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

We apply Generalized Van-Vleck (GVV) perturbation theory to spin-1 atoms confined in a one-dimensional hard-wall trap. This method achieves a relative error of $10^{-6}$ in the ground state energy with a matrix dimension of $\sim 10^3$. With the GVV method, we assess validity of the single-mode approximation (SMA) across static and dynamical regimes 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 tuning $q$, the overlap of the density distribution of spin components in the ground state gradually decreases, leading eventually to the breakdown of the SMA. Quenched spin dynamics further reveals multi-frequency oscillations with enhanced amplitudes, in contrast to the SMA's prediction of single-frequency oscillations. Our results provide a practical numerical method to investigate the spin mixing dynamics beyond the SMA, inspired by recent experiments in spin-1 atomic gases.