Universal bound and scattering properties for two dipoles

The bound state and low-energy scattering properties of two oriented dipoles are investigated for both bosonic and fermionic symmetries. Interestingly, a universal scaling emerges for the expectation value of the angular momentum for deeply bound two-dipole states. This scaling traces to the pendulum motion of two dipoles in the strong dipole regime. The low-energy scattering phase shifts of two dipoles also show universal behavior. These universal observations make connections to the scaling laws reported by Wang et al. [Phys. Rev. Lett. 106, 233201 (2011); Phys. Rev. Lett. 107, 233201 (2011)] for three dipoles.

Figure 1

Number of channels required for converging the dipolar bound states within 1$\%$ in the diabatic (spherical harmonic) and the adiabatic representations.

Figure 2

The scaled angular momentum expectation value $\langle {\widehat{L}}^2\rangle /{(d_{\ell }/r_0)}^{1/2}$ for bosonic dipoles (a) and fermionic dipoles (b).

Figure 3

The cuts of the deeply bound two-dipole wave function along the $x$-$z$ plane for bosons. Different angular and radial excitations $\left(\right|m_l|,\nu ,n)$ are shown: (a) $(0,0,0)$, (b) $(1,0,0)$, (c) $(0,1,0)$, (d) $(0,0,1)$, (e) $(1,0,1)$, and (f) $(0,1,1)$. The dipole length $d_{\ell }=600r_0$ for all these wave functions.

Figure 4

The characteristic size $r_c$ (a) and the characteristic energy $|E_c|$ (see text) (b) for both bosonic and fermionic dipoles with $|m_l|$$=$0 and 1. The irregular jumps come from the avoided crossings between states with different angular momentum character.

Figure 5

The scaling behavior of the binding energy $E_b$ with $d_{\ell }$ and $V_l^{m_l}$ near $p$-wave resonances for fermionic dipoles. The dipole length $d_{\ell }$ is tuned to have one or two bound states (BS) for $m_l=0$ and $m_l=1$.

Figure 6

(a) The $d_{\ell }$ dependence of the coefficient $b_l^{m_l}$ in the low-energy expansion of Re$\left[{\delta }_l^{m_l}\left(k\right)\right]$. (b) The $d_{\ell }$ dependence of the scattering volume $V_l^{m_l}$.