Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1102.0832

Proofs of some simplified characterizations of the ground states of spin-1 Bose-Einstein condensates

Li Li, I‐Liang Chern

Published Feb 4, 2011• 3 citations
arXivHTMLPDF

Abstract

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 $m_F=0$ 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.

DOI
10.48550/arxiv.1102.0832

Related papers

A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates

Liren Lin, I‐Liang Chern

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 $m_F=0$ 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.

Discrete and Continuous Dynamical Systems - B•Jan 1, 2014•11 citations

Phase transition between two-component and three-component ground states of spin-1 Bose-Einstein condensates

Lin, Liren, I‐Liang Chern

For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state $(ψ_1,ψ_0,ψ_{-1})$ undergoes a phase transition from a two-component state ($ψ_0 \equiv 0$) to a three-component state ($ψ_j\ne 0$ 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.

arXiv:1302.0279•Feb 1, 2013•0 citations

Phase transition between two-component and three-component ground states of spin-1 Bose-Einstein condensates

Lin Liren, I‐Liang Chern

For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state $(\psi_1,\psi_0,\psi_{-1})$ undergoes a phase transition from a two-component state ($\psi_0 \equiv 0$) to a three-component state ($\psi_j\ne 0$ 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.

Bulletin of the Institute of Mathematics Academia Sinica NEW SERIES•Jun 1, 2025•0 citations

Single-mode approximation in a spinor-1 atomic condensate

S. Yi, Özgür Esat Müstecaplıoğlu, C. P. Sun +1 more

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 ${}^{87}\mathrm{Rb}$ 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.

arXiv:cond-mat/0201173•Physical Review A•Jul 2, 2002•163 citations

On ground state of spin-1 Bose-Einstein condensates with the external Ioffe-Pitchard magnetic

lv zhongxue

In this paper, we prove the existence of the ground state for the spinor Bose-Einstein condensates with the external Ioffe-Pitchard magnetic field in the one-dimensional case. We also characterize the ground states of spin-1 Bose-Einstein condensates with the external Ioffe-Pitchard magnetic field, i.e., for ferromagnetic systems, we show that under some condition, searching for ground state of ferromagnetic spin-1 BEC with the external Ioffe-Pitchard magnetic field can be reduced to a one-component minimization problem. 10.1017/S0004972712000305

Bulletin of the Australian Mathematical Society•Oct 22, 2012•0 citations

A Mass and Magnetization Conservative and Energy-Diminishing Numerical Method for Computing Ground State of Spin-1 Bose–Einstein Condensates

Weizhu Bao, Hanquan Wang

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.

SIAM Journal on Numerical Analysis•Jan 1, 2007•48 citations
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=0m_F=0mF​=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 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=0m_F=0mF​=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.
For an antiferromagnetic spin-1 Bose-Einstein condensate under an applied uniform magnetic field, its ground state (ψ1,ψ0,ψ−1)(ψ_1,ψ_0,ψ_{-1})(ψ1​,ψ0​,ψ−1​) undergoes a phase transition from a two-component state (ψ0≡0ψ_0 \equiv 0ψ0​≡0) to a three-component state (ψj≠0ψ_j\ne 0ψj​=0 for all jjj) 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)(\psi_1,\psi_0,\psi_{-1})(ψ1​,ψ0​,ψ−1​) undergoes a phase transition from a two-component state (ψ0≡0\psi_0 \equiv 0ψ0​≡0) to a three-component state (ψj≠0\psi_j\ne 0ψj​=0 for all jjj) 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 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{}^{87}\mathrm{Rb}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.
In this paper, we prove the existence of the ground state for the spinor Bose-Einstein condensates with the external Ioffe-Pitchard magnetic field in the one-dimensional case. We also characterize the ground states of spin-1 Bose-Einstein condensates with the external Ioffe-Pitchard magnetic field, i.e., for ferromagnetic systems, we show that under some condition, searching for ground state of ferromagnetic spin-1 BEC with the external Ioffe-Pitchard magnetic field can be reduced to a one-component minimization problem. 10.1017/S0004972712000305
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=1F=1F=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.