The Gross-Pitaevskii Model of Spinor BEC Dongming

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.


Introduction
After the first remarkable experiments concerning the observation of Bose-Einstein condensate BEC in dilute gases of alkali atoms such as 87 Rb 1 , 23 Na 2 , and 7 Li 3 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 Sch ödinger 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 20 .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 15 , Ho shows that in an optical trap the ground states of spin-1 bosons such as 23 Na, 39 K, and 87 Rb can be either ferromagnetic or polar states, depending on the scattering lengths in different angular momentum channels.In 17 , 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 19 , 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.

Gross-Pitaevskii Model for Spinor BEC
In this section we derive GP equation for spinor BEC, that is, the following equation:

2.1
We consider the GP model for F 1 spinor BEC.Particles of F 1 have three quantum states: magnetic quantum number m 1, 0, −1.The corresponding wave function of these three quantum states are denote by Here the physical meaning of |ψ i | 2 is the density of m i particles i 1, 0, −1 .The corresponding Hamilton energy functional of F 1 spinor BEC is as follows: where m is the boson mass, is Planck constant, V • is the external trapping potential, |Ψ| 2 is the density of dilute bosonic atoms, g n is the interaction constant between atoms, and g s is the spin exchange interaction constant a 0 and a 2 are the scattering length, and S S x i S y j S z k is the spin operator: We adopt the following notations: .

2.6
By calculation we can get

2.7
By using Lagrange multiplier theorem, from the Hamilton energy functional E 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 μ 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 i √ −1.From 2.3 and 2.7 , we can obtain the concrete expression of 2.10 as 2.1 .
In the spinor BEC g n and g s we have following physical meaning: ⎩ > 0, corresponding to the repulsive interaction between atoms, < 0, corresponding to the attractive interaction between atoms, ⎩ > 0, corresponding to the antiferromagnetic states, < 0, corresponding to the ferromagnetic states. 2.11

Equivalent Form of Spinor BEC
In this section we will show that GP equation 2.1 is equivalent to the following quantum Hamilton systems see 21 : where , and the energy functional defined as In fact, on the one hand, splitting real and imaginary parts of 2.1 , we can obtain

3.3
On the other hand, it is easy to check that

Infinite Dimensional Hamilton System
In this section, we consider the following infinite-dimensional Hamilton system where 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, Sch ödinger 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 21 .
Then we have the following existence theorem.
Theorem 4.3 see 21 .Assume that F satisfies condition 4.3 and DF : weakly continuous, then for any ϕ, ψ ∈ X 1 × X 2 , there exists one global weak solution of equation 4.1 Furthermore, F u, v is a conservative quantity for weak solution u, v , that is, Proof.We prove the existence of global solution for 4.1 in L ∞ 0, ∞ , X 1 × X 2 by standard Galerkin method.Choose as orthonormal basis of space H. Set X n , X n as follows:

4.7
Consider the ordinary equations as follows: Therefore there exists a subsequence; we still write it as { u n , v n } ∞ n 1 , such that

4.15
According to DF : X 1 × X 2 → X 1 × X 2 * being weakly continuous and 4.10 , 4.15 , we know the following equality Next, we prove F u, v is a conservative quantity for weak solution u, v .From 4.16 , for all h > 0 we have

4.17
Putting in 4.17 , we obtain that

4.19
Therefore, F u, v is a conservative quantity for weak solution u, v .The proof is completed.
Theorem 4.4 see 21 .Let X 1 , X 2 be Hilbert space and F : Proof.Let u, v be a solution of 4.1 .Then we have

4.21
which imply that d/dt G u, v 0 if and only if equality 4.20 holds true.The proof is completed.

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 Ω ⊂ R n 1 ≤ n ≤ 3 is a domain.When Ω R n , then 5.1 become Cauchy problem.By applying Theorem 4.3, we can obtain the following theorem.
Theorem 5.1.Assume that V ∈ L 2 Ω and g n > max{0, −2g s }, then for any Ψ 0 ∈ H 1 Ω, C 3 , there exists one global weak solution of problem 5.1 Remark 5.2.If g s 0, 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. 6.Firstly, we need to verify condition 4.3 in Theorem 4.3.From Section 2, we know that
Therefore, according to Theorem 4.3, there exists a global weak solution of 5.1 .The proof is completed.

The Conservative Quantities of Spinor BEC
In this section we will discuss the conservative quantities of spinor BEC.Let E be defined as 2.3 , N, M 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 E, the total particle number N, and magnetization intensity M are conservative quantities for problem 5.1 .
Proof.Firstly, from 3.1 and 3.2 we can get which imply that the energy E 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 N is a conservative quantity for problem 5.1 .
At last, we show M is a conservative quantity for problem 5.1    which imply that the magnetization intensity M is a conservative quantity for problem 5.1 .
The proof is completed.