For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state
(ψ1,ψ0,ψ−1) undergoes a phase transition from a two-component state (
ψ0≡0) to a three-component state (
ψj=0 for all
j) at a critical value of the magnetic field. This phenomenon has been observed in numerical simulations as well as in experiments. In this paper, we provide a mathematical proof based on a simple principle found by the authors: a redistribution of the mass densities between different components will decrease the kinetic energy.
We justify some characterizations of the ground states of spin-1 Bose-Einstein condensates exhibited from numerical simulations. For ferromagnetic systems, we show the validity of the single-mode approximation (SMA). For an antiferromagnetic system with nonzero magnetization, we prove the vanishing of the
mF=0 component. In the end of the paper some remaining degenerate situations are also discussed. The proofs of the main results are all based on a simple observation, that a redistribution of masses among different components will reduce the kinetic energy.
We study antiferromagnetic spin-1 Bose-Einstein condensates under external uniform magnetic fields. The bifurcation between two-component and three-component regimes of ground states with respect to the magnetization M and the quadratic Zeeman effect q is justified. The proof is based on the technique of “mass redi stribution” introduced in the authors previous work, which gives interesting inequalities and equalities satisfied by ground states. Some open problems arising naturally from our investigation are also discussed in the end.
We justify some characterizations of the ground states of spin-1 Bose-Einsteincondensates exhibited from numerical simulations. For ferromagnetic systems, weshow the validity of the single-mode approximation (SMA). For anantiferromagnetic system with nonzero magnetization, we prove the vanishing ofthe
mF=0 component. In the end of the paper some remaining degeneratesituations are also discussed. The proofs of the main results are all based ona simple observation, that a redistribution of masses among differentcomponents will reduce the kinetic energy.
We develop an analytic theory for the ground state patterns and their phase transitions for spin-1 Bose-Einstein condensates on a bounded domain in the presence of a uniform magnetic field. Within the Thomas-Fermi approximation, these ground state patterns are composed of four basic states: magnetic state, nematic state, two-component state and three-component state, separated by interfaces. A complete phase diagram of the ground state patterns are found analytically with different quadratic Zeeman energy q and total magnetization M for both ferromagnetic and antiferromagnetic systems. Using the Γ-convergence technique, it is found that the semi-classical limits of these ground states minimize an energy functional which consists of interior interface energy plus a boundary contact energy. As a consequence, the interface between two different basic states has constant mean curvature, and the contact angle between the interface and the boundary obeys Young's relation.
Spin-1 Bose-Einstein condensate (BEC) is a special three-component system, written as
Psi=(psi1,psi0,psi−1). Its behavior is described
by an energy functional
E[Psi] with two constraints:
the conservation of the number of atoms and the conservation of total magnetization. That is
int∣Psi∣2 and
intlt(∣psi1∣2−∣psi−1∣2t) are fixed numbers.
And a ground state is a minimizer of
E under the constraints. To explain what we do in this thesis, we remark that according to the sign of a specific parameter in the energy
E, spin-1 BECs are classified into two groups: ferromagnetic ones and antiferromagnetic ones.
The works in this thesis are motivated by the following two phenomena:
1. Any ground state of a ferromagnetic system is of the form
begin{align*}
Psi = (gamma_1 psi,gamma_0 psi,gamma_{-1} psi),
end{align*}
where
gammaj are constants and
psi a function. This is called single-mode approximation.
2. When an external magnetic field is applied, the ground state of an antiferromagnetic system undergoes a bifurcation from
psi0equiv0
to
psi0ne0 as the strength of the magnetic field surpasses a critical value.
Although these phenomena have been well-known from numerical simulations for
quite a long time, there were no rigorous mathematical justifications before
our investigations. In this thesis, our works [16,17]
on their proofs are given, with more details.
The proofs rely on a principle which says that a redistribution of the mass densities
(i.e.
∣psi1∣2,
∣psi0∣2 and
∣psi−1∣2) will decrease the kinetic energy. This principle can be regarded as a simple generalization of a well-known convexity inequality for gradients. We will show how this principle can give a rather unified approach toward our problems.