Spectroscopy for a few atoms harmonically trapped in one dimension

Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3–5$, are provided.

Figure 1

These figures schematically depict the fermionic ground states of four particles in the limit $g\to \infty$. (a) Configuration space is broken into 24 sectors, each with a specific ordering of particles, e.g., $4231\to x_4>x_2>x_3>x_1$. Each of these sectors is represented by a triangle and all wave functions must be zero on the sides of the triangle. The center-of-mass and radial components of the wave functions are not depicted in these diagrams, just the angular part. (b) A state in ${\lfloor 1^4\rfloor }^{+}$: The same function is pasted into each triangle, with the relative phase represented by pluses and minuses. All positive-parity, single component fermionic states, including the ground state, have this form. (c) and (d) The two states (10) in ${\lfloor 2^2\rfloor }^{+}$ reduced by the subgroup ${\mathrm{S}}_2\times {\mathrm{S}}_2$ into the irrep $\lfloor 1^2\rfloor \times \lfloor 1^2\rfloor$. The ground state of the two spin-up, two spin-down configuration adiabatically maps into the space spanned by wave functions with this form. The notation $++$ and $--$ means the amplitude in that section is twice as much, and zero means no amplitude. (e) and (f) The two states (11) in ${\lfloor {21}^2\rfloor }^{-}$ reduced by the subgroup ${\mathrm{S}}_3$ into the irrep $\lfloor 1^3\rfloor$. The ground state of the three spin-up, one spin-down configuration adiabatically maps into the space spanned by wave functions with this form.