Dynamical symmetry in spinor Bose-Einstein condensates

We demonstrate that dynamical symmetry plays a crucial role in determining the structure of the eigenspectra of spinor Bose-Einstein condensates (BECs). In particular, the eigenspectra of spin-1 and spin-2 BECs in the single-mode approximation are shown to be completely determined by dynamical symmetries, where a spin-2 BEC corresponds to the U(5) limit of the interacting boson model in nuclear physics. The eigenspectrum of a spin-3 BEC is solved analytically for a specific class of coupling constants, while it is shown that dynamical symmetry alone is not sufficient to determine the spectrum for arbitrary coupling constants. We also study the low-lying eigenspectra of spin-1 and spin-2 BECs in the absence of external magnetic field, and find, in particular, that the quasidegenerate spectra emerge for antiferromagnetic and cyclic phases. This implies that these systems are highly susceptible to external perturbations and may undergo symmetry-breaking transitions to other states upon increasing the system’s size.

Figure 1

Low-lying eigenspectra of a spin-1 BEC. The angular momentum $F$ of each state is shown to the right. (a) $c_1<0$. (b) $c_1>0$ with even $N$. (c) $c_1>0$ with odd $N$. The low-lying energy-level spacing is of the order of $N$ for the ferromagnetic BEC $(c_1<0)$, while it is of the order of 1 for the antiferromagnetic BEC $(c_1>0)$. ${\Delta }_{F\left(AF\right)}$ denotes the energy gap of the ferromagnetic (antiferromagnetic) ground state.

Figure 2

Low-lying eigenspectra of a spin-2 BEC for the ferromagnetic phases with $N=2k$ and $d_1<0$. The values of $F$, $n_{30}$, and $\tau$ of each state are shown to the left, middle, and right, respectively. (a) $d_5+\frac{2N-3}{2N+1}{d}_1<0$. (b) $-\frac{2N-3}{2N+1}{d}_1<d_5<-\frac{4N-5}{2N+1}{d}_1$. (c) $-\frac{4N-5}{2N+1}{d}_1<d_5<-\frac{8N-14}{2N+1}{d}_1$. (d) $-\frac{8N-14}{2N+1}{d}_1<d_5<-\frac{4N+2}{N+3}{d}_1$. ${\Delta }_F$ denotes the energy gap of the ferromagnetic ground state.

Figure 3

Low-lying eigenspectra of a spin-2 BEC for the antiferromagnetic phases with $N=2k$. The values of $F$, $n_{30}$, and $\tau$ of each state are shown to the left, middle, and right, respectively. (a) $d_1>0$ and $\frac{3}{4}{d}_1<d_5<\frac{5}{2}{d}_1$. (b) $d_1>0$ and $d_5>\frac{5}{2}{d}_1$. (c) $d_1>0$ and $0<d_5<\frac{3}{4}{d}_1$. (d) $d_1<0$ and $d_5>-\frac{4N+2}{N+3}{d}_1$. ${\Delta }_{AF}$ denotes the energy gap of the antiferromagnetic ground state.

Figure 4

Low-lying eigenspectra of a spin-2 BEC for the cyclic phases with $N=3k$ and $d_1>0$. The values of $F$, $n_{30}$, and $\tau$ of each state are shown to the left, middle, and right, respectively. (a) $d_5<-\frac{7}{2N+1}{d}_1$. (b) $-\frac{7}{2N+1}{d}_1<d_5<-\frac{3}{2N+1}{d}_1$. (c) $-\frac{3}{2N+1}{d}_1<d_5<-\frac{2}{2N-3}{d}_1$. (d) $-\frac{2}{2N-3}{d}_1<d_5<0$. ${\Delta }_C$ denotes the energy gap of the cyclic ground state.