applied to the spin-1 ultracold bosons [33]. We consider the ferromagnetic phase of spin-1 BECs, while there are other phases such as polar phase, partially polarized BECs, antiferromagnetic phase (which includes the nematic ordering of the system [34]).…”

“…It leads to the following set of hydrodynamic equations. First, it is the continuity equations for the BEC and for the normal fluid [33]…”

“…where subindex a stands for B and n. Vector v a is the hydrodynamic velocity field. The velocity field is the ratio of the momentum density to the mass m and concentration n. The momentum density is defined via the many-particle wave function similarly to the concentration (1) [33]. The spinor BEC is characterized by the spin vector field…”

“…For the finite temperature it is derived in Ref. [33]. Hydrodynamic model ( 2)-( 11) is equivalent to the set of non-linear Schrodinger equations…”

“…The spin field is the unit field for the ferromagnetic phase s = {cos Φ(r) sin Θ(r), sin Φ(r) sin Θ(r), cos Θ(r)}. (33) Functions Φ(r) and Θ(r) depend in general on the three dimensional coordinate r. They are functions of time as well. However, we are focused on the static regime, where there is plane symmetry.…”