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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 757616, 14 pages
The Gross-Pitaevskii Model of Spinor BEC
Department of Mathematics, Sichuan University, Chengdu 610064, China
Received 4 December 2011; Accepted 15 February 2012
Academic Editor: Wan-Tong Li
Copyright © 2012 Dongming Yan and Tian Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
After the first remarkable experiments concerning the observation of Bose-Einstein condensate (BEC) in dilute gases of alkali atoms such as , , and  the interest in this phenomenon has revived [4, 5]. On the mathematical side, most of the work has concentrated on the Gross-Pitaevskii (GP) model of BEC, which is usually referred to as nonlinear Schödinger equation (NLSE) (cf. [6–14] and references therein). There are also many pieces of the literature on the spinor BEC ([15–19]). In the spinor BEC case, the constituent bosons have internal degrees of freedom, such as spin, the quantum state, and its properties becomes more complex . What has made the alkali spinor BEC particularly interesting is that optical and magnetic fields can be used to probe and manipulate the system.
In , Ho shows that in an optical trap the ground states of spin-1 bosons such as , , and can be either ferromagnetic or polar states, depending on the scattering lengths in different angular momentum channels. In , Pu et al. discuss the energy eigenstates, ground and spin mixing dynamics of a spin-1 spinor BEC for a dilute atomic vapor confined in an optical trap. Their results go beyond the mean field picture and are developed within a fully quantized framework. In , Zou and Mathis propose a three-step scheme for generating the maximally entangled atomic Greenberger-Horne-Zeilinger (GHZ) states in a spinor BEC by using strong classical laser fields to shift atom level and drive single-atom Raman transition. Their scheme can be directly used to generate the maximally entangled states between atoms with hyperfine spin 0 and 1.
In this paper, we want to establish the mathematical theory of the GP model of spinor BEC in which the internal degrees of freedom of atoms are also under consideration. Furthermore, three conservative quantities of spinor BEC, that is, the energy, total particle number, and magnetization intensity are also proved.
2. Gross-Pitaevskii Model for Spinor BEC
In this section we derive GP equation for spinor BEC, that is, the following equation:
We consider the GP model for spinor BEC. Particles of have three quantum states: magnetic quantum number . The corresponding wave function of these three quantum states are denote by Here the physical meaning of is the density of particles . The corresponding Hamilton energy functional of spinor BEC is as follows: where is the boson mass, is Planck constant, is the external trapping potential, is the density of dilute bosonic atoms, is the interaction constant between atoms, and is the spin exchange interaction constant and are the scattering length, and is the spin operator: We adopt the following notations: By calculation we can get
By using Lagrange multiplier theorem, from the Hamilton energy functional (see (2.3) and the total particle number is conservative, and we can obtain the steady state GP equation of spinor BEC as follows: where is the chemical potential. Furthermore, according to general rules of quantum mechanics from steady state GP equation, we can get the dynamical model as follows: where . From ((2.3) and (2.7), we can obtain the concrete expression of (2.10) as (2.1).
In the spinor BEC and we have following physical meaning:
3. Equivalent Form of Spinor BEC
4. Infinite Dimensional Hamilton System
In this section, we consider the following infinite-dimensional Hamilton system where is dense, is linear space, is reflexive Banach space, is Hilbert space, is functional, and is derived operator.
Remark 4.1. Infinite-dimensional Hamilton system (4.1) not only has some kind of beauty in its own form, but also many equations can be written as (4.1). For example, Schödinger equation, Weyl equations, and Dirac equations can be written as (4.1). Hence, it is worth to study the infinite-dimensional Hamilton system (4.1), also see .
Definition 4.2. One says is a weak solution of Hamilton system (4.1), provided
for every .
Let satisfy Then we have the following existence theorem.
Theorem 4.3 (see ). Assume that satisfies condition (4.3) and is weakly continuous, then for any , there exists one global weak solution of equation (4.1) Furthermore, is a conservative quantity for weak solution , that is,
Proof. We prove the existence of global solution for (4.1) in by standard Galerkin method. Choose
as orthonormal basis of space . Set as follows:
Consider the ordinary equations as follows: where .
By the theory of ordinary equations, there exists only one local solution of (4.8):
From (4.8) we can obtain the equality holds true for any . Moreover, equality holds true for any .
Putting in (4.11), we obtain that which implies where
From (4.3) and (4.10), we deduce that is bounded in . Therefore there exists a subsequence; we still write it as , such that
According to being weakly continuous and (4.10), (4.15), we know the following equality holds true for any . Since is dense in and , equality (4.16) holds true for all , which implies that is a global weak solution of (4.1).
Next, we prove is a conservative quantity for weak solution . From (4.16), for all we have Putting in (4.17), we obtain that Therefore, is a conservative quantity for weak solution . The proof is completed.
Theorem 4.4 (see ). Let be Hilbert space and be functional. Then a functional is a conservative quantity for the infinite-dimensional Hamilton system (4.1) if and only if the following equality holds true for any .
5. The Existence of Global Solution of Spinor BEC
In this section we consider the Gross-Pitaevskii equation of spinor BEC (2.10) under the Dirichlet boundary condition, to wit the following initial boundary problem: where is a domain. When , then (5.1) become Cauchy problem. By applying Theorem 4.3, we can obtain the following theorem.
Theorem 5.1. Assume that and , then for any , there exists one global weak solution of problem (5.1)
Remark 5.2. If , then (5.1) reduce to the GP equation of BEC. Theorem 5.1 is also consistent with the experiments in repulsive case. In the situation of repulsive interaction, solutions to the GP equation of BEC are well defined for all times [12, 13, 20], which corresponds to the emergence of the BEC.
Proof. Let . Firstly, we need to verify condition (4.3) in Theorem 4.3. From Section 2, we know that
where , and
Hence, when we have
where . Therefore, we deduce
which implies that condition (4.3) holds true.
Next we need to verify the continuous condition in Theorem 4.3. Let operator : be defined by For any and in , we have Since is dense in , equality (5.8) holds true for all , which implies that is weakly continuous.
Therefore, according to Theorem 4.3, there exists a global weak solution of (5.1). The proof is completed.
6. The Conservative Quantities of Spinor BEC
In this section we will discuss the conservative quantities of spinor BEC. Let be defined as ((2.3), as follows: Then by using the same method as the proof of Theorem 4.4, we will prove the following theorem.
Theorem 6.1. Hamilton energy , the total particle number , and magnetization intensity are conservative quantities for problem (5.1).
Proof. Firstly, from (3.1) and (3.2) we can get
which imply that the energy is a conservative quantity for problem (5.1).
Secondly, by using (3.3) we can get the following equalities: which imply that the total particle number is a conservative quantity for problem (5.1).
At last, we show is a conservative quantity for problem (5.1). Let , then are both functional. Let . It is easy to check that From ((2.3) and (2.7), we have Combining (6.5), (6.6) with (3.1), (3.2), we can get following equalities: which imply that the magnetization intensity is a conservative quantity for problem (5.1). The proof is completed.
The authors would like to thank anonymous reviewers for their careful reading and many valuable comments that greatly improved the presentation of this paper. This work is supported by NSFC 11171236.
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