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Journal of Applied Mathematics

The Gross‐Pitaevskii Model of Spinor BEC

Dongming Yan, Tian Ma

Published Jan 1, 2012• 1 citation
Publisher

Abstract

The Gross‐Pitaevskii model of spinor Bose‐Einstein condensates is studied. Using the abstract results obtained for infinite‐dimensional Hamilton system, we establish the mathematical theory for the model of spinor BEC. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity, are also proved.

DOI
10.1155/2012/757616

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The Gross‐Pitaevskii model of spinor Bose‐Einstein condensates is studied. Using the abstract results obtained for infinite‐dimensional Hamilton system, we establish the mathematical theory for the model of spinor BEC. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity, are also proved.
We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.
We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles.The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction.The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling.In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms.The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case.Quantitative rates of convergence are available, but not made explicit because evidently non-optimal.In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.
In this paper, we systematically review mathematical models, theories and numerical methods for ground states and dynamics of spinor Bose-Einstein condensates (BECs) based on the coupled Gross-Pitaevskii equations (GPEs). We start with a pseudo spin-1/2 BEC system with/without an internal atomic Josephson junction and spin-orbit coupling including (i) existence and uniqueness as well as non-existence of ground states under different parameter regimes, (ii) ground state structures under different limiting parameter regimes, (iii) dynamical properties, and (iv) efficient and accurate numerical methods for computing ground states and dynamics. Then we extend these results to spin-1 BEC and spin-2 BEC. Finally, extensions to dipolar spinor systems and/or general spin-F (F>=3) BEC are discussed.
The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
For spinor Bose-Einstein condensates both the total spin SSS and its ZZZ component SZ{S}_{Z}SZ​ should be conserved. However, in existing theories, only the conservation of SZ{S}_{Z}SZ​ has been taken into account. To remedy this, here we attempt to take the conservation of both SSS and SZ{S}_{Z}SZ​ into account. For this purpose, a total spin state with the good quantum numbers SSS and SZ{S}_{Z}SZ​ is introduced in the trial wave function; thereby a generalized Gross-Pitaevskii equation has been derived. With this equation, the ground bands of the 23Na^{23}\mathrm{Na}23Na and 87Rb^{87}\mathrm{Rb}87Rb condensates have been studied, where the levels that are distinct in SSS split. It was found that the level density is extremely dense in the bottom of the ground band of 23Na^{23}\mathrm{Na}23Na, i.e., in the vicinity of the ground state. On the contrary, for 87Rb^{87}\mathrm{Rb}87Rb, the levels are extremely dense in the top of the ground band.