Two bosonic dipoles under elongated confinement

The behaviors of two particles under harmonic confinement strongly depend on the aspect ratio $\eta$, which is defined as the ratio between the trapping frequency along the $\rho$ and the $z$ directions. It has been shown that the properties of particles interacting through spherically symmetric potentials are, in the extreme limits of very large and very small $\eta$, well described by effective one- and two-dimensional Hamiltonian. This work considers two particles with anisotropic interactions confined in an elongated harmonic trap. Assuming that the dipole moments are aligned along the z-axis, we obtain the eigen spectrum of this system analytically and analyze how it changes as a function of $\eta$. To validate our analytical approach, we compare our results with the eigen spectrum obtained numerically for a short-range shape-dependent potential.