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arXiv:cond-mat/0201173 · Physical Review A

Single-mode approximation in a spinor-1 atomic condensate

S. Yi, Özgür Esat Müstecaplıoğlu, C. P. Sun, Li You

Published Jul 2, 2002• 163 citations
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Abstract

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.

DOI
10.1103/physreva.66.011601

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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.
Compared to single-component Bose-Einstein condensates, spinor Bose-Einstein condensates display much richer dynamics. In addition to density oscillations, spinor Bose-Einstein condensates exhibit intriguing spin dynamics that is associated with population transfer between different hyperfine components. This paper analyzes the validity of the widely employed single-mode approximation when describing the spin dynamics in response to a quench of the system Hamiltonian. The single-mode approximation assumes that the different hyperfine states all share the same time-independent spatial mode. This implies that the resulting spin Hamiltonian only depends on the spin interaction strength and not on the density interaction strength. Taking the spinor sodium Bose-Einstein condensate in the f=1f=1f=1 hyperfine manifold as an example and working within the mean-field theory framework, it is found numerically that the single-mode approximation misses, in some parameter regimes, intricate details of the spin and spatial dynamics. We develop a physical picture that explains the observed phenomenon. Moreover, using that the population oscillations described by the single-mode approximation enter into the effective potential felt by the mean-field spinor, we derive a semiquantitative condition for when dynamical mean-field induced corrections to the single-mode approximation are relevant. Our mean-field results have implications for a variety of published and planned experimental studies.
Abstract. We revisit the topic of the mean field ground state of a spin-1 atomic condensate inside a uniform magnetic field (B) under the constraints that both the total number of atoms (N) and the magnetization (M) are conserved. In the presence of an internal state (spin component) independent trap, we also investigate the dependence of the so-called single spatial mode approximation (SMA) on the magnitude of the magnetic field and M. Our result indicate that the quadratic Zeeman effect is an important factor in balancing the mean field energy from elastic atom-atom collisions that are known to conserve both N and M.Mean field ground state of a spin-1 condensate in a magnetic field 2 1.
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.
With current magnetic-field shielding and high-precision detection in dipolar spinor Bose-Einstein condensates, it is possible to experimentally detect the low- or zero-field nonsecular dipolar dynamics. Here we analytically investigate the zero-field nonsecular magnetic dipolar interaction effect, with an emphasis on magnetization dynamics in a spin-1 Bose-Einstein condensate under the single spatial mode approximation within the mean-field theory. Due to the biaxial nature of the dipolar interaction, a novel resonance occurs in the condensate magnetization oscillation, in contrast to the previous assumption of a conserved magnetization in strong magnetic fields. Furthermore, we propose a dynamical-decoupling detection method for such a resonance, which cancels the stray magnetic fields in experiment but restores the magnetization dynamics. Our results shed light on the dipolar systems and may find potential applications beyond cold atoms.
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