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.
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.
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.
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.
In this paper, a mass (or normalization) and magnetization conservative and energy-diminishing numerical method is presented for computing the ground state of spin-1 (or
F=1 spinor) Bose–Einstein condensates (BECs). We begin with the coupled Gross–Pitaevskii equations, and the ground state is defined as the minimizer of the energy functional under two constraints on the mass and magnetization. By constructing a continuous normalized gradient flow (CNGF) which is mass and magnetization conservative and energy-diminishing, the ground state can be computed as the steady state solution of the CNGF. The CNGF is then discretized by the Crank–Nicolson finite difference method with a proper way to deal with the nonlinear terms, and we prove that the discretization is mass and magnetization conservative and energy-diminishing in the discretized level. Numerical results of the ground state and their energy of spin-1 BECs are reported to demonstrate the efficiency of the numerical method.
We investigate the validity conditions of the single-mode approximation (SMA) in a spinor-1 atomic condensate when effects due to residual magnetic fields are negligible. For atomic interactions of the ferromagnetic type, the SMA is shown to be exact, with a mode function different from what is commonly used. However, the quantitative deviation is small under current experimental conditions (for
87Rb atoms). For antiferromagnetic interactions, we find that the SMA becomes invalid in general. The differences among the mean-field mode functions for the three spin components are shown to depend strongly on the system magnetization. Our results can be important for studies of beyond mean-field quantum correlations, such as fragmentation, spin squeezing, and multipartite entanglement.