Permutation Symmetry in Spinor Quantum Gases: Selection Rules, Conservation Laws, and Correlations

Many-body systems of identical arbitrary-spin particles, with separable spin and spatial degrees of freedom, are considered. Their eigenstates can be classified by Young diagrams, corresponding to nontrivial permutation symmetries (beyond the conventional paradigm of symmetric-antisymmetric states). The present work obtains the following. (a) Selection rules for additional nonseparable (dependent on spins and coordinates) $k$-body interactions: the Young diagrams, associated with the initial and final states of a transition, can differ by relocation of no more than $k$ boxes between their rows. (b) Correlation rules in which eigenstate-averaged local correlations of $k$ particles vanish if $k$ exceeds the number of columns (for bosons) or rows (for fermions) in the associated Young diagram. It also elucidates the physical meaning of the quantities conserved due to permutation symmetry—in 1929, Dirac identified those with characters of the symmetric group—relating them to experimentally observable correlations of several particles. The results provide a way to control the formation of entangled states belonging to multidimensional non-Abelian representations of the symmetric group. These states can find applications in quantum computation and metrology.

Figure 1

Selection rules (4). (a) Six states, which can be coupled to a given state (associated with the central Young diagram) by a one-body interaction ${\widehat{W}}_1$ [Eq. (3)] for $s=1$. (b) An example of the coupling by a two-body interaction ${\widehat{W}}_2$ [Eq. (3)]. The dashed boxes are relocated.

Figure 2

(a) Allowed transitions between $\lambda =[(N-{\lambda }_2+S)/2,{\lambda }_2,(N-{\lambda }_2-S)/2]$ multiplets (red arrows) due to a one-body interaction $W_1$ (3) for $N=18$ bosons with the spin $s=1$. The average energies of the multiplets [circles, see Eq. (10)] for bosons are proportional to the average local two-body correlations (9). The dashed lines connect the points with the same maximal spin $S$. (b) The three- and four-body average local correlations for the same multiplets (black and red, respectively). (c) The three-body average local correlations for the $N=18$ fermions with the spin $s=1$ as functions of ${\lambda }_3$ given the maximal spin $S$ (denoted by numbers).

Figure 3

A scheme of population of $\lambda$ multiplets using spatially homogeneous spin-changing pulses ${\widehat{W}}_{\text{hom}}(t)$ (blue dashed arrows) and a spatially inhomogeneous spin-conserving pulse ${\widehat{W}}_{\text{inh}}(\mathbf{r},t)$ (red solid arrows). Shapes of the Young diagram boxes denote the numbers of occupied atomic spin states.