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arXiv:2610.20096 · Physical Review A

Unconventional quantum spin mixing of two spin-1 atoms beyond the single-mode approximation

Yanhong Liu, Fulin Deng, Haoxue Qiao, Wenxian Zhang

Published Jun 18, 2026• 0 citations
arXivHTMLPDF

Abstract

The single-mode approximation (SMA) has been widely employed in ultracold spinor gases, where all spin components share the same spatial mode. By adopting generalized Van Vleck (GVV) perturbation theory, we achieve a relative error of $10^{-6}$ in the ground-state energy with a reduced effective Hilbert space dimension of $\sim 10^3$ for two spin-1 bosons confined in a one-dimensional hard-wall trap. With the accurate GVV method, we explore the validity of the SMA across ground-state properties and quench spin dynamics for two $^{7}\text{Li}$, $^{87}\text{Rb}$, or $^{23}\text{Na}$ atoms. Numerical results show that the quadratic Zeeman energy $q$ provides a powerful control knob: by increasing $|q|$, the fidelity between the spin components in the ground state gradually decreases, leading eventually to the breakdown of the SMA (due to the competition between the kinetic energy and the quadratic Zeeman energy). Quench spin dynamics with abruptly increased $q$ further reveals multiple resonances, where the SMA-predicted micromotion and beyond-SMA macromotion appear simultaneously. These results point to a direction to investigate the quantum spin-mixing dynamics beyond the SMA, particularly in recent experiments of strongly interacting spin-1 atomic gases.

DOI
10.1103/x9yz-7t6h

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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.
We present the experimental observation of a magnetically tuned resonance phenomenon in the spin mixing dynamics of ultracold atomic gases. In particular, we study the magnetic field dependence of spin conversion in F=2 (87)Rb spinor condensates in the crossover from interaction dominated to quadratic Zeeman dominated dynamics. We discuss the observations in the framework of spin dynamics as well as matter wave four wave mixing. Furthermore, we show that the validity range of the single mode approximation for spin dynamics is significantly extended at high magnetic field.
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Abstract Based on the numerical solutions of the coupled Gross–Pitaevskii equations, the spin-textures of the ground state of a Bose–Einstein condensate with two kinds of spin-1 atoms have been studied. Besides, making use of the fractional parentage coefficients, the probabilities of an atom in spin component μ and of two correlated atoms one in μ and one in ν have been calculated. The previous results under the single spatial mode approximation (SMA) are found to be qualitatively consistent with the present results beyond the SMA. The variation of the probabilities against the input parameters is found to match the corresponding variation of the spin-textures excellently. In particular, once a spin-texture transition occurs, the associated probabilities will change suddenly. Since the probabilities are observable, the observation provides a way for the experimental identification of the spin-textures. Besides, sine the critical points (where a transition occurs) can be accurately identified, information on the input parameters (say, the very weak spin-dependent forces) can be extracted.
We study quantum spin mixing in a mixture of two spin-1 condensates including coherent interspecies mixing processes, using condensates of 87{}^{87}87Rb and 23{}^{23}23Na atoms in the ground lower hyperfine F=1F=1F=1 manifold as prototype examples. Adopting the single spatial mode approximation for each of the two spinor condensates, we find the mixing dynamics reduce to that of three coupled nonrigid pendulums with clear physical interpretation. With suitably prepared initial states, the spin mixing dynamics allow for the determination of interspecies singlet-pairing as well as spin-exchange interaction parameters.