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arXiv:1811.05813 · Physica Scripta

Spin domains in ground state of a trapped spin-1 condensate: a general study under Thomas–Fermi approximation

Projjwal Kanti Kanjilal, A. Bhattacharyay

Published Dec 4, 2019• 4 citations
arXivHTMLPDF

Abstract

Abstract Investigation of ground state structures and phase separation under confinement is of great interest in spinor Bose–Einstein condensates. In this paper we show that, in general, within the Thomas–Fermi approximation, the phase separation scenario of stationary states can be obtained including all the mixed states on an equal footing for a spin-1 condensate under any confinement. According to the process used here, the density of each and every stationary states can be determined exactly in terms of the system parameters. This allows to write the expression for energy density of all states in a form which are not an explicit function of densities of those states. An energy density comparison therefore, among all the allowed stationary states can be done which will reveal domain structures depending on the system parameters. We study here, in details, a particular case of spherically symmetric harmonic confinement as an example and show a wide range of potential phase separation scenario for anti-ferromagnetic and ferromagnetic interactions.

DOI
10.1088/1402-4896/ab5e95

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Abstract Investigation of ground state structures and phase separation under confinement is of great interest in spinor Bose–Einstein condensates. In this paper we show that, in general, within the Thomas–Fermi approximation, the phase separation scenario of stationary states can be obtained including all the mixed states on an equal footing for a spin-1 condensate under any confinement. According to the process used here, the density of each and every stationary states can be determined exactly in terms of the system parameters. This allows to write the expression for energy density of all states in a form which are not an explicit function of densities of those states. An energy density comparison therefore, among all the allowed stationary states can be done which will reveal domain structures depending on the system parameters. We study here, in details, a particular case of spherically symmetric harmonic confinement as an example and show a wide range of potential phase separation scenario for anti-ferromagnetic and ferromagnetic interactions.
We analyze the structure of spin-one Bose-Einstein condensates in the presence of a homogeneous magnetic field. We classify the homogeneous stationary states and study their existence, bifurcations, and energy spectra. We reveal that phase separation can occur in the ground state of antiferromagnetic (polar) condensates while the spin components of the ferromagnetic condensates are always miscible and no phase separation occurs. Our theoretical model, confirmed by numerical simulations, explains that this phenomenon takes place when the energy of the lowest homogeneous state is a concave function of the magnetization. In particular, we predict that phase separation can be observed in a 23Na^{23}\text{N}\text{a}23Na condensate confined in a highly elongated harmonic trap. Finally, we discuss the phenomena of dynamical instability and spin domain formation.
We provide complete phase diagrams describing the ground state of a trapped spinor Bose-Einstein condensate under the combined effects of rotation and a Rashba spin-orbit coupling. The interplay between the different parameters (magnitude of rotation, strength of the spin-orbit coupling, and interaction) leads to a rich ground-state physics that we classify. We explain some features analytically in the Thomas-Fermi approximation, writing the problem in terms of the total density, total phase, and spin. In particular, we analyze the giant skyrmion and find that it is of degree 1 in the strong-segregation case. In some regions of the phase diagrams, we relate the patterns to a ferromagnetic energy.
Bose-Einstein condensates - a state of matter at low temperature in which a macroscopic amount of bosons occupy the quantum-mechanical ground state- were originally trapped magnetically. The drawback of this trapping method is that only the weak-field seeking spin state is trapped and we therefore only have a one component i.e. scalar condensate. In contrast, optical trapping confines atoms of all spin state and therefore allows for spin degrees of freedom, resulting in spinor Bose Einstein condensates. In this thesis, density profiles of these spinor condensates have been calculated in Thomas-Fermi approximation. The profiles show areas with strict phase separation, but also areas with spin mixtures. For sodium atoms we get a strict separation between the mF = ±1 and the mF = 0 spin state in Thomas-Fermi approximation. This however causes discontinuities in the transition between spin domains and thus these interfaces were studied further. We found that the density profile of a spin state behaves roughly like a hyperbolic tangent at such an interface. Finally, a differential equation is proposed that could be solved as a starting point of follow-up research.
We consider a spin-1 Bose-Einstein condensate trapped in a harmonic potential under the influence of a homogeneous magnetic field. We investigate spatial and spin structure of the mean-field ground states under constraints on the number of atoms and the total magnetization. We show that the trapping potential can make the antiferromagnetic condensate separate into three distinct phases and ferromagnetic condensate into two distinct phases. In the ferromagnetic case, the magnetization is located in the center of the harmonic trap, while in the antiferromagnetic case magnetized phases appear in the outer regions. We describe how the transition from the Thomas-Fermi regime to the single-mode approximation regime with decreasing number of atoms results in the disappearance of the domains. We suggest that the ground states can be created in experiment by adiabatically changing the magnetic-field strength.
In the presence of a magnetic field, multicomponent ground states appear in trapped spin-1 Bose-Einstein condensates for both ferromagnetic and antiferromagnetic types of spin-spin interaction. We aim to produce an accurate analytical description of the multicomponent states which is of fundamental importance. Despite being in the so-called regime of the Thomas-Fermi approximation (condensates with large particle number), the scenario of multicomponent states is problematic under this approximation due to the large variation in densities of the subcomponents. We generalize the variational method that we have introduced [P. K. Kanjilal and A. Bhattacharyay, Eur. Phys. J. Plus 137, 547 (2022)] to overcome the limitations of the Thomas-Fermi approximation. We demonstrate that the variational method is crucial in identifying multicomponent ground states. A comparison of the results of the variational method, which is multimodal by construction, with that of the single-mode approximation is also presented in this paper to demonstrate a marked improvement in accuracy over the single-mode approximation. We have also looked into the phase transition between phase-matched and polar states in a trapped condensate using the variational method and have identified substantial change in the phase boundary. The correspondence of the phase diagram of the trapped case with the homogeneous one identifies other limitations of the Thomas-Fermi approximation as opposed to the more accurate variational method.
We demonstrate that the ground state of a trapped spin-1 and spin-2 spinor ferromagnetic Bose-Einstein condensate (BEC) can be well approximated by a single decoupled Gross-Pitaevskii (GP) equation. Useful analytic models for the ground-state densities of ferromagnetic BECs are obtained from the Thomas-Fermi approximation (TFA) to this decoupled equation. Similarly, for the ground states of spin-1 antiferromagnetic and spin-2 antiferromagnetic and cyclic BECs, some of the spin-component densities are zero, which reduces the coupled GP equation to a simple reduced form. Analytic models for ground-state densities are also obtained for antiferromagnetic and cyclic BECs from the TFA to the respective reduced GP equations. The analytic densities are illustrated and compared with the full numerical solution of the GP equation with realistic experimental parameters.